Sizing and Placement of Distributed Generation in Electrical Distribution Systems using Conventional and Heuristic Optimization Methods

by

Mohamad AlHajri

Submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy

at

Dalhousie University Halifax Nova Scotia

June 2009

copy Copyright by Mohamad AlHajri 2009

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To comply with the Canadian Privacy Act the National Library of Canada has requested that the following pages be removed from this copy of the thesis

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DEDICATION PAGE

To my beloved parents my brothers Falah and Abdullah my sisters my wife OmFahad

my daughter Najla and my sons Fahad Falah and Othman

TABLE OF CONTENTS LIST OF TABLES x

LIST OF FIGURES xiii

ABSTRACT xvi

LIST OF ABBREVIATIONS AND SYMBOLS USED xvii

ACKNOWLEDGEMENTS xxiv

Chapter 1 INTRODUCTION 1

11 Motivation 1

12 Distribution Generation - Historic Overview 2

13 Distribution Generation 2

14 Thesis Objectives and Contributions 5

15 Thesis Outline 7

Chapter 2 LITERATURE REVIEW 9

21 Introduction 9

22 Distribution Power Flow 9

23 DG Integration Problem 13

231 Solving the DG Integration Problem via Analytical and Deterministic Methods 14

232 Solving the DG Integration Problem via Metaheuristic Methods 17

24 Summary 20

Chapter 3 FAST AND FLEXIBLE RADIAL POWER FLOW FOR BALANCED AND UNBALANCED THREE-PHASE DISTRIBUTION NETWORKS 21

31 Introduction 21

32 Flexibility and Simplicity of FFRPF in Numbering the RDS Buses and Sections 22

321 Bus Numbering Scheme for Balanced Three-phase RDS 22

322 Unbalanced Three-phase RDS Bus Numbering Scheme 24

33 The Building Block Matrix and its Role in FFRPF 26

v

331 Three-phase Radial Configuration Matrix (RCM) 26

3311 Assessment of the FFRPF Building Block RCM 28

332 Three-phase Section Bus Matrix (SBM) 29

333 Three-phase Bus Section Matrix (BSM) 31

34 FFRPF Approach and Solution Technique 31

341 Unbalanced Multi-phase Impedance Model Calculation 32

342 Load Representation 38

343 Three-phase FFRPF BackwardForward Sweep 40

3431 Three-phase Current Summation Backward Sweep 40

3432 Three-phase Bus Voltage Update Forward Sweep 42

3433 Convergence Criteria 43

3434 Steps of the FFRPF Algorithm 44

344 Modifying the RCM to Accommodate Changes in the RDS 47

35 FFRPF Solution Method for Meshed Three-phase DS 48

351 Meshed Distribution System Corresponding Matrices 50

352 Fundamental Loop Currents 54

353 Meshed Distribution System Section Currents 56

354 Meshed Distribution System BackwardForward Sweep 59

36 Test Results and Discussion 60

361 Three-phase Balanced RDS 60

3611 Case 1 31-Bus with Single Main Feeder RDS 61

3612 Case 2 90-bus RDS with Extreme Radial Topology 70

3613 Case 3 69-bus RDS with Four Main Feeders 71

3614 Case 4 15-bus RDS-Considering Charging Currents 73

362 Three-phase Balanced Meshed Distribution System 74

3621 Case 1 28-bus Weakly Meshed Distribution System 74

3622 Case 2 70-Bus Meshed Distribution System 78

vi

3623 Case 3 201-bus Looped Distribution System 79

363 Three-phase Unbalanced RDS 80

3631 Case 1 10-bus Three-phase Unbalanced RDS 81

3632 Case 2 IEEE 13-bus Three-phase Unbalanced RDS 85

3633 Case 3 26-bus Three-Phase Unbalanced RDS 86

37 Summary 87

Chapter 4 IMPROVED SEQUENTIAL QUADRATIC PROGRAMMING

APPROACH FOR OPTIMAL DG SIZING 89

41 Introduction 89

42 Problem Formulation Overview 89

43 DG Sizing Problem Architecture 90

431 Objective Function 90

432 Equality Constraints 92

433 Inequality Constraints 92

434 DG Modeling 93

44 The DG Sizing Problem A Nonlinear Constrained Optimization Problem 94

45 The Conventional SQP 96

451 Search Direction Determination by Solving the QP Subproblem 96

4511 Satisfying Karush-Khun-Tuker Conditions 98

4512 Newton-KKT Method 101

4513 Hessian Approximation 103

452 Step Size Determination via One-Dimensional Search Method 104

453 Conventional SQP Method Summary 105

46 Fast Sequential Quadratic Programming (FSQP) 108

47 Simulation Results and Discussion 113

471 Case 1 33-busRDS 113

4711 Loss Minimization by Locating Single DG 114

4712 Loss Minimization by Locating Multiple DGs 118

vii

472 Case 2 69-bus RDS 124

4721 Loss Minimization by Locating a Single DG 125

473 Loss Minimization by Locating Multiple DGs 129

474 Computational Time of FSQP vs SQP 134

475 Single DG versus Multiple DG Units Cost Consideration 136

48 Summary 136

Chapter 5 PSO BASED APPROACH FOR OPTIMAL PLANNING OF

MULTIPLE DGS IN DISTRIBUTION NETWORKS 138

51 Introduction 138

52 PSO - The Motivation 138

53 PSO - An Overview 139

531 PSO Applications in Electric Power Systems 141 532 PSO - Pros and Cons 143

54 PSO - Algorithm 144

541 The Velocity Update Formula in Detail 145

5411 The Velocity Update Formula - First Component 146

5412 The Velocity Update Formula - Second Component 148

5413 The Velocity Update Formula-Third Component 149

5414 Cognitive and Social Parameters 150

542 Particle Swarm Optimization-Pseudocode 152

55 PSO Approach for Optimal DG Planning 153

551 Proposed HPSO Constraints Handling Mechanism 155

5511 Inequality Constraints 155

5512 Equality Constraints 157

5513 DG bus Location Variables Treatment 157

56 Simulation Results and Discussion 160

561 Case 1 33-bus RDS 161

viii

5611 33-bus RDS Loss Minimization by Locating a Single DG 161

5612 33-bus RDS Loss Minimization by Locating Multiple

DGs 169

562 Case 2 69-Bus RDS 180

5621 69-bus RDS Loss Minimization by Locating a Single DG 180

5622 69-bus RDS Loss Minimization by Locating Multiple

DGs 187

563 Alternative bus Placements via HP SO 195

57 Summary 196

Chapter 6 CONCLUSION 198

61 Contributions and Conclusions 198

62 Future Work 201

REFERENCES 203

APPENDIX 220

IX

LIST OF TABLES

Table 31 cok rd and De Parameters for Different Operation Conditions 34

Table 32 FFRPF Iteration Results for the 31-Bus RDS 67

Table 33 The 31-bus RDS Section Power Losses Obtained by the FFRPF Method 68

Table 34 FFRPF Voltage Profiles Results for the Three Different Load Models 69

Table 35 Comparisons between 31-Bus RDS Exponential Model Results 70

Table 36 31-bus RDS FFRPF Results vs Other Methods 70

Table 37 90-bus RDS FFRPF Results vs Other Methods 71

Table 38 69-bus RDS FFRPF Results vs Other Methods 73

Table 39 Komamoto 15-bus RDS FFRPF Results vs Other Methods 74

Table 310 Voltage Profiles for Radial and Meshed 28-bus Distribution Network 77 t

Table 311 28-bus Weakly Meshed DS FFRPF Results vs Other Methods 78

Table 312 70-bus Meshed DS FFRPF Results vs Other Methods 79

Table 313 201-bus Meshed DS FFRPF Results vs Other Methods 80

Table 314 10-bus 3(j) RDS FFRPF Results vs Ref [52] and Gauss Zbus Methods 85

Table 315 IEEE 13-bus 3(j) RDS FFRPF Results vs Ref [52] and Gauss Zbus

Methods 86

Table 316 26-bus 34gt RDS FFRPF Results vs Ref [52] and Gauss Zbus Methods 87

Table 41 Single DG Optimal Profile at the 33-bus RDS 115

Table 42 Optimal DG Profiles at all 33 buses 116

Table 43 Multiple DG Installations in the 33-bus RDS with Unspecified Power

Factor 119

Table 44 SQP Method Double-DG Cycled Combinations 121

Table 45 Single and Multiple DG Installations at Pre-specified 085 Power Factor 123

Table 46 Loss Reduction Comparisons for all DG Cases 123

Table 47 69-bus RDS Single DG Optimal Size and Placement Profiles 128

Table 48 Optimal Double DG Profiles in the 69-bus RDS 131

Table 49 Optimal Three DG Units Profiles in the 69-bus RDS 133

Table 410 Loss Reduction Comparison for all DG Installations in the 69-bus RDS 134

Table 411 33-bus RDS CPU Execution Time Comparison 135

Table 412 69-bus RDS CPU Execution Time Comparison 135

x

Table 51 HPSO Parameters for the Single DG Case 162

Table 52 33-bus RDS Single DG Fixedpf Case 20 HPSO Simulations 162

Table 53 Descriptive Statistics for HPSO Results for the Fixedpf Case 163

Table 54 HPSO vs FSQP Results 33-bus RDS-Single DG-FixedDG Case 163

Table 55 33-bus RDS Single DG Unspecified pf Case 20 HPSO Simulations 163

Table 56 Descriptive Statistics for HPSO Results for an Unspecified pf Case 164

Table 57 HPSO vs FSQP Results 33-bus RDS-Single DG-UnspecifiedpCase 164

Table 58 HPSO Parameters for Both Double DG Cases 170

Table 59 33-bus RDS Double DG Fixed pf Case 20 HPSO Simulations 171

Table 510 Descriptive Statistics for HPSO Results for Fixed pf Double DG Case 171

Table 511 HPSO vs FSQP Results 33-bus RDS-Double DGs-FixedCase 172

Table 512 33-bus RDS Double DG Unspecified pf Case 20 HPSO Simulations 172

Table 513 Descriptive Statistics for HPSO Results for UnspecifiedDouble DG

Case 173

Table 514 HPSO vs FSQP Results 33-bus RDS-Double DGs-UnspecifiedCase 173

Table 515 HPSO Parameters for Both Three DG Cases 174

Table 516 33-bus RDS Three DG Fixed pf Case 20 HPSO Simulations 174

Table 517 Descriptive Statistics for HPSO Results for Fixed pf Three DG Case 175

Table 518 HPSO vs FSQP Results 33-bus RDS-Three DG-FixedCase 175

Table 519 33-bus RDS Three DGs Unspecified pf Case 20 HPSO Simulations 175

Table 520 Descriptive Statistics for HPSO Results for the Fixed Three DG Case 176

Table 521 HPSO vs FSQP Results 33-bus RDS-Three DG-UnspecifiedCase 176

Table 522 HPSO Parameters for the Four DG Case 177

Table 523 33-bus RDS Four DG FixedCase 20 HPSO Simulations 178

Table 524 Descriptive Statistics for HPSO Results for Fixed Four DG Cases 178

Table 525 HPSO vs FSQP Results 33-bus RDS-Four DG-FixedCase 179

Table 526 33-bus RDS Four DG Unspecified pf Case 20 HPSO Simulations 179

Table 527 Descriptive Statistics for HPSO Results for Unspecified Four DG

Case 179

Table 528 HPSO vs FSQP 33-bus RDS-Four -Unspecified pf Case 180

Table 529 HPSO Parameters for 69-bus RDS Both Single DG Cases 181

Table 530 69-bus RDS Single DG Fixed pf Case 20 HPSO Simulations 182

xi

Table 531 Descriptive Statistics for HPSO Results for the Fixed Single DG Case 182

Table 532 HPSO vs FSQP Results 69-bus RDS-Single DG-FixedpDG Case 182

Table 533 69-bus RDS Single DG Unspecifiedpf Case 20 HPSO Simulations 183

Table 534 Descriptive Statistics for Unspecified pSingle DG Case 183

Table 535 HPSO vs FSQP Results 69-bus RDS-Single DG-UnspecifiedpDG

Case 184

Table 536 HPSO Parameters for 69-bus RDS the Double DG Cases 188

Table 537 69-bus RDS Double DG Fixed pf Case 20 HPSO Simulations 189

Table 538 Descriptive Statistics for HPSO Results for Fixed pDouble DG Case 189

Table 539 HPSO vs FSQP Results 69-bus RDS-Double DG-FixedpDG Case 190

Table 540 69-bus RDS Double DG UnspecifiedpfCase 20 HPSO Simulations 190

Table 541 Descriptive Statistics for HPSO Results for Unspecified ^Double DG

Case 191

Table 542 HPSO vs FSQP Results 69-bus RDS-Double-UnspecifiedpDG Case 191

Table 543 HPSO Parameters for Both 69-bus RDS Three DG Cases 192

Table 544 69-bus RDS Three DG Fixed pf Case 20 HPSO Simulations 192

Table 545 Descriptive Statistics for HPSO Results for Fixed pf Three DG Case 193

Table 546 HPSO vs FSQP Results 69-bus RDS-Three DGs-FixedCase 193

Table 547 69-bus RDS Three DG Unspecified pf Case 20 HPSO Simulations 194

Table 548 Descriptive Statistics for HPSO Results for Unspecified pf Three DG

Case 194

Table 549 HPSO vs FSQP Results 69-bus RDS-Three DGs-UnspecifiedpCase 195

Table 550 20 HPSO Simulations of the 69-bus RDS Three DG Fixed pf Case with Suboptimal Tuned Parameters 50 Iterations and 50 Swarm Particles 196

xii

LIST OF FIGURES

Figure 31 10-bus RDS 23

Figure 32 Different ways of numbering the system in Fig 31 24

Figure 33 The ease of numbering a modified and augmented RDS 24

Figure 34 Three-phase unbalanced 6-bus RDS representation 25

Figure 35 Three-phase unbalanced 6-bus RDS with missing buses and sections 26

Figure 36 RCM matrices sparsity plots for the 25 and 90 bus RDSs 28

Figure 37 SBM for three-phase unbalanced 6-bus RDS 30

Figure 38 Three-phase section model 32

Figure 39 The final three-phase section model after Kron s reduction 34

Figure 310 Nominal ^-representation for three-phase RDS section 36

Figure 311 Three-phase capacitor bank (a) The schematics (b) The Modeling 40

Figure 312 The FFRPF solution method flow chart 46

Figure 313 10-bus meshed distribution network 50

Figure 314 Fundamental cut-sets for a meshed 10-bus DS 57

Figure 315 31-bus RDS 62

Figure 316 The RCM of the 31-bus RDS 63

Figure 317 The RCM-1 of the 31-bus RDS 64

Figure 318 The SBM of the 31-bus RDS 65

Figure 319 The BSM of the 31-bus RDS 66

Figure 3 20 90-Bus RDS 71

Figure 321 69-bus multi-feeder RDS 72

Figure 322 Komamoto 15-bus RDS 73

Figure 323 28-bus weakly meshed distribution network 75

Figure 324 mRCM for 28-bus weakly meshed distribution network 75

Figure 325 mSBM for 28-bus weakly meshed distribution network 76

Figure 326 C for 28-bus weakly meshed distribution network 76

Figure 327 70-bus meshed distribution system 78

Figure 328 201-bus hybrid augmented test distribution system 80

Figure 329 10-bus three-phase unbalanced RDS 81

Figure 330 The 10-bus three-phase unbalanced RDS RCM 82

xni

Figure 331 21 The 10-bus three-phase unbalanced RDS RCM1 83

Figure 332 The 10-bus three-phase unbalanced RDS SBM 84

Figure 333 The 10-bus three-phase unbalanced RDS BSM 85

Figure 334 IEEE 13-bus 3^ unbalanced RDS 86

Figure 41 The Conventional SQP Algorithm 107

Figure 42 The FSQP Algorithm 112

Figure 43 Case 1 33-busRDS 114

Figure 44 Optimal real power losses for placement of optimal DG size at all the 32

buses using APC method 117

Figure 45 Optimal real power losses vs different DG power factors at bus 30 117

Figure 46 Bus voltages improvement before and after installing a single DG at

bus 30 118

Figure 47 Voltage profiles comparisons of 33-bus RDS cases 120

Figure 48 Voltage improvement of the 33-bus RDS due to three DG installation

compared to pre-DG single and double-DG cases 122

Figure 49 Voltage profiles improvement in 33-bus RDS for all DG cases 124

Figure 410 Case 2 69-bus RDS test case 125

Figure 411 Optimal power losses obtained using APC procedure 126

Figure 412 Real power losses vs DG power factor 69-bus RDS 128

Figure 413 Bus voltage improvements via single DG installation in the 69-bus

RDS 129

Figure 414 Variation in power losses as a function of the DG output at bus 61 129

Figure 415 Bus voltage magnitudes of the original 69-bus RDS and Single DG

and double DGs cases 131

Figure 416 69-bus RDS bus voltage magnitudes improvement in all DG cases 133

Figure 51 Number of publications in IEEEIET and ScienceDirect Databases

since the year 2000 140

Figure 52 Interaction between particles to share the gbest information 150

Figure 53 Illustration of velocity and position updates mechanism for a single particle during iteration k 151

Figure 54 PSO particle i updates its velocity and position vectors during two consecutive iterations A and k+ 152

Figure 55 The proposed HPSO solution methodology 159

xiv

Figure 56 Convergence characteristic of the proposed HPSO in the fixedsingle DG case HPSO proposed number of iterations = 30 164

Figure 57 Convergence characteristic of the proposed HPSO in the fixedsingle

DG case HPSO extended number of iterations = 50 165

Figure 58 Swarm particles on the first HPSO iteration 165

Figure 59 Swarm particles on the fifth HPSO iteration 166

Figure 510 Swarm particles on the tenth HPSO iteration 166

Figure 511 Swarm particles on 15th HPSO iteration 167

Figure 512 Swarm particles on the 20 HPSO iteration 167

Figure 513 Swarm Particles on the 25th HPSO iteration 168

Figure 514 Swarm Particles on the last HPSO iteration 168

Figure 515 A close-up for the particles on the 30th HPSO iteration 169

Figure 516 Convergence characteristics of HPSO in the 69-bus fixedsingle DG case HPSO proposed number of iterations = 15 184

Figure 517 Convergence characteristics of HPSO in the 69-bus fixed single DG

case HPSO proposed number of iterations = 50 185

Figure 518 Swarm particles distribution at the first HPSO iteration 185

Figure 519 Swarm particles distribution at the 5 HPSO iteration 186

Figure 520 Swarm particles distribution at the 10 HPSO iteration 186

Figure 521 Swarm particles distribution at the 15l HPSO iteration 187

Figure 522 Close up of the HSPO particles at iteration 15 187

xv

ABSTRACT

Distribution Generation (DG) has gained increasing popularity as a viable element of electric power systems DG as small scale generation sources located at or near load center is usually deployed within the Distribution System (DS) Deployment of DG has many positive impacts such as reducing transmission and distribution network congestion deferring costly upgrades and improving the overall system performance by reducing power losses and enhancing voltage profiles To achieve the most from DG installation the DG has to be optimally placed and sized In this thesis the DG integration problem for single and multiple installations is handled via deterministic and heuristic methods where the results of the former technique are used to validate and to be compared with the latters outcomes

The unique structure of the radial distribution system is exploited in developing a Fast and Flexible Radial Power Flow (FFRPF) method that accommodates the DS distinct features Only one building block bus-bus oriented data matrix is needed to perform the proposed FFRPF method Two direct descendent matrices are utilized in conducting the backwardforward sweep employed in the FFRPF technique The proposed method was tested using several DSs against other conventional and distribution power flow methods Furthermore the FFRPF method is incorporated within Sequential Quadratic Programming (SQP) method and Particle Swarm Optimization (PSO) metaheuristic method to satisfy the power flow equality constraints

In the deterministic solution method the sizing of the DG is formulated as a constrained nonlinear optimization problem with the distribution active power losses as the objective function to be minimized subject to nonlinear equality and inequality constraints Such a problem is handled by the developed Fast Sequential Quadratic Programming method (FSQP) The proposed deterministic method is an improved version of the conventional SQP that utilizes the FFRPF method in handling the power flow equality constraints Such hybridization resultes in a more robust solution method and drastically reduces the computational time In a subsequent step the placement portion of the DG integration problem is dealt with by using an All Possible Combinations (APC) search method Afterward the FSQP methods outcomes were compared to those of the developed metaheuristic optimization method

The difficult nature of the overall problem poses a serious challenge to most derivative based optimization methods due to the discrete nature associated with the bus location Moreover a major drawback of deterministic methods is that they are highly-dependent on the initial solution point As such a new application of the PSO metaheuristic method in the DG optimal planning area is presented in this thesis The PSO is improved in order to handle both real and integer variables of the DG mixed-integer nonlinear constrained optimization problem The algorithm is utilized to simultaneously search for both the optimal IDG size and bus location The proposed approach hybridizes PSO with the developed FFRPF algorithm to satisfy the equality constraints The inequality constraints handling mechanism is dealt with in the proposed Hybrid PSO (HPSO) by combining the rejecting infeasible solutions method with the preserving feasible solutions method Results signify the potential of the developed algorithms with regard to the addressed problems commonly encountered in DS

xvi

LIST OF ABBREVIATIONS AND SYMBOLS USED

ACO

BFGS

BSM

CHP

CIGRE

CN

DER

DG

DG

DGs

DS

EG

EP

EPAct

EPRI

FD

FFRPF

FSQP

GA

GRG

GS

GWEC

HPSO

IP

KCL

KKT

KVL

LP

Ant Colony Optimization

Quasi-Newton method for Approximating and Updating the Hessian Matrix

Bus Section Matrix

Combined-Heat and Power

The International Council on Large Electric Systems

Condition Number

Distribution Energy Resources

Dispersed Generation

Decentralized Generation

Distribution Generation sources

Distribution System

Embedded Generation

Evolutionary Programming

English Policy Act of 1992

Electric Power Research Institute

Fast Decoupled

Fast and Flexible Radial Power Flow

Fast Sequential Quadratic Programming

Genetic Algorithm

Generalized Reduced Gradient

Gauss-Seidel

Global Wind Energy Council

Hybrid PSO

Interior Point method

Kirchhoff s Current Law

Karush-Khun-Tuker conditions

Kirchhoff s Voltage Law

Linear Programming

xvii

wBSM

mNS

wRCM

mRCM

mSBM

mSBMp

NB

NB

HDG

riL

NR

NS

NS

ftwDG

Pf PSO

PUHCA

PURPA

QP

RCM

RDS

RIT

RPF

SA

SBM

SE Mean

SQP

StDev

TS

UnSpec pf

Meshed BSM

Number of segments in meshed DS

Meshed RCM

Modified mRCM

Meshed SBM

Submatrix of wSBM that correspond to the RDS tree sections

Number of Buses

Number of DS Buses

Total number of DGs

Number of Links or number of the fundamental loops

Newton-Raphson

Number of Sections

Number of Sections in RDS AND in meshed DS tree

Total number of the unspecified pf DGs

power factor

Particle Swarm Optimization

Public Utilities Holding Company Act of 1935

Public Utilities Regulatory Policy Act of 1978

Quadratic Programming

Radial Configuration Matrix

Radial Distribution System

The Reduction in CPU execution Time

Radial Power Flow

Simulated Annealing

Section Bus Matrix

Standard Error of the Mean

Sequential Quadratic Programming

Standard Deviation

Tabu search algorithm

Unspecified power factor DG

xviii

U S P B Unique Set of Phase Buses

USPS Unique Set of Phase Sections

xf Unique set of phase buses

iff Unique set of phase sections

Zsec Section primitive impedance matrix

Z^ (3 X 3) section symmetrical impedance matrix

R D S section length

zu Per unit length self-impedance of conductor i

h Per unit length mutual- impedance be tween conductors a n d

rt Resis tance of conductor i

rd Ear th return conductor resistance

k Inductance multiplying constant

De Dis tance between overhead and its earth return counterpart

GMRj Geometr ic mean radius of conductor i

Dy Dis tance between conductors a n d

Vgbc Three-phase sending end voltages

Vg deg Three-phase receiving end voltages

Ias c Three-phase sending end section currents

lfc Three-phase receiving end section currents

Fscc Three-phase shunt admittance of section k

[]3x3 (3 X 3) identity matrix

[^Lx3 (3x3) zero matrix

^Klc Vol tage drop across three-phase section k

ysect Section k three-phase currents

V0 Nomina l bus voltage

V Operat ing bus voltage

xix

P0 Real power consumed at nominal voltage

Q0 Reactive power consumed at nominal voltage

S Bus load apparent power at single-phase bus sect

YsKus Total three-phase shunt admittance at bus i

Ic Three-phase shunt currents at bus i

IlucSi Bus three-phase currents

jabc Three-phase load current

IltLP Current through single-phase section p and phase ltjgt

its Current at bus and phase ^

Vss Substation voltage magnitude

Vls Substation complex phase voltage

VLt Voltage drop across section k in phase (j)

A and symbol

IMI oo-norm vector II I loo

91 (bull) Real part of complex value

3 (bull) Imaginary part of complex value

C Fundamental loop matrix which is a submatrix of mSBM

Zioop (laquoLx nL) loop-impedance matrix

Csec Upper submatrix of the C matrix with ((NB-1) x (nL) dimension

Zoop Loop-impedance matrix

setrade (NSxNS) meshed DS section-impedance diagonal matrix

ZtradeS (NSxNS) RDS or tree section-impedance diagonal matrix

IL (NB-1 x 1) RDS bus load currents vector

fnlsec (mNS x 1) segments currents column vector of meshed DS network vector

mILL (mNSx 1) meshed DS bus loads and links currents vector

Itrade (NB-1) tree section currents column

xx

( n L x 1) fundamental loop current vector which is also the meshed DS link loop

currents column vector

B ( N B - 1 xmNS) fundamental cut-sets matrix

^ s7 e c f a ( N B - 1 x nL) co-tree cut-set matrix

^ymesh Voltage drops across the tree sections of the meshed DS vector

ymesh j k g messed DS bus voltage profiles vector

PRPL Real power losses

Pj Generated power delivered to DS bus i

PjL Load power supplied by DS bus i

Yjj Magnitude of the if1 element admittance bus matrix

rv Phase angle of Yy = YyZyy

Vi Magnitude of DS bus complex voltage

8 Phase angle of V = ViZSl

bull Transpose of vector or matrix

bull Complex conjugate of vector or matrix

V (1 xNB) DS bus Thevenin voltages

Y (NB xNB) DS admittance matrix

A^ Real power mismatch at bus i

AQt Reactive power mismatch at bus i

|L| Infinity norm where llxll = max (|x|) II llco J II llraquo =l2iVBVI ]gt

bull+ Max imum permissible value

bull Minimum permissible value

bull0 Nominal value

PDG D G operating power factor

S^G D G generated apparent power

SsS Main DS substation apparent power

1 Scalar related to the allowable D G size

xxi

Sy Apparent power flow transmitted from bus to bus j

Stradex Apparent power maximum rating for distribution section if

(x) The objective function

z(x) Equality constraints

g(x) Inequality constraints

(bull) Independent unknown variables lower bounds

(bull) Independent unknown variables upper bounds

x Independent unknown variables vector

RPL ( X ) Distribution system real power losses objective function

d Search direction vector

a Positive step size scalar

WRPL (x ) Gradient of the objective function at point xk)

pound Lagrange function

H^ (nxri) Hessian symmetric matrix at point xw

h^ First-order Taylors expansion of the equality constraints at point xw

Vh(x^) (nxm) Jacobian matrix of the equality constraints at point xw

g ^ First-order Taylors expansion of the inequality constraints at point xw

Vg(x^) (nxp) Jacobian matrix of the inequality constraints at point xw

Xi Individual equality Lagrange multiplier scalar

Pi Individual inequality Lagrange multiplier scalar

k w-dimensional equality Lagrange multiplier vector

P (-dimensional inequality Lagrange multiplier vector

s A predefined small tolerance number

A Active set

m Number of all equality constraints

p Number of all inequality constraints

a Number of the active set equations

xxii

v 2 j6k)

XX

nTgtG

nuDG

y

v Y FFRPFbl

deg FFRPF bl

llAP II II lloo

Vi

Xi

Cj C2

rXgtr2

w

pbestj

gbesti

nk

X

APT Losses

pHPSO Losses

pFSQP Losses

Hessian of the Lagrange function

Total number of DGs

Total number of the unspecified DGs

The change in the Lagrange functions between two successive iterations

Voltage magnitude of bus i obtained by the FFRPF technique

Voltage phase angle of bus obtained by the FFRPF technique

Voltage deviation infinity norm ie II AV = max ( I F - K I ) deg llcD

=UAlaquoVI deg ngt

Particle i velocity

Particle i position vector

Individual and social acceleration positive constants

Random values in the range [0 l] sampled from a uniform distribution

Weight inertia

Personal best position associated with particle own experience

Global best position associated with the whole neighborhood experience

Maximum number of iterations

Constriction factor

The deviation of losses calculated by HPSO method from that determined

by FSQP method

Mean value of HPSO simulation results of real power losses

FSQP deterministic method result of real power losses

xxiii

ACKNOWLEDGEMENTS

All Praise and Thanks be to reg (Allah) Almighty whose countless bounties enabled me to

accomplish this thesis successfully I would like to express my deepest gratitude to my

parents who taught me the value of education and hard work A special note of gratitude

to my brothers Falah and Abdullah deer sisters my wife my daughter Najla and my

sons Fahad Falah and Othamn They endured the long road along with me and

provided me with constant support motivation and encouragement during the course of

my study

I would like to express my sincere gratitude to my advisor Dr M E El-Hawary for

his professional guidance valuable advice continual support and encouragement I also

appreciate the constructive comments of my PhD External Examiner Dr M A Rahman

I am also grateful to my advisory committee members Dr T Little and Dr W Phillips

for spending their valuable time in reading evaluating and discussing my thesis

I would like to acknowledge the academic discussions and the constant

encouragement I received from my dear friend Dr Mohammed AlRashidi- Thank you

Abo Rsheed I wish also to thank a special friend of mine Dr AbdulRahman Al-

Othman for his friendship and for believing in me

I would like to manifest my gratitude to the Public Authority for Applied Education

and Training in Kuwait who sponsored me through my PhD at Dalhousie University

From the Embassy of Kuwait Cultural Attache Office special thanks are due to Shoghig

Sahakyan for her efforts and help to make this work possible

xxiv

CHAPTER 1 INTRODUCTION

11 MOTIVATION

Electric power system networks are composed typically of four major subsystems

generation transmission distribution and utilizations Distribution networks link the

generated power to the end user Transmission and distribution networks share similar

functionality both transfer electric energy at different levels from one point to another

however their network topologies and characteristics are quite different Distribution

networks are well-known for their low XR ratio and significant voltage drop that could

cause substantial power losses along the feeders It is estimated that as much as 13 of

the total power generation is lost in the distribution networks [1] Of the total electric

power system real power losses approximately 70 are associated with the distribution

level [23] In an effort towards manifesting the seriousness of such losses Azim et al

reported that 23 of the total generated power in the Republic of India is lost in the form

of losses in transmission and distribution [4]

Distribution systems usually encompass distribution feeders configured radially and

exclusively fed by a utility substation Incorporating Distribution Generation sources

(DGs) within the distribution level have an overall positive impact towards reducing the

losses as well as improving the network voltage profiles Due to advances in small

generation technologies electric utilities have begun to change their electric

infrastructure and have started adapting on-site multiple small and dispersed DGs In

order to maximize the benefits obtained by integrating DGs within the distribution

system careful attention has to be paid to their placement as well as to the appropriate

amount of power that is injected by the utilized DGs In other words to achieve the best

results of DG deployments the DGs are to be both optimally placed and sized in the

corresponding distribution network

The motivation of this thesis research is to investigate placing and sizing single and

multiple DGs in Radial Distribution Systems (RDSs) The problem investigated involves

two stages finding the optimal DG placements in the distribution network and the

optimal size or rating of such DGs The optimal DG placement and sizing are dealt with

by utilizing deterministic and heuristic optimization methods

12 DISTRIBUTION GENERATION - HISTORIC OVERVIEW

During the first third of the twentieth century there were no restrictions on how many

utility companies could be owned by financial corporations known as utility holding

companies By 1929 80 of US electricity was controlled by 16 holding companies

and three of those corporations controlled 36 of the nations electricity market [5]

During the Great Depression most of these utility holding companies went bankrupt As

a result the US Congress Public Utilities Holding Company Act (PUHCA) of 1935

regulated the gas and electric industries and restricted holding companies to the

ownership of a single integrated utility PUHCA indirectly discouraged wholesale

wheeling of power between different states provinces or even countries The Public

Utilities Regulatory Policy Act (PURPA) of 1978 allowed grid interconnection and

required electric utilities to buy electricity from non-utility-owned entities called

Qualifying Facilities (QF) at each utilitys avoided cost The term QF refers to non-

utility-owned (independent) power generators The term at each utilitys avoided cost

is interpreted to mean that the utility shall buy the generated electricity at a price

equivalent to what it would cost the utility itself if had generated the same amount of

power in its own facility or if it had purchased the power from an open electricity market

ie what the utility saves by not generating the same amount of power This act heralded

the dawn of the DG industry era which paved the way to generate electricity arguably at

a lower cost compared to that of traditional utility companies and consequently have it

delivered to the end user at lower rates The English Policy Act of 1992 (EPAct)

intensified competition in the wholesale electricity market by opening the transmission

system for access by utilities and non-utilities electricity producers [67] entity A could

sell its power to entity B through entity Cs transmission infrastructure

13 DISTRIBUTION GENERATION

DG involves small-scale generation sources scattered within the distribution system level

atnear the load center ie close to where the most energy is consumed [8] The DG

2

generate electricity locally and in a cogeneration case heat can also be generated and

may be utilized in applications such as industrial process heating or space heating DG

generally has better energy efficiency than large-scale power plants The traditional

power stations usually have an efficiency of around 35 whereas the efficiency of DG

such as a Combined Heat and Power (CHP) gas turbine would be in the vicinity of 45-

65 [5]

It seems that there is no universal agreement on the definition of DG size range The

Electric Power Research Institute (EPRI) for example defined the DG size to be up to 5

MW in 1998 [9] in 2001 EPRI redefined the DG capacity to be less than 10 MW [10]

and by 2003 they identified the DG to have a power output ranging from 1 kW to 20 MW

[11] The IEEE published its DG-standards IEEE Std 1547-2003 and IEEE Std 15473-

2007 and emphasized that they are applicable to DGs that have total capacity below 10

MVA [1213] In its 2006 report about the impact of DG Natural Resources Canada

estimated that the DG size starts from few 10s of kW to perhaps 5 MW [14] In 2000

the International Council on Large Electric Systems (CIGRE) referred to the DG as non-

centrally dispatched usually attached to distribution level and smaller than 50-100 MW

[1516]

Many terms referring to DG technology are used in the literature such as Dispersed

Generation (DG) Decentralized Generation (DG) Embedded Generation (EG)

Distribution Energy Resources (DER) and on-site Generation [17] In particular the

term dispersed generation customarily refers to stationary small-scale DG with power

outputs ranging from 1 kW to 500 kW [7]

Late developments and innovations in the DG technology industry liberalization of

the electricity market transmission line congestion and increasing interest in global

warming and environmental issues expedited publicizing their deployment and adoption

world-wide Recent studies suggest that DG will play a vital role in the electric power

system An EPRI study predicts that by the year 2010 25 of the newly installed

generation systems will be DG [18] and a similar study by the Natural Gas Foundation

projects that the share of DG in new generation will be 30 [15] By 2003 around 40

of Denmarks power demand was served by DG while Spain the Netherlands Portugal

and Germany integrated nearly 20) of DG into their distribution networks [19] Of the

3

643 GW generated by the European Union in 2005 approximately 122 GW (19) was

generated by hydro 96 GW (15) of this generated capacity was cogeneration (CHP)

and 53 GW (8) generated by other renewable energy systems Half of the CHP

generated capacity was owned by utility companies and the other half was generated by

independent producers [20]

Globally in 2005 the total installed wind power capacity was 591 GW and the

Global Wind Energy Council (GWEC) expected the wind capacity to reach 1348 GW by

the year 2010 [21] Worldwide wind energy capacity of 19696 MW was added in the

year 2007 [22] and approximately 1400 MW of wind energy capacity was added in the

US during the second quarter of 2008 [23] GWEC also predicted in its Global Wind

Energy Outlook 2008 report that by the year 2020 15 billion tons of CO2 will be saved

every year and by the middle of the 21st century 30 of the worlds electricity will be

supplied by wind energy [24] compared to a total of 13 of the global electricity being

generated by wind at the end of 2007 [22]

DG technologies include a variety of energy sources ie powered by renewable or

by fossil fuel-based prime movers Renewable technologies used in DG include wind

turbines photovoltaic cells small hydro power turbines and solar thermal technologies

while DG based on conventional technologies may involve gas turbines CHP gas

turbines diesel engines fuel cells and micro-turbine technologies Some DGs are

installed by the utility company on the supply side of the consumers meter while some

are installed by the customers themselves on their side of a bi-directional meter thus

enabling them to benefit from the net-metering program offered by utility companies

[25]

Optimal deployment of DG technology would have an overall positive impact

although some negative traits would remain The noise and shadow flicker caused by

large wind blades and the noise caused by the wind turbine gearbox or gas turbines

especially when placed close to residential or populated areas are examples of negative

impacts of widespread use of DG Another drawback from an environmentalist point of

view is that wind DG could disturb bird immigration patterns and cause death to both

birds and bats [26] Renewable-source DGs also could be an indirect source of pollution

by causing the fossil-fuel power plants to shut down and start up more frequently as they

4

attempt to accommodate variable DG power output [27] Some plants have an emission

rate which is inversely proportional to its delivered power Voltage rise as a result of bishy

directional power flow caused by the interconnection of the DG in RDS is another

example of a negative impact caused by DG [28]

The integration of DG into electric power networks has many benefits Some

examples of such benefits could be summarized as follows

bull Improve both the reliability and efficiency of the power supply

bull Release the available capacity of the distribution substation as well as reducing

thermal stresses caused by loaded substations transformers and feeders

bull Improve the system voltage profiles as well as the load factor

bull Decrease the overall system losses

bull Generally DG development and construction have shorter time intervals

bull Delay imminent upgrading of the present system or the need to build newer

infrastructure and subsequently avoid related problems such as right-of-way

concerns

bull Decrease transmission and distribution related costs

bull In general DG tends to be more environmentally friendly when compared to

traditional coal oil or gas fired power plants

The extent of the benefits depends on how the DG is placed and sized in the system In

addition to supplying the system with the power needed to meet certain demands as an

installation incentive the real power losses could be minimal if the DG is optimally sited

and sized

14 THESIS OBJECTIVES AND CONTRIBUTIONS

Optimal integration of single and multiple DG units in the distribution network with

specified and unspecified power factors is thoroughly investigated from a planning

perspective in this thesis The DG problem is handled via deterministic and heuristic

optimization methods where the results of the former method are used to validate and to

be compared with those of the latter

The unique radial distribution structure is exploited in developing a Fast and Flexible

Radial Power Flow (FFRPF) method to deal with a wide class of distribution systems

5

eg radial meshed small large balanced and unbalanced three-phase networks The

proposed FFRPF algorithm starts by developing a Radial Configuration Matrix (RCM)

for radial topology DS or meshed RCM (mRCM) for meshed DS Both matrices consist

of elements with values 1 0 and -1 The RCM (or mRCM) corresponding building

algorithm is simple fast and practical as illustrated in Chapter 3 The RCM is inverted

only once to produce the Section Bus Matrix (SBM) which is then transposed to obtain

the Bus Section Matrix (BSM) For the meshed topology the corresponding resultant

matrices for the mRCM are the meshed Bus Section Matrix (mSBM) and the meshed Bus

Section Matrix (mBSM) The FFRPF technique relies on the backwardforward sweep

that only utilizes the basic electric laws such as Ohms law and both Kirchhoff s voltage

and current laws The backward current sweep is performed via SBM (or mSBM) and

the forward voltage update sweep is executed via the BSM (or mBSM) By utilizing the

two obtained matrices all the bus complex voltages can be obtained and consequently

left to be compared with the immediate previous obtained bus voltages The proposed

approach quickened the iterative process and reduced the CPU time for convergence It

is worth mentioning that the building block matrix is the only input data required by the

FFRPF method besides the DS parameters to perform the proposed distribution power

flow The FFRPF technique is incorporated in both utilized deterministic and

metaheuristic optimization methods to satisfy the power flow equality constraints

requirements

In the deterministic solution method the DG sizing problem is formulated as a

nonlinear optimization problem with the distribution active power losses as the objective

function to be minimized subject to nonlinear equality and inequality constraints

Endeavoring to obtain the optimal DG size an improved version of the Sequential

Quadratic Programming (SQP) methodology is used to solve for the DG size problem

The conventional SQP uses a Newton-like method which consequently utilizes the

Jacobean in handling the nonlinear equality constraints The radial low XR ratio and the

tree-like topology of distribution systems make the system ill-conditioned

A Fast Sequential Quadratic Programming (FSQP) methodology is developed in

order to handle the DG sizing nonlinear optimization problem The FSQP hybrid

approach integrates the FFRPF within the conventional SQP in solving the highly

6

nonlinear equality constraints By utilizing the FFRPF in dealing with equality

constraints instead of the Newton method the burden of calculating the Jacobean and

consequently its inverse as well as the complications of the ill-conditioned Y-matrix of

the RDS is eliminated Another advantage of this hybridization is the drastic reduction

of computational time compared to that consumed by the conventional SQP method

In this thesis a new application of the Particle Swarm Optimization (PSO) method in

the optimal planning of single and multiple DGs in distribution networks is also

presented The algorithm is utilized to simultaneously search for both the optimal DG

size and its corresponding bus location in order to minimize the total network power

losses while satisfying the constraints imposed on the system The proposed approach

hybridizes PSO with the developed distribution radial power flow ie FFRPF to

simultaneously solve the optimal DG placement and sizing problem The difficult nature

of the overall problem poses a serious challenge to most derivative based optimization

methods due to the discrete flavor associated with the bus location in addition to the

subproblem of determining the most suitable DG size Moreover a major drawback of

the deterministic methods is that they are highly-dependent on the initial solution point

The developed PSO is improved in order to handle both real and integer variables of the

DG mixed-integer nonlinear constrained optimization problem Problem constraints are

handled within the proposed approach based on their category The equality constraints

ie power flows are satisfied through the FFRPF subroutine while the inequality bounds

and constraints are treated by exploiting the intrinsic and unique features associated with

each particle The proposed inequality constraint handling technique hybridizes the

rejection of infeasible solutions method in conjunction with the preservation of feasible

solutions method One advantage of this constraint handling mechanism is that it

expedites the solution method converging time of the Hybrid PSO (HPSO)

15 THESIS OUTL INE

This thesis is organized in six chapters The research motivation brief description of the

DG and the thesis objectives are addressed in the first chapter The second chapter deals

with a literature review of the distribution power flow methods and the DG optimal

planning problem In the third chapter development of the proposed FFRPF method

7

utilized in the FSQP and the HPSO methods to satisfy the DG problem of nonlinear

equality constraints is presented The fourth chapter deals with the DG sizing problem

formulation and its solution based on the two deterministic solution methods The

problem is solved via the conventional SQP and the proposed FSQP methods and a

performance comparison between them is presented Basic concepts of the PSO are

presented in chapter five A brief literature review regarding the use of the PSO in

solving the electric power system problems is presented in this chapter In addition it

also addresses the development of the proposed HPSO in solving the DG planning

problem The last chapter provides the thesis concluding remarks and the scope of future

work

8

CHAPTER 2 LITERATURE REVIEW

21 INTRODUCTION

Recent publications in the areas of work relative to this thesis are reviewed and

summarised in this chapter which is organized in two sections as follows

bull The first section reviews the literature on distribution power flow methods A

brief background of conventional power flow methods is presented followed

by a review and summary of the literature on recent developments of the

distribution power flow algorithms

bull The DG integration problem is reviewed in the second section Recent work

on the optimal DG placement and sizing via analytical deterministic and

metaheuristic methods are analyzed and reviewed

22 DISTRIBUTION POWER FLOW

Power flow programs play a vital role in analyzing power systems The problem deals

with calculating unspecified bus voltage angles and magnitudes active and reactive

powers as well as (as a by-product) line loadings and their associated real and reactive

losses for certain operating conditions These values are typically obtained through

iterative numerical methods to analyze the status of a given power system

Since the middle of last century many methods were proposed to solve this problem

Even though Dunstan [29] was the first to demonstrate a digital method for solving the

power flow problem in 1954 Ward and Hale [30] are often credited with the successful

digital formulation and solution of the power flow problem in 1956 Most of the earlier

solution methods were based on both the admittance matrix and the Gauss-Seidel (GS)

iterative method The poor convergence characteristics of GS when large networks

andor ill-conditioned situations are encountered led to the development of the Gaussian

iterative scheme (Zbus) [3132] and later the Newton-Raphson (NR) method [33] as well

as the Decoupled [34] and Fast Decoupled (FD) power flow approaches [35] Though

the NR method generally converges faster than other methods it takes longer

computational time per iteration When Tinney et al [36] introduced the optimally

9

ordered and sparsity-oriented programming techniques Newton-based methods became

the de facto industry standard However the Jacobian matrix for the RDS is

approximately four times the size of the corresponding admittance matrix and it needs to

be evaluated at each iteration

Although conventional power flow methods are well developed in dealing with the

transmission and sub-transmission sections of the power system networks they are

deemed to be inefficient in handling distribution networks This is because the

Distribution System (DS) is different in several ways from its transmission counterpart

DS has a strictly radial topology nature or weakly meshed networks in contrast with

transmission systems which are tightly meshed networks DS is a low voltage system

having low XR ratio sections and a wide range of reactance and resistance values DS

may consist of a tremendously large number of sections and buses spread throughout the

network Sections of the DS could have unbalanced load conditions due to the

unbalanced three-phase loading as well as single and double phase loads in spurred

lateral lines The mutual couplings between phases are not negligible due to rarely

transposed distribution lines [37] All of these characteristics strongly suggest that DS is

to be classified as an ill-conditioned power system

The practical DSs low XR ratio sections may cause both the NR and FD

conventional methods to diverge [38-41] The line impedance angles are small enough to

deteriorate the dominance of the NR Jacobian main diagonal making it prone to

singularity Such a low XR value would also prevent the Jacobian matrix from being

decoupled and simplified

In addition to performance considerations a practical power flow technique needs to

consider all the DS distinctive features and to accommodate the imbalance introduced by

multiphase networks along with the distribution-level loads In the literature a number

of Newton and non-Newton power flow methods designed for distribution systems were

proposed Zhang et al [42] solved the distribution power flow based on the Newton

method although the proposed Jacobian is computed just once the solution converged

with a number of additional iterations more so than the conventional approach

Moreover shunt capacitor banks were ignored in the modeling as well as the line shunt

admittance (JI model) and the constant impedance loads Baran and Wu [43] solved the

10

power flow problem by utilizing three fundamental quadratic equations representing the

real and reactive section powers and the bus voltages in an iterative scheme as a

subroutine during the process of optimizing the capacitor sizing However they

computed the Jacobian using the chain rule within the proposed NR method which is in

turn time consuming Mekhamer et al [44] utilized the three equations developed in [43]

using a different iterative technique without the need for the Jacobian or the NR method

However their process is based on applying a multi-level iterative process on the main

feeder and laterals which makes the speed and the efficiency of their proposed algorithm

a function of the RDS configuration and topology

In [4546] the quadratic equation was also utilized in determining the relation

between the sending and receiving end voltage magnitudes along with the section power

flow They proposed to include the system power losses within their calculation while

solving for the system power flow However the voltage phase angles were ignored

during the solution of the radial power flow in order to speed up the convergence The

latter reference developed work was based on the assumption of balanced RDS and

sophisticated numbering scheme

The radial power flow introduced by [47-49] used a non-Newton power flow techshy

nique based on the ladder network theory This method adds the section currents and

calculates the RDS bus voltages including the substations during a backward sweep If

the difference between the calculated substation voltage value and substation predetershy

mined assigned bus voltage value is acceptable the iterations are concluded If not the

substation bus voltage is reset and the RDS bus voltages are computed for the second

time in the same iteration in the forward sweep Both the ladder and the backshy

wardforward methods are derivative-free instead they employ simple circuit laws

However the ladder method uses many sub-iterations on the laterals and calculates the

system bus voltages twice during a single iteration compared to once in the backshy

wardforward method Thukaram [50] utilized the backwardforward sweep technique to

solve the RDS power flow However the bus numbering procedure was a sophisticated

parent node and child node arrangement which may add some computational overshy

head if the system topology is changed Teng [51] used the backwardforward approach

as the solution procedure through the development of two matrices and multiplied them

11

together in a later stage of the solution process In assembling those matrices all the

system buses and sections have to be inspected carefully In a practical large RDS data

preparation for these matrices will be cumbersome and prone to errors Under continshy

gency situations switching operations or the addition of another feeder to the existing

one are quite common practices in the DSs hence changes in system topology need to be

accommodated by restructuring the corresponding matrices which would add an overshy

head to track modifications The weakly meshed DS was dealt with by adding extra

nodes in the middle of the new links Two equal currents with opposite polarities were

injected into each added node Each injection operation is represented by a two column

matrix which was subsequently added to the first proposed matrix and then the develshy

oped matrices were extended and multiplied together The resultant is a full matrix and

its dimension is reduced by the Kron method in every single iteration That is the

developed full matrix was inverted in each iteration of the solution method and such

procedure is expensive lengthy cumbersome and time consuming

Shirmohammadi et al [39] Cheng and Shirmohammadi [52] and Hague [53]

proposed an iterative solution method for both radial and weakly meshed DSs This

approach necessitates a special numbering scheme in which they number the DS sections

in layers starting from the root node The numbering scheme is to be carried out

carefully by examining the whole system when a new layer is to be numbered The

numbering process is cumbersome and prone to errors For weakly meshed networks

breakpoints are selected opened and consequently the meshed system is converted to a

radial system The loops are broken by adding two fictitious buses In each pair of

dummy buses equal and opposite currents are injected and the new system is evaluated

to produce a reduced order impedance matrix Their proposed method requires that the

breakpoint impedance matrix should be computed cautiously Such a procedure is highly

dependent on the distribution networks topology That is the more links that exist in the

DS the larger the break point impedance matrix and the more time will be consumed in

its computation

Goswami and Basu [38] introduced a direct solution method to solve for radial and

weakly meshed DS They applied a breakpoints method into the meshed DS similar to

that of [39] in order to convert it into RDS In their proposed methodology a restriction

12

was imposed on each of the system buses not to have more than three sections attached to

it Such limitation is a drawback of the method and moreover a difficult node numbering

scheme is a disadvantage

In this thesis the unique structure of the RDS is exploited in order to build up a new

fast flexible power flow technique that deals with radial and looped DSs The numbering

scheme of the DS is simple and straightforward All load types can be accommodated by

the proposed distribution power flow eg spot and distributed loads Unlike

conventional power flow methods no trigonometric functions are used in the proposed

distribution power flow method For weakly meshed and looped DSs the system is dealt

with as it is there is no need for radialization cuts or building breakpoints impedance

matrix The topology of the tested DS whether strictly radial weakly meshed or looped

is represented by a building block matrix which is the only one needed to perform the

backwardforward sweep technique

23 DG INTEGRATION PROBLEM

DG is gaining increasing popularity as a viable element of electric power systems The

presence of DG in power systems may lead to several advantages such as supplying

sensitive loads in case of power outages reducing transmission and distribution networks

congestion and improving the overall system performance by reducing power losses and

enhancing voltage profiles Some of the negative impacts of DG installations are

potential harmonic injections the need to adopt more complex control schemes and the

possibility of encountering reverse power flows in power networks Even though the

concept of DG utilization in electric power grids is not new the importance of such

deployment is presently at its highest levels due to various reasons Recent awareness of

conventionaltraditional thermal power plants harmful impacts on the environment and

the urge to find more environmentally friendly substitutes for electrical power generation

rapid advances made in renewable energy technologies and the attractive and open

electric power market are a few major motives that led to the high penetration of DG in

most industrial nations power grids To achieve the most from DG installation special

attention must be made to DG placement and sizing

13

The problem of optimal DG placement and sizing is divided into two subproblems

where is the optimal location for DG placement and how to select the most suitable size

Many researchers proposed different methods such as analytic procedures as well as

deterministic and heuristic methods to solve the problem

231 Solving the DG Integration Problem via Analytical and Deterministic Methods

In the literature the optimal DG integration problem is solved by means of employing

any analytical or optimization technique that suits the problem formulation Methods and

procedures of optimally sizing and locating the DGs within the DS are varied according

to objectives and solution techniques

Willis [54] presented an application of the famous 23 rule originally developed

for optimal capacitor placement to find a suitable bus candidate for DG placement That

is to install a DG with a rating of 23 of the utilized load at 23 the radial feeder length

down-stream from the source substation However this rule assumes uniformly

distributed loads in a radial configuration and a fixed conductor size throughout the

distribution network In any event the 23 rule was developed for all-reactive load

These assumptions limit its applicability to radial distribution systems and the fact that it

is only suitable for single DG planning

Kashem et al [55] developed an analytical approach to determine the optimal DG

size based on power loss sensitivity analysis Their approach was based on minimizing

the DS power losses The proposed method was tested using a practical distribution

system in Tasmania Australia However it assumes uniformly distributed loads with all

the connected loads along the radial feeder having the same power factor and it also

assumes no external currents injected into the system buses eg capacitors which limits

its practicality

Wang and Nehrir [56] developed an analytical approach to address the optimal DG

placement problem in distribution networks with different continuous load topologies

Minimizing the real power losses was the objective of the proposed method In their

approach the DG units were assumed to have unity power factor and only the overhead

distribution lines with neglected shunt capacitance are considered The candidate bus

was selected based on elements of the admittance matrix power generations and load

14

distribution of the distribution network The issue of DG optimal size was not addressed

in their formulation

Griffin et al [57] analyzed the DG optimal location analytically for two continuous

load distributions types ie uniformly distributed and uniformly increasing loads The

goal of their study was to minimize line losses One of the conclusions of their research

was that the optimal location of DG is highly dependent on the load distribution along the

feeder ie significant loss reduction would take place when placing the DG toward the

end of a uniformly increasing load and in the middle of uniformly distributed load feeder

Acharya et al [58] used the incremental change of the system power losses with

respect to the change of injected real power sensitivity factor developed by Elgerd [59]

This factor was used to determine the bus that would cause the losses to be optimal when

hosting a DG By equating the aforementioned factor to zero the authors solved for the

optimal real value of DG output They proposed an exhaustive search by applying the

sensitivity factor on all the buses and ranked them accordingly The drawback of their

work is the lengthy process of finding the candidate locations and the fact that they

sought to optimize only the DG real power output Furthermore they only considered

planning of a single DG

Popovic et al [60] utilized sensitivity analysis based on the power flow equations to

solve the DG placement and sizing Two indices were used in ranking all the DS buses

for the suitability of hosting the DG The first one is a voltage sensitivity index which is

derived directly from the NR power flow Jacobian inverse the second one exploits the

relation of incremental real power losses with respect to the injected real and reactive

power as developed in [61] Their objective for sizing the DG was to maximize its

capacity subject to boundary constraints such as bus voltage penetration level line flows

and fault current limits To solve the sizing DG problem they gradually increased the

DG capacity at selected most sensitive buses until one of the constraints is violated and

the direct previous installed DG size becomes the one chosen as the optimal rating This

process is a lengthy and impractical procedure and the authors did not elaborate on how

they would deal with multiple DG cases using the proposed scheme

Keane and OMalley [62] solved for the optimal DG size in the Irish system by using

a constrained Linear Programming (LP) approach To cope with the EU regulation which

15

emphasizes that Ireland should provide 132 of its electricity from renewable sources

by 2010 the objective of their proposed method was to maximize the DG generation

The nonlinear constraints were linearized with the goal of utilizing them in the LP

method A DG unit was installed at all the system buses and the candidate buses were

ranked according to their optimal objective function value

Rosehart and Nowicki [63] dealt with only the optimal location portion of the DG

integration problem They developed two formulations to assess the best location for

hosting the DG sources The first is a market based constrained optimal power flow that

minimized the cost of the generated DG power and the second is voltage stability

constrained optimal power flow that maximized the loading factor distance to collapse

Both formulations were solved by utilizing the Interior Point (IP) method The outcomes

of the two formulations were used in ranking the buses for DG installations The optimal

DG size problem was not considered in their paper

Iyer et al [64] employed the primal-dual IP method to find the optimal DG

placement through combined voltage profile improvement and line loss reduction indices

However the proposed approach was based on initially placing DGs at all buses in order

to determine proper locations for DG installations This methodology may not be

realistic for large scale distribution networks

Rau and Wan [65] only treated the DG sizing problem by utilizing the Generalized

Reduced Gradient (GRG) method The DG bus locations were assumed to be provided

by the system planner for the DG units to be installed In their proposed method they

considered minimizing the system active power losses In their formulation only the

power flow equality constraints were considered whereas the boundary conditions and

the inequality constraints were not taken into account

Hedayati et al [66] employed continuous power flow methodology to locate the

buses most sensitive to voltage collapse The sensitive bus set is ranked based on their

severity which is used accordingly to indicate potential bus locations for placement of

single and multiple DG sources An iterative method was proposed for optimally sitting

the DG A certain DG capacity which is known and fixed a priori is added to the DS

and the conventional power flow method was employed to determine the resultant DS

real power losses voltage profiles and power transfer capacity In the subsequent

16

iteration another DG with the same capacity was added to the next sensitive bus and

results were obtained This iterative process would continue until the system outcomes

reached acceptable values The proposed iterative method did not optimize the DG size

232 Solving the DG Integration Problem via Metaheuristic Methods

Metaheuristic techniques have proven their effectiveness in solving optimization

problems with appreciable feasible search space They can be easily modified to cope

with the discrete nature associated with different elements commonly used in power

systems studies Optimization methods such as Genetic Algorithm (GA) [67-71] GA

hybrid methods such as GA and fuzzy set theory [72] and GA and Simulated Annealing

(SA) [73] Tabu search algorithm (TS) [7475] GA-TS hybrid method [76] Ant Colony

Optimization (ACO) [77] Particle Swarm Optimization (PSO) [78] and Evolutionary

Programming (EP) [79] were utilized in the literature to solve for the DG integration

problem

Teng et al [67] developed a value-based method for solving the DG problem The

GA method was utilized in maximizing a DG benefit to cost ratio index subject to only

boundary constraints such as ratio index voltage drop and feeder transfer capacity A

drawback of their procedure is that the candidate DG bus locations were assumed to be

provided by the utility and consequently all combinations of the provided bus locations

were tested for obtaining the optimal DG capacities via the GA method

The proposal set forth by Mithulananthan et al [68] made use of the DS real power

losses as the fitness function to be minimized through GA Their formulation of the DG

size optimization problem is of an unconstrained type Moreover the NR method which

is usually inadequate in dealing with the DS topology was used in calculating the total

power losses Candidate DG bus locations were obtained by placing a DG unit at all

buses of the tested DS which is impractical for large DSs Furthermore the multiple

DGs case was not addressed

Haesen et al [69] and Borges et al [70] solved the DG integration problem by

basically employing the GA method Both utilized the metaheuristic technique in solving

for single and multiple DG sizing and placements Haesen et al used the GA method to

minimize the DS active power flow while the objective for Borges et al was to

17

maximize a DG benefit to total cost ratio index Reference [69] incorporated penalty

factors within the objective function to penalize constraint violations thus adding another

set of variables to be tuned The authors of the latter reference used a PV model for

modeling the DG

Celli et al [71] formulated the DG integration problem as an s-constraint

multiobjective programming problem and solved it using the GA method Their

proposed algorithm divided the set of the objective functions into one master and the rest

are considered as slave objective functions The master is treated as the primary

objective function that is to be minimized while the slaves are regarded as new

inequality constraints that are bounded by a predetermined e value They utilized their

hybrid method to minimize the following objective functions cost of network upgrading

energy losses in the DS sections and purchased energy (from transmission and DG) The

number of the DG sources to be installed was randomly assigned and the units were

randomly located at the network buses Whenever the constraints are violated the

objective function solution is penalized A Pareto set was calculated from this

multiobjective optimization problem to aid the distribution planner in the decision

making process

Kim et al [72] and Gandomkar et al [73] hybridized two methods to solve the DG

sizing problem The former hybridized GA with fuzzy set theory to optimally size the

single DG unit while the latter combined the GA and SA metaheuristic methods to solve

for the optimal DG power output In both references the DG sizing problem was

formulated as a nonlinear optimization problem subject to boundary constraints only

Unlike Gandomkar et al [73] added the nonlinear power flow equality constraints to

their problem formulation The former researchers utilized their methodology to

investigate multiple DG case while the latter solved only the single DG case Both sited

the DG at all DS buses in order to determine the optimal DG location and size

Nara et al [74] assumed that the candidate bus locations for the DG unit to be

installed were pre-assigned by the distribution planner Then they used the TS method in

solving for the optimal DG size The objective of their formulation was to minimize the

system losses The DG size was treated as a discrete variable and the number of the

18

deployed units was considered to be fixed The DS loads were modeled as balanced

uniformly distributed constant current loads with a unity power factor

Golshan and Arefifar [75] applied the TS method to optimally size the DG as well

as the reactive sources (capacitors reactors or both) within the DS They formulated

their constrained nonlinear optimization problem by minimizing an objective function

that sums the total cost of active power losses line loading and the cost of the added

reactive sources The DG locations were not optimized instead a set of locations were

designated to host the proposed DGs and the reactive sources

A hybrid method that combined the GA with the TS technique in order to solve the

DG sizing optimization problem was developed by Gandomkar et al [80] They solved

the DG integration problem by minimizing the distribution real power losses subject to

boundary conditions The authors restricted the number of DGs as well as their gross

capacity to be revealed prior to executing the optimization procedure They augmented

the objective function with penalty terms in their formulation to handle the constraint

violations

Falaghi and Haghifam [77] proposed the ACO methodology as an optimization tool

for solving the DG sizing and placement problems The minimized objective function for

the utilized method was the global network cost ie the summation of the DGs cost their

corresponding operational and maintenance cost the cost of energy bought from the

transmission grid and the cost of the network losses The DG sizes were treated as

discrete values They used a penalty factor to handle the violated constraints ie

infeasible solutions In addition to modeling the DG sources as exclusive constant power

delivering units ie with unity power factor the network loads were all assumed to have

09 power factor Thus it can be stated that such modeling is impractical especially when

real large DSs are encountered

Raj et al [78] dealt with the DG integration in two different steps They employed

the PSO method to optimally determine the size of single and multiple DGs The optimal

location portion of the problem was performed utilizing the NR power flow method to

assign those buses with the lowest voltage profiles as the optimal candidate DG locations

The PSO was used to minimize the system real power losses the voltage profiles

boundary conditions were the only constraints required by the authors to be satisfied

19

Constraint violations were handled via a penalty factor that was augmented with the

objective function The DG units were randomly sited at one or more of the candidate

buses obtained through the NR method and subsequently the PSO was used to find the

optimal size(s)

Rahman et al [79] derived sensitivity indices to identify the most suitable bus for a

single DG installation Subsequently the DG sizing problem was dealt with by

employing an EP approach The objective function of the proposed approach was to

minimize the DS real power losses subject only to the system bus voltage boundary

constraints The formulation of DG sizing in their work was not realistic for a variety of

reasons For instance they ignored the line loading restrictions power flow equality

constraints and DG size limits

In most of the reviewed work on the DG deployment problem the problems of DG

optimal sizing and placement were not simultaneously addressed due to the difficult

nature of the problem as it combines discrete and continuous variables for potential bus

locations and DG sizing in a single optimization problem This combination creates a

major difficulty to most derivative-based optimization techniques and it increases the

feasible search space size considerably In this thesis the DG sizing subproblem is

solved using an improved SQP deterministic method while the two subproblems are

addressed simultaneously via an enhanced PSO metaheuristic algorithm

24 SUMMARY

In this chapter distribution power flow techniques were reviewed in Section 22 The

literature review of DG integration problem solution methods was presented in Section

23 The analytical and deterministic methods that were utilized to handle the DG

integration problem were presented in Subsection 231 Then recent publications that

handled the DG sizing and placement problems via wide-class of metaheuristic methods

were reviewed and summarized

20

CHAPTER 3 FAST AND FLEXIBLE RADIAL POWER FLOW FOR

BALANCED AND UNBALANCED THREE-PHASE DISTRIBUTION

NETWORKS

31 INTRODUCTION

As discussed in Chapter 2 several limitations exist in radial power flow techniques

presently reported in the literature such as complicated bus numbering schemes

convergence related problems and the inability to handle modifications to existing DS in

a straightforward manner This motivated the development of an enhanced distribution

power flow solution method In this thesis the unique structure of the RDS is exploited in

order to build up a Fast Flexible Radial Power Flow (FFRPF) technique The tree-like

RDS configuration is translated into a building block bus-bus oriented data matrix

known as a Radial Configuration Matrix (RCM) which consequently is utilized in the

solution process The developed algorithm is also capable of handeling weakly meshed

and meshed DSs via a meshed RCM (mRCM) RCM or mRCM is the only matrix that

needs to be constructed in order to proceed with the iterative process During the data

preparation stage each RCM (or mRCM) row focuses only on a system bus and its

directly connected buses That is while building such a matrix there is no need to

inspect the entire system buses and sections Moreover no complicated node numbering

scheme is required The building block matrix is designed to have a small condition

number with a determinant and all of its eigenevalues equal to one to ensure its

invertibility By incorporating this matrix and its direct descendant matrices in solving

the power flow problem the CPU execution time is decreased compared with other

methods The FFRPF method is flexible in accommodating any changes that may take

place in an existing radial distribution system since these changes can be exclusively

incorporated within the RCM matrix The proposed power flow solution technique was

tested against other methods in order to judge its overall performance using balanced and

unbalanced DSs

In Chapters 4 and 5 the FFRPF method is incorporated within the developed FSQP

and HPSO algorithms in solving the optimal DG installation problem It is implemented

21

as a subroutine within the proposed algorithms to satisfy the equality constraints ie

solving the radial power flow equations

32 FLEXIBILITY AND SIMPLICITY OF FFRPF I N NUMBERING THE RDS

BUSES AND SECTIONS

The RDS is configured in a unique arborescent structure with the distribution substation

located at its root node from which all the active and reactive power demands as well as

the system losses are supplied The substation feeds one or more main feeders with

spurred out laterals sublaterals and even subsublaterals For this reason the substation is

treated as a swing bus during the power flow iterative procedure

Most radial power flow techniques proposed in the literature assign sophisticated

procedures for numbering the radial distribution networks in order to execute their

algorithms This is cumbersome when expanding andor modifying existing RDSs In

this section a very simple numbering rule for the RDS buses and sections is introduced

A section is defined as part of a feeder lateral or sublateral that connects two buses in the

RDS and the total Number of Sections (NS) is related to the total Number of Buses (NB)

by this relation (NS=NB -1 )

321 Bus Numbering Scheme for Balanced Three-phase RDS

A balanced radial three-phase RDS is represented by a single line diagram In such a

system a feeder or sub level of a feeder having more than one bus is numbered in

sequence and in an ascending order Consequently each section will carry a number

which is less than its receiving end bus number by one as shown in Figure 31

Therefore sections are numbered automatically once the simple numbering rule is

applied

22

Substation

Figure 31 10-busRDS

In numbering the RDS shown in Figure 31 the following was considered buses 1 -

4 form the main feeder and buses 2 - 6 and 3 - 10 are the laterals while the sublateral is

tapped off bus 5 Figure 32 and Figure 33 manifest the ease in renumbering the system

shown previously and the flexibility in adding any portion of RDS to the existing one

respectively In Figure 32a buses 1 -4 have a different path compared to Figure 31 and

are considered to be a main feeder buses 2 - 9 and 3 - 6 are the laterals whereas the

sublateral 7 - 10 is the section spurred off bus 7 Figure 32b shows another numbering

scheme The same system numbered differently would have the same solution when

solved by the FFRPF

Figure 33 illustrates the ease of numbering in the case of a contingency situation or

a switching operation that could cause the existing system to be modified andor to be

augmented with other systems The lateral 3 - 10 in Figure 31 was modified to be

tapped off bus 2 instead and a couple of radial portions were added to be fed from buses

6 and 4 as illustrated in the figure

23

Substation Substation

(a) (b)

