4-EEE - IJEEER - LOAD FLOW - Jithendra Gowd - Paid

LOAD FLOW SOLUTION FOR UNBALANCED RADIAL DISTRIBUTION

SYSTEMS 1K.JITHENDRA GOWD, 2CH.SAI BABU& 3S.SIVANAGARAJU

1JNTUA College of Engineering, Anantapur, Andhra Pradesh, India 2University College of Engineering, JNT University, Kakinada, Andhra Pradesh, India

ABSTRACT

This paper presents a simple three phase load flow method to solve three-phase unbalanced

radial distribution system (RDS). A three phase load flow solution with considering most of load

modeling is presented which has good convergence property for any practical distribution networks with

practical R/X ratio. It solves a simple algebraic recursive expression of voltage magnitude, and all the

data are stored in vector form. The algorithm uses basic principles of circuit theory and can be easily

understood. Mutual coupling between the phases has been included in the mathematical model. The

proposed algorithm has been tested with several unbalanced distribution networks and the result of an

unbalanced RDS is presented in the article. The application of the proposed method is also extended to

find optimum location for reactive power compensation and network reconfiguration for planning and

day-today operation of distribution networks.

KEYWORDS: Radial Distribution Networks, Load Flow, Circuit Model, Three-Phase Four-Wire,

Unbalance.

INTRODUCTION

For any electrical system, the determination of the steady state behavior is the one of the most

fundamental calculation. In power systems, this calculation is the steady state power flow problem, also

called load flow. The majority of power flow algorithms in wide use in industry today, most notably, the

Newton-Raphson method and its variants [1,2] have been developed specifically for transmission

systems which have a meshed structure, with parallel lines and many redundant paths from the

generation points to the load points.

The focus of this paper is on the solution of the power flow problem for the distribution system.

Typically, a distribution system originates at a substation where the electric power is converted from the

high voltage transmission system to a lower voltage for delivery to the customers. Unlike a transmission

system, a distribution system typically has a radial topological structure. Unfortunately, this radial

structure, along with the higher resistance/reactance (R/X) ratio of the lines, makes the fast-decoupled

Newton method unsuitable for most distribution power flow problems. Various efficient distribution

International Journal of Electrical and Electronics Engineering Research (IJEEER) ISSN 2250-155X Vol.2, Issue 3 Sep 2012 37-55 © TJPRC Pvt. Ltd.,

38 K. Jithendra Gowd, CH. Sai Babu & S. Sivanagaraju

power flow algorithms which exploit the radial structure have been proposed in the literature. These

algorithms can be classified into three groups:

• Network reduction methods [3]

• Backward/ forward sweep methods [4-9]

• Fast decoupled methods [10-12]

All of the proposed methods have some limitations. Recently, many researchers presented

techniques, especially to obtain the load flow solution for distribution networks. Das et al. [13] have

proposed a load flow solution method by writing an algebraic equation for bus voltage magnitude.

However, this method is suitable for single-phase analysis. A few researchers have proposed a load flow

solution techniques to analyze unbalanced distribution systems. Zimmerman et al. [12] have formulated

load flow problem as a set of non-linear power mismatch equations as a function of the bus voltages.

These equations have been solved by Newton’s method. Thukaram et al. [14] have proposed three phase

power flow algorithm based on the forward backward walk along the network. The method considers

some aspects of three phase modeling of branches and detailed load modeling. In recent years the three-

phase current injection method (TCIM) has been proposed [15]. TCIM is based on the current injection

equations written in rectangular coordinates and is a full Newton method. As such, it presents quadratic

convergence properties and convergence is obtained for all but some extremely ill-conditioned cases.

Further TCIM developments led to the representation of control devices [16], [17]. Miu et al., [18] have

also proposed method for solving three-phase radial distribution networks.

