Hybrid CLONAL Selection Algorithm with PSO for ...

Hybrid CLONAL Selection Algorithm with PSO for ValvePoint Economic Load Dispatch Sogol Babaei Nejad ,Seyyed Hamid Elyas, Ali Khamseh

Iman Naziri Moghaddam, Mehdi Karrari

Department of Electrical Engineering Semnan University Semnan, Iran {sogol.babai, hamid.elyas, khamseh63}@gmail.com

Department of Electrical Engineering Amir Kabir University of Technology Tehran, Iran {iman_naziri, karrari}@aut.ac.ir

Abstract— Economic load Dispatch (ED) is one of the most important problems in power system operation. It is a nonlinear non-convex problem which stochastic search algorithms seem to be appropriate solutions. This study tries to propose a new method which is derived from the combination of two different algorithms, CLONAL as the basic algorithm and PSO. The proposed method has been tested on two different systems containing thirteen and forty generators and obtained results have been compared with the results of other stochastic search algorithms. The fascinating results obtained from the comparison ensure the efficiency of the new proposed method. Index terms- Economic Load Dispatch, CLONAL Selection Algorithm, Particle Swarm Optimization

I.

INTRODUCTION

As it is clear, Economic Load Dispatch (ED) is one of the most important problems in power system operation. This nonlinear problem is so much effective in the operation of power system which is made of two different parts, generation and control. ED is used to determine the optimal power outputs of all generating units to minimize the total cost. In restructured power systems, assigning of optimum generation with any of the units is very important. Hence; many studies have been made to solve the ED problem which has lead to different methods and algorithms. Heuristic methods which have been proposed during the last few decades such as PSO, GA, IA, EP, QPSO, SA, etc., each contains different advantages and disadvantages. For example, the PSO is no doubt the fast and very efficient global search algorithm. In addition, it is insensitive to the scaling of design variables. It has a simple implementation and can be easily parallelized for concurrent processing but it has a slow convergence in the refined search stage and can be easily trapped in local optimums. One of the improved versions of the PSO, called QPSO, has a better performance in comparison to the pure PSO. It can also obtain better optimums using fewer numbers of particles. It definitely has stronger search ability and quicker convergence speed but it still has the defect of being trapped in local optimums [2, 4]. GA, the other stochastic search algorithm, is one of the

evolutionary algorithms which can quickly scan a vast solution set and bad proposals do not affect the end solution negatively as they are simply discarded[1,5,7]. The disadvantages of the GA include that it is very slow and cannot always find the exact solution. However, it always finds the best solution. The SA, the other algorithm, is a robust and easy to implement technique. It can be used to implement various combinational optimization problems and provides good solutions. However, it involves various parameters which need to be set appropriately to get reasonably good solutions and make the algorithm very slow. Differential Evaluation (DE) algorithm is a population-based algorithm like the GA which has a lot of advantages such as finding the true global optimum regardless of the initial parameter values and it has fast convergence in addition to using few control parameters [8]. Considering this quick review, this paper tries to present a method to solve the non-convex economic dispatch problem. This method is a combination of an Immune based algorithm CLONAL and the PSO which tries to gather all the positive features of both algorithms in order to make a fast reliable hybrid stochastic search technique with high performance in solving the optimization problems. Problem definition is discussed in section 2. Sections 3 and 4 describe the PSO and CLONAL algorithms, and the combination method of these two, respectively. In section 5, the results obtained from the new presented method are discussed. II.

PROBLEM DEFINITIONS

A. Objective Function Economic load dispatch (ED) is a nonlinear optimization problem which tries to minimize the total operating cost of a thermal unit. The problem can be formulized as follows: ∑ Where Fi: is the cost function of each thermal unit;

(1)

proliferated in large scales. Furthermore, on the antibodies that have managed to identify the antigens, mutations are done. This makes antibodies that can recognize the mutant antigens.

N: stands for the number of units; Pi: is the amount of the unit generation; FT: is the total cost. Fuel cost of the thermal units is usually represented as a polynomial function versus power generation (P) of the unit. In this study, it is defined as follows for each generating unit: Fi( Pi ) = ai + bi Pi + ci Pi2

(2)

Where a, b, and c are the related coefficients of each unit. Multi-valve thermal units usually demonstrate a variation in the fuel cost function; therefore, these effects are entered to the cost function by the sinusoidal term: (3) B. Constraints There are two important constraints which should be mentioned during the optimization.

