Scholarly article on topic 'Application of Multi-Objective Teaching Learning based Optimization Algorithm to Optimal Power Flow Problem'

Application of Multi-Objective Teaching Learning based Optimization Algorithm to Optimal Power Flow Problem Academic research paper on "Electrical engineering, electronic engineering, information engineering"

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Abstract of research paper on Electrical engineering, electronic engineering, information engineering, author of scientific article — M.R. Nayak, C.K. Nayak, P.K. Rout

Abstract This paper presents a non-domination based sorting multiobjective teaching-learning-based optimization algorithm, for solving the optimal power flow (OPF) problem. The OPF problem is a nonlinear constrained multi-objective optimization problem where the fuel cost, Transmission losses and L-index are to be minimized. Since the problem is treated as a true multi-objective optimization problem, different trade-off solutions are provided. The decision maker has an option to choose a solution among the different trade-off solutions provided in the pareto-optimal front. The standard IEEE 30-bus test system is used and the results show the effectiveness of MOTLBO and confirm its potential to solve the multi-objective OPF problem. Simulation results clearly show that the proposed method is able to produce true and well distributed Pareto optimal solutions for multiobjective OPF problem and the comparison with the results reported in the literature demonstrates the superiority of the proposed approach and confirms its potential to solve the multi-objective OPF problem.

Academic research paper on topic "Application of Multi-Objective Teaching Learning based Optimization Algorithm to Optimal Power Flow Problem"

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Procedía Technology 6 (2012) 255 - 264

2nd International Conference on Communication, Computing & Security [ICCCS-2012]

Application of Multi-Objective Teaching Learning Based Optimization Algorithm to Optimal Power Flow Problem

M. R. Nayaka, C. K. Nayakb and P.K. Routc *

Deptt. of Electrical Engg, Institute of Technical Education & Research, S'O'A University, Bhubaneswar, 751030, Odisha, India c Deptt. of EEE, Institute of Technical Education & Research, S'O'A University, Bhubaneswar, 751030, Odisha, India

Abstract

This paper presents a non-domination based sorting multiobjective teaching-learning-based optimization algorithm, for solving the optimal power flow (OPF) problem. The OPF problem is a nonlinear constrained multi-objective optimization problem where the fuel cost, Transmission losses and L-index are to be minimized. Since the problem is treated as a true multi-objective optimization problem, different trade-off solutions are provided. The decision maker has an option to choose a solution among the different trade-off solutions provided in the pareto-optimal front. The standard IEEE 30-bus test system is used and the results show the effectiveness of MOTLBO and confirm its potential to solve the multi-objective OPF problem. Simulation results clearly show that the proposed method is able to produce true and well distributed Pareto optimal solutions for multiobjective OPF problem and the comparison with the results reported in the literature demonstrates the superiority of the proposed approach and confirms its potential to solve the multi-objective OPF problem

©2012Elsevier Ltd...Selectionand/orpeer-reviewunderresponsibility oftheDepartmentofComputerScience & Engineering,NationalInstitute of Technology Rourkela

Keywords: Multi-objective Optimal Power Flow; Pareto optimal front; Non-dominated sorting; Teaching - learning-based optimization.

1. Introduction

The optimal power flow (OPF) problem has become an essential for operation, control and planning of

* Corresponding author. Tel.: 09437332558. E-mail address: manasnk72@gmail. com

2212-0173 © 2012 Elsevier Ltd...Selection and/or peer-review under responsibility of the Department of Computer Science & Engineering, National Institute of Technology Rourkela doi: 10.1016/j.protcy.2012.10.031