The objective of this work is to develop a formulation and an efficient solution algorithm for the

distribution power flow problem which takes into account the detailed and extensive modeling necessary

for use in the distribution automation environment of a real world power system which is based on basic

systems analysis method and circuit theory. The proposed method requires lesser computer memory,

computationally fast and involves only recursive algebraic equations to be solved. The algorithm has

been developed considering that all loads draw constant power. However, the algorithm can easily

accommodate composite load modeling, if the composition of load is known. The algorithm has good

convergence property for practical unbalanced radial distribution networks.

MODELLING OF COMPONENTS OF UNBALACED RADIAL DISTRIBUTION SYSTEM Unbalanced Three Phase Line Model A three phase line section model between bus p and q is shown in Fig 1.

Load Flow Solution for Unbalanced Radial Distribution Systems 39

Fig 1 Three phase line section model

A 4X4 matrix, which takes into account the self and mutual coupling effects of the unbalanced

three-phase line section, can be expressed as

= (1)

After Kron’s reduction is applied, the effects of the neutral or ground wire are still included in this model

and (1) can then be rewritten as

= (2)

The relation between branchy voltages and branch currents in Fig.1 can be expressed as

(3)

For any phases fail to present, the corresponding row and column in this matrix will contain null entries.

The general forms of showing the branch voltage and branch current are


Line Shunt Admittance Model

Fig 2 Shunt admittances of the line sections

In the Fig.2, the shunt admittances (capacitances) are modeled. It represents the shunt

admittances connected to each phase and the admittances connected between the phase and ground.

Since the currents are injected into the line, the directions of the injected currents are as represented in

the figure. These current injections for representing the line charging, which should be added to the

respective compensation current injections at nodes p and q are given by

= (4)

Shunt Capacitor Model

Shunt capacitors, often used for reactive power compensation in a distribution network, are

modeled as constant capacitance devices. Capacitors are often placed in distribution networks to regulate

voltage levels and to reduce real power loss. As with loads, they can be connected in a grounded wyes

configuration or an ungrounded delta configuration. In fact, they are treated in exactly the same way as a

purely reactive constant impedance load. It is assumed that shunt capacitors in grounded sections of the

network are wye connected and those in ungrounded sections are three phase and delta connected. The

constant model parameter, in this case, is the admittance which is computed from the given nominal

reactive power injection. The Fig. 3 and Fig 4 represent the capacitors placement in star and delta

connections


Fig 3 Capacitors connected in wye connection.

Ia= -YaaVa

Ib= -YbbVb (5)

Ic= -YccVc

Fig 3 Capacitors connected in delta connection.

Ia = Yab/3 (-2Va+Vb+Vc)

Ia = Yab/3 (-2Va+Vb+Vc) (6)

Ia = Yab/3 (-2Va+Vb+Vc)

Load Model

All the loads are assumed to draw constant complex power (S = P+jQ). It is further assumed

that all three-phase loads are star connected and all double and single-phase loads are connected between

line and neutral. A node in a radial system is connected to several other nodes. However, owing to the

structure, in a radial system, it is obvious that a node is connected to the substation through only one line

that feeds the node. The equations (7) to (9) can be written refer to the power at the receiving end node q.

= (7)

= (8)

= (9)


Star-Connected Loads

In the case of loads which are connected in star is single phase loads connected line-to-neutral,

the load current injections at the Kth bus can be given by

= ,m € [a,b,c ] (10 )

Where and denote real power, reactive power and complex conjugate of the voltage

phasor of each phase, respectively. For simplicity for a constant power consumption at each bus was

assumed during the simulation tests.

Fig 4 Loads connected in star connection

Delta-Connected Loads

The current injection at the Kth bus for three-phase load connected in Delta are single-phase load

connected line-to-line can be expressed by

= - (11)

= - (12)

Ic= - (13)

Fig 5. Loads connected in delta connection


Transformer Model

The impact of the numerous transformers in a distribution system is significant. Transformers

affect system loss, zero sequence current, grounding method, and protection strategy. Although the

transformer is one of the most important components of modern electric power systems, highly

developed transformer models are not employed in system studies. In this thesis a transformer model and

its implementation method are shown, so that large-scale unbalanced distribution system problems such

as power flow, short circuit, system loss, and contingency studies, can be solved.