In mathematical models of the Immune system, each antibody as a candidate for solution is shown with a vector. Each vector component indicates one of the features of the antibody. In optimization problems, the objective function is a criterion to determine the excitability. Therefore, in CLONAL algorithm the objective function is used to determine the suitable particles. According to the above description, the performance steps of the CLONAL algorithm are as follows [16, 17]: • • • • •

B.1 Capacity Constraint

•

The first constraint is the unit generation limit. The generation of each unit should be limited between Pi max and Pi min.

•

Pi min

Pi

Pi max

(4)

B.2 Load Balance Constraint The second constraint is that the total power generation should meet the demand as well as the transmission loss, which is ignored in this study. ∑

(5)

Where PD: is the total load demand; PL: is the transmission loss. Transmission loss (PL) is a function of the units power outputs that can be represented using B matrix coefficients [1]. ∑

∑

∑

(6)

Where Bij: is the ijth element of the loss coefficient square matrix; B0i: is the ith element of the loss coefficient vector; B00: is the loss coefficient constant. Note that the transmission loss (PL) is ignored in this study. III.

P-CLONAL ALGORITHM

A. CLONAL Selection Algorithm The CLONAL selection mechanism is a natural developmental process of the Immune systems. The body defend system depends on the performance of antibodies to recognize and eliminate foreign cells called antigens. The most important immune system cells are lymphocytes. Lymphocyte is divided into two main categories, B cells and T cells. These cells are used to identify the antigens. In the Immune system, after a successful identification of antigens, antibodies that have the ability to identify the antigens are

Generation of initial responses (N). Determination of excitable cells. Reselection of cells from initial set (M). Proliferation of selected cells, proportional to their excitability. Mutation of proliferate cells, proportional to their excitability. Selecting M number of the cells with most excitability (N-M). Randomly selection of P number of the cells and replacement instead of omitted cells.

B. Particle Swarm Optimization Particle Swarm Optimization (PSO) algorithm is a population-based optimization technique inspired by the motion of bird flock, fish schooling, or swarm of bees. Such groups are social organizations whose overall behavior relies on some sort of communication amongst members and cooperation [14]. This algorithm was first introduced by Kennedy and Eberhart in 1995 [13] and was modified by Shi and Eberhart in 1998 [15]. In this algorithm, each individual is called a particle; therefore a flack is a set of particles. The popular term, flying the particles, means the exploration of the search space. Every particle knows its current position and the best position visited since the first fly. The PSO explores by continually sensing the search space at the local level. The connection to search and optimization problems is made by assigning direction vectors and velocities to each point in a multi-dimensional search space. Each point then moves through the search space following its velocity vector, which is influenced by the directions and velocities of other points in its neighborhoods. An individual’s neighborhood may be defined in several ways, configuring somehow the social network of the individual. There exist several neighborhood topologies (full, ring, star, etc.) depending on whether an individual interacts with all, some, or only one particle of the population [14]. How much influence a particular point has on other points is determined by its fitness, a measure assigned to a potential solution, which captures how good it is compared to all other solution points [15]. Here are the steps of the PSO algorithm: • Initialize the population-location and velocities. • Evaluate the fitness of the individual particle (Pbest). • Keep the track of the individual highest fitness (Gbest). • Modify the velocities based on the Pbest and Gbest positions.

(7)

Where

C1 =2.8, C2 = 1.2, Wmin = 0.4, Wmax = 0.9

(10)

Note that the inertia weight is callculated by the following equation:

w : Inertia weight

(11)

c1 , c2 : Cognitive and social acceleration, resppectively.

In proliferation phase, the process mentioned m below is done based on the fitted value. The follo owing equation shows the relation between the fitted value and the proliferation rate:

V : Velocity vector. X : Particle vector. • Update the particles positions.

(12) (8)

• Terminate if the condition is met. • Go to step 2. C. P-CLONAL Algorithm One of the most important characterisstics of the PSO algorithm is its high convergence speed. Thhe combination of the PSO's fast convergence ability with the superior abilities of the CLONAL algorithm such as escaaping from local optimums provides a powerful tool. How tto combine these algorithms is not a clear approach. Thereeupon, the initial generation used in the CLONAL algorithm m becomes more qualified to increase the convergence rate annd the accuracy of the algorithm. The reform process generatioon can be done by the PSO and the selected method dependss on structure of problem and the quality of finding answ wers. Fig.1 shows structure of proposed algorithm. Note that, aat the end of each iteration, the best particles are selected from the initial population, improved population (by PS SO) and mutant population. This strategy prevents losing of thhe best particle. IV.