Figure 32 Different ways of numbering the system in Fig 31

Figure 33 The ease of numbering a modified and augmented RDS

322 Unbalanced Three-phase RDS Bus Numbering Scheme

The three-phase power flow is more comprehensive and realistic when it comes to

finding the three-phase voltage profiles in unbalanced RDSs Figure 34 shows an

unbalanced three-phase RDS The missing sections and buses play a significant role in

the multi-level phase loading and in making the unbalanced state of such a three-phase

DS more pronounced

24

The RDS shown in Figure 34 includes 6 three-phase buses (3(j)NB = 6) 5 three-

phase sections (3(|)NS = 5) no matter how many phase buses or sections exist physically

As such it has 14 single-phase buses (1(|)NB = 14) and 11 single-phase sections (1lt|gtNS =

11) The relations expressed in Eq (31) govern the three-phase and single-phase buses

to their corresponding sections

3^NS = 3^NB-1

l^NS = l^NB-3 (31)

Figure 34 Three-phase unbalanced 6-bus RDS representation

It is simple to implement the numbering process in the three-phase system as was

done in the balanced case Any group of phase buses to be found along a phase feeder or

a sub level of a feeder is to be numbered in a consecutive ascending order Consequently

each phase section number will carry a number which is one less than its receiving end

bus number as shown in Figure 34 In other words the sections are numbered routinely

after the ordering of the three-phase RDS buses has been completed

To develop the building block matrix as will be shown shortly the unbalanced three-

phase system is redrawn by substituting for any missing phase section or bus using dotted

representation as depicted in the 6-bus RDS in Figure 35 By performing this step each

three-phase bussection in the RDS consists of a group of 3 single-phase busessections

a b and c including the missing ones for double and single-phase buses

25

l a

I (1) 2 a | (2) 3 a | (3) 4 a | (4)

Figure 35 Three-phase unbalanced 6-bus RDS with missing buses and sections

33 THE BUILDING BLOCK MATRIX AND ITS ROLE I N FFRPF

The proposed FFRPF procedure starts with a matrix that mimics the radial structure

topology called a system Radial Configuration Matrix (RCM) The inverse of RCM is

then obtained to produce a Section Bus Matrix (SBM) that will be utilized in summing

the section currents during the backward sweep procedure A Bus Section Matrix (BSM)

is next generated by transposing the SBM to sum up the voltage drops in the forward

sweep process Therefore the only input data needed in the solution of an existing

modified or extended RDS other than the system loads and parameters is the RCM

It is worth mentioning that the inversion and transposition operations take place only

once during the whole process of the proposed FFRPF methodology for a tested RDS

whereas other methods like the NR technique invert the Jacobian matrix in every single

iteration The following subsections demonstrate the building of a three-phase RCM and

elucidate the role of both SBM and BSM in solving the radial power flow problem

331 Three-phase Radial Configuration Matrix (RCM)

The only matrix needed to be built for an unbalanced three-phase RDS is the RCM

Whatever changes need to be accommodated as a modification in the existing structure or

an addition to the existing network would be performed through the RCM only The

26

other matrices utilized in the backwardforward sweep are the direct results of the RCM

and no other built matrix is needed to perform the FFRPF

Such a matrix exploits the radial nature of such a system The RCM is of 3(3())NB x

3(|)NB) dimension in which each row and column represents a single-phase bus For a

balanced three-phase RDS represented by a single line diagram the RCM dimension is

(NBxNB) The RCM building algorithm for the unbalanced three-phase RDS case is

illustrated as follows

1 Construct a zero-filled 3(3(|)NB x 3(|)NB) square matrix

2 Change all the diagonal entries to +1 every diagonal entry represents sending

missing or far-end buses

3 In each row if the column index corresponds to an existing receiving single-phase

bus its entry is to be changed to - 1

4 If a single-phase bus is missing or is a far-end bus the only entry in its

corresponding row is the diagonal entry of+1

The above RCM building steps are summarized in the following illustration

Columns Description

RCMbdquo

if is either

a - sending phase bus b - far-end phase bus c - missing phase bus (32)

-1 jkl if jkI are receiving phase buses

connected physically to phase bus 0 otherwise

The [abc] matrix is defined as a zero (3 x 3) matrix with the numbers a b and c as

its diagonal elements eg [ I l l ] is the identity (3 x 3) matrix while [110] is the identity

matrix with the third diagonal element replaced by a zero By following the preceding

steps and utilizing the [abc] definition the RCM for the unbalanced three-phase RDS

shown in Figure 35 is to be constructed as shown in (33)

27

[Ill] [000] [000] [000] [000]

[000]

-[111] [111]

[000] [000] [000]

[000]

[000] -[111]

[111] [000] [000]

[000]

[000] [000]

-[110] [111]

[000] [000]

[000]

[000] [000]

-[010] [111]

[000]

[000] [000]

-[on] [000] [000]

[111]

Because of the nature of the RDS the RCM has three distinctive properties The first

is that the RCM is sparse the second is that RCM is a strictly upper triangular matrix

and thirdly such a matrix is only filled by 0 +1 or - 1 Such a real matrix makes the data

preparation easy to handle and less confusing Figure 36 shows the sparsity pattern plots

of the RCM matrices for unbalanced three-phase 25-bus [48] and balanced 90-bus [38]

radial systems

RCM for 25-Bus unbalanced 3-phase RDS RCM for 90-Bus balanced 3-phase RDS

nz = 131 nz = 179

Figure 36 RCM matrices sparsity plots for the 25 and 90 bus RDSs

3311 Assessment of the FFRPF Building Block RCM

The RCM is well-conditioned and should have a small Condition Number (CN) and a

non-zero determinant The CN measures how far from singularity any matrix is It is

defined as

28

cond(A) = A jjA-l (34)

where ||A|| is any of the 3 types of norm formulations of matrix A 1-norm 2-norm or oo-

norm An ill-conditioned matrix would have a large CN while a CN of 1 represents a

perfectly well-conditioned matrix By definition a singular matrix would have an infinite

CN [81] Having all the matrix eigenvalues to be equal to 1 and a determinant value of 1

safeguard the RCM against singularity For this reason the RCM is not only invertible

but also its inverse is an upper triangular matrix filled only with 0 and +1 digits and no

other numbers would appear in RCM-1

332 Three-phase Section Bus Matrix (SBM)

The SBM for the three-phase RDS is obtained by performing the following steps

1 Remove the corresponding substation rows and columns from the RCM ie the

first three rows and columns The reduced version of the RCM is labeled as

RCM

2 Invert the RCM to obtain the SBM as shown in (35) and more explicitly in

Figure 37

To clarify the two rows and the two columns outside the matrix border shown in

Figure 37 are the three-phase buses and sections ordered respectively The dimension of

the unbalanced three-phase system SBM is 3(3lt|)NS x 3(|gtNS) For the balanced case the

SBM dimension is (NSxNS)

[Ill] [000]

[000] [000] [000]

[111] [111]

[000] [000] [000]

[110] [110]

[111] [000] [000]

[010] [010] [010] [111] [000]

[011] [011] [000] [000] [111]

29

1 a

1 b

1 c

2 a

2 b 2 c

SBM = 3 a

3 b

3 c

4 a

4 b

4 c

5 a

5 b

5 c

2 a

0 0 0 0 0 0 0 0 0 0 0 0 0 0

2 b

0 1 0 0 0 0 0 0 0 0 0 0 0 0 0

2 c

0 0 1 0 0 0 0 0 0 0 0 0 0 0 0

3 3 a b

1 0 0 1 0 0 i o 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

3 c

0 0 1 0 0

1 0 0 o 0 0

o 0 0 0

4 a

1 0 0 1 0

o 1 0

o 0 0

o 0 0 0

4 b

0 1 0 0 1 0 0 1 0 0 0 0 0 0 0

4 c

0 0 0 0 0 0 0 0 1 0 0 0 0 0 0

5 a

0 0 0 0 0 0 0 0 0 1 0 0 0 0 0

5 b

0 1 0 0 1 0 0 1 0 0 1 0 0 0 0

5 c

0 0 0 0 0 o 0 0

o 0 0 1 0 0 0

6 a

0 0 0 0 0

o 0 0

o 0 0 o 1 0 0

6 b

0 1 0 0 1 0 0 0 0 0 0 0 0 1 0

6 c

0 1 0 0 1 0 0 0 0 0 0 0 0 1

Figure 37 SBM for three-phase unbalanced 6-bus RDS

By inspecting Figure 35 it is noted that any single-phase section is connected

downhill to a single-phase far-end bus or buses through a Unique Set of Phase Buses

(USPB) xt bull By inspecting Figure 35 and Figure 37 the USPB for the following

single-phase sections la lb lc 3b 4b are^0 [ gt xlgt x respectively as explained

in (36)

=2 f l 3bdquo4a x=2b3b4b5bA]

Xl=2cA) Xl=) (36)

X=5b]

In the SBM the single-phase section is represented by a row i and will have entries

of ones in all the columns where their indices represent single-phase buses that belong to

the section USPB xf bullgt a s illustrated in (37)

SBMrmt =

Columns Description

c bullgt bull a - Id - buses e r ~ - 1 ijk- ijk-- are either w (37)

lb - diagonal entry 0 other columns otherwise

30

333 Three-phase Bus Section Matrix (BSM)

The BSM is the transpose of the SBM as shown in (38) In the BSM the rows represent

the RDS single-phase buses excluding the substations and all the sections are

represented by the BSM columns Each single-phase bus is connected uphill through a

Unique Set of Phase Sections (USPS) yf to a substation single-phase bus The USPS

for buses 2deg 3a 4b 5b 6C are y ydeg y y yc6 respectively as depicted in Figure 35

and Figure 37 and demonstrated in (39)

BSM

[111] [000] [000] [000] [000]

[111] [111] [000] [000] [000]

[110] [110] [111] [000] [000]

[010] [010] [010] [111] [000]

[011] [011] [000] [000] [111]

(38)

V H U V3=1gt2 y=b2b yb

5=lb2bdquoAb (39)

lt=1C2C5C

In the BSM a single-phase bus i is represented by a row and will have entries of

ones in all the columns where their indices represent single-phase sections that belong to

the bus USPS yf as equivalently shown in (310)

Columns Description

BSMrmi =

( gt - [a-l^-sectionse yf ~ i m

1 ijk ijk are either lt Y Y (310) lb - diagonal entry

0 other columns otherwise

34 FFRPF APPROACH AND SOLUTION TECHNIQUE

The FFRPF methodology is based simply on Kirchhoff s voltage and current laws which

are used in performing the backwardforward sweep iterative process By utilizing the

direct descendant matrices of the RCM ie SBM and BSM the RDS buses complex

31

voltages are calculated in every iteration until convergence criteria are met The next

subsections illustrate the proper usage of such matrices in the proposed FFRPF method

through appropriate modeling of the unbalanced multi-phase RDS section impedances

341 Unbalanced Multi-phase Impedance Model Calculation

Figure 38 shows a three-phase section model that is represented by two buses (sending

and receiving ends) of an overhead three-phase four-wire RDS with multi-grounded

neutral The assumption of a zero voltage drop across the neutral in a three-phase two-

phase and single-phase RDS is found to be valid [4582] Such a configuration is widely

adopted in North Americas distribution networks [8384]

V Sa

_ bull

V Ra

V Sb ab

^VWVVYgt

V Sc Z be

bn

^AAArmdashrYYYV bull

V s n en

ll1 Sa

s b

La

Lb

Lc

bull r Figure 38 Three-phase section model

In the proposed radial power flow solution method each of the three-phase lines is to

be modeled appropriately and mutual coupling effects between phases are not neglected

The primitive impedance matrix for such a four-wire system is a square matrix with a

dimension equal to RDS utilized number of phase and neutral conductors For a system

consisting of three-phase conductors and a neutral wire the section primitive impedance

matrix is expressed as shown in (311)

32

Zaa

ha

Zca

zna

Kb

Kb

Kb

Kb

Ke Kc Ke

nc

art

Zbn

en

nn

where

Z bull primitive impedance matrix

RDS section length

z per unit length self-impedance of conductor i

z per unit length mutual-impedance between conductors andy

zu and zy are calculated according Carsons work [85] and its modifications [86-88] as

illustrated by the following equations

where

k

GMRj

Dbdquo

v GMR

bulli J

zu=rt+rd+ja)k

zv=rd+jltok

resistance of conductor i

earth return conductor resistance

inductance multiplying constant

distance between overhead and its earth return counterpart and it is a

function of both earth resistivity and frequency

geometric mean radius of conductor i

distance between conductors i andj

(312)

(313)

The parameters used in (312) and (313) are shown in Table 31 for both operational

frequencies 50Hz and 60Hz in both metric and imperial units

33

Table 31 cok rj and De Parameters for Different Operation Conditions

De = 2160 Ij (ft)

cok rd

p = 100 Qm

p = 1000 Qm

Metric Units RDS operating frequency 50 Hz 60 Hz

006283km

0049345 QJ km

931 m

29443 m

007539 km

005921412km

850 m

26878 m

Imperial Units RDS operating frequency 50 Hz 60 Hz

010111mile

00794 QI mile

30547f

96598 ft

012134mile

009528 QI mile

27885

88182

Since the neutral is grounded the primitive impedance matrix Zsec can be

transformed into a (3 x 3) symmetrical impedance matrix Zsae

c by utilizing Krons

matrix reduction method The resultant section three-phase impedance matrix is

expressed mathematically in (314) and the three-phase section model is represented

graphically in Figure 39

7 abc

aa

zba Zca

Zab

^bb

Zcb

zac zbc Zee

(314)

VSn

mdash bull i 7 T

i ah bull-sec a

zbdquobdquo bull A V W Y Y Y V

v izK

^WW-rrYYv -+bull

I

bull

vR

Figure 39 The final three-phase section model after Krons reduction

If the RDS section consists of only one or two phase lines its primitive impedance

matrix is transformed by Krons matrix reduction to (laquo-phase x w-phase) symmetrical

impedance matrix Next the corresponding row and column of the missing phase are

replaced by zero entries in the (3x3) section impedance matrices Zsae

c For a two-phase

34

section its impedance matrix Z^c is demonstrated below

Z_a

zci Kron h-gt ZZ za

zbdquo zai

zaa o zac

0 0 0

z_ o zbdquo

Underground lines such as concentric neutral and tape shielded cables are typically

installed in the RDS sections For underground cables with m phases and n additional

neutral conductors the primitive impedance of each section is a (2m+n x 2m+n) matrix

with the entries computed as illustrated in [89-92]

Usually the RDS is modeled as a short line ie less than 80 km and the charging

currents would be neglected by not modeling the line shunt capacitance as depicted in

Figure 38 However under light load conditions and especially in the case of

underground cables the line shunt capacitance needs to be considered in order to obtain

reasonable accuracy ie use nominal 7i-representation The rc-equivalent circuit consists

of a series impedance of the section and one-half the line shunt admittance at each end of

the line Figure 310 (a) (b) and (c) show the RDS 7i-model representation The shunt

admittance matrix for an overhead three-phase section is a full (3x3) symmetrical

admittance matrix while it is a strictly diagonal matrix for the underground RDS cable

section That is the self admittance elements are the only terms computed [92] For the

unbalanced three-phase section eg one or two phases the non-zero elements of shunt

admittances are only those corresponding to the utilized phases

[zic] -AAVmdashrwvgt

T yabc 1 |_ sec J [ yaf tc |

sec J

(a)

35

Lsec J _

2

s

1

Yaa

Yba

Yea

Yab

Ybb

Ycb

Yac

Ybc

Ycc

zaa

zba

tea

zab

zbb

zcb

zac

zbc

zcc

P yabc ~|

lgtlt 2 - =

Yaa

Yba

Yca

Yab

Ybb

Ycb

R

1 1

Yac

Ybc

Ycc

(b)

[ yabc 1 sec J

s 1 1

Yaa

0

0

0

Ybb

0

0

0

Ycc

zaa

zba

zca

Zab

zbb

zcb

zac

zbr

zcc

V yabc ~j

L sec 2 - =

Yaa

0

0

0

Ybb

0

R

1 1

0

0

Ycc

(c)

Figure 310 Nominal 7i-representation for three-phase RDS section

(a) Schematic drawing for the single line diagram of 3(|gt RDS (b) Matrix equivalent for overhead section (c) Matrix equivalent for underground cable section

By applying Kirchhoff s laws to the three-phase system section k the relationship

between the sending and receiving end voltages for medium and short line models and

the voltage drop across the same section in the latter model are expressed in Eq (315)-

(316) and Eq (317) respectively

36

rabc S rabc S

14 L sec Jax3 L rabc

3x3 L secgt J3x3

[C]3 [4 [ yabc~ |~ yabc 1

sec J3x3 L sec J3 [4

zt 1 L sec J3x3

f yabc ~| [~ yaampc 1

L sec J3x3 L sec h

bull R rabc

(315)

rrabc VS rabc

S

1 J3x3 L sec J

[degL [L abc R

(316)

where

TT-afec rrabc S ^ R

rabc rabc S XR

rabc

AK

13x3

aAc sect

rabc see

KrH^ic] three-phase sending and receiving end voltages

three-phase sending and receiving end section currents

three-phase shunt admittance of section k

(3gtlt3) identity matrix

(3gtlt3) zero matrix

voltage drop across three-phase section k

section k three-phase currents

(317)

It is worth noting that Eq (315) is reduced to Eq (316) in the case of short line

modeling since YS^C 1 = 0 The voltage drop across the three-phase section k in the short

line model is expressed in Eq (317) and its corresponding sending end phase voltages

can be expressed in expanded forms as follows

V = V + Ta 7 + 1 7 + Tc 7 rS yR ^1secZjaa ^ Isec^ab ^rlsecpoundjac

v =v +r 7 +17 +r 7 y S y R ^ l sec^ab ^ 1 sec^bb ^ 1 sec^Ac

S mdashyR+ Ktc^ca + sec^cb + sec^a

(318)

(319)

(320)

Equations (317)-(320) show that the voltage drop along any phase in a three-phase

section depends upon all the three-phase currents

37

342 Load Representation Accurate and proper load modeling is of significant concern in power distribution

systems as well in its transmission systems counterpart [8693] Loads in electric power

systems are usually expressed by adequate representations so as to mimic their effects

upon the system The load dependency on the operating bus voltage and on system

frequency is among those representations

Static load models are often utilized in the power flow studies since they relate the

apparent power active and reactive directly to the bus operating voltage A static load

model is used for the static load components ie resistive and lighting load and as an

approximation to the dynamic load components ie motor-driven loads [93] Generally

static loads in DS are assumed to operate at rated and fixed frequency value [94-96]

Loads in the DS are usually expressed as function of the bus operating voltage and

represented by exponential andor polynomial models

The exponential model is shown in (321) and (322)

P = Pbdquo

Q = Q0 vbdquo

(321)

(322)

where

V0 nominal bus voltage

V operating bus voltage

P0 real power consumed at nominal voltage

Q0 reactive power consumed at nominal voltage

Exponents a and fi determine the load characteristics and certain a and values lead to a

specific lode model Therefore

1 If a = P mdash 0 the model represents constant power characteristics ie the load is

constant regardless of the voltage magnitude

2 If a = P = 1 the model represents constant current characteristics ie the load is

proportional to the voltage magnitude

3 If a = P = 2 the model represents constant impedance characteristics ie the load is

38

a quadratic function of the voltage magnitude

As indicated in [97] the exponents could have values larger than 2 or less than 0 and

certain load components would be represented by fractional exponents

The constant current model is considered to be a good approximation for many

distribution circuits since it approximates the overall performance of the mixture of both

constant power and constant impedance models [98] However representing loads with

the constant power model is a conservative approach with regard to voltage drop

consideration [99] and consequently this model will be used in this thesis

Loads can also be represented by a composite model ie the polynomial model The

polynomial model is expressed in (323) and (324)

P = Pbdquo

Q = Q0

(

a p

V

r

V

V

K

V

v0

2

2

V

K

V

+CP

J

)

(323)

(324)

where ap + bp + cp = 1 and aq + b + cq = 1

The polynomial model is also referred to as a ZIP model since it combines all the

three exponential models constant impedance (Z) constant current (I) and constant

power (P) models The ZIP model needs more information and detailed data preparation

The load models can be used in the FFRPF solution method during its iterative

process where flat start values are initially assumed to be the load voltages The three-

phase load voltages are changed during each iteration and consequently the three-phase

currents drawn by the constant current constant impedance andor ZIP three-phase load

models will change accordingly

Different shunt components like spot loads distributed loads and capacitor banks are

customarily spread throughout the RDS In power flow studies spot and distributed

loads are typically dealt with as constant power models while shunt capacitors are

modeled as constant impedances [94 100 101]

The uniformly distributed loads across RDS sections can be modeled equivalently by

either placing a single lumped load at one-half the section length or by placing one-half

the lump-sum of the uniformly distributed loads at each of the section end buses

39

[99 102] The former modeling approach has the disadvantage of increasing the

dimension of the RCM SBM and the BSM since more nodes would be added to the

existing RDS topology In the proposed FFRPF technique the distributed load is

modeled using the latter approach while the three-phase shunt capacitor banks are

modeled as injected three-phase currents [101] as schematically shown in Figure 311

and mathematically represented by Eq (325) and (326)

Qk Cap

^

CCap a Cap

(a) (b)

Figure 311 Three-phase capacitor bank (a) The schematics (b) The Modeling

O3 = poundCap

Qo Capa vbdquo

T-34 _ 1Cap ~

V

a Capbdquo

SQL M Cap

V

filt bullCapo

F

JQ( Cap

(325)

(326)

343 Three-phase FFRPF BackwardForward Sweep

The FFRPF technique employs the SBM in performing the current summation during the

backward sweep and the BSM in updating the RDS bus complex voltages during the

forward sweep as demonstrated in the following subsections

3431 Three-phase Current Summation Backward Sweep DS loads are generally unbalanced due to the unbalanced phase configuration the double

and single-phase loadings as well as the likelihood of unequal load allocation among the

three-phase configuration For the loads they could be represented as constant power

40

constant current constant impedance or any combination of the three models [97 103]

The three-phase load currents at three-phase bus i drawn by a three-phase load eg of a

constant impedance load model is mathematically expressed as shown in Eqs (327) and

(328)

jabc U ~

7e a gt

va

(si) ~

(327)

where

ctabc

o

V

K

2

rft

0

v K

2

K v

2

(328)

where Sf represents the load apparent power at single-phase bus lt|gt As shown in the

preceding equations each load current is a function of its corresponding bus voltage For

Eq (327) if the a phase bus is missing its corresponding phase load current is

eliminated and its corresponding position in the three-phase current vector is replaced by

a zero entry As an illustration and by assuming that there are loads connected to all

existing buses the three-phase load current vector for the system shown in Figure 35 is

expressed as follows

jabc V ja rb jc ra rb re ra rb Q Q rb Q Q rb re l l

The charging currents at the RDS three-phase buses are not to be neglected when

dealing with sections modeled as The shunt admittance at bus is obtained by

applying the following relation

where

Ysh^ bull total three-phase shunt admittance at bus

[l if section k attached to bus i

[0 otherwise

The three-phase shunt currents at bus is as shown in Eq (330)

tabc jrabc 1ch ~~ 1Anbus y i (330)

41

The 3-ltj) bus current is sum of both the 3-sect load current and the 3-cj) charging current as

expressed mathematically in Eq(331)

jabc jabc jabc ) T I 1busl ~ 1Li

+Ich V-gt )U

where 1^ is bus three-phase currents In the case of modeling a three-phase section as

a short line its charging currents are neglected ie I^c = 0 and the bus current will be

represented by the load currents only

The backward sweep sums the phase load currents in the corresponding phase

sections starting from far-end phase buses and moving uphill toward the substation phase

buses The current in phase (j) and section p is computed by utilizing the USPB

principle xp gt during the backward sweep as expressed in (332)

lt = E lt ^here = j 0 ^ J (332)

where

I current through single-phase section and phase ^ (^ =a b or c) SQCp

j current at bus and phase ltb bus x

The SBM is utilized in obtaining the system three-phase section currents in matrix

representation by performing the relation in Eq (333)

[G] = [SBM][lpound] (333)

where the I^ is a 3-(3())NS)-order column vector For a balanced short line RDS

model Eq (333) can be expressed as

[CS] = [SBM][IL] (334)

3432 Three-phase Bus Voltage Update Forward Sweep

The voltage at each phase bus is determined through the forward sweep procedure by

subtracting the sum of the voltage drops across the bus corresponding USPS from the

substation nominal complex voltage The voltage drop across three-phase section k is

calculated as in Eq (317) The voltage drop across all sections of the three-phase RDS

can be obtained by utilizing the SBM as shown in matrix notation via Eq (335)

42

[AKbdquo]=[zr][c] (335)

[ A ^ ] = [ Z s r ] [ 5 5 M ] [ C ]

where ZtradeS J is a diagonal matrix with an 3(3c|)NS x 3ltj)NS) dimension in which its

diagonal entry k corresponds to section k impedance and AV3^ is the computed three-

phase voltage drop values across all the RDS sections as shown below

A ^ = AVa AVb AVC bullbullbull AVb AVC T s e c l s e c l s e c l sec3ltWS sec3tNS J

For calculating the RDS voltage profiles the FFRPF solution method starts by asshy

suming the initial values for all bus voltages to be equal to the substation complex

voltage As a flat start the initial phase voltages at bus will be as follows

2TT 2TT

ya Jdeg vb e~^ VC e+JT V ss e V sisK v sis e (336)

where Vls is the substation complex phase voltage

For the voltage at bus m and phase (j) to be determined at iteration v the calculation is

performed as follows

= amp - pound r A lt wherer = trade lt (337)

The RDS voltage profiles at the system phase buses are obtained by utilizing the BSM as

shown in following matrix representation

[Vi] = [Vsy[BSM][AV^] (338)

where V^fs 1 and V3A are respectively the substation nominal three-phase voltage

column vector and the resultant three-phase bus voltage solution column vector and each

has a dimension of 3(3lt|gtNS)

3433 Convergence Criteria

The bus complex voltage is obtained after every backwardforward sweep After each

iteration all the bus voltage magnitudes and angles are compared with the previous

iteration outcomes The power flow process is concluded and a solution is reached if the

complex voltage real and reactive oo-norm mismatch vector is less than a certain

43

predetermined empirical tolerance value e The convergence criterion is expressed

mathematically as shown in Eq (339)

+i

([gt]w) A a ( |y f lts

where th

i iteration A

(339)

and symbol

||J| vector oo-norm ||x II = max fix IV II lloo l l c 0 1=12NBV I

5H (bull) real part of complex value

3 (bull) imaginary part of complex value

3434 Steps of the FFRPF Algorithm

The FFRPF iterative process can be summarized as follows

Step 1 Begin FFRPF by choosing a test RDS

Step 2 Number and order RDS buses and sections

Step 3 Construct RCM

Step 4 Obtain both SBM and BSM

Step 5 Select load model

Step 6 Start the iterative procedure by assuming flat start voltages for all buses

Step 7 Calculate load currents

Step 8 Start the backward sweep process by calculating section currents using SBM

Step 9 Start the forward sweep process by determining the bus complex voltages

using BSM

Step 10 Compare both magnitudes and angles of the RDS bus voltages between the

current and previous iterations

bull If the co-norm of their difference is lt st

o Solution is reached

44

o Stop and end FFRPF procedure

o Obtain bus voltage profiles section currents and power losses

etc

bull If not utilize the outcome of this iteration (bus complex voltages)

to start a new one by going back to Step 7

The FFRPF solution method is illustrated by the following flow chart shown in Figure

312

45

i laquo - i +1

Calculate Load and leakage

currents

I Start Backward

sweep process by calculating section

currents using SBM

Start Forward sweep process by determining bus

complex voltages V[+1] using BSM

V[+1] Section currents

Section Power Losses Etc

Start FFRPF

Read the test RDS data

Number and order RDS Buses and

Sections

I Construct RCM

Remove the substation

corresponding rows and columns

from RCM to Obtain RCM

Obtain RCM1

To Get SBM

Z Transpose SBM to

get BSM

Calculate RDS section

Impedance and Shunt admittance

Matrices

Select load model

Assuming a flat start voltages for

all buses V[]=10 =0

Figure 312 The FFRPF solution method flow chart

46

344 Modifying the RCM to Accommodate Changes in the RDS The FFRPF technique deals with changes in the network such as the addition of a

transformer by adjusting the original RCM to incorporate its conversion factor (c)

Subsequently the SBM and BSM are obtained accordingly and used in the

backwardforward sweep procedure If a three-phase transformer is incorporated in a

three-phase RDS between buses m and n at section n - 1 the modified BSM entries are

located at the intersection of the matrix rows and columns defined by Eq (340)

BSMZ EzL~-inBSMZ euro lt _ (340)

The affected rows and columns of the modified BSM are those belonging to the

sections USPB and the sending buss USPS respectively For demonstration purposes

the 10-bus balanced RDS shown in Figure 31 will be utilized If two transformers with

conversion factors cfj and cS are to be added within sections 3 and 6 respectively the

original RCM is modified to accommodate such additions as illustrated in (341) Thus

instead of filling -1 for the receiving end bus entry the negative of the conversion factor

is the new entry The process is repeated rc-times for -installed transformers The

corresponding modified SBM and BSM are to be obtained as demonstrated in Section

33

10

RCM^ =

1

2

3

4

5

6

7

8

9

10

1

0

0

0

0

0

0

0

0

0

-1

1

0

0

0

0

0

0

0

0

0

-1

1

0

0

0

0

0

0

0

0

0

-cfi 1

0

0

0

0

0

0

0

-1

0

0

1

0

0

0

0

0

0

0

0

0

-1

1

0

0

0

0

0

0

0

0

-cf2

0

1

0

0

0

0

0

0

0

0

0

-1

1

0

0

0

0

-1

0

0

0

0

0

1

0

0

0

0

0

0

0

0

0

-1

1

(341)

The affected entries of the new BSM are obtained by applying the relation in (340) as

follows

47

[BSMZ eZ s ^nBSMJ euro ^ 3 = 40(12=[(4l)(42)]

(laquoSWe^)nJM6^)=J7 )8)njl I4=gt[(7 )l)(7 )4)(8l) )(M)]

The matrix shown in (342) shows the final B S M after including the transformers in the

10-bus R D S Such a procedure is easily extended to the unbalanced three-phase RDSs

1

1

1

CJ

1

1

cfi

cf2

1

1

2

0

1

ch 0 0

0

0

1

1

It is worth mentioning that by integrating the cf for any transformer configuration

into the RCM building block in the FFRPF technique another light is shed on the

flexibility criterion of the proposed method

35 FFRPF SOLUTION METHOD FOR MESHED THREE-PHASE DS

In practical DS networks alternative paths are typically provided to accommodate for

any contingency incidents that might take place eg feeder failure Therefore it is not

unusual for meshed distribution networks to be part of the DS topology in order to make

the system more reliable The loop analysis approach as well as the graph theory

technique are used to study and analyze the behavior of meshed DS The loop analysis

technique basically applies Kirchhoff s voltage law principle to solve for the fundamental

loop currents in both planar and nonplanar networks while the graph theoretic

formulation preserves the network structure properties [104]

A meshed DS can be viewed from a graph theory perspective as an oriented looped

graph that preserves the network interconnection properties whereas a DS that has no

0

0

0

1

1

cfi

cf2

0

0

0

0

0

0

1

0

0

0

0

0

0

0

0

0

1

1

0

0

0

0

0

0

0

0

1

0

0

0

0

0

0

0

0

0

1

1

0

0

0

0

0

0

0

0

1

(342)

48

loops is considered a tree In graph theory terminology line segments that connect

between buses in a loopless DS tree are called twigs branches or sections (represented by

solid line segments in Figure 313) while those which do not belong to the tree are

known as links (represented by dotted line segments in Figure 313) Links are segments

that close loops in a DS tree thereby composing a co-tree Removal of all co-tree links

results in a strictly radial system Links are usually activated by closing their

corresponding Normally Open (NO) switches Whenever a link is added to a RDS

network a loop is formed and as a result the system will have as many fundamental

loops as the number of links A fundamental loop is a loop that contains only one link

besides one or more sections Segments are used here to name sections and links

together It is noted that the number of fundamental loops is significantly less than the

number of buses in the meshed DS which makes the loop analysis a more appropriate

method in dealing with such systems than other circuit analysis methods like nodal

voltage method [105]

The current directions in the meshed DS sections and links are arbitrarily chosen to

be directed form a lower bus index to a higher one and the positive direction of loop

current is assumed to in the same direction of that of the link as illustrated in Figure 313

The number of segments in a meshed DS is equal to the sum of the total number of its

corresponding graph tree sections and its co-tree links For a meshed DS with NB buses

and mNS segments (total number of sections and links in the meshed DS) the number of

links nL and the number of the fundamental loops as well are obtained according to the

following relation

laquoL=mNS-NB + l (343)

49

Substation 2 Imdash 31 4 1

^ -gtT-gtL- -

Figure 313 10-bus meshed distribution network

351 Meshed Distribution System Corresponding Matrices The FFRPF solution method can deal with the DS meshed networks through modifying

the original RCM Discussion is now focused on the balanced three-phase meshed DS

which can easily be extended to the unbalanced three-phase DS networks

Meshed RCM The meshed RCM (wRCM) order is ((NB + wL) (NB + nVj) and the

mRCM building algorithm is as follows

1 Remove links from the meshed DS and build the RCM for the resulting network

tree as demonstrated earlier in section 331

2 Add nL rows and columns toward the end of the RCM ie each link is represented

by a row and a column attached to the end of the RCM

3 In each link column there are 3 non-zero entries and are to be filled in the following

manner

a -1 at the row which corresponds to the lower index terminal of the link

b +1 at the row which corresponds to the higher index terminal of the link

c +1 at the link diagonal entry

For the 10-bus system shown in Figure 31 three directed links Li L2 and L3 are added

to the DS tree through connecting the following buses 4 - 5 6 - 8 and 4 - 1 0

respectively The system mRCM is constructed as illustrated in (344)

50

10

mRCM (13x13)

1

0

0

0

0

0

0

0

0

0

0

0

0

-1

1

0

0

0

0

0

0

0

0

0

0

0

0

-1

1

0

0

0

0

0

0

0

0

0

0

0

0

-1

1

0

0

0

0

0

0

0

0

0

0

-1

0

0

1

0

0

0

0

0

0

0

0

0

0

0

0

-1

1

0

0

0

0

0

0

0

0

0

0

0

-1

0

1

0

0

0

0

0

0

0

0

0

0

0

0

-1

1

0

0

0

0

0

0

0

-1

0

0

0

0

0

1

0

0

0

0

0

0

0

0

0

0

0

0

-1

1

0

0

0

0

0

0

-1

1

0

0

0

0

0

1

0

0

0

0

0

0

0

-1

0

1

0

0

0

1

0

0

0

0

-1

0

0

0

0

0

1

0

0

1

(344)

Remove the substation corresponding rows and columns from the mRCM to produce the

mRCM The mRCM for the 10-bus system is shown in (345)

10

mRCM (12x12)

1

0

0

0

0

0

0

0

0

0

0

0

-1

1

0

0

0

0

0

0

0

0

0

0

0

-1

1

0

0

0

0

0

0

0

0

0

-1

0

0

1

0

0

0

0

0

0

0

0

0

0

0

-1

1

0

0

0

0

0

0

0

0

0

0

-1

0

1

0

0

0

0

0

0

0

0

0

0

0

-1

1

0

0

0

0

0

0

-1

0

0

0

0

0

1

0

0

0

0

0

0

0

0

0

0

0

-1

1

0

0

0

0

0

-1

1

0

0

0

0

0

1

0

0

0

0

0

0

-1

0

1

0

0

0

1

0

0

0

-1

0

0

0

0

0

1

0

0

1

(345)

Meshed SBM The meshed SBM (mSBM) is then obtained by inverting the mRCM As

51

an illustration the 10-bus meshed network mSBM is obtained as shown in (346)

10

mSBM (12x12)

1

0

0

0

0

0

0

0

0

0

0

0

1

1

0

0

0

0

0

0

0

0

0

0

1

1

1

0

0

0

0

0

0

0

0

0

1

0

0

1

0

0

0

0

0

0

0

0

1

0

0

1

1

0

0

0

0

0

0

0

1

0

0

1

0

1

0

0

0

0

0

0

1

0

0

1

0

1

1

0

0

0

0

0

1

1

0

0

0

0

0

1

0

0

0

0

1

1

0

0

0

0

0

1

1

0

0

0

i deg 1

1

-1

1 deg 0

o 0

0

i

o o

0

0

0

0

1

-1

-1

0

0

0

1

0

0

0

1

0

0

0

0

-1

-1

0

0

1

(346)

Let the mSBM be partitioned into two submatrices mSBMp and C so that the mSBMp

submatrix corresponds to the DS tree sections and the second submatrix C to the

fundamental loops or links as shown in (347)

wSBM = SBM

6 [cl (mNSxnL) = [mSBMp C]

JmNSx(NB-l)

The dotted line shown in the above relation implies matrix partitioning

(347)

Fundamental loop matrix The second submatrix in (347) ie C is the fundamental

loop matrix which governs the direction of currents in each of fundamental loop sections

and links The fundamental loop matrix C can be partitioned into two submatrices [Csec]

and [I] as demonstrated below

M_ where [Csec] is an ((NB - 1) x (laquoL)) matrix and [7] is an nL square identity matrix The

former matrix corresponds to the tree loop sections while the latter corresponds solely to

c-- (348)

52

the co-tree links

By inspecting the fundamental loop matrix C it is noted that each row represents a

section or a link and each column represents a loop Each column entry in the C matrix

CM will have one of the following values

1 Qy = +1 if section k belongs to and is oriented in the same direction of loop

2 Cki = - 1 if section k belongs to and is oriented in the opposite direction of loop

3 Claquo = 0 if section k is not in the loop

By inspecting the 10-bus DS mSBM illustrated in (346) the first loop which is represhy

sented by the tenth column of the matrix is comprised of three sections in addition to the

link The current in two of these sections runs in the same direction as their correspondshy

ing fundamental loop ie sections 2 and 3 while the third one ie section 4 has an

opposite orientation This can be easily verified by tracing the first loop in the meshed

DS single line diagram One can also note that two loop currents pass through the third

section in an additive manner

Meshed BSM The meshed BSM (mBSM) is the transpose of mSBM as illustrated in

(349)

[ B S M I 0](mNS-nL)mNS

L J(nLxmNS)

The second submatrix C[ is the transpose of the fundamental loop matrix as manifested in

(350)

mBSM = [mSBM] = mBSMr

c (349)

C=L

1

0

0

0

2 3

1 1

0 0

0 1

4

-1

0

0

5

0

1

0

6

0

-1

0

7

0

-1

0

g

0

0

-1

9

0

0

-1

h 1

0

0

h 0

1

0

h (f 0

1

(350)

Meshed DS impedance matrices The (laquoLx riL) loop-impedance matrix Zioop is

formulated as follows [106]

KHc]|Xf][c] (35D

53

where [ztradegS ] is defined as a non-singular diagonal (mNS x wNS) segment-impedance

matrix that contains all the meshed DS segment impedances (tree sections and links)

along its main diagonal The matrix [Z^fM is formulated as two diagonal submatrices

as follows

|~ rymDS I

L eg J

Zl

0

^

0

0

7

Zk

0

0

0

raquoL

|gtr ] | o o |[zr] (352)

where Z^S J is the (NB - 1 x NB - 1) loopless tree section-impedance diagonal square

matrix and Z^k 1 is the (raquoL x nL) links impedance diagonal matrix

352 Fundamental Loop Currents The algebraic sum of voltage drops AV around any fundamental loop is zero according

to Kirchhoff s voltage law (KVL) and this can be mathematically expressed in terms of

the fundamental loop matrix C as follows

[C][AF] = 0 (353)

The voltage drop across the meshed DS segments is determined by the following

relations

[W] = [zf][mSBM][mILL]

where

Lm seg J bull (wNS x l) segment currents column vector of the meshed DS network

jW iZJ (mNS x l) meshed DS bus Loads and Link currents column vector

(354)

In order to account for the link currents in the meshed network the segment currents

column vector and the meshed DS bus loads and links currents column vector are

54

respectively partitioned into two subvectors as defined below

[ jtree 1

J(mNSxl)

J((JVB-l)xl)

Jloop[ J(nLxl)

(355)

[mILL l(mNSxl)

L L J((MJ-l)xl)

Jloop J (wLxl)

(356)

where

[Cr J ((NB - 1) x 1) tree section currents column vector

[lL] ((NB - 1) x 1) RDS bus load currents column vector

j 1 (nL x 1) fundamental loop current vector which is also the meshed DS link

currents column vector

By multiplying both sides of Eq (354) by [C] the left-hand-side will equal to zero

according to KVL as shown in (353) and by employing Eqs (351) and (356) Eq (354)

is reformulated as

[C][AV] = [c][z^][mSBM[mILL]

0 = [c f [z f ] [mSBM | C ] L op]

bull = [Cr[zS]([raquoSBM][ l ]+ [C][ 1 ] )

0 = [C] [^ ] [ -raquoSBM] [ I ] + [cr [zbdquor][C][U]

-[c] [zf ][c][J=[c] [ Z - ^ S B M J ]

-[^][^]-[cT[zS][laquoSBM][I] Thus the (nL x 1) fundamental loop currents vector in the meshed DS loop frame of

reference can mathematically be expressed as

[4 J - -K]1 [Cj [z^JmSBMr][lL] (357)

Eq (357) can be expressed in terms of the RDS matrices ie SBM and [ Z ^ J by

performing the following operation

55

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Bien que ces formulaires aient inclus dans la pagination il ny aura aucun contenu manquant

DALHOUSEB UNIVERSITY

To comply with the Canadian Privacy Act the National Library of Canada has requested that the following pages be removed from this copy of the thesis

Prehrmnary Pages Examiners Signature Page Dalhousie Library Copyright Agreement

Appendices Copyright Releases (if applicable)

DEDICATION PAGE

To my beloved parents my brothers Falah and Abdullah my sisters my wife OmFahad

my daughter Najla and my sons Fahad Falah and Othman

TABLE OF CONTENTS LIST OF TABLES x

LIST OF FIGURES xiii

ABSTRACT xvi

LIST OF ABBREVIATIONS AND SYMBOLS USED xvii

ACKNOWLEDGEMENTS xxiv

Chapter 1 INTRODUCTION 1

11 Motivation 1

12 Distribution Generation - Historic Overview 2

13 Distribution Generation 2

14 Thesis Objectives and Contributions 5

15 Thesis Outline 7

Chapter 2 LITERATURE REVIEW 9

21 Introduction 9

22 Distribution Power Flow 9

23 DG Integration Problem 13

231 Solving the DG Integration Problem via Analytical and Deterministic Methods 14

232 Solving the DG Integration Problem via Metaheuristic Methods 17

24 Summary 20

Chapter 3 FAST AND FLEXIBLE RADIAL POWER FLOW FOR BALANCED AND UNBALANCED THREE-PHASE DISTRIBUTION NETWORKS 21

31 Introduction 21

32 Flexibility and Simplicity of FFRPF in Numbering the RDS Buses and Sections 22

321 Bus Numbering Scheme for Balanced Three-phase RDS 22

322 Unbalanced Three-phase RDS Bus Numbering Scheme 24

33 The Building Block Matrix and its Role in FFRPF 26

v

331 Three-phase Radial Configuration Matrix (RCM) 26

3311 Assessment of the FFRPF Building Block RCM 28

332 Three-phase Section Bus Matrix (SBM) 29

333 Three-phase Bus Section Matrix (BSM) 31

34 FFRPF Approach and Solution Technique 31

341 Unbalanced Multi-phase Impedance Model Calculation 32

342 Load Representation 38

343 Three-phase FFRPF BackwardForward Sweep 40

3431 Three-phase Current Summation Backward Sweep 40

3432 Three-phase Bus Voltage Update Forward Sweep 42

3433 Convergence Criteria 43

3434 Steps of the FFRPF Algorithm 44

344 Modifying the RCM to Accommodate Changes in the RDS 47

35 FFRPF Solution Method for Meshed Three-phase DS 48

351 Meshed Distribution System Corresponding Matrices 50

352 Fundamental Loop Currents 54

353 Meshed Distribution System Section Currents 56

354 Meshed Distribution System BackwardForward Sweep 59

36 Test Results and Discussion 60

361 Three-phase Balanced RDS 60

3611 Case 1 31-Bus with Single Main Feeder RDS 61

3612 Case 2 90-bus RDS with Extreme Radial Topology 70

3613 Case 3 69-bus RDS with Four Main Feeders 71

3614 Case 4 15-bus RDS-Considering Charging Currents 73

362 Three-phase Balanced Meshed Distribution System 74

3621 Case 1 28-bus Weakly Meshed Distribution System 74

3622 Case 2 70-Bus Meshed Distribution System 78

vi

3623 Case 3 201-bus Looped Distribution System 79

363 Three-phase Unbalanced RDS 80

3631 Case 1 10-bus Three-phase Unbalanced RDS 81

3632 Case 2 IEEE 13-bus Three-phase Unbalanced RDS 85

3633 Case 3 26-bus Three-Phase Unbalanced RDS 86

37 Summary 87

Chapter 4 IMPROVED SEQUENTIAL QUADRATIC PROGRAMMING

APPROACH FOR OPTIMAL DG SIZING 89

41 Introduction 89

42 Problem Formulation Overview 89

43 DG Sizing Problem Architecture 90

431 Objective Function 90

432 Equality Constraints 92

433 Inequality Constraints 92

434 DG Modeling 93

44 The DG Sizing Problem A Nonlinear Constrained Optimization Problem 94

45 The Conventional SQP 96

451 Search Direction Determination by Solving the QP Subproblem 96

4511 Satisfying Karush-Khun-Tuker Conditions 98

4512 Newton-KKT Method 101

4513 Hessian Approximation 103

452 Step Size Determination via One-Dimensional Search Method 104

453 Conventional SQP Method Summary 105

46 Fast Sequential Quadratic Programming (FSQP) 108

47 Simulation Results and Discussion 113

471 Case 1 33-busRDS 113

4711 Loss Minimization by Locating Single DG 114

4712 Loss Minimization by Locating Multiple DGs 118

vii

472 Case 2 69-bus RDS 124

4721 Loss Minimization by Locating a Single DG 125

473 Loss Minimization by Locating Multiple DGs 129

474 Computational Time of FSQP vs SQP 134

475 Single DG versus Multiple DG Units Cost Consideration 136

48 Summary 136

Chapter 5 PSO BASED APPROACH FOR OPTIMAL PLANNING OF

MULTIPLE DGS IN DISTRIBUTION NETWORKS 138

51 Introduction 138

52 PSO - The Motivation 138

53 PSO - An Overview 139

531 PSO Applications in Electric Power Systems 141 532 PSO - Pros and Cons 143

54 PSO - Algorithm 144

541 The Velocity Update Formula in Detail 145

5411 The Velocity Update Formula - First Component 146

5412 The Velocity Update Formula - Second Component 148

5413 The Velocity Update Formula-Third Component 149

5414 Cognitive and Social Parameters 150

542 Particle Swarm Optimization-Pseudocode 152

55 PSO Approach for Optimal DG Planning 153

551 Proposed HPSO Constraints Handling Mechanism 155

5511 Inequality Constraints 155

5512 Equality Constraints 157

5513 DG bus Location Variables Treatment 157

56 Simulation Results and Discussion 160

561 Case 1 33-bus RDS 161

viii

5611 33-bus RDS Loss Minimization by Locating a Single DG 161

5612 33-bus RDS Loss Minimization by Locating Multiple

DGs 169

562 Case 2 69-Bus RDS 180

5621 69-bus RDS Loss Minimization by Locating a Single DG 180

5622 69-bus RDS Loss Minimization by Locating Multiple

DGs 187

563 Alternative bus Placements via HP SO 195

57 Summary 196

Chapter 6 CONCLUSION 198

61 Contributions and Conclusions 198

62 Future Work 201

REFERENCES 203

APPENDIX 220

IX

LIST OF TABLES

Table 31 cok rd and De Parameters for Different Operation Conditions 34

Table 32 FFRPF Iteration Results for the 31-Bus RDS 67

Table 33 The 31-bus RDS Section Power Losses Obtained by the FFRPF Method 68

Table 34 FFRPF Voltage Profiles Results for the Three Different Load Models 69

Table 35 Comparisons between 31-Bus RDS Exponential Model Results 70

Table 36 31-bus RDS FFRPF Results vs Other Methods 70

Table 37 90-bus RDS FFRPF Results vs Other Methods 71

Table 38 69-bus RDS FFRPF Results vs Other Methods 73

Table 39 Komamoto 15-bus RDS FFRPF Results vs Other Methods 74

Table 310 Voltage Profiles for Radial and Meshed 28-bus Distribution Network 77 t

Table 311 28-bus Weakly Meshed DS FFRPF Results vs Other Methods 78

Table 312 70-bus Meshed DS FFRPF Results vs Other Methods 79

Table 313 201-bus Meshed DS FFRPF Results vs Other Methods 80

Table 314 10-bus 3(j) RDS FFRPF Results vs Ref [52] and Gauss Zbus Methods 85

Table 315 IEEE 13-bus 3(j) RDS FFRPF Results vs Ref [52] and Gauss Zbus

Methods 86

Table 316 26-bus 34gt RDS FFRPF Results vs Ref [52] and Gauss Zbus Methods 87

Table 41 Single DG Optimal Profile at the 33-bus RDS 115

Table 42 Optimal DG Profiles at all 33 buses 116

Table 43 Multiple DG Installations in the 33-bus RDS with Unspecified Power

Factor 119

Table 44 SQP Method Double-DG Cycled Combinations 121

Table 45 Single and Multiple DG Installations at Pre-specified 085 Power Factor 123

Table 46 Loss Reduction Comparisons for all DG Cases 123

Table 47 69-bus RDS Single DG Optimal Size and Placement Profiles 128

Table 48 Optimal Double DG Profiles in the 69-bus RDS 131

Table 49 Optimal Three DG Units Profiles in the 69-bus RDS 133

Table 410 Loss Reduction Comparison for all DG Installations in the 69-bus RDS 134

Table 411 33-bus RDS CPU Execution Time Comparison 135

Table 412 69-bus RDS CPU Execution Time Comparison 135

x

Table 51 HPSO Parameters for the Single DG Case 162

Table 52 33-bus RDS Single DG Fixedpf Case 20 HPSO Simulations 162

Table 53 Descriptive Statistics for HPSO Results for the Fixedpf Case 163

Table 54 HPSO vs FSQP Results 33-bus RDS-Single DG-FixedDG Case 163

Table 55 33-bus RDS Single DG Unspecified pf Case 20 HPSO Simulations 163

Table 56 Descriptive Statistics for HPSO Results for an Unspecified pf Case 164

Table 57 HPSO vs FSQP Results 33-bus RDS-Single DG-UnspecifiedpCase 164

Table 58 HPSO Parameters for Both Double DG Cases 170

Table 59 33-bus RDS Double DG Fixed pf Case 20 HPSO Simulations 171

Table 510 Descriptive Statistics for HPSO Results for Fixed pf Double DG Case 171

Table 511 HPSO vs FSQP Results 33-bus RDS-Double DGs-FixedCase 172

Table 512 33-bus RDS Double DG Unspecified pf Case 20 HPSO Simulations 172

Table 513 Descriptive Statistics for HPSO Results for UnspecifiedDouble DG

Case 173

Table 514 HPSO vs FSQP Results 33-bus RDS-Double DGs-UnspecifiedCase 173

Table 515 HPSO Parameters for Both Three DG Cases 174

Table 516 33-bus RDS Three DG Fixed pf Case 20 HPSO Simulations 174

Table 517 Descriptive Statistics for HPSO Results for Fixed pf Three DG Case 175

Table 518 HPSO vs FSQP Results 33-bus RDS-Three DG-FixedCase 175

Table 519 33-bus RDS Three DGs Unspecified pf Case 20 HPSO Simulations 175

Table 520 Descriptive Statistics for HPSO Results for the Fixed Three DG Case 176

Table 521 HPSO vs FSQP Results 33-bus RDS-Three DG-UnspecifiedCase 176

Table 522 HPSO Parameters for the Four DG Case 177

Table 523 33-bus RDS Four DG FixedCase 20 HPSO Simulations 178

Table 524 Descriptive Statistics for HPSO Results for Fixed Four DG Cases 178

Table 525 HPSO vs FSQP Results 33-bus RDS-Four DG-FixedCase 179

Table 526 33-bus RDS Four DG Unspecified pf Case 20 HPSO Simulations 179

Table 527 Descriptive Statistics for HPSO Results for Unspecified Four DG

Case 179

Table 528 HPSO vs FSQP 33-bus RDS-Four -Unspecified pf Case 180

Table 529 HPSO Parameters for 69-bus RDS Both Single DG Cases 181

Table 530 69-bus RDS Single DG Fixed pf Case 20 HPSO Simulations 182

xi

Table 531 Descriptive Statistics for HPSO Results for the Fixed Single DG Case 182

Table 532 HPSO vs FSQP Results 69-bus RDS-Single DG-FixedpDG Case 182

Table 533 69-bus RDS Single DG Unspecifiedpf Case 20 HPSO Simulations 183

Table 534 Descriptive Statistics for Unspecified pSingle DG Case 183

Table 535 HPSO vs FSQP Results 69-bus RDS-Single DG-UnspecifiedpDG

Case 184

Table 536 HPSO Parameters for 69-bus RDS the Double DG Cases 188

Table 537 69-bus RDS Double DG Fixed pf Case 20 HPSO Simulations 189

Table 538 Descriptive Statistics for HPSO Results for Fixed pDouble DG Case 189

Table 539 HPSO vs FSQP Results 69-bus RDS-Double DG-FixedpDG Case 190

Table 540 69-bus RDS Double DG UnspecifiedpfCase 20 HPSO Simulations 190

Table 541 Descriptive Statistics for HPSO Results for Unspecified ^Double DG

Case 191

Table 542 HPSO vs FSQP Results 69-bus RDS-Double-UnspecifiedpDG Case 191

Table 543 HPSO Parameters for Both 69-bus RDS Three DG Cases 192

Table 544 69-bus RDS Three DG Fixed pf Case 20 HPSO Simulations 192

Table 545 Descriptive Statistics for HPSO Results for Fixed pf Three DG Case 193

Table 546 HPSO vs FSQP Results 69-bus RDS-Three DGs-FixedCase 193

Table 547 69-bus RDS Three DG Unspecified pf Case 20 HPSO Simulations 194

Table 548 Descriptive Statistics for HPSO Results for Unspecified pf Three DG

Case 194

Table 549 HPSO vs FSQP Results 69-bus RDS-Three DGs-UnspecifiedpCase 195

Table 550 20 HPSO Simulations of the 69-bus RDS Three DG Fixed pf Case with Suboptimal Tuned Parameters 50 Iterations and 50 Swarm Particles 196

xii

LIST OF FIGURES

Figure 31 10-bus RDS 23

Figure 32 Different ways of numbering the system in Fig 31 24

Figure 33 The ease of numbering a modified and augmented RDS 24

Figure 34 Three-phase unbalanced 6-bus RDS representation 25

Figure 35 Three-phase unbalanced 6-bus RDS with missing buses and sections 26

Figure 36 RCM matrices sparsity plots for the 25 and 90 bus RDSs 28

Figure 37 SBM for three-phase unbalanced 6-bus RDS 30

Figure 38 Three-phase section model 32

Figure 39 The final three-phase section model after Kron s reduction 34

Figure 310 Nominal ^-representation for three-phase RDS section 36

Figure 311 Three-phase capacitor bank (a) The schematics (b) The Modeling 40

Figure 312 The FFRPF solution method flow chart 46

Figure 313 10-bus meshed distribution network 50

Figure 314 Fundamental cut-sets for a meshed 10-bus DS 57

Figure 315 31-bus RDS 62

Figure 316 The RCM of the 31-bus RDS 63

Figure 317 The RCM-1 of the 31-bus RDS 64

Figure 318 The SBM of the 31-bus RDS 65

Figure 319 The BSM of the 31-bus RDS 66

Figure 3 20 90-Bus RDS 71

Figure 321 69-bus multi-feeder RDS 72

Figure 322 Komamoto 15-bus RDS 73

Figure 323 28-bus weakly meshed distribution network 75

Figure 324 mRCM for 28-bus weakly meshed distribution network 75

Figure 325 mSBM for 28-bus weakly meshed distribution network 76

Figure 326 C for 28-bus weakly meshed distribution network 76

Figure 327 70-bus meshed distribution system 78

Figure 328 201-bus hybrid augmented test distribution system 80

Figure 329 10-bus three-phase unbalanced RDS 81

Figure 330 The 10-bus three-phase unbalanced RDS RCM 82

xni

Figure 331 21 The 10-bus three-phase unbalanced RDS RCM1 83

Figure 332 The 10-bus three-phase unbalanced RDS SBM 84

Figure 333 The 10-bus three-phase unbalanced RDS BSM 85

Figure 334 IEEE 13-bus 3^ unbalanced RDS 86

Figure 41 The Conventional SQP Algorithm 107

Figure 42 The FSQP Algorithm 112

Figure 43 Case 1 33-busRDS 114

Figure 44 Optimal real power losses for placement of optimal DG size at all the 32

buses using APC method 117

Figure 45 Optimal real power losses vs different DG power factors at bus 30 117

Figure 46 Bus voltages improvement before and after installing a single DG at

bus 30 118

Figure 47 Voltage profiles comparisons of 33-bus RDS cases 120

Figure 48 Voltage improvement of the 33-bus RDS due to three DG installation

compared to pre-DG single and double-DG cases 122

Figure 49 Voltage profiles improvement in 33-bus RDS for all DG cases 124

Figure 410 Case 2 69-bus RDS test case 125

Figure 411 Optimal power losses obtained using APC procedure 126

Figure 412 Real power losses vs DG power factor 69-bus RDS 128

Figure 413 Bus voltage improvements via single DG installation in the 69-bus

RDS 129

Figure 414 Variation in power losses as a function of the DG output at bus 61 129

Figure 415 Bus voltage magnitudes of the original 69-bus RDS and Single DG

and double DGs cases 131

Figure 416 69-bus RDS bus voltage magnitudes improvement in all DG cases 133

Figure 51 Number of publications in IEEEIET and ScienceDirect Databases

since the year 2000 140

Figure 52 Interaction between particles to share the gbest information 150

Figure 53 Illustration of velocity and position updates mechanism for a single particle during iteration k 151

Figure 54 PSO particle i updates its velocity and position vectors during two consecutive iterations A and k+ 152

Figure 55 The proposed HPSO solution methodology 159

xiv

Figure 56 Convergence characteristic of the proposed HPSO in the fixedsingle DG case HPSO proposed number of iterations = 30 164

Figure 57 Convergence characteristic of the proposed HPSO in the fixedsingle

DG case HPSO extended number of iterations = 50 165

Figure 58 Swarm particles on the first HPSO iteration 165

Figure 59 Swarm particles on the fifth HPSO iteration 166

Figure 510 Swarm particles on the tenth HPSO iteration 166

Figure 511 Swarm particles on 15th HPSO iteration 167

Figure 512 Swarm particles on the 20 HPSO iteration 167

Figure 513 Swarm Particles on the 25th HPSO iteration 168

Figure 514 Swarm Particles on the last HPSO iteration 168

Figure 515 A close-up for the particles on the 30th HPSO iteration 169

Figure 516 Convergence characteristics of HPSO in the 69-bus fixedsingle DG case HPSO proposed number of iterations = 15 184

Figure 517 Convergence characteristics of HPSO in the 69-bus fixed single DG

case HPSO proposed number of iterations = 50 185

Figure 518 Swarm particles distribution at the first HPSO iteration 185

Figure 519 Swarm particles distribution at the 5 HPSO iteration 186

Figure 520 Swarm particles distribution at the 10 HPSO iteration 186

Figure 521 Swarm particles distribution at the 15l HPSO iteration 187

Figure 522 Close up of the HSPO particles at iteration 15 187

xv

ABSTRACT

Distribution Generation (DG) has gained increasing popularity as a viable element of electric power systems DG as small scale generation sources located at or near load center is usually deployed within the Distribution System (DS) Deployment of DG has many positive impacts such as reducing transmission and distribution network congestion deferring costly upgrades and improving the overall system performance by reducing power losses and enhancing voltage profiles To achieve the most from DG installation the DG has to be optimally placed and sized In this thesis the DG integration problem for single and multiple installations is handled via deterministic and heuristic methods where the results of the former technique are used to validate and to be compared with the latters outcomes

The unique structure of the radial distribution system is exploited in developing a Fast and Flexible Radial Power Flow (FFRPF) method that accommodates the DS distinct features Only one building block bus-bus oriented data matrix is needed to perform the proposed FFRPF method Two direct descendent matrices are utilized in conducting the backwardforward sweep employed in the FFRPF technique The proposed method was tested using several DSs against other conventional and distribution power flow methods Furthermore the FFRPF method is incorporated within Sequential Quadratic Programming (SQP) method and Particle Swarm Optimization (PSO) metaheuristic method to satisfy the power flow equality constraints

In the deterministic solution method the sizing of the DG is formulated as a constrained nonlinear optimization problem with the distribution active power losses as the objective function to be minimized subject to nonlinear equality and inequality constraints Such a problem is handled by the developed Fast Sequential Quadratic Programming method (FSQP) The proposed deterministic method is an improved version of the conventional SQP that utilizes the FFRPF method in handling the power flow equality constraints Such hybridization resultes in a more robust solution method and drastically reduces the computational time In a subsequent step the placement portion of the DG integration problem is dealt with by using an All Possible Combinations (APC) search method Afterward the FSQP methods outcomes were compared to those of the developed metaheuristic optimization method

The difficult nature of the overall problem poses a serious challenge to most derivative based optimization methods due to the discrete nature associated with the bus location Moreover a major drawback of deterministic methods is that they are highly-dependent on the initial solution point As such a new application of the PSO metaheuristic method in the DG optimal planning area is presented in this thesis The PSO is improved in order to handle both real and integer variables of the DG mixed-integer nonlinear constrained optimization problem The algorithm is utilized to simultaneously search for both the optimal IDG size and bus location The proposed approach hybridizes PSO with the developed FFRPF algorithm to satisfy the equality constraints The inequality constraints handling mechanism is dealt with in the proposed Hybrid PSO (HPSO) by combining the rejecting infeasible solutions method with the preserving feasible solutions method Results signify the potential of the developed algorithms with regard to the addressed problems commonly encountered in DS

xvi

LIST OF ABBREVIATIONS AND SYMBOLS USED

ACO

BFGS

BSM

CHP

CIGRE

CN

DER

DG

DG

DGs

DS

EG

EP

EPAct

EPRI

FD

FFRPF

FSQP

GA

GRG

GS

GWEC

HPSO

IP

KCL

KKT

KVL

LP

Ant Colony Optimization

Quasi-Newton method for Approximating and Updating the Hessian Matrix

Bus Section Matrix

Combined-Heat and Power

The International Council on Large Electric Systems

Condition Number

Distribution Energy Resources

Dispersed Generation

Decentralized Generation

Distribution Generation sources

Distribution System

Embedded Generation

Evolutionary Programming

English Policy Act of 1992

Electric Power Research Institute

Fast Decoupled

Fast and Flexible Radial Power Flow

Fast Sequential Quadratic Programming

Genetic Algorithm

Generalized Reduced Gradient

Gauss-Seidel

Global Wind Energy Council

Hybrid PSO

Interior Point method

Kirchhoff s Current Law

Karush-Khun-Tuker conditions

Kirchhoff s Voltage Law

Linear Programming

xvii

wBSM

mNS

wRCM

mRCM

mSBM

mSBMp

NB

NB

HDG

riL

NR

NS

NS

ftwDG

Pf PSO

PUHCA

PURPA

QP

RCM

RDS

RIT

RPF

SA

SBM

SE Mean

SQP

StDev

TS

UnSpec pf

Meshed BSM

Number of segments in meshed DS

Meshed RCM

Modified mRCM

Meshed SBM

Submatrix of wSBM that correspond to the RDS tree sections

Number of Buses

Number of DS Buses

Total number of DGs

Number of Links or number of the fundamental loops

Newton-Raphson

Number of Sections

Number of Sections in RDS AND in meshed DS tree

Total number of the unspecified pf DGs

power factor

Particle Swarm Optimization

Public Utilities Holding Company Act of 1935

Public Utilities Regulatory Policy Act of 1978

Quadratic Programming

Radial Configuration Matrix

Radial Distribution System

The Reduction in CPU execution Time

Radial Power Flow

Simulated Annealing

Section Bus Matrix

Standard Error of the Mean

Sequential Quadratic Programming

Standard Deviation

Tabu search algorithm

Unspecified power factor DG

xviii

U S P B Unique Set of Phase Buses

USPS Unique Set of Phase Sections

xf Unique set of phase buses

iff Unique set of phase sections

Zsec Section primitive impedance matrix

Z^ (3 X 3) section symmetrical impedance matrix

R D S section length

zu Per unit length self-impedance of conductor i

h Per unit length mutual- impedance be tween conductors a n d

rt Resis tance of conductor i

rd Ear th return conductor resistance

k Inductance multiplying constant

De Dis tance between overhead and its earth return counterpart

GMRj Geometr ic mean radius of conductor i

Dy Dis tance between conductors a n d

Vgbc Three-phase sending end voltages

Vg deg Three-phase receiving end voltages

Ias c Three-phase sending end section currents

lfc Three-phase receiving end section currents

Fscc Three-phase shunt admittance of section k

[]3x3 (3 X 3) identity matrix

[^Lx3 (3x3) zero matrix

^Klc Vol tage drop across three-phase section k

ysect Section k three-phase currents

V0 Nomina l bus voltage

V Operat ing bus voltage

xix

P0 Real power consumed at nominal voltage

Q0 Reactive power consumed at nominal voltage

S Bus load apparent power at single-phase bus sect

YsKus Total three-phase shunt admittance at bus i

Ic Three-phase shunt currents at bus i

IlucSi Bus three-phase currents

jabc Three-phase load current

IltLP Current through single-phase section p and phase ltjgt

its Current at bus and phase ^

Vss Substation voltage magnitude

Vls Substation complex phase voltage

VLt Voltage drop across section k in phase (j)

A and symbol

IMI oo-norm vector II I loo

91 (bull) Real part of complex value

3 (bull) Imaginary part of complex value

C Fundamental loop matrix which is a submatrix of mSBM

Zioop (laquoLx nL) loop-impedance matrix

Csec Upper submatrix of the C matrix with ((NB-1) x (nL) dimension

Zoop Loop-impedance matrix

setrade (NSxNS) meshed DS section-impedance diagonal matrix

ZtradeS (NSxNS) RDS or tree section-impedance diagonal matrix

IL (NB-1 x 1) RDS bus load currents vector

fnlsec (mNS x 1) segments currents column vector of meshed DS network vector

mILL (mNSx 1) meshed DS bus loads and links currents vector

Itrade (NB-1) tree section currents column

xx

( n L x 1) fundamental loop current vector which is also the meshed DS link loop

currents column vector

B ( N B - 1 xmNS) fundamental cut-sets matrix

^ s7 e c f a ( N B - 1 x nL) co-tree cut-set matrix

^ymesh Voltage drops across the tree sections of the meshed DS vector

ymesh j k g messed DS bus voltage profiles vector

PRPL Real power losses

Pj Generated power delivered to DS bus i

PjL Load power supplied by DS bus i

Yjj Magnitude of the if1 element admittance bus matrix

rv Phase angle of Yy = YyZyy

Vi Magnitude of DS bus complex voltage

8 Phase angle of V = ViZSl

bull Transpose of vector or matrix

bull Complex conjugate of vector or matrix

V (1 xNB) DS bus Thevenin voltages

Y (NB xNB) DS admittance matrix

A^ Real power mismatch at bus i

AQt Reactive power mismatch at bus i

|L| Infinity norm where llxll = max (|x|) II llco J II llraquo =l2iVBVI ]gt

bull+ Max imum permissible value

bull Minimum permissible value

bull0 Nominal value

PDG D G operating power factor

S^G D G generated apparent power

SsS Main DS substation apparent power

1 Scalar related to the allowable D G size

xxi

Sy Apparent power flow transmitted from bus to bus j

Stradex Apparent power maximum rating for distribution section if

(x) The objective function

z(x) Equality constraints

g(x) Inequality constraints

(bull) Independent unknown variables lower bounds

(bull) Independent unknown variables upper bounds

x Independent unknown variables vector

RPL ( X ) Distribution system real power losses objective function

d Search direction vector

a Positive step size scalar

WRPL (x ) Gradient of the objective function at point xk)

pound Lagrange function

H^ (nxri) Hessian symmetric matrix at point xw

h^ First-order Taylors expansion of the equality constraints at point xw

Vh(x^) (nxm) Jacobian matrix of the equality constraints at point xw

g ^ First-order Taylors expansion of the inequality constraints at point xw

Vg(x^) (nxp) Jacobian matrix of the inequality constraints at point xw

Xi Individual equality Lagrange multiplier scalar

Pi Individual inequality Lagrange multiplier scalar

k w-dimensional equality Lagrange multiplier vector

P (-dimensional inequality Lagrange multiplier vector

s A predefined small tolerance number

A Active set

m Number of all equality constraints

p Number of all inequality constraints

a Number of the active set equations

xxii

v 2 j6k)