modern power systems. In a number of real world optimization problems, multiple competing objectives make us solve them simultaneously instead of solving them separately. This gives rise to a set of optimal solution (Largely known as Pareto optimal solution) rather than a single optimal solution. In the absence of knowledge, it is not possible to find a better solution than others from the Pareto optimal solutions. Because, one can not be better than other without any further information. Therefore, it is necessary to find as many Pareto optimal solutions as possible. Classical methods do convert the multi objective optimization problem to a single objective optimization problem by a suitable scaling/weighting factor method. This results in a single optimal solution. To obtain a Pareto optimal solutions, it should be run as many times as the number of solutions. OPF problem is a nonlinear, constrained optimization problem where many competing objectives are present. Traditionally, OPF problem has been solved for different objectives as a single objective optimization problem (K. Lee et al.,1985, J. Momoh et al.,1999 & Momoh JA et al.,1999). This resulted in a optimal solution which satisfies one objective and not others. Therefore, to satisfy and find a compromise solution between two competing objectives, OPF problem is solved as a multiobjective optimization problem with different constraints.

2. Literature Survey

Traditionally, multiobjective OPF problem has been solved by weighted sum and e -constraint method (C.Coello et al., 1999). The weighted sum method converts multiobjective optimization problem to a single objective optimization problem by giving suitable wei^ts to the objectives. Whereas, £ -constraint method treats most preferred objectives for optimization and non preferred objective as a constraint in the allowable range e . This range is further modified to obtain a Pareto optimal solution. These methods require multiple runs to obtain a Pareto optimal solution and need much computational time resulting in a weekly non-dominated solution.

Recently, multiobjective evolutionary algorithms have been reported to solve environmental/economic dispatch (EED), OPF and VAR dispatch problem (M.A.Abido et al.,2003, M.Abido et al., 2005, M.Abido et al.,2006 & M. Varadarajan et al.,2008 ). These evolutionary algorithms are proved better than traditional method because of their ability to obtain a Pareto optimal solution in a single run. Since evolutionary algorithms use a population of solutions, they can be easily extended to maintain a diverse set of solutions in a single run. Most evolutionary algorithms reported for EED, OPF and VAR problems use non dominated sorting, strength Pareto approach for maintaining diverse Pareto optimal solutions. This paper considers the non dominated sorting and crowding distance method proposed by Deb (K.Deb et al., 2002) to maintain a well distributed Pareto optimal solutions.

Teaching-learning-based optimization (TLBO) is a very new algorithm introduced in (R.V.Rao et al., 2011). This single objective algorithm is based on the effect of influence of a teacher on the output of the learners in the class. The teacher is considered as a highly learned person who does knowledge sharing with the learners in the class. The quality of the teachers exhibits its impact on the outcome of the learners, which is seen from their results or grades. In this work the main objective is to modify the TLBO algorithm to find pareto optimal solutions in a multiobjective problem domain. In this paper a multiobjective teaching-learning-based optimization (MOTLBO) algorithm using non dominated sorting procedure is developed, which is then applied to OPF problem of standard IEEE 30 bus system. Simulation results clearly show the robustness of the MOTLBO method to obtain well distributed optimal solutions.

The rest of this paper is organised as follows: Section 2 gives literature survey, In section 3 the objective is outlined along with the formulations followed in the system modelling. Section 4 presents detail about Multiobjective optimization .Section 5 describes the formulation of non-dominated sorting based MOTLBO algorithm. Section 6 describes the simulation strategy for implementing the solution to OPF problem and the experimental results obtained. Section 7 is the conclusion.

3. Problem formulation

The optimal power flow (OPF) problem is a static, nonlinear and non-convex optimization problem, which is to optimize the setting of control variables from the network state, load data and system parameters for minimizing the certain objective subject to the several equality and inequality constraints. The OPF problem is generally formulated as follows.

A. Objective functions

1) Fuel cost minimization:

This objective is to minimize the total fuel cost FT of the system. The fuel cost curves of the thermal generators are modeled as a quadratic cost curve and can be represented as

FT =£f=cL K Vi + c£) %/hr (1)

Where , bit are the fuel cost coefficients of the /'th generator, P, is real power output of the /th generator and Ng is the total number of generators in the system.

2) Real power loss:

This objective is to minimize the real power transmission line losses PL in the system which can be expressed as follows.

