A solution to the economic dispatch using EP based SA algorithm on large scale power system

Electrical Power and Energy Systems 32 (2010) 583–591

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Electrical Power and Energy Systems journal homepage: www.elsevier.com/locate/ijepes

A solution to the economic dispatch using EP based SA algorithm on large scale power system C. Christober Asir Rajan Department of EEE, Pondicherry Engineering College, Pondicherry 605 014, India

a r t i c l e

i n f o

Article history: Received 23 July 2008 Received in revised form 28 October 2009 Accepted 6 November 2009

Keywords: Economic dispatch Evolutionary Programming Simulated Annealing

a b s t r a c t This paper develops a new approach for solving the Economic Load Dispatch (ELD) using an integrated algorithm based on Evolutionary Programming (EP) and Simulated Annealing (SA) on large scale power system. Classical methods employed for solving Economic Load Dispatch are calculus-based. For generator units having quadratic fuel cost functions, the classical techniques ignore or flatten out the portions of the incremental fuel cost curves and so may be have difficulties in the determination of the global optimum solution for non-differentiable fuel cost functions. To overcome these problems, the intelligent techniques, namely, Evolutionary Programming and Simulated Annealing are employed. The above said optimization techniques are capable of determining the global or near global optimum dispatch solutions. The validity and effectiveness of the proposed integrated algorithm has been tested with 66-bus Indian utility system, IEEE 5-bus, 30-bus, 118-bus system. And the test results are compared with the results obtained from other methods. Numerical results show that the proposed integrated algorithm can provide accurate solutions within reasonable time for any type of fuel cost functions. Ó 2009 Elsevier Ltd. All rights reserved.

1. Introduction Two decades before optimization problem played a vital role in planning large networks. Economic Load Dispatch is the sub-problem of Optimal Power Flow. Until 1990s, Economic Load Dispatch was solved using the fuel cost function as quadratic. In 1993, Walters and Sheble included the effect of valve point loading and formulated the non-smooth quadratic function. Many researchers approximated the non-smooth functions and solved the Economic Load Dispatch problem. In 1997, Fogel et al. introduced the Evolutionary Programming method to solve non-convex functions. This method does not assure any differentiability and solve any type of fuel cost functions within reasonable time. The introduction of dispatching concept has been made to improve its formulation and adopt efficient solution techniques. Conventionally, the optimal operation and planning of power system networks have been an economic criterion. Economic Load Dispatch has been utilized for this purpose and widely accepted by most of the utilities and implemented in their Computer Aided Dispatch centres. Economical Operation is very important for a power system to return a profit on the capital invested. Operational economics involving power generation and delivery can be divided into two parts, one dealing with minimum cost of power production called Economic Load Dispatch and the other dealing with minimum-loss

E-mail address: [email protected] 0142-0615/$ - see front matter Ó 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijepes.2009.11.014

delivery of generated power to the loads. The economic dispatch problem and the minimum-loss problem can be solved by means of the Optimal Power Flow. Optimal Power Flow (OPF) forms a part of modern Energy Management Systems (EMS) and is used to determine the optimal output of the generators in view of minimizing total fuel cost, transmission losses, etc. Simulated Annealing (SA) is the powerful general purpose stochastic search technique to solve hard constrained optimization problems. Though it takes long time, it has many strong features such as, it is easy to implement, requires little expert knowledge and is not memory intensive. Further, it can start with any initial solution and improves on it to find optimal solution with a high probability. The SA optimization technique, begin with a randomly generated solution and then making successive random modifications, until a stopping criterion is satisfied. Research endeavours, therefore, have been focused on; efficient, near-optimal algorithms, which can be applied to large scale, power systems and have reasonable storage and computation time requirements. A survey of existing literature [1–28] on the problem reveals that various numerical optimization techniques have been employed to approach the complicated economic dispatch problem. More specifically, these are the Dynamic Programming method (DP), the Linear Programming method (LP), the Fletcher’s Quadratic Programming method (FQP), the Sequential Programming method (SP), the Newton–Raphson method (NR), the Hop Field method (H), the Fuzzy method (F), the Simulated Annealing method (SA), the Tabu Search (TS), the Genetic Algorithm (GA),

