Particle swarm optimization with crazy particles for nonconvex ...

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Author's personal copy Applied Soft Computing 9 (2009) 962–969

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Particle swarm optimization with crazy particles for nonconvex economic dispatch Krishna Teerth Chaturvedi a, Manjaree Pandit b,*, Laxmi Srivastava b a b

Department of Electrical Engineering, UIT, Rajiv Gandhi University of Technology, Bhopal, India Department of Electrical Engineering, M.I.T.S., Gwalior 474 005, M.P., India

A R T I C L E I N F O

A B S T R A C T

Article history: Received 27 October 2007 Received in revised form 18 November 2008 Accepted 23 November 2008 Available online 30 November 2008

The paper presents an effective evolutionary method for economic power dispatch. The idea is to allocate power demand to the on-line power generators in such a manner that the cost of operation is minimized. Conventional methods assume quadratic or piecewise quadratic cost curves of power generators but modern generating units have non-linearities which make this assumption inaccurate. Evolutionary optimization methods such as genetic algorithms (GA) and particle swarm optimization (PSO) are free from convexity assumptions and succeed in achieving near global solutions due to their excellent parallel search capability. But these methods usually tend to converge prematurely to a local minimum solution, particularly when the search space is irregular. To tackle this problem ‘‘crazy particles’’ are introduced and their velocities are randomized to maintain momentum in the search and avoid saturation. The performance of the PSO with crazy particles has been tested on two model test systems, compared with GA and classical PSO and found to be superior. ß 2008 Elsevier B.V. All rights reserved.

Keywords: Constriction factor Crazy particles Nonconvex economic dispatch (NCED) Particle swarm optimization Prohibited operating zones (POZ) Ramp-rate limits Time-varying inertial weight (TVIW) Valve-point loading effect

1. Introduction The economic dispatch (ED) is one of the important optimization problems in electrical power system in which the objective is to allocate the power demand among available generators in the most economical manner, while satisfying the physical and operational constraints. The cost of power generation, particularly in fossil fuel plants, is very high and economic dispatch helps in saving a significant amount of revenue. Conventional methods like lamda iteration, base point participation factor, gradient methods, etc. rely heavily on the convexity assumption of generator cost curves and hence approximate these curves using quadratic or piecewise quadratic monotonically increasing cost functions [1]. However, practical generators have a variety of non-linearities and discontinuities in their characteristics due to prohibited operating zones, ramp-rate limits and valve-point loading effects [2,3]. Therefore, practical ED problem translates into a non-smooth optimization problem with heavy equality and inequality constraints, having complex and nonconvex characteristics with multiple minima, which make the challenge of obtaining the global minima, very difficult. Traditional methods fail for this

* Corresponding author. Tel.: +91 751 2665962; fax: +91 751 2409380. E-mail address: [email protected] (M. Pandit). 1568-4946/$ – see front matter ß 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.asoc.2008.11.012

NCED problem except dynamic programming [4] in which no restriction is imposed on the shape of cost curves, but this method suffers from dimensionality problem and excessive computational effort. Methods such as evolutionary programming [5,6], tabu search [7], artificial intelligence [8], neural networks [9], genetic algorithm [2,3,10] and particle swarm optimization (PSO) [11– 15] do not depend on convexity assumptions and require very little computational time. These heuristic methods do not always guarantee global best solutions, but are often found to achieve a fast and near global optimal solution. Though GA-based approaches perform well for complex optimization problems, recent research has identified certain deficiencies [16], particularly for problems in which variables are highly correlated. In such cases, the GA crossover and mutation operators do not generate individuals with better fitness of offspring as the chromosomes in the population pool have same structure towards the end of the search. Premature convergence degrades the performance of GA and increases possibility of convergence to a local optimum solution. The PSO, first introduced by Kennedy and Eberhart [17] is a flexible, robust, population-based stochastic search/optimization algorithm with inherent parallelism. In recent years this method has gained popularity over its competitors and is increasingly gaining acceptance for solving economic dispatch [11–15] and a