XX

nTgtG

nuDG

y

v Y FFRPFbl

deg FFRPF bl

llAP II II lloo

Vi

Xi

Cj C2

rXgtr2

w

pbestj

gbesti

nk

X

APT Losses

pHPSO Losses

pFSQP Losses

Hessian of the Lagrange function

Total number of DGs

Total number of the unspecified DGs

The change in the Lagrange functions between two successive iterations

Voltage magnitude of bus i obtained by the FFRPF technique

Voltage phase angle of bus obtained by the FFRPF technique

Voltage deviation infinity norm ie II AV = max ( I F - K I ) deg llcD

=UAlaquoVI deg ngt

Particle i velocity

Particle i position vector

Individual and social acceleration positive constants

Random values in the range [0 l] sampled from a uniform distribution

Weight inertia

Personal best position associated with particle own experience

Global best position associated with the whole neighborhood experience

Maximum number of iterations

Constriction factor

The deviation of losses calculated by HPSO method from that determined

by FSQP method

Mean value of HPSO simulation results of real power losses

FSQP deterministic method result of real power losses

xxiii

ACKNOWLEDGEMENTS

All Praise and Thanks be to reg (Allah) Almighty whose countless bounties enabled me to

accomplish this thesis successfully I would like to express my deepest gratitude to my

parents who taught me the value of education and hard work A special note of gratitude

to my brothers Falah and Abdullah deer sisters my wife my daughter Najla and my

sons Fahad Falah and Othamn They endured the long road along with me and

provided me with constant support motivation and encouragement during the course of

my study

I would like to express my sincere gratitude to my advisor Dr M E El-Hawary for

his professional guidance valuable advice continual support and encouragement I also

appreciate the constructive comments of my PhD External Examiner Dr M A Rahman

I am also grateful to my advisory committee members Dr T Little and Dr W Phillips

for spending their valuable time in reading evaluating and discussing my thesis

I would like to acknowledge the academic discussions and the constant

encouragement I received from my dear friend Dr Mohammed AlRashidi- Thank you

Abo Rsheed I wish also to thank a special friend of mine Dr AbdulRahman Al-

Othman for his friendship and for believing in me

I would like to manifest my gratitude to the Public Authority for Applied Education

and Training in Kuwait who sponsored me through my PhD at Dalhousie University

From the Embassy of Kuwait Cultural Attache Office special thanks are due to Shoghig

Sahakyan for her efforts and help to make this work possible

xxiv

CHAPTER 1 INTRODUCTION

11 MOTIVATION

Electric power system networks are composed typically of four major subsystems

generation transmission distribution and utilizations Distribution networks link the

generated power to the end user Transmission and distribution networks share similar

functionality both transfer electric energy at different levels from one point to another

however their network topologies and characteristics are quite different Distribution

networks are well-known for their low XR ratio and significant voltage drop that could

cause substantial power losses along the feeders It is estimated that as much as 13 of

the total power generation is lost in the distribution networks [1] Of the total electric

power system real power losses approximately 70 are associated with the distribution

level [23] In an effort towards manifesting the seriousness of such losses Azim et al

reported that 23 of the total generated power in the Republic of India is lost in the form

of losses in transmission and distribution [4]

Distribution systems usually encompass distribution feeders configured radially and

exclusively fed by a utility substation Incorporating Distribution Generation sources

(DGs) within the distribution level have an overall positive impact towards reducing the

losses as well as improving the network voltage profiles Due to advances in small

generation technologies electric utilities have begun to change their electric

infrastructure and have started adapting on-site multiple small and dispersed DGs In

order to maximize the benefits obtained by integrating DGs within the distribution

system careful attention has to be paid to their placement as well as to the appropriate

amount of power that is injected by the utilized DGs In other words to achieve the best

results of DG deployments the DGs are to be both optimally placed and sized in the

corresponding distribution network

The motivation of this thesis research is to investigate placing and sizing single and

multiple DGs in Radial Distribution Systems (RDSs) The problem investigated involves

two stages finding the optimal DG placements in the distribution network and the

optimal size or rating of such DGs The optimal DG placement and sizing are dealt with

by utilizing deterministic and heuristic optimization methods

12 DISTRIBUTION GENERATION - HISTORIC OVERVIEW

During the first third of the twentieth century there were no restrictions on how many

utility companies could be owned by financial corporations known as utility holding

companies By 1929 80 of US electricity was controlled by 16 holding companies

and three of those corporations controlled 36 of the nations electricity market [5]

During the Great Depression most of these utility holding companies went bankrupt As

a result the US Congress Public Utilities Holding Company Act (PUHCA) of 1935

regulated the gas and electric industries and restricted holding companies to the

ownership of a single integrated utility PUHCA indirectly discouraged wholesale

wheeling of power between different states provinces or even countries The Public

Utilities Regulatory Policy Act (PURPA) of 1978 allowed grid interconnection and

required electric utilities to buy electricity from non-utility-owned entities called

Qualifying Facilities (QF) at each utilitys avoided cost The term QF refers to non-

utility-owned (independent) power generators The term at each utilitys avoided cost

is interpreted to mean that the utility shall buy the generated electricity at a price

equivalent to what it would cost the utility itself if had generated the same amount of

power in its own facility or if it had purchased the power from an open electricity market

ie what the utility saves by not generating the same amount of power This act heralded

the dawn of the DG industry era which paved the way to generate electricity arguably at

a lower cost compared to that of traditional utility companies and consequently have it

delivered to the end user at lower rates The English Policy Act of 1992 (EPAct)

intensified competition in the wholesale electricity market by opening the transmission

system for access by utilities and non-utilities electricity producers [67] entity A could

sell its power to entity B through entity Cs transmission infrastructure

13 DISTRIBUTION GENERATION

DG involves small-scale generation sources scattered within the distribution system level

atnear the load center ie close to where the most energy is consumed [8] The DG

2

generate electricity locally and in a cogeneration case heat can also be generated and

may be utilized in applications such as industrial process heating or space heating DG

generally has better energy efficiency than large-scale power plants The traditional

power stations usually have an efficiency of around 35 whereas the efficiency of DG

such as a Combined Heat and Power (CHP) gas turbine would be in the vicinity of 45-

65 [5]

It seems that there is no universal agreement on the definition of DG size range The

Electric Power Research Institute (EPRI) for example defined the DG size to be up to 5

MW in 1998 [9] in 2001 EPRI redefined the DG capacity to be less than 10 MW [10]

and by 2003 they identified the DG to have a power output ranging from 1 kW to 20 MW

[11] The IEEE published its DG-standards IEEE Std 1547-2003 and IEEE Std 15473-

2007 and emphasized that they are applicable to DGs that have total capacity below 10

MVA [1213] In its 2006 report about the impact of DG Natural Resources Canada

estimated that the DG size starts from few 10s of kW to perhaps 5 MW [14] In 2000

the International Council on Large Electric Systems (CIGRE) referred to the DG as non-

centrally dispatched usually attached to distribution level and smaller than 50-100 MW

[1516]

Many terms referring to DG technology are used in the literature such as Dispersed

Generation (DG) Decentralized Generation (DG) Embedded Generation (EG)

Distribution Energy Resources (DER) and on-site Generation [17] In particular the

term dispersed generation customarily refers to stationary small-scale DG with power

outputs ranging from 1 kW to 500 kW [7]

Late developments and innovations in the DG technology industry liberalization of

the electricity market transmission line congestion and increasing interest in global

warming and environmental issues expedited publicizing their deployment and adoption

world-wide Recent studies suggest that DG will play a vital role in the electric power

system An EPRI study predicts that by the year 2010 25 of the newly installed

generation systems will be DG [18] and a similar study by the Natural Gas Foundation

projects that the share of DG in new generation will be 30 [15] By 2003 around 40

of Denmarks power demand was served by DG while Spain the Netherlands Portugal

and Germany integrated nearly 20) of DG into their distribution networks [19] Of the

3

643 GW generated by the European Union in 2005 approximately 122 GW (19) was

generated by hydro 96 GW (15) of this generated capacity was cogeneration (CHP)

and 53 GW (8) generated by other renewable energy systems Half of the CHP

generated capacity was owned by utility companies and the other half was generated by

independent producers [20]

Globally in 2005 the total installed wind power capacity was 591 GW and the

Global Wind Energy Council (GWEC) expected the wind capacity to reach 1348 GW by

the year 2010 [21] Worldwide wind energy capacity of 19696 MW was added in the

year 2007 [22] and approximately 1400 MW of wind energy capacity was added in the

US during the second quarter of 2008 [23] GWEC also predicted in its Global Wind

Energy Outlook 2008 report that by the year 2020 15 billion tons of CO2 will be saved

every year and by the middle of the 21st century 30 of the worlds electricity will be

supplied by wind energy [24] compared to a total of 13 of the global electricity being

generated by wind at the end of 2007 [22]

DG technologies include a variety of energy sources ie powered by renewable or

by fossil fuel-based prime movers Renewable technologies used in DG include wind

turbines photovoltaic cells small hydro power turbines and solar thermal technologies

while DG based on conventional technologies may involve gas turbines CHP gas

turbines diesel engines fuel cells and micro-turbine technologies Some DGs are

installed by the utility company on the supply side of the consumers meter while some

are installed by the customers themselves on their side of a bi-directional meter thus

enabling them to benefit from the net-metering program offered by utility companies

[25]

Optimal deployment of DG technology would have an overall positive impact

although some negative traits would remain The noise and shadow flicker caused by

large wind blades and the noise caused by the wind turbine gearbox or gas turbines

especially when placed close to residential or populated areas are examples of negative

impacts of widespread use of DG Another drawback from an environmentalist point of

view is that wind DG could disturb bird immigration patterns and cause death to both

birds and bats [26] Renewable-source DGs also could be an indirect source of pollution

by causing the fossil-fuel power plants to shut down and start up more frequently as they

4

attempt to accommodate variable DG power output [27] Some plants have an emission

rate which is inversely proportional to its delivered power Voltage rise as a result of bishy

directional power flow caused by the interconnection of the DG in RDS is another

example of a negative impact caused by DG [28]

The integration of DG into electric power networks has many benefits Some

examples of such benefits could be summarized as follows

bull Improve both the reliability and efficiency of the power supply

bull Release the available capacity of the distribution substation as well as reducing

thermal stresses caused by loaded substations transformers and feeders

bull Improve the system voltage profiles as well as the load factor

bull Decrease the overall system losses

bull Generally DG development and construction have shorter time intervals

bull Delay imminent upgrading of the present system or the need to build newer

infrastructure and subsequently avoid related problems such as right-of-way

concerns

bull Decrease transmission and distribution related costs

bull In general DG tends to be more environmentally friendly when compared to

traditional coal oil or gas fired power plants

The extent of the benefits depends on how the DG is placed and sized in the system In

addition to supplying the system with the power needed to meet certain demands as an

installation incentive the real power losses could be minimal if the DG is optimally sited

and sized

14 THESIS OBJECTIVES AND CONTRIBUTIONS

Optimal integration of single and multiple DG units in the distribution network with

specified and unspecified power factors is thoroughly investigated from a planning

perspective in this thesis The DG problem is handled via deterministic and heuristic

optimization methods where the results of the former method are used to validate and to

be compared with those of the latter

The unique radial distribution structure is exploited in developing a Fast and Flexible

Radial Power Flow (FFRPF) method to deal with a wide class of distribution systems

5

eg radial meshed small large balanced and unbalanced three-phase networks The

proposed FFRPF algorithm starts by developing a Radial Configuration Matrix (RCM)

for radial topology DS or meshed RCM (mRCM) for meshed DS Both matrices consist

of elements with values 1 0 and -1 The RCM (or mRCM) corresponding building

algorithm is simple fast and practical as illustrated in Chapter 3 The RCM is inverted

only once to produce the Section Bus Matrix (SBM) which is then transposed to obtain

the Bus Section Matrix (BSM) For the meshed topology the corresponding resultant

matrices for the mRCM are the meshed Bus Section Matrix (mSBM) and the meshed Bus

Section Matrix (mBSM) The FFRPF technique relies on the backwardforward sweep

that only utilizes the basic electric laws such as Ohms law and both Kirchhoff s voltage

and current laws The backward current sweep is performed via SBM (or mSBM) and

the forward voltage update sweep is executed via the BSM (or mBSM) By utilizing the

two obtained matrices all the bus complex voltages can be obtained and consequently

left to be compared with the immediate previous obtained bus voltages The proposed

approach quickened the iterative process and reduced the CPU time for convergence It

is worth mentioning that the building block matrix is the only input data required by the

FFRPF method besides the DS parameters to perform the proposed distribution power

flow The FFRPF technique is incorporated in both utilized deterministic and

metaheuristic optimization methods to satisfy the power flow equality constraints

requirements

In the deterministic solution method the DG sizing problem is formulated as a

nonlinear optimization problem with the distribution active power losses as the objective

function to be minimized subject to nonlinear equality and inequality constraints

Endeavoring to obtain the optimal DG size an improved version of the Sequential

Quadratic Programming (SQP) methodology is used to solve for the DG size problem

The conventional SQP uses a Newton-like method which consequently utilizes the

Jacobean in handling the nonlinear equality constraints The radial low XR ratio and the

tree-like topology of distribution systems make the system ill-conditioned

A Fast Sequential Quadratic Programming (FSQP) methodology is developed in

order to handle the DG sizing nonlinear optimization problem The FSQP hybrid

approach integrates the FFRPF within the conventional SQP in solving the highly

6

nonlinear equality constraints By utilizing the FFRPF in dealing with equality

constraints instead of the Newton method the burden of calculating the Jacobean and

consequently its inverse as well as the complications of the ill-conditioned Y-matrix of

the RDS is eliminated Another advantage of this hybridization is the drastic reduction

of computational time compared to that consumed by the conventional SQP method

In this thesis a new application of the Particle Swarm Optimization (PSO) method in

the optimal planning of single and multiple DGs in distribution networks is also

presented The algorithm is utilized to simultaneously search for both the optimal DG

size and its corresponding bus location in order to minimize the total network power

losses while satisfying the constraints imposed on the system The proposed approach

hybridizes PSO with the developed distribution radial power flow ie FFRPF to

simultaneously solve the optimal DG placement and sizing problem The difficult nature

of the overall problem poses a serious challenge to most derivative based optimization

methods due to the discrete flavor associated with the bus location in addition to the

subproblem of determining the most suitable DG size Moreover a major drawback of

the deterministic methods is that they are highly-dependent on the initial solution point

The developed PSO is improved in order to handle both real and integer variables of the

DG mixed-integer nonlinear constrained optimization problem Problem constraints are

handled within the proposed approach based on their category The equality constraints

ie power flows are satisfied through the FFRPF subroutine while the inequality bounds

and constraints are treated by exploiting the intrinsic and unique features associated with

each particle The proposed inequality constraint handling technique hybridizes the

rejection of infeasible solutions method in conjunction with the preservation of feasible

solutions method One advantage of this constraint handling mechanism is that it

expedites the solution method converging time of the Hybrid PSO (HPSO)

15 THESIS OUTL INE

This thesis is organized in six chapters The research motivation brief description of the

DG and the thesis objectives are addressed in the first chapter The second chapter deals

with a literature review of the distribution power flow methods and the DG optimal

planning problem In the third chapter development of the proposed FFRPF method

7

utilized in the FSQP and the HPSO methods to satisfy the DG problem of nonlinear

equality constraints is presented The fourth chapter deals with the DG sizing problem

formulation and its solution based on the two deterministic solution methods The

problem is solved via the conventional SQP and the proposed FSQP methods and a

performance comparison between them is presented Basic concepts of the PSO are

presented in chapter five A brief literature review regarding the use of the PSO in

solving the electric power system problems is presented in this chapter In addition it

also addresses the development of the proposed HPSO in solving the DG planning

problem The last chapter provides the thesis concluding remarks and the scope of future

work

8

CHAPTER 2 LITERATURE REVIEW

21 INTRODUCTION

Recent publications in the areas of work relative to this thesis are reviewed and

summarised in this chapter which is organized in two sections as follows

bull The first section reviews the literature on distribution power flow methods A

brief background of conventional power flow methods is presented followed

by a review and summary of the literature on recent developments of the

distribution power flow algorithms

bull The DG integration problem is reviewed in the second section Recent work

on the optimal DG placement and sizing via analytical deterministic and

metaheuristic methods are analyzed and reviewed

22 DISTRIBUTION POWER FLOW

Power flow programs play a vital role in analyzing power systems The problem deals

with calculating unspecified bus voltage angles and magnitudes active and reactive

powers as well as (as a by-product) line loadings and their associated real and reactive

losses for certain operating conditions These values are typically obtained through

iterative numerical methods to analyze the status of a given power system

Since the middle of last century many methods were proposed to solve this problem

Even though Dunstan [29] was the first to demonstrate a digital method for solving the

power flow problem in 1954 Ward and Hale [30] are often credited with the successful

digital formulation and solution of the power flow problem in 1956 Most of the earlier

solution methods were based on both the admittance matrix and the Gauss-Seidel (GS)

iterative method The poor convergence characteristics of GS when large networks

andor ill-conditioned situations are encountered led to the development of the Gaussian

iterative scheme (Zbus) [3132] and later the Newton-Raphson (NR) method [33] as well

as the Decoupled [34] and Fast Decoupled (FD) power flow approaches [35] Though

the NR method generally converges faster than other methods it takes longer

computational time per iteration When Tinney et al [36] introduced the optimally

9

ordered and sparsity-oriented programming techniques Newton-based methods became

the de facto industry standard However the Jacobian matrix for the RDS is

approximately four times the size of the corresponding admittance matrix and it needs to

be evaluated at each iteration

Although conventional power flow methods are well developed in dealing with the

transmission and sub-transmission sections of the power system networks they are

deemed to be inefficient in handling distribution networks This is because the

Distribution System (DS) is different in several ways from its transmission counterpart

DS has a strictly radial topology nature or weakly meshed networks in contrast with

transmission systems which are tightly meshed networks DS is a low voltage system

having low XR ratio sections and a wide range of reactance and resistance values DS

may consist of a tremendously large number of sections and buses spread throughout the

network Sections of the DS could have unbalanced load conditions due to the

unbalanced three-phase loading as well as single and double phase loads in spurred

lateral lines The mutual couplings between phases are not negligible due to rarely

transposed distribution lines [37] All of these characteristics strongly suggest that DS is

to be classified as an ill-conditioned power system

The practical DSs low XR ratio sections may cause both the NR and FD

conventional methods to diverge [38-41] The line impedance angles are small enough to

deteriorate the dominance of the NR Jacobian main diagonal making it prone to

singularity Such a low XR value would also prevent the Jacobian matrix from being

decoupled and simplified

In addition to performance considerations a practical power flow technique needs to

consider all the DS distinctive features and to accommodate the imbalance introduced by

multiphase networks along with the distribution-level loads In the literature a number

of Newton and non-Newton power flow methods designed for distribution systems were

proposed Zhang et al [42] solved the distribution power flow based on the Newton

method although the proposed Jacobian is computed just once the solution converged

with a number of additional iterations more so than the conventional approach

Moreover shunt capacitor banks were ignored in the modeling as well as the line shunt

admittance (JI model) and the constant impedance loads Baran and Wu [43] solved the

10

power flow problem by utilizing three fundamental quadratic equations representing the

real and reactive section powers and the bus voltages in an iterative scheme as a

subroutine during the process of optimizing the capacitor sizing However they

computed the Jacobian using the chain rule within the proposed NR method which is in

turn time consuming Mekhamer et al [44] utilized the three equations developed in [43]

using a different iterative technique without the need for the Jacobian or the NR method

However their process is based on applying a multi-level iterative process on the main

feeder and laterals which makes the speed and the efficiency of their proposed algorithm

a function of the RDS configuration and topology

In [4546] the quadratic equation was also utilized in determining the relation

between the sending and receiving end voltage magnitudes along with the section power

flow They proposed to include the system power losses within their calculation while

solving for the system power flow However the voltage phase angles were ignored

during the solution of the radial power flow in order to speed up the convergence The

latter reference developed work was based on the assumption of balanced RDS and

sophisticated numbering scheme

The radial power flow introduced by [47-49] used a non-Newton power flow techshy

nique based on the ladder network theory This method adds the section currents and

calculates the RDS bus voltages including the substations during a backward sweep If

the difference between the calculated substation voltage value and substation predetershy

mined assigned bus voltage value is acceptable the iterations are concluded If not the

substation bus voltage is reset and the RDS bus voltages are computed for the second

time in the same iteration in the forward sweep Both the ladder and the backshy

wardforward methods are derivative-free instead they employ simple circuit laws

However the ladder method uses many sub-iterations on the laterals and calculates the

system bus voltages twice during a single iteration compared to once in the backshy

wardforward method Thukaram [50] utilized the backwardforward sweep technique to

solve the RDS power flow However the bus numbering procedure was a sophisticated

parent node and child node arrangement which may add some computational overshy

head if the system topology is changed Teng [51] used the backwardforward approach

as the solution procedure through the development of two matrices and multiplied them

11

together in a later stage of the solution process In assembling those matrices all the

system buses and sections have to be inspected carefully In a practical large RDS data

preparation for these matrices will be cumbersome and prone to errors Under continshy

gency situations switching operations or the addition of another feeder to the existing

one are quite common practices in the DSs hence changes in system topology need to be

accommodated by restructuring the corresponding matrices which would add an overshy

head to track modifications The weakly meshed DS was dealt with by adding extra

nodes in the middle of the new links Two equal currents with opposite polarities were

injected into each added node Each injection operation is represented by a two column

matrix which was subsequently added to the first proposed matrix and then the develshy

oped matrices were extended and multiplied together The resultant is a full matrix and

its dimension is reduced by the Kron method in every single iteration That is the

developed full matrix was inverted in each iteration of the solution method and such

procedure is expensive lengthy cumbersome and time consuming

Shirmohammadi et al [39] Cheng and Shirmohammadi [52] and Hague [53]

proposed an iterative solution method for both radial and weakly meshed DSs This

approach necessitates a special numbering scheme in which they number the DS sections

in layers starting from the root node The numbering scheme is to be carried out

carefully by examining the whole system when a new layer is to be numbered The

numbering process is cumbersome and prone to errors For weakly meshed networks

breakpoints are selected opened and consequently the meshed system is converted to a

radial system The loops are broken by adding two fictitious buses In each pair of

dummy buses equal and opposite currents are injected and the new system is evaluated

to produce a reduced order impedance matrix Their proposed method requires that the

breakpoint impedance matrix should be computed cautiously Such a procedure is highly

dependent on the distribution networks topology That is the more links that exist in the

DS the larger the break point impedance matrix and the more time will be consumed in

its computation

Goswami and Basu [38] introduced a direct solution method to solve for radial and

weakly meshed DS They applied a breakpoints method into the meshed DS similar to

that of [39] in order to convert it into RDS In their proposed methodology a restriction

12

was imposed on each of the system buses not to have more than three sections attached to

it Such limitation is a drawback of the method and moreover a difficult node numbering

scheme is a disadvantage

In this thesis the unique structure of the RDS is exploited in order to build up a new

fast flexible power flow technique that deals with radial and looped DSs The numbering

scheme of the DS is simple and straightforward All load types can be accommodated by

the proposed distribution power flow eg spot and distributed loads Unlike

conventional power flow methods no trigonometric functions are used in the proposed

distribution power flow method For weakly meshed and looped DSs the system is dealt

with as it is there is no need for radialization cuts or building breakpoints impedance

matrix The topology of the tested DS whether strictly radial weakly meshed or looped

is represented by a building block matrix which is the only one needed to perform the

backwardforward sweep technique

23 DG INTEGRATION PROBLEM

DG is gaining increasing popularity as a viable element of electric power systems The

presence of DG in power systems may lead to several advantages such as supplying

sensitive loads in case of power outages reducing transmission and distribution networks

congestion and improving the overall system performance by reducing power losses and

enhancing voltage profiles Some of the negative impacts of DG installations are

potential harmonic injections the need to adopt more complex control schemes and the

possibility of encountering reverse power flows in power networks Even though the

concept of DG utilization in electric power grids is not new the importance of such

deployment is presently at its highest levels due to various reasons Recent awareness of

conventionaltraditional thermal power plants harmful impacts on the environment and

the urge to find more environmentally friendly substitutes for electrical power generation

rapid advances made in renewable energy technologies and the attractive and open

electric power market are a few major motives that led to the high penetration of DG in

most industrial nations power grids To achieve the most from DG installation special

attention must be made to DG placement and sizing

13

The problem of optimal DG placement and sizing is divided into two subproblems

where is the optimal location for DG placement and how to select the most suitable size

Many researchers proposed different methods such as analytic procedures as well as

deterministic and heuristic methods to solve the problem

231 Solving the DG Integration Problem via Analytical and Deterministic Methods

In the literature the optimal DG integration problem is solved by means of employing

any analytical or optimization technique that suits the problem formulation Methods and

procedures of optimally sizing and locating the DGs within the DS are varied according

to objectives and solution techniques

Willis [54] presented an application of the famous 23 rule originally developed

for optimal capacitor placement to find a suitable bus candidate for DG placement That

is to install a DG with a rating of 23 of the utilized load at 23 the radial feeder length

down-stream from the source substation However this rule assumes uniformly

distributed loads in a radial configuration and a fixed conductor size throughout the

distribution network In any event the 23 rule was developed for all-reactive load

These assumptions limit its applicability to radial distribution systems and the fact that it

is only suitable for single DG planning

Kashem et al [55] developed an analytical approach to determine the optimal DG

size based on power loss sensitivity analysis Their approach was based on minimizing

the DS power losses The proposed method was tested using a practical distribution

system in Tasmania Australia However it assumes uniformly distributed loads with all

the connected loads along the radial feeder having the same power factor and it also

assumes no external currents injected into the system buses eg capacitors which limits

its practicality

Wang and Nehrir [56] developed an analytical approach to address the optimal DG

placement problem in distribution networks with different continuous load topologies

Minimizing the real power losses was the objective of the proposed method In their

approach the DG units were assumed to have unity power factor and only the overhead

distribution lines with neglected shunt capacitance are considered The candidate bus

was selected based on elements of the admittance matrix power generations and load

14

distribution of the distribution network The issue of DG optimal size was not addressed

in their formulation

Griffin et al [57] analyzed the DG optimal location analytically for two continuous

load distributions types ie uniformly distributed and uniformly increasing loads The

goal of their study was to minimize line losses One of the conclusions of their research

was that the optimal location of DG is highly dependent on the load distribution along the

feeder ie significant loss reduction would take place when placing the DG toward the

end of a uniformly increasing load and in the middle of uniformly distributed load feeder

Acharya et al [58] used the incremental change of the system power losses with

respect to the change of injected real power sensitivity factor developed by Elgerd [59]

This factor was used to determine the bus that would cause the losses to be optimal when

hosting a DG By equating the aforementioned factor to zero the authors solved for the

optimal real value of DG output They proposed an exhaustive search by applying the

sensitivity factor on all the buses and ranked them accordingly The drawback of their

work is the lengthy process of finding the candidate locations and the fact that they

sought to optimize only the DG real power output Furthermore they only considered

planning of a single DG

Popovic et al [60] utilized sensitivity analysis based on the power flow equations to

solve the DG placement and sizing Two indices were used in ranking all the DS buses

for the suitability of hosting the DG The first one is a voltage sensitivity index which is

derived directly from the NR power flow Jacobian inverse the second one exploits the

relation of incremental real power losses with respect to the injected real and reactive

power as developed in [61] Their objective for sizing the DG was to maximize its

capacity subject to boundary constraints such as bus voltage penetration level line flows

and fault current limits To solve the sizing DG problem they gradually increased the

DG capacity at selected most sensitive buses until one of the constraints is violated and

the direct previous installed DG size becomes the one chosen as the optimal rating This

process is a lengthy and impractical procedure and the authors did not elaborate on how

they would deal with multiple DG cases using the proposed scheme

Keane and OMalley [62] solved for the optimal DG size in the Irish system by using

a constrained Linear Programming (LP) approach To cope with the EU regulation which

15

emphasizes that Ireland should provide 132 of its electricity from renewable sources

by 2010 the objective of their proposed method was to maximize the DG generation

The nonlinear constraints were linearized with the goal of utilizing them in the LP

method A DG unit was installed at all the system buses and the candidate buses were

ranked according to their optimal objective function value

Rosehart and Nowicki [63] dealt with only the optimal location portion of the DG

integration problem They developed two formulations to assess the best location for

hosting the DG sources The first is a market based constrained optimal power flow that

minimized the cost of the generated DG power and the second is voltage stability

constrained optimal power flow that maximized the loading factor distance to collapse

Both formulations were solved by utilizing the Interior Point (IP) method The outcomes

of the two formulations were used in ranking the buses for DG installations The optimal

DG size problem was not considered in their paper

Iyer et al [64] employed the primal-dual IP method to find the optimal DG

placement through combined voltage profile improvement and line loss reduction indices

However the proposed approach was based on initially placing DGs at all buses in order

to determine proper locations for DG installations This methodology may not be

realistic for large scale distribution networks

Rau and Wan [65] only treated the DG sizing problem by utilizing the Generalized

Reduced Gradient (GRG) method The DG bus locations were assumed to be provided

by the system planner for the DG units to be installed In their proposed method they

considered minimizing the system active power losses In their formulation only the

power flow equality constraints were considered whereas the boundary conditions and

the inequality constraints were not taken into account

Hedayati et al [66] employed continuous power flow methodology to locate the

buses most sensitive to voltage collapse The sensitive bus set is ranked based on their

severity which is used accordingly to indicate potential bus locations for placement of

single and multiple DG sources An iterative method was proposed for optimally sitting

the DG A certain DG capacity which is known and fixed a priori is added to the DS

and the conventional power flow method was employed to determine the resultant DS

real power losses voltage profiles and power transfer capacity In the subsequent

16

iteration another DG with the same capacity was added to the next sensitive bus and

results were obtained This iterative process would continue until the system outcomes

reached acceptable values The proposed iterative method did not optimize the DG size

232 Solving the DG Integration Problem via Metaheuristic Methods

Metaheuristic techniques have proven their effectiveness in solving optimization

problems with appreciable feasible search space They can be easily modified to cope

with the discrete nature associated with different elements commonly used in power

systems studies Optimization methods such as Genetic Algorithm (GA) [67-71] GA

hybrid methods such as GA and fuzzy set theory [72] and GA and Simulated Annealing

(SA) [73] Tabu search algorithm (TS) [7475] GA-TS hybrid method [76] Ant Colony

Optimization (ACO) [77] Particle Swarm Optimization (PSO) [78] and Evolutionary

Programming (EP) [79] were utilized in the literature to solve for the DG integration

problem

Teng et al [67] developed a value-based method for solving the DG problem The

GA method was utilized in maximizing a DG benefit to cost ratio index subject to only

boundary constraints such as ratio index voltage drop and feeder transfer capacity A

drawback of their procedure is that the candidate DG bus locations were assumed to be

provided by the utility and consequently all combinations of the provided bus locations

were tested for obtaining the optimal DG capacities via the GA method

The proposal set forth by Mithulananthan et al [68] made use of the DS real power

losses as the fitness function to be minimized through GA Their formulation of the DG

size optimization problem is of an unconstrained type Moreover the NR method which

is usually inadequate in dealing with the DS topology was used in calculating the total

power losses Candidate DG bus locations were obtained by placing a DG unit at all

buses of the tested DS which is impractical for large DSs Furthermore the multiple

DGs case was not addressed

Haesen et al [69] and Borges et al [70] solved the DG integration problem by

basically employing the GA method Both utilized the metaheuristic technique in solving

for single and multiple DG sizing and placements Haesen et al used the GA method to

minimize the DS active power flow while the objective for Borges et al was to

17

maximize a DG benefit to total cost ratio index Reference [69] incorporated penalty

factors within the objective function to penalize constraint violations thus adding another

set of variables to be tuned The authors of the latter reference used a PV model for

modeling the DG

Celli et al [71] formulated the DG integration problem as an s-constraint

multiobjective programming problem and solved it using the GA method Their

proposed algorithm divided the set of the objective functions into one master and the rest

are considered as slave objective functions The master is treated as the primary

objective function that is to be minimized while the slaves are regarded as new

inequality constraints that are bounded by a predetermined e value They utilized their

hybrid method to minimize the following objective functions cost of network upgrading

energy losses in the DS sections and purchased energy (from transmission and DG) The

number of the DG sources to be installed was randomly assigned and the units were

randomly located at the network buses Whenever the constraints are violated the

objective function solution is penalized A Pareto set was calculated from this

multiobjective optimization problem to aid the distribution planner in the decision

making process

Kim et al [72] and Gandomkar et al [73] hybridized two methods to solve the DG

sizing problem The former hybridized GA with fuzzy set theory to optimally size the

single DG unit while the latter combined the GA and SA metaheuristic methods to solve

for the optimal DG power output In both references the DG sizing problem was

formulated as a nonlinear optimization problem subject to boundary constraints only

Unlike Gandomkar et al [73] added the nonlinear power flow equality constraints to

their problem formulation The former researchers utilized their methodology to

investigate multiple DG case while the latter solved only the single DG case Both sited

the DG at all DS buses in order to determine the optimal DG location and size

Nara et al [74] assumed that the candidate bus locations for the DG unit to be

installed were pre-assigned by the distribution planner Then they used the TS method in

solving for the optimal DG size The objective of their formulation was to minimize the

system losses The DG size was treated as a discrete variable and the number of the

18

deployed units was considered to be fixed The DS loads were modeled as balanced

uniformly distributed constant current loads with a unity power factor

Golshan and Arefifar [75] applied the TS method to optimally size the DG as well

as the reactive sources (capacitors reactors or both) within the DS They formulated

their constrained nonlinear optimization problem by minimizing an objective function

that sums the total cost of active power losses line loading and the cost of the added

reactive sources The DG locations were not optimized instead a set of locations were

designated to host the proposed DGs and the reactive sources

A hybrid method that combined the GA with the TS technique in order to solve the

DG sizing optimization problem was developed by Gandomkar et al [80] They solved

the DG integration problem by minimizing the distribution real power losses subject to

boundary conditions The authors restricted the number of DGs as well as their gross

capacity to be revealed prior to executing the optimization procedure They augmented

the objective function with penalty terms in their formulation to handle the constraint

violations

Falaghi and Haghifam [77] proposed the ACO methodology as an optimization tool

for solving the DG sizing and placement problems The minimized objective function for

the utilized method was the global network cost ie the summation of the DGs cost their

corresponding operational and maintenance cost the cost of energy bought from the

transmission grid and the cost of the network losses The DG sizes were treated as

discrete values They used a penalty factor to handle the violated constraints ie

infeasible solutions In addition to modeling the DG sources as exclusive constant power

delivering units ie with unity power factor the network loads were all assumed to have

09 power factor Thus it can be stated that such modeling is impractical especially when

real large DSs are encountered

Raj et al [78] dealt with the DG integration in two different steps They employed

the PSO method to optimally determine the size of single and multiple DGs The optimal

location portion of the problem was performed utilizing the NR power flow method to

assign those buses with the lowest voltage profiles as the optimal candidate DG locations

The PSO was used to minimize the system real power losses the voltage profiles

boundary conditions were the only constraints required by the authors to be satisfied

19

Constraint violations were handled via a penalty factor that was augmented with the

objective function The DG units were randomly sited at one or more of the candidate

buses obtained through the NR method and subsequently the PSO was used to find the

optimal size(s)

Rahman et al [79] derived sensitivity indices to identify the most suitable bus for a

single DG installation Subsequently the DG sizing problem was dealt with by

employing an EP approach The objective function of the proposed approach was to

minimize the DS real power losses subject only to the system bus voltage boundary

constraints The formulation of DG sizing in their work was not realistic for a variety of

reasons For instance they ignored the line loading restrictions power flow equality

constraints and DG size limits

In most of the reviewed work on the DG deployment problem the problems of DG

optimal sizing and placement were not simultaneously addressed due to the difficult

nature of the problem as it combines discrete and continuous variables for potential bus

locations and DG sizing in a single optimization problem This combination creates a

major difficulty to most derivative-based optimization techniques and it increases the

feasible search space size considerably In this thesis the DG sizing subproblem is

solved using an improved SQP deterministic method while the two subproblems are

addressed simultaneously via an enhanced PSO metaheuristic algorithm

24 SUMMARY

In this chapter distribution power flow techniques were reviewed in Section 22 The

literature review of DG integration problem solution methods was presented in Section

23 The analytical and deterministic methods that were utilized to handle the DG

integration problem were presented in Subsection 231 Then recent publications that

handled the DG sizing and placement problems via wide-class of metaheuristic methods

were reviewed and summarized

20

CHAPTER 3 FAST AND FLEXIBLE RADIAL POWER FLOW FOR

BALANCED AND UNBALANCED THREE-PHASE DISTRIBUTION

NETWORKS

31 INTRODUCTION

As discussed in Chapter 2 several limitations exist in radial power flow techniques

presently reported in the literature such as complicated bus numbering schemes

convergence related problems and the inability to handle modifications to existing DS in

a straightforward manner This motivated the development of an enhanced distribution

power flow solution method In this thesis the unique structure of the RDS is exploited in

order to build up a Fast Flexible Radial Power Flow (FFRPF) technique The tree-like

RDS configuration is translated into a building block bus-bus oriented data matrix

known as a Radial Configuration Matrix (RCM) which consequently is utilized in the

solution process The developed algorithm is also capable of handeling weakly meshed

and meshed DSs via a meshed RCM (mRCM) RCM or mRCM is the only matrix that

needs to be constructed in order to proceed with the iterative process During the data

preparation stage each RCM (or mRCM) row focuses only on a system bus and its

directly connected buses That is while building such a matrix there is no need to

inspect the entire system buses and sections Moreover no complicated node numbering

scheme is required The building block matrix is designed to have a small condition

number with a determinant and all of its eigenevalues equal to one to ensure its

invertibility By incorporating this matrix and its direct descendant matrices in solving

the power flow problem the CPU execution time is decreased compared with other

methods The FFRPF method is flexible in accommodating any changes that may take

place in an existing radial distribution system since these changes can be exclusively

incorporated within the RCM matrix The proposed power flow solution technique was

tested against other methods in order to judge its overall performance using balanced and

unbalanced DSs

In Chapters 4 and 5 the FFRPF method is incorporated within the developed FSQP

and HPSO algorithms in solving the optimal DG installation problem It is implemented

21

as a subroutine within the proposed algorithms to satisfy the equality constraints ie

solving the radial power flow equations

32 FLEXIBILITY AND SIMPLICITY OF FFRPF I N NUMBERING THE RDS

BUSES AND SECTIONS

The RDS is configured in a unique arborescent structure with the distribution substation

located at its root node from which all the active and reactive power demands as well as

the system losses are supplied The substation feeds one or more main feeders with

spurred out laterals sublaterals and even subsublaterals For this reason the substation is

treated as a swing bus during the power flow iterative procedure

Most radial power flow techniques proposed in the literature assign sophisticated

procedures for numbering the radial distribution networks in order to execute their

algorithms This is cumbersome when expanding andor modifying existing RDSs In

this section a very simple numbering rule for the RDS buses and sections is introduced

A section is defined as part of a feeder lateral or sublateral that connects two buses in the

RDS and the total Number of Sections (NS) is related to the total Number of Buses (NB)

by this relation (NS=NB -1 )

321 Bus Numbering Scheme for Balanced Three-phase RDS

A balanced radial three-phase RDS is represented by a single line diagram In such a

system a feeder or sub level of a feeder having more than one bus is numbered in

sequence and in an ascending order Consequently each section will carry a number

which is less than its receiving end bus number by one as shown in Figure 31

Therefore sections are numbered automatically once the simple numbering rule is

applied

22

Substation

Figure 31 10-busRDS

In numbering the RDS shown in Figure 31 the following was considered buses 1 -

4 form the main feeder and buses 2 - 6 and 3 - 10 are the laterals while the sublateral is

tapped off bus 5 Figure 32 and Figure 33 manifest the ease in renumbering the system

shown previously and the flexibility in adding any portion of RDS to the existing one

respectively In Figure 32a buses 1 -4 have a different path compared to Figure 31 and

are considered to be a main feeder buses 2 - 9 and 3 - 6 are the laterals whereas the

sublateral 7 - 10 is the section spurred off bus 7 Figure 32b shows another numbering

scheme The same system numbered differently would have the same solution when

solved by the FFRPF

Figure 33 illustrates the ease of numbering in the case of a contingency situation or

a switching operation that could cause the existing system to be modified andor to be

augmented with other systems The lateral 3 - 10 in Figure 31 was modified to be

tapped off bus 2 instead and a couple of radial portions were added to be fed from buses

6 and 4 as illustrated in the figure

23

Substation Substation

(a) (b)

Figure 32 Different ways of numbering the system in Fig 31

Figure 33 The ease of numbering a modified and augmented RDS

322 Unbalanced Three-phase RDS Bus Numbering Scheme

The three-phase power flow is more comprehensive and realistic when it comes to

finding the three-phase voltage profiles in unbalanced RDSs Figure 34 shows an

unbalanced three-phase RDS The missing sections and buses play a significant role in

the multi-level phase loading and in making the unbalanced state of such a three-phase

DS more pronounced

24

The RDS shown in Figure 34 includes 6 three-phase buses (3(j)NB = 6) 5 three-

phase sections (3(|)NS = 5) no matter how many phase buses or sections exist physically

As such it has 14 single-phase buses (1(|)NB = 14) and 11 single-phase sections (1lt|gtNS =

11) The relations expressed in Eq (31) govern the three-phase and single-phase buses

to their corresponding sections

3^NS = 3^NB-1

l^NS = l^NB-3 (31)

Figure 34 Three-phase unbalanced 6-bus RDS representation

It is simple to implement the numbering process in the three-phase system as was

done in the balanced case Any group of phase buses to be found along a phase feeder or

a sub level of a feeder is to be numbered in a consecutive ascending order Consequently

each phase section number will carry a number which is one less than its receiving end

bus number as shown in Figure 34 In other words the sections are numbered routinely

after the ordering of the three-phase RDS buses has been completed

To develop the building block matrix as will be shown shortly the unbalanced three-

phase system is redrawn by substituting for any missing phase section or bus using dotted

representation as depicted in the 6-bus RDS in Figure 35 By performing this step each

three-phase bussection in the RDS consists of a group of 3 single-phase busessections

a b and c including the missing ones for double and single-phase buses

25

l a

I (1) 2 a | (2) 3 a | (3) 4 a | (4)

Figure 35 Three-phase unbalanced 6-bus RDS with missing buses and sections

33 THE BUILDING BLOCK MATRIX AND ITS ROLE I N FFRPF

The proposed FFRPF procedure starts with a matrix that mimics the radial structure

topology called a system Radial Configuration Matrix (RCM) The inverse of RCM is

then obtained to produce a Section Bus Matrix (SBM) that will be utilized in summing

the section currents during the backward sweep procedure A Bus Section Matrix (BSM)

is next generated by transposing the SBM to sum up the voltage drops in the forward

sweep process Therefore the only input data needed in the solution of an existing

modified or extended RDS other than the system loads and parameters is the RCM

It is worth mentioning that the inversion and transposition operations take place only

once during the whole process of the proposed FFRPF methodology for a tested RDS

whereas other methods like the NR technique invert the Jacobian matrix in every single

iteration The following subsections demonstrate the building of a three-phase RCM and

elucidate the role of both SBM and BSM in solving the radial power flow problem

331 Three-phase Radial Configuration Matrix (RCM)

The only matrix needed to be built for an unbalanced three-phase RDS is the RCM

Whatever changes need to be accommodated as a modification in the existing structure or

an addition to the existing network would be performed through the RCM only The

26

other matrices utilized in the backwardforward sweep are the direct results of the RCM

and no other built matrix is needed to perform the FFRPF

Such a matrix exploits the radial nature of such a system The RCM is of 3(3())NB x

3(|)NB) dimension in which each row and column represents a single-phase bus For a

balanced three-phase RDS represented by a single line diagram the RCM dimension is

(NBxNB) The RCM building algorithm for the unbalanced three-phase RDS case is

illustrated as follows

1 Construct a zero-filled 3(3(|)NB x 3(|)NB) square matrix

2 Change all the diagonal entries to +1 every diagonal entry represents sending

missing or far-end buses

3 In each row if the column index corresponds to an existing receiving single-phase

bus its entry is to be changed to - 1

4 If a single-phase bus is missing or is a far-end bus the only entry in its

corresponding row is the diagonal entry of+1

The above RCM building steps are summarized in the following illustration

Columns Description

RCMbdquo

if is either

a - sending phase bus b - far-end phase bus c - missing phase bus (32)

-1 jkl if jkI are receiving phase buses

connected physically to phase bus 0 otherwise

The [abc] matrix is defined as a zero (3 x 3) matrix with the numbers a b and c as

its diagonal elements eg [ I l l ] is the identity (3 x 3) matrix while [110] is the identity

matrix with the third diagonal element replaced by a zero By following the preceding

steps and utilizing the [abc] definition the RCM for the unbalanced three-phase RDS

shown in Figure 35 is to be constructed as shown in (33)

27

[Ill] [000] [000] [000] [000]

[000]

-[111] [111]

[000] [000] [000]

[000]

[000] -[111]

[111] [000] [000]

[000]

[000] [000]

-[110] [111]

[000] [000]

[000]

[000] [000]

-[010] [111]

[000]

[000] [000]

-[on] [000] [000]

[111]

Because of the nature of the RDS the RCM has three distinctive properties The first

is that the RCM is sparse the second is that RCM is a strictly upper triangular matrix

and thirdly such a matrix is only filled by 0 +1 or - 1 Such a real matrix makes the data

preparation easy to handle and less confusing Figure 36 shows the sparsity pattern plots

of the RCM matrices for unbalanced three-phase 25-bus [48] and balanced 90-bus [38]

radial systems

RCM for 25-Bus unbalanced 3-phase RDS RCM for 90-Bus balanced 3-phase RDS

nz = 131 nz = 179

Figure 36 RCM matrices sparsity plots for the 25 and 90 bus RDSs

3311 Assessment of the FFRPF Building Block RCM

The RCM is well-conditioned and should have a small Condition Number (CN) and a

non-zero determinant The CN measures how far from singularity any matrix is It is

defined as

28

cond(A) = A jjA-l (34)

where ||A|| is any of the 3 types of norm formulations of matrix A 1-norm 2-norm or oo-

norm An ill-conditioned matrix would have a large CN while a CN of 1 represents a

perfectly well-conditioned matrix By definition a singular matrix would have an infinite

CN [81] Having all the matrix eigenvalues to be equal to 1 and a determinant value of 1

safeguard the RCM against singularity For this reason the RCM is not only invertible

but also its inverse is an upper triangular matrix filled only with 0 and +1 digits and no

other numbers would appear in RCM-1

332 Three-phase Section Bus Matrix (SBM)

The SBM for the three-phase RDS is obtained by performing the following steps

1 Remove the corresponding substation rows and columns from the RCM ie the

first three rows and columns The reduced version of the RCM is labeled as

RCM

2 Invert the RCM to obtain the SBM as shown in (35) and more explicitly in

Figure 37

To clarify the two rows and the two columns outside the matrix border shown in

Figure 37 are the three-phase buses and sections ordered respectively The dimension of

the unbalanced three-phase system SBM is 3(3lt|)NS x 3(|gtNS) For the balanced case the

SBM dimension is (NSxNS)

[Ill] [000]

[000] [000] [000]

[111] [111]

[000] [000] [000]

[110] [110]

[111] [000] [000]

[010] [010] [010] [111] [000]

[011] [011] [000] [000] [111]

29

1 a

1 b

1 c

2 a

2 b 2 c

SBM = 3 a

3 b

3 c

4 a

4 b

4 c

5 a

5 b

5 c

2 a

0 0 0 0 0 0 0 0 0 0 0 0 0 0

2 b

0 1 0 0 0 0 0 0 0 0 0 0 0 0 0

2 c

0 0 1 0 0 0 0 0 0 0 0 0 0 0 0

3 3 a b

1 0 0 1 0 0 i o 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

3 c

0 0 1 0 0

1 0 0 o 0 0

o 0 0 0

4 a

1 0 0 1 0

o 1 0

o 0 0

o 0 0 0

4 b

0 1 0 0 1 0 0 1 0 0 0 0 0 0 0

4 c

0 0 0 0 0 0 0 0 1 0 0 0 0 0 0

5 a

0 0 0 0 0 0 0 0 0 1 0 0 0 0 0

5 b

0 1 0 0 1 0 0 1 0 0 1 0 0 0 0

5 c

0 0 0 0 0 o 0 0

o 0 0 1 0 0 0

6 a

0 0 0 0 0

o 0 0

o 0 0 o 1 0 0

6 b

0 1 0 0 1 0 0 0 0 0 0 0 0 1 0

6 c

0 1 0 0 1 0 0 0 0 0 0 0 0 1

Figure 37 SBM for three-phase unbalanced 6-bus RDS

By inspecting Figure 35 it is noted that any single-phase section is connected

downhill to a single-phase far-end bus or buses through a Unique Set of Phase Buses

(USPB) xt bull By inspecting Figure 35 and Figure 37 the USPB for the following

single-phase sections la lb lc 3b 4b are^0 [ gt xlgt x respectively as explained

in (36)

=2 f l 3bdquo4a x=2b3b4b5bA]

Xl=2cA) Xl=) (36)

X=5b]

In the SBM the single-phase section is represented by a row i and will have entries

of ones in all the columns where their indices represent single-phase buses that belong to

the section USPB xf bullgt a s illustrated in (37)

SBMrmt =

Columns Description

c bullgt bull a - Id - buses e r ~ - 1 ijk- ijk-- are either w (37)

lb - diagonal entry 0 other columns otherwise

30

333 Three-phase Bus Section Matrix (BSM)

The BSM is the transpose of the SBM as shown in (38) In the BSM the rows represent

the RDS single-phase buses excluding the substations and all the sections are

represented by the BSM columns Each single-phase bus is connected uphill through a

Unique Set of Phase Sections (USPS) yf to a substation single-phase bus The USPS

for buses 2deg 3a 4b 5b 6C are y ydeg y y yc6 respectively as depicted in Figure 35

and Figure 37 and demonstrated in (39)

BSM

[111] [000] [000] [000] [000]

[111] [111] [000] [000] [000]

[110] [110] [111] [000] [000]

[010] [010] [010] [111] [000]

[011] [011] [000] [000] [111]

(38)

V H U V3=1gt2 y=b2b yb

5=lb2bdquoAb (39)

lt=1C2C5C

In the BSM a single-phase bus i is represented by a row and will have entries of

ones in all the columns where their indices represent single-phase sections that belong to

the bus USPS yf as equivalently shown in (310)

Columns Description

BSMrmi =

( gt - [a-l^-sectionse yf ~ i m

1 ijk ijk are either lt Y Y (310) lb - diagonal entry

0 other columns otherwise

34 FFRPF APPROACH AND SOLUTION TECHNIQUE

The FFRPF methodology is based simply on Kirchhoff s voltage and current laws which

are used in performing the backwardforward sweep iterative process By utilizing the

direct descendant matrices of the RCM ie SBM and BSM the RDS buses complex

31

voltages are calculated in every iteration until convergence criteria are met The next

subsections illustrate the proper usage of such matrices in the proposed FFRPF method

through appropriate modeling of the unbalanced multi-phase RDS section impedances

341 Unbalanced Multi-phase Impedance Model Calculation

Figure 38 shows a three-phase section model that is represented by two buses (sending

and receiving ends) of an overhead three-phase four-wire RDS with multi-grounded

neutral The assumption of a zero voltage drop across the neutral in a three-phase two-

phase and single-phase RDS is found to be valid [4582] Such a configuration is widely

adopted in North Americas distribution networks [8384]

V Sa

_ bull

V Ra

V Sb ab

^VWVVYgt

V Sc Z be

bn

^AAArmdashrYYYV bull

V s n en

ll1 Sa

s b

La

Lb

Lc

bull r Figure 38 Three-phase section model

In the proposed radial power flow solution method each of the three-phase lines is to

be modeled appropriately and mutual coupling effects between phases are not neglected

The primitive impedance matrix for such a four-wire system is a square matrix with a

dimension equal to RDS utilized number of phase and neutral conductors For a system

consisting of three-phase conductors and a neutral wire the section primitive impedance

matrix is expressed as shown in (311)

32

Zaa

ha

Zca

zna

Kb

Kb

Kb

Kb

Ke Kc Ke

nc

art

Zbn

en

nn

where

Z bull primitive impedance matrix

RDS section length

z per unit length self-impedance of conductor i

z per unit length mutual-impedance between conductors andy

zu and zy are calculated according Carsons work [85] and its modifications [86-88] as

illustrated by the following equations

where

k

GMRj

Dbdquo

v GMR

bulli J

zu=rt+rd+ja)k

zv=rd+jltok

resistance of conductor i

earth return conductor resistance

inductance multiplying constant

distance between overhead and its earth return counterpart and it is a

function of both earth resistivity and frequency

geometric mean radius of conductor i

distance between conductors i andj

(312)

(313)

The parameters used in (312) and (313) are shown in Table 31 for both operational

frequencies 50Hz and 60Hz in both metric and imperial units

33

Table 31 cok rj and De Parameters for Different Operation Conditions

De = 2160 Ij (ft)

cok rd

p = 100 Qm

p = 1000 Qm

Metric Units RDS operating frequency 50 Hz 60 Hz

006283km

0049345 QJ km

931 m

29443 m

007539 km

005921412km

850 m

26878 m

Imperial Units RDS operating frequency 50 Hz 60 Hz

010111mile

00794 QI mile

30547f

96598 ft

012134mile

009528 QI mile

27885

88182

Since the neutral is grounded the primitive impedance matrix Zsec can be

transformed into a (3 x 3) symmetrical impedance matrix Zsae

c by utilizing Krons

matrix reduction method The resultant section three-phase impedance matrix is

expressed mathematically in (314) and the three-phase section model is represented

graphically in Figure 39

7 abc

aa

zba Zca

Zab

^bb

Zcb

zac zbc Zee

(314)

VSn

mdash bull i 7 T

i ah bull-sec a

zbdquobdquo bull A V W Y Y Y V

v izK

^WW-rrYYv -+bull

I

bull

vR

Figure 39 The final three-phase section model after Krons reduction

If the RDS section consists of only one or two phase lines its primitive impedance

matrix is transformed by Krons matrix reduction to (laquo-phase x w-phase) symmetrical

impedance matrix Next the corresponding row and column of the missing phase are

replaced by zero entries in the (3x3) section impedance matrices Zsae

c For a two-phase

34

section its impedance matrix Z^c is demonstrated below

Z_a

zci Kron h-gt ZZ za

zbdquo zai

zaa o zac

0 0 0

z_ o zbdquo

Underground lines such as concentric neutral and tape shielded cables are typically

installed in the RDS sections For underground cables with m phases and n additional

neutral conductors the primitive impedance of each section is a (2m+n x 2m+n) matrix

with the entries computed as illustrated in [89-92]

Usually the RDS is modeled as a short line ie less than 80 km and the charging

currents would be neglected by not modeling the line shunt capacitance as depicted in

Figure 38 However under light load conditions and especially in the case of

underground cables the line shunt capacitance needs to be considered in order to obtain

reasonable accuracy ie use nominal 7i-representation The rc-equivalent circuit consists

of a series impedance of the section and one-half the line shunt admittance at each end of

the line Figure 310 (a) (b) and (c) show the RDS 7i-model representation The shunt

admittance matrix for an overhead three-phase section is a full (3x3) symmetrical

admittance matrix while it is a strictly diagonal matrix for the underground RDS cable

section That is the self admittance elements are the only terms computed [92] For the

unbalanced three-phase section eg one or two phases the non-zero elements of shunt

admittances are only those corresponding to the utilized phases

[zic] -AAVmdashrwvgt

T yabc 1 |_ sec J [ yaf tc |

sec J

(a)

35

Lsec J _

2

s

1

Yaa

Yba

Yea

Yab

Ybb

Ycb

Yac

Ybc

Ycc

zaa

zba

tea

zab

zbb

zcb

zac

zbc

zcc

P yabc ~|

lgtlt 2 - =

Yaa

Yba

Yca

Yab

Ybb

Ycb

R

1 1

Yac

Ybc

Ycc

(b)

[ yabc 1 sec J

s 1 1

Yaa

0

0

0

Ybb

0

0

0

Ycc

zaa

zba

zca

Zab

zbb

zcb

zac

zbr

zcc

V yabc ~j

L sec 2 - =

Yaa

0

0

0

Ybb

0

R

1 1

0

0

Ycc

(c)

Figure 310 Nominal 7i-representation for three-phase RDS section

(a) Schematic drawing for the single line diagram of 3(|gt RDS (b) Matrix equivalent for overhead section (c) Matrix equivalent for underground cable section

By applying Kirchhoff s laws to the three-phase system section k the relationship

between the sending and receiving end voltages for medium and short line models and

the voltage drop across the same section in the latter model are expressed in Eq (315)-

(316) and Eq (317) respectively

36

rabc S rabc S

14 L sec Jax3 L rabc

3x3 L secgt J3x3

[C]3 [4 [ yabc~ |~ yabc 1

sec J3x3 L sec J3 [4

zt 1 L sec J3x3

f yabc ~| [~ yaampc 1

L sec J3x3 L sec h

bull R rabc

(315)

rrabc VS rabc

S

1 J3x3 L sec J

[degL [L abc R

(316)

where

TT-afec rrabc S ^ R

rabc rabc S XR

rabc

AK

13x3

aAc sect

rabc see

KrH^ic] three-phase sending and receiving end voltages

three-phase sending and receiving end section currents

three-phase shunt admittance of section k

(3gtlt3) identity matrix

(3gtlt3) zero matrix

voltage drop across three-phase section k

section k three-phase currents

(317)

It is worth noting that Eq (315) is reduced to Eq (316) in the case of short line

modeling since YS^C 1 = 0 The voltage drop across the three-phase section k in the short

line model is expressed in Eq (317) and its corresponding sending end phase voltages

can be expressed in expanded forms as follows

V = V + Ta 7 + 1 7 + Tc 7 rS yR ^1secZjaa ^ Isec^ab ^rlsecpoundjac

v =v +r 7 +17 +r 7 y S y R ^ l sec^ab ^ 1 sec^bb ^ 1 sec^Ac

S mdashyR+ Ktc^ca + sec^cb + sec^a

(318)

(319)

(320)

Equations (317)-(320) show that the voltage drop along any phase in a three-phase

section depends upon all the three-phase currents

37

342 Load Representation Accurate and proper load modeling is of significant concern in power distribution

systems as well in its transmission systems counterpart [8693] Loads in electric power

systems are usually expressed by adequate representations so as to mimic their effects

upon the system The load dependency on the operating bus voltage and on system

frequency is among those representations

Static load models are often utilized in the power flow studies since they relate the

apparent power active and reactive directly to the bus operating voltage A static load

model is used for the static load components ie resistive and lighting load and as an

approximation to the dynamic load components ie motor-driven loads [93] Generally

static loads in DS are assumed to operate at rated and fixed frequency value [94-96]

Loads in the DS are usually expressed as function of the bus operating voltage and

represented by exponential andor polynomial models

The exponential model is shown in (321) and (322)

P = Pbdquo

Q = Q0 vbdquo

(321)

(322)

where

V0 nominal bus voltage

V operating bus voltage

P0 real power consumed at nominal voltage

Q0 reactive power consumed at nominal voltage

Exponents a and fi determine the load characteristics and certain a and values lead to a

specific lode model Therefore

1 If a = P mdash 0 the model represents constant power characteristics ie the load is

constant regardless of the voltage magnitude

2 If a = P = 1 the model represents constant current characteristics ie the load is

proportional to the voltage magnitude

3 If a = P = 2 the model represents constant impedance characteristics ie the load is

38

a quadratic function of the voltage magnitude

As indicated in [97] the exponents could have values larger than 2 or less than 0 and

certain load components would be represented by fractional exponents

The constant current model is considered to be a good approximation for many

distribution circuits since it approximates the overall performance of the mixture of both

constant power and constant impedance models [98] However representing loads with

the constant power model is a conservative approach with regard to voltage drop

consideration [99] and consequently this model will be used in this thesis

Loads can also be represented by a composite model ie the polynomial model The

polynomial model is expressed in (323) and (324)

P = Pbdquo

Q = Q0

(

a p

V

r

V

V

K

V

v0

2

2

V

K

V

+CP

J

)

(323)

(324)

where ap + bp + cp = 1 and aq + b + cq = 1

The polynomial model is also referred to as a ZIP model since it combines all the

three exponential models constant impedance (Z) constant current (I) and constant

power (P) models The ZIP model needs more information and detailed data preparation

The load models can be used in the FFRPF solution method during its iterative

process where flat start values are initially assumed to be the load voltages The three-

phase load voltages are changed during each iteration and consequently the three-phase

currents drawn by the constant current constant impedance andor ZIP three-phase load

models will change accordingly

Different shunt components like spot loads distributed loads and capacitor banks are

customarily spread throughout the RDS In power flow studies spot and distributed

loads are typically dealt with as constant power models while shunt capacitors are

modeled as constant impedances [94 100 101]

The uniformly distributed loads across RDS sections can be modeled equivalently by

either placing a single lumped load at one-half the section length or by placing one-half

the lump-sum of the uniformly distributed loads at each of the section end buses

39

[99 102] The former modeling approach has the disadvantage of increasing the

dimension of the RCM SBM and the BSM since more nodes would be added to the

existing RDS topology In the proposed FFRPF technique the distributed load is

modeled using the latter approach while the three-phase shunt capacitor banks are

modeled as injected three-phase currents [101] as schematically shown in Figure 311

and mathematically represented by Eq (325) and (326)

Qk Cap

^

CCap a Cap

(a) (b)

Figure 311 Three-phase capacitor bank (a) The schematics (b) The Modeling

O3 = poundCap

Qo Capa vbdquo

T-34 _ 1Cap ~

V

a Capbdquo

SQL M Cap

V

filt bullCapo

F

JQ( Cap

(325)

(326)

343 Three-phase FFRPF BackwardForward Sweep

The FFRPF technique employs the SBM in performing the current summation during the

backward sweep and the BSM in updating the RDS bus complex voltages during the

forward sweep as demonstrated in the following subsections

3431 Three-phase Current Summation Backward Sweep DS loads are generally unbalanced due to the unbalanced phase configuration the double

and single-phase loadings as well as the likelihood of unequal load allocation among the

three-phase configuration For the loads they could be represented as constant power

40

constant current constant impedance or any combination of the three models [97 103]

The three-phase load currents at three-phase bus i drawn by a three-phase load eg of a

constant impedance load model is mathematically expressed as shown in Eqs (327) and

(328)

jabc U ~

7e a gt

va

(si) ~

(327)

where

ctabc

o

V

K

2

rft

0

v K

2

K v

2

(328)

where Sf represents the load apparent power at single-phase bus lt|gt As shown in the

preceding equations each load current is a function of its corresponding bus voltage For

Eq (327) if the a phase bus is missing its corresponding phase load current is

eliminated and its corresponding position in the three-phase current vector is replaced by

a zero entry As an illustration and by assuming that there are loads connected to all

existing buses the three-phase load current vector for the system shown in Figure 35 is

expressed as follows

jabc V ja rb jc ra rb re ra rb Q Q rb Q Q rb re l l

The charging currents at the RDS three-phase buses are not to be neglected when

dealing with sections modeled as The shunt admittance at bus is obtained by

applying the following relation

where

Ysh^ bull total three-phase shunt admittance at bus

[l if section k attached to bus i

[0 otherwise

The three-phase shunt currents at bus is as shown in Eq (330)

tabc jrabc 1ch ~~ 1Anbus y i (330)

41

The 3-ltj) bus current is sum of both the 3-sect load current and the 3-cj) charging current as

expressed mathematically in Eq(331)

jabc jabc jabc ) T I 1busl ~ 1Li

+Ich V-gt )U

where 1^ is bus three-phase currents In the case of modeling a three-phase section as

a short line its charging currents are neglected ie I^c = 0 and the bus current will be

represented by the load currents only

The backward sweep sums the phase load currents in the corresponding phase

sections starting from far-end phase buses and moving uphill toward the substation phase

buses The current in phase (j) and section p is computed by utilizing the USPB

principle xp gt during the backward sweep as expressed in (332)

lt = E lt ^here = j 0 ^ J (332)

where

I current through single-phase section and phase ^ (^ =a b or c) SQCp

j current at bus and phase ltb bus x

The SBM is utilized in obtaining the system three-phase section currents in matrix

representation by performing the relation in Eq (333)

[G] = [SBM][lpound] (333)

where the I^ is a 3-(3())NS)-order column vector For a balanced short line RDS

model Eq (333) can be expressed as

[CS] = [SBM][IL] (334)

3432 Three-phase Bus Voltage Update Forward Sweep

The voltage at each phase bus is determined through the forward sweep procedure by

subtracting the sum of the voltage drops across the bus corresponding USPS from the

substation nominal complex voltage The voltage drop across three-phase section k is

calculated as in Eq (317) The voltage drop across all sections of the three-phase RDS

can be obtained by utilizing the SBM as shown in matrix notation via Eq (335)

42

[AKbdquo]=[zr][c] (335)

[ A ^ ] = [ Z s r ] [ 5 5 M ] [ C ]

where ZtradeS J is a diagonal matrix with an 3(3c|)NS x 3ltj)NS) dimension in which its

diagonal entry k corresponds to section k impedance and AV3^ is the computed three-

phase voltage drop values across all the RDS sections as shown below

A ^ = AVa AVb AVC bullbullbull AVb AVC T s e c l s e c l s e c l sec3ltWS sec3tNS J

For calculating the RDS voltage profiles the FFRPF solution method starts by asshy

suming the initial values for all bus voltages to be equal to the substation complex

voltage As a flat start the initial phase voltages at bus will be as follows

2TT 2TT

ya Jdeg vb e~^ VC e+JT V ss e V sisK v sis e (336)

where Vls is the substation complex phase voltage

For the voltage at bus m and phase (j) to be determined at iteration v the calculation is

performed as follows

= amp - pound r A lt wherer = trade lt (337)

The RDS voltage profiles at the system phase buses are obtained by utilizing the BSM as

shown in following matrix representation

[Vi] = [Vsy[BSM][AV^] (338)

where V^fs 1 and V3A are respectively the substation nominal three-phase voltage

column vector and the resultant three-phase bus voltage solution column vector and each

has a dimension of 3(3lt|gtNS)

3433 Convergence Criteria

The bus complex voltage is obtained after every backwardforward sweep After each

iteration all the bus voltage magnitudes and angles are compared with the previous

iteration outcomes The power flow process is concluded and a solution is reached if the

complex voltage real and reactive oo-norm mismatch vector is less than a certain

43

predetermined empirical tolerance value e The convergence criterion is expressed

mathematically as shown in Eq (339)

+i

([gt]w) A a ( |y f lts

where th

i iteration A

(339)

and symbol

||J| vector oo-norm ||x II = max fix IV II lloo l l c 0 1=12NBV I

5H (bull) real part of complex value

3 (bull) imaginary part of complex value

3434 Steps of the FFRPF Algorithm

The FFRPF iterative process can be summarized as follows

Step 1 Begin FFRPF by choosing a test RDS

Step 2 Number and order RDS buses and sections

Step 3 Construct RCM

Step 4 Obtain both SBM and BSM

Step 5 Select load model

Step 6 Start the iterative procedure by assuming flat start voltages for all buses

Step 7 Calculate load currents

Step 8 Start the backward sweep process by calculating section currents using SBM

Step 9 Start the forward sweep process by determining the bus complex voltages

using BSM

Step 10 Compare both magnitudes and angles of the RDS bus voltages between the

current and previous iterations

bull If the co-norm of their difference is lt st

o Solution is reached

44

o Stop and end FFRPF procedure

o Obtain bus voltage profiles section currents and power losses

etc

bull If not utilize the outcome of this iteration (bus complex voltages)

to start a new one by going back to Step 7

The FFRPF solution method is illustrated by the following flow chart shown in Figure