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the Hybrid Genetic Algorithm (HGA), the non-convex Economic Power Dispatch (NCED), The Improved Particle Swarm Optimization (IPSO), the Evolutionary Programming (EP), and so on. The major limitations of the numerical techniques are the problem dimensions, large computational time and complexity in programming. Economic Load Dispatch is one of the main functions of the modern Energy Management System which determines the optimal real power settings of generating units with an objective to minimize the total fuel cost of dispatch solutions in cases where the classical methods ceases to be applicable. Traditional dispatch algorithms employ LaGrangian multipliers and require monotonically increasing incremental cost curves. Classical search techniques employed to solve economic dispatch problem produce inaccurate results because the operating units have non-linear incremental cost curves and the conventional procedure either ignores or flattens out the portions of non-linear regions of the curves. But such approximations are not desirable as they may lead to sub-optimal operation and hence huge revenue loss results over time. Hence, there is a demand for techniques that do not impose restrictions on the shape of the fuel cost curves. The DP method [1,2] solution is one of the approaches to solving the inherently non-linear and continuous ED problem however, the problems of ‘curse of dimensionality’ or local optimality with the method are caused in the solution procedure. The LP method [3] for solving the ELD problem, the fuel cost function is chosen to be of quadratic form. However, the fuel cost function becomes more non-linear when valve point leading effects are included. The FQP method [4] to optimal dispatch problem, considering the practical constraints such as transformer taps, voltage, and line flow constraints has been solved. The SP method [5] to solve the combined economic and emission dispatch problem by assigning weighing factors for generation and emission cost functions. The line flows are computed by developing distribution factors using sensitivity parameters from the elements of load flow. The NR method [6] employs the line flow constraints to solve the multiobjective power dispatch problem. The H method [7,8] which selects the weighting factors of the energy function by trails and it also exhibits the ability of attaining power match to any extent. A fuzzy multi-objective optimization technique for the ELD problem was proposed [9]. However the solutions produced are suboptimal and the algorithm does not provide a systematic framework for directing the search toward Pareto-optimal front. The SA [10] based algorithm when implemented alone, is difficult to tune the related control parameters of the annealing schedule and may be too slow when applied to a practical power system. The TS method [11] employs a flexible memory system to avoid the entrapment in a local minimum and developed the ideal distance to the fitness to accelerate optimization. The GA method [12,13] carefully schedules its processes, computational loads and synchronization overhead for the best performance. And it is shown to be viable to online control of constrained ELD due to substantial generator fuel cost savings and fast computing time. The HGA method [14] is proposed to approximate EDP by using a smooth and differentiable function based on the maximum entropy principle in SQP operator. In this way, the performance of the hybrid method is improved. At the same time, to improve rationality of the distribution of initial population, the hybrid technique integrating the uniform design with the genetic algorithm is proposed. The NCED method [15] employs time varying acceleration coefficients (TVAC) in this paper for solving the practical economic dispatch problem with valve point loading effects and prohibited operating zones. The TVAC strategy strikes a proper balance between the cognitive and social component during the initial and latter part of the search and hence is found to avoid premature convergence of the swarm. In the IPSO method [16], a new velocity