Pi = Pi'+lf^cosOWj)] (2)

Where gk is the conductance of a transmission line k connected between /' and / th bus, nl is the total number of transmission lines, I '„ V, A and fyare the voltage magnitudes and pliase angles of /' and /th bus respectively.

3) L-Index:

This objective is to maintain the voltage stability and move the system far away from the voltage collapse point. This can be achieved by minimizing the voltage stability indicator L-index (P.Kessel et al.,1986 & T.Tuan et al.,1994) and can be expresses as

¿me* = max iLx, k = 1,2,..........-ni} (3)

B. Constraints

1) Equality constraints:

These constraints are typical load flow equations which can be described as follows

where Pc. is the real power generation at /th bus, PD. is the real power demand at /'th bus, QG is the reactive power generation at /th bus, QD is the reactive power demand at /th bus, Bv is the suceptance of the line connected between i and jth bus, NB is the total number of buses, NPQ is the number of load buses and NG is the number of generator buses in the system.

2) Inequality constraints: These constraints represent the system operating limits as follows

1) Generation constraints: Generator voltages, real power outputs and reactive power outputs are restricted by their lower and upper bounds as follows:

2) Transformer constraints: Transformer tap settings are restricted by their minimum and maximum limits as follows:

3) Shunt VAR constraints: Reactive power injections at buses are restricted by their minimum and maximum limits as:

4) Security constraints: These include the constraints of voltage magnitudes at load buses and transmission line loadings as follows:

< V?™,i= 1.....Npq

4. Multiobjective optimization

Many real world optimization problem involve simultaneous optimization of several conflicting objectives. Multiobjective optimization problems with such conflicting objectives give rise to a set of optimal solution, rather than a single optimal solution. Because, no solution can be considered to be better than other solutions without a information. These set of optimal solutions are called as a Pareto optimal solutions.

A general multiobjective optimization problem consists of multiple objectives to be optimized simultaneously and the various equality and inequality constraints. This can be generally formulated as

Subjects

j = 1,2.....M

k = 1, 2.....K

Where f. is the /th objective function x is a decision vector that represents a solution N is the number of objective functions, M and K are the number of equality and inequality constraints respectively.

For a mutliobjective optimization problem, any two solutions x1 and x2 can have any one of two possibilities, one dominates other or none dominates other. In a minimization problem, without loss of generality, solution x1 dominates x2 if the following conditions are satisfied.

2. 3;-e {1,2.....AfJ : fjCxD

If any one of the above conditions is violated, then the solution x1 does not dominate x2. If x1 dominates the solution x2, x1 is called as the non dominated solution. The solutions that are non dominated within the entire search space are denoted as Pareto optimal solutions.

3.1. Different cases of Multi- objective OPFproblem

The multi- objective OPF problem is formulated as, simultaneous optimization of Case -1: Fuel cost and loss minimization Case -2: Fuel cost and L-Index

5. Multiobjective Teaching-Learning-Based Optimization (MOTLBO) Algorithm

The TLBO algorithm is a very new algorithm recently introduced in (R.V.Rao et al., 2011). This optimization technique performs based on the dependency of the learners in a class on the quality of teacher in the class. The teacher raises the average performance of the class and shares the knowledge with the rest of the class. The individuals are free to perform on their own and excel after the knowledge is shared. The whole procedure of TLBO is divided in to two phases, the Teacher phase and the Learner phase.

Initialization

Initially, a matrix of N rows and D columns is initialized with randomly generated values within the search space. The value N represents the population size of the "class size" in this case. The value D represents the number of "subjects or courses offered", which is same as the dimensionality of the problem taken. The procedure being iterative is set to run for G number of generations. The jth parameter of the ith vector (learner) in the initial generation is assigned values randomly using the equation

-ran+ ran^x(x^~ xm n)

(i,j )

\i,j )

Where randj represents a uniformly distributed random variable within the range (0,1). The parameters of the ith vector (or learner) for the generation g are given by

Xg = A (i)

, (i,1)' (i,2)'

(i, j )'

The objective values at a given generation form a column vector. In a dual objective scenario, such as this one, two objective values are present for the same row vector. Two objectives (a and b) can be evaluated as

For all the equations used in the algorithm, i=l,2,...,N, j=l,2,...,D and g=l,2,...,G. The random distribution followed by all the rand values is the uniform distribution.

