A Review on Load Flow Studies Final

7/22/2019 A Review on Load Flow Studies Final

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A review on Load flowstudies


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Outline

Introduction Methodology Classical methodsGauss-Seidal methodNewton Raphson methodFast Decoupled method Other methods

Fuzzy Logic application Genetic Algorithm application Particle swarm method (PS0)


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Load/Power Flow studies Load-flow studies are performed to determine the steady-state

operation of an electric power system. It calculates the voltagedrop on each feeder, the voltage at each bus, and the powerflow in all branch and feeder circuits.

Determine if system voltages remain within specified limitsunder various contingency conditions, and whether equipment

such as transformers and conductors are overloaded. Load-flow studies are often used to identify the need for

additional generation, capacitive, or inductive VAR support, orthe placement of capacitors and/or reactors to maintain systemvoltages within specified limits.

Losses in each branch and total system power losses are also

calculated. Necessary for planning, economic scheduling, and control of an

existing system as well as planning its future expansion

Pulse of the system


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Power Flow Equation

Note: Transmission lines arerepresented by their equivalent pi

models (impedance in p.u.)

Applying KCL to this bus results

in

(1)

Fig. 1. A typical bus of the power system.


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(2)

The real and reactive power at bus

iis

Substituting for Ii in (2) yields

Equation (5) is an algebraic non linear equation which must be solved by

iterative techniques


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Gauss-Seidel method

Equation (5) is solved for Visolved iteratively

Where yi jis the actual admittance in p.u.

Pischand Qi

schare the net real and reactive powers in p.u.

In writing the KCL, current entering bus Iwas assumed positive. Thus

for:Generator buses (where real and reactive powers are injected), Pi

schand

Qisch have positive values.

Load buses (real and reactive powers flow away from the bus), Pischand

Qisch have negative values.


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Eqn.5 can be solved forPiand Qi

The power flow equation is usually expressed in terms of the

elements of the bus admittance matrix, Ybus ,shown by upper case

letters, are Yij = -yij, and the diagonal elements are Yii = yij.

Hence eqn. 6 can be written as


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Iterative steps:

Slack bus: both components of the voltage are specified. 2(n-1)

equations to be solved iteratively.

Flat voltage start: initial voltage of 1.0+j0 for unknown voltages.

PQ buses: Pisch and Qisch are known. with flat voltage start, Eqn. 9 issolved for real and imaginary components of Voltage.

PV buses: Pisch and [Vi] are known. Eqn. 11 is solved for Qi

k+1which is

then substituted in Eqn. 9 to solve for Vik+1


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However, since [Vi] is specified, only the imaginary part of Vik+1 is

retained, and its real part is selected in order to satisfy

acceleration factor:the rate of convergence is increased by applying anacceleration factor to the approx. solution obtained from each iteration.

Iteration is continued until

Once a solution is converged, the net real and reactive powers at the slack

bus are computed from Eqns.10 & 11.


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Line flows and Line losses

Considering Ii jpositive in the given direction,

Similarly, considering the line current Ijiin the given direction,

The complex power Si jfrom bus i to j and Sji from bus j to i are


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Newton Raphson Method

Power flow equations formulated in polar form. For the

system in Fig.1, Eqn.2 can be written in terms of busadmittance matrix as

Expressing in polar form;

Note:jalso includes i

Substituting for Iifrom Eqn.21 in Eqn. 4


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Separating the real and imaginary parts,

Expanding Eqns. 23 & 24 in Taylor's series about the initial estimateneglecting h.o.t. we get


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The Jacobian matrix gives the linearized relationship between small changes ini

(k ) and voltage magnitude [Vik] with the small changes in real and reactive

power Pi(k) and Qi

(k)

The diagonal and the off-diagonal elements of J1are:

Similarly we can find the diagonal and off-diagonal elements of J2,J3and J4

The terms Pi(k) and Qi

(k )are the difference between the scheduled

and calculated values, known as the power residuals.


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Procedures:

1. For Load buses (P,Q specified), flat voltage start. For voltage controlledbuses (P,V specified),set equal to 0.

2. For Load buses,Pi(k) and Qi(k )are calculated from Eqns.23 & 24 andPi

(k) and Qi(k )are calculated from Eqns. 29 & 30.

3. For voltage controlled buses, and Pi(k) and Pi

(k )are calculated from

Eqns. 23 & 29 respectively.

4. The elements of the Jacobian matrix are calculated.

5. The linear simultaneous equation 26 is solved directly by optimally

ordered triangle factorization and Gaussian elimination.


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6. The new voltage magnitudes and phase angles are computed from (31)

and (32).7. The process is continued until the residuals Pi

(k) and Qi(k ) are less

than the specified accuracy i.e.

3. Fast Decoupled Method

practical power transmission lines have high X/R ratio.Real power changes are less sensitive to voltage magnitude changes andare most sensitive to changes in phase angle .