APPLICATION OF P-CLONAL ALGORITHM

In this section, details of the algorithm ussed for solving the ED problem is expressed .The population sizze of the proposed algorithm is 20 and the maximum iteration iss fixed to 500. The new population in each iteration is gennerated with the following equation: (9) SO algorithm with Population improving is done through the PS the following settings:

Where β is a multiplying factor ( β = 0.4), and S is the total number of antibodies (S= 100). is done using the In the next phase, the mutation of particles p following equation: 1

(13)

Note that in the abovementioned equation e μ is Gaussian probability distribution function wiith average and standard deviation equal to 0 and 1, respectiveely. V.

POWER SYSTEM ECONOMIC LOAD DISPATCH STUDY

P-CLONAL algorithm, which is proposed in this study for wo power systems (13 and solving ED problem, is tested on tw 40 thermal units). For better realizaation, the final results are compared with the results of other methods. m A. Case 1 The first system consists of 13 thermal units with valve loading effects. In this case, the load d demand is assumed to be PD = 1800 MW. Table (1) shows thee results obtained from the test. Different results obtained from different d algorithms tested on the same system are compared d in Table (2) where the proposed method renders better offerr than other methods [2]. TABLE I. 13-THERMAL UNIT TEST RESULT G

P

G

P

G

P

G

P

1 5

628.31 60

2 6

297.54 60

3 7

224.39 109.86

4 8

109.86 60

9

60

10

40

11

40

12

55

13

55

TABLE II.

AVERAGE AND BEST RESULT T COMPARISON WITH DIFFERENT METHODS

Algorithms CEP

13 Thermal unitt ($) Best result 18048.21

13 thermal units($) Average result 18190.32

FEP

18018.00

18200.79

MFFP

18028.09

18192.00

IFEP

17994.07

18127.06

EGA

18019.15

18144.95

FIA

18014.61

18136.97

SPSO

17988.15

18102.48

P-CLONAL

17972.41

18015.44

B. Case 2 In the same way, the second systtem, which consists of 40 oint and determined load thermal units with valve loading po demand equal to PD= 10500, is testeed with different stochastic ults obtained from the test methods . Table (3) shows the resu Figure 1. Flowchart of algorithm

and as it is clear, the new proposed methood delivers better outages in comparison with other algorithms (Table 4) [3-11]. G 1 5 9 13 17 21 25 29 33 37

TABLE III. P G 114 2 97 6 285.03 10 214.62 14 489.52 18 523.18 22 523.65 26 10 30 190 34 110 38

TABLE IV.

40-THERMAL UNIT TEST R RESULTS G P G P 114 3 97.35 4 140 7 259.63 8 130 11 168.76 12 304.59 15 304.40 16 489.31 19 511.37 20 523.38 23 523.97 24 523.61 27 10 28 93.88 31 190 32 200 35 200 36 110 39 110 40

P 179.81 284.79 168.78 394.21 511.13 523.32 10 190 165.27 511.37

VI.

AVERAGE AND BEST RESULT COMPARIISON WITH DIFFERENT

ES REFERENCE

METHODS

Algorithms

40 Thermal units ($/h) Best result

40 Thermal units ($/h) Average result

CEP MFEP IFEP MPSO ESO PSO-LRS Improved GA HPSOWN IGAMU NPSO HDE NPSO-LRS P-CLONAL

123488.29 122647.57 122624.35 122252.26 122122.16 122035.79 121915.93 121915.30 121819.25 121704.73 121698.51 121664.43 121502.13

124793.48 123489.74 123382.00 N/A 122558.45 122558.45 122811.41 122844.40 N/A 122221.36 122304.30 122209.31 121798.08

In order to realize the ability of proposedd algorithm, Fig.2 and Fig.3 show Convergence process of the total cost for 13Thermal and 40-Thermal units respectivelyy. It is clear that proposed algorithm has better performance thhan PSO.

