Hybrid ant optimization system for multiobjective economic emission load dispatch problem under fuzziness

Swarm and Evolutionary Computation 18 (2014) 11–21

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Swarm and Evolutionary Computation journal homepage: www.elsevier.com/locate/swevo

Regular Paper

Hybrid ant optimization system for multiobjective economic emission load dispatch problem under fuzziness Abd Allah A. Mousa a,b,n a b

Department of Basic Engineering Science, Faculty of Engineering, Shebin El Kom, Menoufia University, Egypt Department of Mathematics and Statistics, Faculty of Sciences, Taif University, Saudi Arabia

art ic l e i nf o

a b s t r a c t

Article history: Received 28 August 2013 Received in revised form 24 March 2014 Accepted 16 June 2014 Available online 5 July 2014

In this paper, a new hybrid optimization system is presented. Our approach integrates the merits of both ant colony optimization and steady state genetic algorithm and it has two characteristic features. Firstly, since there is instabilities in the global market and the rapid fluctuations of prices, a fuzzy representation of the economic emission load dispatch (EELD) problem has been defined, where the input data involve many parameters whose possible values may be assigned by the expert. Secondly, by enhancing ant colony optimization through steady state genetic algorithm, a strong robustness and more effectively algorithm was created. Also, stable Pareto set of solutions has been detected, where in a practical sense only Pareto optimal solutions that are stable are of interest since there are always uncertainties associated with efficiency data. Moreover to help the decision maker DM to extract the best compromise solution from a finite set of alternatives a Technique for Order Performance by Similarity to Ideal Solution (TOPSIS) method is adopted. It is based upon simultaneous minimization of distance from an ideal point (IP) and maximization of distance from a nadir point (NP). The results on the standard IEEE systems demonstrate the capabilities of the proposed approach to generate true and well-distributed Pareto optimal nondominated solutions of the multiobjective EELD. & 2014 Elsevier B.V. All rights reserved.

Keywords: Ant colony optimization Fuzzy numbers Topsis Economic emission load dispatch

1. Introduction Optimal Power Flow (OPF) was the first discussed by Carpentier [1]. In the past two decades, OPF problem has received much attention, because of its ability to determine the dispatch of generators so as to meet the load demand while minimizing the total fuel cost, subject to the satisfaction of all constraints on the system. OPF is a nonlinear, non-convex, large-scale, static optimization problem with both continuous and discrete control variables [2]. More objectives have recently been incorporated into the OPF problem. These include optimization of active/reactive losses, plant emissions, voltage profile and power plant stability. This has extended the definition of the OPF problem from a single objective case to a multiobjective one [3,4]. The Environmental/Economic Dispatch multiobjective problem seeks to simultaneously minimize both fuel cost and the emissions produced by power plants. Environmental concerns on the effect of SO2 and NOX emissions produced by the fossil-fueled power plants led to the inclusion of minimization of emissions as an objective in the OPF formulation. In the previous literatures various mathematical programming and optimization techniques have been used to solve OPF.

n Correspondence address: Department of Basic Engineering Science, Faculty of Engineering, Menoufia University, Egypt.

http://dx.doi.org/10.1016/j.swevo.2014.06.002 2210-6502/& 2014 Elsevier B.V. All rights reserved.

Previously a number of conventional approaches such as the gradient method, linear programming Algorithm, lambda iteration method, quadratic programming, nonlinear programming algorithm, Lagrange relaxation algorithm [5–10], etc. have been applied for solving the EELD problems. These traditional classical methods are based on the assumption that the incremental cost of generator monotonically increases. Also, dynamic programming [10] was proposed as a new algorithm, which does not impose any restrictions on the nature of the cost curves and hence it can solve both the convex and non-convex EELD problems. But this method suffers from the curse of dimensionality in the solution procedure. Nonlinear features of generators in practical aspects is the main reason that generally a classical optimization technique may not be able to find a solution with a significant computational time for medium or large-scale EELD problem and on the other hand these techniques may further being restricted by their lack of robustness and efficiency in a number of practical limitations. Accordingly, these limitations are redounded to introduce the evolutionary algorithms methods [11]. Evolutionary algorithms (EAs) are stochastic search methods that mimic the metaphor of natural biological evolution and/or the social behavior of species. Because of their universality, ease of implementation, and fitness for parallel computing, EAs often take less time to find the optimal solution than classical methods [12,13]. Also, availability of highspeed computer system, more and more interests has been