Similarly, reactive power changes are less sensitive to changes in angle

and are mainly dependent on changes in voltage magnitude.

Therefore the Jacobian matrix in Eqn.26 can be written as


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The diagonal elements of J1 given by Eqn.27 is written as

Replacing the first term of the (37) withQi from (28)

Bi i = sum of susceptances of all the elements incident to bus i.

In a typical power system, Bi i Qi therefore we may neglect Qi


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Furthermore, [Vi]2 [Vi] . Ultimately

In equation (28) assuming ii-i+j ii, the off diagonal elements of J1

becomes

Assuming [Vj] 1 we get

Similarly we can simplify the diagonal and off-diagonal elements of J4 as

With these assumptions, equations (35) & (36) can be written in

the following form


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B and B are the imaginary part of the bus admittance

matrix Ybus. Since the elements of the matrix are constant,need to be triangularized and inverted only once at the

beginning of the iteration.


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Other Methods

Repetitive solution of a large set of linear equations

in LF- time consuming in simulations

Large number of calculations on the Jacobian

matrix.

Jacobian of load flow equation tends to be singular

under heavy loading.

Ill conditioned Jacobian matrix

Doesnt require the formation of the Jacobian matrix

Insensitive to the initial settings of the solutionvariables

Ability to find multiple load-flow solutions.


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Fuzzy Logic application Repetitive solution of a large set of linear equations in the load flow

problem is one of the most time consuming parts of power system

simulations.

Large number of calculations need on account of factorisation,

refactorization and computations of Jacobian matrix.

Fundamentally FL is implemented in a fast decoupled load flow(FDLF) problem.

Mathematical analysis of FDLF

In eqn. 1, the state vector is updated

but state vector V is fixed. Eqn. 2 isused to update the state vector Vwhile state vector is fixed. The

whole calculation will terminate only if

the errors of both these equations are

within acceptable tolerances


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Main idea of FLF Algorithm

FLF algorithm is based on FDLF equation but the repeated update of the state

vector performed via Fuzzy Logic Control instead of using the classical load flow

approach.

The FLF algorithm is illustrated

schematically in Fig. 1.In this Figure

the power parameters FPand FQ

are calculatedand introduced to the

P- FLCP- and Q-V FLCQ-V,

respectively.

The FLCs generate the correction of

the state vector DX namely, the

correction of voltage angle for the

P- cycle and the correction ofvoltage magnitude V for the Q- V

cycle.


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Structure of the fuzzy load flow controller (FLFC)

Calculate and per-unite the power parameters FPand FQat each node

of the system.

The above parameters are elected as crisp input signals. The maximum (orworst) power parameter (FPmax or FQmax) determines the range of scale

mapping that transfers the input signals into corresponding universe ofdiscourse, at every iteration.The input signals are fuzzified into corresponding fuzzy signals (FPfuzor

FQfuzwith seven linguistic variables; largenegative (LN), medium negative

(MN), small negative(SN), zero (ZR), small positive (SP), medium positive

(MP), large positive (LP). They are represented in triangular function.


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The rule base involves seven rules tallying with seven linguistic

variables:Rule 1: if Ffuzis LN then Xfuzis LN

Rule 2: if Ffuzis MN then Xfuzis MN

Rule 3: if

Ffuzis SN thenXfuzis SNRule 4: if Ffuzis ZR then Xfuzis ZR

Rule 5: if Ffuzis SP then Xfuzis SP

Rule 6: if Ffuzis MP then Xfuzis MP

Rule 7: if Ffuzis LP then Xfuzis LP

These fuzzy rules are consistent to that of Eqn.3.


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The maximum corrective action xmaxof state variables determines

the range of scale mapping that transfers the output signals into the

corresponding universe of discourse at every iteration.

where FI expresses the real or reactive power balance

equation at node-I with maximum real or reactive power

mismatch of the system, XI represents the voltage angle or

magnitude at node-I.

The fuzzy signalsffuzare sent to process logic, which generatesthe fuzzy output signals xfuzbased on the previous rule base and

are represented by seven linguistic variables similar to input fuzzy

signals.


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finally the defuzzifier will transform fuzzy output signals into crisp

values for every node of the network. The state vector is updated

as

Index idepicts the number of iterations.


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Case studies


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GA applicationsLoad flow problem

where Gij and Bijare the (I,j)th

element of the admittance

matrix. Ei, and Fi are real and

imaginary parts of the voltage at

node i.

If node i is a PQ node where the load demand is specified, then themismatches in active and reactive powers, Pi and Qi , respectively,

are given by

Pisp

and Qisp

are the specifiedactive and reactive powers at

node i.


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When node i is a PV node, the magnitude of the voltage, Vispand

the active power,Q;P, at i are specified. The mismatch in voltage

magnitude at node i can be defined as

The active power mismatch is as given

in Eqn.3

Objective function H is to be minimized.

Where Npq, Npvare the total numbers of PQ and PV

nodes.