[1]

[2]

[3]

[4]

[5]

[6]

[7]

[8]

[9]

[10]

Figure 2. Convergence process of the total cost foor 13 thermal units [11]

[12]

[13] [14]

[15]

Figure 3. Convergence process of the tottal cost for 40 thermal units

CONCLU USION

A new method derived from the combination of CLONAL selection algorithm and the PSO is proposed p in this study as a solution of the Economic load Disp patch problem with valvepoint effects. The obtained results show that the proposed method is an effective solution for the ED problem in comparison with other stochastic seearch algorithms. The fast convergence and the ability of no ot being trapped in local optimums are no doubt the most imp portant advantages of this new method. Recent surveys sh how that the CLONAL selection algorithm has a lot of fleexibilities in combination with other stochastic search algoriithms; therefore, each of these combined methods can be used d as a solution for the ED problem. N. Amjady, and H. Nasiri-Rad.”N Nonconvex Economic Dispatch with AC Constraints by a New Real Coded Genetic Algorithm”.IEEE. Trans.syst, vol.24, no.3, Aug 2009. Ke Meng, Hong Gang Wang, ZhaoYang Dong, and Kit Po Wong,”Quantum-Inspired Particcle Swarm Optimization for Valve-Point Economic Load L Dispatch,” IEEE Trans.syst,vol.25,no.1,Feb 2010. A. I. Selvakumar and K. Thanushkodi , “A new particle swarm optimization solution to nonconvex economic dispatch problems ,”IEEE Trans. Power Syst., vol. 22 2, no. 1, pp. 42–51, Feb. 2007. C.-L. Chiang, “Genetic-based algo orithm for power economic load dispatch,”IET Gen., Transm. , Disstrib., vol. 1, no. 2, pp. 261–269, Mar.2007 A. Pereira-Neto, C. Unsihuay, and O. R.Saavedra, “Efficient evolutionary strategy optimizatiion procedure to solve the nonconvex economic dispatch h problem with generator constraints,” Proc. Inst. Elect.Eng g., Gen., Transm., Distrib., vol. 152, no. 5, pp. 653–660, Sep. 2005 5. S. H. Ling and F. H. F. Leung, “An “ Improved genetic algorithm with average-bound crossover and d wavelet mutation operations,” Soft Comput ., vol. 11, no. 1, pp. 7–31, 7 Jan. 2007. S.-K. Wang, J.-P. Chiou , an nd C.-W.Liu,“Non-smooth/nonconvex economic dispatch by a novel hybrid differential evolution algorithm,”IET Gen., Transm., T Distrib., vol. 1, no. 5, pp. 793–803, Sep. 2007. han, H. K. Lam, B. C.W. Yeung, JS. H. Ling, H. H. C. Iu, K. Y. Ch and F.H. Leung, “ Hybrid partticle swarm optimization with wavelet mutation and its industrrial applications,” IEEE Trans. Syst., Man., Cybern., vol.38, no. 3, 3 pp. 743–763, Jun. 2008. K. P. Wong and Y. W. Wong, “Genetic “ and genetic/simulatedannealing approaches to econom mic dispatch,” Proc. Inst. Elect. Eng.,Gen., Transm., Distrib., vol.. 141, no. 5, pp. 507–513, Sep. 1994. n, and K. Y. Lee, “A particle J.-B. Park, K.-S. Lee, J.-R. Shin swarm optimization for economicc dispatch with nonsmooth cost functions,”IEEE Trans. Power Sy yst., vol. 20, no. 1, pp. 34–42, Feb. 2005 han, H. K. Lam, B. C.W. Yeung, S. H. Ling, H. H. C. Iu, K. Y. Ch and F.H. Leung, “Hybrid partiicle swarm optimization with wavelet mutation and its industrrial applications,” IEEE Trans. Syst., Man., Cybern., vol.38, no. 3, 3 pp. 743–763, Jun. 2008 J. Kennedy, and R.C.Eberhart, ” particle Swarm Optimization “ proceedings of IEEE Internaatinal Conference on neural networks.iv.pp.1942-1948 J. Kennedy, and R.C.Eberhart,,”Swarm Intelligence. Morgan Kaofmann. ISBN: 1-55860-595-9.. Y.Shi, and R.C.Eberhat “Parameeter selection in Particle swarm optimization“ proceeding of Evo olutionary Pragramming .pp .591.600. hao “Immune Clonal Selection L.Liang, G.Xu, D.Liu, and S.Zh Optimization Method with Mixed d Mutation Strategies”IEEE,9781-4244-4105,2007