MULTI-OBJECTIVE GENETIC ALGORITHM FOR REACTIVE POWER OPTIMIZATION

P. Aruna Jeyanthy et. al. / International Journal of Engineering Science and Technology Vol. 2 (7), 2010, 2715-2729

MULTI-OBJECTIVE GENETIC ALGORITHM

FOR REACTIVE POWER OPTIMIZATION INCLUSING

VOLTAGE STABILITY P. Aruna Jeyanthy

EEE DEPT, N.I.C.E, Kumarakoil. TamilNadu, India, [email protected]

D. Devaraj,

EEE DEPT, Kalasalingam University, TamilNadu, India, [email protected]

Abstract:

Reactive power optimization is a major concern in the operation and control of power systems. In this paper a new multi-objective genetic algorithm method is applied to optimize the reactive power dispatch problem. The objectives of the reactive power optimization problem are minimization of the losses and maximization of the voltage stability margin. The proposed method expands the original GA to tackle the mixed –integer non linear optimization problem with continuous and discrete control variables such as generator terminal voltages, tap position of transformers and reactive power sources. The optimization variables namely, generator voltages, transformer ratios and reactive power sources are taken as floating point numbers in the genetic population. For effective genetic operation, crossover and mutation operators which can directly operate on floating numbers are used. Multi-objective Genetic Algorithm (MOGA) is applied to solve this reactive power dispatch problem. The MOGA emphasize non-dominated solutions and simultaneously maintains diversity in the non-dominated solutions. Thus this technique handles the problem as a true multi-objective optimization problem. In accordance with this, a comparison is made for other evolutionary methods for the real power losses and this method is found to be effective than other methods. The proposed approach has been evaluated on the IEEE 30-bus and IEEE 57- bus test system, and the simulation results show the effectiveness of this approach for solving the multi-objective reactive power optimization problems. Keywords: Multiple objective genetic algorithm (MOGA); Pareto optimal frontier; Reactive power

optimization; Real power loss and Voltage stability margin (VSM)

1. Introduction:

Reactive power optimization is one of the difficult optimization problems in power system operation and control. To improve the voltage profile and to decrease the active power losses along the transmission lines under various operating conditions, power system operator can select a number of control tools such as switching reactive power sources, charging generator voltages and adjusting transformer tap settings. The multi- objective of this paper is to allocate reactive power sources so that the active power transmission loss is to be minimized and the voltage stability margin is to be maximized, while satisfying the number of constraints [1]. Many conventional techniques such as gradient –based search algorithms and various mathematical programming methods are used for the reactive power optimization problems [2]. In the past decade, heuristic methods [3] have been applied to solve the optimal VAR dispatch problem. These methods have many drawbacks such as insecure convergence properties and algorithmic complexity. In general they are not able to locate or identify the global optimum. The studies on evolutionary algorithms, over the past few years, have shown that these methods can be efficiently used to eliminate most of the difficulties of classical methods [4-5].When an optimization problem involves more than one objective function, the task of determining one or more optimum solutions is known as multi-objective optimization. Because of the presence of conflicting multiple objectives, a multi-objective optimization problem results in a number of optimal solutions, known as

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Pareto-optimal solutions. In a multi-objective optimization, effort must be made in determining the set of trade-off optimal solutions by considering all objectives to be important. There are two goals in a multi-objective optimization:

1. To find a set of solutions as close as possible to the Pareto-optimal front.

2. To find a set of solutions as diverse as possible

An interior point [6] is used to solve the reactive power optimization problem with a multi-objective function for maximizing both social benefit and the distance to maximum loading conditions. In [7], linear programming with bounded variables is used to obtain the optimal shift in power dispatch related to contingency states or overload situations in power system operation and planning phases under various objectives such as economy, reliability and environmental conditions. But, these conventional approaches are time consuming and tend to find weak Pareto-optimal solutions. The ability of Evolutionary Computation techniques like genetic algorithm to find multiple optimal solutions in one single simulation run makes them unique in solving multi-objective optimization problems [8].

