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Solution of economic dispatch problems by seeker optimization algorithm
Expert Systems with Applications 39 (2012) 508–519
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Solution of economic dispatch problems by seeker optimization algorithm Binod Shaw a, V. Mukherjee b, S.P. Ghoshal c,⇑ a
Department of Electrical Engineering, Asansol Engineering College, Asansol, West Bengal, India Department of Electrical Engineering, Indian School of Mines, Dhanbad, Jharkhand, India c Department of Electrical Engineering, National Institute of Technology, Durgapur, West Bengal, India b
a r t i c l e
i n f o
Keywords: Economic dispatch Multiple fuel options Seeker optimization algorithm Transmission loss Valve point loading
a b s t r a c t Seeker optimization algorithm (SOA), a novel heuristic population-based search algorithm, is utilized in this paper to solve different economic dispatch (ED) problems of thermal power units. In the SOA, the act of human searching capability and understanding are exploited for the purpose of optimization. In this algorithm, the search direction is based on empirical gradient by evaluating the response to the position changes and the step length is based on uncertainty reasoning by using a simple fuzzy rule. The effectiveness of the algorithm has been tested on four different small, as well as, large scale test power systems to solve the ED problems. The outcome of the present work is to establish the SOA as a promising alternative approach to solve the ED problems in practical power systems. Both the near-optimality of the solution and the convergence speed of the algorithm are promising. The results obtained are compared with those published in the recent literatures. Ó 2011 Published by Elsevier Ltd.
1. Introduction Economic dispatch (ED) allocates generations among the committed generating units in the most economical manner subject to different operational constraints (Wood & Wollenberg, 1984). To date, various investigations on ED problems have been undertaken as better solutions would result in more saving in the operating cost. Increased interests are being paid by the researchers towards the application of artificial intelligence technology for the solution of ED problems. Genetic algorithm (GA) (Chen & Chang, 1995; Gaing, 2003), artificial neural networks (Su & Lin, 2000), simulated annealing (SA) (Wong & Fung, 1993), Tabu search (Lin, Cheng, & Tsay, 2002), evolutionary programing (Jayabarathi, Sadasivam, & Ramachandran, 1999; Yang, Yang, & Huang, 1996), particle swarm optimization (PSO) (Kuo, 2008), ant colony optimization (ACO) (Pothiya, Ngamroo, & Kongprawechnon, 2010), differential evolution (Coelho & Mariani, 2006) are just a few among the numerous techniques adopted for this specific end. PSO is a global search technique originally introduced by Kennedy and Eberhart. It simulates the social evolvement knowledge, probing the optimum by evolving the population which includes candidate solutions. Compared with the other evolutionary algorithms, PSO shows incomparable advantages in searching speed and precision. In order to improve the global search ability, reducing the chance of getting trapped into local optimum in solving ⇑ Corresponding author. Tel.: +91 0326 2235644; fax: +91 0326 2296563. E-mail addresses: [email protected] (B. Shaw), vivek_agamani@ yahoo.com (V. Mukherjee), [email protected] (S.P. Ghoshal). 0957-4174/$ - see front matter Ó 2011 Published by Elsevier Ltd. doi:10.1016/j.eswa.2011.07.041
multimodal problems, many revised versions of PSO have appeared, mainly concentrating in improving the evolution implementations and exploring best parameters’ combinations. These revised and hybrid PSOs have successfully been applied in solving different ED problems by the researchers’ pool and have been reported in the literature time to time. Inspired by the mechanism of the survival of bacteria, e.g., Escherichia coli, an optimization algorithm, called bacterial foraging optimization (BFO) (Passino, 2002) has been developed. The BFO has been applied successfully in several kinds of power system optimization problems including ED (Panigrahi & Pandi, 2008). But this algorithm requires the proper setting of many parameters for its optimal performance. Also, the algorithm executes many different loops, and hence, takes tremendous time to converge to near-global solutions. In 2008, a new optimization algorithm based on biogeography has been proposed by Simon (2008). Biogeography is modeled after the process of natural immigration and emigration of species between islands in search of more friendly habitats. This method has been termed as biogeography-based optimization (BBO). In order to exploit the exploration and the exploitation capabilities of differential evolution (DE) and BBO, a hybrid technique combining DE with BBO, referred to as DE/BBO, has been proposed in the literature. In this method, the migration operator of the BBO along with mutation, crossover and selection operators of DE have been combined together to effectively utilize the goodness of both DE and BBO to enhance the convergence property and to improve the quality of the solution. Both BBO and DE/BBO have been successfully applied in solving ED problems (Bhattacharya & Chattopadhyay, 2010).
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Seeker optimization algorithm (SOA) (Dai, Chen, Zhu, & Zhang, 2009a, 2009b) is, essentially, a novel population based heuristic search algorithm. It is based on human understanding and searching capability for finding an optimum solution. In the SOA, optimum solution is regarded as one which is searched out by a seeker population. The underlying concept of the SOA is very easy to model and relatively easier than other optimization techniques prevailing in the literature. The highlighting characteristic features of this algorithm are the following: (a) Search direction and step length are directly used in this algorithm to update the position. (b) Proportional selection rule is applied for the calculation of search direction, which can improve the population diversity so as to boost the global search ability and decrease the number of control parameters making it simpler to implement. (c) Fuzzy reasoning is used to generate the step length because the uncertain reasoning of human searching could be best described by natural linguistic variables, and a simple if–else control rule. The present work focuses on the performance of the SOA as an optimizing tool in solving the ED problems. In view of the above, the main contribution of this paper can be summarized as follows: (1) Four test systems of the ED problem are solved with the SOA and the best results obtained are presented in this paper. (2) The best results obtained by adopting the SOA are compared with those published in the recent papers. (3) Considering the quality and the improved convergence speed of the solution as obtained, application of the SOA in solving the ED seems to be a promising alternative approach for practical power systems. Table 1 displays the brief descriptions of the four test systems adopted in this paper vis-à-vis different algorithms compared with SOA. The rest of the paper is organized as follows. In Section 2, mathematical modeling of ED problem is done. In Section 3, an objective function is formulated which requires to be optimized. The SOA is narrated in Section 4. Numerical example and simulation results are presented in Section 5 to demonstrate the performance of the algorithm for the ED problem. Section 6 focuses on conclusions of the present work and future research directions. 2. Mathematical modeling of the ED problem The prime objective of the ED problem is to minimize the total generation cost in power system (with an aim to deliver power to the end user at minimal cost) for a given load demand with due regard to the system equality and inequality constraints (Wood & Wollenberg, 1984). 2.1. ED with quadratic cost function and transmission loss The problem of ED is multimodal, non-differentiable and highly non-linear. Mathematically, the problem can be stated as in (1):
Min F T ðPÞ ¼
m X
F i ðPi Þ $=h;
i ¼ 1; . . . ; m;
ð1Þ
i¼1
subject to, (i) Real power balance constraint. The power balance operation can be modeled as in (2): m X i¼1
P i PD PL ¼ 0;
i ¼ 1; . . . ; m:
ð2Þ
The transmission loss (PL) is a function of active power generation of each generating unit for a given load demand (PD). It may be expressed as a quadratic function of generations (using B coefficient matrix) as given by (3):
PL ¼
m X m X i¼1
Pi Bij Pj þ
j¼1
i ¼ 1; . . . ; m;
m X
B0i P i þ B00 ;
i¼1
and j ¼ 1; . . . ; m;
ð3Þ
where Bij, (i j)th element of loss coefficient symmetric matrix, B; Bi0, ith element of the loss coefficient vector; and B00, constant loss coefficient. (ii) Generation capacity constraints. The generating capacity constraints are written as in (4):
P min 6 Pi 6 Pmax ; i i
i ¼ 1; . . . ; m:
ð4Þ
2.2. ED problem with valve point loading For simplicity, the generator cost function was mostly approximated by a single quadratic function (Lin & Viviani, 1984). However, because the cost function of a fossil fired plant, containing inflexions owing to valve point loading, is highly non-linear, and hence, the cost function is realistically denoted as a recurring rectified sinusoidal function (Sinha, Chakrabarti, & Chattopadhyay, 2003). Each generator has multi-valve steam turbines and cost functions comprising of very different input–output curves. To consider the valve-point loadings, a sinusoidal function is introduced into the quadratic function as given in (5):
F i ðPi Þ ¼ ai þ bi Pi þ ci P2i þ ei sin fi Pmin Pi $=h; i i ¼ 1; . . . ; m:
ð5Þ
Ripples are introduced in the input–output curves due to the valve-point loadings, and thereby, the number of local optima is increased. It is to be noted here that the fuel cost coefficients ei and fi are introduced in (5) to model valve point loadings. Variation of fuel cost Fi(Pi) due to valve-point loading with the change of generation value Pi is shown in Fig. 1. 2.3. ED problem with valve point loading and multiple fuel options To model an accurate and practical ED problem, both valve point loading effect and multiple fuel options are also taken into account in Chiang (2005). Units with multiple fuels option utilize ‘‘hybrid cost function’’. Each segment of the hybrid cost function bears some information about the fuel being burnt for the unit’s operation. The single continuous quadratic function of each unit is replaced by several piecewise quadratic functions that reflect the effects of fuel type changes and the generators must identify the most economic fuel to be burnt. To frame both valve point loading effect and multiple fuels, the cost function (Chiang, 2005) may be represented as in (6):
8 > > Pi1 ; ai1 þ bi1 Pi þ ci1 P2i þ ei1 sin fi1 Pmin > i1 > > > > > for fuel 1; Pmin 6 P i 6 Pi1 ; > > i > > > > > a þ b P þ c P2i þ ei2 sin fi2 Pmin Pi2 ; i2 i2 i i2 > i2 < F i ðPi Þ ¼ for fuel 2; Pi1 < Pi 6 Pi ; 2 > > > >. > . > > . > > > 2 min > > P a ; ik þ bik P i þ c ik P i þ eik sin fik P ik ik > > > > : max for fuel k; Pik1 < P i 6 Pi : ð6Þ
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Table 1 Brief descriptions of the four test systems vis-à-vis algorithms considered. Test system
System description
Test system 1
20-generating units without valve point loading
Test system 2
38-Generating units without valve point loading
Sl. no.
Name of the algorithm
Abbreviation used
Reference taken from
1 2 3 4
Biogeography-based optimization Lambda iteration method Hopfield model based approach Chaotic and Gaussian particle swarm optimization
BBO Li HM CGPSO
Bhattacharya and Chattopadhyay (2010a) Su and Lin (2000) Su and Lin (2000) Coelho and Lee (2008)
1
Hybrid differential evolution with biogeography-based optimization Biogeography-based optimization Particle swarm optimization with time varying acceleration coefficients New particle swarm optimization Particle swarm optimization with crazy Simple particle swarm optimization
DE/BBO
Bhattacharya and Chattopadhyay (2010b)
BBO PSO-TVAC
Bhattacharya and Chattopadhyay (2010a) Chaturvedi et al. (2009)
New PSO PSO-Crazy
Chaturvedi et al. (2009) Chaturvedi et al. (2009)
SPSO
Chaturvedi et al. (2009)
DE/BBO
Bhattacharya and Chattopadhyay (2010b)
BBO CCPSO
Bhattacharya and Chattopadhyay (2010a) Park et al. (2010)
QPSO
Meng et al. (2010)
ICA-PSO
Vlachogiannis and Lee (2009)
SOH-PSO
Chaturvedi et al. (2008)
NPSO-LRS
Selvakumar and Thanushkodi (2007)
PSO-LRS
Selvakumar and Thanushkodi (2007)
CBPSO-RVM
Lu et al. (2010)
RCGA IFEP
Amjady and Nasiri-Rad (2009) Sinha et al. (2003)
EP-SQP
Alsumait et al. (2010)
GA-PS-SQP
Alsumait et al. (2010)
BF-NM
Panigrahi and Pandi (2008)
CDE ACO
Coelho and Mariani (2006) Pothiya et al. (2010)
DE/BBO
Bhattacharya and Chattopadhyay (2010b)
BBO NPSO-LRS
Bhattacharya and Chattopadhyay (2010a) Selvakumar and Thanushkodi (2007)
NPSO PSO-LRS
Selvakumar and Thanushkodi (2007) Selvakumar and Thanushkodi (2007)
ACO RCGA IGA-GA
Pothiya et al. (2010) Amjady and Nasiri-Rad (2009) Chiang (2005)
CGA-MU
Chiang (2005)
2 3 4 5 6 Test system 3
40-Generating units with valve point loading
1 2 3 4 5 6 7 8 9
10 11 12 13
14 15 16 Test system 4
10-Generating units with valve point loading and multiple fuel options
1 2 3 4 5 6 7 8 9
Hybrid differential evolution with biogeography-based optimization Biogeography-based optimization PSO with both chaotic sequences and crossover operation Quantum-inspired particle swarm optimization Improved coordinated aggregationbased particle swarm optimization Self-organizing hierarchical particle swarm optimization New particle swarm optimization with local random search Particle swarm optimization with local random search Combined particle swarm optimization with real-valued mutation Real coded genetic algorithm Improved fast evolutionary programing Hybrid evolutionary programing, and sequential quadratic programing Hybrid genetic algorithm, pattern search, and sequential quadratic programing Bacterial foraging with Nelder–Mead algorithm Chaotic differential evolution Ant colony optimization Hybrid differential evolution with biogeography-based optimization Biogeography-based optimization New particle swarm optimization with local random search New particle swarm optimization Particle swarm optimization with local random search Ant colony optimization Real coded genetic algorithm Improved genetic algorithm with multiplier updating Conventional genetic algorithm with multiplier updating
3. Formulation of the objective function We must define the objective function OF() for evaluating the fitness of each individual in the population (Gaing, 2003). In order to emphasize the ‘‘best’’ solution and the speed up convergence of the iteration procedure, the evaluation value is normalized into the range between 0 and 1. The objective function is as given in (7). It is the reciprocal of the sum of generation cost function and power balance constraint. This implies if the values of Fi(Pi) and Ppbc (Pi) of individual Pi were small, then its evaluation value would be large:
OFðÞ ¼
1 ; F cost þ Ppbc
ð7Þ
where
Pm i¼1 F i ðP i Þ F min F cost ¼ 1 þ abs ; ðF max F min Þ !2 m X Pi PD PL ; Ppbc ¼ 1 þ i¼1
ð8Þ
ð9Þ
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bit entirely pro-group behavior. The population then exhibits a selforganized aggregation behavior of which the positive feedback usually takes the form of attraction towards a given signal source. Two optional altruistic directions may be modeled as in (11),(12):
~ g best ðtÞ ~ xi ðtÞÞ; di;alt1 ðtÞ ¼ signð~ ~ ~ di;alt2 ðtÞ ¼ sign lbest ðtÞ ~ xi ðtÞ :
Fig. 1. Input–output curve with valve-point loadings. a, b, c, d, e – valve points.