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variety of power system problems [18], due to its simplicity, superior convergence characteristics and high solution quality. The convergence towards the global best solution is governed by the proper control of global and local exploration capability. The concept of inertia weight was introduced to balance the local and global search. A high inertia weight during initial part of the search ensured global exploration while a lower value at the end facilitates global convergence. With this in view, the concept of time varying inertial weight (TVIW) was introduced in [19]. The TVIW concept improves the performance of the classical PSO but its ability to fine tune the optimum solution remains weak due to the lack of diversity at the end of the search. It has been observed by various researchers [14,20] that the PSO very quickly finds a good local solution but gets stuck there for a number of iterations without further improvement. As a result, it becomes tedious to find global best solutions for complicated nonconvex problems having multiple local minima and an irregular search space. The present paper proposes to solve the NCED problem using PSO with crazy particles to address the premature convergence behavior of classical PSO by re-initialization of the velocity vector. 2. Nonconvex economic dispatch The practical NCED problem with generator non-linearities such as valve-point loading effects, prohibited operating zones and ramp-rate limits, are included in this paper.

Heat-run tests in which the input–output data is measured to cover the operating region are carried out for constructing the generator cost functions. Normally, large turbine generators have a number of fuel admission valves which are opened one by one when the unit is called upon to increase production. When a valve is opened, the throttling losses increase rapidly, as a result of which, the incremental heat rate rises suddenly. The valve-point effects introduce ripples in the heat-rate curves and make the objective function discontinuous, nonconvex and with multiple minima. If valve point effects are neglected, the generator cost curve can be approximated by a quadratic function [1], but for accurate modeling valve point loading effect also needs to be included. One way of including valve point effects is to model the generator cost curve by piecewise quadratic function [8], while a second approach is to use a rectified sinusoidal function in the cost function [2]. In this paper, the latter model has been used for evaluating the fitness function. The fuel input–power output cost function of ith unit is given as (1)

where ai, bi and ci are the fuel-cost coefficients of the ith unit, and ei and fi are the fuel cost-coefficients of the ith unit with valve-point effects. The NCED problem is to determine the generated powers Pi of units for a total load of PD so that the total fuel cost, FT for the N number of generating units is minimized subject to the power balance constraint and unit upper and lower operating limits. The objective is Min F ¼

N X F i ðPi Þ;

subject to the constraints given by :

i¼1 N X Pi  ðPD þ P L Þ ¼ 0

(2)

i¼1

Pimin  Pi  Pimax

i ¼ 1; 2; . . . ; N

For a given total real load PD the system loss PL is a function of active power generation at each generating unit. To calculate system losses, methods based on penalty factors and constant loss formula coefficients or B-coefficients [1] are in use. The latter is adopted in this paper as per which transmission losses are expressed as PL ¼

N X N N X X P i Bi j P j þ Boi Pi þ Boo i¼1 j¼1

(4)

i¼1

2.2. Generator ramp-rate limits Electric utilities are required to find the optimum schedules every hour or half an hour. Sometimes for convenience it is assumed that the unit generation output is adjusted smoothly and instantaneously. But in practice the operating range of on-line generating units is restricted by ramp-rate limits. In practice, the unit output cannot be adjusted instantaneously whenever load changes. Ramp-rate limits, i.e. up-rate limit URi, down-rate limit DRi and previous hour generation Pio restrict the operating region of all the on-line units. When the generator ramp-rate limits are considered, the operating limits of the ith generating unit are modified as follows: MaxðPimin ; Pio  DRi Þ  P i  MinðPimax ; Pio þ URi Þ

(5)

2.3. Prohibited operating zone

2.1. Valve point loading effect

F i ðP i Þ ¼ ai Pi2 þ bi Pi þ ci þ jei  sinð f i  ðPimin  P i ÞÞj

963

(3)

The cost curves of practical generators are discontinuous as whole of the unit operating range is not always available for allocation. In other words, the generating units have prohibited operating zones due to some faults in the shaft bearing or vibration of machines or their accessories such as pumps or boilers, etc. [3]. The prohibited operating zones divide the operating range of a unit between its maximum and minimum generation limits, into a number of disconnected convex sub-zones making the cost curve discontinuous. Best economy is achieved when unit operation is avoided in these areas. Therefore, conventional gradient-based optimization methods cannot be employed to find the solution in such cases. A unit with prohibited operating zones has discontinuous input–output characteristics. This feature can be included in the NCED formulation as follows: 8 min L  Pi  Pi1 < Pi U L Pi 2 Pik1  P i  Pik : U Pizi  P i  Pimax

(6)