strategy equation is formulated suitable for a large system and constriction factor approach (CFA) is incorporated into the velocity equation of PSO which may help assured convergence. This approach considers the base case security constraints such as line flow constraints and bus voltage limits. The PSO algorithm [17] using Gaussian and chaotic signals are powerful strategies to diversify the particle’s swarm in PSO and improve the PSO’s performance in preventing premature convergence to local minima. To enrich the searching behavior and to avoid being trapped into local optimum, a chaotic sequence based on logistic map is incorporated as a randomizer can travel ergodically over the whole search space. In CRAZYPSO method [18], a craziness operator is introduced to ensure that the particle would have a pre-defined craziness probability to maintain the diversity of the particles. The economic CHP dispatch method [19] is formulated as a multi-objective optimization problem and it is treated as a stochastic problem. Both power demand and heat demand are considered random variables. An Improved Particle Swarm Optimization algorithm is used to achieve the reasonable trade-off between multiple objectives including the total generation cost, power generation deviation, and heat generation deviation. Non-dominated sorting genetic algorithm-II (NSGAII) [20] maintains a good spread of solutions and converges near the true Pareto-optimal set. This proposed approach does not require any user-defined parameter for maintaining diversity among population members. Due to the difficulties of binary representation in dealing with continuous search space with large dimensions, the proposed approach has been implemented using real-coded Genetic Algorithm. The EP method [21,23] employs a diversity preserving mechanism to overcome the premature convergence and search bias problems. And it was shown that it is efficient for solving multi-objective optimization where multiple Pareto-optimal solutions can be found in one simulation run. The improved hybrid EP method [24] adjusting the mutation direction of control variables in order to increase the possibility of keeping state variables within bounds. And it was shown that it require much less computation time. The hybrid EP and SQP method [25] provides a good solution quality even the problem has many local optimum solutions at the beginning. Then the local searching property of SQP is used to obtain a final solution. However these methods are typically very slow. In order to achieve optimal trade-off between accuracy and performance, hybrid formulations combining classical optimization methods and GA EP have been recently reported in the literature. From the literature review, it has been observed that there exists a need for evolving simple and effective methods, for obtaining an optimal solution for the ELD. Hence, in this paper, an attempt has been made to couple EP with SA for meeting these requirements of the ELD, which eliminates the above-mentioned drawbacks. In case of SA, the temperature and demand are taken as control parameter. Hence the quality of solution is improved. The algorithm is based on the annealing neural network. Classical optimization methods are a direct means for solving this problem. EP seems to be promising and is still evolving. EP [26–29] has the great advantage of good convergent property and, hence, the computation time is considerably reduced. EP does not suffer from the drawback of handling non-continuous or non-differentiable objective functions as in some plants like Combined Cycle Co-generation plants. Encoding and decoding schemes essential in the GA approach are not needed, considerable computation time is thus saved. The EP combines good solution quality for SA with rapid convergence for EP. The EP Based SA (EPSA) is used to solve the ELD. By doing so, it can help to find the optimum solution rapidly and efficiently. EP is capable of determining the global or near global solution. It is based on the basic genetic operation of human chromosomes. It operates with the stochastic mechanics, which combine offspring creation based on the performance of current trail

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solutions and competition and selection based on the successive generations, form a considerably robust scheme for large – scale real – valued combinational optimization. In this proposed work, the parents are obtained from a pre-defined set of solution’s i.e. each and every solution is adjusted to meet the requirements. And the selection process is done using Evolutionary Strategy [26–29]. The validity and effectiveness of the proposed integrated algorithm has been tested with 66-bus Indian utility system, IEEE 5-bus, 30-bus, 118-bus system. And the test results are compared with the results obtained from other methods.

585

where ai, bi, ci are the cost coefficients of the ith generator, n is the number of units, Pi is the real power output of the ith generator.

rithm, but instead places emphasize on the behavioral linkage between parent and their offspring, rather than seeking to emulate specific genetic operators as observed in nature. Evolutionary Programming is similar to Evolutionary strategies, although the two approaches were developed independently. David Fogel (Fogel, 1988) extended the initial work of his father Larry Fogel (Fogel, 1962) for applications involving real-parameter optimization problems. Combinatorial and real-valued function optimization in which the optimization surface or fitness landscape is ‘rugged’, processing many locally optimum solution, are well suited for Evolutionary Programming. Real-parameter EP is similar in principle to evolution strategy (ES), in that normally distributed mutations are performed in both algorithms. Both algorithms encode mutation strength (or variance of the normal distribution) for each decision variable and a self-adapting rule is used to update the mutation strengths. Several variants of EP have been suggested (Fogel, 1992). The main stages of this technique are initialization, creation of offspring vectors by mutation and competition and selection of best solution. EP is capable of determining the global or near global solution. The ability of the EP method to find the global optimum solution is independent of the size of the discrete load step assigned to each parent vector in the solution process. Also, investigations have shown that Evolutionary Programming was better among other Evolutionary Computation methods such as Genetic Algorithm and micro genetic algorithm [26–28].