where Fmax, the maximum generation cost among all individuals in the initial population; and Fmin, the minimum generation cost among all individuals in the initial population. 4. Seeker optimization algorithm and its application to the ED problem 4.1. Seeker optimization algorithm SOA (Dai et al., 2009a, Dai, Chen, Zhu, & Zhang, 2009b) is a population-based heuristic search algorithm. It regards optimization process as an optimal solution obtained by a seeker population. Each individual of this population is called a seeker. The total population is randomly categorized into three subpopulations. These subpopulations search over several different domains of the search space. All the seekers in the same subpopulation constitute a neighborhood. This neighborhood represents the social component for the social sharing of information. 4.2. Steps of seeker optimization algorithm
ð12Þ
In (11),(12), ~ g best ðtÞ represents neighbors’ historical best position, ~l ðtÞ means neighbors’ current best position. best Moreover, seekers enjoy the properties of pro-activeness; seekers do not simply act in response to their environment; they are able to exhibit goal-directed behavior (Wooldridge & Jennings, 1995). In addition, the future behavior can be predicted and guided by the past behavior (Ajzen, 2002). As a result, the seeker may be pro-active to change his search direction and exhibit goal-directed behavior according to his past behavior. Hence, each seeker is associated with an empirical direction called as pro-activeness direction as given in (13):
~ di;pro ðtÞ ¼ signðx~i ðt1 Þ x~i ðt 2 ÞÞ;
ð13Þ
In (13), t1, t2 2 {t, t 1, t 2} and it is assumed that x~i ðt 1 Þ is better than x~i ðt 2 Þ. Aforementioned four empirical directions as presented in (10)–(13) direct human being to take a rational decision in his search direction. If the jth variable of the ith seeker goes towards the positive direction of the coordinate axis, dij(t) is taken as +1. If the jth variable of the ith seeker goes towards the negative direction of the coordinate axis, dij(t) is assumed as 1. The value of dij(t) is assumed as 0 if the ith seeker stays at the current position. Every variable j of ~ d ðtÞ is selected by applying the following proportional selection i
rule (shown in Fig. 2) as stated in (14):
8 ð0Þ 0; if r j 6 pj ; > > < ð0Þ ðþ1Þ dij ¼ þ1; if pð0Þ 6 r j 6 pj þ pj ; j > > : ð0Þ ðþ1Þ 1; if pj þ pj < r j 6 1;
ð14Þ
ðmÞ
In (14), rj is a uniform random number in [0, 1], pj ðm 2 f0; þ1; 1g
In SOA, a search direction dij(t) and a step length aij(t) are computed separately for each ith seeker on each jth variable at each time step t, where aij(t) P 0 and dij(t) 2 {1, 0, 1}. Here, i represents the population number and j represents the variable number to be optimized. (1) Calculation of search direction, dij(t): It is the natural tendency of the swarms to reciprocate in a cooperative manner while executing their needs and goals. Normally, there are two extreme types of cooperative behavior prevailing in swarm dynamics. One, egotistic, is entirely pro-self and another, altruistic, is entirely pro-group. Every seeker, as a single sophisticated agent, is uniformly egotistic (Eustace, Barnes, & Gray, 1993). He believes that he should go towards his historical best position according to his own judgment. This attitude of ith seeker may be modeled by an empirical direction vector ~ di;ego ðtÞ as shown in (10):
~ di;ego ðtÞ ¼ sign ~ pi;best ðtÞ ~ xi ðtÞ :
ð11Þ
is the percent of the numbers of ‘‘m’’ from the set {dij,ego, dij,alt1, dij,alt2, dij,pro} on each variable j of all the four empirical directions, ðmÞ
i.e., pj
¼ ðthe number of mÞ=4.
(2) Calculation of step length, aij(t): from the view point of human searching behavior, it is understood that one may find the near-optimal solutions in a narrower neighborhood of the point with lower fitness, value and on the other hand, in a wider neighborhood of the point with higher fitness value. A fuzzy system may be an ideal choice to represent the understanding and linguistic behavioral pattern of human searching tendency. Different optimization problems often have different ranges of fitness values. To design a fuzzy system to be applicable to a wide range of optimization problems, the fitness values of
ð10Þ
In (10), sign() is a signum function on each variable of the input vector. On the other hand, in altruistic behavior, seekers want to communicate with each other, cooperate explicitly, and adjust their behaviors in response to the other seeker in the same neighborhood region for achieving the desired goal. That means the seekers exhi-
Fig. 2. The proportional selection rule of search directions.
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B. Shaw et al. / Expert Systems with Applications 39 (2012) 508–519
all the seekers are sorted in descending manner (for minimization problem)/in ascending manner (for maximization problem) and turned into the sequence numbers from 1 to S as the inputs of fuzzy reasoning. The linear membership function is used in the conditional part since the universe of discourse is a given set of numbers, i.e., 1, 2, . . . , S. The expression is presented as in (15):
li ¼ lmax
S Ii l lmin : S 1 max
ð15Þ
In (15), Ii is the sequence number of ~ xi ðtÞ after sorting the fitness values, lmax is the maximum membership degree value which is equal to or a little less than 1.0. Here, the value of lmax is taken as 0.95. A fuzzy system works on the principle of control rule as ‘‘If {the conditional part}, then {the action part}. Bell membership function 2 lðxÞ ¼ ex2 =2d (shown in Fig. 3) is well utilized in the literature to represent the action part. For the convenience, one variable is considered. Thus, the membership degree values of the input variables beyond [ 3d, +3d] are less than 0.0111 (l(±3d) = 0.0111), and the elements beyond [3d, +3d] in the universe of discourse can be neglected for a linguistic atom (Li, 1998). Thus, the minimum value lmin = 0.0111 is set. Moreover, the parameter, ~ d of the Bell membership function is determined by (16):
~ d ¼ x absð~ xbest ~ xrand Þ:
ð16Þ
In (16), the absolute value of the input vector as the corresponding output vector is represented by the symbol abs(). The parameter x is used to decrease the step length with increasing time step so as to gradually improve the search precision. In the present experiments, x is linearly decreased from 0.9 to 0.1 during a run. The ~ xbest and ~ xrand are the best seeker and a randomly selected seeker respectively from the same subpopulation to which the ith seeker belongs. It is to be noted here that ~ xrand is different from ~ xbest and ~ d is shared by all the seekers in the same subpopulation. In order to introduce the randomness in each variable and to improve local search capability, the following equation is introduced to convert li into a vector ~ li with elements as given by (17):
Fig. 3. The action part of the fuzzy reasoning.
Start Real coded initialization of S seekers Set t = 0 Divide the population into K subpopulations randomly
lij ¼ RANDðli ; 1Þ:
ð17Þ
Calculate the objective function value for each seeker
Calculate the personal best position, neighborhood best position and population best position
Compute search direction for each seeker by using (14)
Compute step length for each seeker by using (18)
Update the position of each seeker Calculate the objective function value for each seeker Update the personal best position, neighborhood best position and population best position Subpopulations learn from each other by using (20)
Increment t = t + 1 No Meet stopping criterion? Yes Display the optimal fitness value and optimal solution
Stop Fig. 4. Flowchart of the seeker optimization algorithm.
In (17), RAND(li, 1) returns a uniformly random real number within [li, 1]. Eq. (18) denotes the action part of the fuzzy reasoning and gives the step length (aij) for every variable j:
Table 2 Implementation steps of the SOA algorithm for ED problems. Step 1 Initialization (a) Read input data (b) Read cost curves of machines and B coefficients (c) Set number of run counter (d) Set maximum population number of generator output parameter strings (e) Set lower and upper limits of each generator output (f) Read SOA parameters (g) Set termination criteria (i.e., maximum iteration cycles) Step 2 Initialize the positions of the seekers in the search space randomly and uniformly Step 3 Set the time step t = 0 Step 4 Compute the objective function of the initial positions. The initial historical best position among the population is achieved. Set the personal historical best position of each seeker to his current position Step 5 Let t = t + 1 Step 6 Select the neighbor of each seeker Step 7 Determine the search direction and step length for each seeker, and update his position Step 8 Update the position of each seeker Step 9 Compute the objective function for each seeker Step 10 Update the historical best position among the population and historical best position of each seeker Step 11 Subpopulation learns from each other Step 12 Repeat from Step 6 till the end of the maximum iteration cycles/stopping criterion Step 13 Determine the best string corresponding to optimum objective function value Step 14 Determine the optimal generation string corresponding to the grand optimum objective function value
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B. Shaw et al. / Expert Systems with Applications 39 (2012) 508–519 Table 3 Best results for the test system 1 with PD = 2500 MW.
a
Unit
SOA
BBO (Bhattacharya & Chattopadhyay, 2010a)
Li (Su & Lin, 2000)
HM (Su & Lin, 2000)
CGPSO (Coelho & Lee, 2008)
P1 P2 P3 P4 P5 P6 P7 P8 P9 P10 P11 P12 P13 P14 P15 P16 P17 P18 P19 P20
471.6036 100.9020 123.2770 82.7784 133.2635 43.2814 72.1406 119.1722 153.1067 80.5750 198.8873 444.3575 111.9436 59.1284 77.5241 61.9256 45.2780 58.4592 60.2692 41.3794
513.0892 173.3533 126.9231 103.3292 113.7741 73.06694 114.9843 116.4238 100.6948 99.99979 148.9770 294.0207 119.5754 30.54786 116.4546 36.22787 66.85943 88.54701 100.9802 54.2725
512.7805 169.1033 126.8898 102.8657 113.6386 73.5710 115.2878 116.3994 100.4062 106.0267 150.2394 292.7648 119.1154 30.8340 115.8057 36.2545 66.8590 87.9720 100.8033 54.3050
512.7804 169.1035 126.8897 102.8656 113.6836 73.5709 115.2876 116.3994 100.4063 106.0267 150.2395 292.7647 119.1155 30.8342 115.8056 36.2545 66.8590 87.9720 100.8033 54.3050
563.3155 106.5639 98.7093 117.3171 67.0781 51.4702 47.7261 82.4271 52.0884 106.5097 197.9428 488.3315 99.9464 79.8941 101.525 25.8380 70.0153 53.9530 65.4271 36.2552
Total power output (MW) Total transmission loss (MW) Power mismatch (MW) Total generation cost ($/h) Time/iteration(s)
2539.2530 39.1177 0.1348 59291 0.0234
2592.1011 92.1011 0 62456.77926 0.29282
2591.9670 91.9670 0.000187 62456.6391 0.033757
2591.9670 91.9669 0.000021 62456.6341 0.006355
2512.3343 12.3343 NRa 59804.0500 0.44
NR means not reported in the referred literature.
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
aij ¼ dj ln lij :
ð18Þ
(3) Updating of seekers’ position: in a population of sizeS, for each seeker i(1 6 i 6 S), the position update on each variable j is given by the following equation:
xij ðt þ 1Þ ¼ xij ðtÞ þ aij ðtÞ dij ðtÞ;
ð19Þ
4.3. Implementation of the SOA for the ED problem The steps of the SOA, as implemented for the solution of ED problem of this work, are shown in Table 2.
Table 4 Convergence results (50 trial runs) for the test system 1 with PD = 2500 MW. Algorithms
where xij(t + 1), the position of the jth variable of the ith seeker at time step t + 1; xij(t), the position of the jth variable of the ith seeker at time step t; aij(t), the step length of the jth variable of the ith seeker at time step t; and dij(t), the search direction of the jth variable of the ith seeker at time step t. (4) Subpopulations learn from each other: each subpopulation is searching for the optimal solution using its own information. It hints that the subpopulation may trap into local optima yielding a premature convergence. Subpopulations must learn from each other about the optimum information so far they have acquired in their respective domain. Thus, the position of the worst seeker of each subpopulation is combined with the best one in each of the other subpopulations using the following binomial crossover operator as expressed in (20):
xkn j;worst ¼
xlj;best;
if
xkn j;worst;
else
randj 6 0:5
:
BBO (Bhattacharya & Chattopadhyay, 2010a) Li (Su & Lin, 2000) HM (Su & Lin, 2000) CGPSO (Coelho & Lee, 2008) SOA a
Total generation cost ($/h) Minimum
Maximum
Average
62456.7926
62456.7935
62456.7928
62456.6391 62456.6341 59804.05 59291
a
NR NRa 63184.63 59520
NRa NRa 61171.84 59900
NR means not reported in the referred literature.
ð20Þ
In (20), randj is a uniformly random real number within [0, 1], xkn j;worst is denoted as the jth variable of the nth worst position in the kth subpopulation, xlj,best is the jth variable of the best position in the lth subpopulation. Here, n, k, l = 1, 2, . . . , K 1 and k – l. In order to increase the diversity in the population, good information acquired by each subpopulation is shared among the other subpopulations. The flowchart of the algorithm is depicted in Fig. 4.