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Components in genetic approach

1. Chromosomes: The real and imaginary parts of the

voltages of the nodes in the power system are

encoded using floating-point numbers and are set aselements in the chromosomes.

2. Fitness function:

3. Crossover operation: 2 point crossover method to

bring more diversity in the population of

chromosomes.4. Mutation operation: An element of a chromosome is

selected randomly. The voltage value of the element

is replaced by a value arbitrarily chosen within a

range of voltage values.

M is a constant for amplifying the fitness value.


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Initialize schromosomes in the population

Fitness f(x) of each chromosome (fittest

chromosomes always retained)

Replace the current population with new population

Mutation

Crossover (Pc= crossover

rate/probability)

Selection of chromosomes (roulette

wheel method)

No of

Offspring

=s? Maxnumber of

generation

reached?

End

Fig.1 GA Load flow

Algorithm flowchart

No

Yes

Yes No


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Constraint satisfaction technique for updating candidate nodal

voltages

(a) Satisfying the powers at a PQ node i by updating a PQ node d.

(b) updating the voltage at a PV node to satisfy its voltage and activepower requirements

Constraint satisfaction for PQ nodes

Let the real and imaginary voltages of node d be Eidand Fid. The power

mismatches Pi in eqn. 3 and Q

i in eqn. 4 for node i are now set to zero.

From eqns. 1-4, when d i, Eidand Fid can be calculated according to


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When d = i , the power constraints at PQ node d itself are required to be met.

The constraint equations for calculating Eidand Fid of node d can be derived

from eqns. 1- 4 by the same procedure above and by setting the subscript i

in eqns. 1-4 to d.

Constraint satisfaction for PV nodes

Let the real and imaginary voltages of the PV node d in the chromosome be

Eddand Fdd. The mismatches Pdin eqn. 3 and Vdin eqn. 5 for node d can

now be set to zero. From eqns. 1, 3, 5 and 6, the expressions for Eddand Fddare:


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Methods for enhancing the CGALF Algorithm

a) Dynamic population technique

Diversity of the chromosomes increased by introducing new

chromosomes in the population to escape from local minimum points.

% of existing weaker chromosomes replaced by randomly generatedchromosomes when the values of objective function H are identical for a

specified number of generations or iterations- subject to constraint

satisfaction.

b) Solution acceleration technique

Faster convergence. Modify the constrained candidate solution process such that the revised

solutions in the chromosomes are closer to the candidate solution in the

best or fittest chromosome found so far.

Vk=2Vk,bestVk

c) Nodal voltage updating sequencei. Update the voltages of the PV nodes in the sequence of the node

number using eqns. 12-14.

ii. Then, the PQ node, which has the largest total mismatch, is updated

first using the constraint satisfaction methods.


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(iii) Repeat step (ii) until all the PQ nodes are processed.

In step (i) above, the update operation attempts to meet the

voltage magnitude constraints and active power requirements of

the PV nodes.The strategy employed in step (ii) guarantees a reduction of the

mismatch at the node with the largest total mismatch. The strategy

is applied dynamically during the processing of the nodes as

indicated in step (iii).

Application examples

Klos-kerner 11 node test system.

Two loading condition considered.


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Node 1: Slack node,voltage level=1.05pu. Nodes 5 and 9 are PV nodes

with target voltages of 1.05pu and 1.0375pu.


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Hybrid Particle swarm optimization

application

1. Problem Formulation: The load flow equations, at any given bus(i) inthe system, are as follows:


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The optimization problem is formulated as follows:

2. Hybrid Particle Swarm Optimization.

The PSO model consists of a number of particles moving around

in the search space, each representing a possible solution to a

numerical problem. Each particle has a position vector (xi)and avelocity vector (vi)), the position (pbesti) is the best position

encountered by the particle (i) during its search and the position

(gbest) is that of the best particle in the swarm group.


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In each iteration the velocity of each particle is updated according to its

best-encountered position and the best position encountered among

the group, using the following equation:

The position of each particle is then updated in each iteration by

adding the velocity vector to the position vector.

Inertia weight wcontrol the impact of the previous history of velocities

on the current velocity-it regulates the trade-off between the global and

local exploration abilities of the swarm.

Suitable value for w usually provide balance between global and local

exploration abilities and consequently a reduction on the number of

iterations for optimal solution.


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Ability of breeding, a powerful property of GA is used.


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Numerical Examples

IEEE 14 bus system:


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Thank youReferences:

1. Power System Analysis, Hadi saadat, McGraw Hill International editions.

2. Fuzzy Logic application in load flow studies,J.G.Vlachogiannis,IEE,2001.

3. Development of constrained-Genetic Algorithm load flow method,

K.P.Wong,A.Li,M.Y.Law,IEE,1997.4. Load flow solution using Hybrid Particle Swarm Optimization, Amgad

A.El-Dib et.al, IEEE,2004.

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A Review on Load Flow Studies Final