Recent interest in unbalanced system phenomena has also produced a transformer model

adaptable to the unbalanced problem which is well outlined in [19]. The model developed thus far can be

applied directly to distribution power flow and short-circuit analysis. However, it is still not accurate for

system loss analysis because the transformer core loss contribution to total system loss is significant [20,

21]. To calculate total system loss, the core loss of the transformer must be included in the model. The

complete transformer model combines the unbalanced and loss models from [20] and [21] in order to

integrate system loss analysis in power flow or short-circuit studies.

It is important to note that the unbalanced transformer model derived by Dillon in reference [20] cannot

be applied directly to either the factorized YBus or direct inverse YBus method because of numerical

considerations. For some connections such as grounded wye-delta, delta- grounded wye, these models

make the system YBus singular. Therefore, the application of the factorized or direct inverse methods

becomes impossible. To solve this problem, this thesis introduces an implementation method in which

artificial injection currents are used to make the system YBus nonsingular.

\Derivation of Transformer Models

A three-phase transformer is presented by two blocks shown in Fig. 6.One block represents the

per unit leakage admittance matrix YTabc, and the other block models the core loss as a function of

voltage on the secondary side of the transformer.

Fig 6 Overall Proposed Transformer Model

The presence of the admittance matrix block is the major distinction between the proposed

model and the model used in [14]. In the proposed model, Dillon's model is integrated with the

admittance matrix part. As a result, the copper loss, core loss, system imbalance, and phase shift


characteristics are taken into account. The implementation method is introduced in the following

sections.

Core Loss

The core loss of a transformer is approximated by shunt core loss functions on each phase of the

secondary terminal of the transformer. These core loss approximate functions are based on the results of

EPRI load modeling research which state that real and reactive power losses in the transformer core can

be expressed as functions of the terminal voltage of the transformer. Transformer core loss functions

represented in per unit at the system power base.

Where A = 0.00267 B=0.734 x 10-9 C=13.5

D = 0.00167 E= 0.268 x 10-1 F= 22.7

is the voltage magnitude in per unit.

It must be noted that the coefficients A, B, C, D, E and F are machine dependent constants. For

this work, core losses are represented by the functions and typical constants shown above.

Admittance Matrix

The admittance matrix part of the proposed three-phase transformer models follows the

methodology derived by Dillon, but a novel implementation is introduced here in. For simplification, a

single three-phase transformer is approximated by three identical single-phase transformers connected

appropriately. This assumption is not essential; however, it simplifies the ensuing derivation and

explanation. Based upon this assumption, the characteristic sub matrices used in forming the three- phase

transformer admittance matrices can be developed.

= Where = =

=


SOLUTION METHODOLOGY

The proposed power-flow technique is used for solving radial distribution networks by

calculating the total real and reactive power fed through any node. A unique node, branch and lateral

numbering scheme, which help to evaluate exact real and reactive power loads fed through any node and

receiving-end voltages is proposed. It is assumed that the three-phase radial distribution networks are

balanced and can be represented by their equivalent single-line diagrams.

Identification of Nodes Beyond All The Branches

The detailed flowchart for identifying the nodes beyond all branches is presented in Fig 2.12.

This procedure is very easy in finding the nodes of the system, to find the number of branches and to find

the exact current flowing through all the branches.

Fig 2.12 Flowchart for identifying nodes


Load Flow Calculation

The load flow solution is carried out by considering a branch which consists of three phases in

the network as shown below.

The receiving-end node voltage can be written as

Where =

The equation (1) can be evaluated for p = 1,2 . . . ln.

where ln is the total number of branches.

Current through branch i is equal to the sum of the load currents of all the nodes beyond branch i plus the

sum of the charging currents of all the nodes beyond branch i plus the sum of all injected capacitor

currents of all the nodes, i.e.

(14)

The real and reactive power losses of pth node is given by

(15)


Initially, a constant voltage of all the nodes is assumed and load currents, charging currents and capacitor

currents are computed. After currents have been calculated, the voltage of each node is then calculated.