Here zi are the number of prohibited zones in ith generator curve, k L is the index of prohibited zone of ith generator, Pik is the lower limit U of kth prohibited zone, and Pik is the lower limit of kth prohibited zone of ith generator. 3. Real-coded GA In this optimization method the output of each generating unit is represented by a floating point number, instead of binary coding, resulting in absolute precision, hence dependence of accuracy on string length (number of bits) is eliminated. The outputs of all generators are consolidated to form a solution string called chromosome. A population of chromosomes is initially generated randomly. The population size is an important parameter of GA and its selection needs to be done carefully for each problem. Each chromosome in the population represents a possible solution to the problem. A fitness value, derived from the problem’s objective function is then evaluated for each solution string in the

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population. Strings that have better solutions are awarded higher fitness values, ensuring their survival in the coming generations. 3.1. GA operators The GA searches for better solutions by letting the fitter individuals take over the population through a combined stochastic process of selection and recombination. The main three operators that influence the GA performance are selection, crossover and mutation. Their interaction is highly complex and slight variations in their implementations results in a variety of models. The different models depend on many factors like selection method and mechanism, parent replacement method, crossover and mutation method, serial or parallel implementation, and the type of problem to be solved. The GA model to be used is chosen after a careful analysis of the problem to be solved. 4. Solution of NCED problem using evolutionary techniques A PSO [17] is a population-based modern heuristic search method which is inspired by the movement of a flock of birds searching for food. It is a simple and powerful optimization tool which scatters random particles, i.e. solutions into the problem space. These particles, called swarms collect information from each other through an array constructed by their respective positions. The particles update their positions and the update mode is termed as the velocity of particles. Position and velocity are both updated in a heuristic manner using guidance from particles’ own experience and the experience of its neighbors. The position and velocity vectors of the ith particle of a ddimensional search space can be represented as Xi = (xi1, xi2, . . ., xid) and V i ¼ ðvi1 ; vi2 ; . . . vid Þ respectively. On the basis of the value of the evaluation function, the best previous position of a particle is recorded and represented as pbesti = (pi1, pi2, . . ., pid). If the gth particle is the best among all particles in the group so far, it is represented as pbestg = (pg1, pg2, . . ., pgd). The particle tries to modify its position using the current velocity and the distance from pbest and gbest. The modified velocity and position of each particle for fitness evaluation in the next iteration are calculated using the following equations:

As w increases, the factor C decreases and convergence becomes slower because population diversity is reduced. To handle the problem of premature convergence in classical PSO, the concept of craziness [17] was introduced. The idea was to randomize the velocities of some of the particles, referred to as ‘‘crazy particles’’, selected by applying a certain probability. In [14], the probability of craziness rcr is defined as a function of inertia weight 

rcr ¼ wmin  exp 

wk wmax



Then velocities of particles are randomized as per the following logic:

vkj ¼



randðo; vmax Þ; if rcr  rand ð0; 1Þ otherwise

vkj ;

4.1. Implementation of NCED using PSO with crazy particles The paper presents solution of practical ED problem with valve point loading effect, prohibited operating zones and ramp rate limits. The resulting complex NCED problem is solved using PSO improved with crazy particles. The flowchart for this algorithm is given in Fig. 1. Most of the PSO algorithms suffer from the problem of premature convergence in the early stages of the search and henceforth are unable to locate the global optimum. The crazy particles, whose probability can be controlled, do not allow saturation to set in. The idea is to randomize the velocities to maintain momentum in the optimization process and improve the

(7)

¼ xid þ vkþ1 xkþ1 id id

(8)

Here w is the inertia weight parameter, C is constriction factor, c1, c2 are cognitive and social coefficients, and rand1, rand2 are random numbers between 0 and 1. The inertia weight w controls the global and local exploration capabilities of the particle. The practice is to use larger inertia weight factor during initial exploration and gradual reduction of its value as the search proceeds in further iterations. The concept of time varying inertial weight was introduced in [19] as per which w is given by w ¼ ðwmax  wmin Þ 

ðiter max  iterÞ þ wmin iter max

(9)

where itermax is the maximum number of iterations. Constant c1 pulls the particles towards local best position whereas c2 pulls it towards the global best position. Usually these parameters are selected in the range of 0–4. To improve the convergence of PSO algorithm, the constriction factor is also in use [20] 2 C ¼  pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2  ’  ’2  4’

where 4:1  ’  4:2

(12)

If the PSO algorithm tends to saturate in the beginning a high value of rcr is used to create crazy particles, and a comparatively lower value is used at later stages of search.

vkþ1 ¼ w  vkid þ c1  rand1  ð pbestid  xid Þ þ c2  rand2 id  ðgbestgd  xid Þ

(11)

(10) Fig. 1. Flowchart of NCED solution using PSO_crazy method.