312

45

i laquo - i +1

Calculate Load and leakage

currents

I Start Backward

sweep process by calculating section

currents using SBM

Start Forward sweep process by determining bus

complex voltages V[+1] using BSM

V[+1] Section currents

Section Power Losses Etc

Start FFRPF

Read the test RDS data

Number and order RDS Buses and

Sections

I Construct RCM

Remove the substation

corresponding rows and columns

from RCM to Obtain RCM

Obtain RCM1

To Get SBM

Z Transpose SBM to

get BSM

Calculate RDS section

Impedance and Shunt admittance

Matrices

Select load model

Assuming a flat start voltages for

all buses V[]=10 =0

Figure 312 The FFRPF solution method flow chart

46

344 Modifying the RCM to Accommodate Changes in the RDS The FFRPF technique deals with changes in the network such as the addition of a

transformer by adjusting the original RCM to incorporate its conversion factor (c)

Subsequently the SBM and BSM are obtained accordingly and used in the

backwardforward sweep procedure If a three-phase transformer is incorporated in a

three-phase RDS between buses m and n at section n - 1 the modified BSM entries are

located at the intersection of the matrix rows and columns defined by Eq (340)

BSMZ EzL~-inBSMZ euro lt _ (340)

The affected rows and columns of the modified BSM are those belonging to the

sections USPB and the sending buss USPS respectively For demonstration purposes

the 10-bus balanced RDS shown in Figure 31 will be utilized If two transformers with

conversion factors cfj and cS are to be added within sections 3 and 6 respectively the

original RCM is modified to accommodate such additions as illustrated in (341) Thus

instead of filling -1 for the receiving end bus entry the negative of the conversion factor

is the new entry The process is repeated rc-times for -installed transformers The

corresponding modified SBM and BSM are to be obtained as demonstrated in Section

33

10

RCM^ =

1

2

3

4

5

6

7

8

9

10

1

0

0

0

0

0

0

0

0

0

-1

1

0

0

0

0

0

0

0

0

0

-1

1

0

0

0

0

0

0

0

0

0

-cfi 1

0

0

0

0

0

0

0

-1

0

0

1

0

0

0

0

0

0

0

0

0

-1

1

0

0

0

0

0

0

0

0

-cf2

0

1

0

0

0

0

0

0

0

0

0

-1

1

0

0

0

0

-1

0

0

0

0

0

1

0

0

0

0

0

0

0

0

0

-1

1

(341)

The affected entries of the new BSM are obtained by applying the relation in (340) as

follows

47

[BSMZ eZ s ^nBSMJ euro ^ 3 = 40(12=[(4l)(42)]

(laquoSWe^)nJM6^)=J7 )8)njl I4=gt[(7 )l)(7 )4)(8l) )(M)]

The matrix shown in (342) shows the final B S M after including the transformers in the

10-bus R D S Such a procedure is easily extended to the unbalanced three-phase RDSs

1

1

1

CJ

1

1

cfi

cf2

1

1

2

0

1

ch 0 0

0

0

1

1

It is worth mentioning that by integrating the cf for any transformer configuration

into the RCM building block in the FFRPF technique another light is shed on the

flexibility criterion of the proposed method

35 FFRPF SOLUTION METHOD FOR MESHED THREE-PHASE DS

In practical DS networks alternative paths are typically provided to accommodate for

any contingency incidents that might take place eg feeder failure Therefore it is not

unusual for meshed distribution networks to be part of the DS topology in order to make

the system more reliable The loop analysis approach as well as the graph theory

technique are used to study and analyze the behavior of meshed DS The loop analysis

technique basically applies Kirchhoff s voltage law principle to solve for the fundamental

loop currents in both planar and nonplanar networks while the graph theoretic

formulation preserves the network structure properties [104]

A meshed DS can be viewed from a graph theory perspective as an oriented looped

graph that preserves the network interconnection properties whereas a DS that has no

0

0

0

1

1

cfi

cf2

0

0

0

0

0

0

1

0

0

0

0

0

0

0

0

0

1

1

0

0

0

0

0

0

0

0

1

0

0

0

0

0

0

0

0

0

1

1

0

0

0

0

0

0

0

0

1

(342)

48

loops is considered a tree In graph theory terminology line segments that connect

between buses in a loopless DS tree are called twigs branches or sections (represented by

solid line segments in Figure 313) while those which do not belong to the tree are

known as links (represented by dotted line segments in Figure 313) Links are segments

that close loops in a DS tree thereby composing a co-tree Removal of all co-tree links

results in a strictly radial system Links are usually activated by closing their

corresponding Normally Open (NO) switches Whenever a link is added to a RDS

network a loop is formed and as a result the system will have as many fundamental

loops as the number of links A fundamental loop is a loop that contains only one link

besides one or more sections Segments are used here to name sections and links

together It is noted that the number of fundamental loops is significantly less than the

number of buses in the meshed DS which makes the loop analysis a more appropriate

method in dealing with such systems than other circuit analysis methods like nodal

voltage method [105]

The current directions in the meshed DS sections and links are arbitrarily chosen to

be directed form a lower bus index to a higher one and the positive direction of loop

current is assumed to in the same direction of that of the link as illustrated in Figure 313

The number of segments in a meshed DS is equal to the sum of the total number of its

corresponding graph tree sections and its co-tree links For a meshed DS with NB buses

and mNS segments (total number of sections and links in the meshed DS) the number of

links nL and the number of the fundamental loops as well are obtained according to the

following relation

laquoL=mNS-NB + l (343)

49

Substation 2 Imdash 31 4 1

^ -gtT-gtL- -

Figure 313 10-bus meshed distribution network

351 Meshed Distribution System Corresponding Matrices The FFRPF solution method can deal with the DS meshed networks through modifying

the original RCM Discussion is now focused on the balanced three-phase meshed DS

which can easily be extended to the unbalanced three-phase DS networks

Meshed RCM The meshed RCM (wRCM) order is ((NB + wL) (NB + nVj) and the

mRCM building algorithm is as follows

1 Remove links from the meshed DS and build the RCM for the resulting network

tree as demonstrated earlier in section 331

2 Add nL rows and columns toward the end of the RCM ie each link is represented

by a row and a column attached to the end of the RCM

3 In each link column there are 3 non-zero entries and are to be filled in the following

manner

a -1 at the row which corresponds to the lower index terminal of the link

b +1 at the row which corresponds to the higher index terminal of the link

c +1 at the link diagonal entry

For the 10-bus system shown in Figure 31 three directed links Li L2 and L3 are added

to the DS tree through connecting the following buses 4 - 5 6 - 8 and 4 - 1 0

respectively The system mRCM is constructed as illustrated in (344)

50

10

mRCM (13x13)

1

0

0

0

0

0

0

0

0

0

0

0

0

-1

1

0

0

0

0

0

0

0

0

0

0

0

0

-1

1

0

0

0

0

0

0

0

0

0

0

0

0

-1

1

0

0

0

0

0

0

0

0

0

0

-1

0

0

1

0

0

0

0

0

0

0

0

0

0

0

0

-1

1

0

0

0

0

0

0

0

0

0

0

0

-1

0

1

0

0

0

0

0

0

0

0

0

0

0

0

-1

1

0

0

0

0

0

0

0

-1

0

0

0

0

0

1

0

0

0

0

0

0

0

0

0

0

0

0

-1

1

0

0

0

0

0

0

-1

1

0

0

0

0

0

1

0

0

0

0

0

0

0

-1

0

1

0

0

0

1

0

0

0

0

-1

0

0

0

0

0

1

0

0

1

(344)

Remove the substation corresponding rows and columns from the mRCM to produce the

mRCM The mRCM for the 10-bus system is shown in (345)

10

mRCM (12x12)

1

0

0

0

0

0

0

0

0

0

0

0

-1

1

0

0

0

0

0

0

0

0

0

0

0

-1

1

0

0

0

0

0

0

0

0

0

-1

0

0

1

0

0

0

0

0

0

0

0

0

0

0

-1

1

0

0

0

0

0

0

0

0

0

0

-1

0

1

0

0

0

0

0

0

0

0

0

0

0

-1

1

0

0

0

0

0

0

-1

0

0

0

0

0

1

0

0

0

0

0

0

0

0

0

0

0

-1

1

0

0

0

0

0

-1

1

0

0

0

0

0

1

0

0

0

0

0

0

-1

0

1

0

0

0

1

0

0

0

-1

0

0

0

0

0

1

0

0

1

(345)

Meshed SBM The meshed SBM (mSBM) is then obtained by inverting the mRCM As

51

an illustration the 10-bus meshed network mSBM is obtained as shown in (346)

10

mSBM (12x12)

1

0

0

0

0

0

0

0

0

0

0

0

1

1

0

0

0

0

0

0

0

0

0

0

1

1

1

0

0

0

0

0

0

0

0

0

1

0

0

1

0

0

0

0

0

0

0

0

1

0

0

1

1

0

0

0

0

0

0

0

1

0

0

1

0

1

0

0

0

0

0

0

1

0

0

1

0

1

1

0

0

0

0

0

1

1

0

0

0

0

0

1

0

0

0

0

1

1

0

0

0

0

0

1

1

0

0

0

i deg 1

1

-1

1 deg 0

o 0

0

i

o o

0

0

0

0

1

-1

-1

0

0

0

1

0

0

0

1

0

0

0

0

-1

-1

0

0

1

(346)

Let the mSBM be partitioned into two submatrices mSBMp and C so that the mSBMp

submatrix corresponds to the DS tree sections and the second submatrix C to the

fundamental loops or links as shown in (347)

wSBM = SBM

6 [cl (mNSxnL) = [mSBMp C]

JmNSx(NB-l)

The dotted line shown in the above relation implies matrix partitioning

(347)

Fundamental loop matrix The second submatrix in (347) ie C is the fundamental

loop matrix which governs the direction of currents in each of fundamental loop sections

and links The fundamental loop matrix C can be partitioned into two submatrices [Csec]

and [I] as demonstrated below

M_ where [Csec] is an ((NB - 1) x (laquoL)) matrix and [7] is an nL square identity matrix The

former matrix corresponds to the tree loop sections while the latter corresponds solely to

c-- (348)

52

the co-tree links

By inspecting the fundamental loop matrix C it is noted that each row represents a

section or a link and each column represents a loop Each column entry in the C matrix

CM will have one of the following values

1 Qy = +1 if section k belongs to and is oriented in the same direction of loop

2 Cki = - 1 if section k belongs to and is oriented in the opposite direction of loop

3 Claquo = 0 if section k is not in the loop

By inspecting the 10-bus DS mSBM illustrated in (346) the first loop which is represhy

sented by the tenth column of the matrix is comprised of three sections in addition to the

link The current in two of these sections runs in the same direction as their correspondshy

ing fundamental loop ie sections 2 and 3 while the third one ie section 4 has an

opposite orientation This can be easily verified by tracing the first loop in the meshed

DS single line diagram One can also note that two loop currents pass through the third

section in an additive manner

Meshed BSM The meshed BSM (mBSM) is the transpose of mSBM as illustrated in

(349)

[ B S M I 0](mNS-nL)mNS

L J(nLxmNS)

The second submatrix C[ is the transpose of the fundamental loop matrix as manifested in

(350)

mBSM = [mSBM] = mBSMr

c (349)

C=L

1

0

0

0

2 3

1 1

0 0

0 1

4

-1

0

0

5

0

1

0

6

0

-1

0

7

0

-1

0

g

0

0

-1

9

0

0

-1

h 1

0

0

h 0

1

0

h (f 0

1

(350)

Meshed DS impedance matrices The (laquoLx riL) loop-impedance matrix Zioop is

formulated as follows [106]

KHc]|Xf][c] (35D

53

where [ztradegS ] is defined as a non-singular diagonal (mNS x wNS) segment-impedance

matrix that contains all the meshed DS segment impedances (tree sections and links)

along its main diagonal The matrix [Z^fM is formulated as two diagonal submatrices

as follows

|~ rymDS I

L eg J

Zl

0

^

0

0

7

Zk

0

0

0

raquoL

|gtr ] | o o |[zr] (352)

where Z^S J is the (NB - 1 x NB - 1) loopless tree section-impedance diagonal square

matrix and Z^k 1 is the (raquoL x nL) links impedance diagonal matrix

352 Fundamental Loop Currents The algebraic sum of voltage drops AV around any fundamental loop is zero according

to Kirchhoff s voltage law (KVL) and this can be mathematically expressed in terms of

the fundamental loop matrix C as follows

[C][AF] = 0 (353)

The voltage drop across the meshed DS segments is determined by the following

relations

[W] = [zf][mSBM][mILL]

where

Lm seg J bull (wNS x l) segment currents column vector of the meshed DS network

jW iZJ (mNS x l) meshed DS bus Loads and Link currents column vector

(354)

In order to account for the link currents in the meshed network the segment currents

column vector and the meshed DS bus loads and links currents column vector are

54

respectively partitioned into two subvectors as defined below

[ jtree 1

J(mNSxl)

J((JVB-l)xl)

Jloop[ J(nLxl)

(355)

[mILL l(mNSxl)

L L J((MJ-l)xl)

Jloop J (wLxl)

(356)

where

[Cr J ((NB - 1) x 1) tree section currents column vector

[lL] ((NB - 1) x 1) RDS bus load currents column vector

j 1 (nL x 1) fundamental loop current vector which is also the meshed DS link

currents column vector

By multiplying both sides of Eq (354) by [C] the left-hand-side will equal to zero

according to KVL as shown in (353) and by employing Eqs (351) and (356) Eq (354)

is reformulated as

[C][AV] = [c][z^][mSBM[mILL]

0 = [c f [z f ] [mSBM | C ] L op]

bull = [Cr[zS]([raquoSBM][ l ]+ [C][ 1 ] )

0 = [C] [^ ] [ -raquoSBM] [ I ] + [cr [zbdquor][C][U]

-[c] [zf ][c][J=[c] [ Z - ^ S B M J ]

-[^][^]-[cT[zS][laquoSBM][I] Thus the (nL x 1) fundamental loop currents vector in the meshed DS loop frame of

reference can mathematically be expressed as

[4 J - -K]1 [Cj [z^JmSBMr][lL] (357)

Eq (357) can be expressed in terms of the RDS matrices ie SBM and [ Z ^ J by

performing the following operation

55

DALHOUSEB UNIVERSITY

To comply with the Canadian Privacy Act the National Library of Canada has requested that the following pages be removed from this copy of the thesis

Prehrmnary Pages Examiners Signature Page Dalhousie Library Copyright Agreement

Appendices Copyright Releases (if applicable)

DEDICATION PAGE

To my beloved parents my brothers Falah and Abdullah my sisters my wife OmFahad

my daughter Najla and my sons Fahad Falah and Othman

TABLE OF CONTENTS LIST OF TABLES x

LIST OF FIGURES xiii

ABSTRACT xvi

LIST OF ABBREVIATIONS AND SYMBOLS USED xvii

ACKNOWLEDGEMENTS xxiv

Chapter 1 INTRODUCTION 1

11 Motivation 1

12 Distribution Generation - Historic Overview 2

13 Distribution Generation 2

14 Thesis Objectives and Contributions 5

15 Thesis Outline 7

Chapter 2 LITERATURE REVIEW 9

21 Introduction 9

22 Distribution Power Flow 9

23 DG Integration Problem 13

231 Solving the DG Integration Problem via Analytical and Deterministic Methods 14

232 Solving the DG Integration Problem via Metaheuristic Methods 17

24 Summary 20

Chapter 3 FAST AND FLEXIBLE RADIAL POWER FLOW FOR BALANCED AND UNBALANCED THREE-PHASE DISTRIBUTION NETWORKS 21

31 Introduction 21

32 Flexibility and Simplicity of FFRPF in Numbering the RDS Buses and Sections 22

321 Bus Numbering Scheme for Balanced Three-phase RDS 22

322 Unbalanced Three-phase RDS Bus Numbering Scheme 24

33 The Building Block Matrix and its Role in FFRPF 26

v

331 Three-phase Radial Configuration Matrix (RCM) 26

3311 Assessment of the FFRPF Building Block RCM 28

332 Three-phase Section Bus Matrix (SBM) 29

333 Three-phase Bus Section Matrix (BSM) 31

34 FFRPF Approach and Solution Technique 31

341 Unbalanced Multi-phase Impedance Model Calculation 32

342 Load Representation 38

343 Three-phase FFRPF BackwardForward Sweep 40

3431 Three-phase Current Summation Backward Sweep 40

3432 Three-phase Bus Voltage Update Forward Sweep 42

3433 Convergence Criteria 43

3434 Steps of the FFRPF Algorithm 44

344 Modifying the RCM to Accommodate Changes in the RDS 47

35 FFRPF Solution Method for Meshed Three-phase DS 48

351 Meshed Distribution System Corresponding Matrices 50

352 Fundamental Loop Currents 54

353 Meshed Distribution System Section Currents 56

354 Meshed Distribution System BackwardForward Sweep 59

36 Test Results and Discussion 60

361 Three-phase Balanced RDS 60

3611 Case 1 31-Bus with Single Main Feeder RDS 61

3612 Case 2 90-bus RDS with Extreme Radial Topology 70

3613 Case 3 69-bus RDS with Four Main Feeders 71

3614 Case 4 15-bus RDS-Considering Charging Currents 73

362 Three-phase Balanced Meshed Distribution System 74

3621 Case 1 28-bus Weakly Meshed Distribution System 74

3622 Case 2 70-Bus Meshed Distribution System 78

vi

3623 Case 3 201-bus Looped Distribution System 79

363 Three-phase Unbalanced RDS 80

3631 Case 1 10-bus Three-phase Unbalanced RDS 81

3632 Case 2 IEEE 13-bus Three-phase Unbalanced RDS 85

3633 Case 3 26-bus Three-Phase Unbalanced RDS 86

37 Summary 87

Chapter 4 IMPROVED SEQUENTIAL QUADRATIC PROGRAMMING

APPROACH FOR OPTIMAL DG SIZING 89

41 Introduction 89

42 Problem Formulation Overview 89

43 DG Sizing Problem Architecture 90

431 Objective Function 90

432 Equality Constraints 92

433 Inequality Constraints 92

434 DG Modeling 93

44 The DG Sizing Problem A Nonlinear Constrained Optimization Problem 94

45 The Conventional SQP 96

451 Search Direction Determination by Solving the QP Subproblem 96

4511 Satisfying Karush-Khun-Tuker Conditions 98

4512 Newton-KKT Method 101

4513 Hessian Approximation 103

452 Step Size Determination via One-Dimensional Search Method 104

453 Conventional SQP Method Summary 105

46 Fast Sequential Quadratic Programming (FSQP) 108

47 Simulation Results and Discussion 113

471 Case 1 33-busRDS 113

4711 Loss Minimization by Locating Single DG 114

4712 Loss Minimization by Locating Multiple DGs 118

vii

472 Case 2 69-bus RDS 124

4721 Loss Minimization by Locating a Single DG 125

473 Loss Minimization by Locating Multiple DGs 129

474 Computational Time of FSQP vs SQP 134

475 Single DG versus Multiple DG Units Cost Consideration 136

48 Summary 136

Chapter 5 PSO BASED APPROACH FOR OPTIMAL PLANNING OF

MULTIPLE DGS IN DISTRIBUTION NETWORKS 138

51 Introduction 138

52 PSO - The Motivation 138

53 PSO - An Overview 139

531 PSO Applications in Electric Power Systems 141 532 PSO - Pros and Cons 143

54 PSO - Algorithm 144

541 The Velocity Update Formula in Detail 145

5411 The Velocity Update Formula - First Component 146

5412 The Velocity Update Formula - Second Component 148

5413 The Velocity Update Formula-Third Component 149

5414 Cognitive and Social Parameters 150

542 Particle Swarm Optimization-Pseudocode 152

55 PSO Approach for Optimal DG Planning 153

551 Proposed HPSO Constraints Handling Mechanism 155

5511 Inequality Constraints 155

5512 Equality Constraints 157

5513 DG bus Location Variables Treatment 157

56 Simulation Results and Discussion 160

561 Case 1 33-bus RDS 161

viii

5611 33-bus RDS Loss Minimization by Locating a Single DG 161

5612 33-bus RDS Loss Minimization by Locating Multiple

DGs 169

562 Case 2 69-Bus RDS 180

5621 69-bus RDS Loss Minimization by Locating a Single DG 180

5622 69-bus RDS Loss Minimization by Locating Multiple

DGs 187

563 Alternative bus Placements via HP SO 195

57 Summary 196

Chapter 6 CONCLUSION 198

61 Contributions and Conclusions 198

62 Future Work 201

REFERENCES 203

APPENDIX 220

IX

LIST OF TABLES

Table 31 cok rd and De Parameters for Different Operation Conditions 34

Table 32 FFRPF Iteration Results for the 31-Bus RDS 67

Table 33 The 31-bus RDS Section Power Losses Obtained by the FFRPF Method 68

Table 34 FFRPF Voltage Profiles Results for the Three Different Load Models 69

Table 35 Comparisons between 31-Bus RDS Exponential Model Results 70

Table 36 31-bus RDS FFRPF Results vs Other Methods 70

Table 37 90-bus RDS FFRPF Results vs Other Methods 71

Table 38 69-bus RDS FFRPF Results vs Other Methods 73

Table 39 Komamoto 15-bus RDS FFRPF Results vs Other Methods 74

Table 310 Voltage Profiles for Radial and Meshed 28-bus Distribution Network 77 t

Table 311 28-bus Weakly Meshed DS FFRPF Results vs Other Methods 78

Table 312 70-bus Meshed DS FFRPF Results vs Other Methods 79

Table 313 201-bus Meshed DS FFRPF Results vs Other Methods 80

Table 314 10-bus 3(j) RDS FFRPF Results vs Ref [52] and Gauss Zbus Methods 85

Table 315 IEEE 13-bus 3(j) RDS FFRPF Results vs Ref [52] and Gauss Zbus

Methods 86

Table 316 26-bus 34gt RDS FFRPF Results vs Ref [52] and Gauss Zbus Methods 87

Table 41 Single DG Optimal Profile at the 33-bus RDS 115

Table 42 Optimal DG Profiles at all 33 buses 116

Table 43 Multiple DG Installations in the 33-bus RDS with Unspecified Power

Factor 119

Table 44 SQP Method Double-DG Cycled Combinations 121

Table 45 Single and Multiple DG Installations at Pre-specified 085 Power Factor 123

Table 46 Loss Reduction Comparisons for all DG Cases 123

Table 47 69-bus RDS Single DG Optimal Size and Placement Profiles 128

Table 48 Optimal Double DG Profiles in the 69-bus RDS 131

Table 49 Optimal Three DG Units Profiles in the 69-bus RDS 133

Table 410 Loss Reduction Comparison for all DG Installations in the 69-bus RDS 134

Table 411 33-bus RDS CPU Execution Time Comparison 135

Table 412 69-bus RDS CPU Execution Time Comparison 135

x

Table 51 HPSO Parameters for the Single DG Case 162

Table 52 33-bus RDS Single DG Fixedpf Case 20 HPSO Simulations 162

Table 53 Descriptive Statistics for HPSO Results for the Fixedpf Case 163

Table 54 HPSO vs FSQP Results 33-bus RDS-Single DG-FixedDG Case 163

Table 55 33-bus RDS Single DG Unspecified pf Case 20 HPSO Simulations 163

Table 56 Descriptive Statistics for HPSO Results for an Unspecified pf Case 164

Table 57 HPSO vs FSQP Results 33-bus RDS-Single DG-UnspecifiedpCase 164

Table 58 HPSO Parameters for Both Double DG Cases 170

Table 59 33-bus RDS Double DG Fixed pf Case 20 HPSO Simulations 171

Table 510 Descriptive Statistics for HPSO Results for Fixed pf Double DG Case 171

Table 511 HPSO vs FSQP Results 33-bus RDS-Double DGs-FixedCase 172

Table 512 33-bus RDS Double DG Unspecified pf Case 20 HPSO Simulations 172

Table 513 Descriptive Statistics for HPSO Results for UnspecifiedDouble DG

Case 173

Table 514 HPSO vs FSQP Results 33-bus RDS-Double DGs-UnspecifiedCase 173

Table 515 HPSO Parameters for Both Three DG Cases 174

Table 516 33-bus RDS Three DG Fixed pf Case 20 HPSO Simulations 174

Table 517 Descriptive Statistics for HPSO Results for Fixed pf Three DG Case 175

Table 518 HPSO vs FSQP Results 33-bus RDS-Three DG-FixedCase 175

Table 519 33-bus RDS Three DGs Unspecified pf Case 20 HPSO Simulations 175

Table 520 Descriptive Statistics for HPSO Results for the Fixed Three DG Case 176

Table 521 HPSO vs FSQP Results 33-bus RDS-Three DG-UnspecifiedCase 176

Table 522 HPSO Parameters for the Four DG Case 177

Table 523 33-bus RDS Four DG FixedCase 20 HPSO Simulations 178

Table 524 Descriptive Statistics for HPSO Results for Fixed Four DG Cases 178

Table 525 HPSO vs FSQP Results 33-bus RDS-Four DG-FixedCase 179

Table 526 33-bus RDS Four DG Unspecified pf Case 20 HPSO Simulations 179

Table 527 Descriptive Statistics for HPSO Results for Unspecified Four DG

Case 179

Table 528 HPSO vs FSQP 33-bus RDS-Four -Unspecified pf Case 180

Table 529 HPSO Parameters for 69-bus RDS Both Single DG Cases 181

Table 530 69-bus RDS Single DG Fixed pf Case 20 HPSO Simulations 182

xi

Table 531 Descriptive Statistics for HPSO Results for the Fixed Single DG Case 182

Table 532 HPSO vs FSQP Results 69-bus RDS-Single DG-FixedpDG Case 182

Table 533 69-bus RDS Single DG Unspecifiedpf Case 20 HPSO Simulations 183

Table 534 Descriptive Statistics for Unspecified pSingle DG Case 183

Table 535 HPSO vs FSQP Results 69-bus RDS-Single DG-UnspecifiedpDG

Case 184

Table 536 HPSO Parameters for 69-bus RDS the Double DG Cases 188

Table 537 69-bus RDS Double DG Fixed pf Case 20 HPSO Simulations 189

Table 538 Descriptive Statistics for HPSO Results for Fixed pDouble DG Case 189

Table 539 HPSO vs FSQP Results 69-bus RDS-Double DG-FixedpDG Case 190

Table 540 69-bus RDS Double DG UnspecifiedpfCase 20 HPSO Simulations 190

Table 541 Descriptive Statistics for HPSO Results for Unspecified ^Double DG

Case 191

Table 542 HPSO vs FSQP Results 69-bus RDS-Double-UnspecifiedpDG Case 191

Table 543 HPSO Parameters for Both 69-bus RDS Three DG Cases 192

Table 544 69-bus RDS Three DG Fixed pf Case 20 HPSO Simulations 192

Table 545 Descriptive Statistics for HPSO Results for Fixed pf Three DG Case 193

Table 546 HPSO vs FSQP Results 69-bus RDS-Three DGs-FixedCase 193

Table 547 69-bus RDS Three DG Unspecified pf Case 20 HPSO Simulations 194

Table 548 Descriptive Statistics for HPSO Results for Unspecified pf Three DG

Case 194

Table 549 HPSO vs FSQP Results 69-bus RDS-Three DGs-UnspecifiedpCase 195

Table 550 20 HPSO Simulations of the 69-bus RDS Three DG Fixed pf Case with Suboptimal Tuned Parameters 50 Iterations and 50 Swarm Particles 196

xii

LIST OF FIGURES

Figure 31 10-bus RDS 23

Figure 32 Different ways of numbering the system in Fig 31 24

Figure 33 The ease of numbering a modified and augmented RDS 24

Figure 34 Three-phase unbalanced 6-bus RDS representation 25

Figure 35 Three-phase unbalanced 6-bus RDS with missing buses and sections 26

Figure 36 RCM matrices sparsity plots for the 25 and 90 bus RDSs 28

Figure 37 SBM for three-phase unbalanced 6-bus RDS 30

Figure 38 Three-phase section model 32

Figure 39 The final three-phase section model after Kron s reduction 34

Figure 310 Nominal ^-representation for three-phase RDS section 36

Figure 311 Three-phase capacitor bank (a) The schematics (b) The Modeling 40

Figure 312 The FFRPF solution method flow chart 46

Figure 313 10-bus meshed distribution network 50

Figure 314 Fundamental cut-sets for a meshed 10-bus DS 57

Figure 315 31-bus RDS 62

Figure 316 The RCM of the 31-bus RDS 63

Figure 317 The RCM-1 of the 31-bus RDS 64

Figure 318 The SBM of the 31-bus RDS 65

Figure 319 The BSM of the 31-bus RDS 66

Figure 3 20 90-Bus RDS 71

Figure 321 69-bus multi-feeder RDS 72

Figure 322 Komamoto 15-bus RDS 73

Figure 323 28-bus weakly meshed distribution network 75

Figure 324 mRCM for 28-bus weakly meshed distribution network 75

Figure 325 mSBM for 28-bus weakly meshed distribution network 76

Figure 326 C for 28-bus weakly meshed distribution network 76

Figure 327 70-bus meshed distribution system 78

Figure 328 201-bus hybrid augmented test distribution system 80

Figure 329 10-bus three-phase unbalanced RDS 81

Figure 330 The 10-bus three-phase unbalanced RDS RCM 82

xni

Figure 331 21 The 10-bus three-phase unbalanced RDS RCM1 83

Figure 332 The 10-bus three-phase unbalanced RDS SBM 84

Figure 333 The 10-bus three-phase unbalanced RDS BSM 85

Figure 334 IEEE 13-bus 3^ unbalanced RDS 86

Figure 41 The Conventional SQP Algorithm 107

Figure 42 The FSQP Algorithm 112

Figure 43 Case 1 33-busRDS 114

Figure 44 Optimal real power losses for placement of optimal DG size at all the 32

buses using APC method 117

Figure 45 Optimal real power losses vs different DG power factors at bus 30 117

Figure 46 Bus voltages improvement before and after installing a single DG at

bus 30 118

Figure 47 Voltage profiles comparisons of 33-bus RDS cases 120

Figure 48 Voltage improvement of the 33-bus RDS due to three DG installation

compared to pre-DG single and double-DG cases 122

Figure 49 Voltage profiles improvement in 33-bus RDS for all DG cases 124

Figure 410 Case 2 69-bus RDS test case 125

Figure 411 Optimal power losses obtained using APC procedure 126

Figure 412 Real power losses vs DG power factor 69-bus RDS 128

Figure 413 Bus voltage improvements via single DG installation in the 69-bus

RDS 129

Figure 414 Variation in power losses as a function of the DG output at bus 61 129

Figure 415 Bus voltage magnitudes of the original 69-bus RDS and Single DG

and double DGs cases 131

Figure 416 69-bus RDS bus voltage magnitudes improvement in all DG cases 133

Figure 51 Number of publications in IEEEIET and ScienceDirect Databases

since the year 2000 140

Figure 52 Interaction between particles to share the gbest information 150

Figure 53 Illustration of velocity and position updates mechanism for a single particle during iteration k 151

Figure 54 PSO particle i updates its velocity and position vectors during two consecutive iterations A and k+ 152

Figure 55 The proposed HPSO solution methodology 159

xiv

Figure 56 Convergence characteristic of the proposed HPSO in the fixedsingle DG case HPSO proposed number of iterations = 30 164

Figure 57 Convergence characteristic of the proposed HPSO in the fixedsingle

DG case HPSO extended number of iterations = 50 165

Figure 58 Swarm particles on the first HPSO iteration 165

Figure 59 Swarm particles on the fifth HPSO iteration 166

Figure 510 Swarm particles on the tenth HPSO iteration 166

Figure 511 Swarm particles on 15th HPSO iteration 167

Figure 512 Swarm particles on the 20 HPSO iteration 167

Figure 513 Swarm Particles on the 25th HPSO iteration 168

Figure 514 Swarm Particles on the last HPSO iteration 168

Figure 515 A close-up for the particles on the 30th HPSO iteration 169

Figure 516 Convergence characteristics of HPSO in the 69-bus fixedsingle DG case HPSO proposed number of iterations = 15 184

Figure 517 Convergence characteristics of HPSO in the 69-bus fixed single DG

case HPSO proposed number of iterations = 50 185

Figure 518 Swarm particles distribution at the first HPSO iteration 185

Figure 519 Swarm particles distribution at the 5 HPSO iteration 186

Figure 520 Swarm particles distribution at the 10 HPSO iteration 186

Figure 521 Swarm particles distribution at the 15l HPSO iteration 187

Figure 522 Close up of the HSPO particles at iteration 15 187

xv

ABSTRACT

Distribution Generation (DG) has gained increasing popularity as a viable element of electric power systems DG as small scale generation sources located at or near load center is usually deployed within the Distribution System (DS) Deployment of DG has many positive impacts such as reducing transmission and distribution network congestion deferring costly upgrades and improving the overall system performance by reducing power losses and enhancing voltage profiles To achieve the most from DG installation the DG has to be optimally placed and sized In this thesis the DG integration problem for single and multiple installations is handled via deterministic and heuristic methods where the results of the former technique are used to validate and to be compared with the latters outcomes

The unique structure of the radial distribution system is exploited in developing a Fast and Flexible Radial Power Flow (FFRPF) method that accommodates the DS distinct features Only one building block bus-bus oriented data matrix is needed to perform the proposed FFRPF method Two direct descendent matrices are utilized in conducting the backwardforward sweep employed in the FFRPF technique The proposed method was tested using several DSs against other conventional and distribution power flow methods Furthermore the FFRPF method is incorporated within Sequential Quadratic Programming (SQP) method and Particle Swarm Optimization (PSO) metaheuristic method to satisfy the power flow equality constraints

In the deterministic solution method the sizing of the DG is formulated as a constrained nonlinear optimization problem with the distribution active power losses as the objective function to be minimized subject to nonlinear equality and inequality constraints Such a problem is handled by the developed Fast Sequential Quadratic Programming method (FSQP) The proposed deterministic method is an improved version of the conventional SQP that utilizes the FFRPF method in handling the power flow equality constraints Such hybridization resultes in a more robust solution method and drastically reduces the computational time In a subsequent step the placement portion of the DG integration problem is dealt with by using an All Possible Combinations (APC) search method Afterward the FSQP methods outcomes were compared to those of the developed metaheuristic optimization method

The difficult nature of the overall problem poses a serious challenge to most derivative based optimization methods due to the discrete nature associated with the bus location Moreover a major drawback of deterministic methods is that they are highly-dependent on the initial solution point As such a new application of the PSO metaheuristic method in the DG optimal planning area is presented in this thesis The PSO is improved in order to handle both real and integer variables of the DG mixed-integer nonlinear constrained optimization problem The algorithm is utilized to simultaneously search for both the optimal IDG size and bus location The proposed approach hybridizes PSO with the developed FFRPF algorithm to satisfy the equality constraints The inequality constraints handling mechanism is dealt with in the proposed Hybrid PSO (HPSO) by combining the rejecting infeasible solutions method with the preserving feasible solutions method Results signify the potential of the developed algorithms with regard to the addressed problems commonly encountered in DS

xvi

LIST OF ABBREVIATIONS AND SYMBOLS USED

ACO

BFGS

BSM

CHP

CIGRE

CN

DER

DG

DG

DGs

DS

EG

EP

EPAct

EPRI

FD

FFRPF

FSQP

GA

GRG

GS

GWEC

HPSO

IP

KCL

KKT

KVL

LP

Ant Colony Optimization

Quasi-Newton method for Approximating and Updating the Hessian Matrix

Bus Section Matrix

Combined-Heat and Power

The International Council on Large Electric Systems

Condition Number

Distribution Energy Resources

Dispersed Generation

Decentralized Generation

Distribution Generation sources

Distribution System

Embedded Generation

Evolutionary Programming

English Policy Act of 1992

Electric Power Research Institute

Fast Decoupled

Fast and Flexible Radial Power Flow

Fast Sequential Quadratic Programming

Genetic Algorithm

Generalized Reduced Gradient

Gauss-Seidel

Global Wind Energy Council

Hybrid PSO

Interior Point method

Kirchhoff s Current Law

Karush-Khun-Tuker conditions

Kirchhoff s Voltage Law

Linear Programming

xvii

wBSM

mNS

wRCM

mRCM

mSBM

mSBMp

NB

NB

HDG

riL

NR

NS

NS

ftwDG

Pf PSO

PUHCA

PURPA

QP

RCM

RDS

RIT

RPF

SA

SBM

SE Mean

SQP

StDev

TS

UnSpec pf

Meshed BSM

Number of segments in meshed DS

Meshed RCM

Modified mRCM

Meshed SBM

Submatrix of wSBM that correspond to the RDS tree sections

Number of Buses

Number of DS Buses

Total number of DGs

Number of Links or number of the fundamental loops

Newton-Raphson

Number of Sections

Number of Sections in RDS AND in meshed DS tree

Total number of the unspecified pf DGs

power factor

Particle Swarm Optimization

Public Utilities Holding Company Act of 1935

Public Utilities Regulatory Policy Act of 1978

Quadratic Programming

Radial Configuration Matrix

Radial Distribution System

The Reduction in CPU execution Time

Radial Power Flow

Simulated Annealing

Section Bus Matrix

Standard Error of the Mean

Sequential Quadratic Programming

Standard Deviation

Tabu search algorithm

Unspecified power factor DG

xviii

U S P B Unique Set of Phase Buses

USPS Unique Set of Phase Sections

xf Unique set of phase buses

iff Unique set of phase sections

Zsec Section primitive impedance matrix

Z^ (3 X 3) section symmetrical impedance matrix

R D S section length

zu Per unit length self-impedance of conductor i

h Per unit length mutual- impedance be tween conductors a n d

rt Resis tance of conductor i

rd Ear th return conductor resistance

k Inductance multiplying constant

De Dis tance between overhead and its earth return counterpart

GMRj Geometr ic mean radius of conductor i

Dy Dis tance between conductors a n d

Vgbc Three-phase sending end voltages

Vg deg Three-phase receiving end voltages

Ias c Three-phase sending end section currents

lfc Three-phase receiving end section currents

Fscc Three-phase shunt admittance of section k

[]3x3 (3 X 3) identity matrix

[^Lx3 (3x3) zero matrix

^Klc Vol tage drop across three-phase section k

ysect Section k three-phase currents

V0 Nomina l bus voltage

V Operat ing bus voltage

xix

P0 Real power consumed at nominal voltage

Q0 Reactive power consumed at nominal voltage

S Bus load apparent power at single-phase bus sect

YsKus Total three-phase shunt admittance at bus i

Ic Three-phase shunt currents at bus i

IlucSi Bus three-phase currents

jabc Three-phase load current

IltLP Current through single-phase section p and phase ltjgt

its Current at bus and phase ^

Vss Substation voltage magnitude

Vls Substation complex phase voltage

VLt Voltage drop across section k in phase (j)

A and symbol

IMI oo-norm vector II I loo

91 (bull) Real part of complex value

3 (bull) Imaginary part of complex value

C Fundamental loop matrix which is a submatrix of mSBM

Zioop (laquoLx nL) loop-impedance matrix

Csec Upper submatrix of the C matrix with ((NB-1) x (nL) dimension

Zoop Loop-impedance matrix

setrade (NSxNS) meshed DS section-impedance diagonal matrix

ZtradeS (NSxNS) RDS or tree section-impedance diagonal matrix

IL (NB-1 x 1) RDS bus load currents vector

fnlsec (mNS x 1) segments currents column vector of meshed DS network vector

mILL (mNSx 1) meshed DS bus loads and links currents vector

Itrade (NB-1) tree section currents column

xx

( n L x 1) fundamental loop current vector which is also the meshed DS link loop

currents column vector

B ( N B - 1 xmNS) fundamental cut-sets matrix

^ s7 e c f a ( N B - 1 x nL) co-tree cut-set matrix

^ymesh Voltage drops across the tree sections of the meshed DS vector

ymesh j k g messed DS bus voltage profiles vector

PRPL Real power losses

Pj Generated power delivered to DS bus i

PjL Load power supplied by DS bus i

Yjj Magnitude of the if1 element admittance bus matrix

rv Phase angle of Yy = YyZyy

Vi Magnitude of DS bus complex voltage

8 Phase angle of V = ViZSl

bull Transpose of vector or matrix

bull Complex conjugate of vector or matrix

V (1 xNB) DS bus Thevenin voltages

Y (NB xNB) DS admittance matrix

A^ Real power mismatch at bus i

AQt Reactive power mismatch at bus i

|L| Infinity norm where llxll = max (|x|) II llco J II llraquo =l2iVBVI ]gt

bull+ Max imum permissible value

bull Minimum permissible value

bull0 Nominal value

PDG D G operating power factor

S^G D G generated apparent power

SsS Main DS substation apparent power

1 Scalar related to the allowable D G size

xxi

Sy Apparent power flow transmitted from bus to bus j

Stradex Apparent power maximum rating for distribution section if

(x) The objective function

z(x) Equality constraints

g(x) Inequality constraints

(bull) Independent unknown variables lower bounds

(bull) Independent unknown variables upper bounds

x Independent unknown variables vector

RPL ( X ) Distribution system real power losses objective function

d Search direction vector

a Positive step size scalar

WRPL (x ) Gradient of the objective function at point xk)

pound Lagrange function

H^ (nxri) Hessian symmetric matrix at point xw

h^ First-order Taylors expansion of the equality constraints at point xw

Vh(x^) (nxm) Jacobian matrix of the equality constraints at point xw

g ^ First-order Taylors expansion of the inequality constraints at point xw

Vg(x^) (nxp) Jacobian matrix of the inequality constraints at point xw

Xi Individual equality Lagrange multiplier scalar

Pi Individual inequality Lagrange multiplier scalar

k w-dimensional equality Lagrange multiplier vector

P (-dimensional inequality Lagrange multiplier vector

s A predefined small tolerance number

A Active set

m Number of all equality constraints

p Number of all inequality constraints

a Number of the active set equations

xxii

v 2 j6k)

XX

nTgtG

nuDG

y

v Y FFRPFbl

deg FFRPF bl

llAP II II lloo

Vi

Xi

Cj C2

rXgtr2

w

pbestj

gbesti

nk

X

APT Losses

pHPSO Losses

pFSQP Losses

Hessian of the Lagrange function

Total number of DGs

Total number of the unspecified DGs

The change in the Lagrange functions between two successive iterations

Voltage magnitude of bus i obtained by the FFRPF technique

Voltage phase angle of bus obtained by the FFRPF technique

Voltage deviation infinity norm ie II AV = max ( I F - K I ) deg llcD

=UAlaquoVI deg ngt

Particle i velocity

Particle i position vector

Individual and social acceleration positive constants

Random values in the range [0 l] sampled from a uniform distribution

Weight inertia

Personal best position associated with particle own experience

Global best position associated with the whole neighborhood experience

Maximum number of iterations

Constriction factor

The deviation of losses calculated by HPSO method from that determined

by FSQP method

Mean value of HPSO simulation results of real power losses

FSQP deterministic method result of real power losses

xxiii

ACKNOWLEDGEMENTS

All Praise and Thanks be to reg (Allah) Almighty whose countless bounties enabled me to

accomplish this thesis successfully I would like to express my deepest gratitude to my

parents who taught me the value of education and hard work A special note of gratitude

to my brothers Falah and Abdullah deer sisters my wife my daughter Najla and my

sons Fahad Falah and Othamn They endured the long road along with me and

provided me with constant support motivation and encouragement during the course of

my study

I would like to express my sincere gratitude to my advisor Dr M E El-Hawary for

his professional guidance valuable advice continual support and encouragement I also

appreciate the constructive comments of my PhD External Examiner Dr M A Rahman

I am also grateful to my advisory committee members Dr T Little and Dr W Phillips

for spending their valuable time in reading evaluating and discussing my thesis

I would like to acknowledge the academic discussions and the constant

encouragement I received from my dear friend Dr Mohammed AlRashidi- Thank you

Abo Rsheed I wish also to thank a special friend of mine Dr AbdulRahman Al-

Othman for his friendship and for believing in me

I would like to manifest my gratitude to the Public Authority for Applied Education

and Training in Kuwait who sponsored me through my PhD at Dalhousie University

From the Embassy of Kuwait Cultural Attache Office special thanks are due to Shoghig

Sahakyan for her efforts and help to make this work possible

xxiv

CHAPTER 1 INTRODUCTION

11 MOTIVATION

Electric power system networks are composed typically of four major subsystems

generation transmission distribution and utilizations Distribution networks link the

generated power to the end user Transmission and distribution networks share similar

functionality both transfer electric energy at different levels from one point to another

however their network topologies and characteristics are quite different Distribution

networks are well-known for their low XR ratio and significant voltage drop that could

cause substantial power losses along the feeders It is estimated that as much as 13 of

the total power generation is lost in the distribution networks [1] Of the total electric

power system real power losses approximately 70 are associated with the distribution

level [23] In an effort towards manifesting the seriousness of such losses Azim et al

reported that 23 of the total generated power in the Republic of India is lost in the form

of losses in transmission and distribution [4]

Distribution systems usually encompass distribution feeders configured radially and

exclusively fed by a utility substation Incorporating Distribution Generation sources

(DGs) within the distribution level have an overall positive impact towards reducing the

losses as well as improving the network voltage profiles Due to advances in small

generation technologies electric utilities have begun to change their electric

infrastructure and have started adapting on-site multiple small and dispersed DGs In

order to maximize the benefits obtained by integrating DGs within the distribution

system careful attention has to be paid to their placement as well as to the appropriate

amount of power that is injected by the utilized DGs In other words to achieve the best

results of DG deployments the DGs are to be both optimally placed and sized in the

corresponding distribution network

The motivation of this thesis research is to investigate placing and sizing single and

multiple DGs in Radial Distribution Systems (RDSs) The problem investigated involves

two stages finding the optimal DG placements in the distribution network and the

optimal size or rating of such DGs The optimal DG placement and sizing are dealt with

by utilizing deterministic and heuristic optimization methods

12 DISTRIBUTION GENERATION - HISTORIC OVERVIEW

During the first third of the twentieth century there were no restrictions on how many

utility companies could be owned by financial corporations known as utility holding

companies By 1929 80 of US electricity was controlled by 16 holding companies

and three of those corporations controlled 36 of the nations electricity market [5]

During the Great Depression most of these utility holding companies went bankrupt As

a result the US Congress Public Utilities Holding Company Act (PUHCA) of 1935

regulated the gas and electric industries and restricted holding companies to the

ownership of a single integrated utility PUHCA indirectly discouraged wholesale

wheeling of power between different states provinces or even countries The Public

Utilities Regulatory Policy Act (PURPA) of 1978 allowed grid interconnection and

required electric utilities to buy electricity from non-utility-owned entities called

Qualifying Facilities (QF) at each utilitys avoided cost The term QF refers to non-

utility-owned (independent) power generators The term at each utilitys avoided cost

is interpreted to mean that the utility shall buy the generated electricity at a price

equivalent to what it would cost the utility itself if had generated the same amount of

power in its own facility or if it had purchased the power from an open electricity market

ie what the utility saves by not generating the same amount of power This act heralded

the dawn of the DG industry era which paved the way to generate electricity arguably at

a lower cost compared to that of traditional utility companies and consequently have it

delivered to the end user at lower rates The English Policy Act of 1992 (EPAct)

intensified competition in the wholesale electricity market by opening the transmission

system for access by utilities and non-utilities electricity producers [67] entity A could

sell its power to entity B through entity Cs transmission infrastructure

13 DISTRIBUTION GENERATION

DG involves small-scale generation sources scattered within the distribution system level

atnear the load center ie close to where the most energy is consumed [8] The DG

2

generate electricity locally and in a cogeneration case heat can also be generated and

may be utilized in applications such as industrial process heating or space heating DG

generally has better energy efficiency than large-scale power plants The traditional

power stations usually have an efficiency of around 35 whereas the efficiency of DG

such as a Combined Heat and Power (CHP) gas turbine would be in the vicinity of 45-

65 [5]

It seems that there is no universal agreement on the definition of DG size range The

Electric Power Research Institute (EPRI) for example defined the DG size to be up to 5

MW in 1998 [9] in 2001 EPRI redefined the DG capacity to be less than 10 MW [10]

and by 2003 they identified the DG to have a power output ranging from 1 kW to 20 MW

[11] The IEEE published its DG-standards IEEE Std 1547-2003 and IEEE Std 15473-

2007 and emphasized that they are applicable to DGs that have total capacity below 10

MVA [1213] In its 2006 report about the impact of DG Natural Resources Canada

estimated that the DG size starts from few 10s of kW to perhaps 5 MW [14] In 2000

the International Council on Large Electric Systems (CIGRE) referred to the DG as non-

centrally dispatched usually attached to distribution level and smaller than 50-100 MW

[1516]

Many terms referring to DG technology are used in the literature such as Dispersed

Generation (DG) Decentralized Generation (DG) Embedded Generation (EG)

Distribution Energy Resources (DER) and on-site Generation [17] In particular the

term dispersed generation customarily refers to stationary small-scale DG with power

outputs ranging from 1 kW to 500 kW [7]

Late developments and innovations in the DG technology industry liberalization of

the electricity market transmission line congestion and increasing interest in global

warming and environmental issues expedited publicizing their deployment and adoption

world-wide Recent studies suggest that DG will play a vital role in the electric power

system An EPRI study predicts that by the year 2010 25 of the newly installed

generation systems will be DG [18] and a similar study by the Natural Gas Foundation

projects that the share of DG in new generation will be 30 [15] By 2003 around 40

of Denmarks power demand was served by DG while Spain the Netherlands Portugal

and Germany integrated nearly 20) of DG into their distribution networks [19] Of the

3

643 GW generated by the European Union in 2005 approximately 122 GW (19) was

generated by hydro 96 GW (15) of this generated capacity was cogeneration (CHP)

and 53 GW (8) generated by other renewable energy systems Half of the CHP

generated capacity was owned by utility companies and the other half was generated by

independent producers [20]

Globally in 2005 the total installed wind power capacity was 591 GW and the

Global Wind Energy Council (GWEC) expected the wind capacity to reach 1348 GW by

the year 2010 [21] Worldwide wind energy capacity of 19696 MW was added in the

year 2007 [22] and approximately 1400 MW of wind energy capacity was added in the

US during the second quarter of 2008 [23] GWEC also predicted in its Global Wind

Energy Outlook 2008 report that by the year 2020 15 billion tons of CO2 will be saved

every year and by the middle of the 21st century 30 of the worlds electricity will be

supplied by wind energy [24] compared to a total of 13 of the global electricity being

generated by wind at the end of 2007 [22]

DG technologies include a variety of energy sources ie powered by renewable or

by fossil fuel-based prime movers Renewable technologies used in DG include wind

turbines photovoltaic cells small hydro power turbines and solar thermal technologies

while DG based on conventional technologies may involve gas turbines CHP gas

turbines diesel engines fuel cells and micro-turbine technologies Some DGs are

installed by the utility company on the supply side of the consumers meter while some

are installed by the customers themselves on their side of a bi-directional meter thus

enabling them to benefit from the net-metering program offered by utility companies

[25]

Optimal deployment of DG technology would have an overall positive impact

although some negative traits would remain The noise and shadow flicker caused by

large wind blades and the noise caused by the wind turbine gearbox or gas turbines

especially when placed close to residential or populated areas are examples of negative

impacts of widespread use of DG Another drawback from an environmentalist point of

view is that wind DG could disturb bird immigration patterns and cause death to both

birds and bats [26] Renewable-source DGs also could be an indirect source of pollution

by causing the fossil-fuel power plants to shut down and start up more frequently as they

4

attempt to accommodate variable DG power output [27] Some plants have an emission

rate which is inversely proportional to its delivered power Voltage rise as a result of bishy

directional power flow caused by the interconnection of the DG in RDS is another

example of a negative impact caused by DG [28]

The integration of DG into electric power networks has many benefits Some

examples of such benefits could be summarized as follows

bull Improve both the reliability and efficiency of the power supply

bull Release the available capacity of the distribution substation as well as reducing

thermal stresses caused by loaded substations transformers and feeders

bull Improve the system voltage profiles as well as the load factor

bull Decrease the overall system losses

bull Generally DG development and construction have shorter time intervals

bull Delay imminent upgrading of the present system or the need to build newer

infrastructure and subsequently avoid related problems such as right-of-way

concerns

bull Decrease transmission and distribution related costs

bull In general DG tends to be more environmentally friendly when compared to

traditional coal oil or gas fired power plants

The extent of the benefits depends on how the DG is placed and sized in the system In

addition to supplying the system with the power needed to meet certain demands as an

installation incentive the real power losses could be minimal if the DG is optimally sited

and sized

14 THESIS OBJECTIVES AND CONTRIBUTIONS

Optimal integration of single and multiple DG units in the distribution network with

specified and unspecified power factors is thoroughly investigated from a planning

perspective in this thesis The DG problem is handled via deterministic and heuristic

optimization methods where the results of the former method are used to validate and to

be compared with those of the latter

The unique radial distribution structure is exploited in developing a Fast and Flexible

Radial Power Flow (FFRPF) method to deal with a wide class of distribution systems

5

eg radial meshed small large balanced and unbalanced three-phase networks The

proposed FFRPF algorithm starts by developing a Radial Configuration Matrix (RCM)

for radial topology DS or meshed RCM (mRCM) for meshed DS Both matrices consist

of elements with values 1 0 and -1 The RCM (or mRCM) corresponding building

algorithm is simple fast and practical as illustrated in Chapter 3 The RCM is inverted

only once to produce the Section Bus Matrix (SBM) which is then transposed to obtain

the Bus Section Matrix (BSM) For the meshed topology the corresponding resultant

matrices for the mRCM are the meshed Bus Section Matrix (mSBM) and the meshed Bus

Section Matrix (mBSM) The FFRPF technique relies on the backwardforward sweep

that only utilizes the basic electric laws such as Ohms law and both Kirchhoff s voltage

and current laws The backward current sweep is performed via SBM (or mSBM) and

the forward voltage update sweep is executed via the BSM (or mBSM) By utilizing the

two obtained matrices all the bus complex voltages can be obtained and consequently

left to be compared with the immediate previous obtained bus voltages The proposed

approach quickened the iterative process and reduced the CPU time for convergence It

is worth mentioning that the building block matrix is the only input data required by the

FFRPF method besides the DS parameters to perform the proposed distribution power

flow The FFRPF technique is incorporated in both utilized deterministic and

metaheuristic optimization methods to satisfy the power flow equality constraints

requirements

In the deterministic solution method the DG sizing problem is formulated as a

nonlinear optimization problem with the distribution active power losses as the objective

function to be minimized subject to nonlinear equality and inequality constraints

Endeavoring to obtain the optimal DG size an improved version of the Sequential

Quadratic Programming (SQP) methodology is used to solve for the DG size problem

The conventional SQP uses a Newton-like method which consequently utilizes the

Jacobean in handling the nonlinear equality constraints The radial low XR ratio and the

tree-like topology of distribution systems make the system ill-conditioned

A Fast Sequential Quadratic Programming (FSQP) methodology is developed in

order to handle the DG sizing nonlinear optimization problem The FSQP hybrid

approach integrates the FFRPF within the conventional SQP in solving the highly

6

nonlinear equality constraints By utilizing the FFRPF in dealing with equality

constraints instead of the Newton method the burden of calculating the Jacobean and

consequently its inverse as well as the complications of the ill-conditioned Y-matrix of

the RDS is eliminated Another advantage of this hybridization is the drastic reduction

of computational time compared to that consumed by the conventional SQP method

In this thesis a new application of the Particle Swarm Optimization (PSO) method in

the optimal planning of single and multiple DGs in distribution networks is also

presented The algorithm is utilized to simultaneously search for both the optimal DG

size and its corresponding bus location in order to minimize the total network power

losses while satisfying the constraints imposed on the system The proposed approach

hybridizes PSO with the developed distribution radial power flow ie FFRPF to

simultaneously solve the optimal DG placement and sizing problem The difficult nature

of the overall problem poses a serious challenge to most derivative based optimization

methods due to the discrete flavor associated with the bus location in addition to the

subproblem of determining the most suitable DG size Moreover a major drawback of

the deterministic methods is that they are highly-dependent on the initial solution point

The developed PSO is improved in order to handle both real and integer variables of the

DG mixed-integer nonlinear constrained optimization problem Problem constraints are

handled within the proposed approach based on their category The equality constraints

ie power flows are satisfied through the FFRPF subroutine while the inequality bounds

and constraints are treated by exploiting the intrinsic and unique features associated with

each particle The proposed inequality constraint handling technique hybridizes the

rejection of infeasible solutions method in conjunction with the preservation of feasible

solutions method One advantage of this constraint handling mechanism is that it

expedites the solution method converging time of the Hybrid PSO (HPSO)

15 THESIS OUTL INE

This thesis is organized in six chapters The research motivation brief description of the

DG and the thesis objectives are addressed in the first chapter The second chapter deals

with a literature review of the distribution power flow methods and the DG optimal

planning problem In the third chapter development of the proposed FFRPF method

7

utilized in the FSQP and the HPSO methods to satisfy the DG problem of nonlinear

equality constraints is presented The fourth chapter deals with the DG sizing problem

formulation and its solution based on the two deterministic solution methods The

problem is solved via the conventional SQP and the proposed FSQP methods and a

performance comparison between them is presented Basic concepts of the PSO are

presented in chapter five A brief literature review regarding the use of the PSO in

solving the electric power system problems is presented in this chapter In addition it

also addresses the development of the proposed HPSO in solving the DG planning

problem The last chapter provides the thesis concluding remarks and the scope of future

work

8

CHAPTER 2 LITERATURE REVIEW

21 INTRODUCTION

Recent publications in the areas of work relative to this thesis are reviewed and

summarised in this chapter which is organized in two sections as follows

bull The first section reviews the literature on distribution power flow methods A

brief background of conventional power flow methods is presented followed

by a review and summary of the literature on recent developments of the

distribution power flow algorithms

bull The DG integration problem is reviewed in the second section Recent work

on the optimal DG placement and sizing via analytical deterministic and

metaheuristic methods are analyzed and reviewed

22 DISTRIBUTION POWER FLOW

Power flow programs play a vital role in analyzing power systems The problem deals

with calculating unspecified bus voltage angles and magnitudes active and reactive

powers as well as (as a by-product) line loadings and their associated real and reactive

losses for certain operating conditions These values are typically obtained through

iterative numerical methods to analyze the status of a given power system

Since the middle of last century many methods were proposed to solve this problem

Even though Dunstan [29] was the first to demonstrate a digital method for solving the

power flow problem in 1954 Ward and Hale [30] are often credited with the successful

digital formulation and solution of the power flow problem in 1956 Most of the earlier

solution methods were based on both the admittance matrix and the Gauss-Seidel (GS)

iterative method The poor convergence characteristics of GS when large networks

andor ill-conditioned situations are encountered led to the development of the Gaussian

iterative scheme (Zbus) [3132] and later the Newton-Raphson (NR) method [33] as well

as the Decoupled [34] and Fast Decoupled (FD) power flow approaches [35] Though

the NR method generally converges faster than other methods it takes longer

computational time per iteration When Tinney et al [36] introduced the optimally

9

ordered and sparsity-oriented programming techniques Newton-based methods became

the de facto industry standard However the Jacobian matrix for the RDS is

approximately four times the size of the corresponding admittance matrix and it needs to

be evaluated at each iteration

Although conventional power flow methods are well developed in dealing with the

transmission and sub-transmission sections of the power system networks they are

deemed to be inefficient in handling distribution networks This is because the

Distribution System (DS) is different in several ways from its transmission counterpart

DS has a strictly radial topology nature or weakly meshed networks in contrast with

transmission systems which are tightly meshed networks DS is a low voltage system

having low XR ratio sections and a wide range of reactance and resistance values DS

may consist of a tremendously large number of sections and buses spread throughout the

network Sections of the DS could have unbalanced load conditions due to the

unbalanced three-phase loading as well as single and double phase loads in spurred

lateral lines The mutual couplings between phases are not negligible due to rarely

transposed distribution lines [37] All of these characteristics strongly suggest that DS is

to be classified as an ill-conditioned power system

The practical DSs low XR ratio sections may cause both the NR and FD

conventional methods to diverge [38-41] The line impedance angles are small enough to

deteriorate the dominance of the NR Jacobian main diagonal making it prone to

singularity Such a low XR value would also prevent the Jacobian matrix from being

decoupled and simplified

In addition to performance considerations a practical power flow technique needs to

consider all the DS distinctive features and to accommodate the imbalance introduced by

multiphase networks along with the distribution-level loads In the literature a number

of Newton and non-Newton power flow methods designed for distribution systems were

proposed Zhang et al [42] solved the distribution power flow based on the Newton

method although the proposed Jacobian is computed just once the solution converged

with a number of additional iterations more so than the conventional approach

Moreover shunt capacitor banks were ignored in the modeling as well as the line shunt

admittance (JI model) and the constant impedance loads Baran and Wu [43] solved the

10

power flow problem by utilizing three fundamental quadratic equations representing the

real and reactive section powers and the bus voltages in an iterative scheme as a

subroutine during the process of optimizing the capacitor sizing However they

computed the Jacobian using the chain rule within the proposed NR method which is in

turn time consuming Mekhamer et al [44] utilized the three equations developed in [43]

using a different iterative technique without the need for the Jacobian or the NR method

However their process is based on applying a multi-level iterative process on the main

feeder and laterals which makes the speed and the efficiency of their proposed algorithm

a function of the RDS configuration and topology

In [4546] the quadratic equation was also utilized in determining the relation

between the sending and receiving end voltage magnitudes along with the section power

flow They proposed to include the system power losses within their calculation while

solving for the system power flow However the voltage phase angles were ignored

during the solution of the radial power flow in order to speed up the convergence The

latter reference developed work was based on the assumption of balanced RDS and

sophisticated numbering scheme

The radial power flow introduced by [47-49] used a non-Newton power flow techshy

nique based on the ladder network theory This method adds the section currents and

calculates the RDS bus voltages including the substations during a backward sweep If

the difference between the calculated substation voltage value and substation predetershy

mined assigned bus voltage value is acceptable the iterations are concluded If not the

substation bus voltage is reset and the RDS bus voltages are computed for the second

time in the same iteration in the forward sweep Both the ladder and the backshy

wardforward methods are derivative-free instead they employ simple circuit laws

However the ladder method uses many sub-iterations on the laterals and calculates the

system bus voltages twice during a single iteration compared to once in the backshy

wardforward method Thukaram [50] utilized the backwardforward sweep technique to

solve the RDS power flow However the bus numbering procedure was a sophisticated

parent node and child node arrangement which may add some computational overshy

head if the system topology is changed Teng [51] used the backwardforward approach

as the solution procedure through the development of two matrices and multiplied them

11

together in a later stage of the solution process In assembling those matrices all the

system buses and sections have to be inspected carefully In a practical large RDS data

preparation for these matrices will be cumbersome and prone to errors Under continshy

gency situations switching operations or the addition of another feeder to the existing

one are quite common practices in the DSs hence changes in system topology need to be

accommodated by restructuring the corresponding matrices which would add an overshy

head to track modifications The weakly meshed DS was dealt with by adding extra

nodes in the middle of the new links Two equal currents with opposite polarities were

injected into each added node Each injection operation is represented by a two column

matrix which was subsequently added to the first proposed matrix and then the develshy

oped matrices were extended and multiplied together The resultant is a full matrix and

its dimension is reduced by the Kron method in every single iteration That is the

developed full matrix was inverted in each iteration of the solution method and such

procedure is expensive lengthy cumbersome and time consuming

Shirmohammadi et al [39] Cheng and Shirmohammadi [52] and Hague [53]

proposed an iterative solution method for both radial and weakly meshed DSs This

approach necessitates a special numbering scheme in which they number the DS sections

in layers starting from the root node The numbering scheme is to be carried out

carefully by examining the whole system when a new layer is to be numbered The

numbering process is cumbersome and prone to errors For weakly meshed networks

breakpoints are selected opened and consequently the meshed system is converted to a

radial system The loops are broken by adding two fictitious buses In each pair of

dummy buses equal and opposite currents are injected and the new system is evaluated

to produce a reduced order impedance matrix Their proposed method requires that the

breakpoint impedance matrix should be computed cautiously Such a procedure is highly

dependent on the distribution networks topology That is the more links that exist in the

DS the larger the break point impedance matrix and the more time will be consumed in

its computation

Goswami and Basu [38] introduced a direct solution method to solve for radial and

weakly meshed DS They applied a breakpoints method into the meshed DS similar to

that of [39] in order to convert it into RDS In their proposed methodology a restriction

12

was imposed on each of the system buses not to have more than three sections attached to

it Such limitation is a drawback of the method and moreover a difficult node numbering

scheme is a disadvantage

In this thesis the unique structure of the RDS is exploited in order to build up a new

fast flexible power flow technique that deals with radial and looped DSs The numbering

scheme of the DS is simple and straightforward All load types can be accommodated by

the proposed distribution power flow eg spot and distributed loads Unlike

conventional power flow methods no trigonometric functions are used in the proposed

distribution power flow method For weakly meshed and looped DSs the system is dealt

with as it is there is no need for radialization cuts or building breakpoints impedance

matrix The topology of the tested DS whether strictly radial weakly meshed or looped

is represented by a building block matrix which is the only one needed to perform the

backwardforward sweep technique

23 DG INTEGRATION PROBLEM

DG is gaining increasing popularity as a viable element of electric power systems The

presence of DG in power systems may lead to several advantages such as supplying

sensitive loads in case of power outages reducing transmission and distribution networks

congestion and improving the overall system performance by reducing power losses and

enhancing voltage profiles Some of the negative impacts of DG installations are

potential harmonic injections the need to adopt more complex control schemes and the

possibility of encountering reverse power flows in power networks Even though the

concept of DG utilization in electric power grids is not new the importance of such

deployment is presently at its highest levels due to various reasons Recent awareness of

conventionaltraditional thermal power plants harmful impacts on the environment and

the urge to find more environmentally friendly substitutes for electrical power generation

rapid advances made in renewable energy technologies and the attractive and open

electric power market are a few major motives that led to the high penetration of DG in

most industrial nations power grids To achieve the most from DG installation special

attention must be made to DG placement and sizing

13

The problem of optimal DG placement and sizing is divided into two subproblems

where is the optimal location for DG placement and how to select the most suitable size

Many researchers proposed different methods such as analytic procedures as well as

deterministic and heuristic methods to solve the problem

231 Solving the DG Integration Problem via Analytical and Deterministic Methods

In the literature the optimal DG integration problem is solved by means of employing

any analytical or optimization technique that suits the problem formulation Methods and

procedures of optimally sizing and locating the DGs within the DS are varied according

to objectives and solution techniques

Willis [54] presented an application of the famous 23 rule originally developed

for optimal capacitor placement to find a suitable bus candidate for DG placement That

is to install a DG with a rating of 23 of the utilized load at 23 the radial feeder length

down-stream from the source substation However this rule assumes uniformly

distributed loads in a radial configuration and a fixed conductor size throughout the

distribution network In any event the 23 rule was developed for all-reactive load

These assumptions limit its applicability to radial distribution systems and the fact that it

is only suitable for single DG planning

Kashem et al [55] developed an analytical approach to determine the optimal DG

size based on power loss sensitivity analysis Their approach was based on minimizing

the DS power losses The proposed method was tested using a practical distribution

system in Tasmania Australia However it assumes uniformly distributed loads with all

the connected loads along the radial feeder having the same power factor and it also

assumes no external currents injected into the system buses eg capacitors which limits

its practicality

Wang and Nehrir [56] developed an analytical approach to address the optimal DG

placement problem in distribution networks with different continuous load topologies

Minimizing the real power losses was the objective of the proposed method In their

approach the DG units were assumed to have unity power factor and only the overhead

distribution lines with neglected shunt capacitance are considered The candidate bus

was selected based on elements of the admittance matrix power generations and load

14

distribution of the distribution network The issue of DG optimal size was not addressed

in their formulation

Griffin et al [57] analyzed the DG optimal location analytically for two continuous

load distributions types ie uniformly distributed and uniformly increasing loads The

goal of their study was to minimize line losses One of the conclusions of their research

was that the optimal location of DG is highly dependent on the load distribution along the

feeder ie significant loss reduction would take place when placing the DG toward the

end of a uniformly increasing load and in the middle of uniformly distributed load feeder

Acharya et al [58] used the incremental change of the system power losses with

respect to the change of injected real power sensitivity factor developed by Elgerd [59]