Fig. 5. Convergence profile of the total generation cost for 20-generating units.
c d e f
Unit
SOA
DE/BBO (Bhattacharya & Chattopadhyay, 2010b)
BBO (Bhattacharya & Chattopadhyay, 2010a)
PSO-TVAC (Chaturvedi et al., 2009)
New-PSO (Chaturvedi et al., 2009)
P1 P2 P3 P4 P5 P6 P7 P8 P9 P10 P11 P12 P13 P14 P15 P16 P17 P18 P19 P20 P21 P22 P23 P24 P25 P26 P27 P28 P29 P30 P31 P32 P33 P34 P35 P36 P37 P38
398.4807 384.9040 338.4664 316.8407 271.2609 270.3114 259.7542 441.9667 193.5191 276.9221 203.3973 170.1541 229.5496 145.4604 253.1107 239.3751 100.5346 123.2491 109.7293 202.3414 188.1623 177.4071 104.8227 62.1276 79.1108 86.7702 46.7746 58.3767 39.5597 48.9387 58.8491 53.4548 45.2953 33.6909 18.6810 49.7400 28.3711 20.0464
426.606060 426.606054 429.663164 429.663181 429.663193 429.663164 429.663185 429.663168 114.000000 114.000000 119.768032 127.072817 110.000000 90.0000000 82.0000000 120.000000 159.598036 65.0000000 65.0000000 272.000000 272.000000 260.000000 130.648618 10.0000000 113.305034 88.0669159 37.5051018 20.0000000 20.0000000 20.0000000 20.0000000 20.0000000 25.0000000 18.0000000 8.00000000 25.0000000 21.7820891 21.0621792
422.230586 422.117933 435.779411 445.481950 428.475752 428.649254 428.115368 429.900663 115.904947 114.115368 115.418662 127.511404 110.000948 90.0217671 82.0000000 120.038496 160.303835 65.0001141 65.0001370 271.999591 271.872680 259.732054 125.993076 10.4134771 109.417723 89.3772664 36.4110655 20.0098880 20.0089554 20.0000000 20.0000000 20.0033959 25.0066586 18.0222107 8.00004260 25.0060660 22.0005641 20.6076309
443.659 342.956 433.117 500.00 410.539 492.864 409.483 446.079 119.566 137.274 138.933 155.401 121.719 90.924 97.941 128.106 189.108 65.00 65.00 267.422 221.383 130.804 124.269 11.535 77.103 55.018 75.000 21.682 29.829 20.326 20.000 21.840 25.620 24.261 9.667 25.000 31.642 29.935
550.000 512.263 485.733 391.083 443.846 358.398 415.729 320.816 115.347 204.422 114.000 249.197 118.886 102.802 89.039 120.000 156.562 84.265 65.041 151.104 226.344 209.298 85.719 10.000 60.000 90.489 39.670 20.000 20.995 22.810 20.000 20.416 25.000 21.319 9.122 25.184 20.000 25.104
TGa TTLb PMc TGCd TIe
6129.507 129.497 0.01 9.0940e + 06 0.19
NRf NRf NRf 9417235.78639 NRf
NRf NRf NRf 9417633.637644 NRf
NRf NRf NRf 9500448.307 NRf
NRf NRf NRf 9516448.312 NRf
TG means total generation (MW). TTL means total transmission loss (MW). PM means power mismatch (MW). TGC means total generation cost ($/h). TI means time/iteration. NR means not reported in the referred literature.
B. Shaw et al. / Expert Systems with Applications 39 (2012) 508–519
a b
514
Table 5 Best results for the test system 2 with PD = 6000 MW.
B. Shaw et al. / Expert Systems with Applications 39 (2012) 508–519
515
5. Test systems and solution results SOA has been applied to solve ED problems of four different test systems for investigating its optimization capability. The software has been written in MATLAB-7.3 language and executed on a 3.0GHz Pentium IV personal computer with 512-MB RAM. 5.1. Description of the test systems The following four test systems are considered in this work. (A) Test system 1: 20-generating units without valve point loading: A system with 20 generators is taken as the test system 1. The system input data are available in Su & Lin (2000); Coelho & Lee (2008). The valve point loading effect is not considered for this system but transmission loss is considered. For this test system load demand is 2500 MW. The results reported in the literature like BBO (Bhattacharya & Chattopadhyay, 2010a), Li (Su & Lin, 2000), HM (Su & Lin, 2000), and CGPSO (Coelho & Lee, 2008) are compared with the SOA-based results and the potential benefit of the SOA as an optimizing algorithm for this specific application is established. The best solutions of the generation schedules, the total optimal generation cost etc. as obtained from 50 trial runs of the SOA and the other afore-mentioned algorithms are presented in Table 3. Convergence results for the different algorithms with the same PD are also shown in Table 4. The convergence profile of the cost function is depicted in Fig. 5. (B) Test system 2: 38-generating units without valve point loading: A system with 38 generators is taken as the test system 2. Fuel cost characteristics are quadratic. Transmission loss is considered for this test system. The input data of the system are taken from Sydulu (1999). The load demand is 6000 MW. The best results obtained by using the SOA have been compared with those by using DE/BBO (Bhattacharya & Chattopadhyay, 2010b), BBO (Bhattacharya & Chattopadhyay, 2010a), PSO-TVAC (Chaturvedi, Pandit, & Srivastava, 2009), New PSO (Chaturvedi et al., 2009), PSO-Crazy (Chaturvedi et al., 2009), and SPSO (Chaturvedi et al., 2009). The best solutions of the generation schedules and the total generation cost etc. as obtained from 100 trial runs of the different algorithms are shown in Table 5. Convergence results for the algorithms with the same PD are also presented in Table 6. The convergence profile of the cost function is depicted in Fig. 6.
Table 6 Convergence results (100 trial runs) for the test system 2 with PD = 6000 MW. Algorithms
Total generation cost ($/h) Minimum
a
Maximum a
Average
SPSO (Chaturvedi et al., 2009) PSO-Crazy (Chaturvedi et al., 2009) New PSO (Chaturvedi et al., 2009) PSO-TVAC (Chaturvedi et al., 2009) BBO (Bhattacharya & Chattopadhyay, 2010a) DE/BBO (Bhattacharya & Chattopadhyay, 2010b)
9543984.777
NR
NRa
9520024.601
NRa
NRa
9516448.312
NRa
NRa
9500448.307
NRa
NRa
9417633.637
NRa
NRa
9417235.786
NRa
NRa
SOA
9.0940e + 006
9.1812e + 006
9.0912e + 006
NR means not reported in the referred literature.
Fig. 6. Convergence profile of the total generation cost for 38-generating units.
(C) Test system 3: 40-generating units with valve point loading: A system with 40 generators with valve point loadings is considered as the test system 3. The input data are given in Sinha et al. (2003). The load demand is 10500 MW. The best results obtained from the SOA are compared to those obtained by using DE/BBO (Bhattacharya & Chattopadhyay, 2010b), BBO (Bhattacharya & Chattopadhyay, 2010a), CCPSO (Park, Jeong, Shin, & Lee, 2010), QPSO (Meng, Wang, Dong, & Wong, 2010), ICA-PSO (Vlachogiannis & Lee, 2009), SOH-PSO (Chaturvedi, Pandit, & Srivastava, 2008), NPSO-LRS (Selvakumar & Thanushkodi, 2007), NPSO (Selvakumar & Thanushkodi, 2007), PSO-LRS (Selvakumar & Thanushkodi, 2007), CBPSO-RVM (Lu, Sriyanyong, Song, & Dillon, 2010), RCGA (Amjady & Nasiri-Rad, 2009), IFEP (Sinha et al., 2003), EPSQP (Alsumait, Sykulski, & Al-Othman, 2010), GA-PS-SQP (Alsumait et al., 2010), BF-NM (Panigrahi & Pandi, 2008), CDE (Coelho & Mariani, 2006), and ACO (Pothiya et al., 2010). The best solutions of the generation schedules and the total generation cost etc. as obtained from 50 trial runs are presented in Table 7. Convergence results for the different algorithms with the same PD are also presented in Table 8. Table 9 shows the frequency of attaining minimum cost within different ranges for this test system out of 50 independent trials. The convergence profile of the cost function is depicted in Fig. 7. (D) Test system 4: 10-generating units with valve point loading and multiple fuel options: A system comprising of 10 thermal units with valve point loading and multiple fuels option is considered as the test system 4. The input data are taken from Chiang (2005). The load demand is 2700 MW. Transmission loss is not considered for this case. The best results obtained by the SOA are compared to those obtained by the combined DE-BBO (Bhattacharya & Chattopadhyay, 2010b); BBO (Bhattacharya & Chattopadhyay, 2010a); different PSO techniques namely NPSO-LRS (Selvakumar & Thanushkodi, 2007), NPSO (Selvakumar & Thanushkodi, 2007), PSO-LRS (Selvakumar & Thanushkodi, 2007); ACO (Pothiya et al., 2010), and different GA methods like are RCGA (Amjady & Nasiri-Rad, 2009), IGA-MU (Chiang, 2005), CGA-MU (Chiang, 2005). The best solutions of the generation schedules and the total generation cost etc. as obtained from 100 trial runs of the algorithms are shown in Table 10. Convergence results for the algorithms with the same PD are presented in
Unit
c d e f
BBO (Bhattacharya & Chattopadhyay, 2010a)
CCPSO (Park et al., 2010)
QPSO (Meng et al., 2010)
ICA-PSO (Vlachogiannis & Lee, 2009)
SOH-PSO (Chaturvedi et al., 2008)
NPSO-LRS (Selvakumar & Thanushkodi, 2007)
104.5681 89.5109 102.6168 170.8105 81.9546 127.5261 245.8990 256.1526 259.9806 258.5663
110.7998 110.7998 97.3999 179.7331 87.9576 140.00 259.5997 284.5997 284.5997 130.00
111.0465 111.5915 97.60077 179.7095 88.30605 139.9992 259.6313 284.7366 284.7801 130.2484
110.7998 110.7999 97.3999 179.7331 87.7999 140.0000 259.5997 284.5997 284.5997 130.0000
111.20 111.70 97.40 179.73 90.14 140.00 259.60 284.80 284.84 130.00
110.80 110.80 97.41 179.74 88.52 140.00 259.60 284.60 284.60 130.00
110.80 110.80 97.40 179.73 87.80 140.00 259.60 284.60 284.60 130.00
113.9761 113.9986 97.4241 179.7327 89.6511 105.0444 259.7502 288.4534 284.6460 204.8120
P11 P12 P13 P14 P15 P16 P17 P18 P19 P20
279.0242 295.6307 414.2454 418.8108 399.6508 422.0261 407.6866 446.4102 486.8343 424.1161
168.7998 94.00 214.7598 394.2794 394.2794 304.5196 489.2794 489.2794 511.2794 511.2794
168.8461 168.8239 214.7038 304.5894 394.2461 394.2409 489.2919 489.4188 511.2997 511.3073
94.0000 94.0000 214.7598 394.2794 394.2794 394.2794 489.2794 489.2794 511.2794 511.2794
168.80 168.8 214.76 304.53 394.28 394.28 489.28 489.28 511.28 511.28
168.80 94.00 214.76 394.28 394.28 304.52 498.28 489.28 511.28 511.28
94.00 94.00 304.52 304.52 394.28 398.28 489.28 489.28 511.28 511.27
168.8311 94.0000 214.7663 394.2852 304.5187 394.2811 489.2807 489.2832 511.2845 511.3049
P21 P22 P23 P24 P25 P26 P27 P28 P29 P30
514.9760 520.6876 500.4430 485.1688 514.0632 512.7867 112.0454 135.0547 103.1277 81.5614
523.2794 523.2794 523.2794 523.2794 523.2794 523.2794 10.00 10.00 10.00 97.00
523.417 523.2795 523.3793 523.3225 523.3661 523.4262 10.05316 10.01135 10.00302 88.47754
523.2794 523.2794 523.2794 523.2794 523.2794 523.2794 10.0000 10.0000 10.0000 87.8000
523.28 523.28 523.29 523.28 523.29 523.28 10.01 10.01 10.00 88.47
523.28 523.28 523.28 523.28 523.28 523.28 10.00 10.00 10.00 96.39
523.28 523.28 523.28 523.28 523.28 523.28 10.00 10.00 10.00 97.00
523.2916 523.2853 523.2797 523.2994 523.2865 523.2936 10.0000 10.0001 10.0000 89.0139
P31 P32 P33 P34 P35 P36 P37 P38 P39 P40
160.4972 149.1974 147.2893 171.3613 179.2939 176.9985 85.5451 85.4917 81.6582 347.7701
190.00 190.00 190.00 164.7998 200.00 200.00 110.00 110.00 110.00 511.2794
189.9983 189.9881 189.9663 164.8054 165.1267 165.7695 109.9059 109.9971 109.9695 511.2794
190.0000 190.0000 190.0000 164.7998 194.3976 200.0000 110.0000 110.0000 110.0000 511.2794
190.00 190.00 190.00 164.91 165.36 167.19 110.00 107.01 110.00 511.36
190.00 190.00 190.00 164.82 200.00 200.00 110.00 110.00 110.00 511.28
190.00 190.00 190.00 185.20 164.80 200.00 110.00 110.00 110.00 511.28
190.0000 190.0000 190.0000 199.9998 165.1397 172.0275 110.0000 110.0000 93.0962 511.2996
NRf NRf NRf 121403.5362 NRf
NRf NRf NRf 121448.21 NRf
NRf NRf NRf 121413.2 0.22
NRf NRf NRf 121501.14 NRf
NRf NRf NRf 121664.4308 NRf
10757.04 257.03 0.01 114060 0.05
NRf NRf NRf 121420.89 0.06
TG means total generation (MW). TTL means total transmission loss (MW). PM means power mismatch (MW). TGC means total generation cost ($/h). TI means time/iteration. NR means not reported in the referred literature.