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solution quality. Its implementation consists of the following steps: Step 1: Initialization of the swarm: For a population size M, the particles are randomly generated and normalized between the maximum and the minimum operating limits of the generators. If there are N units, the ith particle is represented as n n n n Pi ¼ ðPi1 ; Pi2 ; Pi3 ; . . . ; PiN Þ. The jth dimension if the ith particle is normalized to Pinj as given below to satisfy the constraint given by (3). Here, r 2 [0, 1]. Pinj ¼ P i jmin þ rðPi jmax  P i jmin Þ

(13)

The modeling strategy adopted here is such that the generation values (solutions) never lie within generation limits or POZ and always follow ramp rate limits. The particles are initialized as per Eqs. (3), (5) and (6). For generators with ramp rate limits the initialization is based on (5) and where prohibited zones are present particles are clamped at the respective lower or upper zone limit whichever is nearer to the particle position as per (6). Again when particles are updated these limits are imposed. Hence none of the solutions (not only optimal solutions but other particles of the swarm too) lie within these zones/limits. Step 2: Defining the evaluation function: The merit of each individual particle in the swarm, is found using a fitness function called evaluation function. The evaluation function should be such that cost is minimized while constraints are satisfied. The popular penalty function method employs functions composed of squared or absolute violations to reduce the fitness of the particle in proportion to the magnitude of the violation. Large values for penalty parameters ill condition the penalty function while very small values do not allow the violations to contribute effectively in penalizing a particle. Therefore, the penalty parameters are chosen carefully to distinguish between feasible and infeasible solution. Hence, the penalty parameters, are chosen such that an infeasible solution is awarded fitness worse than the weakest feasible string. Since two infeasible strings are not treated equally, the string further away from the feasibility boundary, is more heavily penalized. Thus, a constrained optimization problem is converted to unconstrained optimization problem

calculated for the updated positions of the particles. If the new value is better than the previous pbest, the new value is set to pbest. Similarly, value of gbest is also updated if the best pbest is better than the stored value of gbest. Step 6: Stopping criteria: A stochastic optimization algorithm is stopped either based on the tolerance limit or maximum number of iterations. In this paper maximum number of iterations is adapted as the stopping criterion after which the positions of gbest are stored as the optimal solution. 4.2. Implementation of NCED using GA For the implementation of NCED through RGA the flowchart in Fig. 2 is used. The initial population is generated as in Eq. (13). The objective of NCED is to minimize operating cost while demand and other constraints are satisfied. RGA works through the population to maximize fitness; hence the fitness function is evaluated as reciprocal of the function (14). The initial population evolves using the basic genetic operators of selection, crossover and mutation. The chromosomes from the parent population are copied into the mating pool with a probability proportional to their fitness to form the offspring population using Roulette–Wheel selection [21]. For a population size M the probability of selection of the ith string is given by F

ri ¼ PM i

j¼1

(16)

Fj

So far, the strings were only getting replicas without any new addition. The crossover operation is used to create new individuals with higher fitness. In binary GA a part of the parents is exchanged with each other at randomly selected sites; the crossover probability controls the frequency of crossover in a generation. This is achieved by swapping of two parent strings at randomly selected locations. For continuous real coded GA if two chromosomes Pin and P nj are randomly selected from the population for

The evaluation function f(Pi) is defined to minimize the non smooth cost function given by Eq. (1) for a given load demand PD while satisfying the constraints given by Eqs. (2) and (3) as " #2 N N X X f ðP i Þ ¼ F i ðPi Þ þ a P i  ðP D þ P L Þ i¼1

þb

"

i¼1 ni X

P i ðviolationÞk

#2 (14)

k¼1

where a is the penalty parameter for not satisfying load demand and b represents the penalty for a unit loading falling within a prohibited operating zone. Step 3: Initialization of pbest and gbest: The fitness values obtained above for the initial particles of the swarm are set as the initial pbest values of the particles. The best value among all the pbest values is identified as gbest. Step 4: Evaluation of velocity: To control excessive roaming of particles, velocity is restricted between V max and þV max . Here, j j V max was set between 15% and 20%. The maximum velocity j limit for the jth generating unit is computed as follows: V max ¼ j

P jmax  P jmin R

(15)

Step 5: Update the swarm: The particle position vector is updated using Eq. (8). The values of the evaluation function are

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Fig. 2. Flowchart of real-coded genetic algorithm.