This factor was used to determine the bus that would cause the losses to be optimal when

hosting a DG By equating the aforementioned factor to zero the authors solved for the

optimal real value of DG output They proposed an exhaustive search by applying the

sensitivity factor on all the buses and ranked them accordingly The drawback of their

work is the lengthy process of finding the candidate locations and the fact that they

sought to optimize only the DG real power output Furthermore they only considered

planning of a single DG

Popovic et al [60] utilized sensitivity analysis based on the power flow equations to

solve the DG placement and sizing Two indices were used in ranking all the DS buses

for the suitability of hosting the DG The first one is a voltage sensitivity index which is

derived directly from the NR power flow Jacobian inverse the second one exploits the

relation of incremental real power losses with respect to the injected real and reactive

power as developed in [61] Their objective for sizing the DG was to maximize its

capacity subject to boundary constraints such as bus voltage penetration level line flows

and fault current limits To solve the sizing DG problem they gradually increased the

DG capacity at selected most sensitive buses until one of the constraints is violated and

the direct previous installed DG size becomes the one chosen as the optimal rating This

process is a lengthy and impractical procedure and the authors did not elaborate on how

they would deal with multiple DG cases using the proposed scheme

Keane and OMalley [62] solved for the optimal DG size in the Irish system by using

a constrained Linear Programming (LP) approach To cope with the EU regulation which

15

emphasizes that Ireland should provide 132 of its electricity from renewable sources

by 2010 the objective of their proposed method was to maximize the DG generation

The nonlinear constraints were linearized with the goal of utilizing them in the LP

method A DG unit was installed at all the system buses and the candidate buses were

ranked according to their optimal objective function value

Rosehart and Nowicki [63] dealt with only the optimal location portion of the DG

integration problem They developed two formulations to assess the best location for

hosting the DG sources The first is a market based constrained optimal power flow that

minimized the cost of the generated DG power and the second is voltage stability

constrained optimal power flow that maximized the loading factor distance to collapse

Both formulations were solved by utilizing the Interior Point (IP) method The outcomes

of the two formulations were used in ranking the buses for DG installations The optimal

DG size problem was not considered in their paper

Iyer et al [64] employed the primal-dual IP method to find the optimal DG

placement through combined voltage profile improvement and line loss reduction indices

However the proposed approach was based on initially placing DGs at all buses in order

to determine proper locations for DG installations This methodology may not be

realistic for large scale distribution networks

Rau and Wan [65] only treated the DG sizing problem by utilizing the Generalized

Reduced Gradient (GRG) method The DG bus locations were assumed to be provided

by the system planner for the DG units to be installed In their proposed method they

considered minimizing the system active power losses In their formulation only the

power flow equality constraints were considered whereas the boundary conditions and

the inequality constraints were not taken into account

Hedayati et al [66] employed continuous power flow methodology to locate the

buses most sensitive to voltage collapse The sensitive bus set is ranked based on their

severity which is used accordingly to indicate potential bus locations for placement of

single and multiple DG sources An iterative method was proposed for optimally sitting

the DG A certain DG capacity which is known and fixed a priori is added to the DS

and the conventional power flow method was employed to determine the resultant DS

real power losses voltage profiles and power transfer capacity In the subsequent

16

iteration another DG with the same capacity was added to the next sensitive bus and

results were obtained This iterative process would continue until the system outcomes

reached acceptable values The proposed iterative method did not optimize the DG size

232 Solving the DG Integration Problem via Metaheuristic Methods

Metaheuristic techniques have proven their effectiveness in solving optimization

problems with appreciable feasible search space They can be easily modified to cope

with the discrete nature associated with different elements commonly used in power

systems studies Optimization methods such as Genetic Algorithm (GA) [67-71] GA

hybrid methods such as GA and fuzzy set theory [72] and GA and Simulated Annealing

(SA) [73] Tabu search algorithm (TS) [7475] GA-TS hybrid method [76] Ant Colony

Optimization (ACO) [77] Particle Swarm Optimization (PSO) [78] and Evolutionary

Programming (EP) [79] were utilized in the literature to solve for the DG integration

problem

Teng et al [67] developed a value-based method for solving the DG problem The

GA method was utilized in maximizing a DG benefit to cost ratio index subject to only

boundary constraints such as ratio index voltage drop and feeder transfer capacity A

drawback of their procedure is that the candidate DG bus locations were assumed to be

provided by the utility and consequently all combinations of the provided bus locations

were tested for obtaining the optimal DG capacities via the GA method

The proposal set forth by Mithulananthan et al [68] made use of the DS real power

losses as the fitness function to be minimized through GA Their formulation of the DG

size optimization problem is of an unconstrained type Moreover the NR method which

is usually inadequate in dealing with the DS topology was used in calculating the total

power losses Candidate DG bus locations were obtained by placing a DG unit at all

buses of the tested DS which is impractical for large DSs Furthermore the multiple

DGs case was not addressed

Haesen et al [69] and Borges et al [70] solved the DG integration problem by

basically employing the GA method Both utilized the metaheuristic technique in solving

for single and multiple DG sizing and placements Haesen et al used the GA method to

minimize the DS active power flow while the objective for Borges et al was to

17

maximize a DG benefit to total cost ratio index Reference [69] incorporated penalty

factors within the objective function to penalize constraint violations thus adding another

set of variables to be tuned The authors of the latter reference used a PV model for

modeling the DG

Celli et al [71] formulated the DG integration problem as an s-constraint

multiobjective programming problem and solved it using the GA method Their

proposed algorithm divided the set of the objective functions into one master and the rest

are considered as slave objective functions The master is treated as the primary

objective function that is to be minimized while the slaves are regarded as new

inequality constraints that are bounded by a predetermined e value They utilized their

hybrid method to minimize the following objective functions cost of network upgrading

energy losses in the DS sections and purchased energy (from transmission and DG) The

number of the DG sources to be installed was randomly assigned and the units were

randomly located at the network buses Whenever the constraints are violated the

objective function solution is penalized A Pareto set was calculated from this

multiobjective optimization problem to aid the distribution planner in the decision

making process

Kim et al [72] and Gandomkar et al [73] hybridized two methods to solve the DG

sizing problem The former hybridized GA with fuzzy set theory to optimally size the

single DG unit while the latter combined the GA and SA metaheuristic methods to solve

for the optimal DG power output In both references the DG sizing problem was

formulated as a nonlinear optimization problem subject to boundary constraints only

Unlike Gandomkar et al [73] added the nonlinear power flow equality constraints to

their problem formulation The former researchers utilized their methodology to

investigate multiple DG case while the latter solved only the single DG case Both sited

the DG at all DS buses in order to determine the optimal DG location and size

Nara et al [74] assumed that the candidate bus locations for the DG unit to be

installed were pre-assigned by the distribution planner Then they used the TS method in

solving for the optimal DG size The objective of their formulation was to minimize the

system losses The DG size was treated as a discrete variable and the number of the

18

deployed units was considered to be fixed The DS loads were modeled as balanced

uniformly distributed constant current loads with a unity power factor

Golshan and Arefifar [75] applied the TS method to optimally size the DG as well

as the reactive sources (capacitors reactors or both) within the DS They formulated

their constrained nonlinear optimization problem by minimizing an objective function

that sums the total cost of active power losses line loading and the cost of the added

reactive sources The DG locations were not optimized instead a set of locations were

designated to host the proposed DGs and the reactive sources

A hybrid method that combined the GA with the TS technique in order to solve the

DG sizing optimization problem was developed by Gandomkar et al [80] They solved

the DG integration problem by minimizing the distribution real power losses subject to

boundary conditions The authors restricted the number of DGs as well as their gross

capacity to be revealed prior to executing the optimization procedure They augmented

the objective function with penalty terms in their formulation to handle the constraint

violations

Falaghi and Haghifam [77] proposed the ACO methodology as an optimization tool

for solving the DG sizing and placement problems The minimized objective function for

the utilized method was the global network cost ie the summation of the DGs cost their

corresponding operational and maintenance cost the cost of energy bought from the

transmission grid and the cost of the network losses The DG sizes were treated as

discrete values They used a penalty factor to handle the violated constraints ie

infeasible solutions In addition to modeling the DG sources as exclusive constant power

delivering units ie with unity power factor the network loads were all assumed to have

09 power factor Thus it can be stated that such modeling is impractical especially when

real large DSs are encountered

Raj et al [78] dealt with the DG integration in two different steps They employed

the PSO method to optimally determine the size of single and multiple DGs The optimal

location portion of the problem was performed utilizing the NR power flow method to

assign those buses with the lowest voltage profiles as the optimal candidate DG locations

The PSO was used to minimize the system real power losses the voltage profiles

boundary conditions were the only constraints required by the authors to be satisfied

19

Constraint violations were handled via a penalty factor that was augmented with the

objective function The DG units were randomly sited at one or more of the candidate

buses obtained through the NR method and subsequently the PSO was used to find the

optimal size(s)

Rahman et al [79] derived sensitivity indices to identify the most suitable bus for a

single DG installation Subsequently the DG sizing problem was dealt with by

employing an EP approach The objective function of the proposed approach was to

minimize the DS real power losses subject only to the system bus voltage boundary

constraints The formulation of DG sizing in their work was not realistic for a variety of

reasons For instance they ignored the line loading restrictions power flow equality

constraints and DG size limits

In most of the reviewed work on the DG deployment problem the problems of DG

optimal sizing and placement were not simultaneously addressed due to the difficult

nature of the problem as it combines discrete and continuous variables for potential bus

locations and DG sizing in a single optimization problem This combination creates a

major difficulty to most derivative-based optimization techniques and it increases the

feasible search space size considerably In this thesis the DG sizing subproblem is

solved using an improved SQP deterministic method while the two subproblems are

addressed simultaneously via an enhanced PSO metaheuristic algorithm

24 SUMMARY

In this chapter distribution power flow techniques were reviewed in Section 22 The

literature review of DG integration problem solution methods was presented in Section

23 The analytical and deterministic methods that were utilized to handle the DG

integration problem were presented in Subsection 231 Then recent publications that

handled the DG sizing and placement problems via wide-class of metaheuristic methods

were reviewed and summarized

20

CHAPTER 3 FAST AND FLEXIBLE RADIAL POWER FLOW FOR

BALANCED AND UNBALANCED THREE-PHASE DISTRIBUTION

NETWORKS

31 INTRODUCTION

As discussed in Chapter 2 several limitations exist in radial power flow techniques

presently reported in the literature such as complicated bus numbering schemes

convergence related problems and the inability to handle modifications to existing DS in

a straightforward manner This motivated the development of an enhanced distribution

power flow solution method In this thesis the unique structure of the RDS is exploited in

order to build up a Fast Flexible Radial Power Flow (FFRPF) technique The tree-like

RDS configuration is translated into a building block bus-bus oriented data matrix

known as a Radial Configuration Matrix (RCM) which consequently is utilized in the

solution process The developed algorithm is also capable of handeling weakly meshed

and meshed DSs via a meshed RCM (mRCM) RCM or mRCM is the only matrix that

needs to be constructed in order to proceed with the iterative process During the data

preparation stage each RCM (or mRCM) row focuses only on a system bus and its

directly connected buses That is while building such a matrix there is no need to

inspect the entire system buses and sections Moreover no complicated node numbering

scheme is required The building block matrix is designed to have a small condition

number with a determinant and all of its eigenevalues equal to one to ensure its

invertibility By incorporating this matrix and its direct descendant matrices in solving

the power flow problem the CPU execution time is decreased compared with other

methods The FFRPF method is flexible in accommodating any changes that may take

place in an existing radial distribution system since these changes can be exclusively

incorporated within the RCM matrix The proposed power flow solution technique was

tested against other methods in order to judge its overall performance using balanced and

unbalanced DSs

In Chapters 4 and 5 the FFRPF method is incorporated within the developed FSQP

and HPSO algorithms in solving the optimal DG installation problem It is implemented

21

as a subroutine within the proposed algorithms to satisfy the equality constraints ie

solving the radial power flow equations

32 FLEXIBILITY AND SIMPLICITY OF FFRPF I N NUMBERING THE RDS

BUSES AND SECTIONS

The RDS is configured in a unique arborescent structure with the distribution substation

located at its root node from which all the active and reactive power demands as well as

the system losses are supplied The substation feeds one or more main feeders with

spurred out laterals sublaterals and even subsublaterals For this reason the substation is

treated as a swing bus during the power flow iterative procedure

Most radial power flow techniques proposed in the literature assign sophisticated

procedures for numbering the radial distribution networks in order to execute their

algorithms This is cumbersome when expanding andor modifying existing RDSs In

this section a very simple numbering rule for the RDS buses and sections is introduced

A section is defined as part of a feeder lateral or sublateral that connects two buses in the

RDS and the total Number of Sections (NS) is related to the total Number of Buses (NB)

by this relation (NS=NB -1 )

321 Bus Numbering Scheme for Balanced Three-phase RDS

A balanced radial three-phase RDS is represented by a single line diagram In such a

system a feeder or sub level of a feeder having more than one bus is numbered in

sequence and in an ascending order Consequently each section will carry a number

which is less than its receiving end bus number by one as shown in Figure 31

Therefore sections are numbered automatically once the simple numbering rule is

applied

22

Substation

Figure 31 10-busRDS

In numbering the RDS shown in Figure 31 the following was considered buses 1 -

4 form the main feeder and buses 2 - 6 and 3 - 10 are the laterals while the sublateral is

tapped off bus 5 Figure 32 and Figure 33 manifest the ease in renumbering the system

shown previously and the flexibility in adding any portion of RDS to the existing one

respectively In Figure 32a buses 1 -4 have a different path compared to Figure 31 and

are considered to be a main feeder buses 2 - 9 and 3 - 6 are the laterals whereas the

sublateral 7 - 10 is the section spurred off bus 7 Figure 32b shows another numbering

scheme The same system numbered differently would have the same solution when

solved by the FFRPF

Figure 33 illustrates the ease of numbering in the case of a contingency situation or

a switching operation that could cause the existing system to be modified andor to be

augmented with other systems The lateral 3 - 10 in Figure 31 was modified to be

tapped off bus 2 instead and a couple of radial portions were added to be fed from buses

6 and 4 as illustrated in the figure

23

Substation Substation

(a) (b)

Figure 32 Different ways of numbering the system in Fig 31

Figure 33 The ease of numbering a modified and augmented RDS

322 Unbalanced Three-phase RDS Bus Numbering Scheme

The three-phase power flow is more comprehensive and realistic when it comes to

finding the three-phase voltage profiles in unbalanced RDSs Figure 34 shows an

unbalanced three-phase RDS The missing sections and buses play a significant role in

the multi-level phase loading and in making the unbalanced state of such a three-phase

DS more pronounced

24

The RDS shown in Figure 34 includes 6 three-phase buses (3(j)NB = 6) 5 three-

phase sections (3(|)NS = 5) no matter how many phase buses or sections exist physically

As such it has 14 single-phase buses (1(|)NB = 14) and 11 single-phase sections (1lt|gtNS =

11) The relations expressed in Eq (31) govern the three-phase and single-phase buses

to their corresponding sections

3^NS = 3^NB-1

l^NS = l^NB-3 (31)

Figure 34 Three-phase unbalanced 6-bus RDS representation

It is simple to implement the numbering process in the three-phase system as was

done in the balanced case Any group of phase buses to be found along a phase feeder or

a sub level of a feeder is to be numbered in a consecutive ascending order Consequently

each phase section number will carry a number which is one less than its receiving end

bus number as shown in Figure 34 In other words the sections are numbered routinely

after the ordering of the three-phase RDS buses has been completed

To develop the building block matrix as will be shown shortly the unbalanced three-

phase system is redrawn by substituting for any missing phase section or bus using dotted

representation as depicted in the 6-bus RDS in Figure 35 By performing this step each

three-phase bussection in the RDS consists of a group of 3 single-phase busessections

a b and c including the missing ones for double and single-phase buses

25

l a

I (1) 2 a | (2) 3 a | (3) 4 a | (4)

Figure 35 Three-phase unbalanced 6-bus RDS with missing buses and sections

33 THE BUILDING BLOCK MATRIX AND ITS ROLE I N FFRPF

The proposed FFRPF procedure starts with a matrix that mimics the radial structure

topology called a system Radial Configuration Matrix (RCM) The inverse of RCM is

then obtained to produce a Section Bus Matrix (SBM) that will be utilized in summing

the section currents during the backward sweep procedure A Bus Section Matrix (BSM)

is next generated by transposing the SBM to sum up the voltage drops in the forward

sweep process Therefore the only input data needed in the solution of an existing

modified or extended RDS other than the system loads and parameters is the RCM

It is worth mentioning that the inversion and transposition operations take place only

once during the whole process of the proposed FFRPF methodology for a tested RDS

whereas other methods like the NR technique invert the Jacobian matrix in every single

iteration The following subsections demonstrate the building of a three-phase RCM and

elucidate the role of both SBM and BSM in solving the radial power flow problem

331 Three-phase Radial Configuration Matrix (RCM)

The only matrix needed to be built for an unbalanced three-phase RDS is the RCM

Whatever changes need to be accommodated as a modification in the existing structure or

an addition to the existing network would be performed through the RCM only The

26

other matrices utilized in the backwardforward sweep are the direct results of the RCM

and no other built matrix is needed to perform the FFRPF

Such a matrix exploits the radial nature of such a system The RCM is of 3(3())NB x

3(|)NB) dimension in which each row and column represents a single-phase bus For a

balanced three-phase RDS represented by a single line diagram the RCM dimension is

(NBxNB) The RCM building algorithm for the unbalanced three-phase RDS case is

illustrated as follows

1 Construct a zero-filled 3(3(|)NB x 3(|)NB) square matrix

2 Change all the diagonal entries to +1 every diagonal entry represents sending

missing or far-end buses

3 In each row if the column index corresponds to an existing receiving single-phase

bus its entry is to be changed to - 1

4 If a single-phase bus is missing or is a far-end bus the only entry in its

corresponding row is the diagonal entry of+1

The above RCM building steps are summarized in the following illustration

Columns Description

RCMbdquo

if is either

a - sending phase bus b - far-end phase bus c - missing phase bus (32)

-1 jkl if jkI are receiving phase buses

connected physically to phase bus 0 otherwise

The [abc] matrix is defined as a zero (3 x 3) matrix with the numbers a b and c as

its diagonal elements eg [ I l l ] is the identity (3 x 3) matrix while [110] is the identity

matrix with the third diagonal element replaced by a zero By following the preceding

steps and utilizing the [abc] definition the RCM for the unbalanced three-phase RDS

shown in Figure 35 is to be constructed as shown in (33)

27

[Ill] [000] [000] [000] [000]

[000]

-[111] [111]

[000] [000] [000]

[000]

[000] -[111]

[111] [000] [000]

[000]

[000] [000]

-[110] [111]

[000] [000]

[000]

[000] [000]

-[010] [111]

[000]

[000] [000]

-[on] [000] [000]

[111]

Because of the nature of the RDS the RCM has three distinctive properties The first

is that the RCM is sparse the second is that RCM is a strictly upper triangular matrix

and thirdly such a matrix is only filled by 0 +1 or - 1 Such a real matrix makes the data

preparation easy to handle and less confusing Figure 36 shows the sparsity pattern plots

of the RCM matrices for unbalanced three-phase 25-bus [48] and balanced 90-bus [38]

radial systems

RCM for 25-Bus unbalanced 3-phase RDS RCM for 90-Bus balanced 3-phase RDS

nz = 131 nz = 179

Figure 36 RCM matrices sparsity plots for the 25 and 90 bus RDSs

3311 Assessment of the FFRPF Building Block RCM

The RCM is well-conditioned and should have a small Condition Number (CN) and a

non-zero determinant The CN measures how far from singularity any matrix is It is

defined as

28

cond(A) = A jjA-l (34)

where ||A|| is any of the 3 types of norm formulations of matrix A 1-norm 2-norm or oo-

norm An ill-conditioned matrix would have a large CN while a CN of 1 represents a

perfectly well-conditioned matrix By definition a singular matrix would have an infinite

CN [81] Having all the matrix eigenvalues to be equal to 1 and a determinant value of 1

safeguard the RCM against singularity For this reason the RCM is not only invertible

but also its inverse is an upper triangular matrix filled only with 0 and +1 digits and no

other numbers would appear in RCM-1

332 Three-phase Section Bus Matrix (SBM)

The SBM for the three-phase RDS is obtained by performing the following steps

1 Remove the corresponding substation rows and columns from the RCM ie the

first three rows and columns The reduced version of the RCM is labeled as

RCM

2 Invert the RCM to obtain the SBM as shown in (35) and more explicitly in

Figure 37

To clarify the two rows and the two columns outside the matrix border shown in

Figure 37 are the three-phase buses and sections ordered respectively The dimension of

the unbalanced three-phase system SBM is 3(3lt|)NS x 3(|gtNS) For the balanced case the

SBM dimension is (NSxNS)

[Ill] [000]

[000] [000] [000]

[111] [111]

[000] [000] [000]

[110] [110]

[111] [000] [000]

[010] [010] [010] [111] [000]

[011] [011] [000] [000] [111]

29

1 a

1 b

1 c

2 a

2 b 2 c

SBM = 3 a

3 b

3 c

4 a

4 b

4 c

5 a

5 b

5 c

2 a

0 0 0 0 0 0 0 0 0 0 0 0 0 0

2 b

0 1 0 0 0 0 0 0 0 0 0 0 0 0 0

2 c

0 0 1 0 0 0 0 0 0 0 0 0 0 0 0

3 3 a b

1 0 0 1 0 0 i o 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

3 c

0 0 1 0 0

1 0 0 o 0 0

o 0 0 0

4 a

1 0 0 1 0

o 1 0

o 0 0

o 0 0 0

4 b

0 1 0 0 1 0 0 1 0 0 0 0 0 0 0

4 c

0 0 0 0 0 0 0 0 1 0 0 0 0 0 0

5 a

0 0 0 0 0 0 0 0 0 1 0 0 0 0 0

5 b

0 1 0 0 1 0 0 1 0 0 1 0 0 0 0

5 c

0 0 0 0 0 o 0 0

o 0 0 1 0 0 0

6 a

0 0 0 0 0

o 0 0

o 0 0 o 1 0 0

6 b

0 1 0 0 1 0 0 0 0 0 0 0 0 1 0

6 c

0 1 0 0 1 0 0 0 0 0 0 0 0 1

Figure 37 SBM for three-phase unbalanced 6-bus RDS

By inspecting Figure 35 it is noted that any single-phase section is connected

downhill to a single-phase far-end bus or buses through a Unique Set of Phase Buses

(USPB) xt bull By inspecting Figure 35 and Figure 37 the USPB for the following

single-phase sections la lb lc 3b 4b are^0 [ gt xlgt x respectively as explained

in (36)

=2 f l 3bdquo4a x=2b3b4b5bA]

Xl=2cA) Xl=) (36)

X=5b]

In the SBM the single-phase section is represented by a row i and will have entries

of ones in all the columns where their indices represent single-phase buses that belong to

the section USPB xf bullgt a s illustrated in (37)

SBMrmt =

Columns Description

c bullgt bull a - Id - buses e r ~ - 1 ijk- ijk-- are either w (37)

lb - diagonal entry 0 other columns otherwise

30

333 Three-phase Bus Section Matrix (BSM)

The BSM is the transpose of the SBM as shown in (38) In the BSM the rows represent

the RDS single-phase buses excluding the substations and all the sections are

represented by the BSM columns Each single-phase bus is connected uphill through a

Unique Set of Phase Sections (USPS) yf to a substation single-phase bus The USPS

for buses 2deg 3a 4b 5b 6C are y ydeg y y yc6 respectively as depicted in Figure 35

and Figure 37 and demonstrated in (39)

BSM

[111] [000] [000] [000] [000]

[111] [111] [000] [000] [000]

[110] [110] [111] [000] [000]

[010] [010] [010] [111] [000]

[011] [011] [000] [000] [111]

(38)

V H U V3=1gt2 y=b2b yb

5=lb2bdquoAb (39)

lt=1C2C5C

In the BSM a single-phase bus i is represented by a row and will have entries of

ones in all the columns where their indices represent single-phase sections that belong to

the bus USPS yf as equivalently shown in (310)

Columns Description

BSMrmi =

( gt - [a-l^-sectionse yf ~ i m

1 ijk ijk are either lt Y Y (310) lb - diagonal entry

0 other columns otherwise

34 FFRPF APPROACH AND SOLUTION TECHNIQUE

The FFRPF methodology is based simply on Kirchhoff s voltage and current laws which

are used in performing the backwardforward sweep iterative process By utilizing the

direct descendant matrices of the RCM ie SBM and BSM the RDS buses complex

31

voltages are calculated in every iteration until convergence criteria are met The next

subsections illustrate the proper usage of such matrices in the proposed FFRPF method

through appropriate modeling of the unbalanced multi-phase RDS section impedances

341 Unbalanced Multi-phase Impedance Model Calculation

Figure 38 shows a three-phase section model that is represented by two buses (sending

and receiving ends) of an overhead three-phase four-wire RDS with multi-grounded

neutral The assumption of a zero voltage drop across the neutral in a three-phase two-

phase and single-phase RDS is found to be valid [4582] Such a configuration is widely

adopted in North Americas distribution networks [8384]

V Sa

_ bull

V Ra

V Sb ab

^VWVVYgt

V Sc Z be

bn

^AAArmdashrYYYV bull

V s n en

ll1 Sa

s b

La

Lb

Lc

bull r Figure 38 Three-phase section model

In the proposed radial power flow solution method each of the three-phase lines is to

be modeled appropriately and mutual coupling effects between phases are not neglected

The primitive impedance matrix for such a four-wire system is a square matrix with a

dimension equal to RDS utilized number of phase and neutral conductors For a system

consisting of three-phase conductors and a neutral wire the section primitive impedance

matrix is expressed as shown in (311)

32

Zaa

ha

Zca

zna

Kb

Kb

Kb

Kb

Ke Kc Ke

nc

art

Zbn

en

nn

where

Z bull primitive impedance matrix

RDS section length

z per unit length self-impedance of conductor i

z per unit length mutual-impedance between conductors andy

zu and zy are calculated according Carsons work [85] and its modifications [86-88] as

illustrated by the following equations

where

k

GMRj

Dbdquo

v GMR

bulli J

zu=rt+rd+ja)k

zv=rd+jltok

resistance of conductor i

earth return conductor resistance

inductance multiplying constant

distance between overhead and its earth return counterpart and it is a

function of both earth resistivity and frequency

geometric mean radius of conductor i

distance between conductors i andj

(312)

(313)

The parameters used in (312) and (313) are shown in Table 31 for both operational

frequencies 50Hz and 60Hz in both metric and imperial units

33

Table 31 cok rj and De Parameters for Different Operation Conditions

De = 2160 Ij (ft)

cok rd

p = 100 Qm

p = 1000 Qm

Metric Units RDS operating frequency 50 Hz 60 Hz

006283km

0049345 QJ km

931 m

29443 m

007539 km

005921412km

850 m

26878 m

Imperial Units RDS operating frequency 50 Hz 60 Hz

010111mile

00794 QI mile

30547f

96598 ft

012134mile

009528 QI mile

27885

88182

Since the neutral is grounded the primitive impedance matrix Zsec can be

transformed into a (3 x 3) symmetrical impedance matrix Zsae

c by utilizing Krons

matrix reduction method The resultant section three-phase impedance matrix is

expressed mathematically in (314) and the three-phase section model is represented

graphically in Figure 39

7 abc

aa

zba Zca

Zab

^bb

Zcb

zac zbc Zee

(314)

VSn

mdash bull i 7 T

i ah bull-sec a

zbdquobdquo bull A V W Y Y Y V

v izK

^WW-rrYYv -+bull

I

bull

vR

Figure 39 The final three-phase section model after Krons reduction

If the RDS section consists of only one or two phase lines its primitive impedance

matrix is transformed by Krons matrix reduction to (laquo-phase x w-phase) symmetrical

impedance matrix Next the corresponding row and column of the missing phase are

replaced by zero entries in the (3x3) section impedance matrices Zsae

c For a two-phase

34

section its impedance matrix Z^c is demonstrated below

Z_a

zci Kron h-gt ZZ za

zbdquo zai

zaa o zac

0 0 0

z_ o zbdquo

Underground lines such as concentric neutral and tape shielded cables are typically

installed in the RDS sections For underground cables with m phases and n additional

neutral conductors the primitive impedance of each section is a (2m+n x 2m+n) matrix

with the entries computed as illustrated in [89-92]

Usually the RDS is modeled as a short line ie less than 80 km and the charging

currents would be neglected by not modeling the line shunt capacitance as depicted in

Figure 38 However under light load conditions and especially in the case of

underground cables the line shunt capacitance needs to be considered in order to obtain

reasonable accuracy ie use nominal 7i-representation The rc-equivalent circuit consists

of a series impedance of the section and one-half the line shunt admittance at each end of

the line Figure 310 (a) (b) and (c) show the RDS 7i-model representation The shunt

admittance matrix for an overhead three-phase section is a full (3x3) symmetrical

admittance matrix while it is a strictly diagonal matrix for the underground RDS cable

section That is the self admittance elements are the only terms computed [92] For the

unbalanced three-phase section eg one or two phases the non-zero elements of shunt

admittances are only those corresponding to the utilized phases

[zic] -AAVmdashrwvgt

T yabc 1 |_ sec J [ yaf tc |

sec J

(a)

35

Lsec J _

2

s

1

Yaa

Yba

Yea

Yab

Ybb

Ycb

Yac

Ybc

Ycc

zaa

zba

tea

zab

zbb

zcb

zac

zbc

zcc

P yabc ~|

lgtlt 2 - =

Yaa

Yba

Yca

Yab

Ybb

Ycb

R

1 1

Yac

Ybc

Ycc

(b)

[ yabc 1 sec J

s 1 1

Yaa

0

0

0

Ybb

0

0

0

Ycc

zaa

zba

zca

Zab

zbb

zcb

zac

zbr

zcc

V yabc ~j

L sec 2 - =

Yaa

0

0

0

Ybb

0

R

1 1

0

0

Ycc

(c)

Figure 310 Nominal 7i-representation for three-phase RDS section

(a) Schematic drawing for the single line diagram of 3(|gt RDS (b) Matrix equivalent for overhead section (c) Matrix equivalent for underground cable section

By applying Kirchhoff s laws to the three-phase system section k the relationship

between the sending and receiving end voltages for medium and short line models and

the voltage drop across the same section in the latter model are expressed in Eq (315)-

(316) and Eq (317) respectively

36

rabc S rabc S

14 L sec Jax3 L rabc

3x3 L secgt J3x3

[C]3 [4 [ yabc~ |~ yabc 1

sec J3x3 L sec J3 [4

zt 1 L sec J3x3

f yabc ~| [~ yaampc 1

L sec J3x3 L sec h

bull R rabc

(315)

rrabc VS rabc

S

1 J3x3 L sec J

[degL [L abc R

(316)

where

TT-afec rrabc S ^ R

rabc rabc S XR

rabc

AK

13x3

aAc sect

rabc see

KrH^ic] three-phase sending and receiving end voltages

three-phase sending and receiving end section currents

three-phase shunt admittance of section k

(3gtlt3) identity matrix

(3gtlt3) zero matrix

voltage drop across three-phase section k

section k three-phase currents

(317)

It is worth noting that Eq (315) is reduced to Eq (316) in the case of short line

modeling since YS^C 1 = 0 The voltage drop across the three-phase section k in the short

line model is expressed in Eq (317) and its corresponding sending end phase voltages

can be expressed in expanded forms as follows

V = V + Ta 7 + 1 7 + Tc 7 rS yR ^1secZjaa ^ Isec^ab ^rlsecpoundjac

v =v +r 7 +17 +r 7 y S y R ^ l sec^ab ^ 1 sec^bb ^ 1 sec^Ac

S mdashyR+ Ktc^ca + sec^cb + sec^a

(318)

(319)

(320)

Equations (317)-(320) show that the voltage drop along any phase in a three-phase

section depends upon all the three-phase currents

37

342 Load Representation Accurate and proper load modeling is of significant concern in power distribution

systems as well in its transmission systems counterpart [8693] Loads in electric power

systems are usually expressed by adequate representations so as to mimic their effects

upon the system The load dependency on the operating bus voltage and on system

frequency is among those representations

Static load models are often utilized in the power flow studies since they relate the

apparent power active and reactive directly to the bus operating voltage A static load

model is used for the static load components ie resistive and lighting load and as an

approximation to the dynamic load components ie motor-driven loads [93] Generally

static loads in DS are assumed to operate at rated and fixed frequency value [94-96]

Loads in the DS are usually expressed as function of the bus operating voltage and

represented by exponential andor polynomial models

The exponential model is shown in (321) and (322)

P = Pbdquo

Q = Q0 vbdquo

(321)

(322)

where

V0 nominal bus voltage

V operating bus voltage

P0 real power consumed at nominal voltage

Q0 reactive power consumed at nominal voltage

Exponents a and fi determine the load characteristics and certain a and values lead to a

specific lode model Therefore

1 If a = P mdash 0 the model represents constant power characteristics ie the load is

constant regardless of the voltage magnitude

2 If a = P = 1 the model represents constant current characteristics ie the load is

proportional to the voltage magnitude

3 If a = P = 2 the model represents constant impedance characteristics ie the load is

38

a quadratic function of the voltage magnitude

As indicated in [97] the exponents could have values larger than 2 or less than 0 and

certain load components would be represented by fractional exponents

The constant current model is considered to be a good approximation for many

distribution circuits since it approximates the overall performance of the mixture of both

constant power and constant impedance models [98] However representing loads with

the constant power model is a conservative approach with regard to voltage drop

consideration [99] and consequently this model will be used in this thesis

Loads can also be represented by a composite model ie the polynomial model The

polynomial model is expressed in (323) and (324)

P = Pbdquo

Q = Q0

(

a p

V

r

V

V

K

V

v0

2

2

V

K

V

+CP

J

)

(323)

(324)

where ap + bp + cp = 1 and aq + b + cq = 1

The polynomial model is also referred to as a ZIP model since it combines all the

three exponential models constant impedance (Z) constant current (I) and constant

power (P) models The ZIP model needs more information and detailed data preparation

The load models can be used in the FFRPF solution method during its iterative

process where flat start values are initially assumed to be the load voltages The three-

phase load voltages are changed during each iteration and consequently the three-phase

currents drawn by the constant current constant impedance andor ZIP three-phase load

models will change accordingly

Different shunt components like spot loads distributed loads and capacitor banks are

customarily spread throughout the RDS In power flow studies spot and distributed

loads are typically dealt with as constant power models while shunt capacitors are

modeled as constant impedances [94 100 101]

The uniformly distributed loads across RDS sections can be modeled equivalently by

either placing a single lumped load at one-half the section length or by placing one-half

the lump-sum of the uniformly distributed loads at each of the section end buses

39

[99 102] The former modeling approach has the disadvantage of increasing the

dimension of the RCM SBM and the BSM since more nodes would be added to the

existing RDS topology In the proposed FFRPF technique the distributed load is

modeled using the latter approach while the three-phase shunt capacitor banks are

modeled as injected three-phase currents [101] as schematically shown in Figure 311

and mathematically represented by Eq (325) and (326)

Qk Cap

^

CCap a Cap

(a) (b)

Figure 311 Three-phase capacitor bank (a) The schematics (b) The Modeling

O3 = poundCap

Qo Capa vbdquo

T-34 _ 1Cap ~

V

a Capbdquo

SQL M Cap

V

filt bullCapo

F

JQ( Cap

(325)

(326)

343 Three-phase FFRPF BackwardForward Sweep

The FFRPF technique employs the SBM in performing the current summation during the

backward sweep and the BSM in updating the RDS bus complex voltages during the

forward sweep as demonstrated in the following subsections

3431 Three-phase Current Summation Backward Sweep DS loads are generally unbalanced due to the unbalanced phase configuration the double

and single-phase loadings as well as the likelihood of unequal load allocation among the

three-phase configuration For the loads they could be represented as constant power

40

constant current constant impedance or any combination of the three models [97 103]

The three-phase load currents at three-phase bus i drawn by a three-phase load eg of a

constant impedance load model is mathematically expressed as shown in Eqs (327) and

(328)

jabc U ~

7e a gt

va

(si) ~

(327)

where

ctabc

o

V

K

2

rft

0

v K

2

K v

2

(328)

where Sf represents the load apparent power at single-phase bus lt|gt As shown in the

preceding equations each load current is a function of its corresponding bus voltage For

Eq (327) if the a phase bus is missing its corresponding phase load current is

eliminated and its corresponding position in the three-phase current vector is replaced by

a zero entry As an illustration and by assuming that there are loads connected to all

existing buses the three-phase load current vector for the system shown in Figure 35 is

expressed as follows

jabc V ja rb jc ra rb re ra rb Q Q rb Q Q rb re l l

The charging currents at the RDS three-phase buses are not to be neglected when

dealing with sections modeled as The shunt admittance at bus is obtained by

applying the following relation

where

Ysh^ bull total three-phase shunt admittance at bus

[l if section k attached to bus i

[0 otherwise

The three-phase shunt currents at bus is as shown in Eq (330)

tabc jrabc 1ch ~~ 1Anbus y i (330)

41

The 3-ltj) bus current is sum of both the 3-sect load current and the 3-cj) charging current as

expressed mathematically in Eq(331)

jabc jabc jabc ) T I 1busl ~ 1Li

+Ich V-gt )U

where 1^ is bus three-phase currents In the case of modeling a three-phase section as

a short line its charging currents are neglected ie I^c = 0 and the bus current will be

represented by the load currents only

The backward sweep sums the phase load currents in the corresponding phase

sections starting from far-end phase buses and moving uphill toward the substation phase

buses The current in phase (j) and section p is computed by utilizing the USPB

principle xp gt during the backward sweep as expressed in (332)

lt = E lt ^here = j 0 ^ J (332)

where

I current through single-phase section and phase ^ (^ =a b or c) SQCp

j current at bus and phase ltb bus x

The SBM is utilized in obtaining the system three-phase section currents in matrix

representation by performing the relation in Eq (333)

[G] = [SBM][lpound] (333)

where the I^ is a 3-(3())NS)-order column vector For a balanced short line RDS

model Eq (333) can be expressed as

[CS] = [SBM][IL] (334)

3432 Three-phase Bus Voltage Update Forward Sweep

The voltage at each phase bus is determined through the forward sweep procedure by

subtracting the sum of the voltage drops across the bus corresponding USPS from the

substation nominal complex voltage The voltage drop across three-phase section k is

calculated as in Eq (317) The voltage drop across all sections of the three-phase RDS

can be obtained by utilizing the SBM as shown in matrix notation via Eq (335)

42

[AKbdquo]=[zr][c] (335)

[ A ^ ] = [ Z s r ] [ 5 5 M ] [ C ]

where ZtradeS J is a diagonal matrix with an 3(3c|)NS x 3ltj)NS) dimension in which its

diagonal entry k corresponds to section k impedance and AV3^ is the computed three-

phase voltage drop values across all the RDS sections as shown below

A ^ = AVa AVb AVC bullbullbull AVb AVC T s e c l s e c l s e c l sec3ltWS sec3tNS J

For calculating the RDS voltage profiles the FFRPF solution method starts by asshy

suming the initial values for all bus voltages to be equal to the substation complex

voltage As a flat start the initial phase voltages at bus will be as follows

2TT 2TT

ya Jdeg vb e~^ VC e+JT V ss e V sisK v sis e (336)

where Vls is the substation complex phase voltage

For the voltage at bus m and phase (j) to be determined at iteration v the calculation is

performed as follows

= amp - pound r A lt wherer = trade lt (337)

The RDS voltage profiles at the system phase buses are obtained by utilizing the BSM as

shown in following matrix representation

[Vi] = [Vsy[BSM][AV^] (338)

where V^fs 1 and V3A are respectively the substation nominal three-phase voltage

column vector and the resultant three-phase bus voltage solution column vector and each

has a dimension of 3(3lt|gtNS)

3433 Convergence Criteria

The bus complex voltage is obtained after every backwardforward sweep After each

iteration all the bus voltage magnitudes and angles are compared with the previous

iteration outcomes The power flow process is concluded and a solution is reached if the

complex voltage real and reactive oo-norm mismatch vector is less than a certain

43

predetermined empirical tolerance value e The convergence criterion is expressed

mathematically as shown in Eq (339)

+i

([gt]w) A a ( |y f lts

where th

i iteration A

(339)

and symbol

||J| vector oo-norm ||x II = max fix IV II lloo l l c 0 1=12NBV I

5H (bull) real part of complex value

3 (bull) imaginary part of complex value

3434 Steps of the FFRPF Algorithm

The FFRPF iterative process can be summarized as follows

Step 1 Begin FFRPF by choosing a test RDS

Step 2 Number and order RDS buses and sections

Step 3 Construct RCM

Step 4 Obtain both SBM and BSM

Step 5 Select load model

Step 6 Start the iterative procedure by assuming flat start voltages for all buses

Step 7 Calculate load currents

Step 8 Start the backward sweep process by calculating section currents using SBM

Step 9 Start the forward sweep process by determining the bus complex voltages

using BSM

Step 10 Compare both magnitudes and angles of the RDS bus voltages between the

current and previous iterations

bull If the co-norm of their difference is lt st

o Solution is reached

44

o Stop and end FFRPF procedure

o Obtain bus voltage profiles section currents and power losses

etc

bull If not utilize the outcome of this iteration (bus complex voltages)

to start a new one by going back to Step 7

The FFRPF solution method is illustrated by the following flow chart shown in Figure

312

45

i laquo - i +1

Calculate Load and leakage

currents

I Start Backward

sweep process by calculating section

currents using SBM

Start Forward sweep process by determining bus

complex voltages V[+1] using BSM

V[+1] Section currents

Section Power Losses Etc

Start FFRPF

Read the test RDS data

Number and order RDS Buses and

Sections

I Construct RCM

Remove the substation

corresponding rows and columns

from RCM to Obtain RCM

Obtain RCM1

To Get SBM

Z Transpose SBM to

get BSM

Calculate RDS section

Impedance and Shunt admittance

Matrices

Select load model

Assuming a flat start voltages for

all buses V[]=10 =0

Figure 312 The FFRPF solution method flow chart

46

344 Modifying the RCM to Accommodate Changes in the RDS The FFRPF technique deals with changes in the network such as the addition of a

transformer by adjusting the original RCM to incorporate its conversion factor (c)

Subsequently the SBM and BSM are obtained accordingly and used in the

backwardforward sweep procedure If a three-phase transformer is incorporated in a

three-phase RDS between buses m and n at section n - 1 the modified BSM entries are

located at the intersection of the matrix rows and columns defined by Eq (340)

BSMZ EzL~-inBSMZ euro lt _ (340)

The affected rows and columns of the modified BSM are those belonging to the

sections USPB and the sending buss USPS respectively For demonstration purposes

the 10-bus balanced RDS shown in Figure 31 will be utilized If two transformers with

conversion factors cfj and cS are to be added within sections 3 and 6 respectively the

original RCM is modified to accommodate such additions as illustrated in (341) Thus

instead of filling -1 for the receiving end bus entry the negative of the conversion factor

is the new entry The process is repeated rc-times for -installed transformers The

corresponding modified SBM and BSM are to be obtained as demonstrated in Section

33

10

RCM^ =

1

2

3

4

5

6

7

8

9

10

1

0

0

0

0

0

0

0

0

0

-1

1

0

0

0

0

0

0

0

0

0

-1

1

0

0

0

0

0

0

0

0

0

-cfi 1

0

0

0

0

0

0

0

-1

0

0

1

0

0

0

0

0

0

0

0

0

-1

1

0

0

0

0

0

0

0

0

-cf2

0

1

0

0

0

0

0

0

0

0

0

-1

1

0

0

0

0

-1

0

0

0

0

0

1

0

0

0

0

0

0

0

0

0

-1

1

(341)

The affected entries of the new BSM are obtained by applying the relation in (340) as

follows

47

[BSMZ eZ s ^nBSMJ euro ^ 3 = 40(12=[(4l)(42)]

(laquoSWe^)nJM6^)=J7 )8)njl I4=gt[(7 )l)(7 )4)(8l) )(M)]

The matrix shown in (342) shows the final B S M after including the transformers in the

10-bus R D S Such a procedure is easily extended to the unbalanced three-phase RDSs

1

1

1

CJ

1

1

cfi

cf2

1

1

2

0

1

ch 0 0

0

0

1

1

It is worth mentioning that by integrating the cf for any transformer configuration

into the RCM building block in the FFRPF technique another light is shed on the

flexibility criterion of the proposed method

35 FFRPF SOLUTION METHOD FOR MESHED THREE-PHASE DS

In practical DS networks alternative paths are typically provided to accommodate for

any contingency incidents that might take place eg feeder failure Therefore it is not

unusual for meshed distribution networks to be part of the DS topology in order to make

the system more reliable The loop analysis approach as well as the graph theory

technique are used to study and analyze the behavior of meshed DS The loop analysis

technique basically applies Kirchhoff s voltage law principle to solve for the fundamental

loop currents in both planar and nonplanar networks while the graph theoretic

formulation preserves the network structure properties [104]

A meshed DS can be viewed from a graph theory perspective as an oriented looped

graph that preserves the network interconnection properties whereas a DS that has no

0

0

0

1

1

cfi

cf2

0

0

0

0

0

0

1

0

0

0

0

0

0

0

0

0

1

1

0

0

0

0

0

0

0

0

1

0

0

0

0

0

0

0

0

0

1

1

0

0

0

0

0

0

0

0

1

(342)

48

loops is considered a tree In graph theory terminology line segments that connect

between buses in a loopless DS tree are called twigs branches or sections (represented by

solid line segments in Figure 313) while those which do not belong to the tree are

known as links (represented by dotted line segments in Figure 313) Links are segments

that close loops in a DS tree thereby composing a co-tree Removal of all co-tree links

results in a strictly radial system Links are usually activated by closing their

corresponding Normally Open (NO) switches Whenever a link is added to a RDS

network a loop is formed and as a result the system will have as many fundamental

loops as the number of links A fundamental loop is a loop that contains only one link

besides one or more sections Segments are used here to name sections and links

together It is noted that the number of fundamental loops is significantly less than the

number of buses in the meshed DS which makes the loop analysis a more appropriate

method in dealing with such systems than other circuit analysis methods like nodal

voltage method [105]

The current directions in the meshed DS sections and links are arbitrarily chosen to

be directed form a lower bus index to a higher one and the positive direction of loop

current is assumed to in the same direction of that of the link as illustrated in Figure 313

The number of segments in a meshed DS is equal to the sum of the total number of its

corresponding graph tree sections and its co-tree links For a meshed DS with NB buses

and mNS segments (total number of sections and links in the meshed DS) the number of

links nL and the number of the fundamental loops as well are obtained according to the

following relation

laquoL=mNS-NB + l (343)

49

Substation 2 Imdash 31 4 1

^ -gtT-gtL- -

Figure 313 10-bus meshed distribution network

351 Meshed Distribution System Corresponding Matrices The FFRPF solution method can deal with the DS meshed networks through modifying

the original RCM Discussion is now focused on the balanced three-phase meshed DS

which can easily be extended to the unbalanced three-phase DS networks

Meshed RCM The meshed RCM (wRCM) order is ((NB + wL) (NB + nVj) and the

mRCM building algorithm is as follows

1 Remove links from the meshed DS and build the RCM for the resulting network

tree as demonstrated earlier in section 331

2 Add nL rows and columns toward the end of the RCM ie each link is represented

by a row and a column attached to the end of the RCM

3 In each link column there are 3 non-zero entries and are to be filled in the following

manner

a -1 at the row which corresponds to the lower index terminal of the link

b +1 at the row which corresponds to the higher index terminal of the link

c +1 at the link diagonal entry

For the 10-bus system shown in Figure 31 three directed links Li L2 and L3 are added

to the DS tree through connecting the following buses 4 - 5 6 - 8 and 4 - 1 0

respectively The system mRCM is constructed as illustrated in (344)

50

10

mRCM (13x13)

1

0

0

0

0

0

0

0

0

0

0

0

0

-1

1

0

0

0

0

0

0

0

0

0

0

0

0

-1

1

0

0

0

0

0

0

0

0

0

0

0

0

-1

1

0

0

0

0

0

0

0

0

0

0

-1

0

0

1

0

0

0

0

0

0

0

0

0

0

0

0

-1

1

0

0

0

0

0

0

0

0

0

0

0

-1

0

1

0

0

0

0

0

0

0

0

0

0

0

0

-1

1

0

0

0

0

0

0

0

-1

0

0

0

0

0

1

0

0

0

0

0

0

0

0

0

0

0

0

-1

1

0

0

0

0

0

0

-1

1

0

0

0

0

0

1

0

0

0

0

0

0

0

-1

0

1

0

0

0

1

0

0

0

0

-1

0

0

0

0

0

1

0

0

1

(344)

Remove the substation corresponding rows and columns from the mRCM to produce the

mRCM The mRCM for the 10-bus system is shown in (345)

10

mRCM (12x12)

1

0

0

0

0

0

0

0

0

0

0

0

-1

1

0

0

0

0

0

0

0

0

0

0

0

-1

1

0

0

0

0

0

0

0

0

0

-1

0

0

1

0

0

0

0

0

0

0

0

0

0

0

-1

1

0

0

0

0

0

0

0

0

0

0

-1

0

1

0

0

0

0

0

0

0

0

0

0

0

-1

1

0

0

0

0

0

0

-1

0

0

0

0

0

1

0

0

0

0

0

0

0

0

0

0

0

-1

1

0

0

0

0

0

-1

1

0

0

0

0

0

1

0

0

0

0

0

0

-1

0

1

0

0

0

1

0

0

0

-1

0

0

0

0

0

1

0

0

1

(345)

Meshed SBM The meshed SBM (mSBM) is then obtained by inverting the mRCM As

51

an illustration the 10-bus meshed network mSBM is obtained as shown in (346)

10

mSBM (12x12)

1

0

0

0

0

0

0

0

0

0

0

0

1

1

0

0

0

0

0

0

0

0

0

0

1

1

1

0

0

0

0

0

0

0

0

0

1

0

0

1

0

0

0

0

0

0

0

0

1

0

0

1

1

0

0

0

0

0

0

0

1

0

0

1

0

1

0

0

0

0

0

0

1

0

0

1

0

1

1

0

0

0

0

0

1

1

0

0

0

0

0

1

0

0

0

0

1

1

0

0

0

0

0

1

1

0

0

0

i deg 1

1

-1

1 deg 0

o 0

0

i

o o

0

0

0

0

1

-1

-1

0

0

0

1

0

0

0

1

0

0

0

0

-1

-1

0

0

1

(346)

Let the mSBM be partitioned into two submatrices mSBMp and C so that the mSBMp

submatrix corresponds to the DS tree sections and the second submatrix C to the

fundamental loops or links as shown in (347)

wSBM = SBM

6 [cl (mNSxnL) = [mSBMp C]

JmNSx(NB-l)

The dotted line shown in the above relation implies matrix partitioning

(347)

Fundamental loop matrix The second submatrix in (347) ie C is the fundamental

loop matrix which governs the direction of currents in each of fundamental loop sections

and links The fundamental loop matrix C can be partitioned into two submatrices [Csec]

and [I] as demonstrated below

M_ where [Csec] is an ((NB - 1) x (laquoL)) matrix and [7] is an nL square identity matrix The

former matrix corresponds to the tree loop sections while the latter corresponds solely to

c-- (348)

52

the co-tree links

By inspecting the fundamental loop matrix C it is noted that each row represents a

section or a link and each column represents a loop Each column entry in the C matrix

CM will have one of the following values

1 Qy = +1 if section k belongs to and is oriented in the same direction of loop

2 Cki = - 1 if section k belongs to and is oriented in the opposite direction of loop

3 Claquo = 0 if section k is not in the loop

By inspecting the 10-bus DS mSBM illustrated in (346) the first loop which is represhy

sented by the tenth column of the matrix is comprised of three sections in addition to the

link The current in two of these sections runs in the same direction as their correspondshy

ing fundamental loop ie sections 2 and 3 while the third one ie section 4 has an

opposite orientation This can be easily verified by tracing the first loop in the meshed

DS single line diagram One can also note that two loop currents pass through the third

section in an additive manner

Meshed BSM The meshed BSM (mBSM) is the transpose of mSBM as illustrated in

(349)

[ B S M I 0](mNS-nL)mNS

L J(nLxmNS)

The second submatrix C[ is the transpose of the fundamental loop matrix as manifested in

(350)

mBSM = [mSBM] = mBSMr

c (349)

C=L

1

0

0

0

2 3

1 1

0 0

0 1

4

-1

0

0

5

0

1

0

6

0

-1

0

7

0

-1

0

g

0

0

-1

9

0

0

-1

h 1

0

0

h 0

1

0

h (f 0

1

(350)

Meshed DS impedance matrices The (laquoLx riL) loop-impedance matrix Zioop is

formulated as follows [106]

KHc]|Xf][c] (35D

53

where [ztradegS ] is defined as a non-singular diagonal (mNS x wNS) segment-impedance

matrix that contains all the meshed DS segment impedances (tree sections and links)

along its main diagonal The matrix [Z^fM is formulated as two diagonal submatrices

as follows

|~ rymDS I

L eg J

Zl

0

^

0

0

7

Zk

0

0

0

raquoL

|gtr ] | o o |[zr] (352)

where Z^S J is the (NB - 1 x NB - 1) loopless tree section-impedance diagonal square

matrix and Z^k 1 is the (raquoL x nL) links impedance diagonal matrix

352 Fundamental Loop Currents The algebraic sum of voltage drops AV around any fundamental loop is zero according

to Kirchhoff s voltage law (KVL) and this can be mathematically expressed in terms of

the fundamental loop matrix C as follows

[C][AF] = 0 (353)

The voltage drop across the meshed DS segments is determined by the following

relations

[W] = [zf][mSBM][mILL]

where

Lm seg J bull (wNS x l) segment currents column vector of the meshed DS network

jW iZJ (mNS x l) meshed DS bus Loads and Link currents column vector

(354)

In order to account for the link currents in the meshed network the segment currents

column vector and the meshed DS bus loads and links currents column vector are

54

respectively partitioned into two subvectors as defined below

[ jtree 1

J(mNSxl)

J((JVB-l)xl)

Jloop[ J(nLxl)

(355)

[mILL l(mNSxl)

L L J((MJ-l)xl)

Jloop J (wLxl)

(356)

where

[Cr J ((NB - 1) x 1) tree section currents column vector

[lL] ((NB - 1) x 1) RDS bus load currents column vector

j 1 (nL x 1) fundamental loop current vector which is also the meshed DS link

currents column vector

By multiplying both sides of Eq (354) by [C] the left-hand-side will equal to zero

according to KVL as shown in (353) and by employing Eqs (351) and (356) Eq (354)

is reformulated as

[C][AV] = [c][z^][mSBM[mILL]

0 = [c f [z f ] [mSBM | C ] L op]

bull = [Cr[zS]([raquoSBM][ l ]+ [C][ 1 ] )

0 = [C] [^ ] [ -raquoSBM] [ I ] + [cr [zbdquor][C][U]

-[c] [zf ][c][J=[c] [ Z - ^ S B M J ]

-[^][^]-[cT[zS][laquoSBM][I] Thus the (nL x 1) fundamental loop currents vector in the meshed DS loop frame of

reference can mathematically be expressed as

[4 J - -K]1 [Cj [z^JmSBMr][lL] (357)

Eq (357) can be expressed in terms of the RDS matrices ie SBM and [ Z ^ J by

performing the following operation

55

DEDICATION PAGE

To my beloved parents my brothers Falah and Abdullah my sisters my wife OmFahad

my daughter Najla and my sons Fahad Falah and Othman

TABLE OF CONTENTS LIST OF TABLES x

LIST OF FIGURES xiii

ABSTRACT xvi

LIST OF ABBREVIATIONS AND SYMBOLS USED xvii

ACKNOWLEDGEMENTS xxiv

Chapter 1 INTRODUCTION 1

11 Motivation 1

12 Distribution Generation - Historic Overview 2

13 Distribution Generation 2

14 Thesis Objectives and Contributions 5

15 Thesis Outline 7

Chapter 2 LITERATURE REVIEW 9

21 Introduction 9

22 Distribution Power Flow 9

23 DG Integration Problem 13

231 Solving the DG Integration Problem via Analytical and Deterministic Methods 14

232 Solving the DG Integration Problem via Metaheuristic Methods 17

24 Summary 20

Chapter 3 FAST AND FLEXIBLE RADIAL POWER FLOW FOR BALANCED AND UNBALANCED THREE-PHASE DISTRIBUTION NETWORKS 21

31 Introduction 21

32 Flexibility and Simplicity of FFRPF in Numbering the RDS Buses and Sections 22

321 Bus Numbering Scheme for Balanced Three-phase RDS 22

322 Unbalanced Three-phase RDS Bus Numbering Scheme 24

33 The Building Block Matrix and its Role in FFRPF 26

v

331 Three-phase Radial Configuration Matrix (RCM) 26

3311 Assessment of the FFRPF Building Block RCM 28

332 Three-phase Section Bus Matrix (SBM) 29

333 Three-phase Bus Section Matrix (BSM) 31

34 FFRPF Approach and Solution Technique 31

341 Unbalanced Multi-phase Impedance Model Calculation 32

342 Load Representation 38

343 Three-phase FFRPF BackwardForward Sweep 40

3431 Three-phase Current Summation Backward Sweep 40

3432 Three-phase Bus Voltage Update Forward Sweep 42

3433 Convergence Criteria 43

3434 Steps of the FFRPF Algorithm 44

344 Modifying the RCM to Accommodate Changes in the RDS 47

35 FFRPF Solution Method for Meshed Three-phase DS 48

351 Meshed Distribution System Corresponding Matrices 50

352 Fundamental Loop Currents 54

353 Meshed Distribution System Section Currents 56

354 Meshed Distribution System BackwardForward Sweep 59

36 Test Results and Discussion 60

361 Three-phase Balanced RDS 60

3611 Case 1 31-Bus with Single Main Feeder RDS 61

3612 Case 2 90-bus RDS with Extreme Radial Topology 70

3613 Case 3 69-bus RDS with Four Main Feeders 71

3614 Case 4 15-bus RDS-Considering Charging Currents 73

362 Three-phase Balanced Meshed Distribution System 74

3621 Case 1 28-bus Weakly Meshed Distribution System 74

3622 Case 2 70-Bus Meshed Distribution System 78

vi

3623 Case 3 201-bus Looped Distribution System 79

363 Three-phase Unbalanced RDS 80

3631 Case 1 10-bus Three-phase Unbalanced RDS 81

3632 Case 2 IEEE 13-bus Three-phase Unbalanced RDS 85

3633 Case 3 26-bus Three-Phase Unbalanced RDS 86

37 Summary 87

Chapter 4 IMPROVED SEQUENTIAL QUADRATIC PROGRAMMING

APPROACH FOR OPTIMAL DG SIZING 89

41 Introduction 89

42 Problem Formulation Overview 89

43 DG Sizing Problem Architecture 90

431 Objective Function 90

432 Equality Constraints 92

433 Inequality Constraints 92

434 DG Modeling 93

44 The DG Sizing Problem A Nonlinear Constrained Optimization Problem 94

45 The Conventional SQP 96

451 Search Direction Determination by Solving the QP Subproblem 96

4511 Satisfying Karush-Khun-Tuker Conditions 98

4512 Newton-KKT Method 101

4513 Hessian Approximation 103

452 Step Size Determination via One-Dimensional Search Method 104

453 Conventional SQP Method Summary 105

46 Fast Sequential Quadratic Programming (FSQP) 108

47 Simulation Results and Discussion 113

471 Case 1 33-busRDS 113

4711 Loss Minimization by Locating Single DG 114

4712 Loss Minimization by Locating Multiple DGs 118

vii

472 Case 2 69-bus RDS 124

4721 Loss Minimization by Locating a Single DG 125

473 Loss Minimization by Locating Multiple DGs 129

474 Computational Time of FSQP vs SQP 134

475 Single DG versus Multiple DG Units Cost Consideration 136

48 Summary 136

Chapter 5 PSO BASED APPROACH FOR OPTIMAL PLANNING OF

MULTIPLE DGS IN DISTRIBUTION NETWORKS 138

51 Introduction 138

52 PSO - The Motivation 138

53 PSO - An Overview 139

531 PSO Applications in Electric Power Systems 141 532 PSO - Pros and Cons 143

54 PSO - Algorithm 144

541 The Velocity Update Formula in Detail 145

5411 The Velocity Update Formula - First Component 146

5412 The Velocity Update Formula - Second Component 148

5413 The Velocity Update Formula-Third Component 149

5414 Cognitive and Social Parameters 150

542 Particle Swarm Optimization-Pseudocode 152

55 PSO Approach for Optimal DG Planning 153

551 Proposed HPSO Constraints Handling Mechanism 155

5511 Inequality Constraints 155

5512 Equality Constraints 157

5513 DG bus Location Variables Treatment 157

56 Simulation Results and Discussion 160

561 Case 1 33-bus RDS 161

viii

5611 33-bus RDS Loss Minimization by Locating a Single DG 161

5612 33-bus RDS Loss Minimization by Locating Multiple

DGs 169

562 Case 2 69-Bus RDS 180

5621 69-bus RDS Loss Minimization by Locating a Single DG 180

5622 69-bus RDS Loss Minimization by Locating Multiple

DGs 187

563 Alternative bus Placements via HP SO 195

57 Summary 196

Chapter 6 CONCLUSION 198

61 Contributions and Conclusions 198

62 Future Work 201

REFERENCES 203

APPENDIX 220

IX

LIST OF TABLES

Table 31 cok rd and De Parameters for Different Operation Conditions 34

Table 32 FFRPF Iteration Results for the 31-Bus RDS 67

Table 33 The 31-bus RDS Section Power Losses Obtained by the FFRPF Method 68

Table 34 FFRPF Voltage Profiles Results for the Three Different Load Models 69

Table 35 Comparisons between 31-Bus RDS Exponential Model Results 70

Table 36 31-bus RDS FFRPF Results vs Other Methods 70

Table 37 90-bus RDS FFRPF Results vs Other Methods 71

Table 38 69-bus RDS FFRPF Results vs Other Methods 73

Table 39 Komamoto 15-bus RDS FFRPF Results vs Other Methods 74

Table 310 Voltage Profiles for Radial and Meshed 28-bus Distribution Network 77 t

Table 311 28-bus Weakly Meshed DS FFRPF Results vs Other Methods 78

Table 312 70-bus Meshed DS FFRPF Results vs Other Methods 79

Table 313 201-bus Meshed DS FFRPF Results vs Other Methods 80

Table 314 10-bus 3(j) RDS FFRPF Results vs Ref [52] and Gauss Zbus Methods 85

Table 315 IEEE 13-bus 3(j) RDS FFRPF Results vs Ref [52] and Gauss Zbus

Methods 86

Table 316 26-bus 34gt RDS FFRPF Results vs Ref [52] and Gauss Zbus Methods 87

Table 41 Single DG Optimal Profile at the 33-bus RDS 115

Table 42 Optimal DG Profiles at all 33 buses 116

Table 43 Multiple DG Installations in the 33-bus RDS with Unspecified Power

Factor 119

Table 44 SQP Method Double-DG Cycled Combinations 121

Table 45 Single and Multiple DG Installations at Pre-specified 085 Power Factor 123

Table 46 Loss Reduction Comparisons for all DG Cases 123

Table 47 69-bus RDS Single DG Optimal Size and Placement Profiles 128

Table 48 Optimal Double DG Profiles in the 69-bus RDS 131

Table 49 Optimal Three DG Units Profiles in the 69-bus RDS 133

Table 410 Loss Reduction Comparison for all DG Installations in the 69-bus RDS 134

Table 411 33-bus RDS CPU Execution Time Comparison 135

Table 412 69-bus RDS CPU Execution Time Comparison 135

x

Table 51 HPSO Parameters for the Single DG Case 162

Table 52 33-bus RDS Single DG Fixedpf Case 20 HPSO Simulations 162

Table 53 Descriptive Statistics for HPSO Results for the Fixedpf Case 163

Table 54 HPSO vs FSQP Results 33-bus RDS-Single DG-FixedDG Case 163

Table 55 33-bus RDS Single DG Unspecified pf Case 20 HPSO Simulations 163

Table 56 Descriptive Statistics for HPSO Results for an Unspecified pf Case 164

Table 57 HPSO vs FSQP Results 33-bus RDS-Single DG-UnspecifiedpCase 164

Table 58 HPSO Parameters for Both Double DG Cases 170

Table 59 33-bus RDS Double DG Fixed pf Case 20 HPSO Simulations 171

Table 510 Descriptive Statistics for HPSO Results for Fixed pf Double DG Case 171

Table 511 HPSO vs FSQP Results 33-bus RDS-Double DGs-FixedCase 172

Table 512 33-bus RDS Double DG Unspecified pf Case 20 HPSO Simulations 172

Table 513 Descriptive Statistics for HPSO Results for UnspecifiedDouble DG

Case 173

Table 514 HPSO vs FSQP Results 33-bus RDS-Double DGs-UnspecifiedCase 173

Table 515 HPSO Parameters for Both Three DG Cases 174

Table 516 33-bus RDS Three DG Fixed pf Case 20 HPSO Simulations 174

Table 517 Descriptive Statistics for HPSO Results for Fixed pf Three DG Case 175

Table 518 HPSO vs FSQP Results 33-bus RDS-Three DG-FixedCase 175

Table 519 33-bus RDS Three DGs Unspecified pf Case 20 HPSO Simulations 175

Table 520 Descriptive Statistics for HPSO Results for the Fixed Three DG Case 176

Table 521 HPSO vs FSQP Results 33-bus RDS-Three DG-UnspecifiedCase 176

Table 522 HPSO Parameters for the Four DG Case 177

Table 523 33-bus RDS Four DG FixedCase 20 HPSO Simulations 178

Table 524 Descriptive Statistics for HPSO Results for Fixed Four DG Cases 178

Table 525 HPSO vs FSQP Results 33-bus RDS-Four DG-FixedCase 179

Table 526 33-bus RDS Four DG Unspecified pf Case 20 HPSO Simulations 179

Table 527 Descriptive Statistics for HPSO Results for Unspecified Four DG

Case 179

Table 528 HPSO vs FSQP 33-bus RDS-Four -Unspecified pf Case 180

Table 529 HPSO Parameters for 69-bus RDS Both Single DG Cases 181

Table 530 69-bus RDS Single DG Fixed pf Case 20 HPSO Simulations 182

xi

Table 531 Descriptive Statistics for HPSO Results for the Fixed Single DG Case 182

