Solution of nonconvex and nonsmooth economic dispatch by a new Adaptive Real Coded Genetic Algorithm

Expert Systems with Applications 37 (2010) 5239–5245

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Expert Systems with Applications journal homepage: www.elsevier.com/locate/eswa

Solution of nonconvex and nonsmooth economic dispatch by a new Adaptive Real Coded Genetic Algorithm Nima Amjady, Hadi Nasiri-Rad * Department of Electrical Engineering, Semnan University, Semnan, Iran

a r t i c l e

i n f o

Keywords: Adaptive Real Coded Genetic Algorithm Economic dispatch Valve loading effects Multiple fuel source option

a b s t r a c t This paper proposes a novel Adaptive Real Coded Genetic Algorithm (ARCGA) to solve the nonconvex and nonsmooth economic dispatch (ED) problem considering valve loading effects and multiple fuel source options. Considering valve effects and multiple fuel options change ED into nonlinear, nonconvex and nonsmooth optimization problem with multiple minima. These characteristics challenge analytical and heuristic methods in finding optimal solution in reasonable time. The proposed ARCGA technique is composed of new genetic operators including arithmetic-average-bound crossover (AABX) and B-Spline wavelet mutation (BWM). Moreover, to enhance the computational efficiency of the suggested solution method, an adaptation process is also included in the ARCGA. To show the superiority of the ARCGA, it is compared with several most recently published methods proposed to solve the ED problem. Ó 2009 Elsevier Ltd. All rights reserved.

1. Introduction Power systems should be operated under a high degree of economy. Economic dispatch is an important optimization task addressing this vital concern for power system operations. ED is defined as the process of allocating generation levels to the generating units, so that the system load is supplied entirely and most economically (Wood & Wollenberg, 1996). ED is subproblem of unit commitment (UC) (Patra, Goswami, & Goswami, 2009; Yuan, Nie, Su, Wang, & Yuan, 2009) and determines the generation level of each committed unit. Practical economic dispatch has complex and nonlinear characteristics with many equality and inequality constraints (Li, 1998; Li & Aggarwal, 2000). Traditionally, in the ED problem, the cost function of each generator has been approximated by a single quadratic function, and the valve-point effects and multi-fuel source options were ignored. This would often introduce inaccuracy into the resulting dispatch. Large modern generating units with multi-valve steam turbines have a number of steam admission valves that are opened sequentially to obtain ever-increasing output of units and the valve-point effects produce a ripple-like heat rate curve. However, since the cost curve of a generator is highly nonlinear, containing discontinuities owing to valve-point loadings, the cost function is more realistically denoted as a segmented piecewise nonlinear function (Sinha, Chakrabarti, & Chattopadhyay, 2003) rather than a single quadratic function. Moreover, many generating units, specifically those which are supplied with multi-fuel source lead to the problem of determining * Corresponding author. Tel.: +98 912 5399305; fax: +98 021 88880098. E-mail address: [email protected] (H. Nasiri-Rad). 0957-4174/$ - see front matter Ó 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.eswa.2009.12.084

the most economic fuel to burn (Lio & Cai, 2005). Solution spaces of these problems have multiple minima due to valve-point effects and multi-fuel options. Recently several stochastic techniques are proposed to solve ED problem (Altun & Yalcinoz, 2008; Kuo, 2008; Yuan, Wang, Zhang, & Yuan, 2009). Some methods such as hybrid differential evolution (HDE) (Yuan et al., 2009), genetic algorithm (GA) (Chiang, 2007; Ling & Leung, 2007), modified PSO (MPSO) with a dynamic search space reduction strategy (Park, Lee, Shin, & Lee, 2005), evolutionary strategy optimization (ESO) (Pereira-Neto, Unsihuay, & Saavedra, 2005), quantum evolutionary algorithm (Babu, Das, & Patvardhan, 2008), partition approach algorithm (PAA) (Lin, Gow, & Tsay, 2007), pattern search (PS) method (Al-Sumait, AL-Othman, & Sykulski, 2007), self-tuning hybrid differential evolutionary (ST-HDE) (Wang, Chiou, & Liu, 2007), hybrid genetic algorithm (HGA) (He, Wang, & Mao, 2008), hybrid bacterial foraging (BF) technique (Panigrahi & Pandi, 2008), self-organizing hierarchical particle swarm optimization (SOH-PSO) (Chaturvedi, Pandit, & Srivastava, 2008) and society-civilization algorithm (SCA) combined with particle swarm optimization (PSO) called CSO (Selvakumar & Thanushkodi, 2009) have been proposed to solve ED problem in light of the valve-point effects. Moreover, a number of heuristic techniques such as Taguchi method (TM) (Yuan et al., 2009), Hopfield neural network (HNN) (Park, Kim, Eom, & Lee, 1993), adaptive Hopfield neural network (AHNN) (Lee, Sode-Yome, & Park, 1998), and evolutionary programming (EP) (Sadasivam & Sadasivam, 2000) have been applied to solve ED problem with the consideration of multiple fuel source options. Recently, a few modern approaches such as improved genetic algorithm with multiplier updating (IGA-MU) (Chiang, 2005), new particle swarm