The real and reactive power losses are calculated. Once the new values of voltages of all the nodes are

computed, convergence of the solution is checked. If it does not converge, then the load and charging

currents are computed using the recent values of the voltages and the whole process is repeated. The

convergence criterion of the proposed method is that if, in successive iterations the maximum difference

in voltage magnitude (Dvmax) is less than 0.0001 p.u., the solution has then converged. This solution

method is known recursive voltage computation method

The power flow calculation explained can be easily understood by representing in the form of

flow chart shown in Fig 2.13

2.13 Flowchart for load flow solution

RESULTS AND ANALYSIS

The proposed method is illustrated with two IEEE test systems consisting of 13 bus and 37 bus

Unbalanced Radial Distribution Systems on on P-IV computer with 2.4 GHz frequency


Example 1: The proposed algorithm is tested on 13 bus URDS. For the load flow, the base voltage and

base kVA are chosen as 4.16 kV and 100 kVA respectively.

Table 2.2 Voltages and phase angles of a 13 node radial distribution system

Bus No

Phase A Phase B Phase C Va deg Vb deg Vc deg

1 1 0 1 -120 1 120 2

0.99734 -

0.14 1 -

120.2 0.9984 119.84 3

0.99172 -

0.88 1 -

120.2 0.9887 119.26 4

0.99172 -

0.88 1 -

120.2 0.9887 119.26 5

0.99655 -

0.16 1 -

120.2 0.9978 119.82 6

0.99655 -

0.16 1 -

120.2 0.9978 119.82 7

0 0 0.99 -

120.3 0.9997 119.84 8

0 0 0.99 -

120.4 1.0003 119.84 9

0.99172 -

0.88 1 -

120.2 0.9887 119.26 10

0.99013 -

0.94 1 -

120.2 0.9882 119.27 11

0.99121 -

0.89 0 0 0.9882 119.24 12 0 0 0 0 0.9877 119.2 13

0.98979 -

0.87 0 0 0 0

Table 2.3 Power losses of 13 node radial distribution system

Description Phase A Phase B Phase C Total Active Power Loss (kW)

13.4527 18.816 16.4604

Total Reactive Power Loss (kVar)

12.7443 35.0432 32.3409


Table 2.4 Summary of test results of 13 bus radial distribution system

Proposed method Branch Phase A Phase B Phase C

kW kVar kW kVar kW kVar

1 2.2522 0.8919 0.9955 8.954 3.96 8.8135 2 1.5601 0.9494 7.9322 17.69 0.28 16.941 3 0 0 0 0 0 0 4 0.1966 0.0324 0.2581 0.75 0.24 0.094 5 7.8779 5.8727 7.6779 5.872 7.98 5.8727

6 0 3.449 0 0 2.48 -1E-04

7 0 0.9825 0 0 0.9 0 8 0 0 0 0 0 0 9 0.9846 0.5664 0.7641 0.842 0.61 -0.003 10 0.2712 0 0.2923 0.459 0 0.3965 11 0 0 0.8959 0 0 0.2262 12 0.3101 0 0 0.48 0 0

The Total Active Power losses are 48.72911 kW

The Total Reactive Power losses are 80.1294 kVar

For the proposed method, the maximum deviation of voltage and its phase angle from the

Forward backward sweep method is 0.0001 p.u and 0.01 deg.

The load flow is converged in 2 iterations for the tolerance of 0.001 p.u.. When the tolerance

limit is set as 0.0001, the number of iterations required for the convergence is 3 for Forward backward

sweep method and 2 for proposed method. The execution time is 0.048 seconds for the Forward

backward sweep method and 0.016 seconds.

Example – 2:The proposed algorithm is also tested

on IEEE 37 bus URDS . The feeder consists of

three-wire delta operating at a nominal voltage of 4.8

kV.


Table 2.5 Voltages and phase angles of a 37 node radial distribution system

Bus No.