NRf NRf NRf 121426.95 0.11
B. Shaw et al. / Expert Systems with Applications 39 (2012) 508–519
a
DE/BBO (Bhattacharya & Chattopadhyay, 2010b)
P1 P2 P3 P4 P5 P6 P7 P8 P9 P10
TGa TTLb PMc TGCd TIe b
SOA
516
Table 7 Best results for the test system 3 with PD = 10500 MW.
517
B. Shaw et al. / Expert Systems with Applications 39 (2012) 508–519 Table 8 Convergence results (50 trial runs) for the test system 3 with PD = 10500 MW. Algorithms
Total generation cost ($/h)
IFEP (Sinha et al., 2003) EP-SQP (Alsumait et al., 2010) PSO-LRS (Selvakumar & Thanushkodi, 2007) CDE (Coelho & Mariani, 2006) NPSO (Selvakumar & Thanushkodi, 2007) NPSO-LRS (Selvakumar & Thanushkodi, 2007) CBPSO-RVM (Lu et al., 2010) ACO (Pothiya et al., 2010) SOH-PSO (Chaturvedi et al., 2008) GA-PS-SQP (Alsumait et al., 2010) QPSO (Meng et al., 2010) BBO (Bhattacharya & Chattopadhyay, 2010a) BF-NM (Panigrahi & Pandi, 2008) DE/BBO (Bhattacharya & Chattopadhyay, 2010b) RCGA (Amjady & Nasiri-Rad, 2009) ICA-PSO (Vlachogiannis & Lee, 2009) CCPSO (Park et al., 2010) SOA a
Minimum
Maximum
Average
122624.3500 122324 122035.7946 121741.9793 121704.7391 121664.4308 121555.32 121532.41 121501.14 121458.14 121448.21 121426.953 121423.63792 121420.8948 121418.5425 121413.20 121403.5362 114060
125740.6300 NRa 123461.6794 NRa 122995.0976 122981.5913 123094.98 121679.64 122446.30 NRa NRa 121688.6634 NRa 121420.8968 121628.5987 121453.56 121525.4934 116000
123382.0000 122379 122558.4565 121814.9465 122221.3697 122209.3186 122281.14 121606.45 121853.57 122039 122225.07 121508.0325 122295.1278 121420.8952 121504.1169 121428.14 121445.3269 114850
NR means not reported in the referred literature.
Table 9 Frequency of convergence in 50 trial runs for 40-generating units with PD = 10500 MW. Algorithms
SOA DE/BBO (Bhattacharya & Chattopadhyay, 2010b) BBO (Bhattacharya & Chattopadhyay, 2010a) QPSO (Meng et al., 2010) SOH-PSO (Chaturvedi et al., 2008) NPSO-LRS (Selvakumar & Thanushkodi, 2007) NPSO (Selvakumar & Thanushkodi, 2007) PSO-LRS (Selvakumar & Thanushkodi, 2007) CBPSO-RVM (Lu et al., 2010) IFEP (Sinha et al., 2003)
Range of total generation cost (103, $/h) <120.0
120.0– 121.5
121.5– 122.5
122.5– 123.0
50 0
0 50
0 0
0 0
0
38
12
0 0 0
2 0 0
0
123.0– 123.5
123.5– 124.0
124.0– 124.5
124.5– 125.0
125.0– 125.5
125.5– 126.0
>126.0
0 0
0 0
0 0
0 0
0 0
0 0
0 0
0
0
0
0
0
0
0
0
27 50 40
20 10
1 0 0
0 0 0
0 0 0
0 0 0
0 0 0
0 0 0
0 0 0
0
37
13
0
0
0
0
0
0
0
0
0
26
17
7
0
0
0
0
0
0
0 0
41 0
8 0
1 11
0 25
0 9
0 2
0 2
0 0
0 0
0 0
of 100 independent trials. The convergence profile of the cost function is depicted in Fig. 8. The results of interest are bold faced in the respective tables. Discussions on the results of the different test systems are made in the following subsection. 5.2. Discussions on the results of the test systems
Fig. 7. Convergence profile of the total generation cost for the 40-generating units.
Table 11. Table 12 shows the frequency of attaining minimum cost within different ranges for this test system out
(A) Solution quality: It is noticed from Tables 3, 5, 7, and 10 that the minimum cost achieved by applying the SOA is the least one as compared to those achieved by earlier reported algorithms as mentioned in the respective tables. It may also be noted from Tables 4, 6, 8, and 11 that the minimum, the maximum, and as well as, the average costs achieved by the SOA are the least once among all the other methods taken into consideration. It emphasizes on the fact that the SOA offers the best near-optimal solution for the ED problems considered. (B) Comparison of best generation costs: It may be observed from Tables 3, 5, 7, and 10 that the minimum costs achieved by the SOA based method for test systems 1–4, are 59291 $/h, 9.0940e + 06 $/h, 114060 $/h, and 598.1652 $/h, respectively. Again, power mismatches are the least ones in the
518
B. Shaw et al. / Expert Systems with Applications 39 (2012) 508–519
Table 10 Best results for the test system 4 with PD = 2700 MW. Unit
SOA
DE/BBO (Bhattacharya & Chattopadhyay, 2010b)
P1 P2 P3 P4 P5
174.5461 122.3390 408.7926 200.2129 330.2403
1 2 1 1 1
213.4589 209.4836 332.0000 238.0269 269.1423
2 1 3 3 1
212.9 209.4 332.0 238.3 269.2
2 1 3 3 1
223.335 212.195 276.216 239.418 274.647
2 1 1 3 1
219.126 211.164 280.657 238.477 276.417
2 1 1 3 1
P6 P7 P8 P9 P10
235.0207 384.7870 206.2955 343.1772 294.5888
2 1 1 3 1
238.0269 280.6144 238.1613 414.7001 266.3850
3 1 3 3 1
237.6 280.6 238.4 414.8 266.3
3 1 3 3 1
239.797 285.538 240.632 429.263 278.954
3 1 3 3 1
240.467 287.739 240.761 429.337 275.851
3 1 3 3 1
Generation
Total generation (MW) Total transmission loss (MW) Power mismatch (MW) Total generation cost ($/h) Time/iteration(s)
2700 0 0 598.1652 0.15
BBO (Bhattacharya & Chattopadhyay, 2010a)
Fuel type
Generation
2700 0 0 605.6230127 0.48
Fuel type
NPSO-LRS (Selvakumar & Thanushkodi, 2007) Generation
2700 0 0 605.6387 0.80
Fuel type
IGA-MU (Chiang, 2005) Generation
2700 0 0 624.127 0.52
Fuel type
2700 0 0 624.517 7.25
NR⁄ means not reported in the referred literature.
Table 11 Convergence results (100 trial runs) for the test system 4 with PD = 2700 MW. Algorithms
Total generation cost ($/h)
IGA-GA (Chiang, 2005) CGA-MU (Chiang, 2005) PSO-LRS (Selvakumar & Thanushkodi, 2007) NPSO (Selvakumar & Thanushkodi, 2007) NPSO-LRS (Selvakumar & Thanushkodi, 2007) RCGA (Amjady & Nasiri-Rad, 2009) ACO (Pothiya et al., 2010) BBO (Bhattacharya & Chattopadhyay, 2010a) DE/BBO (Bhattacharya & Chattopadhyay, 2010b) SOA
Minimum
Maximum
Average
627.5178 624.7193 624.2297 624.1624 624.1273 623.8281 623.70 605.6387 605.6230 598.1652
630.8705 633.8652 628.3214 627.4237 626.9981 623.8814 624.09 605.9103 605.6231 601.2624
625.8692 627.6087 625.7887 625.2180 624.9985 623.8495 623.90 605.8622 605.6252 600.0214
Table 12 Frequency of convergence in 100 trial runs for 10-generating units with PD = 2700 MW. Algorithms
SOA DE/BBO (Bhattacharya & Chattopadhyay, 2010b) BBO (Bhattacharya & Chattopadhyay, 2010a) NPSO-LRS (Selvakumar & Thanushkodi, 2007) NPSO (Selvakumar & Thanushkodi, 2007) PSO-LRS (Selvakumar & Thanushkodi, 2007) IGA-MU (Chiang, 2005) CGA-MU (Chiang, 2005)
Range of total generation cost ($/h) <605.5
605.5– 623.5
623.5– 624.5
624.5– 625.5
625.5– 626.5
629.5– 630.5
630.5– 631.5
631.5– 632.5
632.5– 633.5
633.5– 634.5
100 0
0 100
0 0
0 0
0 0
0 0
0 0
0 0
0 0
0 0
0 0
0 0
0 0
0
100
0
0
0
0
0
0
0
0
0
0
0
0
0
20
58
17
5
0
0
0
0
0
0
0
0
0
18
54
16
12
0
0
0
0
0
0
0
0
0
5
37
36
17
5
0
0
0
0
0
0
0 0
0 0
0 0
39 5
45 20
11 31
2 21
2 10
0 7
1 3
0 2
0 0
0 1
SOA as compared to those offered by the other algorithms. Hence, it can be concluded that for all the four test systems the performance of the SOA is found to be the best one. (C) Testing of robustness: The performance of any heuristic search based optimization algorithm is best judged through repetitive trial runs so as to compare the robustness/consistency of the algorithm. For this specific goal, the frequency of convergence to the minimum cost at different ranges of generation cost with fixed load demand is recorded and pre-
626.5– 627.5
627.5– 628.5
628.5– 629.5
sented in Table 9 for the test system 3 and in Table 12 those for the test system 4, respectively. The same for the test systems 1 and 2 are not included in the referred literatures, and hence, are not reported in the present work. But it is experimented by the authors of the present work with the SOA for the test systems 1 and 2 and it is noticed that the convergence to the minimum value of the cost function with a minimal variation is achieved. Tables 9 and 12 show that the frequency of converging to a better solution is higher
B. Shaw et al. / Expert Systems with Applications 39 (2012) 508–519
Fig. 8. Convergence profile of the total generation cost for 10-generating units.
in the SOA as compared to the other methods. Thus, it may be inferred that the SOA method is the most consistent in achieving the lowest cost in all the runs. (D) Computational efficiency: Apart from yielding minimum cost by the SOA, it may also be noted that the SOA yields minimum cost at comparatively lesser time of execution of the program. Thus, this approach is also efficient as far as the computational time is concerned. 6. Conclusion In this paper, the SOA has been successfully implemented to solve different ED problems. It has been observed that the SOA has the ability to converge to a better quality near-optimal solution and possesses better convergence characteristics and robustness than other prevailing techniques reported in the recent literatures. It is also clear from the results obtained by different trials that the SOA is free from the shortcoming of premature convergence exhibited by the other optimization algorithms. Thus, this algorithm may become very promising for solving some more complex engineering optimization problems for future researchers. References Ajzen, A. (2002). Residual effects of past on later behavior: Habituation and reasoned action perspectives. Personality and Social Psychology Review, 6(2), 107–122. Alsumait, J. S., Sykulski, J. K., & Al-Othman, A. K. (2010). A hybrid GA-PS-SQP method to solve power system valve point economic dispatch problems. Applied Energy, 87(5), 1773–1781. Amjady, N., & Nasiri-Rad, H. (2009). Nonconvex economic dispatch with AC constraints by a new real coded genetic algorithm. IEEE Transactions on Power Systems, 24(3), 1489–1502. Bhattacharya, A., & Chattopadhyay, P. K. (2010a). Biogeography-based optimization for different economic load dispatch problems. IEEE Transactions on Power Systems, 25(2), 1064–1077. Bhattacharya, A., & Chattopadhyay, P. K. (2010b). Hybrid differential evolution with biogeography-based optimization for solution of economic load dispatch. IEEE Transactions on Power Systems, 25(4), 1955–1964. Chaturvedi, K. T., Pandit, M., & Srivastava, L. (2008). Self-organizing hierarchical particle swarm optimization for nonconvex economic dispatch. IEEE Transactions on Power Systems, 23(3), 1079–1087. Chaturvedi, K. T., Pandit, M., & Srivastava, L. (2009). Particle swarm optimization with time varying acceleration coefficients for non-convex economic power dispatch. International Journal of Electrical Power & Energy Systems, 31(6), 249–257. Chen, P. H., & Chang, H. C. (1995). Large-scale economic dispatch by genetic algorithm. IEEE Transactions on Power Systems, 10(4), 1919–1926.