2.1. Constraints

3.1. EP algorithm to solve ELD problem

2.1.1. Inequality constraints The maximum active power generation of a source is limited by thermal consideration and also minimum power generation is limited by the flame instability of a boiler. If the power output of a generator for optimum scheduling of the system is less than a pre-specified value Pmin, the unit is not synchronized with the bus bar because it is not possible to generate that low value of power from that unit. Hence the generator power cannot be outside the range stated by the inequality

Evolutionary Programming is a probabilistic search technique, which generates initial parent vectors distributed uniformly in intervals within the limits and obtains global optimum solution over number of iterations. The main stages of this technique are initialization, creation of offspring vectors by mutation and competition and selection of best vectors. The step-by-step algorithm is given below:-

2. Problem formulation The objective of Economic Load Dispatch [29] is to allocate the most optimum real power generation level for all the available generating units in the power station that satisfies the load demand at the same time meeting all the operating constraints. The main objective function of the thermal ELD problem is the fuel cost function of the thermal units expressed as

Ci ¼

n X

ai P2i þ bi Pi þ ci

ð1Þ

i¼1

Pi;min 6 Pi 6 Pi;max

for i ¼ 1; 2; . . . ; n

ð2Þ

where Pi,min – lower real power generation limit of unit ‘i’ (MW), Pi,max – upper real power generation limit of unit ‘i’ (MW). 2.1.2. Generating constraints In order to satisfy the load demand, the sum of all the generating units on line must equal the system load plus the transmission losses. The system power balance constraint is, n X

Pi  PL  PD ¼ 0

ð3Þ

i¼1

where PD – load demand, Pi – real power output produced by unit ‘i’ (MW), PL – total loss in the transmission network give by B-matrix coefficient.

PL ¼ Pt BP þ Pt Bo þ B1

ð4Þ

where Pt – the vector generator loading, B – loss coefficient matrix, Bo – loss coefficient vector, B1 – loss constant. 3. Evolutionary Programming Evolutionary Programming [26], originally conceived by Lawrence J. Fogel in 1960, is a stochastic optimization strategy. Evolutionary Programming is a mutation-based evolutionary algorithm applied to discrete search spaces similar to Genetic Algo-

(1) Prepare the data base for the line data, bus data and generator data. Line data includes the information of the lines such as resistance and reactance. Bus data includes the information of generators, loads at each and every bus. The generator data includes the cost coefficient of the generators including real and reactive generation limits. (2) Formation of admittance matrix using line resistance, reactance, shunt elements. (3) Initialization: Assume a maximum population and population size. (4) Generate parent vector Pi ¼ ½½P1 ; ½P2 ; . . . ; ½P n1  of the generators depending on population size and satisfying its minimum and maximum generation limits. (5) The slack bus generator vector Ps will be calculated using load flow by Newton–Raphson methods for the above generations. (6) Check the minimum and maximum limits of Ps, if not, got to step 4. (7) Check the total numbers of generations if it is equal to maximum generations then go to step 18. (8) Calculate the cost of generation Fi (Pi). (9) Find the cost for all the parent vectors, and then store it as Fi (Pi). It is the value of the objective function (cost) associated with the parent vectors in the population. Store the optimal cost (fmin) for the corresponding population. (10) Mutation: (a) Calculate standard deviation

ri ¼ b  f pi =fmin ðPi;max  Pi;min Þ

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START

Preparation of Database using line, bus and generator data

Generate parent vector population within the generator limits Compute Slack Bus generator vector(P s ) ,losses , line flows using N-R method for the above generations

Ps

Check for the limits of and line flows

No

Yes

Compute the cost of all parent vectors(F i) and find out the minimum cost f min among the parent population

Creation of

Offspring vectors

Check for the limits of generators and line flows Yes Compute the cost of generations with offspring vectors(Fi')

Competition among parent and off-spring for survival

Selection of best population for new parents

Check for maximum generations

STOP

Fig. 1. Flowchart for EP based ELD.

where fpi is the fuel cost of the ith parent vector. b is the scaling factor and this value depends on the maximum and minimum limits of generation.