Yag fa\X g)|

Ybg AX (o).

Teacher Phase

The mean vector containing the mean of the learners in the class for each subject is computed. The mean vector M is given as

mean^n)x^,..., x( N J

mean^x}),...,x§, X(N,j)j

^ean \|_X(1,D), ■ • •, X(i,D),-~, X( N ,D) \

which effectively gives us

Mg = |mg,mg,...,mg ,...,mg^

The best vector with the minimum objective function value is taken as the teacher (Xgeacher) for that iteration.

The algorithm proceeds by shifting the mean of the learners towards its teacher. A randomly weighted differential vector is formed from the current mean and the desired mean vectors and added to the existing population of learners to get a new set of improved learners.

Xnew(gl)= Xg) + randg x(X?eaCher-TFMg)

where TF is a teaching factor which is randomly taken at each iteration to be either 1 or 2. The superior learners in the matrix Xnew replace the inferior learners in the matrix X using the non-dominated sorting algorithm [20].

Learner Phase

This phase consists of the interaction of learners with one another. The process of mutual interaction tends to increase the knowledge of the learner. Each learner interacts randomly with other learners and hence

facilitates knowledge sharing. For a given learner, Xg) another learner X(gr) is randomly selected (i#). The ith

vector of the matrix Xnew in the learner phase is given as

Xg)+ randg)X Xg)- Xg)) ifYg<V)

Xg + randg)X (x gr) — X g)J otherwise

Xnewg) =

The MOTLBO algorithm, due to the multiobjective requirements, adapts to the scenario by having multiple Xnew matrices in the learner phase, one for each objective. So, the learner phase operations for a dual objective problem are as shown in equations below.

Xnewg)

Xnewg)

X g)+ rand gx (X g0- X g)) X g)+ rand fax (X grX *)

X g)+ rand g)X (X g)-X g)) X g)+ rand g^x (X g)-X g))

ifYag <Yag) otherwise

if(Ybg<Ybg) otherwise

The X matrix and the Xnew matrices are passed together to the non-dominated sorting algorithm and only N best learners are selected for the next iteration. Algorithm Termination

The algorithm is terminated after G iterations are completed. The final set of learners represents the pareto

curve through their objective values.

Best Compromise Solution

For the purpose of decision making, a best compromise solution is computed. In this work, a technique based on fuzzy set theory was applied to extract the best compromise solution. The procedure of this technique can be explained as follows:

• Search through all solutions, to find Fmax and Fmin corresponding to each objective function.

• Use the following linear membership function to calculate a membership function for each objective.

• The above equation gives a measure of the degree of satisfaction for each objective function for a particular solution and also map the objectives into the rage 1 ~ 0.

• The corresponding membership function for the non dominated solution ' k, is calculated as follows:

where, M: # of Pareto solutions; NO: # of objectives. Finally, the best compromise solution is the one achieving the maximum member ship function ( " ).

6. Simulation Results

The simulations have been done using MATLAB software package on a Core2 Duo Intel processor with 3 GHz clock speed and supported by 3 GB of random access volatile memory. The proposed procedure was executed a few times out of which the best solution set is presented here. As an advantage, this algorithm has no parameters to be tuned and hence exhibits homogeneous behavior in all cases, without any requirement of parametric study of the algorithm.

The simulation strategy constitutes of estimating the solutions contributing to the extreme points in the expected pareto curve and then performing the multiobjective evaluations with those solutions points as the initial search points in the MOTLBO algorithm. The optimization capability of the algorithm makes the estimated pareto move nearer to the desired pareto as the iteration progresses. In this current study, only two points, one from each extreme of the pareto curve, are estimated by the MOTLBO algorithm and used as the initial seeds for the MOTLBO algorithm. The rest of the learners are initialized randomly.