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(2002). An improved Tabu search for economic dispatch with multiple minima. IEEE Transactions on Power Systems, 17(1), 108–112. Lin, C. E., & Viviani, G. L. (1984). Hierarchical economic dispatch for piecewise quadratic cost functions. IEEE Transactions on Power Apparatus and Systems, 103(6), 1170–1175. Lu, H., Sriyanyong, P., Song, Y. H., & Dillon, T. (2010). Experimental study of a new hybrid PSO with mutation for economic dispatch with non-smooth cost function. International Journal of Electrical Power & Energy System, 32(9), 921–935. Meng, Ke., Wang, H. G., Dong, Z. Y., & Wong, Kit Po. (2010). Quantum-inspired particle swarm optimization for valve-point economic load dispatch. IEEE Transactions on Power Systems, 25(1), 215–222. Panigrahi, B. K., & Pandi, V. R. (2008). Bacterial foraging optimisation: Nelder–Mead hybrid algorithm for economic load dispatch. IET Proceedings Generation Transmission and Distribution, 2(4), 556–565. Park, J.-B., Jeong, Y.-W., Shin, J.-R., & Lee, K. Y. (2010). An improved particle swarm optimization for non-convex economic dispatch problems. IEEE Transactions on Power Systems, 25(1), 156–166. Passino, K. M. (2002). Biomimicry of bacterial foraging for distributed optimization and control. IEEE Transactions on Control Systems, 22(3), 52–67. Pothiya, S., Ngamroo, I., & Kongprawechnon, W. (2010). Ant colony optimization for economic dispatch problem with non-smooth cost functions. International Journal of Electrical Power & Energy Systems, 32(5), 478–487. Selvakumar, A. I., & Thanushkodi, K. (2007). A new particle swarm optimization solution to non-convex economic dispatch problems. IEEE Transactions on Power Systems, 22(1), 42–51. Simon, D. (2008). Biogeography-based optimization. IEEE Transactions on Evolutionary Computation, 12(6), 702–713. Sinha, N., Chakrabarti, R., & Chattopadhyay, P. K. (2003). Evolutionary programming techniques for economic load dispatch. IEEE Transactions on Evolutionary Computation, 7(1), 83–94. Su, C. 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Contents lists available at ScienceDirect
Expert Systems with Applications journal homepage: www.elsevier.com/locate/eswa
Solution of economic dispatch problems by seeker optimization algorithm Binod Shaw a, V. Mukherjee b, S.P. Ghoshal c,⇑ a
Department of Electrical Engineering, Asansol Engineering College, Asansol, West Bengal, India Department of Electrical Engineering, Indian School of Mines, Dhanbad, Jharkhand, India c Department of Electrical Engineering, National Institute of Technology, Durgapur, West Bengal, India b
a r t i c l e
i n f o
Keywords: Economic dispatch Multiple fuel options Seeker optimization algorithm Transmission loss Valve point loading
a b s t r a c t Seeker optimization algorithm (SOA), a novel heuristic population-based search algorithm, is utilized in this paper to solve different economic dispatch (ED) problems of thermal power units. In the SOA, the act of human searching capability and understanding are exploited for the purpose of optimization. In this algorithm, the search direction is based on empirical gradient by evaluating the response to the position changes and the step length is based on uncertainty reasoning by using a simple fuzzy rule. The effectiveness of the algorithm has been tested on four different small, as well as, large scale test power systems to solve the ED problems. The outcome of the present work is to establish the SOA as a promising alternative approach to solve the ED problems in practical power systems. Both the near-optimality of the solution and the convergence speed of the algorithm are promising. The results obtained are compared with those published in the recent literatures. Ó 2011 Published by Elsevier Ltd.
1. Introduction Economic dispatch (ED) allocates generations among the committed generating units in the most economical manner subject to different operational constraints (Wood & Wollenberg, 1984). To date, various investigations on ED problems have been undertaken as better solutions would result in more saving in the operating cost. Increased interests are being paid by the researchers towards the application of artificial intelligence technology for the solution of ED problems. Genetic algorithm (GA) (Chen & Chang, 1995; Gaing, 2003), artificial neural networks (Su & Lin, 2000), simulated annealing (SA) (Wong & Fung, 1993), Tabu search (Lin, Cheng, & Tsay, 2002), evolutionary programing (Jayabarathi, Sadasivam, & Ramachandran, 1999; Yang, Yang, & Huang, 1996), particle swarm optimization (PSO) (Kuo, 2008), ant colony optimization (ACO) (Pothiya, Ngamroo, & Kongprawechnon, 2010), differential evolution (Coelho & Mariani, 2006) are just a few among the numerous techniques adopted for this specific end. PSO is a global search technique originally introduced by Kennedy and Eberhart. It simulates the social evolvement knowledge, probing the optimum by evolving the population which includes candidate solutions. Compared with the other evolutionary algorithms, PSO shows incomparable advantages in searching speed and precision. In order to improve the global search ability, reducing the chance of getting trapped into local optimum in solving ⇑ Corresponding author. Tel.: +91 0326 2235644; fax: +91 0326 2296563. E-mail addresses: [email protected] (B. Shaw), vivek_agamani@ yahoo.com (V. Mukherjee), [email protected] (S.P. Ghoshal). 0957-4174/$ - see front matter Ó 2011 Published by Elsevier Ltd. doi:10.1016/j.eswa.2011.07.041
multimodal problems, many revised versions of PSO have appeared, mainly concentrating in improving the evolution implementations and exploring best parameters’ combinations. These revised and hybrid PSOs have successfully been applied in solving different ED problems by the researchers’ pool and have been reported in the literature time to time. Inspired by the mechanism of the survival of bacteria, e.g., Escherichia coli, an optimization algorithm, called bacterial foraging optimization (BFO) (Passino, 2002) has been developed. The BFO has been applied successfully in several kinds of power system optimization problems including ED (Panigrahi & Pandi, 2008). But this algorithm requires the proper setting of many parameters for its optimal performance. Also, the algorithm executes many different loops, and hence, takes tremendous time to converge to near-global solutions. In 2008, a new optimization algorithm based on biogeography has been proposed by Simon (2008). Biogeography is modeled after the process of natural immigration and emigration of species between islands in search of more friendly habitats. This method has been termed as biogeography-based optimization (BBO). In order to exploit the exploration and the exploitation capabilities of differential evolution (DE) and BBO, a hybrid technique combining DE with BBO, referred to as DE/BBO, has been proposed in the literature. In this method, the migration operator of the BBO along with mutation, crossover and selection operators of DE have been combined together to effectively utilize the goodness of both DE and BBO to enhance the convergence property and to improve the quality of the solution. Both BBO and DE/BBO have been successfully applied in solving ED problems (Bhattacharya & Chattopadhyay, 2010).
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Seeker optimization algorithm (SOA) (Dai, Chen, Zhu, & Zhang, 2009a, 2009b) is, essentially, a novel population based heuristic search algorithm. It is based on human understanding and searching capability for finding an optimum solution. In the SOA, optimum solution is regarded as one which is searched out by a seeker population. The underlying concept of the SOA is very easy to model and relatively easier than other optimization techniques prevailing in the literature. The highlighting characteristic features of this algorithm are the following: (a) Search direction and step length are directly used in this algorithm to update the position. (b) Proportional selection rule is applied for the calculation of search direction, which can improve the population diversity so as to boost the global search ability and decrease the number of control parameters making it simpler to implement. (c) Fuzzy reasoning is used to generate the step length because the uncertain reasoning of human searching could be best described by natural linguistic variables, and a simple if–else control rule. The present work focuses on the performance of the SOA as an optimizing tool in solving the ED problems. In view of the above, the main contribution of this paper can be summarized as follows: (1) Four test systems of the ED problem are solved with the SOA and the best results obtained are presented in this paper. (2) The best results obtained by adopting the SOA are compared with those published in the recent papers. (3) Considering the quality and the improved convergence speed of the solution as obtained, application of the SOA in solving the ED seems to be a promising alternative approach for practical power systems. Table 1 displays the brief descriptions of the four test systems adopted in this paper vis-à-vis different algorithms compared with SOA. The rest of the paper is organized as follows. In Section 2, mathematical modeling of ED problem is done. In Section 3, an objective function is formulated which requires to be optimized. The SOA is narrated in Section 4. Numerical example and simulation results are presented in Section 5 to demonstrate the performance of the algorithm for the ED problem. Section 6 focuses on conclusions of the present work and future research directions. 2. Mathematical modeling of the ED problem The prime objective of the ED problem is to minimize the total generation cost in power system (with an aim to deliver power to the end user at minimal cost) for a given load demand with due regard to the system equality and inequality constraints (Wood & Wollenberg, 1984). 2.1. ED with quadratic cost function and transmission loss The problem of ED is multimodal, non-differentiable and highly non-linear. Mathematically, the problem can be stated as in (1):
Min F T ðPÞ ¼
m X
F i ðPi Þ $=h;
i ¼ 1; . . . ; m;
ð1Þ
i¼1
subject to, (i) Real power balance constraint. The power balance operation can be modeled as in (2): m X i¼1
P i PD PL ¼ 0;
i ¼ 1; . . . ; m:
ð2Þ
The transmission loss (PL) is a function of active power generation of each generating unit for a given load demand (PD). It may be expressed as a quadratic function of generations (using B coefficient matrix) as given by (3):
PL ¼
m X m X i¼1
Pi Bij Pj þ
j¼1
i ¼ 1; . . . ; m;
m X
B0i P i þ B00 ;
i¼1
and j ¼ 1; . . . ; m;
ð3Þ
where Bij, (i j)th element of loss coefficient symmetric matrix, B; Bi0, ith element of the loss coefficient vector; and B00, constant loss coefficient. (ii) Generation capacity constraints. The generating capacity constraints are written as in (4):
P min 6 Pi 6 Pmax ; i i
i ¼ 1; . . . ; m:
ð4Þ
2.2. ED problem with valve point loading For simplicity, the generator cost function was mostly approximated by a single quadratic function (Lin & Viviani, 1984). However, because the cost function of a fossil fired plant, containing inflexions owing to valve point loading, is highly non-linear, and hence, the cost function is realistically denoted as a recurring rectified sinusoidal function (Sinha, Chakrabarti, & Chattopadhyay, 2003). Each generator has multi-valve steam turbines and cost functions comprising of very different input–output curves. To consider the valve-point loadings, a sinusoidal function is introduced into the quadratic function as given in (5):
F i ðPi Þ ¼ ai þ bi Pi þ ci P2i þ ei sin fi Pmin Pi $=h; i i ¼ 1; . . . ; m:
ð5Þ
Ripples are introduced in the input–output curves due to the valve-point loadings, and thereby, the number of local optima is increased. It is to be noted here that the fuel cost coefficients ei and fi are introduced in (5) to model valve point loadings. Variation of fuel cost Fi(Pi) due to valve-point loading with the change of generation value Pi is shown in Fig. 1. 2.3. ED problem with valve point loading and multiple fuel options To model an accurate and practical ED problem, both valve point loading effect and multiple fuel options are also taken into account in Chiang (2005). Units with multiple fuels option utilize ‘‘hybrid cost function’’. Each segment of the hybrid cost function bears some information about the fuel being burnt for the unit’s operation. The single continuous quadratic function of each unit is replaced by several piecewise quadratic functions that reflect the effects of fuel type changes and the generators must identify the most economic fuel to be burnt. To frame both valve point loading effect and multiple fuels, the cost function (Chiang, 2005) may be represented as in (6):
8 > > Pi1 ; ai1 þ bi1 Pi þ ci1 P2i þ ei1 sin fi1 Pmin > i1 > > > > > for fuel 1; Pmin 6 P i 6 Pi1 ; > > i > > > > > a þ b P þ c P2i þ ei2 sin fi2 Pmin Pi2 ; i2 i2 i i2 > i2 < F i ðPi Þ ¼ for fuel 2; Pi1 < Pi 6 Pi ; 2 > > > >. > . > > . > > > 2 min > > P a ; ik þ bik P i þ c ik P i þ eik sin fik P ik ik > > > > : max for fuel k; Pik1 < P i 6 Pi : ð6Þ
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Table 1 Brief descriptions of the four test systems vis-à-vis algorithms considered. Test system
System description
Test system 1
20-generating units without valve point loading
Test system 2
38-Generating units without valve point loading
Sl. no.
Name of the algorithm
Abbreviation used
Reference taken from
1 2 3 4
Biogeography-based optimization Lambda iteration method Hopfield model based approach Chaotic and Gaussian particle swarm optimization
BBO Li HM CGPSO
Bhattacharya and Chattopadhyay (2010a) Su and Lin (2000) Su and Lin (2000) Coelho and Lee (2008)
1
Hybrid differential evolution with biogeography-based optimization Biogeography-based optimization Particle swarm optimization with time varying acceleration coefficients New particle swarm optimization Particle swarm optimization with crazy Simple particle swarm optimization
DE/BBO
Bhattacharya and Chattopadhyay (2010b)
BBO PSO-TVAC
Bhattacharya and Chattopadhyay (2010a) Chaturvedi et al. (2009)
New PSO PSO-Crazy
Chaturvedi et al. (2009) Chaturvedi et al. (2009)
SPSO
Chaturvedi et al. (2009)
DE/BBO
Bhattacharya and Chattopadhyay (2010b)
BBO CCPSO
Bhattacharya and Chattopadhyay (2010a) Park et al. (2010)
QPSO
Meng et al. (2010)
ICA-PSO
Vlachogiannis and Lee (2009)
SOH-PSO
Chaturvedi et al. (2008)
NPSO-LRS
Selvakumar and Thanushkodi (2007)
PSO-LRS
Selvakumar and Thanushkodi (2007)
CBPSO-RVM
Lu et al. (2010)
RCGA IFEP
Amjady and Nasiri-Rad (2009) Sinha et al. (2003)
EP-SQP
Alsumait et al. (2010)
GA-PS-SQP
Alsumait et al. (2010)
BF-NM
Panigrahi and Pandi (2008)
CDE ACO
Coelho and Mariani (2006) Pothiya et al. (2010)
DE/BBO
Bhattacharya and Chattopadhyay (2010b)
BBO NPSO-LRS
Bhattacharya and Chattopadhyay (2010a) Selvakumar and Thanushkodi (2007)
NPSO PSO-LRS
Selvakumar and Thanushkodi (2007) Selvakumar and Thanushkodi (2007)
ACO RCGA IGA-GA
Pothiya et al. (2010) Amjady and Nasiri-Rad (2009) Chiang (2005)
CGA-MU
Chiang (2005)
2 3 4 5 6 Test system 3
40-Generating units with valve point loading
1 2 3 4 5 6 7 8 9
10 11 12 13
14 15 16 Test system 4
10-Generating units with valve point loading and multiple fuel options
1 2 3 4 5 6 7 8 9
Hybrid differential evolution with biogeography-based optimization Biogeography-based optimization PSO with both chaotic sequences and crossover operation Quantum-inspired particle swarm optimization Improved coordinated aggregationbased particle swarm optimization Self-organizing hierarchical particle swarm optimization New particle swarm optimization with local random search Particle swarm optimization with local random search Combined particle swarm optimization with real-valued mutation Real coded genetic algorithm Improved fast evolutionary programing Hybrid evolutionary programing, and sequential quadratic programing Hybrid genetic algorithm, pattern search, and sequential quadratic programing Bacterial foraging with Nelder–Mead algorithm Chaotic differential evolution Ant colony optimization Hybrid differential evolution with biogeography-based optimization Biogeography-based optimization New particle swarm optimization with local random search New particle swarm optimization Particle swarm optimization with local random search Ant colony optimization Real coded genetic algorithm Improved genetic algorithm with multiplier updating Conventional genetic algorithm with multiplier updating
3. Formulation of the objective function We must define the objective function OF() for evaluating the fitness of each individual in the population (Gaing, 2003). In order to emphasize the ‘‘best’’ solution and the speed up convergence of the iteration procedure, the evaluation value is normalized into the range between 0 and 1. The objective function is as given in (7). It is the reciprocal of the sum of generation cost function and power balance constraint. This implies if the values of Fi(Pi) and Ppbc (Pi) of individual Pi were small, then its evaluation value would be large:
OFðÞ ¼
1 ; F cost þ Ppbc
ð7Þ
where
Pm i¼1 F i ðP i Þ F min F cost ¼ 1 þ abs ; ðF max F min Þ !2 m X Pi PD PL ; Ppbc ¼ 1 þ i¼1
ð8Þ
ð9Þ
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bit entirely pro-group behavior. The population then exhibits a selforganized aggregation behavior of which the positive feedback usually takes the form of attraction towards a given signal source. Two optional altruistic directions may be modeled as in (11),(12):
~ g best ðtÞ ~ xi ðtÞÞ; di;alt1 ðtÞ ¼ signð~ ~ ~ di;alt2 ðtÞ ¼ sign lbest ðtÞ ~ xi ðtÞ :
Fig. 1. Input–output curve with valve-point loadings. a, b, c, d, e – valve points.