(b)

Add a Gaussian random variable Nð0; r2i Þ to all the components of parent vectors to get offspring vector p0i .

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587

Fig. 2. Flowchart for Simulated Annealing algorithm.

(11) An offspring vector P0i ¼ ½½P 01 ; ½P 02 ; . . . ; ½P0n1  is created from each parent vector by P 01 ; P 02 ; . . . ; P0n1 addition to each component of parent a Gaussian random variable with mean and a standard deviation Nð0; r2i Þ proportional to the scaled cost values of the parent trial solutionP0i ¼ Pi þ Nð0; r2i Þ for i ¼ 1; 2; . . . ; n  1. (12) Check offspring vector P01 ; P02 ; . . . ; P 0n1 of the generators within its minimum and maximum generation limits otherwise go to step 10 (13) The slack bus generator vector P0s will be calculated using load flow by Newton–Raphson method for the above offspring generations.

(14) Check P0s is within minimum and maximum limits otherwise go to step 10. (15) Calculate cost of generation with offspring generation vectors and store it as Fi (Pi). (16) Competition and selection: The parent trial vectors Pi and the corresponding offspring P0i contend for survival with each other within the competing pool by comparing the cost of parent vectors Fi (Pi) with the corresponding cost of offspring vectors Fi ðP i Þ0 in this population (17) The best vectors having minimum cost, whether parent vector Pi (or) offspring vector P 0i are selected for new parents for the next generation, go to step7.

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B

A

START Generation of Parent Vectors within the generator limits

Accept or reject the neighborhood solution based on Probability acceptance function

Evolutionary Programming Starts

No Check if m has reached max

Evaluation of Objective Fuel Cost function fpi and find minimum cost fmin

Yes Decrease T

Creation of Off spring vectors from parent population by Gaussian Mutation

No Is T minimum?

Competition and Selection of survivor vectors

Use trial vectors as parents for EP

Simulated Annealing starts

Compute Fuel Cost fu for survivor vectors

Yes Check for max generations

Take temperature T as Control Parameter ’m’ as loop count Make a move to Neighborhood by Gaussian mutation

Evaluate Objective Fuel Cost function fu for neighborhood solutions

A

Yes

No Go to EP part

Print the optimal schedule and Fuel Cost

B Fig. 3. Flowchart for proposed EPSA method.

Table 1 Comparison of results for IEEE 5-bus system. Technique

P1 (MW)

P2 (MW)

P3 (MW)

Fuel cost (Rs/h)

CPU time (s)

Hybrid EPSA Evolutionary Programming Simulated Annealing k – iteration method

42.019 35.070 33.253 23.558

50.529 64.843 60.125 69.561

57.452 50.087 56.622 56.881

1470.35 1496.58 1500.23 1503.24

5.4 5.9 9.1 11.0

(18) Find the optimal solution among all population groups. (19) Initialization and mutation are repeated until there is no appreciable improvement in the fitness value. The diagrammatic representation of solving Economic Load Dispatch using Evolutionary Programming is shown in the flowchart Fig. 1.

4. Simulated Annealing The Simulated Annealing technique is an algorithm [10], which exploits the resemblance between the annealing of a metal and a minimization process. A metal is a system of many atoms. The total internal energy of the metal depends on its state in the form of relative position, orientation and motion of the atoms in the metal.

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1.55

x 10

4

EPSA Convergence for 50 parents

1.54

Operating Cost Rs/MWhr

1.53 1.52 1.51 1.5

Min Cost = 15008.22 Rs/MWH r 1.49 1.48 0

5

10 15 No of Generations

20

25 Fig. 5. EPSA convergence characteristics with constant, step, non-linear scaling factor for IEEE 26-bus system.

Fig. 4. EPSA convergence for IEEE 26-bus with 50 parents.