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966 0

0

crossover, then the two off springs Pin and P nj are evaluated as 0

Pin ¼ rPin þ ð1  rÞP nj 0 P nj ¼ rPnj þ ð1  rÞPin

(17) (18)

where r is a random positive number; r 2 [0, 1]. The above two operators, crossover and reproduction efficiently recombined existing chromosomes but no new genetic information is added to the pool. The mutation operator changes the alleles (individual members in the chromosome, i.e. outputs of generators) at random. If the kth variable of the mth chromosome is randomly selected from a population of M for mutation, then the n0 new chromosome produced is given by Pm such that Pkn assumes Max Min random value between Pk and Pk 0

n Pm ¼ ½P1n ; P2n . . . Pkn ; . . . ; PNn 

Fig. 3. Comparison of convergence characteristics (3-unit system).

(19)

A simple RGA treats mutation as a secondary operator. The mutation operator introduces new genetic structures in the population by randomly modifying some of its building blocks. It helps the search algorithm to escape from the local minima. The probability of crossover and mutation occurring in an execution is selected suitably. These parameters have a dramatic effect on RGA convergence. While the population moves through the search space, guided by the genetic operators, it is likely that the best solution strings, i.e. strings with high fitness values may be lost. Elitism ensures that a few best strings of previous generations are copied to the new generation without alteration. The above operations are repeated till the stopping criterion is met. 5. Test results and analysis The PSO algorithm with crazy particles for practical nonconvex ED problem is tested on two test systems.

Fig. 4. Comparison of convergence characteristics (6-unit system).

objective functions are nonconvex and multimodal for which conventional gradient-based methods either fail or give suboptimal/infeasible solutions. 5.2. Convergence characteristics

I. The first system has 3-generating units [2] has a total load of 850 MW, and cost function includes the valve-point effects in addition to the constraints given by Eqs. (2) and (3). The data is given in the Appendix. II. The second test case has 6-generating units [11], a total load of 1263 MW, all the units have prohibited zones and ramp-rate limit constraints. Power losses have also been considered for this system using the B-matrix from Ref. [11] listed in the Appendix.

The convergence behavior of the three methods was tested employing the same evaluation function, same initial population and velocity for same iterations. The results for all three methods for one trial are shown in Figs. 3 and 4 for the two test systems. When search advances and reaches a certain iteration count, RGA is the first to saturate and converge to a local optimum solution. The next to saturate is the classical PSO. But the PSO-crazy shows superior performance and does not saturate due to the generation of crazy particles; whose velocities are reinitialized to fight premature convergence. For the 3-unit system, which is not very

The results have been compared and validated with previously published results [5,7,11,12]. The Performance of both the systems using PSO_crazy is compared with real coded GA (RGA) and classical PSO and it is observed that PSO_crazy produces superior results and outperforms RGA and classical PSO methods in terms of convergence behavior, solution quality, consistency and computational efficiency.

Table 1 Comparison of different methods for 3-unit system (50 trials). S. no

Method

Minimum cost ($/h)

Maximum cost ($/h)

Average cost ($/h)

5.1. PSO and GA parameters

1 2 3

PSO PSO-crazy RGA

8234.0718 8234.0717 8234.0725

8421.5231 8382.0081 8432.1571

8330.8512 8279.1650 8337.0334

The number of iterations used is 110, w is varied from 0.9 to 0.4 using (9), constriction factor is also varied from 0.73 to 0.64 (for, 4.1  w  4.2) as search proceeds. The value of c1 = c2 = 2.0 is found to be most suitable. In RGA, the crossover probability was set as 0.64 and mutation rate was taken as 0.001 as these values were found to give the best results. Population size was set at 50 for both systems. The limitations of evolutionary optimization techniques like GA and PSO are (i) they are sensitive to parameter variations and (ii) they do not guarantee global convergence every time as the swarm converges to different (but near global) solutions in every run. However their parallel random search capability makes these approaches very attractive for complex problems where the

The results of the best out of the three methods, are highlighted by presenting in bold. Table 2 Comparison of different methods for 6-unit system (50 trials). S. no.