where Fmax, the maximum generation cost among all individuals in the initial population; and Fmin, the minimum generation cost among all individuals in the initial population. 4. Seeker optimization algorithm and its application to the ED problem 4.1. Seeker optimization algorithm SOA (Dai et al., 2009a, Dai, Chen, Zhu, & Zhang, 2009b) is a population-based heuristic search algorithm. It regards optimization process as an optimal solution obtained by a seeker population. Each individual of this population is called a seeker. The total population is randomly categorized into three subpopulations. These subpopulations search over several different domains of the search space. All the seekers in the same subpopulation constitute a neighborhood. This neighborhood represents the social component for the social sharing of information. 4.2. Steps of seeker optimization algorithm
ð12Þ
In (11),(12), ~ g best ðtÞ represents neighbors’ historical best position, ~l ðtÞ means neighbors’ current best position. best Moreover, seekers enjoy the properties of pro-activeness; seekers do not simply act in response to their environment; they are able to exhibit goal-directed behavior (Wooldridge & Jennings, 1995). In addition, the future behavior can be predicted and guided by the past behavior (Ajzen, 2002). As a result, the seeker may be pro-active to change his search direction and exhibit goal-directed behavior according to his past behavior. Hence, each seeker is associated with an empirical direction called as pro-activeness direction as given in (13):
~ di;pro ðtÞ ¼ signðx~i ðt1 Þ x~i ðt 2 ÞÞ;
ð13Þ
In (13), t1, t2 2 {t, t 1, t 2} and it is assumed that x~i ðt 1 Þ is better than x~i ðt 2 Þ. Aforementioned four empirical directions as presented in (10)–(13) direct human being to take a rational decision in his search direction. If the jth variable of the ith seeker goes towards the positive direction of the coordinate axis, dij(t) is taken as +1. If the jth variable of the ith seeker goes towards the negative direction of the coordinate axis, dij(t) is assumed as 1. The value of dij(t) is assumed as 0 if the ith seeker stays at the current position. Every variable j of ~ d ðtÞ is selected by applying the following proportional selection i
rule (shown in Fig. 2) as stated in (14):
8 ð0Þ 0; if r j 6 pj ; > > < ð0Þ ðþ1Þ dij ¼ þ1; if pð0Þ 6 r j 6 pj þ pj ; j > > : ð0Þ ðþ1Þ 1; if pj þ pj < r j 6 1;
ð14Þ
ðmÞ
In (14), rj is a uniform random number in [0, 1], pj ðm 2 f0; þ1; 1g
In SOA, a search direction dij(t) and a step length aij(t) are computed separately for each ith seeker on each jth variable at each time step t, where aij(t) P 0 and dij(t) 2 {1, 0, 1}. Here, i represents the population number and j represents the variable number to be optimized. (1) Calculation of search direction, dij(t): It is the natural tendency of the swarms to reciprocate in a cooperative manner while executing their needs and goals. Normally, there are two extreme types of cooperative behavior prevailing in swarm dynamics. One, egotistic, is entirely pro-self and another, altruistic, is entirely pro-group. Every seeker, as a single sophisticated agent, is uniformly egotistic (Eustace, Barnes, & Gray, 1993). He believes that he should go towards his historical best position according to his own judgment. This attitude of ith seeker may be modeled by an empirical direction vector ~ di;ego ðtÞ as shown in (10):
~ di;ego ðtÞ ¼ sign ~ pi;best ðtÞ ~ xi ðtÞ :
ð11Þ
is the percent of the numbers of ‘‘m’’ from the set {dij,ego, dij,alt1, dij,alt2, dij,pro} on each variable j of all the four empirical directions, ðmÞ
i.e., pj
¼ ðthe number of mÞ=4.
(2) Calculation of step length, aij(t): from the view point of human searching behavior, it is understood that one may find the near-optimal solutions in a narrower neighborhood of the point with lower fitness, value and on the other hand, in a wider neighborhood of the point with higher fitness value. A fuzzy system may be an ideal choice to represent the understanding and linguistic behavioral pattern of human searching tendency. Different optimization problems often have different ranges of fitness values. To design a fuzzy system to be applicable to a wide range of optimization problems, the fitness values of
ð10Þ
In (10), sign() is a signum function on each variable of the input vector. On the other hand, in altruistic behavior, seekers want to communicate with each other, cooperate explicitly, and adjust their behaviors in response to the other seeker in the same neighborhood region for achieving the desired goal. That means the seekers exhi-
Fig. 2. The proportional selection rule of search directions.
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all the seekers are sorted in descending manner (for minimization problem)/in ascending manner (for maximization problem) and turned into the sequence numbers from 1 to S as the inputs of fuzzy reasoning. The linear membership function is used in the conditional part since the universe of discourse is a given set of numbers, i.e., 1, 2, . . . , S. The expression is presented as in (15):
li ¼ lmax
S Ii l lmin : S 1 max
ð15Þ
In (15), Ii is the sequence number of ~ xi ðtÞ after sorting the fitness values, lmax is the maximum membership degree value which is equal to or a little less than 1.0. Here, the value of lmax is taken as 0.95. A fuzzy system works on the principle of control rule as ‘‘If {the conditional part}, then {the action part}. Bell membership function 2 lðxÞ ¼ ex2 =2d (shown in Fig. 3) is well utilized in the literature to represent the action part. For the convenience, one variable is considered. Thus, the membership degree values of the input variables beyond [ 3d, +3d] are less than 0.0111 (l(±3d) = 0.0111), and the elements beyond [3d, +3d] in the universe of discourse can be neglected for a linguistic atom (Li, 1998). Thus, the minimum value lmin = 0.0111 is set. Moreover, the parameter, ~ d of the Bell membership function is determined by (16):
~ d ¼ x absð~ xbest ~ xrand Þ:
ð16Þ
In (16), the absolute value of the input vector as the corresponding output vector is represented by the symbol abs(). The parameter x is used to decrease the step length with increasing time step so as to gradually improve the search precision. In the present experiments, x is linearly decreased from 0.9 to 0.1 during a run. The ~ xbest and ~ xrand are the best seeker and a randomly selected seeker respectively from the same subpopulation to which the ith seeker belongs. It is to be noted here that ~ xrand is different from ~ xbest and ~ d is shared by all the seekers in the same subpopulation. In order to introduce the randomness in each variable and to improve local search capability, the following equation is introduced to convert li into a vector ~ li with elements as given by (17):
Fig. 3. The action part of the fuzzy reasoning.
Start Real coded initialization of S seekers Set t = 0 Divide the population into K subpopulations randomly
lij ¼ RANDðli ; 1Þ:
ð17Þ
Calculate the objective function value for each seeker
Calculate the personal best position, neighborhood best position and population best position
Compute search direction for each seeker by using (14)
Compute step length for each seeker by using (18)
Update the position of each seeker Calculate the objective function value for each seeker Update the personal best position, neighborhood best position and population best position Subpopulations learn from each other by using (20)
Increment t = t + 1 No Meet stopping criterion? Yes Display the optimal fitness value and optimal solution
Stop Fig. 4. Flowchart of the seeker optimization algorithm.
In (17), RAND(li, 1) returns a uniformly random real number within [li, 1]. Eq. (18) denotes the action part of the fuzzy reasoning and gives the step length (aij) for every variable j:
Table 2 Implementation steps of the SOA algorithm for ED problems. Step 1 Initialization (a) Read input data (b) Read cost curves of machines and B coefficients (c) Set number of run counter (d) Set maximum population number of generator output parameter strings (e) Set lower and upper limits of each generator output (f) Read SOA parameters (g) Set termination criteria (i.e., maximum iteration cycles) Step 2 Initialize the positions of the seekers in the search space randomly and uniformly Step 3 Set the time step t = 0 Step 4 Compute the objective function of the initial positions. The initial historical best position among the population is achieved. Set the personal historical best position of each seeker to his current position Step 5 Let t = t + 1 Step 6 Select the neighbor of each seeker Step 7 Determine the search direction and step length for each seeker, and update his position Step 8 Update the position of each seeker Step 9 Compute the objective function for each seeker Step 10 Update the historical best position among the population and historical best position of each seeker Step 11 Subpopulation learns from each other Step 12 Repeat from Step 6 till the end of the maximum iteration cycles/stopping criterion Step 13 Determine the best string corresponding to optimum objective function value Step 14 Determine the optimal generation string corresponding to the grand optimum objective function value
513
B. Shaw et al. / Expert Systems with Applications 39 (2012) 508–519 Table 3 Best results for the test system 1 with PD = 2500 MW.
a
Unit
SOA
BBO (Bhattacharya & Chattopadhyay, 2010a)
Li (Su & Lin, 2000)
HM (Su & Lin, 2000)
CGPSO (Coelho & Lee, 2008)
P1 P2 P3 P4 P5 P6 P7 P8 P9 P10 P11 P12 P13 P14 P15 P16 P17 P18 P19 P20
471.6036 100.9020 123.2770 82.7784 133.2635 43.2814 72.1406 119.1722 153.1067 80.5750 198.8873 444.3575 111.9436 59.1284 77.5241 61.9256 45.2780 58.4592 60.2692 41.3794
513.0892 173.3533 126.9231 103.3292 113.7741 73.06694 114.9843 116.4238 100.6948 99.99979 148.9770 294.0207 119.5754 30.54786 116.4546 36.22787 66.85943 88.54701 100.9802 54.2725
512.7805 169.1033 126.8898 102.8657 113.6386 73.5710 115.2878 116.3994 100.4062 106.0267 150.2394 292.7648 119.1154 30.8340 115.8057 36.2545 66.8590 87.9720 100.8033 54.3050
512.7804 169.1035 126.8897 102.8656 113.6836 73.5709 115.2876 116.3994 100.4063 106.0267 150.2395 292.7647 119.1155 30.8342 115.8056 36.2545 66.8590 87.9720 100.8033 54.3050
563.3155 106.5639 98.7093 117.3171 67.0781 51.4702 47.7261 82.4271 52.0884 106.5097 197.9428 488.3315 99.9464 79.8941 101.525 25.8380 70.0153 53.9530 65.4271 36.2552
Total power output (MW) Total transmission loss (MW) Power mismatch (MW) Total generation cost ($/h) Time/iteration(s)
2539.2530 39.1177 0.1348 59291 0.0234
2592.1011 92.1011 0 62456.77926 0.29282
2591.9670 91.9670 0.000187 62456.6391 0.033757
2591.9670 91.9669 0.000021 62456.6341 0.006355
2512.3343 12.3343 NRa 59804.0500 0.44
NR means not reported in the referred literature.
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
aij ¼ dj ln lij :
ð18Þ
(3) Updating of seekers’ position: in a population of sizeS, for each seeker i(1 6 i 6 S), the position update on each variable j is given by the following equation:
xij ðt þ 1Þ ¼ xij ðtÞ þ aij ðtÞ dij ðtÞ;
ð19Þ
4.3. Implementation of the SOA for the ED problem The steps of the SOA, as implemented for the solution of ED problem of this work, are shown in Table 2.