In order to validate the robustness of the proposed MOTLBO algorithm method, a standard IEEE 30 bus system has been considered. This system consists of 6 generators at buses 1, 2, 5, 8, 11 and 13, 4 transformers with off-nominal tap ratio in the lines 6-9,6-10, 4-12 and 27-28 and reactive power injection at the buses 10, 12, 15, 17, 20, 21, 23, 24 and 29. The complete system data is given in (K .Lee et al.,1985). The total system demand is PLoad = 283.4 MW and QLoad= 126.2MVAR at 100MVA base. Transformer taps and reactive compensation devices are discrete variables with the changes step of 0.01p.u. The maximum and minimum operating limits of the control variables are given in Table 1.In this paper, three objectives, namely, fuel cost , Transmission losses and L-index ( M.Sailaja Kumari et al., 2010 & S.Sivasubramanai et al., 2011) have been considered. The Fuel cost coefficients are given in Table 2. Here the initial parameters for the MOTLBO algorithm are set as Population size (N) =50, maximum number of iterations (GEN) =500.

Table-1 [Operating limits of the control variables of IEEE 30 Bus Test System ]

Active Power Generation Limits (MW)

Reactive Power Generation Limits (MVAR)

1 2 5 8 11 13 1 2 5 8 11 13

PGmax 200 80 50 35 30 40 QGmax 200 100 80 60 50 60

PGmin 50 20 15 10 10 12 Qg™ -20 -20 -15 -15 -10 -15

Voltage (P.U.) and Tap Setting Limits

T 7 max Vg t 7 min Vg t 7 max t 7 min Vload Vload ^max ^min

1.1 0.95 1.05 0.95 1.1 0.90

Qcmax 0.05

Reactive Compensation and Voltage limits (P. U.) Qcmin Vcmax

0 1.05

Table-2 [ Fuel cost coefficients ]

Bus no. Cost coefficients

1 0.00 2.00 0.00375

2 0.00 1.75 0.01750 5 0.00 1.00 0.06250 8 0.00 3.25 0.00834 11 0.00 3.00 0.02500 13 0.00 3.00 0.02500

A. Case 1: Fuel cost Vs Transmission Losses

In this case, two competing objectives, i.e., fuel cost and losses, were considered. This multiobjective optimization problem was solved by the proposed approach. The Pareto optimal solution obtained using the proposed MOTLBO algorithm is shown in Fig. 1. From the Pareto optimal solution, it is clear that the proposed MOTLBO method is giving well distributed solutions. The compromise solution was found using the fuzzy membership approach. The best solution for minimum cost, minimum loss and the compromise solution

Fuel cost ($/h)

Fig. 1. Pareto Optimal Solutions for Case 1

Table - 3 [Optimal control variables for IEEE 30 -bus power system, Case-1, Case-2]

Best Cost Best Losses Best Comp. Best Cost Best L-Index Best Comp.