Voltage Stability is becoming an increasing source of concern in stability operation of present day power systems. The problem of voltage instability is mainly considered as the inability of the network to meet the load demand imposed interms of inadequate reactive power support or active power transmission capability or both. Voltage collapse is a local load bus problem and depends mostly on load conditions in the system. Thus the reactive power support and voltage problems are intrinsically related. Hence this paper formulates the reactive power optimization as a multi-objective optimization problem with loss minimization and voltage stability margin maximization objectives. The static voltage stability margin is primarily associated with the reactive power support. Several tools have been presented in the literature for the analysis of the static voltage stability of a system. Here the modal analysis is used as the indicator of voltage stability margin. This technique provides voltage stability critical areas and gives information about the best corrective/preventive actions for maximizing static voltage stability margins. It is done by evaluating the Jacobian matrix, the critical eigen values/vector [9, 10].

Multi-objective genetic algorithm (MOGA) [11] is applied for solving the multi-objective reactive power optimization problem. This method was first introduced by Fonseca and Fleming (1993) [12]. Generally, binary strings are used to represent the decision variables of the optimization problem in the genetic population irrespective of the nature of the decision variables. This binary coded GA has Hamming cliff problems [13] which sometimes may cause difficulties in the case of coding continuous variables. Also, for discrete variables with total number of permissible choices not equal to 2k (where k is an integer) it becomes difficult to use a fixed length binary coding to represent all permissible values. This problem is one of the combinatorial optimization problems with multi- extremism and non-linear property. To overcome the difficulties, in this paper, the optimization variables namely, generator voltages, transformer ratios and VARS are taken as floating numbers in the genetic population. For effective genetic operation, crossover and mutation operators which can directly operate on floating numbers are used. The effectiveness and potential of the proposed approach to solve the multi-objective reactive power optimization problem has been evaluated using IEEE 30-bus and IEEE 57-bus systems.

2. Problem formulation:

Power systems are expected to operate economically (minimize losses) and technically (good stability).Therefore reactive power optimization is formulated as a multi-objective search which includes the technical and economic functions.

2.1 Economic function:

The economic function is mainly to minimize the active power transmission loss and it is stated as [14]

F1 = Minimize lossP =

ENk

ijjijik VVVVg )cos2( 22 (1)

The reactive power optimization problem is subjected to the following constraints.

Equality Constraints:

These constraints represent load flow equation such as

1

)sincos(j

ijijijijjiDiGi BGVVPP 1,2,1 BNi (2)

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1

)cossin(j

ijijijijjiDiGi BGVVQQ PQNi ,2,1 (3)

The equality constraints are satisfied by running the power flow program.

Inequality constraints:

These constraints represent the system operating constraints. Generator terminal bus voltages, transformers tap setting, reactive power generated by the capacitor bank is control variables and they are self –restricted by the optimization algorithm. The active power generation at the slack bus , load bus voltages, reactive power generation and line flow limit are state variables .The state variables are satisfied by adding a penalty terms in the objective function. These constraints are formulates as: Voltage constraints:

maxmin

iii VVV BNi (4)

Transformer tap-setting limit:

maxmin

kkk TTT TNk (5)

Generator reactive power capability limit:

maxminGiGi QQQ

Gi GNi (6)

Capacitive reactive power capability limit:

CCiCiCi NiQQQ maxmin (7)

Transmission line flow limit:

maxll SS lNl (8)

Where GV is the generator voltage magnitude at bus i (continuous)

kT is the transformer tap setting (integer)

cQ is the shunt capacitor/ inductor (integer)

LV is the load bus voltage

GQ is the generator reactive power

kgiB NJNijik ,,),,( is the conductance of branch k

ij is the voltage angle difference between bus i and j

GiP is the injected active power at bus i

DiP is the demanded active power at bus i

ijG is the transfer conductance between bus i and j

ijB is the transfer susceptance between bus I and j

GiQ is the injected reactive power at bus i

DiQ is the demanded reactive power at bus i

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EN is the set of numbers of network branches

PQN is the set of number of PQ buses

BN is the set of numbers of total buses

iN is the set of numbers of buses adjacent to bus i (including bus i )

cN is the set of numbers of possible reactive power source installation buses

TN is the set of numbers of transformer branches

lS is the power flow in branch l

The subscripts ‘min’ and “max” in Eq. (4-8) denote the corresponding lower and upper limits

respectively.