Table 4 Convergence results (50 trial runs) for the test system 1 with PD = 2500 MW. Algorithms
where xij(t + 1), the position of the jth variable of the ith seeker at time step t + 1; xij(t), the position of the jth variable of the ith seeker at time step t; aij(t), the step length of the jth variable of the ith seeker at time step t; and dij(t), the search direction of the jth variable of the ith seeker at time step t. (4) Subpopulations learn from each other: each subpopulation is searching for the optimal solution using its own information. It hints that the subpopulation may trap into local optima yielding a premature convergence. Subpopulations must learn from each other about the optimum information so far they have acquired in their respective domain. Thus, the position of the worst seeker of each subpopulation is combined with the best one in each of the other subpopulations using the following binomial crossover operator as expressed in (20):
xkn j;worst ¼
xlj;best;
if
xkn j;worst;
else
randj 6 0:5
:
BBO (Bhattacharya & Chattopadhyay, 2010a) Li (Su & Lin, 2000) HM (Su & Lin, 2000) CGPSO (Coelho & Lee, 2008) SOA a
Total generation cost ($/h) Minimum
Maximum
Average
62456.7926
62456.7935
62456.7928
62456.6391 62456.6341 59804.05 59291
a
NR NRa 63184.63 59520
NRa NRa 61171.84 59900
NR means not reported in the referred literature.
ð20Þ
In (20), randj is a uniformly random real number within [0, 1], xkn j;worst is denoted as the jth variable of the nth worst position in the kth subpopulation, xlj,best is the jth variable of the best position in the lth subpopulation. Here, n, k, l = 1, 2, . . . , K 1 and k – l. In order to increase the diversity in the population, good information acquired by each subpopulation is shared among the other subpopulations. The flowchart of the algorithm is depicted in Fig. 4.
Fig. 5. Convergence profile of the total generation cost for 20-generating units.
c d e f
Unit
SOA
DE/BBO (Bhattacharya & Chattopadhyay, 2010b)
BBO (Bhattacharya & Chattopadhyay, 2010a)
PSO-TVAC (Chaturvedi et al., 2009)
New-PSO (Chaturvedi et al., 2009)
P1 P2 P3 P4 P5 P6 P7 P8 P9 P10 P11 P12 P13 P14 P15 P16 P17 P18 P19 P20 P21 P22 P23 P24 P25 P26 P27 P28 P29 P30 P31 P32 P33 P34 P35 P36 P37 P38
398.4807 384.9040 338.4664 316.8407 271.2609 270.3114 259.7542 441.9667 193.5191 276.9221 203.3973 170.1541 229.5496 145.4604 253.1107 239.3751 100.5346 123.2491 109.7293 202.3414 188.1623 177.4071 104.8227 62.1276 79.1108 86.7702 46.7746 58.3767 39.5597 48.9387 58.8491 53.4548 45.2953 33.6909 18.6810 49.7400 28.3711 20.0464
426.606060 426.606054 429.663164 429.663181 429.663193 429.663164 429.663185 429.663168 114.000000 114.000000 119.768032 127.072817 110.000000 90.0000000 82.0000000 120.000000 159.598036 65.0000000 65.0000000 272.000000 272.000000 260.000000 130.648618 10.0000000 113.305034 88.0669159 37.5051018 20.0000000 20.0000000 20.0000000 20.0000000 20.0000000 25.0000000 18.0000000 8.00000000 25.0000000 21.7820891 21.0621792
422.230586 422.117933 435.779411 445.481950 428.475752 428.649254 428.115368 429.900663 115.904947 114.115368 115.418662 127.511404 110.000948 90.0217671 82.0000000 120.038496 160.303835 65.0001141 65.0001370 271.999591 271.872680 259.732054 125.993076 10.4134771 109.417723 89.3772664 36.4110655 20.0098880 20.0089554 20.0000000 20.0000000 20.0033959 25.0066586 18.0222107 8.00004260 25.0060660 22.0005641 20.6076309
443.659 342.956 433.117 500.00 410.539 492.864 409.483 446.079 119.566 137.274 138.933 155.401 121.719 90.924 97.941 128.106 189.108 65.00 65.00 267.422 221.383 130.804 124.269 11.535 77.103 55.018 75.000 21.682 29.829 20.326 20.000 21.840 25.620 24.261 9.667 25.000 31.642 29.935
550.000 512.263 485.733 391.083 443.846 358.398 415.729 320.816 115.347 204.422 114.000 249.197 118.886 102.802 89.039 120.000 156.562 84.265 65.041 151.104 226.344 209.298 85.719 10.000 60.000 90.489 39.670 20.000 20.995 22.810 20.000 20.416 25.000 21.319 9.122 25.184 20.000 25.104
TGa TTLb PMc TGCd TIe
6129.507 129.497 0.01 9.0940e + 06 0.19
NRf NRf NRf 9417235.78639 NRf
NRf NRf NRf 9417633.637644 NRf
NRf NRf NRf 9500448.307 NRf
NRf NRf NRf 9516448.312 NRf
TG means total generation (MW). TTL means total transmission loss (MW). PM means power mismatch (MW). TGC means total generation cost ($/h). TI means time/iteration. NR means not reported in the referred literature.
B. Shaw et al. / Expert Systems with Applications 39 (2012) 508–519
a b
514
Table 5 Best results for the test system 2 with PD = 6000 MW.
B. Shaw et al. / Expert Systems with Applications 39 (2012) 508–519
515
5. Test systems and solution results SOA has been applied to solve ED problems of four different test systems for investigating its optimization capability. The software has been written in MATLAB-7.3 language and executed on a 3.0GHz Pentium IV personal computer with 512-MB RAM. 5.1. Description of the test systems The following four test systems are considered in this work. (A) Test system 1: 20-generating units without valve point loading: A system with 20 generators is taken as the test system 1. The system input data are available in Su & Lin (2000); Coelho & Lee (2008). The valve point loading effect is not considered for this system but transmission loss is considered. For this test system load demand is 2500 MW. The results reported in the literature like BBO (Bhattacharya & Chattopadhyay, 2010a), Li (Su & Lin, 2000), HM (Su & Lin, 2000), and CGPSO (Coelho & Lee, 2008) are compared with the SOA-based results and the potential benefit of the SOA as an optimizing algorithm for this specific application is established. The best solutions of the generation schedules, the total optimal generation cost etc. as obtained from 50 trial runs of the SOA and the other afore-mentioned algorithms are presented in Table 3. Convergence results for the different algorithms with the same PD are also shown in Table 4. The convergence profile of the cost function is depicted in Fig. 5. (B) Test system 2: 38-generating units without valve point loading: A system with 38 generators is taken as the test system 2. Fuel cost characteristics are quadratic. Transmission loss is considered for this test system. The input data of the system are taken from Sydulu (1999). The load demand is 6000 MW. The best results obtained by using the SOA have been compared with those by using DE/BBO (Bhattacharya & Chattopadhyay, 2010b), BBO (Bhattacharya & Chattopadhyay, 2010a), PSO-TVAC (Chaturvedi, Pandit, & Srivastava, 2009), New PSO (Chaturvedi et al., 2009), PSO-Crazy (Chaturvedi et al., 2009), and SPSO (Chaturvedi et al., 2009). The best solutions of the generation schedules and the total generation cost etc. as obtained from 100 trial runs of the different algorithms are shown in Table 5. Convergence results for the algorithms with the same PD are also presented in Table 6. The convergence profile of the cost function is depicted in Fig. 6.
Table 6 Convergence results (100 trial runs) for the test system 2 with PD = 6000 MW. Algorithms
Total generation cost ($/h) Minimum
a
Maximum a
Average
SPSO (Chaturvedi et al., 2009) PSO-Crazy (Chaturvedi et al., 2009) New PSO (Chaturvedi et al., 2009) PSO-TVAC (Chaturvedi et al., 2009) BBO (Bhattacharya & Chattopadhyay, 2010a) DE/BBO (Bhattacharya & Chattopadhyay, 2010b)
9543984.777
NR
NRa
9520024.601
NRa
NRa
9516448.312
NRa
NRa
9500448.307
NRa
NRa
9417633.637
NRa
NRa
9417235.786
NRa
NRa
SOA
9.0940e + 006
9.1812e + 006
9.0912e + 006
NR means not reported in the referred literature.
Fig. 6. Convergence profile of the total generation cost for 38-generating units.
(C) Test system 3: 40-generating units with valve point loading: A system with 40 generators with valve point loadings is considered as the test system 3. The input data are given in Sinha et al. (2003). The load demand is 10500 MW. The best results obtained from the SOA are compared to those obtained by using DE/BBO (Bhattacharya & Chattopadhyay, 2010b), BBO (Bhattacharya & Chattopadhyay, 2010a), CCPSO (Park, Jeong, Shin, & Lee, 2010), QPSO (Meng, Wang, Dong, & Wong, 2010), ICA-PSO (Vlachogiannis & Lee, 2009), SOH-PSO (Chaturvedi, Pandit, & Srivastava, 2008), NPSO-LRS (Selvakumar & Thanushkodi, 2007), NPSO (Selvakumar & Thanushkodi, 2007), PSO-LRS (Selvakumar & Thanushkodi, 2007), CBPSO-RVM (Lu, Sriyanyong, Song, & Dillon, 2010), RCGA (Amjady & Nasiri-Rad, 2009), IFEP (Sinha et al., 2003), EPSQP (Alsumait, Sykulski, & Al-Othman, 2010), GA-PS-SQP (Alsumait et al., 2010), BF-NM (Panigrahi & Pandi, 2008), CDE (Coelho & Mariani, 2006), and ACO (Pothiya et al., 2010). The best solutions of the generation schedules and the total generation cost etc. as obtained from 50 trial runs are presented in Table 7. Convergence results for the different algorithms with the same PD are also presented in Table 8. Table 9 shows the frequency of attaining minimum cost within different ranges for this test system out of 50 independent trials. The convergence profile of the cost function is depicted in Fig. 7. (D) Test system 4: 10-generating units with valve point loading and multiple fuel options: A system comprising of 10 thermal units with valve point loading and multiple fuels option is considered as the test system 4. The input data are taken from Chiang (2005). The load demand is 2700 MW. Transmission loss is not considered for this case. The best results obtained by the SOA are compared to those obtained by the combined DE-BBO (Bhattacharya & Chattopadhyay, 2010b); BBO (Bhattacharya & Chattopadhyay, 2010a); different PSO techniques namely NPSO-LRS (Selvakumar & Thanushkodi, 2007), NPSO (Selvakumar & Thanushkodi, 2007), PSO-LRS (Selvakumar & Thanushkodi, 2007); ACO (Pothiya et al., 2010), and different GA methods like are RCGA (Amjady & Nasiri-Rad, 2009), IGA-MU (Chiang, 2005), CGA-MU (Chiang, 2005). The best solutions of the generation schedules and the total generation cost etc. as obtained from 100 trial runs of the algorithms are shown in Table 10. Convergence results for the algorithms with the same PD are presented in
Unit
c d e f
BBO (Bhattacharya & Chattopadhyay, 2010a)
CCPSO (Park et al., 2010)
QPSO (Meng et al., 2010)
ICA-PSO (Vlachogiannis & Lee, 2009)
SOH-PSO (Chaturvedi et al., 2008)
NPSO-LRS (Selvakumar & Thanushkodi, 2007)
104.5681 89.5109 102.6168 170.8105 81.9546 127.5261 245.8990 256.1526 259.9806 258.5663
110.7998 110.7998 97.3999 179.7331 87.9576 140.00 259.5997 284.5997 284.5997 130.00
111.0465 111.5915 97.60077 179.7095 88.30605 139.9992 259.6313 284.7366 284.7801 130.2484
110.7998 110.7999 97.3999 179.7331 87.7999 140.0000 259.5997 284.5997 284.5997 130.0000
111.20 111.70 97.40 179.73 90.14 140.00 259.60 284.80 284.84 130.00
110.80 110.80 97.41 179.74 88.52 140.00 259.60 284.60 284.60 130.00
110.80 110.80 97.40 179.73 87.80 140.00 259.60 284.60 284.60 130.00
113.9761 113.9986 97.4241 179.7327 89.6511 105.0444 259.7502 288.4534 284.6460 204.8120
P11 P12 P13 P14 P15 P16 P17 P18 P19 P20
279.0242 295.6307 414.2454 418.8108 399.6508 422.0261 407.6866 446.4102 486.8343 424.1161
168.7998 94.00 214.7598 394.2794 394.2794 304.5196 489.2794 489.2794 511.2794 511.2794
168.8461 168.8239 214.7038 304.5894 394.2461 394.2409 489.2919 489.4188 511.2997 511.3073
94.0000 94.0000 214.7598 394.2794 394.2794 394.2794 489.2794 489.2794 511.2794 511.2794
168.80 168.8 214.76 304.53 394.28 394.28 489.28 489.28 511.28 511.28
168.80 94.00 214.76 394.28 394.28 304.52 498.28 489.28 511.28 511.28
94.00 94.00 304.52 304.52 394.28 398.28 489.28 489.28 511.28 511.27
168.8311 94.0000 214.7663 394.2852 304.5187 394.2811 489.2807 489.2832 511.2845 511.3049
P21 P22 P23 P24 P25 P26 P27 P28 P29 P30
514.9760 520.6876 500.4430 485.1688 514.0632 512.7867 112.0454 135.0547 103.1277 81.5614
523.2794 523.2794 523.2794 523.2794 523.2794 523.2794 10.00 10.00 10.00 97.00
523.417 523.2795 523.3793 523.3225 523.3661 523.4262 10.05316 10.01135 10.00302 88.47754
523.2794 523.2794 523.2794 523.2794 523.2794 523.2794 10.0000 10.0000 10.0000 87.8000
523.28 523.28 523.29 523.28 523.29 523.28 10.01 10.01 10.00 88.47
523.28 523.28 523.28 523.28 523.28 523.28 10.00 10.00 10.00 96.39
523.28 523.28 523.28 523.28 523.28 523.28 10.00 10.00 10.00 97.00
523.2916 523.2853 523.2797 523.2994 523.2865 523.2936 10.0000 10.0001 10.0000 89.0139
P31 P32 P33 P34 P35 P36 P37 P38 P39 P40
160.4972 149.1974 147.2893 171.3613 179.2939 176.9985 85.5451 85.4917 81.6582 347.7701
190.00 190.00 190.00 164.7998 200.00 200.00 110.00 110.00 110.00 511.2794
189.9983 189.9881 189.9663 164.8054 165.1267 165.7695 109.9059 109.9971 109.9695 511.2794
190.0000 190.0000 190.0000 164.7998 194.3976 200.0000 110.0000 110.0000 110.0000 511.2794
190.00 190.00 190.00 164.91 165.36 167.19 110.00 107.01 110.00 511.36
190.00 190.00 190.00 164.82 200.00 200.00 110.00 110.00 110.00 511.28
190.00 190.00 190.00 185.20 164.80 200.00 110.00 110.00 110.00 511.28
190.0000 190.0000 190.0000 199.9998 165.1397 172.0275 110.0000 110.0000 93.0962 511.2996
NRf NRf NRf 121403.5362 NRf
NRf NRf NRf 121448.21 NRf
NRf NRf NRf 121413.2 0.22
NRf NRf NRf 121501.14 NRf
NRf NRf NRf 121664.4308 NRf
10757.04 257.03 0.01 114060 0.05
NRf NRf NRf 121420.89 0.06
TG means total generation (MW). TTL means total transmission loss (MW). PM means power mismatch (MW). TGC means total generation cost ($/h). TI means time/iteration. NR means not reported in the referred literature.