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optimization (NPSO) with local random search (NPSO-LRS) (Selvakumar & Thanushkodi, 2007), anti-predatory particle swarm optimization (APSO) (Selvakumar & Thanushkodi, 2008), and various evolutionary algorithms (EA) (Manoharan, Kannan, Baskar, & Iruthayarajan, 2008) proposed to solve ED considering both valve loading effects and multi-fuel options together. To obtain an accurate and practical economic dispatch solution, the realistic operation of the ED problem should take both valvepoint effects and multiple fuels into account, which usually are found in practical power systems simultaneously. To solve this ED problem, a new Adaptive Real Coded Genetic Algorithm (ARCGA) with new genetic operators including arithmetic-averagebound crossover (AABX) and B-Spline wavelet mutation (BWM) is proposed. The proposed ARCGA is executed adaptively and can provide a more diverse search of the solution space. So, better optimum solutions with lower computation burden can be found compared with the previous stochastic search techniques proposed to solve the ED problem. The remaining parts of the paper are organized as follows. In Section 2, the formulation of the practical ED problem including valve loading effect and multiple fuel option is presented. The proposed solution method, i.e. ARCGA, is introduced in the third section. Application of the proposed ARCGA to solve the ED problem is described in Section 4. Obtained numerical results are presented and discussed in Section 5. Section 6 concludes the paper. 2. Problem formulation ED problem can be formulated as follows (Wood & Wollenberg, 1996):

Min

FT ¼

n X

! F i ðPi Þ

ð1Þ

i¼1

where F T is the total generation cost ($/hr), n is the number of dispatchable units, P i is the generation output of ith dispatchable unit, and F i ðPi Þ is the fuel cost function of ith dispatchable unit ($/hr). The objective function of (1) is subject to power balance and generating capacity constraints:

Subject to

n X

Pi ¼ Pd þ PLoss

ð2Þ

i¼1

6 Pi 6 Pmax P min i i

i ¼ 1; . . . ; n

ð3Þ

where P d is the total active power demand, PLoss is the total active is the minimum operating limit of ith unit, loss of network, Pmin i is the maximum operating limit of ith unit, and PLoss is not conPmax i sidered in this work; however it can be easily calculated by the B matrix loss formula (Selvakumar & Thanushkodi, 2007; Wang et al., 2007). The fuel cost of thermal generation units is usually approximated by a quadratic function as follows:

F i ðPi Þ ¼ ai þ bi P i þ ci P 2i

ð4Þ

where ai ; bi , and ci represent fuel cost coefficients of ith unit. However, a generator with multi-valve steam turbine has a different input–output curve compared with the smooth cost function of (4). Typically, the valve-point effect results in the ripples in fuel cost curve as each steam valve starts to open. To take into account the valve-point effect, a recurring rectifying sinusoidal function is added to the quadratic fuel cost function as follows (Chaturvedi et al., 2008; Panigrahi & Pandi, 2008):

    F i ðPi Þ ¼ ai þ bi P i þ ci P 2i þ ei sinðfi ðPmin  Pi ÞÞ i

ð5Þ

where ai ; bi ; ci ; ei , and fi indicate fuel cost coefficients of ith unit with valve loading effects. In addition, usually there are many generating

units supplied with multiple fuels. In the case of these units, unlike the conventional cost function, the cost function of each unit should be presented by a few piecewise functions reflecting the effects of fuel type changes and each segment of the hybrid cost function implies some information about the type of fuel being burned or the operational characteristics of the unit. Then the fuel cost function for ith unit reflecting the multiple fuel option is as follows (Lio & Cai, 2005; Sadasivam & Sadasivam, 2000):

8 a þ bi1 Pi þ ci1 P2i > > > i1 > > < ai2 þ bi2 Pi þ ci2 P2i F i ðPi Þ ¼ .. > > . > > > : aij þ bij Pi þ cij P2i

fuel1; Pmin 6 Pi 6 Pmax i i1 fuel2; Pmin 6 Pi 6 Pmax i2 i2 .. .