Table 532 HPSO vs FSQP Results 69-bus RDS-Single DG-FixedpDG Case 182

Table 533 69-bus RDS Single DG Unspecifiedpf Case 20 HPSO Simulations 183

Table 534 Descriptive Statistics for Unspecified pSingle DG Case 183

Table 535 HPSO vs FSQP Results 69-bus RDS-Single DG-UnspecifiedpDG

Case 184

Table 536 HPSO Parameters for 69-bus RDS the Double DG Cases 188

Table 537 69-bus RDS Double DG Fixed pf Case 20 HPSO Simulations 189

Table 538 Descriptive Statistics for HPSO Results for Fixed pDouble DG Case 189

Table 539 HPSO vs FSQP Results 69-bus RDS-Double DG-FixedpDG Case 190

Table 540 69-bus RDS Double DG UnspecifiedpfCase 20 HPSO Simulations 190

Table 541 Descriptive Statistics for HPSO Results for Unspecified ^Double DG

Case 191

Table 542 HPSO vs FSQP Results 69-bus RDS-Double-UnspecifiedpDG Case 191

Table 543 HPSO Parameters for Both 69-bus RDS Three DG Cases 192

Table 544 69-bus RDS Three DG Fixed pf Case 20 HPSO Simulations 192

Table 545 Descriptive Statistics for HPSO Results for Fixed pf Three DG Case 193

Table 546 HPSO vs FSQP Results 69-bus RDS-Three DGs-FixedCase 193

Table 547 69-bus RDS Three DG Unspecified pf Case 20 HPSO Simulations 194

Table 548 Descriptive Statistics for HPSO Results for Unspecified pf Three DG

Case 194

Table 549 HPSO vs FSQP Results 69-bus RDS-Three DGs-UnspecifiedpCase 195

Table 550 20 HPSO Simulations of the 69-bus RDS Three DG Fixed pf Case with Suboptimal Tuned Parameters 50 Iterations and 50 Swarm Particles 196

xii

LIST OF FIGURES

Figure 31 10-bus RDS 23

Figure 32 Different ways of numbering the system in Fig 31 24

Figure 33 The ease of numbering a modified and augmented RDS 24

Figure 34 Three-phase unbalanced 6-bus RDS representation 25

Figure 35 Three-phase unbalanced 6-bus RDS with missing buses and sections 26

Figure 36 RCM matrices sparsity plots for the 25 and 90 bus RDSs 28

Figure 37 SBM for three-phase unbalanced 6-bus RDS 30

Figure 38 Three-phase section model 32

Figure 39 The final three-phase section model after Kron s reduction 34

Figure 310 Nominal ^-representation for three-phase RDS section 36

Figure 311 Three-phase capacitor bank (a) The schematics (b) The Modeling 40

Figure 312 The FFRPF solution method flow chart 46

Figure 313 10-bus meshed distribution network 50

Figure 314 Fundamental cut-sets for a meshed 10-bus DS 57

Figure 315 31-bus RDS 62

Figure 316 The RCM of the 31-bus RDS 63

Figure 317 The RCM-1 of the 31-bus RDS 64

Figure 318 The SBM of the 31-bus RDS 65

Figure 319 The BSM of the 31-bus RDS 66

Figure 3 20 90-Bus RDS 71

Figure 321 69-bus multi-feeder RDS 72

Figure 322 Komamoto 15-bus RDS 73

Figure 323 28-bus weakly meshed distribution network 75

Figure 324 mRCM for 28-bus weakly meshed distribution network 75

Figure 325 mSBM for 28-bus weakly meshed distribution network 76

Figure 326 C for 28-bus weakly meshed distribution network 76

Figure 327 70-bus meshed distribution system 78

Figure 328 201-bus hybrid augmented test distribution system 80

Figure 329 10-bus three-phase unbalanced RDS 81

Figure 330 The 10-bus three-phase unbalanced RDS RCM 82

xni

Figure 331 21 The 10-bus three-phase unbalanced RDS RCM1 83

Figure 332 The 10-bus three-phase unbalanced RDS SBM 84

Figure 333 The 10-bus three-phase unbalanced RDS BSM 85

Figure 334 IEEE 13-bus 3^ unbalanced RDS 86

Figure 41 The Conventional SQP Algorithm 107

Figure 42 The FSQP Algorithm 112

Figure 43 Case 1 33-busRDS 114

Figure 44 Optimal real power losses for placement of optimal DG size at all the 32

buses using APC method 117

Figure 45 Optimal real power losses vs different DG power factors at bus 30 117

Figure 46 Bus voltages improvement before and after installing a single DG at

bus 30 118

Figure 47 Voltage profiles comparisons of 33-bus RDS cases 120

Figure 48 Voltage improvement of the 33-bus RDS due to three DG installation

compared to pre-DG single and double-DG cases 122

Figure 49 Voltage profiles improvement in 33-bus RDS for all DG cases 124

Figure 410 Case 2 69-bus RDS test case 125

Figure 411 Optimal power losses obtained using APC procedure 126

Figure 412 Real power losses vs DG power factor 69-bus RDS 128

Figure 413 Bus voltage improvements via single DG installation in the 69-bus

RDS 129

Figure 414 Variation in power losses as a function of the DG output at bus 61 129

Figure 415 Bus voltage magnitudes of the original 69-bus RDS and Single DG

and double DGs cases 131

Figure 416 69-bus RDS bus voltage magnitudes improvement in all DG cases 133

Figure 51 Number of publications in IEEEIET and ScienceDirect Databases

since the year 2000 140

Figure 52 Interaction between particles to share the gbest information 150

Figure 53 Illustration of velocity and position updates mechanism for a single particle during iteration k 151

Figure 54 PSO particle i updates its velocity and position vectors during two consecutive iterations A and k+ 152

Figure 55 The proposed HPSO solution methodology 159

xiv

Figure 56 Convergence characteristic of the proposed HPSO in the fixedsingle DG case HPSO proposed number of iterations = 30 164

Figure 57 Convergence characteristic of the proposed HPSO in the fixedsingle

DG case HPSO extended number of iterations = 50 165

Figure 58 Swarm particles on the first HPSO iteration 165

Figure 59 Swarm particles on the fifth HPSO iteration 166

Figure 510 Swarm particles on the tenth HPSO iteration 166

Figure 511 Swarm particles on 15th HPSO iteration 167

Figure 512 Swarm particles on the 20 HPSO iteration 167

Figure 513 Swarm Particles on the 25th HPSO iteration 168

Figure 514 Swarm Particles on the last HPSO iteration 168

Figure 515 A close-up for the particles on the 30th HPSO iteration 169

Figure 516 Convergence characteristics of HPSO in the 69-bus fixedsingle DG case HPSO proposed number of iterations = 15 184

Figure 517 Convergence characteristics of HPSO in the 69-bus fixed single DG

case HPSO proposed number of iterations = 50 185

Figure 518 Swarm particles distribution at the first HPSO iteration 185

Figure 519 Swarm particles distribution at the 5 HPSO iteration 186

Figure 520 Swarm particles distribution at the 10 HPSO iteration 186

Figure 521 Swarm particles distribution at the 15l HPSO iteration 187

Figure 522 Close up of the HSPO particles at iteration 15 187

xv

ABSTRACT

Distribution Generation (DG) has gained increasing popularity as a viable element of electric power systems DG as small scale generation sources located at or near load center is usually deployed within the Distribution System (DS) Deployment of DG has many positive impacts such as reducing transmission and distribution network congestion deferring costly upgrades and improving the overall system performance by reducing power losses and enhancing voltage profiles To achieve the most from DG installation the DG has to be optimally placed and sized In this thesis the DG integration problem for single and multiple installations is handled via deterministic and heuristic methods where the results of the former technique are used to validate and to be compared with the latters outcomes

The unique structure of the radial distribution system is exploited in developing a Fast and Flexible Radial Power Flow (FFRPF) method that accommodates the DS distinct features Only one building block bus-bus oriented data matrix is needed to perform the proposed FFRPF method Two direct descendent matrices are utilized in conducting the backwardforward sweep employed in the FFRPF technique The proposed method was tested using several DSs against other conventional and distribution power flow methods Furthermore the FFRPF method is incorporated within Sequential Quadratic Programming (SQP) method and Particle Swarm Optimization (PSO) metaheuristic method to satisfy the power flow equality constraints

In the deterministic solution method the sizing of the DG is formulated as a constrained nonlinear optimization problem with the distribution active power losses as the objective function to be minimized subject to nonlinear equality and inequality constraints Such a problem is handled by the developed Fast Sequential Quadratic Programming method (FSQP) The proposed deterministic method is an improved version of the conventional SQP that utilizes the FFRPF method in handling the power flow equality constraints Such hybridization resultes in a more robust solution method and drastically reduces the computational time In a subsequent step the placement portion of the DG integration problem is dealt with by using an All Possible Combinations (APC) search method Afterward the FSQP methods outcomes were compared to those of the developed metaheuristic optimization method

The difficult nature of the overall problem poses a serious challenge to most derivative based optimization methods due to the discrete nature associated with the bus location Moreover a major drawback of deterministic methods is that they are highly-dependent on the initial solution point As such a new application of the PSO metaheuristic method in the DG optimal planning area is presented in this thesis The PSO is improved in order to handle both real and integer variables of the DG mixed-integer nonlinear constrained optimization problem The algorithm is utilized to simultaneously search for both the optimal IDG size and bus location The proposed approach hybridizes PSO with the developed FFRPF algorithm to satisfy the equality constraints The inequality constraints handling mechanism is dealt with in the proposed Hybrid PSO (HPSO) by combining the rejecting infeasible solutions method with the preserving feasible solutions method Results signify the potential of the developed algorithms with regard to the addressed problems commonly encountered in DS

xvi

LIST OF ABBREVIATIONS AND SYMBOLS USED

ACO

BFGS

BSM

CHP

CIGRE

CN

DER

DG

DG

DGs

DS

EG

EP

EPAct

EPRI

FD

FFRPF

FSQP

GA

GRG

GS

GWEC

HPSO

IP

KCL

KKT

KVL

LP

Ant Colony Optimization

Quasi-Newton method for Approximating and Updating the Hessian Matrix

Bus Section Matrix

Combined-Heat and Power

The International Council on Large Electric Systems

Condition Number

Distribution Energy Resources

Dispersed Generation

Decentralized Generation

Distribution Generation sources

Distribution System

Embedded Generation

Evolutionary Programming

English Policy Act of 1992

Electric Power Research Institute

Fast Decoupled

Fast and Flexible Radial Power Flow

Fast Sequential Quadratic Programming

Genetic Algorithm

Generalized Reduced Gradient

Gauss-Seidel

Global Wind Energy Council

Hybrid PSO

Interior Point method

Kirchhoff s Current Law

Karush-Khun-Tuker conditions

Kirchhoff s Voltage Law

Linear Programming

xvii

wBSM

mNS

wRCM

mRCM

mSBM

mSBMp

NB

NB

HDG

riL

NR

NS

NS

ftwDG

Pf PSO

PUHCA

PURPA

QP

RCM

RDS

RIT

RPF

SA

SBM

SE Mean

SQP

StDev

TS

UnSpec pf

Meshed BSM

Number of segments in meshed DS

Meshed RCM

Modified mRCM

Meshed SBM

Submatrix of wSBM that correspond to the RDS tree sections

Number of Buses

Number of DS Buses

Total number of DGs

Number of Links or number of the fundamental loops

Newton-Raphson

Number of Sections

Number of Sections in RDS AND in meshed DS tree

Total number of the unspecified pf DGs

power factor

Particle Swarm Optimization

Public Utilities Holding Company Act of 1935

Public Utilities Regulatory Policy Act of 1978

Quadratic Programming

Radial Configuration Matrix

Radial Distribution System

The Reduction in CPU execution Time

Radial Power Flow

Simulated Annealing

Section Bus Matrix

Standard Error of the Mean

Sequential Quadratic Programming

Standard Deviation

Tabu search algorithm

Unspecified power factor DG

xviii

U S P B Unique Set of Phase Buses

USPS Unique Set of Phase Sections

xf Unique set of phase buses

iff Unique set of phase sections

Zsec Section primitive impedance matrix

Z^ (3 X 3) section symmetrical impedance matrix

R D S section length

zu Per unit length self-impedance of conductor i

h Per unit length mutual- impedance be tween conductors a n d

rt Resis tance of conductor i

rd Ear th return conductor resistance

k Inductance multiplying constant

De Dis tance between overhead and its earth return counterpart

GMRj Geometr ic mean radius of conductor i

Dy Dis tance between conductors a n d

Vgbc Three-phase sending end voltages

Vg deg Three-phase receiving end voltages

Ias c Three-phase sending end section currents

lfc Three-phase receiving end section currents

Fscc Three-phase shunt admittance of section k

[]3x3 (3 X 3) identity matrix

[^Lx3 (3x3) zero matrix

^Klc Vol tage drop across three-phase section k

ysect Section k three-phase currents

V0 Nomina l bus voltage

V Operat ing bus voltage

xix

P0 Real power consumed at nominal voltage

Q0 Reactive power consumed at nominal voltage

S Bus load apparent power at single-phase bus sect

YsKus Total three-phase shunt admittance at bus i

Ic Three-phase shunt currents at bus i

IlucSi Bus three-phase currents

jabc Three-phase load current

IltLP Current through single-phase section p and phase ltjgt

its Current at bus and phase ^

Vss Substation voltage magnitude

Vls Substation complex phase voltage

VLt Voltage drop across section k in phase (j)

A and symbol

IMI oo-norm vector II I loo

91 (bull) Real part of complex value

3 (bull) Imaginary part of complex value

C Fundamental loop matrix which is a submatrix of mSBM

Zioop (laquoLx nL) loop-impedance matrix

Csec Upper submatrix of the C matrix with ((NB-1) x (nL) dimension

Zoop Loop-impedance matrix

setrade (NSxNS) meshed DS section-impedance diagonal matrix

ZtradeS (NSxNS) RDS or tree section-impedance diagonal matrix

IL (NB-1 x 1) RDS bus load currents vector

fnlsec (mNS x 1) segments currents column vector of meshed DS network vector

mILL (mNSx 1) meshed DS bus loads and links currents vector

Itrade (NB-1) tree section currents column

xx

( n L x 1) fundamental loop current vector which is also the meshed DS link loop

currents column vector

B ( N B - 1 xmNS) fundamental cut-sets matrix

^ s7 e c f a ( N B - 1 x nL) co-tree cut-set matrix

^ymesh Voltage drops across the tree sections of the meshed DS vector

ymesh j k g messed DS bus voltage profiles vector

PRPL Real power losses

Pj Generated power delivered to DS bus i

PjL Load power supplied by DS bus i

Yjj Magnitude of the if1 element admittance bus matrix

rv Phase angle of Yy = YyZyy

Vi Magnitude of DS bus complex voltage

8 Phase angle of V = ViZSl

bull Transpose of vector or matrix

bull Complex conjugate of vector or matrix

V (1 xNB) DS bus Thevenin voltages

Y (NB xNB) DS admittance matrix

A^ Real power mismatch at bus i

AQt Reactive power mismatch at bus i

|L| Infinity norm where llxll = max (|x|) II llco J II llraquo =l2iVBVI ]gt

bull+ Max imum permissible value

bull Minimum permissible value

bull0 Nominal value

PDG D G operating power factor

S^G D G generated apparent power

SsS Main DS substation apparent power

1 Scalar related to the allowable D G size

xxi

Sy Apparent power flow transmitted from bus to bus j

Stradex Apparent power maximum rating for distribution section if

(x) The objective function

z(x) Equality constraints

g(x) Inequality constraints

(bull) Independent unknown variables lower bounds

(bull) Independent unknown variables upper bounds

x Independent unknown variables vector

RPL ( X ) Distribution system real power losses objective function

d Search direction vector

a Positive step size scalar

WRPL (x ) Gradient of the objective function at point xk)

pound Lagrange function

H^ (nxri) Hessian symmetric matrix at point xw

h^ First-order Taylors expansion of the equality constraints at point xw

Vh(x^) (nxm) Jacobian matrix of the equality constraints at point xw

g ^ First-order Taylors expansion of the inequality constraints at point xw

Vg(x^) (nxp) Jacobian matrix of the inequality constraints at point xw

Xi Individual equality Lagrange multiplier scalar

Pi Individual inequality Lagrange multiplier scalar

k w-dimensional equality Lagrange multiplier vector

P (-dimensional inequality Lagrange multiplier vector

s A predefined small tolerance number

A Active set

m Number of all equality constraints

p Number of all inequality constraints

a Number of the active set equations

xxii

v 2 j6k)

XX

nTgtG

nuDG

y

v Y FFRPFbl

deg FFRPF bl

llAP II II lloo

Vi

Xi

Cj C2

rXgtr2

w

pbestj

gbesti

nk

X

APT Losses

pHPSO Losses

pFSQP Losses

Hessian of the Lagrange function

Total number of DGs

Total number of the unspecified DGs

The change in the Lagrange functions between two successive iterations

Voltage magnitude of bus i obtained by the FFRPF technique

Voltage phase angle of bus obtained by the FFRPF technique

Voltage deviation infinity norm ie II AV = max ( I F - K I ) deg llcD

=UAlaquoVI deg ngt

Particle i velocity

Particle i position vector

Individual and social acceleration positive constants

Random values in the range [0 l] sampled from a uniform distribution

Weight inertia

Personal best position associated with particle own experience

Global best position associated with the whole neighborhood experience

Maximum number of iterations

Constriction factor

The deviation of losses calculated by HPSO method from that determined

by FSQP method

Mean value of HPSO simulation results of real power losses

FSQP deterministic method result of real power losses

xxiii

ACKNOWLEDGEMENTS

All Praise and Thanks be to reg (Allah) Almighty whose countless bounties enabled me to

accomplish this thesis successfully I would like to express my deepest gratitude to my

parents who taught me the value of education and hard work A special note of gratitude

to my brothers Falah and Abdullah deer sisters my wife my daughter Najla and my

sons Fahad Falah and Othamn They endured the long road along with me and

provided me with constant support motivation and encouragement during the course of

my study

I would like to express my sincere gratitude to my advisor Dr M E El-Hawary for

his professional guidance valuable advice continual support and encouragement I also

appreciate the constructive comments of my PhD External Examiner Dr M A Rahman

I am also grateful to my advisory committee members Dr T Little and Dr W Phillips

for spending their valuable time in reading evaluating and discussing my thesis

I would like to acknowledge the academic discussions and the constant

encouragement I received from my dear friend Dr Mohammed AlRashidi- Thank you

Abo Rsheed I wish also to thank a special friend of mine Dr AbdulRahman Al-

Othman for his friendship and for believing in me

I would like to manifest my gratitude to the Public Authority for Applied Education

and Training in Kuwait who sponsored me through my PhD at Dalhousie University

From the Embassy of Kuwait Cultural Attache Office special thanks are due to Shoghig

Sahakyan for her efforts and help to make this work possible

xxiv

CHAPTER 1 INTRODUCTION

11 MOTIVATION

Electric power system networks are composed typically of four major subsystems

generation transmission distribution and utilizations Distribution networks link the

generated power to the end user Transmission and distribution networks share similar

functionality both transfer electric energy at different levels from one point to another

however their network topologies and characteristics are quite different Distribution

networks are well-known for their low XR ratio and significant voltage drop that could

cause substantial power losses along the feeders It is estimated that as much as 13 of

the total power generation is lost in the distribution networks [1] Of the total electric

power system real power losses approximately 70 are associated with the distribution

level [23] In an effort towards manifesting the seriousness of such losses Azim et al

reported that 23 of the total generated power in the Republic of India is lost in the form

of losses in transmission and distribution [4]

Distribution systems usually encompass distribution feeders configured radially and

exclusively fed by a utility substation Incorporating Distribution Generation sources

(DGs) within the distribution level have an overall positive impact towards reducing the

losses as well as improving the network voltage profiles Due to advances in small

generation technologies electric utilities have begun to change their electric

infrastructure and have started adapting on-site multiple small and dispersed DGs In

order to maximize the benefits obtained by integrating DGs within the distribution

system careful attention has to be paid to their placement as well as to the appropriate

amount of power that is injected by the utilized DGs In other words to achieve the best

results of DG deployments the DGs are to be both optimally placed and sized in the

corresponding distribution network

The motivation of this thesis research is to investigate placing and sizing single and

multiple DGs in Radial Distribution Systems (RDSs) The problem investigated involves

two stages finding the optimal DG placements in the distribution network and the

optimal size or rating of such DGs The optimal DG placement and sizing are dealt with

by utilizing deterministic and heuristic optimization methods

12 DISTRIBUTION GENERATION - HISTORIC OVERVIEW

During the first third of the twentieth century there were no restrictions on how many

utility companies could be owned by financial corporations known as utility holding

companies By 1929 80 of US electricity was controlled by 16 holding companies

and three of those corporations controlled 36 of the nations electricity market [5]

During the Great Depression most of these utility holding companies went bankrupt As

a result the US Congress Public Utilities Holding Company Act (PUHCA) of 1935

regulated the gas and electric industries and restricted holding companies to the

ownership of a single integrated utility PUHCA indirectly discouraged wholesale

wheeling of power between different states provinces or even countries The Public

Utilities Regulatory Policy Act (PURPA) of 1978 allowed grid interconnection and

required electric utilities to buy electricity from non-utility-owned entities called

Qualifying Facilities (QF) at each utilitys avoided cost The term QF refers to non-

utility-owned (independent) power generators The term at each utilitys avoided cost

is interpreted to mean that the utility shall buy the generated electricity at a price

equivalent to what it would cost the utility itself if had generated the same amount of

power in its own facility or if it had purchased the power from an open electricity market

ie what the utility saves by not generating the same amount of power This act heralded

the dawn of the DG industry era which paved the way to generate electricity arguably at

a lower cost compared to that of traditional utility companies and consequently have it

delivered to the end user at lower rates The English Policy Act of 1992 (EPAct)

intensified competition in the wholesale electricity market by opening the transmission

system for access by utilities and non-utilities electricity producers [67] entity A could

sell its power to entity B through entity Cs transmission infrastructure

13 DISTRIBUTION GENERATION

DG involves small-scale generation sources scattered within the distribution system level

atnear the load center ie close to where the most energy is consumed [8] The DG

2

generate electricity locally and in a cogeneration case heat can also be generated and

may be utilized in applications such as industrial process heating or space heating DG

generally has better energy efficiency than large-scale power plants The traditional

power stations usually have an efficiency of around 35 whereas the efficiency of DG

such as a Combined Heat and Power (CHP) gas turbine would be in the vicinity of 45-

65 [5]

It seems that there is no universal agreement on the definition of DG size range The

Electric Power Research Institute (EPRI) for example defined the DG size to be up to 5

MW in 1998 [9] in 2001 EPRI redefined the DG capacity to be less than 10 MW [10]

and by 2003 they identified the DG to have a power output ranging from 1 kW to 20 MW

[11] The IEEE published its DG-standards IEEE Std 1547-2003 and IEEE Std 15473-

2007 and emphasized that they are applicable to DGs that have total capacity below 10

MVA [1213] In its 2006 report about the impact of DG Natural Resources Canada

estimated that the DG size starts from few 10s of kW to perhaps 5 MW [14] In 2000

the International Council on Large Electric Systems (CIGRE) referred to the DG as non-

centrally dispatched usually attached to distribution level and smaller than 50-100 MW

[1516]

Many terms referring to DG technology are used in the literature such as Dispersed

Generation (DG) Decentralized Generation (DG) Embedded Generation (EG)

Distribution Energy Resources (DER) and on-site Generation [17] In particular the

term dispersed generation customarily refers to stationary small-scale DG with power

outputs ranging from 1 kW to 500 kW [7]

Late developments and innovations in the DG technology industry liberalization of

the electricity market transmission line congestion and increasing interest in global

warming and environmental issues expedited publicizing their deployment and adoption

world-wide Recent studies suggest that DG will play a vital role in the electric power

system An EPRI study predicts that by the year 2010 25 of the newly installed

generation systems will be DG [18] and a similar study by the Natural Gas Foundation

projects that the share of DG in new generation will be 30 [15] By 2003 around 40

of Denmarks power demand was served by DG while Spain the Netherlands Portugal

and Germany integrated nearly 20) of DG into their distribution networks [19] Of the

3

643 GW generated by the European Union in 2005 approximately 122 GW (19) was

generated by hydro 96 GW (15) of this generated capacity was cogeneration (CHP)

and 53 GW (8) generated by other renewable energy systems Half of the CHP

generated capacity was owned by utility companies and the other half was generated by

independent producers [20]

Globally in 2005 the total installed wind power capacity was 591 GW and the

Global Wind Energy Council (GWEC) expected the wind capacity to reach 1348 GW by

the year 2010 [21] Worldwide wind energy capacity of 19696 MW was added in the

year 2007 [22] and approximately 1400 MW of wind energy capacity was added in the

US during the second quarter of 2008 [23] GWEC also predicted in its Global Wind

Energy Outlook 2008 report that by the year 2020 15 billion tons of CO2 will be saved

every year and by the middle of the 21st century 30 of the worlds electricity will be

supplied by wind energy [24] compared to a total of 13 of the global electricity being

generated by wind at the end of 2007 [22]

DG technologies include a variety of energy sources ie powered by renewable or

by fossil fuel-based prime movers Renewable technologies used in DG include wind

turbines photovoltaic cells small hydro power turbines and solar thermal technologies

while DG based on conventional technologies may involve gas turbines CHP gas

turbines diesel engines fuel cells and micro-turbine technologies Some DGs are

installed by the utility company on the supply side of the consumers meter while some

are installed by the customers themselves on their side of a bi-directional meter thus

enabling them to benefit from the net-metering program offered by utility companies

[25]

Optimal deployment of DG technology would have an overall positive impact

although some negative traits would remain The noise and shadow flicker caused by

large wind blades and the noise caused by the wind turbine gearbox or gas turbines

especially when placed close to residential or populated areas are examples of negative

impacts of widespread use of DG Another drawback from an environmentalist point of

view is that wind DG could disturb bird immigration patterns and cause death to both

birds and bats [26] Renewable-source DGs also could be an indirect source of pollution

by causing the fossil-fuel power plants to shut down and start up more frequently as they

4

attempt to accommodate variable DG power output [27] Some plants have an emission

rate which is inversely proportional to its delivered power Voltage rise as a result of bishy

directional power flow caused by the interconnection of the DG in RDS is another

example of a negative impact caused by DG [28]

The integration of DG into electric power networks has many benefits Some

examples of such benefits could be summarized as follows

bull Improve both the reliability and efficiency of the power supply

bull Release the available capacity of the distribution substation as well as reducing

thermal stresses caused by loaded substations transformers and feeders

bull Improve the system voltage profiles as well as the load factor

bull Decrease the overall system losses

bull Generally DG development and construction have shorter time intervals

bull Delay imminent upgrading of the present system or the need to build newer

infrastructure and subsequently avoid related problems such as right-of-way

concerns

bull Decrease transmission and distribution related costs

bull In general DG tends to be more environmentally friendly when compared to

traditional coal oil or gas fired power plants

The extent of the benefits depends on how the DG is placed and sized in the system In

addition to supplying the system with the power needed to meet certain demands as an

installation incentive the real power losses could be minimal if the DG is optimally sited

and sized

14 THESIS OBJECTIVES AND CONTRIBUTIONS

Optimal integration of single and multiple DG units in the distribution network with

specified and unspecified power factors is thoroughly investigated from a planning

perspective in this thesis The DG problem is handled via deterministic and heuristic

optimization methods where the results of the former method are used to validate and to

be compared with those of the latter

The unique radial distribution structure is exploited in developing a Fast and Flexible

Radial Power Flow (FFRPF) method to deal with a wide class of distribution systems

5

eg radial meshed small large balanced and unbalanced three-phase networks The

proposed FFRPF algorithm starts by developing a Radial Configuration Matrix (RCM)

for radial topology DS or meshed RCM (mRCM) for meshed DS Both matrices consist

of elements with values 1 0 and -1 The RCM (or mRCM) corresponding building

algorithm is simple fast and practical as illustrated in Chapter 3 The RCM is inverted

only once to produce the Section Bus Matrix (SBM) which is then transposed to obtain

the Bus Section Matrix (BSM) For the meshed topology the corresponding resultant

matrices for the mRCM are the meshed Bus Section Matrix (mSBM) and the meshed Bus

Section Matrix (mBSM) The FFRPF technique relies on the backwardforward sweep

that only utilizes the basic electric laws such as Ohms law and both Kirchhoff s voltage

and current laws The backward current sweep is performed via SBM (or mSBM) and

the forward voltage update sweep is executed via the BSM (or mBSM) By utilizing the

two obtained matrices all the bus complex voltages can be obtained and consequently

left to be compared with the immediate previous obtained bus voltages The proposed

approach quickened the iterative process and reduced the CPU time for convergence It

is worth mentioning that the building block matrix is the only input data required by the

FFRPF method besides the DS parameters to perform the proposed distribution power

flow The FFRPF technique is incorporated in both utilized deterministic and

metaheuristic optimization methods to satisfy the power flow equality constraints

requirements

In the deterministic solution method the DG sizing problem is formulated as a

nonlinear optimization problem with the distribution active power losses as the objective

function to be minimized subject to nonlinear equality and inequality constraints

Endeavoring to obtain the optimal DG size an improved version of the Sequential

Quadratic Programming (SQP) methodology is used to solve for the DG size problem

The conventional SQP uses a Newton-like method which consequently utilizes the

Jacobean in handling the nonlinear equality constraints The radial low XR ratio and the

tree-like topology of distribution systems make the system ill-conditioned

A Fast Sequential Quadratic Programming (FSQP) methodology is developed in

order to handle the DG sizing nonlinear optimization problem The FSQP hybrid

approach integrates the FFRPF within the conventional SQP in solving the highly

6

nonlinear equality constraints By utilizing the FFRPF in dealing with equality

constraints instead of the Newton method the burden of calculating the Jacobean and

consequently its inverse as well as the complications of the ill-conditioned Y-matrix of

the RDS is eliminated Another advantage of this hybridization is the drastic reduction

of computational time compared to that consumed by the conventional SQP method

In this thesis a new application of the Particle Swarm Optimization (PSO) method in

the optimal planning of single and multiple DGs in distribution networks is also

presented The algorithm is utilized to simultaneously search for both the optimal DG

size and its corresponding bus location in order to minimize the total network power

losses while satisfying the constraints imposed on the system The proposed approach

hybridizes PSO with the developed distribution radial power flow ie FFRPF to

simultaneously solve the optimal DG placement and sizing problem The difficult nature

of the overall problem poses a serious challenge to most derivative based optimization

methods due to the discrete flavor associated with the bus location in addition to the

subproblem of determining the most suitable DG size Moreover a major drawback of

the deterministic methods is that they are highly-dependent on the initial solution point

The developed PSO is improved in order to handle both real and integer variables of the

DG mixed-integer nonlinear constrained optimization problem Problem constraints are

handled within the proposed approach based on their category The equality constraints

ie power flows are satisfied through the FFRPF subroutine while the inequality bounds

and constraints are treated by exploiting the intrinsic and unique features associated with

each particle The proposed inequality constraint handling technique hybridizes the

rejection of infeasible solutions method in conjunction with the preservation of feasible

solutions method One advantage of this constraint handling mechanism is that it

expedites the solution method converging time of the Hybrid PSO (HPSO)

15 THESIS OUTL INE

This thesis is organized in six chapters The research motivation brief description of the

DG and the thesis objectives are addressed in the first chapter The second chapter deals

with a literature review of the distribution power flow methods and the DG optimal

planning problem In the third chapter development of the proposed FFRPF method

7

utilized in the FSQP and the HPSO methods to satisfy the DG problem of nonlinear

equality constraints is presented The fourth chapter deals with the DG sizing problem

formulation and its solution based on the two deterministic solution methods The

problem is solved via the conventional SQP and the proposed FSQP methods and a

performance comparison between them is presented Basic concepts of the PSO are

presented in chapter five A brief literature review regarding the use of the PSO in

solving the electric power system problems is presented in this chapter In addition it

also addresses the development of the proposed HPSO in solving the DG planning

problem The last chapter provides the thesis concluding remarks and the scope of future

work

8

CHAPTER 2 LITERATURE REVIEW

21 INTRODUCTION

Recent publications in the areas of work relative to this thesis are reviewed and

summarised in this chapter which is organized in two sections as follows

bull The first section reviews the literature on distribution power flow methods A

brief background of conventional power flow methods is presented followed

by a review and summary of the literature on recent developments of the

distribution power flow algorithms

bull The DG integration problem is reviewed in the second section Recent work

on the optimal DG placement and sizing via analytical deterministic and

metaheuristic methods are analyzed and reviewed

22 DISTRIBUTION POWER FLOW

Power flow programs play a vital role in analyzing power systems The problem deals

with calculating unspecified bus voltage angles and magnitudes active and reactive

powers as well as (as a by-product) line loadings and their associated real and reactive

losses for certain operating conditions These values are typically obtained through

iterative numerical methods to analyze the status of a given power system

Since the middle of last century many methods were proposed to solve this problem

Even though Dunstan [29] was the first to demonstrate a digital method for solving the

power flow problem in 1954 Ward and Hale [30] are often credited with the successful

digital formulation and solution of the power flow problem in 1956 Most of the earlier

solution methods were based on both the admittance matrix and the Gauss-Seidel (GS)

iterative method The poor convergence characteristics of GS when large networks

andor ill-conditioned situations are encountered led to the development of the Gaussian

iterative scheme (Zbus) [3132] and later the Newton-Raphson (NR) method [33] as well

as the Decoupled [34] and Fast Decoupled (FD) power flow approaches [35] Though

the NR method generally converges faster than other methods it takes longer

computational time per iteration When Tinney et al [36] introduced the optimally

9

ordered and sparsity-oriented programming techniques Newton-based methods became

the de facto industry standard However the Jacobian matrix for the RDS is

approximately four times the size of the corresponding admittance matrix and it needs to

be evaluated at each iteration

Although conventional power flow methods are well developed in dealing with the

transmission and sub-transmission sections of the power system networks they are

deemed to be inefficient in handling distribution networks This is because the

Distribution System (DS) is different in several ways from its transmission counterpart

DS has a strictly radial topology nature or weakly meshed networks in contrast with

transmission systems which are tightly meshed networks DS is a low voltage system

having low XR ratio sections and a wide range of reactance and resistance values DS

may consist of a tremendously large number of sections and buses spread throughout the

network Sections of the DS could have unbalanced load conditions due to the

unbalanced three-phase loading as well as single and double phase loads in spurred

lateral lines The mutual couplings between phases are not negligible due to rarely

transposed distribution lines [37] All of these characteristics strongly suggest that DS is

to be classified as an ill-conditioned power system

The practical DSs low XR ratio sections may cause both the NR and FD

conventional methods to diverge [38-41] The line impedance angles are small enough to

deteriorate the dominance of the NR Jacobian main diagonal making it prone to

singularity Such a low XR value would also prevent the Jacobian matrix from being

decoupled and simplified

In addition to performance considerations a practical power flow technique needs to

consider all the DS distinctive features and to accommodate the imbalance introduced by

multiphase networks along with the distribution-level loads In the literature a number

of Newton and non-Newton power flow methods designed for distribution systems were

proposed Zhang et al [42] solved the distribution power flow based on the Newton

method although the proposed Jacobian is computed just once the solution converged

with a number of additional iterations more so than the conventional approach

Moreover shunt capacitor banks were ignored in the modeling as well as the line shunt

admittance (JI model) and the constant impedance loads Baran and Wu [43] solved the

10

power flow problem by utilizing three fundamental quadratic equations representing the

real and reactive section powers and the bus voltages in an iterative scheme as a

subroutine during the process of optimizing the capacitor sizing However they

computed the Jacobian using the chain rule within the proposed NR method which is in

turn time consuming Mekhamer et al [44] utilized the three equations developed in [43]

using a different iterative technique without the need for the Jacobian or the NR method

However their process is based on applying a multi-level iterative process on the main

feeder and laterals which makes the speed and the efficiency of their proposed algorithm

a function of the RDS configuration and topology

In [4546] the quadratic equation was also utilized in determining the relation

between the sending and receiving end voltage magnitudes along with the section power

flow They proposed to include the system power losses within their calculation while

solving for the system power flow However the voltage phase angles were ignored

during the solution of the radial power flow in order to speed up the convergence The

latter reference developed work was based on the assumption of balanced RDS and

sophisticated numbering scheme

The radial power flow introduced by [47-49] used a non-Newton power flow techshy

nique based on the ladder network theory This method adds the section currents and

calculates the RDS bus voltages including the substations during a backward sweep If

the difference between the calculated substation voltage value and substation predetershy

mined assigned bus voltage value is acceptable the iterations are concluded If not the

substation bus voltage is reset and the RDS bus voltages are computed for the second

time in the same iteration in the forward sweep Both the ladder and the backshy

wardforward methods are derivative-free instead they employ simple circuit laws

However the ladder method uses many sub-iterations on the laterals and calculates the

system bus voltages twice during a single iteration compared to once in the backshy

wardforward method Thukaram [50] utilized the backwardforward sweep technique to

solve the RDS power flow However the bus numbering procedure was a sophisticated

parent node and child node arrangement which may add some computational overshy

head if the system topology is changed Teng [51] used the backwardforward approach

as the solution procedure through the development of two matrices and multiplied them

11

together in a later stage of the solution process In assembling those matrices all the

system buses and sections have to be inspected carefully In a practical large RDS data

preparation for these matrices will be cumbersome and prone to errors Under continshy

gency situations switching operations or the addition of another feeder to the existing

one are quite common practices in the DSs hence changes in system topology need to be

accommodated by restructuring the corresponding matrices which would add an overshy

head to track modifications The weakly meshed DS was dealt with by adding extra

nodes in the middle of the new links Two equal currents with opposite polarities were

injected into each added node Each injection operation is represented by a two column

matrix which was subsequently added to the first proposed matrix and then the develshy

oped matrices were extended and multiplied together The resultant is a full matrix and

its dimension is reduced by the Kron method in every single iteration That is the

developed full matrix was inverted in each iteration of the solution method and such

procedure is expensive lengthy cumbersome and time consuming

Shirmohammadi et al [39] Cheng and Shirmohammadi [52] and Hague [53]

proposed an iterative solution method for both radial and weakly meshed DSs This

approach necessitates a special numbering scheme in which they number the DS sections

in layers starting from the root node The numbering scheme is to be carried out

carefully by examining the whole system when a new layer is to be numbered The

numbering process is cumbersome and prone to errors For weakly meshed networks

breakpoints are selected opened and consequently the meshed system is converted to a

radial system The loops are broken by adding two fictitious buses In each pair of

dummy buses equal and opposite currents are injected and the new system is evaluated

to produce a reduced order impedance matrix Their proposed method requires that the

breakpoint impedance matrix should be computed cautiously Such a procedure is highly

dependent on the distribution networks topology That is the more links that exist in the

DS the larger the break point impedance matrix and the more time will be consumed in

its computation

Goswami and Basu [38] introduced a direct solution method to solve for radial and

weakly meshed DS They applied a breakpoints method into the meshed DS similar to

that of [39] in order to convert it into RDS In their proposed methodology a restriction

12

was imposed on each of the system buses not to have more than three sections attached to

it Such limitation is a drawback of the method and moreover a difficult node numbering

scheme is a disadvantage

In this thesis the unique structure of the RDS is exploited in order to build up a new

fast flexible power flow technique that deals with radial and looped DSs The numbering

scheme of the DS is simple and straightforward All load types can be accommodated by

the proposed distribution power flow eg spot and distributed loads Unlike

conventional power flow methods no trigonometric functions are used in the proposed

distribution power flow method For weakly meshed and looped DSs the system is dealt

with as it is there is no need for radialization cuts or building breakpoints impedance

matrix The topology of the tested DS whether strictly radial weakly meshed or looped

is represented by a building block matrix which is the only one needed to perform the

backwardforward sweep technique

23 DG INTEGRATION PROBLEM

DG is gaining increasing popularity as a viable element of electric power systems The

presence of DG in power systems may lead to several advantages such as supplying

sensitive loads in case of power outages reducing transmission and distribution networks

congestion and improving the overall system performance by reducing power losses and

enhancing voltage profiles Some of the negative impacts of DG installations are

potential harmonic injections the need to adopt more complex control schemes and the

possibility of encountering reverse power flows in power networks Even though the

concept of DG utilization in electric power grids is not new the importance of such

deployment is presently at its highest levels due to various reasons Recent awareness of

conventionaltraditional thermal power plants harmful impacts on the environment and

the urge to find more environmentally friendly substitutes for electrical power generation

rapid advances made in renewable energy technologies and the attractive and open

electric power market are a few major motives that led to the high penetration of DG in

most industrial nations power grids To achieve the most from DG installation special

attention must be made to DG placement and sizing

13

The problem of optimal DG placement and sizing is divided into two subproblems

where is the optimal location for DG placement and how to select the most suitable size

Many researchers proposed different methods such as analytic procedures as well as

deterministic and heuristic methods to solve the problem

231 Solving the DG Integration Problem via Analytical and Deterministic Methods

In the literature the optimal DG integration problem is solved by means of employing

any analytical or optimization technique that suits the problem formulation Methods and

procedures of optimally sizing and locating the DGs within the DS are varied according

to objectives and solution techniques

Willis [54] presented an application of the famous 23 rule originally developed

for optimal capacitor placement to find a suitable bus candidate for DG placement That

is to install a DG with a rating of 23 of the utilized load at 23 the radial feeder length

down-stream from the source substation However this rule assumes uniformly

distributed loads in a radial configuration and a fixed conductor size throughout the

distribution network In any event the 23 rule was developed for all-reactive load

These assumptions limit its applicability to radial distribution systems and the fact that it

is only suitable for single DG planning

Kashem et al [55] developed an analytical approach to determine the optimal DG

size based on power loss sensitivity analysis Their approach was based on minimizing

the DS power losses The proposed method was tested using a practical distribution

system in Tasmania Australia However it assumes uniformly distributed loads with all

the connected loads along the radial feeder having the same power factor and it also

assumes no external currents injected into the system buses eg capacitors which limits

its practicality

Wang and Nehrir [56] developed an analytical approach to address the optimal DG

placement problem in distribution networks with different continuous load topologies

Minimizing the real power losses was the objective of the proposed method In their

approach the DG units were assumed to have unity power factor and only the overhead

distribution lines with neglected shunt capacitance are considered The candidate bus

was selected based on elements of the admittance matrix power generations and load

14

distribution of the distribution network The issue of DG optimal size was not addressed

in their formulation

Griffin et al [57] analyzed the DG optimal location analytically for two continuous

load distributions types ie uniformly distributed and uniformly increasing loads The

goal of their study was to minimize line losses One of the conclusions of their research

was that the optimal location of DG is highly dependent on the load distribution along the

feeder ie significant loss reduction would take place when placing the DG toward the

end of a uniformly increasing load and in the middle of uniformly distributed load feeder

Acharya et al [58] used the incremental change of the system power losses with

respect to the change of injected real power sensitivity factor developed by Elgerd [59]

This factor was used to determine the bus that would cause the losses to be optimal when

hosting a DG By equating the aforementioned factor to zero the authors solved for the

optimal real value of DG output They proposed an exhaustive search by applying the

sensitivity factor on all the buses and ranked them accordingly The drawback of their

work is the lengthy process of finding the candidate locations and the fact that they

sought to optimize only the DG real power output Furthermore they only considered

planning of a single DG

Popovic et al [60] utilized sensitivity analysis based on the power flow equations to

solve the DG placement and sizing Two indices were used in ranking all the DS buses

for the suitability of hosting the DG The first one is a voltage sensitivity index which is

derived directly from the NR power flow Jacobian inverse the second one exploits the

relation of incremental real power losses with respect to the injected real and reactive

power as developed in [61] Their objective for sizing the DG was to maximize its

capacity subject to boundary constraints such as bus voltage penetration level line flows

and fault current limits To solve the sizing DG problem they gradually increased the

DG capacity at selected most sensitive buses until one of the constraints is violated and

the direct previous installed DG size becomes the one chosen as the optimal rating This

process is a lengthy and impractical procedure and the authors did not elaborate on how

they would deal with multiple DG cases using the proposed scheme

Keane and OMalley [62] solved for the optimal DG size in the Irish system by using

a constrained Linear Programming (LP) approach To cope with the EU regulation which

15

emphasizes that Ireland should provide 132 of its electricity from renewable sources

by 2010 the objective of their proposed method was to maximize the DG generation

The nonlinear constraints were linearized with the goal of utilizing them in the LP

method A DG unit was installed at all the system buses and the candidate buses were

ranked according to their optimal objective function value

Rosehart and Nowicki [63] dealt with only the optimal location portion of the DG

integration problem They developed two formulations to assess the best location for

hosting the DG sources The first is a market based constrained optimal power flow that

minimized the cost of the generated DG power and the second is voltage stability

constrained optimal power flow that maximized the loading factor distance to collapse

Both formulations were solved by utilizing the Interior Point (IP) method The outcomes

of the two formulations were used in ranking the buses for DG installations The optimal

DG size problem was not considered in their paper

Iyer et al [64] employed the primal-dual IP method to find the optimal DG

placement through combined voltage profile improvement and line loss reduction indices

However the proposed approach was based on initially placing DGs at all buses in order

to determine proper locations for DG installations This methodology may not be

realistic for large scale distribution networks

Rau and Wan [65] only treated the DG sizing problem by utilizing the Generalized

Reduced Gradient (GRG) method The DG bus locations were assumed to be provided

by the system planner for the DG units to be installed In their proposed method they

considered minimizing the system active power losses In their formulation only the

power flow equality constraints were considered whereas the boundary conditions and

the inequality constraints were not taken into account

Hedayati et al [66] employed continuous power flow methodology to locate the

buses most sensitive to voltage collapse The sensitive bus set is ranked based on their

severity which is used accordingly to indicate potential bus locations for placement of

single and multiple DG sources An iterative method was proposed for optimally sitting

the DG A certain DG capacity which is known and fixed a priori is added to the DS

and the conventional power flow method was employed to determine the resultant DS

real power losses voltage profiles and power transfer capacity In the subsequent

16

iteration another DG with the same capacity was added to the next sensitive bus and

results were obtained This iterative process would continue until the system outcomes

reached acceptable values The proposed iterative method did not optimize the DG size

232 Solving the DG Integration Problem via Metaheuristic Methods

Metaheuristic techniques have proven their effectiveness in solving optimization

problems with appreciable feasible search space They can be easily modified to cope

with the discrete nature associated with different elements commonly used in power

systems studies Optimization methods such as Genetic Algorithm (GA) [67-71] GA

hybrid methods such as GA and fuzzy set theory [72] and GA and Simulated Annealing

(SA) [73] Tabu search algorithm (TS) [7475] GA-TS hybrid method [76] Ant Colony

Optimization (ACO) [77] Particle Swarm Optimization (PSO) [78] and Evolutionary

Programming (EP) [79] were utilized in the literature to solve for the DG integration

problem

Teng et al [67] developed a value-based method for solving the DG problem The

GA method was utilized in maximizing a DG benefit to cost ratio index subject to only

boundary constraints such as ratio index voltage drop and feeder transfer capacity A

drawback of their procedure is that the candidate DG bus locations were assumed to be

provided by the utility and consequently all combinations of the provided bus locations

were tested for obtaining the optimal DG capacities via the GA method

The proposal set forth by Mithulananthan et al [68] made use of the DS real power

losses as the fitness function to be minimized through GA Their formulation of the DG

size optimization problem is of an unconstrained type Moreover the NR method which

is usually inadequate in dealing with the DS topology was used in calculating the total

power losses Candidate DG bus locations were obtained by placing a DG unit at all

buses of the tested DS which is impractical for large DSs Furthermore the multiple

DGs case was not addressed

Haesen et al [69] and Borges et al [70] solved the DG integration problem by

basically employing the GA method Both utilized the metaheuristic technique in solving

for single and multiple DG sizing and placements Haesen et al used the GA method to

minimize the DS active power flow while the objective for Borges et al was to

17

maximize a DG benefit to total cost ratio index Reference [69] incorporated penalty

factors within the objective function to penalize constraint violations thus adding another

set of variables to be tuned The authors of the latter reference used a PV model for

modeling the DG

Celli et al [71] formulated the DG integration problem as an s-constraint

multiobjective programming problem and solved it using the GA method Their

proposed algorithm divided the set of the objective functions into one master and the rest

are considered as slave objective functions The master is treated as the primary

objective function that is to be minimized while the slaves are regarded as new

inequality constraints that are bounded by a predetermined e value They utilized their

hybrid method to minimize the following objective functions cost of network upgrading

energy losses in the DS sections and purchased energy (from transmission and DG) The

number of the DG sources to be installed was randomly assigned and the units were

randomly located at the network buses Whenever the constraints are violated the

objective function solution is penalized A Pareto set was calculated from this

multiobjective optimization problem to aid the distribution planner in the decision

making process

Kim et al [72] and Gandomkar et al [73] hybridized two methods to solve the DG

sizing problem The former hybridized GA with fuzzy set theory to optimally size the

single DG unit while the latter combined the GA and SA metaheuristic methods to solve

for the optimal DG power output In both references the DG sizing problem was

formulated as a nonlinear optimization problem subject to boundary constraints only

Unlike Gandomkar et al [73] added the nonlinear power flow equality constraints to

their problem formulation The former researchers utilized their methodology to

investigate multiple DG case while the latter solved only the single DG case Both sited

the DG at all DS buses in order to determine the optimal DG location and size

Nara et al [74] assumed that the candidate bus locations for the DG unit to be

installed were pre-assigned by the distribution planner Then they used the TS method in

solving for the optimal DG size The objective of their formulation was to minimize the

system losses The DG size was treated as a discrete variable and the number of the

18

deployed units was considered to be fixed The DS loads were modeled as balanced

uniformly distributed constant current loads with a unity power factor

Golshan and Arefifar [75] applied the TS method to optimally size the DG as well

as the reactive sources (capacitors reactors or both) within the DS They formulated

their constrained nonlinear optimization problem by minimizing an objective function

that sums the total cost of active power losses line loading and the cost of the added

reactive sources The DG locations were not optimized instead a set of locations were

designated to host the proposed DGs and the reactive sources

A hybrid method that combined the GA with the TS technique in order to solve the

DG sizing optimization problem was developed by Gandomkar et al [80] They solved

the DG integration problem by minimizing the distribution real power losses subject to

boundary conditions The authors restricted the number of DGs as well as their gross

capacity to be revealed prior to executing the optimization procedure They augmented

the objective function with penalty terms in their formulation to handle the constraint

violations

Falaghi and Haghifam [77] proposed the ACO methodology as an optimization tool

for solving the DG sizing and placement problems The minimized objective function for

the utilized method was the global network cost ie the summation of the DGs cost their

corresponding operational and maintenance cost the cost of energy bought from the

transmission grid and the cost of the network losses The DG sizes were treated as

discrete values They used a penalty factor to handle the violated constraints ie

infeasible solutions In addition to modeling the DG sources as exclusive constant power

delivering units ie with unity power factor the network loads were all assumed to have

09 power factor Thus it can be stated that such modeling is impractical especially when

real large DSs are encountered

Raj et al [78] dealt with the DG integration in two different steps They employed

the PSO method to optimally determine the size of single and multiple DGs The optimal

location portion of the problem was performed utilizing the NR power flow method to

assign those buses with the lowest voltage profiles as the optimal candidate DG locations

The PSO was used to minimize the system real power losses the voltage profiles

boundary conditions were the only constraints required by the authors to be satisfied

19

Constraint violations were handled via a penalty factor that was augmented with the

objective function The DG units were randomly sited at one or more of the candidate

buses obtained through the NR method and subsequently the PSO was used to find the

optimal size(s)

Rahman et al [79] derived sensitivity indices to identify the most suitable bus for a

single DG installation Subsequently the DG sizing problem was dealt with by

employing an EP approach The objective function of the proposed approach was to

minimize the DS real power losses subject only to the system bus voltage boundary

constraints The formulation of DG sizing in their work was not realistic for a variety of

reasons For instance they ignored the line loading restrictions power flow equality

constraints and DG size limits

In most of the reviewed work on the DG deployment problem the problems of DG

optimal sizing and placement were not simultaneously addressed due to the difficult

nature of the problem as it combines discrete and continuous variables for potential bus

locations and DG sizing in a single optimization problem This combination creates a

major difficulty to most derivative-based optimization techniques and it increases the

feasible search space size considerably In this thesis the DG sizing subproblem is

solved using an improved SQP deterministic method while the two subproblems are

addressed simultaneously via an enhanced PSO metaheuristic algorithm

24 SUMMARY

In this chapter distribution power flow techniques were reviewed in Section 22 The

literature review of DG integration problem solution methods was presented in Section

23 The analytical and deterministic methods that were utilized to handle the DG

integration problem were presented in Subsection 231 Then recent publications that

handled the DG sizing and placement problems via wide-class of metaheuristic methods

were reviewed and summarized

20

CHAPTER 3 FAST AND FLEXIBLE RADIAL POWER FLOW FOR

BALANCED AND UNBALANCED THREE-PHASE DISTRIBUTION

NETWORKS

31 INTRODUCTION

As discussed in Chapter 2 several limitations exist in radial power flow techniques

presently reported in the literature such as complicated bus numbering schemes

convergence related problems and the inability to handle modifications to existing DS in

a straightforward manner This motivated the development of an enhanced distribution

power flow solution method In this thesis the unique structure of the RDS is exploited in

order to build up a Fast Flexible Radial Power Flow (FFRPF) technique The tree-like

RDS configuration is translated into a building block bus-bus oriented data matrix

known as a Radial Configuration Matrix (RCM) which consequently is utilized in the

solution process The developed algorithm is also capable of handeling weakly meshed

and meshed DSs via a meshed RCM (mRCM) RCM or mRCM is the only matrix that

needs to be constructed in order to proceed with the iterative process During the data

preparation stage each RCM (or mRCM) row focuses only on a system bus and its

directly connected buses That is while building such a matrix there is no need to

inspect the entire system buses and sections Moreover no complicated node numbering

scheme is required The building block matrix is designed to have a small condition

number with a determinant and all of its eigenevalues equal to one to ensure its

invertibility By incorporating this matrix and its direct descendant matrices in solving

the power flow problem the CPU execution time is decreased compared with other

methods The FFRPF method is flexible in accommodating any changes that may take

place in an existing radial distribution system since these changes can be exclusively

incorporated within the RCM matrix The proposed power flow solution technique was

tested against other methods in order to judge its overall performance using balanced and

unbalanced DSs

In Chapters 4 and 5 the FFRPF method is incorporated within the developed FSQP

and HPSO algorithms in solving the optimal DG installation problem It is implemented

21

as a subroutine within the proposed algorithms to satisfy the equality constraints ie

solving the radial power flow equations

32 FLEXIBILITY AND SIMPLICITY OF FFRPF I N NUMBERING THE RDS

BUSES AND SECTIONS

The RDS is configured in a unique arborescent structure with the distribution substation

located at its root node from which all the active and reactive power demands as well as

the system losses are supplied The substation feeds one or more main feeders with

spurred out laterals sublaterals and even subsublaterals For this reason the substation is

treated as a swing bus during the power flow iterative procedure

Most radial power flow techniques proposed in the literature assign sophisticated

procedures for numbering the radial distribution networks in order to execute their

algorithms This is cumbersome when expanding andor modifying existing RDSs In

this section a very simple numbering rule for the RDS buses and sections is introduced

A section is defined as part of a feeder lateral or sublateral that connects two buses in the

RDS and the total Number of Sections (NS) is related to the total Number of Buses (NB)

by this relation (NS=NB -1 )

321 Bus Numbering Scheme for Balanced Three-phase RDS

A balanced radial three-phase RDS is represented by a single line diagram In such a

system a feeder or sub level of a feeder having more than one bus is numbered in

sequence and in an ascending order Consequently each section will carry a number

which is less than its receiving end bus number by one as shown in Figure 31

Therefore sections are numbered automatically once the simple numbering rule is

applied

22

Substation

Figure 31 10-busRDS

In numbering the RDS shown in Figure 31 the following was considered buses 1 -

4 form the main feeder and buses 2 - 6 and 3 - 10 are the laterals while the sublateral is

tapped off bus 5 Figure 32 and Figure 33 manifest the ease in renumbering the system

shown previously and the flexibility in adding any portion of RDS to the existing one

respectively In Figure 32a buses 1 -4 have a different path compared to Figure 31 and

are considered to be a main feeder buses 2 - 9 and 3 - 6 are the laterals whereas the

sublateral 7 - 10 is the section spurred off bus 7 Figure 32b shows another numbering

scheme The same system numbered differently would have the same solution when

solved by the FFRPF

Figure 33 illustrates the ease of numbering in the case of a contingency situation or

a switching operation that could cause the existing system to be modified andor to be

augmented with other systems The lateral 3 - 10 in Figure 31 was modified to be

tapped off bus 2 instead and a couple of radial portions were added to be fed from buses

6 and 4 as illustrated in the figure

23

Substation Substation

(a) (b)

Figure 32 Different ways of numbering the system in Fig 31

Figure 33 The ease of numbering a modified and augmented RDS

322 Unbalanced Three-phase RDS Bus Numbering Scheme

The three-phase power flow is more comprehensive and realistic when it comes to

finding the three-phase voltage profiles in unbalanced RDSs Figure 34 shows an

unbalanced three-phase RDS The missing sections and buses play a significant role in

the multi-level phase loading and in making the unbalanced state of such a three-phase

DS more pronounced

24

The RDS shown in Figure 34 includes 6 three-phase buses (3(j)NB = 6) 5 three-

phase sections (3(|)NS = 5) no matter how many phase buses or sections exist physically

As such it has 14 single-phase buses (1(|)NB = 14) and 11 single-phase sections (1lt|gtNS =

11) The relations expressed in Eq (31) govern the three-phase and single-phase buses

to their corresponding sections

3^NS = 3^NB-1

l^NS = l^NB-3 (31)

Figure 34 Three-phase unbalanced 6-bus RDS representation

It is simple to implement the numbering process in the three-phase system as was

done in the balanced case Any group of phase buses to be found along a phase feeder or

a sub level of a feeder is to be numbered in a consecutive ascending order Consequently

each phase section number will carry a number which is one less than its receiving end

bus number as shown in Figure 34 In other words the sections are numbered routinely

after the ordering of the three-phase RDS buses has been completed

To develop the building block matrix as will be shown shortly the unbalanced three-

phase system is redrawn by substituting for any missing phase section or bus using dotted

representation as depicted in the 6-bus RDS in Figure 35 By performing this step each

three-phase bussection in the RDS consists of a group of 3 single-phase busessections

a b and c including the missing ones for double and single-phase buses

25

l a

I (1) 2 a | (2) 3 a | (3) 4 a | (4)

Figure 35 Three-phase unbalanced 6-bus RDS with missing buses and sections

33 THE BUILDING BLOCK MATRIX AND ITS ROLE I N FFRPF

The proposed FFRPF procedure starts with a matrix that mimics the radial structure

topology called a system Radial Configuration Matrix (RCM) The inverse of RCM is

then obtained to produce a Section Bus Matrix (SBM) that will be utilized in summing

the section currents during the backward sweep procedure A Bus Section Matrix (BSM)

is next generated by transposing the SBM to sum up the voltage drops in the forward

sweep process Therefore the only input data needed in the solution of an existing

modified or extended RDS other than the system loads and parameters is the RCM

It is worth mentioning that the inversion and transposition operations take place only

once during the whole process of the proposed FFRPF methodology for a tested RDS

whereas other methods like the NR technique invert the Jacobian matrix in every single

iteration The following subsections demonstrate the building of a three-phase RCM and

elucidate the role of both SBM and BSM in solving the radial power flow problem

331 Three-phase Radial Configuration Matrix (RCM)

The only matrix needed to be built for an unbalanced three-phase RDS is the RCM

Whatever changes need to be accommodated as a modification in the existing structure or

an addition to the existing network would be performed through the RCM only The

26

other matrices utilized in the backwardforward sweep are the direct results of the RCM

and no other built matrix is needed to perform the FFRPF

Such a matrix exploits the radial nature of such a system The RCM is of 3(3())NB x

3(|)NB) dimension in which each row and column represents a single-phase bus For a

balanced three-phase RDS represented by a single line diagram the RCM dimension is

(NBxNB) The RCM building algorithm for the unbalanced three-phase RDS case is

illustrated as follows

1 Construct a zero-filled 3(3(|)NB x 3(|)NB) square matrix

2 Change all the diagonal entries to +1 every diagonal entry represents sending

missing or far-end buses

3 In each row if the column index corresponds to an existing receiving single-phase

bus its entry is to be changed to - 1

4 If a single-phase bus is missing or is a far-end bus the only entry in its

corresponding row is the diagonal entry of+1

The above RCM building steps are summarized in the following illustration

Columns Description

RCMbdquo

if is either

a - sending phase bus b - far-end phase bus c - missing phase bus (32)

-1 jkl if jkI are receiving phase buses

connected physically to phase bus 0 otherwise

The [abc] matrix is defined as a zero (3 x 3) matrix with the numbers a b and c as

its diagonal elements eg [ I l l ] is the identity (3 x 3) matrix while [110] is the identity

matrix with the third diagonal element replaced by a zero By following the preceding

steps and utilizing the [abc] definition the RCM for the unbalanced three-phase RDS

shown in Figure 35 is to be constructed as shown in (33)

27

[Ill] [000] [000] [000] [000]

[000]

-[111] [111]

[000] [000] [000]

[000]

[000] -[111]

[111] [000] [000]

[000]

[000] [000]

-[110] [111]

[000] [000]

[000]

[000] [000]

-[010] [111]

[000]

[000] [000]

-[on] [000] [000]

[111]

Because of the nature of the RDS the RCM has three distinctive properties The first

is that the RCM is sparse the second is that RCM is a strictly upper triangular matrix

and thirdly such a matrix is only filled by 0 +1 or - 1 Such a real matrix makes the data

preparation easy to handle and less confusing Figure 36 shows the sparsity pattern plots

of the RCM matrices for unbalanced three-phase 25-bus [48] and balanced 90-bus [38]

radial systems

RCM for 25-Bus unbalanced 3-phase RDS RCM for 90-Bus balanced 3-phase RDS

nz = 131 nz = 179

Figure 36 RCM matrices sparsity plots for the 25 and 90 bus RDSs

3311 Assessment of the FFRPF Building Block RCM

The RCM is well-conditioned and should have a small Condition Number (CN) and a

non-zero determinant The CN measures how far from singularity any matrix is It is

defined as

28

cond(A) = A jjA-l (34)

where ||A|| is any of the 3 types of norm formulations of matrix A 1-norm 2-norm or oo-

norm An ill-conditioned matrix would have a large CN while a CN of 1 represents a

perfectly well-conditioned matrix By definition a singular matrix would have an infinite

CN [81] Having all the matrix eigenvalues to be equal to 1 and a determinant value of 1

safeguard the RCM against singularity For this reason the RCM is not only invertible

but also its inverse is an upper triangular matrix filled only with 0 and +1 digits and no

other numbers would appear in RCM-1

332 Three-phase Section Bus Matrix (SBM)

The SBM for the three-phase RDS is obtained by performing the following steps

1 Remove the corresponding substation rows and columns from the RCM ie the

first three rows and columns The reduced version of the RCM is labeled as

RCM

2 Invert the RCM to obtain the SBM as shown in (35) and more explicitly in

Figure 37

To clarify the two rows and the two columns outside the matrix border shown in

Figure 37 are the three-phase buses and sections ordered respectively The dimension of

the unbalanced three-phase system SBM is 3(3lt|)NS x 3(|gtNS) For the balanced case the

SBM dimension is (NSxNS)

[Ill] [000]

[000] [000] [000]

[111] [111]

[000] [000] [000]

[110] [110]

[111] [000] [000]

[010] [010] [010] [111] [000]

[011] [011] [000] [000] [111]

29

1 a

1 b

1 c

2 a

2 b 2 c

SBM = 3 a

3 b

3 c

4 a

4 b

4 c

5 a

5 b

5 c

2 a

0 0 0 0 0 0 0 0 0 0 0 0 0 0

2 b

0 1 0 0 0 0 0 0 0 0 0 0 0 0 0

2 c

0 0 1 0 0 0 0 0 0 0 0 0 0 0 0

3 3 a b

1 0 0 1 0 0 i o 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

3 c

0 0 1 0 0

1 0 0 o 0 0

o 0 0 0

4 a

1 0 0 1 0

o 1 0

o 0 0

o 0 0 0

4 b

0 1 0 0 1 0 0 1 0 0 0 0 0 0 0

4 c

0 0 0 0 0 0 0 0 1 0 0 0 0 0 0

5 a

0 0 0 0 0 0 0 0 0 1 0 0 0 0 0

5 b

0 1 0 0 1 0 0 1 0 0 1 0 0 0 0

5 c

0 0 0 0 0 o 0 0

o 0 0 1 0 0 0

6 a

0 0 0 0 0

o 0 0

o 0 0 o 1 0 0

6 b

0 1 0 0 1 0 0 0 0 0 0 0 0 1 0

6 c

0 1 0 0 1 0 0 0 0 0 0 0 0 1

Figure 37 SBM for three-phase unbalanced 6-bus RDS

By inspecting Figure 35 it is noted that any single-phase section is connected

downhill to a single-phase far-end bus or buses through a Unique Set of Phase Buses

(USPB) xt bull By inspecting Figure 35 and Figure 37 the USPB for the following

single-phase sections la lb lc 3b 4b are^0 [ gt xlgt x respectively as explained

in (36)

=2 f l 3bdquo4a x=2b3b4b5bA]

Xl=2cA) Xl=) (36)

X=5b]

In the SBM the single-phase section is represented by a row i and will have entries

of ones in all the columns where their indices represent single-phase buses that belong to

the section USPB xf bullgt a s illustrated in (37)

SBMrmt =

Columns Description

c bullgt bull a - Id - buses e r ~ - 1 ijk- ijk-- are either w (37)

lb - diagonal entry 0 other columns otherwise

30

333 Three-phase Bus Section Matrix (BSM)

The BSM is the transpose of the SBM as shown in (38) In the BSM the rows represent

the RDS single-phase buses excluding the substations and all the sections are

represented by the BSM columns Each single-phase bus is connected uphill through a

Unique Set of Phase Sections (USPS) yf to a substation single-phase bus The USPS

for buses 2deg 3a 4b 5b 6C are y ydeg y y yc6 respectively as depicted in Figure 35

and Figure 37 and demonstrated in (39)

BSM

[111] [000] [000] [000] [000]

[111] [111] [000] [000] [000]

[110] [110] [111] [000] [000]

[010] [010] [010] [111] [000]

[011] [011] [000] [000] [111]

(38)

V H U V3=1gt2 y=b2b yb

5=lb2bdquoAb (39)

lt=1C2C5C

In the BSM a single-phase bus i is represented by a row and will have entries of

ones in all the columns where their indices represent single-phase sections that belong to

the bus USPS yf as equivalently shown in (310)

Columns Description

BSMrmi =

( gt - [a-l^-sectionse yf ~ i m

1 ijk ijk are either lt Y Y (310) lb - diagonal entry

0 other columns otherwise

34 FFRPF APPROACH AND SOLUTION TECHNIQUE

The FFRPF methodology is based simply on Kirchhoff s voltage and current laws which

are used in performing the backwardforward sweep iterative process By utilizing the

direct descendant matrices of the RCM ie SBM and BSM the RDS buses complex

31

voltages are calculated in every iteration until convergence criteria are met The next

subsections illustrate the proper usage of such matrices in the proposed FFRPF method

through appropriate modeling of the unbalanced multi-phase RDS section impedances

341 Unbalanced Multi-phase Impedance Model Calculation

Figure 38 shows a three-phase section model that is represented by two buses (sending

and receiving ends) of an overhead three-phase four-wire RDS with multi-grounded

neutral The assumption of a zero voltage drop across the neutral in a three-phase two-

phase and single-phase RDS is found to be valid [4582] Such a configuration is widely

adopted in North Americas distribution networks [8384]

V Sa

_ bull

V Ra

V Sb ab

^VWVVYgt

V Sc Z be

bn

^AAArmdashrYYYV bull

V s n en

ll1 Sa

s b

La

Lb

Lc

bull r Figure 38 Three-phase section model

In the proposed radial power flow solution method each of the three-phase lines is to

be modeled appropriately and mutual coupling effects between phases are not neglected

The primitive impedance matrix for such a four-wire system is a square matrix with a

dimension equal to RDS utilized number of phase and neutral conductors For a system

consisting of three-phase conductors and a neutral wire the section primitive impedance

matrix is expressed as shown in (311)

32

Zaa

ha

Zca

zna

Kb

Kb

Kb

Kb

Ke Kc Ke

nc

art

Zbn

en

nn

where

Z bull primitive impedance matrix

RDS section length

z per unit length self-impedance of conductor i

z per unit length mutual-impedance between conductors andy

zu and zy are calculated according Carsons work [85] and its modifications [86-88] as

illustrated by the following equations

where

k

GMRj

Dbdquo

v GMR

bulli J

zu=rt+rd+ja)k

zv=rd+jltok

resistance of conductor i

earth return conductor resistance

inductance multiplying constant

distance between overhead and its earth return counterpart and it is a

function of both earth resistivity and frequency

geometric mean radius of conductor i

distance between conductors i andj

(312)

(313)

The parameters used in (312) and (313) are shown in Table 31 for both operational

frequencies 50Hz and 60Hz in both metric and imperial units

33

Table 31 cok rj and De Parameters for Different Operation Conditions

De = 2160 Ij (ft)

cok rd

p = 100 Qm

p = 1000 Qm

Metric Units RDS operating frequency 50 Hz 60 Hz

006283km

0049345 QJ km

931 m

29443 m

007539 km

005921412km

850 m

26878 m

Imperial Units RDS operating frequency 50 Hz 60 Hz

010111mile

00794 QI mile

30547f

96598 ft

012134mile

009528 QI mile

27885

88182

Since the neutral is grounded the primitive impedance matrix Zsec can be

transformed into a (3 x 3) symmetrical impedance matrix Zsae

c by utilizing Krons

matrix reduction method The resultant section three-phase impedance matrix is

expressed mathematically in (314) and the three-phase section model is represented

graphically in Figure 39

7 abc

aa

zba Zca

Zab

^bb

Zcb

zac zbc Zee

(314)