Method

Minimum cost ($/h)

Maximum cost ($/h)

Average cost ($/h)

1 2 3

PSO PSO-crazy RGA

15,451.3106 15,449.3394 15,461.3992

15,635.3768 15,611.2724 15,642.5462

15,517.3926 15,499.3423 15,527.8342

The results of the best out of the three methods, are highlighted by presenting in bold.

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Fig. 8. Mean value of different PSO strategies (6-unit system). Fig. 5. Comparison of standard deviation (3-unit system).

PS is the population size here and f(Pi) is the evaluation function defined in (14). Figs. 5 and 6 plot the standard deviation and Figs. 7 and 8 show the mean, for 3-unit and 6-unit system respectively, obtained in one trial of 110 iterations. The PSO_crazy method records a clear superiority over the other methods and produces better dynamic convergence as the mean cost and the standard deviation of the swarm reduces continuously. The other methods show premature convergence and do not achieve minima. 5.4. Computational efficiency

Fig. 6. Comparison of standard deviation (6-unit system).

complex, all three methods converge to global solutions, but for the 6-unit system having more complexities, the superiority of PSO_crazy becomes obvious. 5.3. Solution quality The minimum, maximum and average costs obtained out of 50 trials for RGA, classical PSO and PSO_crazy are given in Tables 1 and 2. It can be seen that the minimum cost as well as average cost produced by PSO_crazy is least compared with other methods emphasizing the better solution quality of the method. The dynamic convergence behavior of the three methods was also studied by calculating the mean and standard deviation of each individual in the swarm after each iteration. The mean value m and standard deviation s are defined as PPS

f ðP i Þ PS

(20)

vffiffiffiffiffiffiffiffiffiffiffiffiffiffi u PS u1 X 2 s¼t ð f ðP i Þ  mÞ PS i¼1

(21)

m¼

i¼1

Tables 3 and 4 present the best cost achieved by the different PSO algorithms for the two test cases, while satisfying the constraints. The results of economic power dispatch of 6-unit system are given in Table 4. The previous hour load Pio is given in Table A.5. The prohibited operating zones are listed too. Using data from Appendix Tables A.3 and A.5 it can be seen that the solution listed in Table 4 fulfills the constraints of ramp rate limits and operating min–max limits for all the units. Similarly the solutions do not lie within the prohibited operating zones listed in Table A.5. It can be seen from Tables 4 and 5 that the PSO with crazy particles is computationally quite efficient as the cpu time required is almost comparable to the classical PSO method but the results are much superior. The global minimum cost reported for the 3-unit system without considering losses is $8234.07 [5,7,12]. These tables show that all three strategies achieve global minimum Table 3 Generator output for least cost (3-unit system; 50 trials). Unit power output

PSO

PSO_crazy

RGA

P1 (MW) P2 (MW) P3 (MW) Total power output (MW) Total generation cost ($/h) CPU time (s)

400.000 300.2667 149.7333 850 8234.0718 0.0590

400.000 300.2668 149.7332 850 8234.0717 0.0598

400.000 300.2653 149.7347 850 8234.0725 0.0659

The results of the best out of the three methods, are highlighted by presenting in bold. Table 4 Generator output for least cost (6-unit system; 50 trials).

Fig. 7. Mean value of different PSO strategies (3-unit system).

Unit power output

PSO

PSO_crazy

RGA

P1 (MW) P2 (MW) P3 (MW) P4 (MW) P5 (MW) P6 (MW) Total power output (MW) Total loss Total cost ($/h) CPU time (s)

469.9415 175.5558 246.5108 138.7732 152.3809 92.1599 1263.0000 12.3222 15451.3106 0.0603

464.5764 177.8071 265.0000 120.9708 156.7055 90.6358 1263.0000 12.6956 15449.3394 0.0612

420.2342 199.4412 263.7234 120.0030 167.2319 105.1250 1263.0000 12.7588 15461.3992 0.0813

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Table A.1 Generator operating limits and cost coefficients (3-unit system). Generator

Variable

Unit 1 Unit 2 Unit 3

ai ($)

bi ($/MW)

ci ($/MW2)

ei

fi

Pimax

Pimin

0.00156 0.00194 0.00482

7.92 7.85 7.97

561 310 78

300 200 150

0.031 0.042 0.063

600 400 200

100 100 50

Table A.2 B-loss coefficients of 3-unit system. Bij=

Fig. 9. Best results of different PSO algorithms (6-unit system).