2.2 Technical function:

The technical function is to minimize the bus voltage deviation from the ideal voltage and to improve the voltage stability margin (VSM) and it is stated as [15] F2 = Max VSM= Max(min|eig(jacobi)|) (9)

Where jacobi is the load flow jacobian matrix , eig (jacobi) returns all the eigen values of the Jacobian matrix, min(eig(Jacobi)) is the minimum value of eig (Jacobi) , Max ( min ( eig (Jacobi))) is to maximize the minimal eigen value in the Jacobian matrix.

2.3 Multi-objective function:

Aggregating the objectives and constraints, the problem can be mathematically formulated as a non linear constrained multi-objective optimization problem as follows: Minimize F= [F1, F2] (10)

Subject to the constraints (2) – (9)

3. Multi-objective Genetic Algorithm :

3.1 Overview:

Genetic Algorithms [16] are generalized search algorithms based on the mechanics of natural genetics. GA maintains a population of individuals that represent the candidate solutions to the given problem. Each individual in the population is evaluated to give some measure to its fitness to the problem from the objective function. Genetic Algorithms combine solution evaluation with stochastic operators namely, selection, crossover and mutation to obtain optimality. Being a population –based approach, GA is well suited to solve multi-objective optimization problems.

Many real world problems involve simultaneous optimization of several objective functions. Generally,

these objective functions are non-commensurable and often conflicting [17]. Multi-objective optimization with such conflicting objective functions give rise to a set of optimal solutions, instead of one optimal solution. The reason for the optimality of many solutions is that no one can be considered to be better than any other with respect to all objective functions. These optimal solutions are known as Pareto-optimal solutions. A set of Pareto solutions is called Pareto-set and its image on the objective space is called Pareto-front. A dominated solution contained within a problem search space is a solution that is inferior to at least one other solution with respect to all defined objectives. Non-dominated solutions are said to be Pareto Optimal solutions. The set of all non-dominated solutions form the Pareto Frontier which depicts the optimal tradeoffs that exist between competing objectives. There are different approaches to solve multi-objective optimization problems e.g., aggregating, population based non-Pareto, and Pareto-based techniques. In aggregating techniques, the different objectives are generally combined into one using weighting or goal-based method. Vector evaluated genetic algorithm (VEGA) is a technique in the population-based non-Pareto approach in which different subpopulations are used for the different objectives. Multiple objective GA (MOGA), non-dominated sorting GA (NSGA), and niched Pareto GA (NPGA) constitute a number of techniques under the Pareto-based non-elitist approaches [18-19].

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The present paper implements MOGA as the multi-objective optimization algorithm. The MOGA differs from the standard GA in the way fitness is assigned to each solution in the population. The rest of the algorithm is similar to the original GA. Figure: 1 below shows the MOGA algorithm. MOGA is a commonly used multi-objective optimization technique well suited to solve highly constrained optimization problems. At its heart, the MOGA is essentially a simple genetic algorithm tuned to solve multi-objective problems. The proposed optimization algorithm is based on MOGA, since it is the efficient and most simple method in the evolutionary algorithms. Here the fitness value of an individual is proportional to the number of other individual it dominates.

Figure: 1 Flow chart of MOGA Algorithm

Initialize the population

Run the load flow program

Evolution (fitness assignment)

Set i=1

Selection, crossover and mutation

Form the new population

Run the load flow program

Evolution (Fitness Assignment)

Meeting the

stopping rule

Output

i=i+1

No

Yes

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3.2 Algorithm of MOGA:

In MOGA, first each solution is checked for its domination in the population. To a solution “i”,a rank equal to one plus the number of solutions that dominate solution “i” is assigned .In this way, non-dominated solutions are assigned a rank equal to 1, since no solution would dominates a non-dominated solution in a population. Once the ranking is performed, a raw fitness to a solution is assigned based on its rank. To perform this, first the ranks are sorted in ascending order of magnitude. Then a raw fitness is assigned to each solution by using a linear (or any other) mapping function. Usually, the mapping function is chosen so as to assign fitness between N (for the best rank solution) and 1 (for the worst rank solution).Thereafter, solutions of each rank are considered at a time and their raw fitnesses are averaged. This average fitness is now called the assigned fitness to each solution of the rank. This emphasizes non-dominated solutions in the population. In order to maintain diversity among non-dominated solutions, niching among solutions of each rank are introduced. The algorithm used in MOGA is as follows:

Step 1 Set i=1 .Initialize 0)( j for all possible ranks j= 1, 2,… N.