ð6Þ

fuelj; Pmin 6 Pi 6 Pmax ij i

min It is noted thatP max ij1 ¼ P ij . By integrating valve-point effect and multiple fuel option, the cost function of ith unit becomes as follows:

    F i ðPi Þ ¼ aij þ bij Pi þ cij P2i þ eij sinðfij ðPmin  Pi ÞÞ if Pmin i i;j 6 Pi 6 Pmax i;j

j ¼ 1; . . . ; nf

i ¼ 1; . . . ; n

ð7Þ

where j is the type of fuel source, Pmin is the minimum power geni;j is the maximum power generation of unit i with fuel option j, Pmax i;j eration of unit i with fuel option j, and nf is the number of fuel types for each unit. The incorporated fuel cost function of (7) is substituted in (1) to obtain the realistic objective function F T of the practical ED problem. Therefore, the objective function F T is really composed of a set of nonsmooth and nonconvex fuel cost functions F i ðP i Þ, which each one integrates the valve loading effects and multi-fuel options (Manoharan et al., 2008; Selvakumar & Thanushkodi, 2007). Such ED problem is a nonlinear, nonconvex and nonsmooth optimization problem with multiple minima, which is hard, if not impossible, to solve using traditionally deterministic optimization algorithms. So, a new stochastic search technique, i.e. ARCGA, is proposed in this paper to solve the ED problem. 3. Proposed Adaptive Real Coded Genetic Algorithm solution GAs are search and optimization procedures that inspired by the natural genetics. GA begins with a population of randomly generated chromosomes, which each one represents a potential solution for the optimization problem. The evolution of a GA population is generally based on the selection of the parents according to their fitness values, generation of the offspring chromosomes (e.g. by crossover and mutation operators) and survival of the fittest. For more details about GAs, the interested reader can refer to Michalewicz (1996). In the following, at first the proposed RCGA (without adaptation) is introduced. Then the adaptation process is added to the RCGA to convert it to ARCGA. The proposed RCGA has new genetic operators including AABX and BWM, which are introduced in the following, respectively. 3.1. Crossover operator The crossover operation of GA is a method of sharing information among selected chromosomes (parents) which has usually been regarded as the main search operation in GAs as it exploits the available information in previous samples to influence future searches. The crossover operator of the proposed RCGA, i.e. AABX, combines the arithmetic, average and bound crossovers. The arithmetic crossover operation produces some children with their parent’s features; average crossover manipulates the genes of the selected parents and the minimum and maximum possible values

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of the genes; bound crossover is capable of moving the offspring near the domain boundary. Therefore, the offspring spread over the domain so that a higher chance of reaching the global optimum can be obtained. The selection of chromosomes for the AABX is based on the roulette-wheel parent selection mechanism (Michalewicz, 1996). The probability of crossover should be given as an input to this mechanism, which in this work it is set randomly in the interval of [0.3, 0.7] to avoid trapping in local minima. In other words, in each generation, 30–70% of the chromosomes of the population are selected by the roulette-wheel mechanism for the AABX operation. Suppose two vectors C tv ¼ ½P t1;v ; Pt2;v ; . . . ; P tm;v  and C tw ¼ ½Pt1;w ; P t2;w ; . . . ; Ptm;w  are two selected chromosomes in the generation t of the RCGA evolution. Each chromosome has m genes, which are real numbers (the generation outputs of dispatchable units). The AABX operator, including the arithmetic, average and bound crossovers, creates eight children (generation t + 1) from the parents C tv and C tw as follows:

1

0.8 0.6 0.4 0.2 0

-0.2 -0.4 -0.6 -0.8 -1 -8

-6

-4

-2

0

2

4

6

8

Fig. 1. Real part of complex frequency B-Spline wavelet.