VSn

mdash bull i 7 T

i ah bull-sec a

zbdquobdquo bull A V W Y Y Y V

v izK

^WW-rrYYv -+bull

I

bull

vR

Figure 39 The final three-phase section model after Krons reduction

If the RDS section consists of only one or two phase lines its primitive impedance

matrix is transformed by Krons matrix reduction to (laquo-phase x w-phase) symmetrical

impedance matrix Next the corresponding row and column of the missing phase are

replaced by zero entries in the (3x3) section impedance matrices Zsae

c For a two-phase

34

section its impedance matrix Z^c is demonstrated below

Z_a

zci Kron h-gt ZZ za

zbdquo zai

zaa o zac

0 0 0

z_ o zbdquo

Underground lines such as concentric neutral and tape shielded cables are typically

installed in the RDS sections For underground cables with m phases and n additional

neutral conductors the primitive impedance of each section is a (2m+n x 2m+n) matrix

with the entries computed as illustrated in [89-92]

Usually the RDS is modeled as a short line ie less than 80 km and the charging

currents would be neglected by not modeling the line shunt capacitance as depicted in

Figure 38 However under light load conditions and especially in the case of

underground cables the line shunt capacitance needs to be considered in order to obtain

reasonable accuracy ie use nominal 7i-representation The rc-equivalent circuit consists

of a series impedance of the section and one-half the line shunt admittance at each end of

the line Figure 310 (a) (b) and (c) show the RDS 7i-model representation The shunt

admittance matrix for an overhead three-phase section is a full (3x3) symmetrical

admittance matrix while it is a strictly diagonal matrix for the underground RDS cable

section That is the self admittance elements are the only terms computed [92] For the

unbalanced three-phase section eg one or two phases the non-zero elements of shunt

admittances are only those corresponding to the utilized phases

[zic] -AAVmdashrwvgt

T yabc 1 |_ sec J [ yaf tc |

sec J

(a)

35

Lsec J _

2

s

1

Yaa

Yba

Yea

Yab

Ybb

Ycb

Yac

Ybc

Ycc

zaa

zba

tea

zab

zbb

zcb

zac

zbc

zcc

P yabc ~|

lgtlt 2 - =

Yaa

Yba

Yca

Yab

Ybb

Ycb

R

1 1

Yac

Ybc

Ycc

(b)

[ yabc 1 sec J

s 1 1

Yaa

0

0

0

Ybb

0

0

0

Ycc

zaa

zba

zca

Zab

zbb

zcb

zac

zbr

zcc

V yabc ~j

L sec 2 - =

Yaa

0

0

0

Ybb

0

R

1 1

0

0

Ycc

(c)

Figure 310 Nominal 7i-representation for three-phase RDS section

(a) Schematic drawing for the single line diagram of 3(|gt RDS (b) Matrix equivalent for overhead section (c) Matrix equivalent for underground cable section

By applying Kirchhoff s laws to the three-phase system section k the relationship

between the sending and receiving end voltages for medium and short line models and

the voltage drop across the same section in the latter model are expressed in Eq (315)-

(316) and Eq (317) respectively

36

rabc S rabc S

14 L sec Jax3 L rabc

3x3 L secgt J3x3

[C]3 [4 [ yabc~ |~ yabc 1

sec J3x3 L sec J3 [4

zt 1 L sec J3x3

f yabc ~| [~ yaampc 1

L sec J3x3 L sec h

bull R rabc

(315)

rrabc VS rabc

S

1 J3x3 L sec J

[degL [L abc R

(316)

where

TT-afec rrabc S ^ R

rabc rabc S XR

rabc

AK

13x3

aAc sect

rabc see

KrH^ic] three-phase sending and receiving end voltages

three-phase sending and receiving end section currents

three-phase shunt admittance of section k

(3gtlt3) identity matrix

(3gtlt3) zero matrix

voltage drop across three-phase section k

section k three-phase currents

(317)

It is worth noting that Eq (315) is reduced to Eq (316) in the case of short line

modeling since YS^C 1 = 0 The voltage drop across the three-phase section k in the short

line model is expressed in Eq (317) and its corresponding sending end phase voltages

can be expressed in expanded forms as follows

V = V + Ta 7 + 1 7 + Tc 7 rS yR ^1secZjaa ^ Isec^ab ^rlsecpoundjac

v =v +r 7 +17 +r 7 y S y R ^ l sec^ab ^ 1 sec^bb ^ 1 sec^Ac

S mdashyR+ Ktc^ca + sec^cb + sec^a

(318)

(319)

(320)

Equations (317)-(320) show that the voltage drop along any phase in a three-phase

section depends upon all the three-phase currents

37

342 Load Representation Accurate and proper load modeling is of significant concern in power distribution

systems as well in its transmission systems counterpart [8693] Loads in electric power

systems are usually expressed by adequate representations so as to mimic their effects

upon the system The load dependency on the operating bus voltage and on system

frequency is among those representations

Static load models are often utilized in the power flow studies since they relate the

apparent power active and reactive directly to the bus operating voltage A static load

model is used for the static load components ie resistive and lighting load and as an

approximation to the dynamic load components ie motor-driven loads [93] Generally

static loads in DS are assumed to operate at rated and fixed frequency value [94-96]

Loads in the DS are usually expressed as function of the bus operating voltage and

represented by exponential andor polynomial models

The exponential model is shown in (321) and (322)

P = Pbdquo

Q = Q0 vbdquo

(321)

(322)

where

V0 nominal bus voltage

V operating bus voltage

P0 real power consumed at nominal voltage

Q0 reactive power consumed at nominal voltage

Exponents a and fi determine the load characteristics and certain a and values lead to a

specific lode model Therefore

1 If a = P mdash 0 the model represents constant power characteristics ie the load is

constant regardless of the voltage magnitude

2 If a = P = 1 the model represents constant current characteristics ie the load is

proportional to the voltage magnitude

3 If a = P = 2 the model represents constant impedance characteristics ie the load is

38

a quadratic function of the voltage magnitude

As indicated in [97] the exponents could have values larger than 2 or less than 0 and

certain load components would be represented by fractional exponents

The constant current model is considered to be a good approximation for many

distribution circuits since it approximates the overall performance of the mixture of both

constant power and constant impedance models [98] However representing loads with

the constant power model is a conservative approach with regard to voltage drop

consideration [99] and consequently this model will be used in this thesis

Loads can also be represented by a composite model ie the polynomial model The

polynomial model is expressed in (323) and (324)

P = Pbdquo

Q = Q0

(

a p

V

r

V

V

K

V

v0

2

2

V

K

V

+CP

J

)

(323)

(324)

where ap + bp + cp = 1 and aq + b + cq = 1

The polynomial model is also referred to as a ZIP model since it combines all the

three exponential models constant impedance (Z) constant current (I) and constant

power (P) models The ZIP model needs more information and detailed data preparation

The load models can be used in the FFRPF solution method during its iterative

process where flat start values are initially assumed to be the load voltages The three-

phase load voltages are changed during each iteration and consequently the three-phase

currents drawn by the constant current constant impedance andor ZIP three-phase load

models will change accordingly

Different shunt components like spot loads distributed loads and capacitor banks are

customarily spread throughout the RDS In power flow studies spot and distributed

loads are typically dealt with as constant power models while shunt capacitors are

modeled as constant impedances [94 100 101]

The uniformly distributed loads across RDS sections can be modeled equivalently by

either placing a single lumped load at one-half the section length or by placing one-half

the lump-sum of the uniformly distributed loads at each of the section end buses

39

[99 102] The former modeling approach has the disadvantage of increasing the

dimension of the RCM SBM and the BSM since more nodes would be added to the

existing RDS topology In the proposed FFRPF technique the distributed load is

modeled using the latter approach while the three-phase shunt capacitor banks are

modeled as injected three-phase currents [101] as schematically shown in Figure 311

and mathematically represented by Eq (325) and (326)

Qk Cap

^

CCap a Cap

(a) (b)

Figure 311 Three-phase capacitor bank (a) The schematics (b) The Modeling

O3 = poundCap

Qo Capa vbdquo

T-34 _ 1Cap ~

V

a Capbdquo

SQL M Cap

V

filt bullCapo

F

JQ( Cap

(325)

(326)

343 Three-phase FFRPF BackwardForward Sweep

The FFRPF technique employs the SBM in performing the current summation during the

backward sweep and the BSM in updating the RDS bus complex voltages during the

forward sweep as demonstrated in the following subsections

3431 Three-phase Current Summation Backward Sweep DS loads are generally unbalanced due to the unbalanced phase configuration the double

and single-phase loadings as well as the likelihood of unequal load allocation among the

three-phase configuration For the loads they could be represented as constant power

40

constant current constant impedance or any combination of the three models [97 103]

The three-phase load currents at three-phase bus i drawn by a three-phase load eg of a

constant impedance load model is mathematically expressed as shown in Eqs (327) and

(328)

jabc U ~

7e a gt

va

(si) ~

(327)

where

ctabc

o

V

K

2

rft

0

v K

2

K v

2

(328)

where Sf represents the load apparent power at single-phase bus lt|gt As shown in the

preceding equations each load current is a function of its corresponding bus voltage For

Eq (327) if the a phase bus is missing its corresponding phase load current is

eliminated and its corresponding position in the three-phase current vector is replaced by

a zero entry As an illustration and by assuming that there are loads connected to all

existing buses the three-phase load current vector for the system shown in Figure 35 is

expressed as follows

jabc V ja rb jc ra rb re ra rb Q Q rb Q Q rb re l l

The charging currents at the RDS three-phase buses are not to be neglected when

dealing with sections modeled as The shunt admittance at bus is obtained by

applying the following relation

where

Ysh^ bull total three-phase shunt admittance at bus

[l if section k attached to bus i

[0 otherwise

The three-phase shunt currents at bus is as shown in Eq (330)

tabc jrabc 1ch ~~ 1Anbus y i (330)

41

The 3-ltj) bus current is sum of both the 3-sect load current and the 3-cj) charging current as

expressed mathematically in Eq(331)

jabc jabc jabc ) T I 1busl ~ 1Li

+Ich V-gt )U

where 1^ is bus three-phase currents In the case of modeling a three-phase section as

a short line its charging currents are neglected ie I^c = 0 and the bus current will be

represented by the load currents only

The backward sweep sums the phase load currents in the corresponding phase

sections starting from far-end phase buses and moving uphill toward the substation phase

buses The current in phase (j) and section p is computed by utilizing the USPB

principle xp gt during the backward sweep as expressed in (332)

lt = E lt ^here = j 0 ^ J (332)

where

I current through single-phase section and phase ^ (^ =a b or c) SQCp

j current at bus and phase ltb bus x

The SBM is utilized in obtaining the system three-phase section currents in matrix

representation by performing the relation in Eq (333)

[G] = [SBM][lpound] (333)

where the I^ is a 3-(3())NS)-order column vector For a balanced short line RDS

model Eq (333) can be expressed as

[CS] = [SBM][IL] (334)

3432 Three-phase Bus Voltage Update Forward Sweep

The voltage at each phase bus is determined through the forward sweep procedure by

subtracting the sum of the voltage drops across the bus corresponding USPS from the

substation nominal complex voltage The voltage drop across three-phase section k is

calculated as in Eq (317) The voltage drop across all sections of the three-phase RDS

can be obtained by utilizing the SBM as shown in matrix notation via Eq (335)

42

[AKbdquo]=[zr][c] (335)

[ A ^ ] = [ Z s r ] [ 5 5 M ] [ C ]

where ZtradeS J is a diagonal matrix with an 3(3c|)NS x 3ltj)NS) dimension in which its

diagonal entry k corresponds to section k impedance and AV3^ is the computed three-

phase voltage drop values across all the RDS sections as shown below

A ^ = AVa AVb AVC bullbullbull AVb AVC T s e c l s e c l s e c l sec3ltWS sec3tNS J

For calculating the RDS voltage profiles the FFRPF solution method starts by asshy

suming the initial values for all bus voltages to be equal to the substation complex

voltage As a flat start the initial phase voltages at bus will be as follows

2TT 2TT

ya Jdeg vb e~^ VC e+JT V ss e V sisK v sis e (336)

where Vls is the substation complex phase voltage

For the voltage at bus m and phase (j) to be determined at iteration v the calculation is

performed as follows

= amp - pound r A lt wherer = trade lt (337)

The RDS voltage profiles at the system phase buses are obtained by utilizing the BSM as

shown in following matrix representation

[Vi] = [Vsy[BSM][AV^] (338)

where V^fs 1 and V3A are respectively the substation nominal three-phase voltage

column vector and the resultant three-phase bus voltage solution column vector and each

has a dimension of 3(3lt|gtNS)

3433 Convergence Criteria

The bus complex voltage is obtained after every backwardforward sweep After each

iteration all the bus voltage magnitudes and angles are compared with the previous

iteration outcomes The power flow process is concluded and a solution is reached if the

complex voltage real and reactive oo-norm mismatch vector is less than a certain

43

predetermined empirical tolerance value e The convergence criterion is expressed

mathematically as shown in Eq (339)

+i

([gt]w) A a ( |y f lts

where th

i iteration A

(339)

and symbol

||J| vector oo-norm ||x II = max fix IV II lloo l l c 0 1=12NBV I

5H (bull) real part of complex value

3 (bull) imaginary part of complex value

3434 Steps of the FFRPF Algorithm

The FFRPF iterative process can be summarized as follows

Step 1 Begin FFRPF by choosing a test RDS

Step 2 Number and order RDS buses and sections

Step 3 Construct RCM

Step 4 Obtain both SBM and BSM

Step 5 Select load model

Step 6 Start the iterative procedure by assuming flat start voltages for all buses

Step 7 Calculate load currents

Step 8 Start the backward sweep process by calculating section currents using SBM

Step 9 Start the forward sweep process by determining the bus complex voltages

using BSM

Step 10 Compare both magnitudes and angles of the RDS bus voltages between the

current and previous iterations

bull If the co-norm of their difference is lt st

o Solution is reached

44

o Stop and end FFRPF procedure

o Obtain bus voltage profiles section currents and power losses

etc

bull If not utilize the outcome of this iteration (bus complex voltages)

to start a new one by going back to Step 7

The FFRPF solution method is illustrated by the following flow chart shown in Figure

312

45

i laquo - i +1

Calculate Load and leakage

currents

I Start Backward

sweep process by calculating section

currents using SBM

Start Forward sweep process by determining bus

complex voltages V[+1] using BSM

V[+1] Section currents

Section Power Losses Etc

Start FFRPF

Read the test RDS data

Number and order RDS Buses and

Sections

I Construct RCM

Remove the substation

corresponding rows and columns

from RCM to Obtain RCM

Obtain RCM1

To Get SBM

Z Transpose SBM to

get BSM

Calculate RDS section

Impedance and Shunt admittance

Matrices

Select load model

Assuming a flat start voltages for

all buses V[]=10 =0

Figure 312 The FFRPF solution method flow chart

46

344 Modifying the RCM to Accommodate Changes in the RDS The FFRPF technique deals with changes in the network such as the addition of a

transformer by adjusting the original RCM to incorporate its conversion factor (c)

Subsequently the SBM and BSM are obtained accordingly and used in the

backwardforward sweep procedure If a three-phase transformer is incorporated in a

three-phase RDS between buses m and n at section n - 1 the modified BSM entries are

located at the intersection of the matrix rows and columns defined by Eq (340)

BSMZ EzL~-inBSMZ euro lt _ (340)

The affected rows and columns of the modified BSM are those belonging to the

sections USPB and the sending buss USPS respectively For demonstration purposes

the 10-bus balanced RDS shown in Figure 31 will be utilized If two transformers with

conversion factors cfj and cS are to be added within sections 3 and 6 respectively the

original RCM is modified to accommodate such additions as illustrated in (341) Thus

instead of filling -1 for the receiving end bus entry the negative of the conversion factor

is the new entry The process is repeated rc-times for -installed transformers The

corresponding modified SBM and BSM are to be obtained as demonstrated in Section

33

10

RCM^ =

1

2

3

4

5

6

7

8

9

10

1

0

0

0

0

0

0

0

0

0

-1

1

0

0

0

0

0

0

0

0

0

-1

1

0

0

0

0

0

0

0

0

0

-cfi 1

0

0

0

0

0

0

0

-1

0

0

1

0

0

0

0

0

0

0

0

0

-1

1

0

0

0

0

0

0

0

0

-cf2

0

1

0

0

0

0

0

0

0

0

0

-1

1

0

0

0

0

-1

0

0

0

0

0

1

0

0

0

0

0

0

0

0

0

-1

1

(341)

The affected entries of the new BSM are obtained by applying the relation in (340) as

follows

47

[BSMZ eZ s ^nBSMJ euro ^ 3 = 40(12=[(4l)(42)]

(laquoSWe^)nJM6^)=J7 )8)njl I4=gt[(7 )l)(7 )4)(8l) )(M)]

The matrix shown in (342) shows the final B S M after including the transformers in the

10-bus R D S Such a procedure is easily extended to the unbalanced three-phase RDSs

1

1

1

CJ

1

1

cfi

cf2

1

1

2

0

1

ch 0 0

0

0

1

1

It is worth mentioning that by integrating the cf for any transformer configuration

into the RCM building block in the FFRPF technique another light is shed on the

flexibility criterion of the proposed method

35 FFRPF SOLUTION METHOD FOR MESHED THREE-PHASE DS

In practical DS networks alternative paths are typically provided to accommodate for

any contingency incidents that might take place eg feeder failure Therefore it is not

unusual for meshed distribution networks to be part of the DS topology in order to make

the system more reliable The loop analysis approach as well as the graph theory

technique are used to study and analyze the behavior of meshed DS The loop analysis

technique basically applies Kirchhoff s voltage law principle to solve for the fundamental

loop currents in both planar and nonplanar networks while the graph theoretic

formulation preserves the network structure properties [104]

A meshed DS can be viewed from a graph theory perspective as an oriented looped

graph that preserves the network interconnection properties whereas a DS that has no

0

0

0

1

1

cfi

cf2

0

0

0

0

0

0

1

0

0

0

0

0

0

0

0

0

1

1

0

0

0

0

0

0

0

0

1

0

0

0

0

0

0

0

0

0

1

1

0

0

0

0

0

0

0

0

1

(342)

48

loops is considered a tree In graph theory terminology line segments that connect

between buses in a loopless DS tree are called twigs branches or sections (represented by

solid line segments in Figure 313) while those which do not belong to the tree are

known as links (represented by dotted line segments in Figure 313) Links are segments

that close loops in a DS tree thereby composing a co-tree Removal of all co-tree links

results in a strictly radial system Links are usually activated by closing their

corresponding Normally Open (NO) switches Whenever a link is added to a RDS

network a loop is formed and as a result the system will have as many fundamental

loops as the number of links A fundamental loop is a loop that contains only one link

besides one or more sections Segments are used here to name sections and links

together It is noted that the number of fundamental loops is significantly less than the

number of buses in the meshed DS which makes the loop analysis a more appropriate

method in dealing with such systems than other circuit analysis methods like nodal

voltage method [105]

The current directions in the meshed DS sections and links are arbitrarily chosen to

be directed form a lower bus index to a higher one and the positive direction of loop

current is assumed to in the same direction of that of the link as illustrated in Figure 313

The number of segments in a meshed DS is equal to the sum of the total number of its

corresponding graph tree sections and its co-tree links For a meshed DS with NB buses

and mNS segments (total number of sections and links in the meshed DS) the number of

links nL and the number of the fundamental loops as well are obtained according to the

following relation

laquoL=mNS-NB + l (343)

49

Substation 2 Imdash 31 4 1

^ -gtT-gtL- -

Figure 313 10-bus meshed distribution network

351 Meshed Distribution System Corresponding Matrices The FFRPF solution method can deal with the DS meshed networks through modifying

the original RCM Discussion is now focused on the balanced three-phase meshed DS

which can easily be extended to the unbalanced three-phase DS networks

Meshed RCM The meshed RCM (wRCM) order is ((NB + wL) (NB + nVj) and the

mRCM building algorithm is as follows

1 Remove links from the meshed DS and build the RCM for the resulting network

tree as demonstrated earlier in section 331

2 Add nL rows and columns toward the end of the RCM ie each link is represented

by a row and a column attached to the end of the RCM

3 In each link column there are 3 non-zero entries and are to be filled in the following

manner

a -1 at the row which corresponds to the lower index terminal of the link

b +1 at the row which corresponds to the higher index terminal of the link

c +1 at the link diagonal entry

For the 10-bus system shown in Figure 31 three directed links Li L2 and L3 are added

to the DS tree through connecting the following buses 4 - 5 6 - 8 and 4 - 1 0

respectively The system mRCM is constructed as illustrated in (344)

50

10

mRCM (13x13)

1

0

0

0

0

0

0

0

0

0

0

0

0

-1

1

0

0

0

0

0

0

0

0

0

0

0

0

-1

1

0

0

0

0

0

0

0

0

0

0

0

0

-1

1

0

0

0

0

0

0

0

0

0

0

-1

0

0

1

0

0

0

0

0

0

0

0

0

0

0

0

-1

1

0

0

0

0

0

0

0

0

0

0

0

-1

0

1

0

0

0

0

0

0

0

0

0

0

0

0

-1

1

0

0

0

0

0

0

0

-1

0

0

0

0

0

1

0

0

0

0

0

0

0

0

0

0

0

0

-1

1

0

0

0

0

0

0

-1

1

0

0

0

0

0

1

0

0

0

0

0

0

0

-1

0

1

0

0

0

1

0

0

0

0

-1

0

0

0

0

0

1

0

0

1

(344)

Remove the substation corresponding rows and columns from the mRCM to produce the

mRCM The mRCM for the 10-bus system is shown in (345)

10

mRCM (12x12)

1

0

0

0

0

0

0

0

0

0

0

0

-1

1

0

0

0

0

0

0

0

0

0

0

0

-1

1

0

0

0

0

0

0

0

0

0

-1

0

0

1

0

0

0

0

0

0

0

0

0

0

0

-1

1

0

0

0

0

0

0

0

0

0

0

-1

0

1

0

0

0

0

0

0

0

0

0

0

0

-1

1

0

0

0

0

0

0

-1

0

0

0

0

0

1

0

0

0

0

0

0

0

0

0

0

0

-1

1

0

0

0

0

0

-1

1

0

0

0

0

0

1

0

0

0

0

0

0

-1

0

1

0

0

0

1

0

0

0

-1

0

0

0

0

0

1

0

0

1

(345)

Meshed SBM The meshed SBM (mSBM) is then obtained by inverting the mRCM As

51

an illustration the 10-bus meshed network mSBM is obtained as shown in (346)

10

mSBM (12x12)

1

0

0

0

0

0

0

0

0

0

0

0

1

1

0

0

0

0

0

0

0

0

0

0

1

1

1

0

0

0

0

0

0

0

0

0

1

0

0

1

0

0

0

0

0

0

0

0

1

0

0

1

1

0

0

0

0

0

0

0

1

0

0

1

0

1

0

0

0

0

0

0

1

0

0

1

0

1

1

0

0

0

0

0

1

1

0

0

0

0

0

1

0

0

0

0

1

1

0

0

0

0

0

1

1

0

0

0

i deg 1

1

-1

1 deg 0

o 0

0

i

o o

0

0

0

0

1

-1

-1

0

0

0

1

0

0

0

1

0

0

0

0

-1

-1

0

0

1

(346)

Let the mSBM be partitioned into two submatrices mSBMp and C so that the mSBMp

submatrix corresponds to the DS tree sections and the second submatrix C to the

fundamental loops or links as shown in (347)

wSBM = SBM

6 [cl (mNSxnL) = [mSBMp C]

JmNSx(NB-l)

The dotted line shown in the above relation implies matrix partitioning

(347)

Fundamental loop matrix The second submatrix in (347) ie C is the fundamental

loop matrix which governs the direction of currents in each of fundamental loop sections

and links The fundamental loop matrix C can be partitioned into two submatrices [Csec]

and [I] as demonstrated below

M_ where [Csec] is an ((NB - 1) x (laquoL)) matrix and [7] is an nL square identity matrix The

former matrix corresponds to the tree loop sections while the latter corresponds solely to

c-- (348)

52

the co-tree links

By inspecting the fundamental loop matrix C it is noted that each row represents a

section or a link and each column represents a loop Each column entry in the C matrix

CM will have one of the following values

1 Qy = +1 if section k belongs to and is oriented in the same direction of loop

2 Cki = - 1 if section k belongs to and is oriented in the opposite direction of loop

3 Claquo = 0 if section k is not in the loop

By inspecting the 10-bus DS mSBM illustrated in (346) the first loop which is represhy

sented by the tenth column of the matrix is comprised of three sections in addition to the

link The current in two of these sections runs in the same direction as their correspondshy

ing fundamental loop ie sections 2 and 3 while the third one ie section 4 has an

opposite orientation This can be easily verified by tracing the first loop in the meshed

DS single line diagram One can also note that two loop currents pass through the third

section in an additive manner

Meshed BSM The meshed BSM (mBSM) is the transpose of mSBM as illustrated in

(349)

[ B S M I 0](mNS-nL)mNS

L J(nLxmNS)

The second submatrix C[ is the transpose of the fundamental loop matrix as manifested in

(350)

mBSM = [mSBM] = mBSMr

c (349)

C=L

1

0

0

0

2 3

1 1

0 0

0 1

4

-1

0

0

5

0

1

0

6

0

-1

0

7

0

-1

0

g

0

0

-1

9

0

0

-1

h 1

0

0

h 0

1

0

h (f 0

1

(350)

Meshed DS impedance matrices The (laquoLx riL) loop-impedance matrix Zioop is

formulated as follows [106]

KHc]|Xf][c] (35D

53

where [ztradegS ] is defined as a non-singular diagonal (mNS x wNS) segment-impedance

matrix that contains all the meshed DS segment impedances (tree sections and links)

along its main diagonal The matrix [Z^fM is formulated as two diagonal submatrices

as follows

|~ rymDS I

L eg J

Zl

0

^

0

0

7

Zk

0

0

0

raquoL

|gtr ] | o o |[zr] (352)

where Z^S J is the (NB - 1 x NB - 1) loopless tree section-impedance diagonal square

matrix and Z^k 1 is the (raquoL x nL) links impedance diagonal matrix

352 Fundamental Loop Currents The algebraic sum of voltage drops AV around any fundamental loop is zero according

to Kirchhoff s voltage law (KVL) and this can be mathematically expressed in terms of

the fundamental loop matrix C as follows

[C][AF] = 0 (353)

The voltage drop across the meshed DS segments is determined by the following

relations

[W] = [zf][mSBM][mILL]

where

Lm seg J bull (wNS x l) segment currents column vector of the meshed DS network

jW iZJ (mNS x l) meshed DS bus Loads and Link currents column vector

(354)

In order to account for the link currents in the meshed network the segment currents

column vector and the meshed DS bus loads and links currents column vector are

54

respectively partitioned into two subvectors as defined below

[ jtree 1

J(mNSxl)

J((JVB-l)xl)

Jloop[ J(nLxl)

(355)

[mILL l(mNSxl)

L L J((MJ-l)xl)

Jloop J (wLxl)

(356)

where

[Cr J ((NB - 1) x 1) tree section currents column vector

[lL] ((NB - 1) x 1) RDS bus load currents column vector

j 1 (nL x 1) fundamental loop current vector which is also the meshed DS link

currents column vector

By multiplying both sides of Eq (354) by [C] the left-hand-side will equal to zero

according to KVL as shown in (353) and by employing Eqs (351) and (356) Eq (354)

is reformulated as

[C][AV] = [c][z^][mSBM[mILL]

0 = [c f [z f ] [mSBM | C ] L op]

bull = [Cr[zS]([raquoSBM][ l ]+ [C][ 1 ] )

0 = [C] [^ ] [ -raquoSBM] [ I ] + [cr [zbdquor][C][U]

-[c] [zf ][c][J=[c] [ Z - ^ S B M J ]

-[^][^]-[cT[zS][laquoSBM][I] Thus the (nL x 1) fundamental loop currents vector in the meshed DS loop frame of

reference can mathematically be expressed as

[4 J - -K]1 [Cj [z^JmSBMr][lL] (357)

Eq (357) can be expressed in terms of the RDS matrices ie SBM and [ Z ^ J by

performing the following operation

55

TABLE OF CONTENTS LIST OF TABLES x

LIST OF FIGURES xiii

ABSTRACT xvi

LIST OF ABBREVIATIONS AND SYMBOLS USED xvii

ACKNOWLEDGEMENTS xxiv

Chapter 1 INTRODUCTION 1

11 Motivation 1

12 Distribution Generation - Historic Overview 2

13 Distribution Generation 2

14 Thesis Objectives and Contributions 5

15 Thesis Outline 7

Chapter 2 LITERATURE REVIEW 9

21 Introduction 9

22 Distribution Power Flow 9

23 DG Integration Problem 13

231 Solving the DG Integration Problem via Analytical and Deterministic Methods 14

232 Solving the DG Integration Problem via Metaheuristic Methods 17

24 Summary 20

Chapter 3 FAST AND FLEXIBLE RADIAL POWER FLOW FOR BALANCED AND UNBALANCED THREE-PHASE DISTRIBUTION NETWORKS 21

31 Introduction 21

32 Flexibility and Simplicity of FFRPF in Numbering the RDS Buses and Sections 22

321 Bus Numbering Scheme for Balanced Three-phase RDS 22

322 Unbalanced Three-phase RDS Bus Numbering Scheme 24

33 The Building Block Matrix and its Role in FFRPF 26

v

331 Three-phase Radial Configuration Matrix (RCM) 26

3311 Assessment of the FFRPF Building Block RCM 28

332 Three-phase Section Bus Matrix (SBM) 29

333 Three-phase Bus Section Matrix (BSM) 31

34 FFRPF Approach and Solution Technique 31

341 Unbalanced Multi-phase Impedance Model Calculation 32

342 Load Representation 38

343 Three-phase FFRPF BackwardForward Sweep 40

3431 Three-phase Current Summation Backward Sweep 40

3432 Three-phase Bus Voltage Update Forward Sweep 42

3433 Convergence Criteria 43

3434 Steps of the FFRPF Algorithm 44

344 Modifying the RCM to Accommodate Changes in the RDS 47

35 FFRPF Solution Method for Meshed Three-phase DS 48

351 Meshed Distribution System Corresponding Matrices 50

352 Fundamental Loop Currents 54

353 Meshed Distribution System Section Currents 56

354 Meshed Distribution System BackwardForward Sweep 59

36 Test Results and Discussion 60

361 Three-phase Balanced RDS 60

3611 Case 1 31-Bus with Single Main Feeder RDS 61

3612 Case 2 90-bus RDS with Extreme Radial Topology 70

3613 Case 3 69-bus RDS with Four Main Feeders 71

3614 Case 4 15-bus RDS-Considering Charging Currents 73

362 Three-phase Balanced Meshed Distribution System 74

3621 Case 1 28-bus Weakly Meshed Distribution System 74

3622 Case 2 70-Bus Meshed Distribution System 78

vi

3623 Case 3 201-bus Looped Distribution System 79

363 Three-phase Unbalanced RDS 80

3631 Case 1 10-bus Three-phase Unbalanced RDS 81

3632 Case 2 IEEE 13-bus Three-phase Unbalanced RDS 85

3633 Case 3 26-bus Three-Phase Unbalanced RDS 86

37 Summary 87

Chapter 4 IMPROVED SEQUENTIAL QUADRATIC PROGRAMMING

APPROACH FOR OPTIMAL DG SIZING 89

41 Introduction 89

42 Problem Formulation Overview 89

43 DG Sizing Problem Architecture 90

431 Objective Function 90

432 Equality Constraints 92

433 Inequality Constraints 92

434 DG Modeling 93

44 The DG Sizing Problem A Nonlinear Constrained Optimization Problem 94

45 The Conventional SQP 96

451 Search Direction Determination by Solving the QP Subproblem 96

4511 Satisfying Karush-Khun-Tuker Conditions 98

4512 Newton-KKT Method 101

4513 Hessian Approximation 103

452 Step Size Determination via One-Dimensional Search Method 104

453 Conventional SQP Method Summary 105

46 Fast Sequential Quadratic Programming (FSQP) 108

47 Simulation Results and Discussion 113

471 Case 1 33-busRDS 113

4711 Loss Minimization by Locating Single DG 114

4712 Loss Minimization by Locating Multiple DGs 118

vii

472 Case 2 69-bus RDS 124

4721 Loss Minimization by Locating a Single DG 125

473 Loss Minimization by Locating Multiple DGs 129

474 Computational Time of FSQP vs SQP 134

475 Single DG versus Multiple DG Units Cost Consideration 136

48 Summary 136

Chapter 5 PSO BASED APPROACH FOR OPTIMAL PLANNING OF

MULTIPLE DGS IN DISTRIBUTION NETWORKS 138

51 Introduction 138

52 PSO - The Motivation 138

53 PSO - An Overview 139

531 PSO Applications in Electric Power Systems 141 532 PSO - Pros and Cons 143

54 PSO - Algorithm 144

541 The Velocity Update Formula in Detail 145

5411 The Velocity Update Formula - First Component 146

5412 The Velocity Update Formula - Second Component 148

5413 The Velocity Update Formula-Third Component 149

5414 Cognitive and Social Parameters 150

542 Particle Swarm Optimization-Pseudocode 152

55 PSO Approach for Optimal DG Planning 153

551 Proposed HPSO Constraints Handling Mechanism 155

5511 Inequality Constraints 155

5512 Equality Constraints 157

5513 DG bus Location Variables Treatment 157

56 Simulation Results and Discussion 160

561 Case 1 33-bus RDS 161

viii

5611 33-bus RDS Loss Minimization by Locating a Single DG 161

5612 33-bus RDS Loss Minimization by Locating Multiple

DGs 169

562 Case 2 69-Bus RDS 180

5621 69-bus RDS Loss Minimization by Locating a Single DG 180

5622 69-bus RDS Loss Minimization by Locating Multiple

DGs 187

563 Alternative bus Placements via HP SO 195

57 Summary 196

Chapter 6 CONCLUSION 198

61 Contributions and Conclusions 198

62 Future Work 201

REFERENCES 203

APPENDIX 220

IX

LIST OF TABLES

Table 31 cok rd and De Parameters for Different Operation Conditions 34

Table 32 FFRPF Iteration Results for the 31-Bus RDS 67

Table 33 The 31-bus RDS Section Power Losses Obtained by the FFRPF Method 68

Table 34 FFRPF Voltage Profiles Results for the Three Different Load Models 69

Table 35 Comparisons between 31-Bus RDS Exponential Model Results 70

Table 36 31-bus RDS FFRPF Results vs Other Methods 70

Table 37 90-bus RDS FFRPF Results vs Other Methods 71

Table 38 69-bus RDS FFRPF Results vs Other Methods 73

Table 39 Komamoto 15-bus RDS FFRPF Results vs Other Methods 74

Table 310 Voltage Profiles for Radial and Meshed 28-bus Distribution Network 77 t

Table 311 28-bus Weakly Meshed DS FFRPF Results vs Other Methods 78

Table 312 70-bus Meshed DS FFRPF Results vs Other Methods 79

Table 313 201-bus Meshed DS FFRPF Results vs Other Methods 80

Table 314 10-bus 3(j) RDS FFRPF Results vs Ref [52] and Gauss Zbus Methods 85

Table 315 IEEE 13-bus 3(j) RDS FFRPF Results vs Ref [52] and Gauss Zbus

Methods 86

Table 316 26-bus 34gt RDS FFRPF Results vs Ref [52] and Gauss Zbus Methods 87

Table 41 Single DG Optimal Profile at the 33-bus RDS 115

Table 42 Optimal DG Profiles at all 33 buses 116

Table 43 Multiple DG Installations in the 33-bus RDS with Unspecified Power

Factor 119

Table 44 SQP Method Double-DG Cycled Combinations 121

Table 45 Single and Multiple DG Installations at Pre-specified 085 Power Factor 123

Table 46 Loss Reduction Comparisons for all DG Cases 123

Table 47 69-bus RDS Single DG Optimal Size and Placement Profiles 128

Table 48 Optimal Double DG Profiles in the 69-bus RDS 131

Table 49 Optimal Three DG Units Profiles in the 69-bus RDS 133

Table 410 Loss Reduction Comparison for all DG Installations in the 69-bus RDS 134

Table 411 33-bus RDS CPU Execution Time Comparison 135

Table 412 69-bus RDS CPU Execution Time Comparison 135

x

Table 51 HPSO Parameters for the Single DG Case 162

Table 52 33-bus RDS Single DG Fixedpf Case 20 HPSO Simulations 162

Table 53 Descriptive Statistics for HPSO Results for the Fixedpf Case 163

Table 54 HPSO vs FSQP Results 33-bus RDS-Single DG-FixedDG Case 163

Table 55 33-bus RDS Single DG Unspecified pf Case 20 HPSO Simulations 163

Table 56 Descriptive Statistics for HPSO Results for an Unspecified pf Case 164

Table 57 HPSO vs FSQP Results 33-bus RDS-Single DG-UnspecifiedpCase 164

Table 58 HPSO Parameters for Both Double DG Cases 170

Table 59 33-bus RDS Double DG Fixed pf Case 20 HPSO Simulations 171

Table 510 Descriptive Statistics for HPSO Results for Fixed pf Double DG Case 171

Table 511 HPSO vs FSQP Results 33-bus RDS-Double DGs-FixedCase 172

Table 512 33-bus RDS Double DG Unspecified pf Case 20 HPSO Simulations 172

Table 513 Descriptive Statistics for HPSO Results for UnspecifiedDouble DG

Case 173

Table 514 HPSO vs FSQP Results 33-bus RDS-Double DGs-UnspecifiedCase 173

Table 515 HPSO Parameters for Both Three DG Cases 174

Table 516 33-bus RDS Three DG Fixed pf Case 20 HPSO Simulations 174

Table 517 Descriptive Statistics for HPSO Results for Fixed pf Three DG Case 175

Table 518 HPSO vs FSQP Results 33-bus RDS-Three DG-FixedCase 175

Table 519 33-bus RDS Three DGs Unspecified pf Case 20 HPSO Simulations 175

Table 520 Descriptive Statistics for HPSO Results for the Fixed Three DG Case 176

Table 521 HPSO vs FSQP Results 33-bus RDS-Three DG-UnspecifiedCase 176

Table 522 HPSO Parameters for the Four DG Case 177

Table 523 33-bus RDS Four DG FixedCase 20 HPSO Simulations 178

Table 524 Descriptive Statistics for HPSO Results for Fixed Four DG Cases 178

Table 525 HPSO vs FSQP Results 33-bus RDS-Four DG-FixedCase 179

Table 526 33-bus RDS Four DG Unspecified pf Case 20 HPSO Simulations 179

Table 527 Descriptive Statistics for HPSO Results for Unspecified Four DG

Case 179

Table 528 HPSO vs FSQP 33-bus RDS-Four -Unspecified pf Case 180

Table 529 HPSO Parameters for 69-bus RDS Both Single DG Cases 181

Table 530 69-bus RDS Single DG Fixed pf Case 20 HPSO Simulations 182

xi

Table 531 Descriptive Statistics for HPSO Results for the Fixed Single DG Case 182

Table 532 HPSO vs FSQP Results 69-bus RDS-Single DG-FixedpDG Case 182

Table 533 69-bus RDS Single DG Unspecifiedpf Case 20 HPSO Simulations 183

Table 534 Descriptive Statistics for Unspecified pSingle DG Case 183

Table 535 HPSO vs FSQP Results 69-bus RDS-Single DG-UnspecifiedpDG

Case 184

Table 536 HPSO Parameters for 69-bus RDS the Double DG Cases 188

Table 537 69-bus RDS Double DG Fixed pf Case 20 HPSO Simulations 189

Table 538 Descriptive Statistics for HPSO Results for Fixed pDouble DG Case 189

Table 539 HPSO vs FSQP Results 69-bus RDS-Double DG-FixedpDG Case 190

Table 540 69-bus RDS Double DG UnspecifiedpfCase 20 HPSO Simulations 190

Table 541 Descriptive Statistics for HPSO Results for Unspecified ^Double DG

Case 191

Table 542 HPSO vs FSQP Results 69-bus RDS-Double-UnspecifiedpDG Case 191

Table 543 HPSO Parameters for Both 69-bus RDS Three DG Cases 192

Table 544 69-bus RDS Three DG Fixed pf Case 20 HPSO Simulations 192

Table 545 Descriptive Statistics for HPSO Results for Fixed pf Three DG Case 193

Table 546 HPSO vs FSQP Results 69-bus RDS-Three DGs-FixedCase 193

Table 547 69-bus RDS Three DG Unspecified pf Case 20 HPSO Simulations 194

Table 548 Descriptive Statistics for HPSO Results for Unspecified pf Three DG

Case 194

Table 549 HPSO vs FSQP Results 69-bus RDS-Three DGs-UnspecifiedpCase 195

Table 550 20 HPSO Simulations of the 69-bus RDS Three DG Fixed pf Case with Suboptimal Tuned Parameters 50 Iterations and 50 Swarm Particles 196

xii

LIST OF FIGURES

Figure 31 10-bus RDS 23

Figure 32 Different ways of numbering the system in Fig 31 24

Figure 33 The ease of numbering a modified and augmented RDS 24

Figure 34 Three-phase unbalanced 6-bus RDS representation 25

Figure 35 Three-phase unbalanced 6-bus RDS with missing buses and sections 26

Figure 36 RCM matrices sparsity plots for the 25 and 90 bus RDSs 28

Figure 37 SBM for three-phase unbalanced 6-bus RDS 30

Figure 38 Three-phase section model 32

Figure 39 The final three-phase section model after Kron s reduction 34

Figure 310 Nominal ^-representation for three-phase RDS section 36

Figure 311 Three-phase capacitor bank (a) The schematics (b) The Modeling 40

Figure 312 The FFRPF solution method flow chart 46

Figure 313 10-bus meshed distribution network 50

Figure 314 Fundamental cut-sets for a meshed 10-bus DS 57

Figure 315 31-bus RDS 62

Figure 316 The RCM of the 31-bus RDS 63

Figure 317 The RCM-1 of the 31-bus RDS 64

Figure 318 The SBM of the 31-bus RDS 65

Figure 319 The BSM of the 31-bus RDS 66

Figure 3 20 90-Bus RDS 71

Figure 321 69-bus multi-feeder RDS 72

Figure 322 Komamoto 15-bus RDS 73

Figure 323 28-bus weakly meshed distribution network 75

Figure 324 mRCM for 28-bus weakly meshed distribution network 75

Figure 325 mSBM for 28-bus weakly meshed distribution network 76

Figure 326 C for 28-bus weakly meshed distribution network 76

Figure 327 70-bus meshed distribution system 78

Figure 328 201-bus hybrid augmented test distribution system 80

Figure 329 10-bus three-phase unbalanced RDS 81

Figure 330 The 10-bus three-phase unbalanced RDS RCM 82

xni

Figure 331 21 The 10-bus three-phase unbalanced RDS RCM1 83

Figure 332 The 10-bus three-phase unbalanced RDS SBM 84

Figure 333 The 10-bus three-phase unbalanced RDS BSM 85

Figure 334 IEEE 13-bus 3^ unbalanced RDS 86

Figure 41 The Conventional SQP Algorithm 107

Figure 42 The FSQP Algorithm 112

Figure 43 Case 1 33-busRDS 114

Figure 44 Optimal real power losses for placement of optimal DG size at all the 32

buses using APC method 117

Figure 45 Optimal real power losses vs different DG power factors at bus 30 117

Figure 46 Bus voltages improvement before and after installing a single DG at

bus 30 118

Figure 47 Voltage profiles comparisons of 33-bus RDS cases 120

Figure 48 Voltage improvement of the 33-bus RDS due to three DG installation

compared to pre-DG single and double-DG cases 122

Figure 49 Voltage profiles improvement in 33-bus RDS for all DG cases 124

Figure 410 Case 2 69-bus RDS test case 125

Figure 411 Optimal power losses obtained using APC procedure 126

Figure 412 Real power losses vs DG power factor 69-bus RDS 128

Figure 413 Bus voltage improvements via single DG installation in the 69-bus

RDS 129

Figure 414 Variation in power losses as a function of the DG output at bus 61 129

Figure 415 Bus voltage magnitudes of the original 69-bus RDS and Single DG

and double DGs cases 131

Figure 416 69-bus RDS bus voltage magnitudes improvement in all DG cases 133

Figure 51 Number of publications in IEEEIET and ScienceDirect Databases

since the year 2000 140

Figure 52 Interaction between particles to share the gbest information 150

Figure 53 Illustration of velocity and position updates mechanism for a single particle during iteration k 151

Figure 54 PSO particle i updates its velocity and position vectors during two consecutive iterations A and k+ 152

Figure 55 The proposed HPSO solution methodology 159

xiv

Figure 56 Convergence characteristic of the proposed HPSO in the fixedsingle DG case HPSO proposed number of iterations = 30 164

Figure 57 Convergence characteristic of the proposed HPSO in the fixedsingle

DG case HPSO extended number of iterations = 50 165

Figure 58 Swarm particles on the first HPSO iteration 165

Figure 59 Swarm particles on the fifth HPSO iteration 166

Figure 510 Swarm particles on the tenth HPSO iteration 166

Figure 511 Swarm particles on 15th HPSO iteration 167

Figure 512 Swarm particles on the 20 HPSO iteration 167

Figure 513 Swarm Particles on the 25th HPSO iteration 168

Figure 514 Swarm Particles on the last HPSO iteration 168

Figure 515 A close-up for the particles on the 30th HPSO iteration 169

Figure 516 Convergence characteristics of HPSO in the 69-bus fixedsingle DG case HPSO proposed number of iterations = 15 184

Figure 517 Convergence characteristics of HPSO in the 69-bus fixed single DG

case HPSO proposed number of iterations = 50 185

Figure 518 Swarm particles distribution at the first HPSO iteration 185

Figure 519 Swarm particles distribution at the 5 HPSO iteration 186

Figure 520 Swarm particles distribution at the 10 HPSO iteration 186

Figure 521 Swarm particles distribution at the 15l HPSO iteration 187

Figure 522 Close up of the HSPO particles at iteration 15 187

xv

ABSTRACT

Distribution Generation (DG) has gained increasing popularity as a viable element of electric power systems DG as small scale generation sources located at or near load center is usually deployed within the Distribution System (DS) Deployment of DG has many positive impacts such as reducing transmission and distribution network congestion deferring costly upgrades and improving the overall system performance by reducing power losses and enhancing voltage profiles To achieve the most from DG installation the DG has to be optimally placed and sized In this thesis the DG integration problem for single and multiple installations is handled via deterministic and heuristic methods where the results of the former technique are used to validate and to be compared with the latters outcomes

The unique structure of the radial distribution system is exploited in developing a Fast and Flexible Radial Power Flow (FFRPF) method that accommodates the DS distinct features Only one building block bus-bus oriented data matrix is needed to perform the proposed FFRPF method Two direct descendent matrices are utilized in conducting the backwardforward sweep employed in the FFRPF technique The proposed method was tested using several DSs against other conventional and distribution power flow methods Furthermore the FFRPF method is incorporated within Sequential Quadratic Programming (SQP) method and Particle Swarm Optimization (PSO) metaheuristic method to satisfy the power flow equality constraints

In the deterministic solution method the sizing of the DG is formulated as a constrained nonlinear optimization problem with the distribution active power losses as the objective function to be minimized subject to nonlinear equality and inequality constraints Such a problem is handled by the developed Fast Sequential Quadratic Programming method (FSQP) The proposed deterministic method is an improved version of the conventional SQP that utilizes the FFRPF method in handling the power flow equality constraints Such hybridization resultes in a more robust solution method and drastically reduces the computational time In a subsequent step the placement portion of the DG integration problem is dealt with by using an All Possible Combinations (APC) search method Afterward the FSQP methods outcomes were compared to those of the developed metaheuristic optimization method

The difficult nature of the overall problem poses a serious challenge to most derivative based optimization methods due to the discrete nature associated with the bus location Moreover a major drawback of deterministic methods is that they are highly-dependent on the initial solution point As such a new application of the PSO metaheuristic method in the DG optimal planning area is presented in this thesis The PSO is improved in order to handle both real and integer variables of the DG mixed-integer nonlinear constrained optimization problem The algorithm is utilized to simultaneously search for both the optimal IDG size and bus location The proposed approach hybridizes PSO with the developed FFRPF algorithm to satisfy the equality constraints The inequality constraints handling mechanism is dealt with in the proposed Hybrid PSO (HPSO) by combining the rejecting infeasible solutions method with the preserving feasible solutions method Results signify the potential of the developed algorithms with regard to the addressed problems commonly encountered in DS

xvi

LIST OF ABBREVIATIONS AND SYMBOLS USED

ACO

BFGS

BSM

CHP

CIGRE

CN

DER

DG

DG

DGs

DS

EG

EP

EPAct

EPRI

FD

FFRPF

FSQP

GA

GRG

GS

GWEC

HPSO

IP

KCL

KKT

KVL

LP

Ant Colony Optimization

Quasi-Newton method for Approximating and Updating the Hessian Matrix

Bus Section Matrix

Combined-Heat and Power

The International Council on Large Electric Systems

Condition Number

Distribution Energy Resources

Dispersed Generation

Decentralized Generation

Distribution Generation sources

Distribution System

Embedded Generation

Evolutionary Programming

English Policy Act of 1992

Electric Power Research Institute

Fast Decoupled

Fast and Flexible Radial Power Flow

Fast Sequential Quadratic Programming

Genetic Algorithm

Generalized Reduced Gradient

Gauss-Seidel

Global Wind Energy Council

Hybrid PSO

Interior Point method

Kirchhoff s Current Law

Karush-Khun-Tuker conditions

Kirchhoff s Voltage Law

Linear Programming

xvii

wBSM

mNS

wRCM

mRCM

mSBM

mSBMp

NB

NB

HDG

riL

NR

NS

NS

ftwDG

Pf PSO

PUHCA

PURPA

QP

RCM

RDS

RIT

RPF

SA

SBM

SE Mean

SQP

StDev

TS

UnSpec pf

Meshed BSM

Number of segments in meshed DS

Meshed RCM

Modified mRCM

Meshed SBM

Submatrix of wSBM that correspond to the RDS tree sections

Number of Buses

Number of DS Buses

Total number of DGs

Number of Links or number of the fundamental loops

Newton-Raphson

Number of Sections

Number of Sections in RDS AND in meshed DS tree

Total number of the unspecified pf DGs

power factor

Particle Swarm Optimization

Public Utilities Holding Company Act of 1935

Public Utilities Regulatory Policy Act of 1978

Quadratic Programming

Radial Configuration Matrix

Radial Distribution System

The Reduction in CPU execution Time

Radial Power Flow

Simulated Annealing

Section Bus Matrix

Standard Error of the Mean

Sequential Quadratic Programming

Standard Deviation

Tabu search algorithm

Unspecified power factor DG

xviii

U S P B Unique Set of Phase Buses

USPS Unique Set of Phase Sections

xf Unique set of phase buses

iff Unique set of phase sections

Zsec Section primitive impedance matrix

Z^ (3 X 3) section symmetrical impedance matrix

R D S section length

zu Per unit length self-impedance of conductor i

h Per unit length mutual- impedance be tween conductors a n d

rt Resis tance of conductor i

rd Ear th return conductor resistance

k Inductance multiplying constant

De Dis tance between overhead and its earth return counterpart

GMRj Geometr ic mean radius of conductor i

Dy Dis tance between conductors a n d

Vgbc Three-phase sending end voltages

Vg deg Three-phase receiving end voltages

Ias c Three-phase sending end section currents

lfc Three-phase receiving end section currents

Fscc Three-phase shunt admittance of section k

[]3x3 (3 X 3) identity matrix

[^Lx3 (3x3) zero matrix

^Klc Vol tage drop across three-phase section k

ysect Section k three-phase currents

V0 Nomina l bus voltage

V Operat ing bus voltage

xix

P0 Real power consumed at nominal voltage

Q0 Reactive power consumed at nominal voltage

S Bus load apparent power at single-phase bus sect

YsKus Total three-phase shunt admittance at bus i

Ic Three-phase shunt currents at bus i

IlucSi Bus three-phase currents

jabc Three-phase load current

IltLP Current through single-phase section p and phase ltjgt

its Current at bus and phase ^

Vss Substation voltage magnitude

Vls Substation complex phase voltage

VLt Voltage drop across section k in phase (j)

A and symbol

IMI oo-norm vector II I loo

91 (bull) Real part of complex value

3 (bull) Imaginary part of complex value

C Fundamental loop matrix which is a submatrix of mSBM

Zioop (laquoLx nL) loop-impedance matrix

Csec Upper submatrix of the C matrix with ((NB-1) x (nL) dimension

Zoop Loop-impedance matrix

setrade (NSxNS) meshed DS section-impedance diagonal matrix

ZtradeS (NSxNS) RDS or tree section-impedance diagonal matrix

IL (NB-1 x 1) RDS bus load currents vector

fnlsec (mNS x 1) segments currents column vector of meshed DS network vector

mILL (mNSx 1) meshed DS bus loads and links currents vector

Itrade (NB-1) tree section currents column

xx

( n L x 1) fundamental loop current vector which is also the meshed DS link loop

currents column vector

B ( N B - 1 xmNS) fundamental cut-sets matrix

^ s7 e c f a ( N B - 1 x nL) co-tree cut-set matrix

^ymesh Voltage drops across the tree sections of the meshed DS vector

ymesh j k g messed DS bus voltage profiles vector

PRPL Real power losses

Pj Generated power delivered to DS bus i

PjL Load power supplied by DS bus i

Yjj Magnitude of the if1 element admittance bus matrix

rv Phase angle of Yy = YyZyy

Vi Magnitude of DS bus complex voltage

8 Phase angle of V = ViZSl

bull Transpose of vector or matrix

bull Complex conjugate of vector or matrix

V (1 xNB) DS bus Thevenin voltages

Y (NB xNB) DS admittance matrix

A^ Real power mismatch at bus i

AQt Reactive power mismatch at bus i

|L| Infinity norm where llxll = max (|x|) II llco J II llraquo =l2iVBVI ]gt

bull+ Max imum permissible value

bull Minimum permissible value

bull0 Nominal value

PDG D G operating power factor

S^G D G generated apparent power

SsS Main DS substation apparent power

1 Scalar related to the allowable D G size

xxi

Sy Apparent power flow transmitted from bus to bus j

Stradex Apparent power maximum rating for distribution section if

(x) The objective function

z(x) Equality constraints

g(x) Inequality constraints

(bull) Independent unknown variables lower bounds

(bull) Independent unknown variables upper bounds

x Independent unknown variables vector

RPL ( X ) Distribution system real power losses objective function

d Search direction vector

a Positive step size scalar

WRPL (x ) Gradient of the objective function at point xk)

pound Lagrange function

H^ (nxri) Hessian symmetric matrix at point xw

h^ First-order Taylors expansion of the equality constraints at point xw

Vh(x^) (nxm) Jacobian matrix of the equality constraints at point xw

g ^ First-order Taylors expansion of the inequality constraints at point xw

Vg(x^) (nxp) Jacobian matrix of the inequality constraints at point xw

Xi Individual equality Lagrange multiplier scalar

Pi Individual inequality Lagrange multiplier scalar

k w-dimensional equality Lagrange multiplier vector

P (-dimensional inequality Lagrange multiplier vector

s A predefined small tolerance number

A Active set

m Number of all equality constraints

p Number of all inequality constraints

a Number of the active set equations

xxii

v 2 j6k)

XX

nTgtG

nuDG

y

v Y FFRPFbl

deg FFRPF bl

llAP II II lloo

Vi

Xi

Cj C2

rXgtr2

w

pbestj

gbesti

nk

X

APT Losses

pHPSO Losses

pFSQP Losses

Hessian of the Lagrange function

Total number of DGs

Total number of the unspecified DGs

The change in the Lagrange functions between two successive iterations

Voltage magnitude of bus i obtained by the FFRPF technique

Voltage phase angle of bus obtained by the FFRPF technique

Voltage deviation infinity norm ie II AV = max ( I F - K I ) deg llcD

=UAlaquoVI deg ngt

Particle i velocity

Particle i position vector

Individual and social acceleration positive constants

Random values in the range [0 l] sampled from a uniform distribution

Weight inertia

Personal best position associated with particle own experience

Global best position associated with the whole neighborhood experience

Maximum number of iterations

Constriction factor

The deviation of losses calculated by HPSO method from that determined

by FSQP method

Mean value of HPSO simulation results of real power losses

FSQP deterministic method result of real power losses

xxiii

ACKNOWLEDGEMENTS

All Praise and Thanks be to reg (Allah) Almighty whose countless bounties enabled me to

accomplish this thesis successfully I would like to express my deepest gratitude to my

parents who taught me the value of education and hard work A special note of gratitude

to my brothers Falah and Abdullah deer sisters my wife my daughter Najla and my

sons Fahad Falah and Othamn They endured the long road along with me and

provided me with constant support motivation and encouragement during the course of

my study

I would like to express my sincere gratitude to my advisor Dr M E El-Hawary for

his professional guidance valuable advice continual support and encouragement I also

appreciate the constructive comments of my PhD External Examiner Dr M A Rahman

I am also grateful to my advisory committee members Dr T Little and Dr W Phillips

for spending their valuable time in reading evaluating and discussing my thesis

I would like to acknowledge the academic discussions and the constant

encouragement I received from my dear friend Dr Mohammed AlRashidi- Thank you

Abo Rsheed I wish also to thank a special friend of mine Dr AbdulRahman Al-

Othman for his friendship and for believing in me

I would like to manifest my gratitude to the Public Authority for Applied Education

and Training in Kuwait who sponsored me through my PhD at Dalhousie University

From the Embassy of Kuwait Cultural Attache Office special thanks are due to Shoghig

Sahakyan for her efforts and help to make this work possible

xxiv

CHAPTER 1 INTRODUCTION

11 MOTIVATION

Electric power system networks are composed typically of four major subsystems

generation transmission distribution and utilizations Distribution networks link the

generated power to the end user Transmission and distribution networks share similar

functionality both transfer electric energy at different levels from one point to another

however their network topologies and characteristics are quite different Distribution

networks are well-known for their low XR ratio and significant voltage drop that could

cause substantial power losses along the feeders It is estimated that as much as 13 of

the total power generation is lost in the distribution networks [1] Of the total electric

power system real power losses approximately 70 are associated with the distribution

level [23] In an effort towards manifesting the seriousness of such losses Azim et al

reported that 23 of the total generated power in the Republic of India is lost in the form

of losses in transmission and distribution [4]

Distribution systems usually encompass distribution feeders configured radially and

exclusively fed by a utility substation Incorporating Distribution Generation sources

(DGs) within the distribution level have an overall positive impact towards reducing the

losses as well as improving the network voltage profiles Due to advances in small

generation technologies electric utilities have begun to change their electric

infrastructure and have started adapting on-site multiple small and dispersed DGs In

order to maximize the benefits obtained by integrating DGs within the distribution

system careful attention has to be paid to their placement as well as to the appropriate

amount of power that is injected by the utilized DGs In other words to achieve the best

results of DG deployments the DGs are to be both optimally placed and sized in the

corresponding distribution network

The motivation of this thesis research is to investigate placing and sizing single and

multiple DGs in Radial Distribution Systems (RDSs) The problem investigated involves

two stages finding the optimal DG placements in the distribution network and the

optimal size or rating of such DGs The optimal DG placement and sizing are dealt with

by utilizing deterministic and heuristic optimization methods

12 DISTRIBUTION GENERATION - HISTORIC OVERVIEW

During the first third of the twentieth century there were no restrictions on how many

utility companies could be owned by financial corporations known as utility holding

companies By 1929 80 of US electricity was controlled by 16 holding companies

and three of those corporations controlled 36 of the nations electricity market [5]

During the Great Depression most of these utility holding companies went bankrupt As

a result the US Congress Public Utilities Holding Company Act (PUHCA) of 1935

regulated the gas and electric industries and restricted holding companies to the

ownership of a single integrated utility PUHCA indirectly discouraged wholesale

wheeling of power between different states provinces or even countries The Public

Utilities Regulatory Policy Act (PURPA) of 1978 allowed grid interconnection and

required electric utilities to buy electricity from non-utility-owned entities called

Qualifying Facilities (QF) at each utilitys avoided cost The term QF refers to non-

utility-owned (independent) power generators The term at each utilitys avoided cost

is interpreted to mean that the utility shall buy the generated electricity at a price

equivalent to what it would cost the utility itself if had generated the same amount of

power in its own facility or if it had purchased the power from an open electricity market

ie what the utility saves by not generating the same amount of power This act heralded

the dawn of the DG industry era which paved the way to generate electricity arguably at

a lower cost compared to that of traditional utility companies and consequently have it

delivered to the end user at lower rates The English Policy Act of 1992 (EPAct)

intensified competition in the wholesale electricity market by opening the transmission

system for access by utilities and non-utilities electricity producers [67] entity A could

sell its power to entity B through entity Cs transmission infrastructure

13 DISTRIBUTION GENERATION

DG involves small-scale generation sources scattered within the distribution system level

atnear the load center ie close to where the most energy is consumed [8] The DG

2

generate electricity locally and in a cogeneration case heat can also be generated and

may be utilized in applications such as industrial process heating or space heating DG

generally has better energy efficiency than large-scale power plants The traditional

power stations usually have an efficiency of around 35 whereas the efficiency of DG

such as a Combined Heat and Power (CHP) gas turbine would be in the vicinity of 45-

65 [5]

It seems that there is no universal agreement on the definition of DG size range The

Electric Power Research Institute (EPRI) for example defined the DG size to be up to 5

MW in 1998 [9] in 2001 EPRI redefined the DG capacity to be less than 10 MW [10]

and by 2003 they identified the DG to have a power output ranging from 1 kW to 20 MW

[11] The IEEE published its DG-standards IEEE Std 1547-2003 and IEEE Std 15473-

2007 and emphasized that they are applicable to DGs that have total capacity below 10

MVA [1213] In its 2006 report about the impact of DG Natural Resources Canada

estimated that the DG size starts from few 10s of kW to perhaps 5 MW [14] In 2000

the International Council on Large Electric Systems (CIGRE) referred to the DG as non-

centrally dispatched usually attached to distribution level and smaller than 50-100 MW

[1516]

Many terms referring to DG technology are used in the literature such as Dispersed

Generation (DG) Decentralized Generation (DG) Embedded Generation (EG)

Distribution Energy Resources (DER) and on-site Generation [17] In particular the

term dispersed generation customarily refers to stationary small-scale DG with power

outputs ranging from 1 kW to 500 kW [7]

Late developments and innovations in the DG technology industry liberalization of

the electricity market transmission line congestion and increasing interest in global

warming and environmental issues expedited publicizing their deployment and adoption

world-wide Recent studies suggest that DG will play a vital role in the electric power

system An EPRI study predicts that by the year 2010 25 of the newly installed

generation systems will be DG [18] and a similar study by the Natural Gas Foundation

projects that the share of DG in new generation will be 30 [15] By 2003 around 40

of Denmarks power demand was served by DG while Spain the Netherlands Portugal

and Germany integrated nearly 20) of DG into their distribution networks [19] Of the

3

643 GW generated by the European Union in 2005 approximately 122 GW (19) was

generated by hydro 96 GW (15) of this generated capacity was cogeneration (CHP)

and 53 GW (8) generated by other renewable energy systems Half of the CHP

generated capacity was owned by utility companies and the other half was generated by

independent producers [20]

Globally in 2005 the total installed wind power capacity was 591 GW and the

Global Wind Energy Council (GWEC) expected the wind capacity to reach 1348 GW by

the year 2010 [21] Worldwide wind energy capacity of 19696 MW was added in the

year 2007 [22] and approximately 1400 MW of wind energy capacity was added in the

US during the second quarter of 2008 [23] GWEC also predicted in its Global Wind

Energy Outlook 2008 report that by the year 2020 15 billion tons of CO2 will be saved

every year and by the middle of the 21st century 30 of the worlds electricity will be

supplied by wind energy [24] compared to a total of 13 of the global electricity being

generated by wind at the end of 2007 [22]

DG technologies include a variety of energy sources ie powered by renewable or

by fossil fuel-based prime movers Renewable technologies used in DG include wind

turbines photovoltaic cells small hydro power turbines and solar thermal technologies

while DG based on conventional technologies may involve gas turbines CHP gas

turbines diesel engines fuel cells and micro-turbine technologies Some DGs are

installed by the utility company on the supply side of the consumers meter while some

are installed by the customers themselves on their side of a bi-directional meter thus

enabling them to benefit from the net-metering program offered by utility companies

[25]

Optimal deployment of DG technology would have an overall positive impact

although some negative traits would remain The noise and shadow flicker caused by

large wind blades and the noise caused by the wind turbine gearbox or gas turbines

especially when placed close to residential or populated areas are examples of negative

impacts of widespread use of DG Another drawback from an environmentalist point of

view is that wind DG could disturb bird immigration patterns and cause death to both

birds and bats [26] Renewable-source DGs also could be an indirect source of pollution

by causing the fossil-fuel power plants to shut down and start up more frequently as they

4

attempt to accommodate variable DG power output [27] Some plants have an emission

rate which is inversely proportional to its delivered power Voltage rise as a result of bishy

directional power flow caused by the interconnection of the DG in RDS is another

example of a negative impact caused by DG [28]

The integration of DG into electric power networks has many benefits Some

examples of such benefits could be summarized as follows

bull Improve both the reliability and efficiency of the power supply

bull Release the available capacity of the distribution substation as well as reducing

thermal stresses caused by loaded substations transformers and feeders

bull Improve the system voltage profiles as well as the load factor

bull Decrease the overall system losses

bull Generally DG development and construction have shorter time intervals

bull Delay imminent upgrading of the present system or the need to build newer

infrastructure and subsequently avoid related problems such as right-of-way

concerns

bull Decrease transmission and distribution related costs

bull In general DG tends to be more environmentally friendly when compared to

traditional coal oil or gas fired power plants

The extent of the benefits depends on how the DG is placed and sized in the system In

addition to supplying the system with the power needed to meet certain demands as an

installation incentive the real power losses could be minimal if the DG is optimally sited

and sized

14 THESIS OBJECTIVES AND CONTRIBUTIONS

Optimal integration of single and multiple DG units in the distribution network with

specified and unspecified power factors is thoroughly investigated from a planning

perspective in this thesis The DG problem is handled via deterministic and heuristic

optimization methods where the results of the former method are used to validate and to

be compared with those of the latter

The unique radial distribution structure is exploited in developing a Fast and Flexible

Radial Power Flow (FFRPF) method to deal with a wide class of distribution systems

5

eg radial meshed small large balanced and unbalanced three-phase networks The

proposed FFRPF algorithm starts by developing a Radial Configuration Matrix (RCM)

for radial topology DS or meshed RCM (mRCM) for meshed DS Both matrices consist

of elements with values 1 0 and -1 The RCM (or mRCM) corresponding building

algorithm is simple fast and practical as illustrated in Chapter 3 The RCM is inverted

only once to produce the Section Bus Matrix (SBM) which is then transposed to obtain

the Bus Section Matrix (BSM) For the meshed topology the corresponding resultant

matrices for the mRCM are the meshed Bus Section Matrix (mSBM) and the meshed Bus

Section Matrix (mBSM) The FFRPF technique relies on the backwardforward sweep

that only utilizes the basic electric laws such as Ohms law and both Kirchhoff s voltage

and current laws The backward current sweep is performed via SBM (or mSBM) and

the forward voltage update sweep is executed via the BSM (or mBSM) By utilizing the

two obtained matrices all the bus complex voltages can be obtained and consequently

left to be compared with the immediate previous obtained bus voltages The proposed

approach quickened the iterative process and reduced the CPU time for convergence It

is worth mentioning that the building block matrix is the only input data required by the

FFRPF method besides the DS parameters to perform the proposed distribution power

flow The FFRPF technique is incorporated in both utilized deterministic and

metaheuristic optimization methods to satisfy the power flow equality constraints

requirements

In the deterministic solution method the DG sizing problem is formulated as a

nonlinear optimization problem with the distribution active power losses as the objective

function to be minimized subject to nonlinear equality and inequality constraints