Step 2 Calculate the number of solutions (n i) that dominate solution i. Compute the rank of the i-th solution as r i=1+ ni . Increment the count for the number of solutions in rank ri by one, that is .1)()( ii rr

Step 3 If i<N, increment i by one and go to step 1. Otherwise go to step 4.

Step 4 Identify the maximum rank r* by checking the largest r i which has .0( ) ir the sorting

according to rank and fitness –averaging yields the following assignment of the average fitness to any solution i=1, …N.

1

1

)1)((5.0)(ir

kii rkNF (11)

Step 5 For each solution i in rank r, calculate the niche count nci with other solutions of the same rank using equation (16) - (18).

)(

1

)(ir

jiji dShnc

(12)

Where )( ir is the number of solutions in rank ir

sharing function is

otherwise

difd

dSh shareijshare

ij

ij

0

)(1)(

(13)

share is the maximum distance allowed between any two individuals to become a

number of niche . is a scaling factor less than or equal to 1. ijd is the normalized distance between two solutions i and j

2

1

1

2minmax

)()(

)(

M

k kk

jk

ik

ijff

ffd (14)

where fkmax and fk

min are the maximum and minimum objective function value of the kth objective. The

shared function takes a value in [0, 1] depending on the values of dij and σshare .The shared fitness value is calculated by dividing the assigned fitness to a solution by its niche count. Although all solutions of any particular rank have the identical fitness, the shared fitness value of a solution residing in a less crowded region has a better shared fitness. This produces a large selection pressure for poorly represented solutions in any rank.

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Calculate the shared fitness usingj

jj nc

FF ' . To preserve the same average fitness, scale the shared

fitness as follows:

')(

1

'

')(

jr

kk

jj F

F

rFF

(15)

Step 6 If r < r*, increment r by one and go to step 5. Otherwise the process is complete. Thus the procedure is continued until all ranks are processed. Thereafter, selection, crossover and mutation operators are applied to create a new population.

Based on the above algorithm steps the ranking, fitness and sharing the fitness are computed. Here dividing the assigned fitness values by the niche count reduces the fitness of each solution. In order to keep the average fitness of the solutions in a rank the same as that before sharing, these fitness values are scaled so that their average shared fitness value is the same as the average assigned fitness value. After these calculations, the solution to the next rank is executed. Thus this process is repeated until all the ranks are processed. Then the stochastic universal selection with shared fitness values, crossover and the mutation operators are created to a new population.

4. Implementation of Multi-objective Genetic Algorithm:

When applying MOGA for solving reactive power optimization problem, the following issues need to be

addressed:

(1) Solution Representation (2) Fitness evaluation and (3) Application of genetic operators

4.1: Solution Representation:

Implementation of GA for a problem starts with the parameter encoding (i.e. the representation of the problem). Each individual in the genetic population represents a candidate solution. The elements of the solution consist of all the decision variables in the system.Figure:2 represent the elements of the solution. The decision variables of the reactive power optimization problem include generator bus voltage magnitude, transformer tap settings and reactive power sources. The solution variables are represented as floating point numbers. With direct representation of the solution variables, the computer memory required to store the population is reduced.

v1 v2 …… vg T1 T2 ….. Tt λ1 λ2 ….. λt

Elements of the solution

Figure 2: Representation of elements of the solution of a MOGA

The length of the elements is equal to the total number of control variables. Here the generator terminal voltages and the transformer tap setting are used as the control variables. These variables are represented in the natural form. With this mixed form of representation, a typical chromosome of the reactive power optimization problem looks like the following:

kkkGnGG TTTVVV

81281305.1970.0981.022121

The use of floating numbers in MOGA representation has a number of advantages like lesser memory requirement and no loss in precision by discrimination to binary or other values.