Proposed method Phase A Phase B Phase C Va deg Vb deg Vc deg

1 1 0 1 -120

1 120

2 0.99862 -0.026 0.999 -120

1 119.95

3 0.99703 -0.126 0.99656 -120

1 119.87

4 0.99489 0.268 0.99325 -120

0.99 119.76

5 0.9924 -0.314 0.99185 -120

0.99 119.71

6 0.99148 -0.325 0.99138 -120

0.99 119.71

7 0.99148 -0.325 0.99138 -120

0.99 119.71

8 0.99808 -0.037 0.9983 -120

1 119.95

9 0.99721 -0.049 0.99729 -120

1 119.96

10 0.99732 -0.025 0.99556 -120

1 119.95

11 0.9986 -0.011 0.99784 -120

1 119.93

12 0.99853 0.002 0.99692 -120

1 119.92

13 0.99445 -0.274 0.99317 -120

0.99 119.77

14 0.99412 -0.281 0.99315 -120

0.99 119.77

15 0.9939 -0.286 0.9932 -120

0.99 119.77

16 0.99148 -0.3 0.99039 -120

0.99 119.7

17 0.99055 -0.361 0.99149 -120

0.99 119.71

18 0.99058 -0.362 0.99149 -120

0.99 119.7

19 0.9973 0.003 0.99244 -120

0.99 120

20 0.99728 0.021 0.9914 -120

1 119.99

21 0.99731 -0.019 0.99538 -120

1 119.94

22 0.99731 -0.017 0.9953 -120

1 119.94

23 0.99698 -0.054 0.9973 -120

1 119.96

24 0.99557 -0.092 0.99766 -120

1 119.98

25 0.99865 -0.013 0.99784 -120

1 119.92


26 0.99399 -0.28 0.99302 -120

0.99 119.77

27 0.98961 -0.396 0.9916 -120

0.99 119.7

28 0.9885 -0.439 0.99161 -120

0.99 119.69

29 0.98711 -0.499 0.99216 -120

0.99 119.7

30 0.9973 0.03 0.9922 -120

0.99 120

31 0.98858 -0.426 0.99088 -120

0.99 119.67

32 0.98688 -0.514 0.99229 -120

0.99 119.69

33 0.987 -0.519 0.99229 -120

0.99 119.68

34 0.98706 -0.524 0.99229 -120

0.99 119.67

35 0.98854 -0.393 0.98913 -120

0.99 119.66

36 0.98862 -0.428 0.99088 -120

0.99 119.67

37 0.98704 -0.52 0.99229 -120

0.99 119.67

Table 2.6 Power losses of 37 node radial distribution system

Branch Proposed method Phase A Phase B Phase C

kW kVar kW kVar kW kVar

1 1.2301 0.8762 1.2541 0.9863 0.9989 1.4117

2 0.6697 0.8688 0.7869 1.3336 0.7275 0.8825

3 0.9163 1.1762 1.0888 1.639 1.0071 1.2059

4 1.6254 0.4763 1.5154 0.7072 0.2596 0.5594

5 0.4339 0.1498 0.1259 0.2107 0.1372 0.0568

6 0.6433 13.9297 0.6233 13.9297 0.6433 13.9297

7 0.096 0.2098 0.136 0.0394 0.1268 0.0752

8 0.1546 0.3033 0.1241 0.0461 0.2184 0.0479

9 0.0071 0.4697 0.1599 -0.0017 0.1739 0.1682

10 0.0011 0.1979 0.0055 -0.0039 0.0562 0.0169

11 0.0011 0.1573 0 -0.0036 0.0586 0

12 0.0491 0.0036 0.0184 0.0213 0.0011 0.0016

13 0.0356 0 0.005 0.0206 0.0028 -0.0011

14 0.0106 0 0 0.0039 0 0

15 0.0008 0.1677 0 0 0.0618 0

16 0.3015 0.0083 0.1071 0.2098 0.0117 0.008

17 0 0 0.0041 0 0 0.0015

18 0.0031 0.754 0.2824 0.0001 0.2191 0.101

19 0.0006 0.086 0 0 0.0305 0


20 0.0003 0.0049 0 0 0.0025 0

21 0 0.0022 0 0 0.0011 0

22 0.0438 0.0007 0 0.0144 0.0009 0

23 0.2353 0 0 0.0871 0 0

24 0 0 0.0122 0 0 0.0045

25 0.0061 0.0058 0.0056 0.0019 0.0017 0.002

26 0.3106 0.0082 0.0768 0.2035 0.0145 -0.0021

27 0.2578 0.0004 0.1428 0.1937 0.2143 0.0338

28 0.3378 0 0.055 0.2387 0 -0.023

29 0 0.0381 0.0446 0 0.0092 0.0164

30 0.2019 0.0592 0.2156 -0.0006 0.0108 0.0163

31 0.2188 0 0.2402 0.2227 0 0.0102

32 0.0004 0 0.0404 -0.0005 0 0.0231

33 0 0 0.0093 -0.0001 0 0.0049

34 0.0019 0.1446 0 0.0001 0.0517 0

35 0 0 0.0103 0 0 0.0038

36 0 0 0.0103 0 0 0.0038

Table 2.7 Summary of test results of 37 bus radial distribution system

Description Phase A Phase B Phase C