Although the prediction of a specific state is almost impossible due to the extremely rapid microscopic movement of the atoms, the statistical properties of many replica systems of atoms in their (thermodynamic) equilibrium can be characterized. It has been observed that when a metal is annealed, or cooled slowly, the energy of the metal tends to assume the global minimal value. Motivated by this observation, Simulated Annealing generates feasible solutions of a minimization problem which correspond to the states of a metal, with the cost of a feasible solution corresponding to the energy of the metal in a state. By moving among the feasible solutions the way the states of a metal under annealing would evolve, the global optimum of the problem can be approached with high probabilities. Simulated Annealing has been successfully applied to many difficult combinatorial optimization problems. The method assumes no special problem structure and is highly flexible with respect to various constraints.

Table 3 Comparison of results for Indian 66-bus utility system and IEEE 118-bus system. System

Technique

Fuel cost (Rs/h)

CPU time (s)

66 – bus Indian utility system

Classical EP [16] with security constraints Fast EP [16] with security constraints Hybrid EPSA EP SA k – iteration method

806970

35.4

781720 467121 467688 467912 470014

33.2 32.1 32.8 45.2 53.2

IEEE 118 – bus system

Hybrid EPSA EP SA k – iteration method

831375 831755 831900 840029

31.1 31.7 46.8 57.9

(b)

4.1. SA algorithm to solve ELD problem The algorithm is as follows: (1) Initialize the power loadings or parent vectors P ¼ ½P 1 ; P2 ; . . . ; P N ; i ¼ 1; 2; . . . ; N p , such that each element in the vector is determined by Pj  random (P j;min ; Pj;max ; j ¼ 1; 2; . . . ; N), with one generator as dependent generator. (2) Evaluate the fuel cost F objective function. Using the trial vector Pi and find the minimum of F. (3) Initialize the temperature T and local loop counter m. (4) Generate neighbourhood solutions P0 for the initial power loadings by adding the Gaussian random variables to the current solution, keeping one solution dependant. These solutions P’s are created as follows: (a) Calculate the standard deviation

Add a Gaussian random variable, Nð0; r2i Þ to all the component of Pi, to get P0i .

(5) Evaluate objective function using P0 . Find the difference DF in F and F0 . (6) Evaluate the probability of acceptance q as q = 1/(1 + exp (DF/T)). (7) Accept P0 as the new solution if q > random (0, 1), otherwise new solution is P. (8) Increment the counter m and repeat from step 3 until m reaches its maximum value. (9) If m reaches its maximum value decrease T by T = dT, where d < 1, is the scaling factor, and repeat from step 3 until T reaches its minimum limit. (10) The final P is the required optimum value of power generation satisfying all constraints.

ri ¼ b  f pi =fmin ðPi;max  Pi;min Þ

Table 2 Comparison of results for IEEE 26-bus system. Technique

P1 (MW)

P2 (MW)

P3 (MW)

P4 (MW)

P5 (MW)

P26 (MW)

Fuel cost (Rs/h)

CPU time (s)

Hybrid EPSA EP SA k – iteration method

470.67 472.21 483.02 474.11

183.09 145.45 180.30 173.07

282.55 296.50 285.56 290.09

148.57 141.40 143.45 150.00

181.80 185.44 190.36 196.72

119.44 106.33 103.74 103.05

15463.1 15474.3 15493.1 16760.0

11.1 11.8 15.7 19.3

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C. Christober Asir Rajan / Electrical Power and Energy Systems 32 (2010) 583–591

The complete flowchart for Simulated Annealing algorithm is shown in Fig. 2.

(3)

5. Epsa algorithm to solve eld problem

(4)

In this proposed work the EP is assisted by the SA technique to rapidly converge towards the optimum point. The search is basically done with EP but the Simulated Annealing is used to escape the search path from the local optimum region.