Endeavoring to obtain the optimal DG size an improved version of the Sequential

Quadratic Programming (SQP) methodology is used to solve for the DG size problem

The conventional SQP uses a Newton-like method which consequently utilizes the

Jacobean in handling the nonlinear equality constraints The radial low XR ratio and the

tree-like topology of distribution systems make the system ill-conditioned

A Fast Sequential Quadratic Programming (FSQP) methodology is developed in

order to handle the DG sizing nonlinear optimization problem The FSQP hybrid

approach integrates the FFRPF within the conventional SQP in solving the highly

6

nonlinear equality constraints By utilizing the FFRPF in dealing with equality

constraints instead of the Newton method the burden of calculating the Jacobean and

consequently its inverse as well as the complications of the ill-conditioned Y-matrix of

the RDS is eliminated Another advantage of this hybridization is the drastic reduction

of computational time compared to that consumed by the conventional SQP method

In this thesis a new application of the Particle Swarm Optimization (PSO) method in

the optimal planning of single and multiple DGs in distribution networks is also

presented The algorithm is utilized to simultaneously search for both the optimal DG

size and its corresponding bus location in order to minimize the total network power

losses while satisfying the constraints imposed on the system The proposed approach

hybridizes PSO with the developed distribution radial power flow ie FFRPF to

simultaneously solve the optimal DG placement and sizing problem The difficult nature

of the overall problem poses a serious challenge to most derivative based optimization

methods due to the discrete flavor associated with the bus location in addition to the

subproblem of determining the most suitable DG size Moreover a major drawback of

the deterministic methods is that they are highly-dependent on the initial solution point

The developed PSO is improved in order to handle both real and integer variables of the

DG mixed-integer nonlinear constrained optimization problem Problem constraints are

handled within the proposed approach based on their category The equality constraints

ie power flows are satisfied through the FFRPF subroutine while the inequality bounds

and constraints are treated by exploiting the intrinsic and unique features associated with

each particle The proposed inequality constraint handling technique hybridizes the

rejection of infeasible solutions method in conjunction with the preservation of feasible

solutions method One advantage of this constraint handling mechanism is that it

expedites the solution method converging time of the Hybrid PSO (HPSO)

15 THESIS OUTL INE

This thesis is organized in six chapters The research motivation brief description of the

DG and the thesis objectives are addressed in the first chapter The second chapter deals

with a literature review of the distribution power flow methods and the DG optimal

planning problem In the third chapter development of the proposed FFRPF method

7

utilized in the FSQP and the HPSO methods to satisfy the DG problem of nonlinear

equality constraints is presented The fourth chapter deals with the DG sizing problem

formulation and its solution based on the two deterministic solution methods The

problem is solved via the conventional SQP and the proposed FSQP methods and a

performance comparison between them is presented Basic concepts of the PSO are

presented in chapter five A brief literature review regarding the use of the PSO in

solving the electric power system problems is presented in this chapter In addition it

also addresses the development of the proposed HPSO in solving the DG planning

problem The last chapter provides the thesis concluding remarks and the scope of future

work

8

CHAPTER 2 LITERATURE REVIEW

21 INTRODUCTION

Recent publications in the areas of work relative to this thesis are reviewed and

summarised in this chapter which is organized in two sections as follows

bull The first section reviews the literature on distribution power flow methods A

brief background of conventional power flow methods is presented followed

by a review and summary of the literature on recent developments of the

distribution power flow algorithms

bull The DG integration problem is reviewed in the second section Recent work

on the optimal DG placement and sizing via analytical deterministic and

metaheuristic methods are analyzed and reviewed

22 DISTRIBUTION POWER FLOW

Power flow programs play a vital role in analyzing power systems The problem deals

with calculating unspecified bus voltage angles and magnitudes active and reactive

powers as well as (as a by-product) line loadings and their associated real and reactive

losses for certain operating conditions These values are typically obtained through

iterative numerical methods to analyze the status of a given power system

Since the middle of last century many methods were proposed to solve this problem

Even though Dunstan [29] was the first to demonstrate a digital method for solving the

power flow problem in 1954 Ward and Hale [30] are often credited with the successful

digital formulation and solution of the power flow problem in 1956 Most of the earlier

solution methods were based on both the admittance matrix and the Gauss-Seidel (GS)

iterative method The poor convergence characteristics of GS when large networks

andor ill-conditioned situations are encountered led to the development of the Gaussian

iterative scheme (Zbus) [3132] and later the Newton-Raphson (NR) method [33] as well

as the Decoupled [34] and Fast Decoupled (FD) power flow approaches [35] Though

the NR method generally converges faster than other methods it takes longer

computational time per iteration When Tinney et al [36] introduced the optimally

9

ordered and sparsity-oriented programming techniques Newton-based methods became

the de facto industry standard However the Jacobian matrix for the RDS is

approximately four times the size of the corresponding admittance matrix and it needs to

be evaluated at each iteration

Although conventional power flow methods are well developed in dealing with the

transmission and sub-transmission sections of the power system networks they are

deemed to be inefficient in handling distribution networks This is because the

Distribution System (DS) is different in several ways from its transmission counterpart

DS has a strictly radial topology nature or weakly meshed networks in contrast with

transmission systems which are tightly meshed networks DS is a low voltage system

having low XR ratio sections and a wide range of reactance and resistance values DS

may consist of a tremendously large number of sections and buses spread throughout the

network Sections of the DS could have unbalanced load conditions due to the

unbalanced three-phase loading as well as single and double phase loads in spurred

lateral lines The mutual couplings between phases are not negligible due to rarely

transposed distribution lines [37] All of these characteristics strongly suggest that DS is

to be classified as an ill-conditioned power system

The practical DSs low XR ratio sections may cause both the NR and FD

conventional methods to diverge [38-41] The line impedance angles are small enough to

deteriorate the dominance of the NR Jacobian main diagonal making it prone to

singularity Such a low XR value would also prevent the Jacobian matrix from being

decoupled and simplified

In addition to performance considerations a practical power flow technique needs to

consider all the DS distinctive features and to accommodate the imbalance introduced by

multiphase networks along with the distribution-level loads In the literature a number

of Newton and non-Newton power flow methods designed for distribution systems were

proposed Zhang et al [42] solved the distribution power flow based on the Newton

method although the proposed Jacobian is computed just once the solution converged

with a number of additional iterations more so than the conventional approach

Moreover shunt capacitor banks were ignored in the modeling as well as the line shunt

admittance (JI model) and the constant impedance loads Baran and Wu [43] solved the

10

power flow problem by utilizing three fundamental quadratic equations representing the

real and reactive section powers and the bus voltages in an iterative scheme as a

subroutine during the process of optimizing the capacitor sizing However they

computed the Jacobian using the chain rule within the proposed NR method which is in

turn time consuming Mekhamer et al [44] utilized the three equations developed in [43]

using a different iterative technique without the need for the Jacobian or the NR method

However their process is based on applying a multi-level iterative process on the main

feeder and laterals which makes the speed and the efficiency of their proposed algorithm

a function of the RDS configuration and topology

In [4546] the quadratic equation was also utilized in determining the relation

between the sending and receiving end voltage magnitudes along with the section power

flow They proposed to include the system power losses within their calculation while

solving for the system power flow However the voltage phase angles were ignored

during the solution of the radial power flow in order to speed up the convergence The

latter reference developed work was based on the assumption of balanced RDS and

sophisticated numbering scheme

The radial power flow introduced by [47-49] used a non-Newton power flow techshy

nique based on the ladder network theory This method adds the section currents and

calculates the RDS bus voltages including the substations during a backward sweep If

the difference between the calculated substation voltage value and substation predetershy

mined assigned bus voltage value is acceptable the iterations are concluded If not the

substation bus voltage is reset and the RDS bus voltages are computed for the second

time in the same iteration in the forward sweep Both the ladder and the backshy

wardforward methods are derivative-free instead they employ simple circuit laws

However the ladder method uses many sub-iterations on the laterals and calculates the

system bus voltages twice during a single iteration compared to once in the backshy

wardforward method Thukaram [50] utilized the backwardforward sweep technique to

solve the RDS power flow However the bus numbering procedure was a sophisticated

parent node and child node arrangement which may add some computational overshy

head if the system topology is changed Teng [51] used the backwardforward approach

as the solution procedure through the development of two matrices and multiplied them

11

together in a later stage of the solution process In assembling those matrices all the

system buses and sections have to be inspected carefully In a practical large RDS data

preparation for these matrices will be cumbersome and prone to errors Under continshy

gency situations switching operations or the addition of another feeder to the existing

one are quite common practices in the DSs hence changes in system topology need to be

accommodated by restructuring the corresponding matrices which would add an overshy

head to track modifications The weakly meshed DS was dealt with by adding extra

nodes in the middle of the new links Two equal currents with opposite polarities were

injected into each added node Each injection operation is represented by a two column

matrix which was subsequently added to the first proposed matrix and then the develshy

oped matrices were extended and multiplied together The resultant is a full matrix and

its dimension is reduced by the Kron method in every single iteration That is the

developed full matrix was inverted in each iteration of the solution method and such

procedure is expensive lengthy cumbersome and time consuming

Shirmohammadi et al [39] Cheng and Shirmohammadi [52] and Hague [53]

proposed an iterative solution method for both radial and weakly meshed DSs This

approach necessitates a special numbering scheme in which they number the DS sections

in layers starting from the root node The numbering scheme is to be carried out

carefully by examining the whole system when a new layer is to be numbered The

numbering process is cumbersome and prone to errors For weakly meshed networks

breakpoints are selected opened and consequently the meshed system is converted to a

radial system The loops are broken by adding two fictitious buses In each pair of

dummy buses equal and opposite currents are injected and the new system is evaluated

to produce a reduced order impedance matrix Their proposed method requires that the

breakpoint impedance matrix should be computed cautiously Such a procedure is highly

dependent on the distribution networks topology That is the more links that exist in the

DS the larger the break point impedance matrix and the more time will be consumed in

its computation

Goswami and Basu [38] introduced a direct solution method to solve for radial and

weakly meshed DS They applied a breakpoints method into the meshed DS similar to

that of [39] in order to convert it into RDS In their proposed methodology a restriction

12

was imposed on each of the system buses not to have more than three sections attached to

it Such limitation is a drawback of the method and moreover a difficult node numbering

scheme is a disadvantage

In this thesis the unique structure of the RDS is exploited in order to build up a new

fast flexible power flow technique that deals with radial and looped DSs The numbering

scheme of the DS is simple and straightforward All load types can be accommodated by

the proposed distribution power flow eg spot and distributed loads Unlike

conventional power flow methods no trigonometric functions are used in the proposed

distribution power flow method For weakly meshed and looped DSs the system is dealt

with as it is there is no need for radialization cuts or building breakpoints impedance

matrix The topology of the tested DS whether strictly radial weakly meshed or looped

is represented by a building block matrix which is the only one needed to perform the

backwardforward sweep technique

23 DG INTEGRATION PROBLEM

DG is gaining increasing popularity as a viable element of electric power systems The

presence of DG in power systems may lead to several advantages such as supplying

sensitive loads in case of power outages reducing transmission and distribution networks

congestion and improving the overall system performance by reducing power losses and

enhancing voltage profiles Some of the negative impacts of DG installations are

potential harmonic injections the need to adopt more complex control schemes and the

possibility of encountering reverse power flows in power networks Even though the

concept of DG utilization in electric power grids is not new the importance of such

deployment is presently at its highest levels due to various reasons Recent awareness of

conventionaltraditional thermal power plants harmful impacts on the environment and

the urge to find more environmentally friendly substitutes for electrical power generation

rapid advances made in renewable energy technologies and the attractive and open

electric power market are a few major motives that led to the high penetration of DG in

most industrial nations power grids To achieve the most from DG installation special

attention must be made to DG placement and sizing

13

The problem of optimal DG placement and sizing is divided into two subproblems

where is the optimal location for DG placement and how to select the most suitable size

Many researchers proposed different methods such as analytic procedures as well as

deterministic and heuristic methods to solve the problem

231 Solving the DG Integration Problem via Analytical and Deterministic Methods

In the literature the optimal DG integration problem is solved by means of employing

any analytical or optimization technique that suits the problem formulation Methods and

procedures of optimally sizing and locating the DGs within the DS are varied according

to objectives and solution techniques

Willis [54] presented an application of the famous 23 rule originally developed

for optimal capacitor placement to find a suitable bus candidate for DG placement That

is to install a DG with a rating of 23 of the utilized load at 23 the radial feeder length

down-stream from the source substation However this rule assumes uniformly

distributed loads in a radial configuration and a fixed conductor size throughout the

distribution network In any event the 23 rule was developed for all-reactive load

These assumptions limit its applicability to radial distribution systems and the fact that it

is only suitable for single DG planning

Kashem et al [55] developed an analytical approach to determine the optimal DG

size based on power loss sensitivity analysis Their approach was based on minimizing

the DS power losses The proposed method was tested using a practical distribution

system in Tasmania Australia However it assumes uniformly distributed loads with all

the connected loads along the radial feeder having the same power factor and it also

assumes no external currents injected into the system buses eg capacitors which limits

its practicality

Wang and Nehrir [56] developed an analytical approach to address the optimal DG

placement problem in distribution networks with different continuous load topologies

Minimizing the real power losses was the objective of the proposed method In their

approach the DG units were assumed to have unity power factor and only the overhead

distribution lines with neglected shunt capacitance are considered The candidate bus

was selected based on elements of the admittance matrix power generations and load

14

distribution of the distribution network The issue of DG optimal size was not addressed

in their formulation

Griffin et al [57] analyzed the DG optimal location analytically for two continuous

load distributions types ie uniformly distributed and uniformly increasing loads The

goal of their study was to minimize line losses One of the conclusions of their research

was that the optimal location of DG is highly dependent on the load distribution along the

feeder ie significant loss reduction would take place when placing the DG toward the

end of a uniformly increasing load and in the middle of uniformly distributed load feeder

Acharya et al [58] used the incremental change of the system power losses with

respect to the change of injected real power sensitivity factor developed by Elgerd [59]

This factor was used to determine the bus that would cause the losses to be optimal when

hosting a DG By equating the aforementioned factor to zero the authors solved for the

optimal real value of DG output They proposed an exhaustive search by applying the

sensitivity factor on all the buses and ranked them accordingly The drawback of their

work is the lengthy process of finding the candidate locations and the fact that they

sought to optimize only the DG real power output Furthermore they only considered

planning of a single DG

Popovic et al [60] utilized sensitivity analysis based on the power flow equations to

solve the DG placement and sizing Two indices were used in ranking all the DS buses

for the suitability of hosting the DG The first one is a voltage sensitivity index which is

derived directly from the NR power flow Jacobian inverse the second one exploits the

relation of incremental real power losses with respect to the injected real and reactive

power as developed in [61] Their objective for sizing the DG was to maximize its

capacity subject to boundary constraints such as bus voltage penetration level line flows

and fault current limits To solve the sizing DG problem they gradually increased the

DG capacity at selected most sensitive buses until one of the constraints is violated and

the direct previous installed DG size becomes the one chosen as the optimal rating This

process is a lengthy and impractical procedure and the authors did not elaborate on how

they would deal with multiple DG cases using the proposed scheme

Keane and OMalley [62] solved for the optimal DG size in the Irish system by using

a constrained Linear Programming (LP) approach To cope with the EU regulation which

15

emphasizes that Ireland should provide 132 of its electricity from renewable sources

by 2010 the objective of their proposed method was to maximize the DG generation

The nonlinear constraints were linearized with the goal of utilizing them in the LP

method A DG unit was installed at all the system buses and the candidate buses were

ranked according to their optimal objective function value

Rosehart and Nowicki [63] dealt with only the optimal location portion of the DG

integration problem They developed two formulations to assess the best location for

hosting the DG sources The first is a market based constrained optimal power flow that

minimized the cost of the generated DG power and the second is voltage stability

constrained optimal power flow that maximized the loading factor distance to collapse

Both formulations were solved by utilizing the Interior Point (IP) method The outcomes

of the two formulations were used in ranking the buses for DG installations The optimal

DG size problem was not considered in their paper

Iyer et al [64] employed the primal-dual IP method to find the optimal DG

placement through combined voltage profile improvement and line loss reduction indices

However the proposed approach was based on initially placing DGs at all buses in order

to determine proper locations for DG installations This methodology may not be

realistic for large scale distribution networks

Rau and Wan [65] only treated the DG sizing problem by utilizing the Generalized

Reduced Gradient (GRG) method The DG bus locations were assumed to be provided

by the system planner for the DG units to be installed In their proposed method they

considered minimizing the system active power losses In their formulation only the

power flow equality constraints were considered whereas the boundary conditions and

the inequality constraints were not taken into account

Hedayati et al [66] employed continuous power flow methodology to locate the

buses most sensitive to voltage collapse The sensitive bus set is ranked based on their

severity which is used accordingly to indicate potential bus locations for placement of

single and multiple DG sources An iterative method was proposed for optimally sitting

the DG A certain DG capacity which is known and fixed a priori is added to the DS

and the conventional power flow method was employed to determine the resultant DS

real power losses voltage profiles and power transfer capacity In the subsequent

16

iteration another DG with the same capacity was added to the next sensitive bus and

results were obtained This iterative process would continue until the system outcomes

reached acceptable values The proposed iterative method did not optimize the DG size

232 Solving the DG Integration Problem via Metaheuristic Methods

Metaheuristic techniques have proven their effectiveness in solving optimization

problems with appreciable feasible search space They can be easily modified to cope

with the discrete nature associated with different elements commonly used in power

systems studies Optimization methods such as Genetic Algorithm (GA) [67-71] GA

hybrid methods such as GA and fuzzy set theory [72] and GA and Simulated Annealing

(SA) [73] Tabu search algorithm (TS) [7475] GA-TS hybrid method [76] Ant Colony

Optimization (ACO) [77] Particle Swarm Optimization (PSO) [78] and Evolutionary

Programming (EP) [79] were utilized in the literature to solve for the DG integration

problem

Teng et al [67] developed a value-based method for solving the DG problem The

GA method was utilized in maximizing a DG benefit to cost ratio index subject to only

boundary constraints such as ratio index voltage drop and feeder transfer capacity A

drawback of their procedure is that the candidate DG bus locations were assumed to be

provided by the utility and consequently all combinations of the provided bus locations

were tested for obtaining the optimal DG capacities via the GA method

The proposal set forth by Mithulananthan et al [68] made use of the DS real power

losses as the fitness function to be minimized through GA Their formulation of the DG

size optimization problem is of an unconstrained type Moreover the NR method which

is usually inadequate in dealing with the DS topology was used in calculating the total

power losses Candidate DG bus locations were obtained by placing a DG unit at all

buses of the tested DS which is impractical for large DSs Furthermore the multiple

DGs case was not addressed

Haesen et al [69] and Borges et al [70] solved the DG integration problem by

basically employing the GA method Both utilized the metaheuristic technique in solving

for single and multiple DG sizing and placements Haesen et al used the GA method to

minimize the DS active power flow while the objective for Borges et al was to

17

maximize a DG benefit to total cost ratio index Reference [69] incorporated penalty

factors within the objective function to penalize constraint violations thus adding another

set of variables to be tuned The authors of the latter reference used a PV model for

modeling the DG

Celli et al [71] formulated the DG integration problem as an s-constraint

multiobjective programming problem and solved it using the GA method Their

proposed algorithm divided the set of the objective functions into one master and the rest

are considered as slave objective functions The master is treated as the primary

objective function that is to be minimized while the slaves are regarded as new

inequality constraints that are bounded by a predetermined e value They utilized their

hybrid method to minimize the following objective functions cost of network upgrading

energy losses in the DS sections and purchased energy (from transmission and DG) The

number of the DG sources to be installed was randomly assigned and the units were

randomly located at the network buses Whenever the constraints are violated the

objective function solution is penalized A Pareto set was calculated from this

multiobjective optimization problem to aid the distribution planner in the decision

making process

Kim et al [72] and Gandomkar et al [73] hybridized two methods to solve the DG

sizing problem The former hybridized GA with fuzzy set theory to optimally size the

single DG unit while the latter combined the GA and SA metaheuristic methods to solve

for the optimal DG power output In both references the DG sizing problem was

formulated as a nonlinear optimization problem subject to boundary constraints only

Unlike Gandomkar et al [73] added the nonlinear power flow equality constraints to

their problem formulation The former researchers utilized their methodology to

investigate multiple DG case while the latter solved only the single DG case Both sited

the DG at all DS buses in order to determine the optimal DG location and size

Nara et al [74] assumed that the candidate bus locations for the DG unit to be

installed were pre-assigned by the distribution planner Then they used the TS method in

solving for the optimal DG size The objective of their formulation was to minimize the

system losses The DG size was treated as a discrete variable and the number of the

18

deployed units was considered to be fixed The DS loads were modeled as balanced

uniformly distributed constant current loads with a unity power factor

Golshan and Arefifar [75] applied the TS method to optimally size the DG as well

as the reactive sources (capacitors reactors or both) within the DS They formulated

their constrained nonlinear optimization problem by minimizing an objective function

that sums the total cost of active power losses line loading and the cost of the added

reactive sources The DG locations were not optimized instead a set of locations were

designated to host the proposed DGs and the reactive sources

A hybrid method that combined the GA with the TS technique in order to solve the

DG sizing optimization problem was developed by Gandomkar et al [80] They solved

the DG integration problem by minimizing the distribution real power losses subject to

boundary conditions The authors restricted the number of DGs as well as their gross

capacity to be revealed prior to executing the optimization procedure They augmented

the objective function with penalty terms in their formulation to handle the constraint

violations

Falaghi and Haghifam [77] proposed the ACO methodology as an optimization tool

for solving the DG sizing and placement problems The minimized objective function for

the utilized method was the global network cost ie the summation of the DGs cost their

corresponding operational and maintenance cost the cost of energy bought from the

transmission grid and the cost of the network losses The DG sizes were treated as

discrete values They used a penalty factor to handle the violated constraints ie

infeasible solutions In addition to modeling the DG sources as exclusive constant power

delivering units ie with unity power factor the network loads were all assumed to have

09 power factor Thus it can be stated that such modeling is impractical especially when

real large DSs are encountered

Raj et al [78] dealt with the DG integration in two different steps They employed

the PSO method to optimally determine the size of single and multiple DGs The optimal

location portion of the problem was performed utilizing the NR power flow method to

assign those buses with the lowest voltage profiles as the optimal candidate DG locations

The PSO was used to minimize the system real power losses the voltage profiles

boundary conditions were the only constraints required by the authors to be satisfied

19

Constraint violations were handled via a penalty factor that was augmented with the

objective function The DG units were randomly sited at one or more of the candidate

buses obtained through the NR method and subsequently the PSO was used to find the

optimal size(s)

Rahman et al [79] derived sensitivity indices to identify the most suitable bus for a

single DG installation Subsequently the DG sizing problem was dealt with by

employing an EP approach The objective function of the proposed approach was to

minimize the DS real power losses subject only to the system bus voltage boundary

constraints The formulation of DG sizing in their work was not realistic for a variety of

reasons For instance they ignored the line loading restrictions power flow equality

constraints and DG size limits

In most of the reviewed work on the DG deployment problem the problems of DG

optimal sizing and placement were not simultaneously addressed due to the difficult

nature of the problem as it combines discrete and continuous variables for potential bus

locations and DG sizing in a single optimization problem This combination creates a

major difficulty to most derivative-based optimization techniques and it increases the

feasible search space size considerably In this thesis the DG sizing subproblem is

solved using an improved SQP deterministic method while the two subproblems are

addressed simultaneously via an enhanced PSO metaheuristic algorithm

24 SUMMARY

In this chapter distribution power flow techniques were reviewed in Section 22 The

literature review of DG integration problem solution methods was presented in Section

23 The analytical and deterministic methods that were utilized to handle the DG

integration problem were presented in Subsection 231 Then recent publications that

handled the DG sizing and placement problems via wide-class of metaheuristic methods

were reviewed and summarized

20

CHAPTER 3 FAST AND FLEXIBLE RADIAL POWER FLOW FOR

BALANCED AND UNBALANCED THREE-PHASE DISTRIBUTION

NETWORKS

31 INTRODUCTION

As discussed in Chapter 2 several limitations exist in radial power flow techniques

presently reported in the literature such as complicated bus numbering schemes

convergence related problems and the inability to handle modifications to existing DS in

a straightforward manner This motivated the development of an enhanced distribution

power flow solution method In this thesis the unique structure of the RDS is exploited in

order to build up a Fast Flexible Radial Power Flow (FFRPF) technique The tree-like

RDS configuration is translated into a building block bus-bus oriented data matrix

known as a Radial Configuration Matrix (RCM) which consequently is utilized in the

solution process The developed algorithm is also capable of handeling weakly meshed

and meshed DSs via a meshed RCM (mRCM) RCM or mRCM is the only matrix that

needs to be constructed in order to proceed with the iterative process During the data

preparation stage each RCM (or mRCM) row focuses only on a system bus and its

directly connected buses That is while building such a matrix there is no need to

inspect the entire system buses and sections Moreover no complicated node numbering

scheme is required The building block matrix is designed to have a small condition

number with a determinant and all of its eigenevalues equal to one to ensure its

invertibility By incorporating this matrix and its direct descendant matrices in solving

the power flow problem the CPU execution time is decreased compared with other

methods The FFRPF method is flexible in accommodating any changes that may take

place in an existing radial distribution system since these changes can be exclusively

incorporated within the RCM matrix The proposed power flow solution technique was

tested against other methods in order to judge its overall performance using balanced and

unbalanced DSs

In Chapters 4 and 5 the FFRPF method is incorporated within the developed FSQP

and HPSO algorithms in solving the optimal DG installation problem It is implemented

21

as a subroutine within the proposed algorithms to satisfy the equality constraints ie

solving the radial power flow equations

32 FLEXIBILITY AND SIMPLICITY OF FFRPF I N NUMBERING THE RDS

BUSES AND SECTIONS

The RDS is configured in a unique arborescent structure with the distribution substation

located at its root node from which all the active and reactive power demands as well as

the system losses are supplied The substation feeds one or more main feeders with

spurred out laterals sublaterals and even subsublaterals For this reason the substation is

treated as a swing bus during the power flow iterative procedure

Most radial power flow techniques proposed in the literature assign sophisticated

procedures for numbering the radial distribution networks in order to execute their

algorithms This is cumbersome when expanding andor modifying existing RDSs In

this section a very simple numbering rule for the RDS buses and sections is introduced

A section is defined as part of a feeder lateral or sublateral that connects two buses in the

RDS and the total Number of Sections (NS) is related to the total Number of Buses (NB)

by this relation (NS=NB -1 )

321 Bus Numbering Scheme for Balanced Three-phase RDS

A balanced radial three-phase RDS is represented by a single line diagram In such a

system a feeder or sub level of a feeder having more than one bus is numbered in

sequence and in an ascending order Consequently each section will carry a number

which is less than its receiving end bus number by one as shown in Figure 31

Therefore sections are numbered automatically once the simple numbering rule is

applied

22

Substation

Figure 31 10-busRDS

In numbering the RDS shown in Figure 31 the following was considered buses 1 -

4 form the main feeder and buses 2 - 6 and 3 - 10 are the laterals while the sublateral is

tapped off bus 5 Figure 32 and Figure 33 manifest the ease in renumbering the system

shown previously and the flexibility in adding any portion of RDS to the existing one

respectively In Figure 32a buses 1 -4 have a different path compared to Figure 31 and

are considered to be a main feeder buses 2 - 9 and 3 - 6 are the laterals whereas the

sublateral 7 - 10 is the section spurred off bus 7 Figure 32b shows another numbering

scheme The same system numbered differently would have the same solution when

solved by the FFRPF

Figure 33 illustrates the ease of numbering in the case of a contingency situation or

a switching operation that could cause the existing system to be modified andor to be

augmented with other systems The lateral 3 - 10 in Figure 31 was modified to be

tapped off bus 2 instead and a couple of radial portions were added to be fed from buses

6 and 4 as illustrated in the figure

23

Substation Substation

(a) (b)

Figure 32 Different ways of numbering the system in Fig 31

Figure 33 The ease of numbering a modified and augmented RDS

322 Unbalanced Three-phase RDS Bus Numbering Scheme

The three-phase power flow is more comprehensive and realistic when it comes to

finding the three-phase voltage profiles in unbalanced RDSs Figure 34 shows an

unbalanced three-phase RDS The missing sections and buses play a significant role in

the multi-level phase loading and in making the unbalanced state of such a three-phase

DS more pronounced

24

The RDS shown in Figure 34 includes 6 three-phase buses (3(j)NB = 6) 5 three-

phase sections (3(|)NS = 5) no matter how many phase buses or sections exist physically

As such it has 14 single-phase buses (1(|)NB = 14) and 11 single-phase sections (1lt|gtNS =

11) The relations expressed in Eq (31) govern the three-phase and single-phase buses

to their corresponding sections

3^NS = 3^NB-1

l^NS = l^NB-3 (31)

Figure 34 Three-phase unbalanced 6-bus RDS representation

It is simple to implement the numbering process in the three-phase system as was

done in the balanced case Any group of phase buses to be found along a phase feeder or

a sub level of a feeder is to be numbered in a consecutive ascending order Consequently

each phase section number will carry a number which is one less than its receiving end

bus number as shown in Figure 34 In other words the sections are numbered routinely

after the ordering of the three-phase RDS buses has been completed

To develop the building block matrix as will be shown shortly the unbalanced three-

phase system is redrawn by substituting for any missing phase section or bus using dotted

representation as depicted in the 6-bus RDS in Figure 35 By performing this step each

three-phase bussection in the RDS consists of a group of 3 single-phase busessections

a b and c including the missing ones for double and single-phase buses

25

l a

I (1) 2 a | (2) 3 a | (3) 4 a | (4)

Figure 35 Three-phase unbalanced 6-bus RDS with missing buses and sections

33 THE BUILDING BLOCK MATRIX AND ITS ROLE I N FFRPF

The proposed FFRPF procedure starts with a matrix that mimics the radial structure

topology called a system Radial Configuration Matrix (RCM) The inverse of RCM is

then obtained to produce a Section Bus Matrix (SBM) that will be utilized in summing

the section currents during the backward sweep procedure A Bus Section Matrix (BSM)

is next generated by transposing the SBM to sum up the voltage drops in the forward

sweep process Therefore the only input data needed in the solution of an existing

modified or extended RDS other than the system loads and parameters is the RCM

It is worth mentioning that the inversion and transposition operations take place only

once during the whole process of the proposed FFRPF methodology for a tested RDS

whereas other methods like the NR technique invert the Jacobian matrix in every single

iteration The following subsections demonstrate the building of a three-phase RCM and

elucidate the role of both SBM and BSM in solving the radial power flow problem

331 Three-phase Radial Configuration Matrix (RCM)

The only matrix needed to be built for an unbalanced three-phase RDS is the RCM

Whatever changes need to be accommodated as a modification in the existing structure or

an addition to the existing network would be performed through the RCM only The

26

other matrices utilized in the backwardforward sweep are the direct results of the RCM

and no other built matrix is needed to perform the FFRPF

Such a matrix exploits the radial nature of such a system The RCM is of 3(3())NB x

3(|)NB) dimension in which each row and column represents a single-phase bus For a

balanced three-phase RDS represented by a single line diagram the RCM dimension is

(NBxNB) The RCM building algorithm for the unbalanced three-phase RDS case is

illustrated as follows

1 Construct a zero-filled 3(3(|)NB x 3(|)NB) square matrix

2 Change all the diagonal entries to +1 every diagonal entry represents sending

missing or far-end buses

3 In each row if the column index corresponds to an existing receiving single-phase

bus its entry is to be changed to - 1

4 If a single-phase bus is missing or is a far-end bus the only entry in its

corresponding row is the diagonal entry of+1

The above RCM building steps are summarized in the following illustration

Columns Description

RCMbdquo

if is either

a - sending phase bus b - far-end phase bus c - missing phase bus (32)

-1 jkl if jkI are receiving phase buses

connected physically to phase bus 0 otherwise

The [abc] matrix is defined as a zero (3 x 3) matrix with the numbers a b and c as

its diagonal elements eg [ I l l ] is the identity (3 x 3) matrix while [110] is the identity

matrix with the third diagonal element replaced by a zero By following the preceding

steps and utilizing the [abc] definition the RCM for the unbalanced three-phase RDS

shown in Figure 35 is to be constructed as shown in (33)

27

[Ill] [000] [000] [000] [000]

[000]

-[111] [111]

[000] [000] [000]

[000]

[000] -[111]

[111] [000] [000]

[000]

[000] [000]

-[110] [111]

[000] [000]

[000]

[000] [000]

-[010] [111]

[000]

[000] [000]

-[on] [000] [000]

[111]

Because of the nature of the RDS the RCM has three distinctive properties The first

is that the RCM is sparse the second is that RCM is a strictly upper triangular matrix

and thirdly such a matrix is only filled by 0 +1 or - 1 Such a real matrix makes the data

preparation easy to handle and less confusing Figure 36 shows the sparsity pattern plots

of the RCM matrices for unbalanced three-phase 25-bus [48] and balanced 90-bus [38]

radial systems

RCM for 25-Bus unbalanced 3-phase RDS RCM for 90-Bus balanced 3-phase RDS

nz = 131 nz = 179

Figure 36 RCM matrices sparsity plots for the 25 and 90 bus RDSs

3311 Assessment of the FFRPF Building Block RCM

The RCM is well-conditioned and should have a small Condition Number (CN) and a

non-zero determinant The CN measures how far from singularity any matrix is It is

defined as

28

cond(A) = A jjA-l (34)

where ||A|| is any of the 3 types of norm formulations of matrix A 1-norm 2-norm or oo-

norm An ill-conditioned matrix would have a large CN while a CN of 1 represents a

perfectly well-conditioned matrix By definition a singular matrix would have an infinite

CN [81] Having all the matrix eigenvalues to be equal to 1 and a determinant value of 1

safeguard the RCM against singularity For this reason the RCM is not only invertible

but also its inverse is an upper triangular matrix filled only with 0 and +1 digits and no

other numbers would appear in RCM-1

332 Three-phase Section Bus Matrix (SBM)

The SBM for the three-phase RDS is obtained by performing the following steps

1 Remove the corresponding substation rows and columns from the RCM ie the

first three rows and columns The reduced version of the RCM is labeled as

RCM

2 Invert the RCM to obtain the SBM as shown in (35) and more explicitly in

Figure 37

To clarify the two rows and the two columns outside the matrix border shown in

Figure 37 are the three-phase buses and sections ordered respectively The dimension of

the unbalanced three-phase system SBM is 3(3lt|)NS x 3(|gtNS) For the balanced case the

SBM dimension is (NSxNS)

[Ill] [000]

[000] [000] [000]

[111] [111]

[000] [000] [000]

[110] [110]

[111] [000] [000]

[010] [010] [010] [111] [000]

[011] [011] [000] [000] [111]

29

1 a

1 b

1 c

2 a

2 b 2 c

SBM = 3 a

3 b

3 c

4 a

4 b

4 c

5 a

5 b

5 c

2 a

0 0 0 0 0 0 0 0 0 0 0 0 0 0

2 b

0 1 0 0 0 0 0 0 0 0 0 0 0 0 0

2 c

0 0 1 0 0 0 0 0 0 0 0 0 0 0 0

3 3 a b

1 0 0 1 0 0 i o 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

3 c

0 0 1 0 0

1 0 0 o 0 0

o 0 0 0

4 a

1 0 0 1 0

o 1 0

o 0 0

o 0 0 0

4 b

0 1 0 0 1 0 0 1 0 0 0 0 0 0 0

4 c

0 0 0 0 0 0 0 0 1 0 0 0 0 0 0

5 a

0 0 0 0 0 0 0 0 0 1 0 0 0 0 0

5 b

0 1 0 0 1 0 0 1 0 0 1 0 0 0 0

5 c

0 0 0 0 0 o 0 0

o 0 0 1 0 0 0

6 a

0 0 0 0 0

o 0 0

o 0 0 o 1 0 0

6 b

0 1 0 0 1 0 0 0 0 0 0 0 0 1 0

6 c

0 1 0 0 1 0 0 0 0 0 0 0 0 1

Figure 37 SBM for three-phase unbalanced 6-bus RDS

By inspecting Figure 35 it is noted that any single-phase section is connected

downhill to a single-phase far-end bus or buses through a Unique Set of Phase Buses

(USPB) xt bull By inspecting Figure 35 and Figure 37 the USPB for the following

single-phase sections la lb lc 3b 4b are^0 [ gt xlgt x respectively as explained

in (36)

=2 f l 3bdquo4a x=2b3b4b5bA]

Xl=2cA) Xl=) (36)

X=5b]

In the SBM the single-phase section is represented by a row i and will have entries

of ones in all the columns where their indices represent single-phase buses that belong to

the section USPB xf bullgt a s illustrated in (37)

SBMrmt =

Columns Description

c bullgt bull a - Id - buses e r ~ - 1 ijk- ijk-- are either w (37)

lb - diagonal entry 0 other columns otherwise

30

333 Three-phase Bus Section Matrix (BSM)

The BSM is the transpose of the SBM as shown in (38) In the BSM the rows represent

the RDS single-phase buses excluding the substations and all the sections are

represented by the BSM columns Each single-phase bus is connected uphill through a

Unique Set of Phase Sections (USPS) yf to a substation single-phase bus The USPS

for buses 2deg 3a 4b 5b 6C are y ydeg y y yc6 respectively as depicted in Figure 35

and Figure 37 and demonstrated in (39)

BSM

[111] [000] [000] [000] [000]

[111] [111] [000] [000] [000]

[110] [110] [111] [000] [000]

[010] [010] [010] [111] [000]

[011] [011] [000] [000] [111]

(38)

V H U V3=1gt2 y=b2b yb

5=lb2bdquoAb (39)

lt=1C2C5C

In the BSM a single-phase bus i is represented by a row and will have entries of

ones in all the columns where their indices represent single-phase sections that belong to

the bus USPS yf as equivalently shown in (310)

Columns Description

BSMrmi =

( gt - [a-l^-sectionse yf ~ i m

1 ijk ijk are either lt Y Y (310) lb - diagonal entry

0 other columns otherwise

34 FFRPF APPROACH AND SOLUTION TECHNIQUE

The FFRPF methodology is based simply on Kirchhoff s voltage and current laws which

are used in performing the backwardforward sweep iterative process By utilizing the

direct descendant matrices of the RCM ie SBM and BSM the RDS buses complex

31

voltages are calculated in every iteration until convergence criteria are met The next

subsections illustrate the proper usage of such matrices in the proposed FFRPF method

through appropriate modeling of the unbalanced multi-phase RDS section impedances

341 Unbalanced Multi-phase Impedance Model Calculation

Figure 38 shows a three-phase section model that is represented by two buses (sending

and receiving ends) of an overhead three-phase four-wire RDS with multi-grounded

neutral The assumption of a zero voltage drop across the neutral in a three-phase two-

phase and single-phase RDS is found to be valid [4582] Such a configuration is widely

adopted in North Americas distribution networks [8384]

V Sa

_ bull

V Ra

V Sb ab

^VWVVYgt

V Sc Z be

bn

^AAArmdashrYYYV bull

V s n en

ll1 Sa

s b

La

Lb

Lc

bull r Figure 38 Three-phase section model

In the proposed radial power flow solution method each of the three-phase lines is to

be modeled appropriately and mutual coupling effects between phases are not neglected

The primitive impedance matrix for such a four-wire system is a square matrix with a

dimension equal to RDS utilized number of phase and neutral conductors For a system

consisting of three-phase conductors and a neutral wire the section primitive impedance

matrix is expressed as shown in (311)

32

Zaa

ha

Zca

zna

Kb

Kb

Kb

Kb

Ke Kc Ke

nc

art

Zbn

en

nn

where

Z bull primitive impedance matrix

RDS section length

z per unit length self-impedance of conductor i

z per unit length mutual-impedance between conductors andy

zu and zy are calculated according Carsons work [85] and its modifications [86-88] as

illustrated by the following equations

where

k

GMRj

Dbdquo

v GMR

bulli J

zu=rt+rd+ja)k

zv=rd+jltok

resistance of conductor i

earth return conductor resistance

inductance multiplying constant

distance between overhead and its earth return counterpart and it is a

function of both earth resistivity and frequency

geometric mean radius of conductor i

distance between conductors i andj

(312)

(313)

The parameters used in (312) and (313) are shown in Table 31 for both operational

frequencies 50Hz and 60Hz in both metric and imperial units

33

Table 31 cok rj and De Parameters for Different Operation Conditions

De = 2160 Ij (ft)

cok rd

p = 100 Qm

p = 1000 Qm

Metric Units RDS operating frequency 50 Hz 60 Hz

006283km

0049345 QJ km

931 m

29443 m

007539 km

005921412km

850 m

26878 m

Imperial Units RDS operating frequency 50 Hz 60 Hz

010111mile

00794 QI mile

30547f

96598 ft

012134mile

009528 QI mile

27885

88182

Since the neutral is grounded the primitive impedance matrix Zsec can be

transformed into a (3 x 3) symmetrical impedance matrix Zsae

c by utilizing Krons

matrix reduction method The resultant section three-phase impedance matrix is

expressed mathematically in (314) and the three-phase section model is represented

graphically in Figure 39

7 abc

aa

zba Zca

Zab

^bb

Zcb

zac zbc Zee

(314)

VSn

mdash bull i 7 T

i ah bull-sec a

zbdquobdquo bull A V W Y Y Y V

v izK

^WW-rrYYv -+bull

I

bull

vR

Figure 39 The final three-phase section model after Krons reduction

If the RDS section consists of only one or two phase lines its primitive impedance

matrix is transformed by Krons matrix reduction to (laquo-phase x w-phase) symmetrical

impedance matrix Next the corresponding row and column of the missing phase are

replaced by zero entries in the (3x3) section impedance matrices Zsae

c For a two-phase

34

section its impedance matrix Z^c is demonstrated below

Z_a

zci Kron h-gt ZZ za

zbdquo zai

zaa o zac

0 0 0

z_ o zbdquo

Underground lines such as concentric neutral and tape shielded cables are typically

installed in the RDS sections For underground cables with m phases and n additional

neutral conductors the primitive impedance of each section is a (2m+n x 2m+n) matrix

with the entries computed as illustrated in [89-92]

Usually the RDS is modeled as a short line ie less than 80 km and the charging

currents would be neglected by not modeling the line shunt capacitance as depicted in

Figure 38 However under light load conditions and especially in the case of

underground cables the line shunt capacitance needs to be considered in order to obtain

reasonable accuracy ie use nominal 7i-representation The rc-equivalent circuit consists

of a series impedance of the section and one-half the line shunt admittance at each end of

the line Figure 310 (a) (b) and (c) show the RDS 7i-model representation The shunt

admittance matrix for an overhead three-phase section is a full (3x3) symmetrical

admittance matrix while it is a strictly diagonal matrix for the underground RDS cable

section That is the self admittance elements are the only terms computed [92] For the

unbalanced three-phase section eg one or two phases the non-zero elements of shunt

admittances are only those corresponding to the utilized phases

[zic] -AAVmdashrwvgt

T yabc 1 |_ sec J [ yaf tc |

sec J

(a)

35

Lsec J _

2

s

1

Yaa

Yba

Yea

Yab

Ybb

Ycb

Yac

Ybc

Ycc

zaa

zba

tea

zab

zbb

zcb

zac

zbc

zcc

P yabc ~|

lgtlt 2 - =

Yaa

Yba

Yca

Yab

Ybb

Ycb

R

1 1

Yac

Ybc

Ycc

(b)

[ yabc 1 sec J

s 1 1

Yaa

0

0

0

Ybb

0

0

0

Ycc

zaa

zba

zca

Zab

zbb

zcb

zac

zbr

zcc

V yabc ~j

L sec 2 - =

Yaa

0

0

0

Ybb

0

R

1 1

0

0

Ycc

(c)

Figure 310 Nominal 7i-representation for three-phase RDS section

(a) Schematic drawing for the single line diagram of 3(|gt RDS (b) Matrix equivalent for overhead section (c) Matrix equivalent for underground cable section

By applying Kirchhoff s laws to the three-phase system section k the relationship

between the sending and receiving end voltages for medium and short line models and

the voltage drop across the same section in the latter model are expressed in Eq (315)-

(316) and Eq (317) respectively

36

rabc S rabc S

14 L sec Jax3 L rabc

3x3 L secgt J3x3

[C]3 [4 [ yabc~ |~ yabc 1

sec J3x3 L sec J3 [4

zt 1 L sec J3x3

f yabc ~| [~ yaampc 1

L sec J3x3 L sec h

bull R rabc

(315)

rrabc VS rabc

S

1 J3x3 L sec J

[degL [L abc R

(316)

where

TT-afec rrabc S ^ R

rabc rabc S XR

rabc

AK

13x3

aAc sect

rabc see

KrH^ic] three-phase sending and receiving end voltages

three-phase sending and receiving end section currents

three-phase shunt admittance of section k

(3gtlt3) identity matrix

(3gtlt3) zero matrix

voltage drop across three-phase section k

section k three-phase currents

(317)

It is worth noting that Eq (315) is reduced to Eq (316) in the case of short line

modeling since YS^C 1 = 0 The voltage drop across the three-phase section k in the short

line model is expressed in Eq (317) and its corresponding sending end phase voltages

can be expressed in expanded forms as follows

V = V + Ta 7 + 1 7 + Tc 7 rS yR ^1secZjaa ^ Isec^ab ^rlsecpoundjac

v =v +r 7 +17 +r 7 y S y R ^ l sec^ab ^ 1 sec^bb ^ 1 sec^Ac

S mdashyR+ Ktc^ca + sec^cb + sec^a

(318)

(319)

(320)

Equations (317)-(320) show that the voltage drop along any phase in a three-phase

section depends upon all the three-phase currents

37

342 Load Representation Accurate and proper load modeling is of significant concern in power distribution

systems as well in its transmission systems counterpart [8693] Loads in electric power

systems are usually expressed by adequate representations so as to mimic their effects

upon the system The load dependency on the operating bus voltage and on system

frequency is among those representations

Static load models are often utilized in the power flow studies since they relate the

apparent power active and reactive directly to the bus operating voltage A static load

model is used for the static load components ie resistive and lighting load and as an

approximation to the dynamic load components ie motor-driven loads [93] Generally

static loads in DS are assumed to operate at rated and fixed frequency value [94-96]

Loads in the DS are usually expressed as function of the bus operating voltage and

represented by exponential andor polynomial models

The exponential model is shown in (321) and (322)

P = Pbdquo

Q = Q0 vbdquo

(321)

(322)

where

V0 nominal bus voltage

V operating bus voltage

P0 real power consumed at nominal voltage

Q0 reactive power consumed at nominal voltage

Exponents a and fi determine the load characteristics and certain a and values lead to a

specific lode model Therefore

1 If a = P mdash 0 the model represents constant power characteristics ie the load is

constant regardless of the voltage magnitude

2 If a = P = 1 the model represents constant current characteristics ie the load is

proportional to the voltage magnitude

3 If a = P = 2 the model represents constant impedance characteristics ie the load is

38

a quadratic function of the voltage magnitude

As indicated in [97] the exponents could have values larger than 2 or less than 0 and

certain load components would be represented by fractional exponents

The constant current model is considered to be a good approximation for many

distribution circuits since it approximates the overall performance of the mixture of both

constant power and constant impedance models [98] However representing loads with

the constant power model is a conservative approach with regard to voltage drop

consideration [99] and consequently this model will be used in this thesis

Loads can also be represented by a composite model ie the polynomial model The

polynomial model is expressed in (323) and (324)

P = Pbdquo

Q = Q0

(

a p

V

r

V

V

K

V

v0

2

2

V

K

V

+CP

J

)

(323)

(324)

where ap + bp + cp = 1 and aq + b + cq = 1

The polynomial model is also referred to as a ZIP model since it combines all the

three exponential models constant impedance (Z) constant current (I) and constant

power (P) models The ZIP model needs more information and detailed data preparation

The load models can be used in the FFRPF solution method during its iterative

process where flat start values are initially assumed to be the load voltages The three-

phase load voltages are changed during each iteration and consequently the three-phase

currents drawn by the constant current constant impedance andor ZIP three-phase load

models will change accordingly

Different shunt components like spot loads distributed loads and capacitor banks are

customarily spread throughout the RDS In power flow studies spot and distributed

loads are typically dealt with as constant power models while shunt capacitors are

modeled as constant impedances [94 100 101]

The uniformly distributed loads across RDS sections can be modeled equivalently by

either placing a single lumped load at one-half the section length or by placing one-half

the lump-sum of the uniformly distributed loads at each of the section end buses

39

[99 102] The former modeling approach has the disadvantage of increasing the

dimension of the RCM SBM and the BSM since more nodes would be added to the

existing RDS topology In the proposed FFRPF technique the distributed load is

modeled using the latter approach while the three-phase shunt capacitor banks are

modeled as injected three-phase currents [101] as schematically shown in Figure 311

and mathematically represented by Eq (325) and (326)

Qk Cap

^

CCap a Cap

(a) (b)

Figure 311 Three-phase capacitor bank (a) The schematics (b) The Modeling

O3 = poundCap

Qo Capa vbdquo

T-34 _ 1Cap ~

V

a Capbdquo

SQL M Cap

V

filt bullCapo

F

JQ( Cap

(325)

(326)

343 Three-phase FFRPF BackwardForward Sweep

The FFRPF technique employs the SBM in performing the current summation during the

backward sweep and the BSM in updating the RDS bus complex voltages during the

forward sweep as demonstrated in the following subsections

3431 Three-phase Current Summation Backward Sweep DS loads are generally unbalanced due to the unbalanced phase configuration the double

and single-phase loadings as well as the likelihood of unequal load allocation among the

three-phase configuration For the loads they could be represented as constant power

40

constant current constant impedance or any combination of the three models [97 103]

The three-phase load currents at three-phase bus i drawn by a three-phase load eg of a

constant impedance load model is mathematically expressed as shown in Eqs (327) and

(328)

jabc U ~

7e a gt

va

(si) ~

(327)

where

ctabc

o

V

K

2

rft

0

v K

2

K v

2

(328)

where Sf represents the load apparent power at single-phase bus lt|gt As shown in the

preceding equations each load current is a function of its corresponding bus voltage For

Eq (327) if the a phase bus is missing its corresponding phase load current is

eliminated and its corresponding position in the three-phase current vector is replaced by

a zero entry As an illustration and by assuming that there are loads connected to all

existing buses the three-phase load current vector for the system shown in Figure 35 is

expressed as follows

jabc V ja rb jc ra rb re ra rb Q Q rb Q Q rb re l l

The charging currents at the RDS three-phase buses are not to be neglected when

dealing with sections modeled as The shunt admittance at bus is obtained by

applying the following relation

where

Ysh^ bull total three-phase shunt admittance at bus

[l if section k attached to bus i

[0 otherwise

The three-phase shunt currents at bus is as shown in Eq (330)

tabc jrabc 1ch ~~ 1Anbus y i (330)

41

The 3-ltj) bus current is sum of both the 3-sect load current and the 3-cj) charging current as

expressed mathematically in Eq(331)

jabc jabc jabc ) T I 1busl ~ 1Li

+Ich V-gt )U

where 1^ is bus three-phase currents In the case of modeling a three-phase section as

a short line its charging currents are neglected ie I^c = 0 and the bus current will be

represented by the load currents only

The backward sweep sums the phase load currents in the corresponding phase

sections starting from far-end phase buses and moving uphill toward the substation phase

buses The current in phase (j) and section p is computed by utilizing the USPB

principle xp gt during the backward sweep as expressed in (332)

lt = E lt ^here = j 0 ^ J (332)

where

I current through single-phase section and phase ^ (^ =a b or c) SQCp

j current at bus and phase ltb bus x

The SBM is utilized in obtaining the system three-phase section currents in matrix

representation by performing the relation in Eq (333)

[G] = [SBM][lpound] (333)

where the I^ is a 3-(3())NS)-order column vector For a balanced short line RDS

model Eq (333) can be expressed as

[CS] = [SBM][IL] (334)

3432 Three-phase Bus Voltage Update Forward Sweep

The voltage at each phase bus is determined through the forward sweep procedure by

subtracting the sum of the voltage drops across the bus corresponding USPS from the

substation nominal complex voltage The voltage drop across three-phase section k is

calculated as in Eq (317) The voltage drop across all sections of the three-phase RDS

can be obtained by utilizing the SBM as shown in matrix notation via Eq (335)

42

[AKbdquo]=[zr][c] (335)

[ A ^ ] = [ Z s r ] [ 5 5 M ] [ C ]

where ZtradeS J is a diagonal matrix with an 3(3c|)NS x 3ltj)NS) dimension in which its

diagonal entry k corresponds to section k impedance and AV3^ is the computed three-

phase voltage drop values across all the RDS sections as shown below

A ^ = AVa AVb AVC bullbullbull AVb AVC T s e c l s e c l s e c l sec3ltWS sec3tNS J

For calculating the RDS voltage profiles the FFRPF solution method starts by asshy

suming the initial values for all bus voltages to be equal to the substation complex

voltage As a flat start the initial phase voltages at bus will be as follows

2TT 2TT

ya Jdeg vb e~^ VC e+JT V ss e V sisK v sis e (336)

where Vls is the substation complex phase voltage

For the voltage at bus m and phase (j) to be determined at iteration v the calculation is

performed as follows

= amp - pound r A lt wherer = trade lt (337)

The RDS voltage profiles at the system phase buses are obtained by utilizing the BSM as

shown in following matrix representation

[Vi] = [Vsy[BSM][AV^] (338)

where V^fs 1 and V3A are respectively the substation nominal three-phase voltage

column vector and the resultant three-phase bus voltage solution column vector and each

has a dimension of 3(3lt|gtNS)

3433 Convergence Criteria

The bus complex voltage is obtained after every backwardforward sweep After each

iteration all the bus voltage magnitudes and angles are compared with the previous

iteration outcomes The power flow process is concluded and a solution is reached if the

complex voltage real and reactive oo-norm mismatch vector is less than a certain

43

predetermined empirical tolerance value e The convergence criterion is expressed

mathematically as shown in Eq (339)

+i

([gt]w) A a ( |y f lts

where th

i iteration A

(339)

and symbol

||J| vector oo-norm ||x II = max fix IV II lloo l l c 0 1=12NBV I

5H (bull) real part of complex value

3 (bull) imaginary part of complex value

3434 Steps of the FFRPF Algorithm

The FFRPF iterative process can be summarized as follows

Step 1 Begin FFRPF by choosing a test RDS

Step 2 Number and order RDS buses and sections

Step 3 Construct RCM

Step 4 Obtain both SBM and BSM

Step 5 Select load model

Step 6 Start the iterative procedure by assuming flat start voltages for all buses

Step 7 Calculate load currents

Step 8 Start the backward sweep process by calculating section currents using SBM

Step 9 Start the forward sweep process by determining the bus complex voltages

using BSM

Step 10 Compare both magnitudes and angles of the RDS bus voltages between the

current and previous iterations

bull If the co-norm of their difference is lt st

o Solution is reached

44

o Stop and end FFRPF procedure

o Obtain bus voltage profiles section currents and power losses

etc

bull If not utilize the outcome of this iteration (bus complex voltages)

to start a new one by going back to Step 7

The FFRPF solution method is illustrated by the following flow chart shown in Figure

312

45

i laquo - i +1

Calculate Load and leakage

currents

I Start Backward

sweep process by calculating section

currents using SBM

Start Forward sweep process by determining bus

complex voltages V[+1] using BSM

V[+1] Section currents

Section Power Losses Etc

Start FFRPF

Read the test RDS data

Number and order RDS Buses and

Sections

I Construct RCM

Remove the substation

corresponding rows and columns

from RCM to Obtain RCM

Obtain RCM1

To Get SBM

Z Transpose SBM to

get BSM

Calculate RDS section

Impedance and Shunt admittance

Matrices

Select load model

Assuming a flat start voltages for

all buses V[]=10 =0

Figure 312 The FFRPF solution method flow chart

46

344 Modifying the RCM to Accommodate Changes in the RDS The FFRPF technique deals with changes in the network such as the addition of a

transformer by adjusting the original RCM to incorporate its conversion factor (c)

Subsequently the SBM and BSM are obtained accordingly and used in the

backwardforward sweep procedure If a three-phase transformer is incorporated in a

three-phase RDS between buses m and n at section n - 1 the modified BSM entries are

located at the intersection of the matrix rows and columns defined by Eq (340)

BSMZ EzL~-inBSMZ euro lt _ (340)

The affected rows and columns of the modified BSM are those belonging to the

sections USPB and the sending buss USPS respectively For demonstration purposes

the 10-bus balanced RDS shown in Figure 31 will be utilized If two transformers with

conversion factors cfj and cS are to be added within sections 3 and 6 respectively the

original RCM is modified to accommodate such additions as illustrated in (341) Thus

instead of filling -1 for the receiving end bus entry the negative of the conversion factor

is the new entry The process is repeated rc-times for -installed transformers The

corresponding modified SBM and BSM are to be obtained as demonstrated in Section

33

10

RCM^ =

1

2

3

4

5

6

7

8

9

10

1

0

0

0

0

0

0

0

0

0

-1

1

0

0

0

0

0

0

0

0

0

-1

1

0

0

0

0

0

0

0

0

0

-cfi 1

0

0

0

0

0

0

0

-1

0

0

1

0

0

0

0

0

0

0

0

0

-1

1

0

0

0

0

0

0

0

0

-cf2

0

1

0

0

0

0

0

0

0

0

0

-1

1

0

0

0

0

-1

0

0

0

0

0

1

0

0

0

0

0

0

0

0

0

-1

1

(341)

The affected entries of the new BSM are obtained by applying the relation in (340) as

follows

47

[BSMZ eZ s ^nBSMJ euro ^ 3 = 40(12=[(4l)(42)]

(laquoSWe^)nJM6^)=J7 )8)njl I4=gt[(7 )l)(7 )4)(8l) )(M)]

The matrix shown in (342) shows the final B S M after including the transformers in the

10-bus R D S Such a procedure is easily extended to the unbalanced three-phase RDSs

1

1

1

CJ

1

1

cfi

cf2

1

1

2

0

1

ch 0 0

0

0

1

1

It is worth mentioning that by integrating the cf for any transformer configuration

into the RCM building block in the FFRPF technique another light is shed on the

flexibility criterion of the proposed method

35 FFRPF SOLUTION METHOD FOR MESHED THREE-PHASE DS

In practical DS networks alternative paths are typically provided to accommodate for

any contingency incidents that might take place eg feeder failure Therefore it is not

unusual for meshed distribution networks to be part of the DS topology in order to make

the system more reliable The loop analysis approach as well as the graph theory

technique are used to study and analyze the behavior of meshed DS The loop analysis

technique basically applies Kirchhoff s voltage law principle to solve for the fundamental

loop currents in both planar and nonplanar networks while the graph theoretic

formulation preserves the network structure properties [104]

A meshed DS can be viewed from a graph theory perspective as an oriented looped

graph that preserves the network interconnection properties whereas a DS that has no

0

0

0

1

1

cfi

cf2

0

0

0

0

0

0

1

0

0

0

0

0

0

0

0

0

1

1

0

0

0

0

0

0

0

0

1

0

0

0

0

0

0

0

0

0

1

1

0

0

0

0

0

0

0

0

1

(342)

48

loops is considered a tree In graph theory terminology line segments that connect

between buses in a loopless DS tree are called twigs branches or sections (represented by

solid line segments in Figure 313) while those which do not belong to the tree are

known as links (represented by dotted line segments in Figure 313) Links are segments

that close loops in a DS tree thereby composing a co-tree Removal of all co-tree links

results in a strictly radial system Links are usually activated by closing their

corresponding Normally Open (NO) switches Whenever a link is added to a RDS

network a loop is formed and as a result the system will have as many fundamental

loops as the number of links A fundamental loop is a loop that contains only one link

besides one or more sections Segments are used here to name sections and links

together It is noted that the number of fundamental loops is significantly less than the

number of buses in the meshed DS which makes the loop analysis a more appropriate

method in dealing with such systems than other circuit analysis methods like nodal

voltage method [105]

The current directions in the meshed DS sections and links are arbitrarily chosen to

be directed form a lower bus index to a higher one and the positive direction of loop

current is assumed to in the same direction of that of the link as illustrated in Figure 313

The number of segments in a meshed DS is equal to the sum of the total number of its

corresponding graph tree sections and its co-tree links For a meshed DS with NB buses

and mNS segments (total number of sections and links in the meshed DS) the number of

links nL and the number of the fundamental loops as well are obtained according to the

following relation

laquoL=mNS-NB + l (343)

49

Substation 2 Imdash 31 4 1

^ -gtT-gtL- -

Figure 313 10-bus meshed distribution network

351 Meshed Distribution System Corresponding Matrices The FFRPF solution method can deal with the DS meshed networks through modifying

the original RCM Discussion is now focused on the balanced three-phase meshed DS

which can easily be extended to the unbalanced three-phase DS networks

Meshed RCM The meshed RCM (wRCM) order is ((NB + wL) (NB + nVj) and the

mRCM building algorithm is as follows

1 Remove links from the meshed DS and build the RCM for the resulting network

tree as demonstrated earlier in section 331

2 Add nL rows and columns toward the end of the RCM ie each link is represented

by a row and a column attached to the end of the RCM

3 In each link column there are 3 non-zero entries and are to be filled in the following

manner

a -1 at the row which corresponds to the lower index terminal of the link

b +1 at the row which corresponds to the higher index terminal of the link

c +1 at the link diagonal entry

For the 10-bus system shown in Figure 31 three directed links Li L2 and L3 are added

to the DS tree through connecting the following buses 4 - 5 6 - 8 and 4 - 1 0

respectively The system mRCM is constructed as illustrated in (344)

50

10

mRCM (13x13)

1

0

0

0

0

0

0

0

0

0

0

0

0

-1

1

0

0

0

0

0

0

0

0

0

0

0

0

-1

1

0

0

0

0

0

0

0

0

0

0

0

0

-1

1

0

0

0

0

0

0

0

0

0

0

-1

0

0

1

0

0

0

0

0

0

0

0

0

0

0

0

-1

1

0

0

0

0

0

0

0

0

0

0

0

-1

0

1

0

0

0

0

0

0

0

0

0

0

0

0

-1

1

0

0

0

0

0

0

0

-1

0

0

0

0

0

1

0

0

0

0

0

0

0

0

0

0

0

0

-1

1

0

0

0

0

0

0

-1

1

0

0

0

0

0

1

0

0

0

0

0

0

0

-1

0

1

0

0

0

1

0

0

0

0

-1

0

0

0

0

0

1

0

0

1

(344)

Remove the substation corresponding rows and columns from the mRCM to produce the

mRCM The mRCM for the 10-bus system is shown in (345)

10

mRCM (12x12)

1

0

0

0

0

0

0

0

0

0

0

0

-1

1

0

0

0

0

0

0

0

0

0

0

0

-1

1

0

0

0

0

0

0

0

0

0

-1

0

0

1

0

0

0

0

0

0

0

0

0

0

0

-1

1

0

0

0

0

0

0

0

0

0

0

-1

0

1

0

0

0

0

0

0

0

0

0

0

0

-1

1

0

0

0

0

0

0

-1

0

0

0

0

0

1

0

0

0

0

0

0

0

0

0

0

0

-1

1

0

0

0

0

0

-1

1

0

0

0

0

0

1

0

0

0

0

0

0

-1

0

1

0

0

0

1

0

0

0

-1

0

0

0

0

0

1

0

0

1

(345)

Meshed SBM The meshed SBM (mSBM) is then obtained by inverting the mRCM As

51

an illustration the 10-bus meshed network mSBM is obtained as shown in (346)

10

mSBM (12x12)

1

0

0

0

0

0

0

0

0

0

0

0

1

1

0

0

0

0

0

0

0

0

0

0

1

1

1

0

0

0

0

0

0

0

0

0

1

0

0

1

0

0

0

0

0

0

0

0

1

0

0

1

1

0

0

0

0

0

0

0

1

0

0

1

0

1

0

0

0

0

0

0

1

0

0

1

0

1

1

0

0

0

0

0

1

1

0

0

0

0

0

1

0

0

0

0

1

1

0

0

0

0

0

1

1

0

0

0

i deg 1

1

-1

1 deg 0

o 0

0

i

o o

0

0

0

0

1

-1

-1

0

0

0

1

0

0

0

1

0

0

0

0

-1

-1

0

0

1

(346)

Let the mSBM be partitioned into two submatrices mSBMp and C so that the mSBMp

submatrix corresponds to the DS tree sections and the second submatrix C to the

fundamental loops or links as shown in (347)

wSBM = SBM

6 [cl (mNSxnL) = [mSBMp C]

JmNSx(NB-l)

The dotted line shown in the above relation implies matrix partitioning

(347)

Fundamental loop matrix The second submatrix in (347) ie C is the fundamental

loop matrix which governs the direction of currents in each of fundamental loop sections

and links The fundamental loop matrix C can be partitioned into two submatrices [Csec]

and [I] as demonstrated below

M_ where [Csec] is an ((NB - 1) x (laquoL)) matrix and [7] is an nL square identity matrix The

former matrix corresponds to the tree loop sections while the latter corresponds solely to

c-- (348)

52

the co-tree links

By inspecting the fundamental loop matrix C it is noted that each row represents a

section or a link and each column represents a loop Each column entry in the C matrix

CM will have one of the following values

1 Qy = +1 if section k belongs to and is oriented in the same direction of loop

2 Cki = - 1 if section k belongs to and is oriented in the opposite direction of loop

3 Claquo = 0 if section k is not in the loop

By inspecting the 10-bus DS mSBM illustrated in (346) the first loop which is represhy

sented by the tenth column of the matrix is comprised of three sections in addition to the

link The current in two of these sections runs in the same direction as their correspondshy

ing fundamental loop ie sections 2 and 3 while the third one ie section 4 has an

opposite orientation This can be easily verified by tracing the first loop in the meshed

DS single line diagram One can also note that two loop currents pass through the third

section in an additive manner

Meshed BSM The meshed BSM (mBSM) is the transpose of mSBM as illustrated in

(349)

[ B S M I 0](mNS-nL)mNS

L J(nLxmNS)

The second submatrix C[ is the transpose of the fundamental loop matrix as manifested in

(350)

mBSM = [mSBM] = mBSMr

c (349)

C=L

1

0

0

0

2 3

1 1

0 0

0 1

4

-1

0

0

5

0

1

0

6

0

-1

0

7

0

-1

0

g

0

0

-1

9

0

0

-1

h 1

0

0

h 0

1

0

h (f 0

1

(350)

Meshed DS impedance matrices The (laquoLx riL) loop-impedance matrix Zioop is

formulated as follows [106]

KHc]|Xf][c] (35D

53

where [ztradegS ] is defined as a non-singular diagonal (mNS x wNS) segment-impedance

matrix that contains all the meshed DS segment impedances (tree sections and links)

along its main diagonal The matrix [Z^fM is formulated as two diagonal submatrices

as follows

|~ rymDS I

L eg J

Zl

0

^

0

0

7

Zk

0

0

0

raquoL

|gtr ] | o o |[zr] (352)

where Z^S J is the (NB - 1 x NB - 1) loopless tree section-impedance diagonal square

matrix and Z^k 1 is the (raquoL x nL) links impedance diagonal matrix

352 Fundamental Loop Currents The algebraic sum of voltage drops AV around any fundamental loop is zero according

to Kirchhoff s voltage law (KVL) and this can be mathematically expressed in terms of

the fundamental loop matrix C as follows

[C][AF] = 0 (353)

The voltage drop across the meshed DS segments is determined by the following

relations

[W] = [zf][mSBM][mILL]

where

Lm seg J bull (wNS x l) segment currents column vector of the meshed DS network

jW iZJ (mNS x l) meshed DS bus Loads and Link currents column vector

(354)

In order to account for the link currents in the meshed network the segment currents

column vector and the meshed DS bus loads and links currents column vector are

54

respectively partitioned into two subvectors as defined below

[ jtree 1

J(mNSxl)

J((JVB-l)xl)

Jloop[ J(nLxl)

(355)

[mILL l(mNSxl)

L L J((MJ-l)xl)

Jloop J (wLxl)

(356)

where

[Cr J ((NB - 1) x 1) tree section currents column vector

[lL] ((NB - 1) x 1) RDS bus load currents column vector

j 1 (nL x 1) fundamental loop current vector which is also the meshed DS link

currents column vector

By multiplying both sides of Eq (354) by [C] the left-hand-side will equal to zero

according to KVL as shown in (353) and by employing Eqs (351) and (356) Eq (354)

is reformulated as

[C][AV] = [c][z^][mSBM[mILL]

0 = [c f [z f ] [mSBM | C ] L op]

bull = [Cr[zS]([raquoSBM][ l ]+ [C][ 1 ] )

0 = [C] [^ ] [ -raquoSBM] [ I ] + [cr [zbdquor][C][U]

-[c] [zf ][c][J=[c] [ Z - ^ S B M J ]

-[^][^]-[cT[zS][laquoSBM][I] Thus the (nL x 1) fundamental loop currents vector in the meshed DS loop frame of

reference can mathematically be expressed as

[4 J - -K]1 [Cj [z^JmSBMr][lL] (357)

Eq (357) can be expressed in terms of the RDS matrices ie SBM and [ Z ^ J by

performing the following operation

55