(1) Arithmetic crossover

C 1tþ1 C 2tþ1 C 3tþ1 C 4tþ1

t

¼ xa C v þ ð1  x

t a ÞC w t aCw

¼ ð1  xa ÞC tv þ x   ¼ min C tv ; C tw   ¼ max C tv ; C tw

ð10Þ

Hereafter, we refer to f(x), defined in (16), as the B-Spline wavelet function. In (16), the parameters m; fc and fb represent order, central frequency and bandwidth frequency of the B-Spline wavelet (Teolis, 1998). In order to control the magnitude of the B-Spline wavelet function, fd ðxÞ is defined as:

ð11Þ

 pffiffiffi fd ðxÞ ¼ 1= d  f ðxÞ

ð18Þ

ð12Þ

where d is the dilation parameter of the wavelet and fd ðxÞ is the amplitude-scaled version of the B-Spline wavelet. The amplitude of the wavelet will be scaled down as the dilation parameter d increases. In (18), the argument x is as follows:

ð8Þ ð9Þ

(2) Average crossover

C 5tþ1 ¼ ðC tv þ C tw Þ=2 C 6tþ1

t

¼ ½xb ðC v þ

C tw Þ

þ ð1  xb ÞðC min þ C max Þ=2

ð13Þ

x ¼ RNDðlb; ubÞ

(3) Bound crossover

  C 7tþ1 ¼ xc  max C tv ; C tw þ ð1  xc Þ  C max   C 8tþ1 ¼ xc  min C tv ; C tw þ ð1  xc Þ  C min

ð14Þ

where lb and ub are lower bound and upper bound of x. The dilation parameter d is determined as follows:

ð15Þ

d ¼ expðlnðgÞ  ð1  ð1  t=TÞn ÞÞ

    where min C tv ; C tw and max C tv ; C tw denotes the vectors with each element obtained by taking minimum and maximum between the corresponding elements of chromosomes C tv and C tw , respectively; C min and C max are the minimum and maximum limits of chromosome vectors, respectively; xa ; xb and xc are constant weights in the interval [0, 1], which in our work, are set to 0.3, 0.5 and 0.5, respectively. Among these eight children presented in (8)–(15), the two children that have the highest fitness values are selected as the offspring chromosomes of the crossover operation. These two offspring chromosomes are added to the previous population including the parents. So, after the execution of the AABX operator the previous population will be enlarged, which is considered for the BWM operator. 3.2. Mutation operator The mutation operator of the proposed RCGA, i.e. BWM, is based on the wavelet theory (Teolis, 1998). We use the real part of complex frequency B-Spline wavelet (Teolis, 1998), shown in Fig. 1, in the mutation operation. This wavelet function is described as follows:

hpffiffiffiffi i f ðxÞ ¼ Real fb ðsincðfb  x=mÞÞm  expð2pi  fc  xÞ

ð16Þ

where Real [.] extracts real part of a complex variable and

sincðxÞ ¼ sinðxÞ=x

ð19Þ

ð17Þ

ð20Þ

where n is the shape parameter of the monotonic increasing function of d, g is the upper limit of the dilation parameter, t is the current generation number, and T is the maximum number of generations. In this work, the shape parameter n and upper limit g are set at 0.05 and 1000, respectively. The dilation parameter d is set to vary with the value of t/T, giving the adaptive search capability to the proposed RCGA. For the adjustable parameters of the B-Spline wavelet, the default values usually used in the signal processing applications are used (Teolis, 1998): m = 2; fb = 1; fc = 0.5; lb = 8; ub = 8. The details of the BWM operation are as follows. Every gene of the chromosomes will have a chance to mutate governed by the probability of the mutation Pm , which is set randomly in the interval [0.1, 0.3]. For each gene of the chromosome, a random number in the range of [0, 1] is generated. If the random number is less than Pm , that gene is selected for the mutation, otherwise it is not selected. If C v ¼ ½P 1;v ; P2;v ; . . . ; P m;v  is a chromosome vector and its gene P i;v is chosen by the mutation probability to be muted, the new gene Pmut i;v will be after mutation:

(

Pmut i;v

¼

if f d ðxÞ P 0 Pi;v þ fd ðxÞ  ðPmax i;v  P i;v Þ Pi;v þ fd ðxÞ  ðPi;v  Pmin i;v Þ

if f d ðxÞ < 0

ð21Þ

where fd ðxÞ is computed from (18). In the initial generations of the RCGA (when t is small compared with T), the dilation parameter d takes small values according to (20). Thus, fd ðxÞ takes large values based on (18), which makes large changes in the mutating genes