(5) (6) (7)

5.1. EPSA algorithm (8) The proposed integrated algorithm combines EP and SA techniques [23–26] to solve the ELD problem. The EP technique, hold the main responsibility of finding the optimal point and SA assists EP to converge towards the optimum point. The search is basically done with EP, but the SA is used to escape the search path from local optimum point. The algorithm for the proposed method is as follows: i. Initialize the parent vectors P i ¼ ½P 1 ; P2 ; . . . ; P n1 , where i ¼ 1; 2; . . . ; N p , such that each element in the vector is determined by Pj  random (Pj;min ; P j;max ) where j ¼ 1; 2; . . . ; N  1 with one generator as a dependant generator. ii. Evaluate the fuel cost function fpi of the objective function using the trial vector pi and find the minimum of fpi. iii. Create the offspring trial solutions using p0i using the following steps, (a) Calculate standard deviation

ri ¼ b  f pi =fmin ðPi;max  Pi;min Þ

(b)

where fpi is the fuel cost of the ith parent vector. b is the scaling factor and this value depends on the maximum and minimum generation limits. Add a Gaussian random variable Nð0; r2i Þ to all the components of parent vectors to get offspring vector p0i .

iv. Select the first Np individuals from the total of 2Np individuals of both parents and off springs based on a fitness score assigned to each of the 2Np individuals. (a) A reference competitor is selected at random from among the 2Np trial solutions r ¼ ð2N p randomð01Þ þ 1Þ (b) Assign a fitness score to each trial vector according to

W pi ¼

2N p X

wt ; t ¼ 1; 2; . . . ; Np

i¼1

wt ¼ 1; if q  ðf

fpi

pi þfpr Þ

¼ 0;

otherwise

where q is a uniform random number ranging over [0,1]. v. Sort the Wpi in descending order and the first Np individuals will survive and are transcribed along with their objective function fpi into the survivor set as feasible states for SA algorithm. vi. The SA algorithm is as follows: (1) From the individuals selected from EP, P 0u calculate the objective fuel cost function Fu. (2) Initialize the control parameter, temperature T and local loop counter m.

Generate the neighbourhood solutions P0u for the power loading Pu by adding the Gaussian random variable Nð0; r2i Þ keeping one solution as dependant. Evaluate the fuel cost function F 0u , for the neighbourhood solution, P 0u and find the difference DF ¼ F u  F 0u . Evaluate the probability of acceptance, q as q = 1/ (1 + exp (DF/T)). Accept P 0u as a new solution if q > random (0, 1), otherwise new solution is P0u Increment the counter m and repeat from step 3 until m reaches its maximum value. If m reaches its maximum value, decrease T by T = dT where d < 1 is the scaling factor. Repeat steps from 3 until T reaches its minimum limit.