0.0000676 0.00000953 0.00000507

Boi=

0.00000953 0.00005210 0.00000901

0.0007760

0.0000057 0.00000901 0.00029400 0.01890

0.0000342

Boo=

Table 5 Frequency distribution table for 3-unit system.

0.040357

Table A.3 Generator operating limits and cost coefficiets (6-unit system).

Applied method

Ranges of minimum cost ($) 8200–8250

8250–8300

8300–8350

8350–8400

8400–8450

PSO PSO_crazy RGA

25 29 24

19 16 14

3 3 7

2 2 3

1 0 2

solution for the 3-unit systems, but PSO_crazy performs better for the 6-unit system which is more complex.

Unit

Pimin

Pimax

ai ($)

bi ($/MW)

ci ($/MW2)

1 2 3 4 5 6

100 50 80 50 50 50

500 200 300 150 200 120

240 200 220 200 220 190

7 10 8.5 11 10.5 12

0.0070 0.0095 0.0090 0.0090 0.0080 0.0075

5.5. Robustness

Table A.4 B-cofficient of 6-unit system.

The performance of heuristic search based optimization algorithms is judged through many trials with different initial populations to compare the robustness/consistency of PSO_crazy with RGA and PSO. The frequency of convergence for the 3-unit systems out of 50 trials is presented in Table 5. The table shows that the frequency of converging to a better solution is higher in classical PSO as compared to RGA, while it is highest in PSO-crazy. Fig. 9. plots the best results achieved by the three methods for the 6-unit system out of 50-trials. It can be observed that the PSO_crazy method is most consistent in achieving the lowest cost in all runs.

Bij=

6. Conclusion The PSO_crazy strategy is proposed for solving the complex problem of nonconvex economic power dispatch with multiple minima. The performance of this method is compared with RGA and classical PSO. It is found that PSO_crazy approach handles the problem of premature convergence found in GA and classical PSO very effectively by generating crazy particles whose velocities are reinitialized with a certain probability. The superiority of PSOcrazy becomes more evident for more complex systems. It has been clearly demonstrated that PSO_crazy is capable of achieving global solutions. This method outperforms other reported methods in terms of solution quality, computational efficiency, dynamic convergence, robustness and stability. Acknowledgements The authors sincerely acknowledge the financial support provided by Department of Science and Technology (DST), Government of India, New Delhi, India under research project entitled Integrated fuzzy neural network approach for power system voltage security assessment of Madhya Pradesh State Electricity Board System vide letter no. SR/S3/EECE/14/2003-SERC dated 11/5/2004, and AICTE New Delhi for financial assistance under RPS project F No. 8023/RID/BOR/RPS-45/2005-06 dated 10/

0.0017 0.0012 0.0007 0.0001 0.0005 0.0002

0.0012 0.0014 0.0009 0.0001 0.0006 0.0001

0.0007 0.0009 0.0031 0.0000 0.0010 0.0006

0.0001 0.0001 0.000 0.0024 0.0006 0.0008

0.0005 0.0006 0.0010 0.0006 0.0129 0.0002

.0002 0.0001 0.0006 0.0008 0.0002 0.0150

Boi= 0.0003908 0.0001297 0.0007047189 0.0000591 0.0002161 0.0006635 Boo=

0.056

Table A.5 Data of 6-unit system for prohibited zones and ramp-rate limits. Unit

Pio

URi (MW/h)

DRi (MW/h)

Prohibited zone (MW)

1 2 3 4 5 6

440 170 200 150 190 110

80 50 65 50 50 50

120 90 100 90 90 90

[210,240] [90,110] [150,170] [8,090] [90,110] [7,585]

[350,380] [140,160] [210,240] [110,120] [140,150] [100,105]

03/2006. The authors also thank the Director, M.I.T.S. Gwalior for providing facilities for carrying out this work.

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