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Pmut i;v according to (21). On the other hand, when the RCGA proceeds, d becomes larger and so smaller changes in the mutating genes are made by fd ðxÞ. In this way the mutation operator performs a wider search in the solution space at the early stages of the evolution, and at the later stages the search is restricted around the local area of the parameter (the selected gene), resembling a hill-climbing operator. As another important characteristic, it is seen that fd ðxÞ takes its values from the B-Spline wavelet which has diverse outputs in the range of [-1,+1], as shown in Fig. 1. This characteristic gives more diversity to the mutation operation in searching the solution space and so the chance of finding the optimum solution of the problem increases, which is another advantage of the proposed RCGA. This characteristic plus its local search ability is the motivation behind using the B-Spline wavelet in the mutation operation. As a result, the proposed mutation operation, i.e. BWM, is a powerful tool for fine tuning the genes to search the solution space locally. At the same time, as described in the previous subsection, the offspring chromosomes produced by the AABX crossover spread over the domain, which gives the global search capability to the RCGA. So, the proposed RCGA including the AABX and BWM operators can benefit from both global and local search capabilities, which enhance the searching performance and provide a faster convergence of the RCGA. The B-Spline mutation is performed on the population obtained from the AABX operator. When the mutation is performed on the gene(s) of a chromosome, the mutated chromosome is compared with its parent and each one owning better fitness value is selected and returned to the population. The other chromosome is removed from the population. However, still the population size is larger than the initial one due to the AABX operation (described in the previous subsection). So after the execution of the AABX and then B-Spline mutations, the poorest chromosomes with the smallest fitness values are removed from the population such that the population size returns to its initial value. 3.3. The adaptation process To decrease the computation burden and increase the solution quality in solving the ED problem, the RCGA is adaptively executed. In the previous subsections, the proposed RCGA has been introduced. In this section, the adaptation process is added to the RCGA to convert it to ARCGA. Suppose that N C indicates number of chromosomes of the RCGA. In the initial generations of the RCGA, since the search process is in the initial stages, the RCGA should be executed with high N C . In other words, in the initial generations of the RCGA, high diversity in its search process is required to cover different regions of the solution space. On the other hand, by evolving the RCGA along the evolution process, it becomes closer to the convergence point and so poor RCGA chromosomes can be omitted resulting in lower N C values. In other words, when the RCGA approaches the optimum point, poor chromosomes (which are far from the optimum) have no effect on the convergence of the RCGA and so can be eliminated from the RCGA population. Considering the above explanations, instead of fixed N C , it is adaptively changed along the evolution process of the RCGA as follows (converting the RCGA to ARCGA):

 h i NC ¼ int N C1 exp lnðNC2 =NC1 Þð1  ð1  t=TÞb Þ

ð22Þ

where int(.) is a function returning the closest integer to its real variable argument. NC1 and N C2 indicate the initial and final values of N C along the evolution process, respectively. In (22), b is a parameter that controls the slope of the adaptation process. Higher values of b cause that NC more quickly goes toward its final value N C2 . For the adaptation process, we should have NC1 > N C2 . To solve the ED

problem by the proposed ARCGA, NC1 ¼ 100; N C2 ¼ 10 and b = 5 are considered, which are tuned by a few experiments.

4. Application of the proposed ARCGA to solve the ED problem Application of proposed ARCGA to solve the ED problem can be summarized as the following step by step algorithm: (1) At first the initial population of the ARCGA is randomly produced where each gene of the chromosomes represents output of a dispatchable generating unit. In the production of the initial population, the output of (n  1) dispatchable units can be chosen arbitrary within their respective generating capacity constraints (3) while the output of reference unit is constrained by the system power balance (2), such that the power balance of the system is satisfied (Damousis, Bakirtzis, & Dokopoulos, 2003). In other words, in the step by step algorithm, constraint (3), including generating capacity limits, is implemented by the preservation method for (n  1) dispatchable units (excluding reference unit). In this method, solutions are initially placed in the feasible search space and remain within by adapting an update mechanism that generates only feasible solutions. (2) To handle the generation capacity constraint of reference unit, we use the adaptive penalty function method. In this method, deviation from the constraint is added to the objective function such that each unfeasible solution is penalized by a penalty term proportional to the deviation. Suppose that nth unit is reference unit. The penalty term for the generation capacity constraint of the reference unit, denoted by X, can be computed as follows:

8 min > < ð1  Pn =Pn Þ  F max max X ¼ ðPn =P n  1Þ  F max > : 0

if Pn < Pmin n if Pn > Pmax n

ð23Þ

else

where X is in proportion to the reference unit limits violation and zero in case of no violation of these limits. The penalty term is based on the deviation from the constraint and it should be chosen high enough to make constraint violations prohibitive in the final solution. So, F max is selected as the maximum generation cost, which is calculated as below:

F max ¼

n X

F i ðP max Þ i

ð24Þ

i¼1

(3) The fitness function FF considering both the cost function F T and penalty term X can be computed as follows:

FF ¼

C FT þ X

ð25Þ

In other words, a chromosome with higher value of generation cost F T or larger deviation from the constraint (higher value of the penalty term X) will have a lower fitness function value. It is noted that the generation cost F T is obtained from (1) with the fuel cost function F i ðPi Þ computed based on (7). In (25), C is a constant used in order to prevent from obtaining too small values for the fitness function FF and its magnitude should be in the order of the system maximum generating cost F max . (4) After computation of the fitness function (FF) value for each chromosome, check the generation number. If the generation number reaches to its maximum limit (defined by user) then the ARCGA stops and the best chromosome of the last generation owning the highest fitness function value is returned as the final solution, else go to step 5. It is noted

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that, as described in the previous section, the poorest chromosomes of each population are removed and so the best chromosome of each generation is automatically transferred to the next one. So, the best chromosome of the last generation is the best chromosome among all generations. (5) The next generation is produced by the AABX crossover and BWM mutation as described in the previous section. Also, generation number is incremented by 1. (6) The adaptation process is performed and N C is changed according to (22). Go back to step 2.

Table 2 Detailed results of the best solution of the proposed ARCGA for the first test system.

5. Numerical results and analysis The proposed algorithm was implemented in the MATLAB software package on a simple Pentium IV personal computer with 1.8 GHz CPU with 256 MB RAM. Owing to the randomness of the stochastic algorithms, their performance cannot be judged by the result of a single run. Many trials with independent population initializations should be made to acquire a useful conclusion of the performance of the algorithm. In this paper, the statistical indices such as the best, mean and worst results (generation costs) plus mean computation time over 20 independent trials are reported. To find the effectiveness of the proposed ARCGA, the test results are also compared with the results already reported by the most recently published methods for solving the ED problem. 5.1. Case 1: 40-unit test system The first test system has 40 generating units with nonconvex fuel cost function addressing valve loading effects. Its load demand is 10,500 MW. Data of this test system can be obtained from (Sinha et al., 2003). The solution space of this test case has many local minima, and the global minimum is hard to obtain. The obtained results of the proposed ARCGA to solve the ED problem for this test system are shown in Table 1. In this table, the comparison of the worst, average and best solutions of the proposed ARCGA and most recently published ED solution methods is shown. As seen from Table 1, the worst, average and best solutions of the proposed ARCGA are better than those of all other methods indicating more efficiency of the ARCGA to solve the ED problem than the other methods. Also, the worst, average and best solutions of the ARCGA are very close to each other representing the robustness of the proposed solution method. Detailed results of the best solution of the proposed ARCGA, including generation output of each unit for this test system, are shown in Table 2.

Unit

P (MW)

Unit

P (MW)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

110.8252 113.9112 97.4000 179.7331 88.6454 140.0000 259.6000 284.6000 284.6000 130.0000 168.7985 168.7994 214.7600 394.2800 304.5200 394.2800 489.2798 489.2800 511.2806 511.2800

21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40

523.2803 523.2800 523.2800 523.2800 523.2800 523.2801 10.0000 10.0000 10.0000 88.7611 190.0000 190.0000 190.0000 164.8000 164.8000 164.8054 110.0000 110.0000 110.0000 511.2800

5.2. Case 2: 10-unit test system The second test system consists of 10 generating units with practical cost function including both valve loading effects and multiple fuel source option. The ED problem for this test system includes nonsmooth and nonconvex cost functions. Data of this test system can be found in Chiang (2005). Table 3 shows the comparison of the best solution obtained by the proposed ARCGA and Differential Evolution (DE) (Manoharan et al., 2008), Real-coded GA (RGA) (Manoharan et al., 2008) and particle swarm optimization (PSO) (Manoharan et al., 2008) for different power demands ðP d Þ of the test system. As seen from this table, the proposed ARCGA gives a lower generation cost than the other methods for all load levels. The comparison of the worst, average and best solutions of the proposed ARCGA and most recently published ED solution methods for the second test system with 2700 MW power demand is shown in Table 4. This table validates the superiority of the proposed ARCGA over all other methods to solve the ED problem with practical fuel cost function. Despite the previous test system which only included the valve loading effect, this test system incorporates both valve loading effect and multiple fuel source option. For this more complex test case with nonconvex and nonsmooth fuel cost functions, the efficiency of the proposed ARCGA compared with the other ED solution methods is better reflected. Not only the best,