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4.2: Evaluation Function:

The power system operator has the control to vary a few variables in the system only. Some of these independent variables called control variables will be represented as the solution variables in the population. From these candidate solutions (control variables), the dependent variables (state variables) of the system have to be derived and those candidate solutions which result in violation of upper or lower limits of the state variables have to be penalized to discourage the infeasible solutions. GA searches for the optimal solution by maximizing a given fitness function, and therefore an evaluation function which provides a measure of the quality of the problem solution must be provided. In the reactive power optimization problem under consideration, the objective is to minimize the losses and maximize the voltage stability margin satisfying the constraints. The equality constraints are satisfied by running the power flow program. Generator terminal bus voltages, transformers tap setting, reactive power generated by the capacitor bank is control variables and they are self –restricted by the optimization algorithm. The active power generation at the slack bus , load bus voltages, reactive power generation and line flow limit are state variables .The state variables are satisfied by adding a penalty terms in the objective function. With the inclusion of the penalty function, the new multi-objective objective function with the economic and technical function from equation (10) is generalized as follows:

lim lim

1

1

2max

22 )(maxmin

V QN N

N

LiisGQLvloss SSQVVSMPimizef (16)

where Qv , ,λs are the penalty factors. limVN is the set of numbers of load buses on which voltage outside

limits . limQN is the set of number of generator buses on which injected reactive power outside limits . LV and

GQ are defined as:

minmin

maxmaxLLLL

LLLL

VVifVVVVifVVLV

(17)

minmin

maxmaxGGGG

GGGG

QQifQQ

QQifQQGQ

(18)

Thus MOGA is usually designed to maximize the fitness function, which is a measure of the quality of each candidate solution. Hence in this work, the fitness is taken as the inverse of the new objective function.

4.3: Genetic Operators:

4.3.1: Selection Strategy:

Selection plays an important role in GA, it determines the direction of search in the search space. It emphasizes good solutions and eliminates bad solutions while keeping the population size constant. The goal is to allow the “fittest” individuals to be selected more often to reproduce. In this work we use “tournament selection” for this purpose. In tournament selection,”n” individuals are selected at random from the population, and the best of the “n” is inserted into the new population foe further genetic processing. This procedure is repeated until the matting pool is filled. Tournaments are often held between pairs of individuals (tournament size-2), although larger tournaments can be used. 4.3.2: Crossover:

The crossover operator is a method for sharing information between chromosomes. Generally, it combines the features of two parent chromosomes to form two offspring, with the possibility that good chromosomes may generate better ones. In this work, BLX-α crossover is applied on the selected individuals. Figure: 3 illustrate the BLX- α crossover operation for the one dimensional case. In this figure, u1 and u2 are the selected individuals and umin and umax are selected variable’s lower and upper limit respectively.

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Figure 3: Representation of BLX-α Crossover

In the BLX-α crossover, the offspring (y) is sampled from the space [e1, e2] as follows:

y =

otherwisesamplingrepeat

uyuifeere

;

; maxmin121

(19)

where ,

e1=u1-α(u2-u1) and e2=u1+α(u2-u1) and

r = uniform random number [0 1]

For two parent solutions u1 and u2 (assuming u1<u2) BLX- α randomly picks new solutions in the range [e1, e2].

It is to be noted that e1 and e2 will lie between umin and umax, the variable’s lower and upper bound respectively. In a number of test problems, the investigators have observed that α = 0.5 provides good results. In this crossover operator the location of the off spring depends on the difference in parent solutions. If both parents are close to each other, the new point will also be close to the parents. On the other hand, if the parents are far from each other, the search is more like a random search. This property of a search operator allows us to constitute an adaptive search.

4.3.3: Mutation:

The mutation operator is used to inject new genetic material into the population. Mutation changes randomly the new offspring. In this work, “Non Uniform Mutation” operator is applied to the mixed variables with some modifications. First a variable is selected from an individual randomly. If the selected variable is uk with the range |uk

min , uk

max| , two random numbers are generated and the result u1k is calculated as

5.01.

5.01.