Pgi (MW) 173.0630 50.0000 121.8494 168.7574 168.0218 169.0025

Pg2 (MW) 47.5580 80.0000 52.0364 48.4579 37.6116 41.6078

Pg5 (MW) 20.4393 50.0000 32.5781 19.3814 19.5037 19.4754

Pgs (MW) 27.7019 35.0000 34.2330 30.1614 30.0622 30.0920

Pgii(MW) 11.2125 30.0000 24.3696 11.7149 29.9990 18.0536

Pgib(MW) 12.0000 40.0000 23.5560 13.0556 13.3035 13.3372

Vi ( P.U. ) 1.0996 1.1000 1.0968 1.1000 1.1000 1.1000

V2( P.U. ) 1.0467 1.0495 1.0494 1.0909 1.1000 1. 1000

V5 ( P.U. ) 1.0788 1.0850 1.0644 1.0667 1.1000 1. 1000

Vs( P.U. ) 1.0839 1.1000 1.0960 1.0998 1.1000 1. 1000

V11 ( P.U. ) 1.0670 1.0889 1.0862 1.1000 1. 1000 1.1000

V13 ( P.U. ) 1.0911 1.0718 1.0983 1.0997 1.0958 1.0952

Tr(6-9) 1.1000 1.0597 1.0259 0.9001 0.9000 0.9012

Tr(6-10) 0.9645 0.9012 0.9184 0.9013 0.9000 0. 9000

Tr(4-12) 1.0620 0.9900 1.0024 0.9006 0.9000 0. 9000

Tr(28-27) 0.9925 1.0022 0.9728 0.9029 0.9000 0. 9000

Q c10 (P.U.) 2.4135 5.0000 3.4650 4.4589 5.0000 4.9929

Q c12 (P.U.) 5.0000 5.0000 5.0000 3.0032 5.0000 4.9978

Q c15 (P.U. ) 2.9637 3.5430 5.0000 4.7934 5.0000 4.9894

Q c17 (P.U.) 0.0000 5.0000 3.9050 4.6118 5.0000 4.9814

Q c20 (P.U.) 5.0000 4.5173 5.0000 4.7587 5.0000 4.9274

Q c21 (P.U.) 4.9574 5.0000 5.0000 4.8633 5.0000 5. 0000

Q c23 (P.U.) 3.7549 0.7625 0.4530 4.9242 5.0000 4.9618

Q c24 (P.U.) 2.5109 5.0000 3.4200 4.8570 5.0000 4.9984

Q c29 (P.U.) 3.0173 3.1191 0.3139 4.9693 5.0000 4.9581

Fuel cost ($/h) 800.7257 977.5925 830.7813 800.6797 914.0426 803.6317

Tr. loss (MW) 8.5632 2.9501 5.2742 - - -

L - Index (P.U.) - - - 0.1050 0.1019 0.1020

B. Case 2: Fuel cost Vs L-index

In this case, L-index is considered in place of transmission losses. L-index gives a scalar number to each load bus This index uses information on a normal power flow and is in the range zero (no load case) to one (voltage collapse). To maintain the voltage stability and move away from voltage collapse point, maximum value of L-index among load buses ( Lmax) must be minimized. These two competing objective functions were optimized by the proposed MOTLBO method. The Pareto optimal solution for this case is shown in Fig.2.The best solution vectors are also given for minimum cost, minimum L-index and compromise case in Table 3.

Fuel cost ($/h) F

Fig. 2. Pareto Optimal Solutions for Case 2

Table - 4 [Comparison of best compromise solution obtained for multiobjective optimization, Case-1, & Case-2]

Fuel cost ($ / h) losses (MW)

Fuel cost ($ / h) L - Index (P.U.)

MOBA [23] 848.229

MOPSO - Fuzzy [21] 847.01

5.2061 5.666 5.3143 5.2742

MOPSO - Fuzzy [21] MOTLBO

809.79 803.63

0.1146

0.1020

MOHS [22] MOTLBO

832.6709 830.7813

Table 4 provides comparison between the results obtained with TLBO and other methodology. The results reveal that MOTLBO model provides better optimal solution compared to other methodology for multi-objective OPF solution.

7. Conclusions

The paper has employed a multiobjective teaching-learning-based optimization (MOTLBO) algorithm to solve the MOOPF problem with many constraints in IEEE 30-bus system. A clustering technique is implemented to provide the operator with manageable Pareto- optimal set. The results show that the proposed approach is efficient and high quality for solving MOOPF problem, which multiple Pareto- optimal solutions can be found in a single run. In addition, the non-dominated solutions are well - distributed and have satisfactory diversity characteristics.A fuzzy set theory approach has been used to identify the best compromise solution. In the future, efforts will be made to incorporate with more objective functions to the problem structure,which will be attempted by the proposed methodology.

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