Table 1 Worst, average and best generation costs for the first test system. Methods

IFEP Sinha et al. (2003) PAA Lin et al. (2007) ESO Pereira-Neto et al. (2005) PSO-LRS Selvakumar and Thanushkodi (2007) IGA Ling and Leung (2007) GA Chiang (2007) PSO Selvakumar and Thanushkodi (2008) NPSO Selvakumar and Thanushkodi (2007) ST-HDE Chaturvedi et al. (2008) NPSO-LRS Selvakumar and Thanushkodi (2007) APSO Selvakumar and Thanushkodi (2008) SOH-PSO Chaturvedi et al. (2008) CSO Selvakumar and Thanushkodi (2009) BF Panigrahi and Pandi (2008) PS Al-Sumait et al. (2007) Proposed ARCGA

Total generation cost ($/h) Worst

Average

Best

125740.6300 122243.189 123143.0700 123461.6794 123334.0000 – 123467.4086 122995.0976 – 122981.5913 122912.3958 122446.3000 122844.5391 – 125486.2900 121536.8745

123382.0000 122243.189 122558.4565 122558.4565 122811.4100 – 122513.9175 122221.3697 122304.3000 122209.3185 122153.6730 121853.5700 121936.1926 121814.9465 122332.6500 121462.1502

122624.3500 122243.189 122122.1600 122035.7946 121915.9300 121819.2521 121735.4736 121704.7391 121698.5100 121664.4308 121663.5222 121501.1400 121461.6707 121423.6379 121415.1400 121410.10382

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Table 3 Comparison of the best solution of the DE, RGA, PSO and proposed ARCGA for the second test system. Methods

Total generation cost ($/h)

DE Manoharan et al. (2008) RGA Manoharan et al. (2008) PSO Manoharan et al. (2008) Proposed ARCGA

P d ¼ 2400 (MW)

P d ¼ 2500 (MW)

P d ¼ 2600 (MW)

P d ¼ 2700 (MW)

482.5114 482.5275 482.5088 481.7434

527.0189 527.0360 527.0185 526.2589

575.1610 575.1753 575.1606 574.4054

624.5081 624.5146 624.5074 623.8281

Table 4 Comparison of worst, average and best generation costs for the second test system ðPd ¼ 2700 MWÞ. Methods

Total generation cost ($/h) Worst

Average

Best

CGA-MU Chiang (2005) IGA-MU Chiang (2005) DE Manoharan et al. (2008) RGA Manoharan et al. (2008) PSO Manoharan et al. (2008) PSO-LRS Lee et al. (1998) NPSO Selvakumar and Thanushkodi (2007) NPSO-LRS Selvakumar and Thanushkodi (2007) APSO Selvakumar and Thanushkodi (2008) Proposed ARCGA

633.8652 630.8705 624.5458 624.5088 624.5074 628.3214 627.4237 626.9981 627.3049 623.8550

627.6087 625.8692 624.5246 624.5079 624.5074 625.7887 625.218 624.9985 624.8185 623.8431

624.7193 624.5178 624.5146 624.5081 624.5074 624.2297 624.1624 624.1273 624.0145 623.8281

average and worst solutions of the proposed ARCGA are better than those of the other methods for the second test system, but also even the worst solution of the ARCGA is better than the best solution of all other methods of Table 4. Moreover, the worst, average and best solutions of the ARCGA for this test system are very close to each other indicating robustness of the proposed solution method to solve the ED problem. Detailed results, including generation output and fuel type, of the best solution of the ARCGA for this test system with various power demands are shown in Table 5. As seen from Tables 2 and 5 for the first and second test systems, respectively, summation of generation outputs obtained by the ARCGA in each case is equal to the corresponding power demand. Also, each generation output in Tables 2 and 5 is in its respective allowable range. In other words, the obtained ED solutions by the ARCGA satisfy both power balance constraint (2) and generation capacity limits (3). So, the ED solutions of the ARCGA are feasible. In order to show the effect of the adaptation process on the performance of the proposed solution method, the evolution of the ARCGA (with the adaptation process) and RCGA (without the adaptation process) for a random trial run in 100 generations is shown in Fig. 2. As seen, although the ARCGA begins from a worse random initial point than the RCGA, however the total cost of the ARCGA decreases much more quickly than the RCGA. In other words, the adaptation process improves the convergence behavior of the pro-

690 RCGA ARCGA

680

Total Cost ($/h)

670 660 650 640 630 620

0

10

20

30

40

50

60

70

80

90

100

Generation Number Fig. 2. The evolution of the proposed ARCGA (with the adaptation process) and RCGA (without the adaptation process).

posed ARCGA and therefore, better solution quality with lower computation burden can be obtained.