1

1

1min

1

1

1max

1

rifruuu

rifruuu

uq

q

M

pk

kk

M

p

kk

k

k

(20)

Where p is the generation number, q is a non uniform mutation parameter and M is the maximum generation number. If the selected variable is an integer then the randomly generated floating point number is truncated to the nearest integer. After mutation, the new generation is complete and the algorithm begins again with the fitness evaluation of the population. 5. Simulation results:

The proposed MOGA approach for solving the reactive power optimization was applied to IEEE 30-bus and IEEE 57-bus test system. The real power settings are taken from [20-21]. The proposed algorithm was run with

I

u max

e2

u2 u1

e1

u min

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minimization of real power loss and maximization of voltage stability margin as the objectives. To demonstrate the effectiveness of the proposed approach, three different cases have been considered as follows:

Case 1: Multi-objective reactive power optimization for voltage stability margin in IEEE 30-bus system.

Case 2: Multi-objective reactive power optimization for voltage stability margin in IEEE 57-bus system.

Case 3: Contingency constrained reactive power optimization for IEEE 30-bus and IEEE 57-bus system. Case 1: Multi-objective reactive power optimization for voltage stability margin in IEEE 30-bus system:

The IEEE 30 bus system has 6 generator buses,24 load buses and 41 transmission lines of which four branches are (6-9),(6,10),(4,12) and (28-27) are with tap setting transformers. This is shown in figure: 4, the upper and lower voltage limits at all buses except slack bus are taken as 1.10 p.u and 0.95 p.u respectively. The slack bus voltage is fixed to its specified value of 1.06 p.u. Generator terminal voltages, transformer tap settings and reactive power sources were taken as the optimization variables. The possible locations for reactive power sources are buses 10, 12, 15, 17, 20, 21, 23, 24 and 29. The optimization variables are represented as a mixture of floating point numbers in the MOGA population. The initial population was randomly generated between the variable’s lower and upper limits. Tournament selection was applied to select the members of the new population. Blend crossover and uniform mutation were applied on the selected individuals. The performance of MOGA generally depends on the MOGA parameter used, in particular, the crossover and mutation probabilities respectively. The performance of MOGA for various crossover and mutation probabilities in the range 0.6-0.9 and 0.001-0.01 respectively was therefore evaluated. It was applied by considering several sets of parameters inorder to prove its capability to provide acceptable trade-offs close to the Pareto optimal front. The optimal settings of the MOGA were obtained by the following parameters are given below:

Generations : 50

Population size: 50

Crossover rate: 0.8

Mutation rate: 0.01

Variable : 19

It is worth mentioning that the proposed approach produces nearly 15 Pareto optimal solutions in a single run that have satisfactory diversity characteristics and span over the entire Pareto optimal front. Figure: 5 represent the Pareto-optimal front curve. From the Pareto front, two optimal solutions which are the extreme points of figure: 5 represent the minimum real power loss and maximum voltage stability margin. The optimal values of the control variable are given in Table: 1. The minimum loss obtained by the proposed algorithm is compared with other evolutionary methods and the results are presented in Table: 5. The minimum loss obtained by this method is less than the other methods. This shows the effectiveness of the proposed approach in solving the reactive power optimization problem.

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Figure: 4 IEEE 30 bus systems

Table I Pareto optimal solutions of control variables for IEEE 30 bus system

Control variables Minimum real power loss solution Maximum voltage stability margin solution

V1

V2

V5

V8

V11

V13

T11

T12

T15

T36

QC10

QC12

QC15

QC17

QC20

QC21

QC23

QC24

QC29

Real Power Loss

VSM

0.9554

0.9695

1.0151

0.9531

0.9985

0.9585

1.05

1.05

1.1

1.1750

2

2

4

4

0

3

1

5

2

4.483

0.1872

0.9667

0.9567

0.9790

0.9512

0.9720

0.9949

1.025

1.1

1.750

1.0750

0

1

0

0

5

6

0

2

4

3.235

0.205

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Figure 5 Pareto-optimal front using MOGA for IEEE 30 bus system

Case 2: Multi-objective reactive power optimization for voltage stability margin in IEEE 57-bus system:

The IEEE 57-bus system has 7 generators, 50 load buses, 80 transmission lines of which 17 branches are with tap setting transformers. The placements of reactive power sources for installation are buses 25, 30, 32, 34, 35 and 53 to supply reactive power. The MOGA algorithm was tested with different parameter settings and the best results are obtained with the following setting:

Generations : 50

Population size : 50

Crossover rate : 0.8

Mutation rate : 0.01

Variable : 29

The optimal values and the Pareto front diagram are shown in Table: III and figure: 6 respectively. The algorithm reaches a minimum loss of 25.667 MW. The loss obtained by the proposed algorithm is compared with evolutionary computations. The loss obtained is less than the value reported in [22].This shows the effectiveness of the proposed approach in solving the reactive power optimization.