vii. The final Pu is the trial solution for the EP algorithm and is Parent vector, pi of step i. viii. The search process is stopped as the count of generations reaches Nm, a maximum integer. The diagrammatic description of the proposed hybrid EPSA algorithm is shown in Fig. 3. 6. Numerical example and results To investigate the effectiveness of the EPSA hybrid algorithm, it is applied on two sample systems such as 66-bus Indian utility system, IEEE 5-bus, 26-bus and 118-bus system. The actual transmission losses for both the systems are determined by Newton– Raphson load flow. The proposed EPSA hybrid algorithm was developed in P4 system in MATLAB environment. The IEEE 5-bus system has 3 generators with 7 lines. The total load demand on this system is 150 MW. After considering the line flow using Newton–Raphson method, the losses were found to be 2.15 MW. The line data, bus data, generator data are the necessary data’s for solving this problem. The optimal or minimum solutions for the system are given in Table 1. The IEEE 26-bus system has 6 generators with 46 lines. The total load demand on this system is 1200 MW. After considering the line flow using Newton–Raphson method, the losses were found to be 8 MW. The convergence curves for 26-bus system with 50 parents were shown in Fig. 4. The optimal or minimum solutions for the system are given in Table 2. Fig. 5 shows the EPSA convergence characteristics with constant, step, non-linear scaling factor for IEEE 26-bus system. The IEEE 118-bus system has 54 generators, 186 branches, 9 transformers, 2 reactors and 12 capacitors. The total demand on the system is 3668 MW. After considering the line flow using Newton–Raphson method, the losses were found to be 132.5 MW. The 66-bus Indian utility system has 12 generators with 93 lines. The total demand on this system is 1700 MW. After considering the line flow using Newton–Raphson method, the losses were found to be 71 MW. The optimal or minimum solutions for the 66-bus Indian utility and IEEE 118-bus system are given in Table 3. By comparing the ELD results obtained by various optimization methods for three test systems, the results of proposed EPSA method are in close with classical methods. From the comparison, the EPSA method had lesser total fuel cost in all the power systems considered. As indicated in this paper, the EP algorithm has also proved to be an efficient tool for solving the important economic dispatch problem for units with ‘‘non-smooth” fuel cost functions as referred in [26–28]. Such functions may be included in the proposed EP search for practical problem solving. There is no obvious limitation on the size of the problem that must be addressed, for its data structure is such that the search space is reduced to a minimum; no ‘‘relaxation of constraints” is required; instead, popula-

C. Christober Asir Rajan / Electrical Power and Energy Systems 32 (2010) 583–591

tions of feasible solutions are produced at each generation and through out the evolution process. The main advantages of the proposed algorithm are speed. The proposed EPSA approach was compared to the related methods in the references indented to serve this purpose, such as the DP with a zoom feature, the SA, and the GA approaches. Further SA can start with any initial solution and improves on it to find optimal solution with a high probability. By means of stochastically searching multiple points at one time and considering trail solutions of successive generations, the EPSA approach avoids entrapping in local optimum solutions. Also, disadvantages of huge memory size required by the SA method are eliminated. Moreover, intellectual schemes of encoding and decoding entailed by the GA approach are not needed in the proposed EPSA approach. The problem of power unbalance previously existing in the solution of GA is circumvented as well in this paper. In comparison with the results produced by the referenced techniques, the EPSA method obviously displays a satisfactory performance with respect to the quality of its evolved solutions and to its computational requirements.

7. Conclusion This paper is concerned with obtaining a better efficient, fast and robust solution for Economic Load Dispatch problem through three different optimization techniques, namely, Evolutionary Programming, Simulated Annealing and a hybrid combination of Evolutionary Programming and Simulated Annealing algorithm viz., Evolutionary Programming Based Simulated Annealing Technique. The results obtained through these methods are satisfactory. In the Evolutionary Programming approach to the ELD problem, the essential processes simulated are mutation, competition and selection. The mutation rate is computed as a function of the ratio of the total cost by the cost of best schedule in the current population. Competition and selection are applied to select from among the parents and the offspring, the best solutions, to form the basis of the subsequent generation. The Simulated Annealing algorithm is incorporated for improving the performance of Evolutionary Programming. Its prevents Evolutionary Programming from getting into local optimum region, thereby Simulated Annealing helps in reducing considerable amount of CPU time. In Evolutionary Programming based Simulated Annealing, SA is employed in the mutation part of Evolutionary Programming and the solution obtained through Evolutionary Programming is fed as initial solution to Simulated Annealing. SA is also used to verify the constraints, which is time consuming when done by EP alone. On comparing the results attained by the proposed techniques (Simulated Annealing and Evolutionary Programming based Simulated Annealing), the Evolutionary Programming based Simulated Annealing technique obviously displays a satisfactory performance. In EP, there is no obvious limitation on the size of the problem that must be addressed, for its data structure is such that the search space is reduced to a minimum. The population of feasible solutions is produced ate ach generation and throughout the evolution process. No relaxation of constraints is required. Simulated Annealing is known to converge to the global minimum with a probability of unity and thus improves any given solution. Thus, the solution obtained through Evolutionary Programming Based Simulated Annealing is having better quality and in terms of economy and computation time.

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