NRf NRf NRf 121426.95 0.11
B. Shaw et al. / Expert Systems with Applications 39 (2012) 508–519
a
DE/BBO (Bhattacharya & Chattopadhyay, 2010b)
P1 P2 P3 P4 P5 P6 P7 P8 P9 P10
TGa TTLb PMc TGCd TIe b
SOA
516
Table 7 Best results for the test system 3 with PD = 10500 MW.
517
B. Shaw et al. / Expert Systems with Applications 39 (2012) 508–519 Table 8 Convergence results (50 trial runs) for the test system 3 with PD = 10500 MW. Algorithms
Total generation cost ($/h)
IFEP (Sinha et al., 2003) EP-SQP (Alsumait et al., 2010) PSO-LRS (Selvakumar & Thanushkodi, 2007) CDE (Coelho & Mariani, 2006) NPSO (Selvakumar & Thanushkodi, 2007) NPSO-LRS (Selvakumar & Thanushkodi, 2007) CBPSO-RVM (Lu et al., 2010) ACO (Pothiya et al., 2010) SOH-PSO (Chaturvedi et al., 2008) GA-PS-SQP (Alsumait et al., 2010) QPSO (Meng et al., 2010) BBO (Bhattacharya & Chattopadhyay, 2010a) BF-NM (Panigrahi & Pandi, 2008) DE/BBO (Bhattacharya & Chattopadhyay, 2010b) RCGA (Amjady & Nasiri-Rad, 2009) ICA-PSO (Vlachogiannis & Lee, 2009) CCPSO (Park et al., 2010) SOA a
Minimum
Maximum
Average
122624.3500 122324 122035.7946 121741.9793 121704.7391 121664.4308 121555.32 121532.41 121501.14 121458.14 121448.21 121426.953 121423.63792 121420.8948 121418.5425 121413.20 121403.5362 114060
125740.6300 NRa 123461.6794 NRa 122995.0976 122981.5913 123094.98 121679.64 122446.30 NRa NRa 121688.6634 NRa 121420.8968 121628.5987 121453.56 121525.4934 116000
123382.0000 122379 122558.4565 121814.9465 122221.3697 122209.3186 122281.14 121606.45 121853.57 122039 122225.07 121508.0325 122295.1278 121420.8952 121504.1169 121428.14 121445.3269 114850
NR means not reported in the referred literature.
Table 9 Frequency of convergence in 50 trial runs for 40-generating units with PD = 10500 MW. Algorithms
SOA DE/BBO (Bhattacharya & Chattopadhyay, 2010b) BBO (Bhattacharya & Chattopadhyay, 2010a) QPSO (Meng et al., 2010) SOH-PSO (Chaturvedi et al., 2008) NPSO-LRS (Selvakumar & Thanushkodi, 2007) NPSO (Selvakumar & Thanushkodi, 2007) PSO-LRS (Selvakumar & Thanushkodi, 2007) CBPSO-RVM (Lu et al., 2010) IFEP (Sinha et al., 2003)
Range of total generation cost (103, $/h) <120.0
120.0– 121.5
121.5– 122.5
122.5– 123.0
50 0
0 50
0 0
0 0
0
38
12
0 0 0
2 0 0
0
123.0– 123.5
123.5– 124.0
124.0– 124.5
124.5– 125.0
125.0– 125.5
125.5– 126.0
>126.0
0 0
0 0
0 0
0 0
0 0
0 0
0 0
0
0
0
0
0
0
0
0
27 50 40
20 10
1 0 0
0 0 0
0 0 0
0 0 0
0 0 0
0 0 0
0 0 0
0
37
13
0
0
0
0
0
0
0
0
0
26
17
7
0
0
0
0
0
0
0 0
41 0
8 0
1 11
0 25
0 9
0 2
0 2
0 0
0 0
0 0
of 100 independent trials. The convergence profile of the cost function is depicted in Fig. 8. The results of interest are bold faced in the respective tables. Discussions on the results of the different test systems are made in the following subsection. 5.2. Discussions on the results of the test systems
Fig. 7. Convergence profile of the total generation cost for the 40-generating units.
Table 11. Table 12 shows the frequency of attaining minimum cost within different ranges for this test system out
(A) Solution quality: It is noticed from Tables 3, 5, 7, and 10 that the minimum cost achieved by applying the SOA is the least one as compared to those achieved by earlier reported algorithms as mentioned in the respective tables. It may also be noted from Tables 4, 6, 8, and 11 that the minimum, the maximum, and as well as, the average costs achieved by the SOA are the least once among all the other methods taken into consideration. It emphasizes on the fact that the SOA offers the best near-optimal solution for the ED problems considered. (B) Comparison of best generation costs: It may be observed from Tables 3, 5, 7, and 10 that the minimum costs achieved by the SOA based method for test systems 1–4, are 59291 $/h, 9.0940e + 06 $/h, 114060 $/h, and 598.1652 $/h, respectively. Again, power mismatches are the least ones in the
518
B. Shaw et al. / Expert Systems with Applications 39 (2012) 508–519
Table 10 Best results for the test system 4 with PD = 2700 MW. Unit
SOA
DE/BBO (Bhattacharya & Chattopadhyay, 2010b)
P1 P2 P3 P4 P5
174.5461 122.3390 408.7926 200.2129 330.2403
1 2 1 1 1
213.4589 209.4836 332.0000 238.0269 269.1423
2 1 3 3 1
212.9 209.4 332.0 238.3 269.2
2 1 3 3 1
223.335 212.195 276.216 239.418 274.647
2 1 1 3 1
219.126 211.164 280.657 238.477 276.417
2 1 1 3 1
P6 P7 P8 P9 P10
235.0207 384.7870 206.2955 343.1772 294.5888
2 1 1 3 1
238.0269 280.6144 238.1613 414.7001 266.3850
3 1 3 3 1
237.6 280.6 238.4 414.8 266.3
3 1 3 3 1
239.797 285.538 240.632 429.263 278.954
3 1 3 3 1
240.467 287.739 240.761 429.337 275.851
3 1 3 3 1
Generation
Total generation (MW) Total transmission loss (MW) Power mismatch (MW) Total generation cost ($/h) Time/iteration(s)
2700 0 0 598.1652 0.15
BBO (Bhattacharya & Chattopadhyay, 2010a)
Fuel type
Generation
2700 0 0 605.6230127 0.48
Fuel type
NPSO-LRS (Selvakumar & Thanushkodi, 2007) Generation
2700 0 0 605.6387 0.80
Fuel type
IGA-MU (Chiang, 2005) Generation
2700 0 0 624.127 0.52
Fuel type
2700 0 0 624.517 7.25
NR⁄ means not reported in the referred literature.
Table 11 Convergence results (100 trial runs) for the test system 4 with PD = 2700 MW. Algorithms
Total generation cost ($/h)
IGA-GA (Chiang, 2005) CGA-MU (Chiang, 2005) PSO-LRS (Selvakumar & Thanushkodi, 2007) NPSO (Selvakumar & Thanushkodi, 2007) NPSO-LRS (Selvakumar & Thanushkodi, 2007) RCGA (Amjady & Nasiri-Rad, 2009) ACO (Pothiya et al., 2010) BBO (Bhattacharya & Chattopadhyay, 2010a) DE/BBO (Bhattacharya & Chattopadhyay, 2010b) SOA
Minimum
Maximum
Average
627.5178 624.7193 624.2297 624.1624 624.1273 623.8281 623.70 605.6387 605.6230 598.1652
630.8705 633.8652 628.3214 627.4237 626.9981 623.8814 624.09 605.9103 605.6231 601.2624
625.8692 627.6087 625.7887 625.2180 624.9985 623.8495 623.90 605.8622 605.6252 600.0214
Table 12 Frequency of convergence in 100 trial runs for 10-generating units with PD = 2700 MW. Algorithms
SOA DE/BBO (Bhattacharya & Chattopadhyay, 2010b) BBO (Bhattacharya & Chattopadhyay, 2010a) NPSO-LRS (Selvakumar & Thanushkodi, 2007) NPSO (Selvakumar & Thanushkodi, 2007) PSO-LRS (Selvakumar & Thanushkodi, 2007) IGA-MU (Chiang, 2005) CGA-MU (Chiang, 2005)
Range of total generation cost ($/h) <605.5
605.5– 623.5
623.5– 624.5
624.5– 625.5
625.5– 626.5
629.5– 630.5
630.5– 631.5
631.5– 632.5
632.5– 633.5
633.5– 634.5
100 0
0 100
0 0
0 0
0 0
0 0
0 0
0 0
0 0
0 0
0 0
0 0
0 0
0
100
0
0
0
0
0
0
0
0
0
0
0
0
0
20
58
17
5
0
0
0
0
0
0
0
0
0
18
54
16
12
0
0
0
0
0
0
0
0
0
5
37
36
17
5
0
0
0
0
0
0
0 0
0 0
0 0
39 5
45 20
11 31
2 21
2 10
0 7
1 3
0 2
0 0
0 1
SOA as compared to those offered by the other algorithms. Hence, it can be concluded that for all the four test systems the performance of the SOA is found to be the best one. (C) Testing of robustness: The performance of any heuristic search based optimization algorithm is best judged through repetitive trial runs so as to compare the robustness/consistency of the algorithm. For this specific goal, the frequency of convergence to the minimum cost at different ranges of generation cost with fixed load demand is recorded and pre-
626.5– 627.5
627.5– 628.5
628.5– 629.5
sented in Table 9 for the test system 3 and in Table 12 those for the test system 4, respectively. The same for the test systems 1 and 2 are not included in the referred literatures, and hence, are not reported in the present work. But it is experimented by the authors of the present work with the SOA for the test systems 1 and 2 and it is noticed that the convergence to the minimum value of the cost function with a minimal variation is achieved. Tables 9 and 12 show that the frequency of converging to a better solution is higher
B. Shaw et al. / Expert Systems with Applications 39 (2012) 508–519
Fig. 8. Convergence profile of the total generation cost for 10-generating units.
in the SOA as compared to the other methods. Thus, it may be inferred that the SOA method is the most consistent in achieving the lowest cost in all the runs. (D) Computational efficiency: Apart from yielding minimum cost by the SOA, it may also be noted that the SOA yields minimum cost at comparatively lesser time of execution of the program. Thus, this approach is also efficient as far as the computational time is concerned. 6. Conclusion In this paper, the SOA has been successfully implemented to solve different ED problems. It has been observed that the SOA has the ability to converge to a better quality near-optimal solution and possesses better convergence characteristics and robustness than other prevailing techniques reported in the recent literatures. It is also clear from the results obtained by different trials that the SOA is free from the shortcoming of premature convergence exhibited by the other optimization algorithms. Thus, this algorithm may become very promising for solving some more complex engineering optimization problems for future researchers. References Ajzen, A. (2002). Residual effects of past on later behavior: Habituation and reasoned action perspectives. Personality and Social Psychology Review, 6(2), 107–122. Alsumait, J. S., Sykulski, J. K., & Al-Othman, A. K. (2010). A hybrid GA-PS-SQP method to solve power system valve point economic dispatch problems. Applied Energy, 87(5), 1773–1781. Amjady, N., & Nasiri-Rad, H. (2009). Nonconvex economic dispatch with AC constraints by a new real coded genetic algorithm. IEEE Transactions on Power Systems, 24(3), 1489–1502. Bhattacharya, A., & Chattopadhyay, P. K. (2010a). Biogeography-based optimization for different economic load dispatch problems. IEEE Transactions on Power Systems, 25(2), 1064–1077. Bhattacharya, A., & Chattopadhyay, P. K. (2010b). Hybrid differential evolution with biogeography-based optimization for solution of economic load dispatch. IEEE Transactions on Power Systems, 25(4), 1955–1964. Chaturvedi, K. T., Pandit, M., & Srivastava, L. (2008). Self-organizing hierarchical particle swarm optimization for nonconvex economic dispatch. IEEE Transactions on Power Systems, 23(3), 1079–1087. Chaturvedi, K. T., Pandit, M., & Srivastava, L. (2009). Particle swarm optimization with time varying acceleration coefficients for non-convex economic power dispatch. International Journal of Electrical Power & Energy Systems, 31(6), 249–257. Chen, P. H., & Chang, H. C. (1995). Large-scale economic dispatch by genetic algorithm. IEEE Transactions on Power Systems, 10(4), 1919–1926.
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