Table III Optimal control variables in IEEE – 57 bus system

Control Variables Variable Setting Variable Setting including the VSM

VI V2 V3 V6 V8 V9

V12 T19

0.9421 1.0232 0.9638 0.9512 1.0639 0.9410 0.9880 0.9508

0.9451 1.0161 0.9968 0.9965 0.9575 0.9810 0.9516 0.9491

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T20 T31 T35 T36 T37 T41 T46 T54 T58 T59 T65 T66 T71 T73 T76 T80 Q30 Q32 Q31 Q33 Q34

Real power Loss VSM

1.1 1.0250 1.0740 0.95

1.0250 1.0772 1.1951 1.0577 1.0163 1.0934 0.9755 1.0834

1 1.0063 0.9501 0.9697

0 0 1 0 5

25.5667 0.1256

1.05 0.9

1.1281 0.9789 0.9244 1.0250 1.0956 1.0573 1.0892

0.9 0.9990 1.0847 0.9750 0.9807 1.0014 1.0287

4 3 1 1 0

25.6507 0.1297

Figure 4 Pareto-optimal front using MOGA for IEEE 57 bus system

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Case 3: Contingency constrained reactive power optimization for IEEE 30-bus and IEEE 57-bus system:

In case 1 and case 2, the proposed MOGA algorithm is applied to minimize the real power losses and to maximize the voltage stability margin without including the contingency constraint. To access the voltage security of the system, contingency analysis was conducted using the control variable setting obtained in case 1 and case 2.From the contingency analysis, the line outages 28-27, 1-2, 4-12 ,3-4 are identified as the most severe critical lines in IEEE 30-bus system. Similarly the lines 25-30, 34-32, 37-38 are identified as most severe in IEEE 57-bus system .The maximum voltage stability margin values corresponding to these contingencies are given in Table 3 and Table 4. From this result it is observed that the minimum eigen value has increased appreciably for all contingencies in the second case (MOGA-VSM).This shows that the proposed algorithm has helped to improve the voltage security of the system. Table 2 contingency analysis for IEEE 30 bus system:

Line VSM MOGA-VSM

28-27

1-2

4-12

3-4

0.0704

0.0926

0.1254

0.1436

0.0931

0.0951

0.1293

0.1474

Table IV contingency analysis for IEEE 57 bus system:

Line VSM MOGA-VSM

25-30

34-32

37-38

0.0049

0.0017

0.0239

0.1298

0.0092

0.0137

Table V Comparison of optional result obtained by different methods for

IEEE 30 – bus and IEEE 57 – bus system

No Evolutionary methods Minimum power loss (MW) 1 2 3 4

IEEE 30 – bus Evolutionary programming [23]

Simple GA [3] Real coded GA [1]

IEEE 57 – bus Genetic Algorithm [23]

5.0159 4.98 4.568

25.9654

6. Conclusion

This paper presents a MOGA algorithm approach to obtain the optimum values of the reactive power variables including the voltage stability constraint. The effectiveness of the proposed method is demonstrated on IEEE-30 and IEEE-57 bus system with promising results. The performance of the proposed algorithm is demonstrated through its voltage stability enhancement by modal analysis. From this multi-objective reactive power optimization solution, the application of MOGA has performed well when it was used to characterize Pareto optimal front and leads to global search with fast convergence rate and a feature of robust computation. From the simulation work, it is concluded that MOGA performs better results than other evolutionary methods. This approach is found to generate high quality solutions with more stable convergence than simple genetic algorithms.

References:

[1] D.Devaraj, “Improved genetic algorithm for multi – objective reactive power dispatch problem,” European Transactions on Electrical Power, 2007; 17; 569-581.

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MULTI-OBJECTIVE GENETIC ALGORITHM FOR REACTIVE POWER OPTIMIZATION

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