Total Active Power Loss (kW) 7.7946 7.1 5.0412

Total Reactive Power Loss (kVar) 20.0987 20.0994 18.5589

The Total Active Power losses are 19.9358 kW

The Total Reactive Power losses are 58.7570 kVar

Table 2.5 shows comparison of the voltage magnitudes obtained by Forward backward sweep

method [21] and proposed method. For proposed method, the maximum deviation of voltage and its

phase angle from the Forward backward sweep method is 0.0001 p.u and 0.01 deg. Thus, the two

discussed methods are quite accurate. For both the methods, load flow converged in 2 iterations for the

tolerance of 0.001 p.u. When the tolerance limit is set as 0.0001, the number of iterations required for the

convergence is 4 for Forward backward sweep method and 3 for proposed method. The summary of test

results is given in Table 2. The execution time is 0.016 seconds for the Forward backward sweep method

and 0.00659 seconds for the proposed method on P-IV computer with 2.4 GHz frequency.

CONCLUSIONS


In this chapter, a simple and efficient computer algorithm has been presented to solve

unbalanced radial distribution networks. A three phase load flow solution is proposed considering most

of load modeling. The proposed method has good convergence property for any practical distribution

networks with practical R/X ratio. Computationally, this method is extremely efficient, as it solves simple

algebraic recursive equations for voltage phasors. Using the proposed method, the node identification

will be easy where as it is difficult for existing method. Another advantage of the proposed method is all

the data is stored in vector form, thus saving enormous amount of computer memory when tested for

large systems.

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AUTHOR BIBLIOGRAPHY


K. Jithendra Gowd completed is B.Tech from JNT University, Hyderabad in 2002, M.Tech from JNT

University, Kakinada in 2006 and pursuing his Ph.D. Presently working as Assistant Professor in Dept.

of EEE, JNTUA College of Engineering, Anantapur . His areas of interest are Distribution Systems,

HVDC Transmission.

Dr. Ch. Sai Babu received the B.E from Andhra University (Electrical & Electronics Engineering),

M.Tech in Electrical Machines and Industrial Drives from REC, Warangal and Ph.D in Reliability

Studies of HVDC Converters from JNTU, Hyderabad. Currently he is working as a Professor in Dept. of

EEE in University College of Engineering, JNT University, Kakinada. He has published several National

and International Journals and Conferences. His area of interest is Power Electronics and Drives, Power

System Reliability, HVDC Converter Reliability, Optimization of Electrical Systems and Real Time

Energy Management.

Dr. S.Sivanagaraju is graduated in 1998, Masters in 2000 fron IIT Kharaghpur and did his Ph.D in JNT

University, Hyderabad in 2004 and working as a Associate Professor in Department of Electrical

Engineering, University College of Engineering, Kakinada. He received two national awards (Pandit

Madhan Mohan memorial Prize and best paper) for the year 2003-04. His areas of interest are

Distribution and Automation.

Documents

4-EEE - IJEEER - LOAD FLOW - Jithendra Gowd - Paid