Table 5 Detailed results of the best solution of the ARCGA, including generation output and fuel type, for the second test system with different load levels. Unit

P (MW)

Fuel type

P (MW)

Fuel type

P (MW)

Fuel type

P (MW)

Fuel type

1 2 3 4 5 6 7 8 9 10

189.3053 202.5519 255.4481 232.9209 240.3603 232.6514 252.2638 233.0552 320.3954 241.0479

1 1 1 3 1 3 1 3 1 1

207.297 206.5129 267.5501 235.877 258.7206 235.3396 268.8669 235.7427 330.3599 253.7333

2 1 1 3 1 3 1 3 1 1

215.5124 210.2262 277.6316 238.2957 276.5234 238.9676 287.7275 238.1613 344.1996 272.7547

2 1 1 3 1 3 1 3 1 1

218.594 211.2166 280.6571 239.3707 279.9345 239.3707 287.7275 239.5051 427.7553 275.8685

2 1 1 3 1 3 1 3 3 1

P d (MW) Total cost ($/h)

2400 481.7434

2500 526.2589

2600 574.4054

2700 623.8281

N. Amjady, H. Nasiri-Rad / Expert Systems with Applications 37 (2010) 5239–5245 Table 6 CPU time comparison (‘–’ means the CPU time is not available in the respective reference). Methods

CGA-MU Chiang (2007) and Chiang (2005) PS Al-Sumait et al. (2007) IGA-MU Chiang (2007) and Chiang (2005) RGA Manoharan et al. (2008) PSO Manoharan et al. (2008) DE Manoharan et al. (2008) Proposed ARCGA

Mean time (s) First test system

Second test system

61.42 42.98 27.03 – – – 15.67

26.64 – 7.32 4.1340 3.3852 2.8236 0.85

In order to show the computational efficiency of the proposed ARCGA, its average CPU time for the two test systems are shown in Table 6 and compared with that of the other ED solution methods. This table shows less computation times of the proposed ARCGA than all other techniques. In other words, the proposed ARCGA not only can find better solutions for the ED problem, but also has lower computation burden than the other ED solution methods. 6. Conclusion In this paper, a new stochastic search technique is proposed to solve the nonconvex and nonsmooth ED problem including both valve loading effects and multiple fuel source options of units. The proposed ARCGA has new genetic operators including AABX crossover and BWM mutation. By the proposed operators, the ARCGA can benefit from both global and local search abilities. Also, to enhance the convergence behavior of the ARCGA, an adaptation process is incorporated in its evolution. To show the effectiveness of the proposed ARCGA to solve the nonconvex and nonsmooth ED problem, it is compared with several of most recently published ED solution methods. The results show that the proposed ARCGA has higher efficiency and robustness with lower computation burden to solve the practical ED problem. The research work is under way in order to incorporate security and emission constraints of the power system in the ED model. References Al-Sumait, J. S., AL-Othman, A. K., & Sykulski, J. K. (2007). Application of pattern search method to power system valve-point economic load dispatch. Electric Power and Energy System, 29, 720–730. Altun, H., & Yalcinoz, T. (2008). Implementing soft computing techniques to solve economic dispatch problem in power systems. Expert Systems with Applications, 35, 1668–1678. Babu, G. S. S., Das, D. B., & Patvardhan, C. (2008). Real-parameter quantum evolutionary algorithm for economic load dispatch. IET Generation, Transmission and Distribution, 2, 22–31. Chaturvedi, K. T., Pandit, M., & Srivastava, L. (2008). Self-organizing hierarchical particle swarm optimization for nonconvex economic dispatch. IEEE Transaction on Power Systems, 3, 1079–1087. Chiang, C.-L. (2005). Improved genetic algorithm for power economic dispatch of units with valve-point effects and multiple fuels. IEEE Transaction on Power Systems, 20, 1690–1699. Chiang, C.-L. (2007). Genetic-based algorithm for power economic load dispatch. IET Generation, Transmission and Distribution, 1, 261–269.

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