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focused on the application of artificial intelligence technology for solution of EELD problems. Recently, there has been a boom in applying evolutionary algorithms to solve EELD problems. Several artificial intelligence methods, such as the genetic algorithm [12– 15]; artificial neural networks [16]; simulated annealing, Tabu search [17]; evolutionary programming [18]; swarm optimization [19–22]; differential evolution [23], have been developed and applied successfully to ELD problems. Hopfield neural networks have also implemented [24] to solve EELD problems for units with piecewise quadratic fuel cost functions and prohibited zones constraint. In order to meet the ever increasing demands in the design problems, a new evolutionary algorithm called ant colony optimization algorithm have all been used successfully to mimic the corresponding natural, or physical, or social phenomena[25– 27]. Ant colony optimization (ACO) is a metaheuristic inspired by the shortest path searching behavior of various ant species. Since the initial work of Dorigo et al. on the first ACO algorithm, the ant system [28], several researchers have designed ACO systems to deal with multiobjective problems. Recently, other powerful techniques called hybridization methods have been suggested, The hybrid methods are applied to handle more complicated constraints, including fuzzy adaptive PSO algorithm with Nelder–Mead simplex search (FAPSO–NM) [29], hybrid PSO and sequential quadratic programming (PSO– SQP) [30], hybrid PSO and local serach scheme (PSO–LS) [31], hybrid EP and sequential quadratic programming (EP–SQP) [32], hybrid DE and sequential quadratic programming (DE–SQP) [33], multiobjective evolutionary algorithm based on decomposition (MOEA/D) [34], and Combining ACO with EA based on decomposition MOEA/D [35]. This paper intends to present a new optimization algorithm for solving EELD under fuzziness. The proposed approach integrates the merits of both ACO and steady state Genetic algorithm SSGA [36]. Since there is instabilities in the global market, implications of global financial crisis and the rapid fluctuations of prices, for this reasons a fuzzy representation of the multiobjective EELD has been defined, where the input data involve many parameters whose possible values may be assigned by the experts. In practice, it is natural to consider that the possible values of these parameters as fuzzy numerical data which can be represented by means of fuzzy subsets of the real line known as fuzzy numbers. The proposed approach has two characteristic features. Firstly, a fuzzy representation of the optimal power flow problem has been defined. Secondly, by enhancing ACO through SSGA, a strong robustness and more effectively optimization system was created. Moreover to help the DM to extract the best compromise solution from a finite set of alternatives a TOPSIS method is adopted. Several optimization runs of the proposed approach will be carry out on the standard IEEE systems to verify the validity of the proposed approach. Section 2 provides a brief description on multiobjective optimization. The mathematical formulation of EELD problem is discussed in Section 3. Section 4 reviews the standard ACO metaheuristic The original optimization system is described in Section 5 along with a short description of the algorithm used in this test system. The parameter settings for the test system to evaluate the performance of the optimization system and the simulation studies are discussed in Section 6. Results and discussion are given in Section 7. Finally, the conclusions is drawn in Section 8.

to minimize a vector of objective functions that usually conflict with each other. The following fuzzy vector minimization problem (FVMP) involving fuzzy parameters in the objective functions and constraints such a problem takes the form: ) ~ f 2 ðX; aÞ; ~ :::::; f m ðX; aÞg ~ Min ff 1 ðX; aÞ; ð1Þ ~ r0 subject to gðX; aÞ ~ is the ith objective function; and gðX; aÞ ~ is constraint where f 1 ðX; aÞ vector, X is vector of decision variables; and a~ ¼ ða~ 1 ; a~ 2 ; ::::a~ n Þ represented a vector of fuzzy parameters in the problem. Fuzzy parameters are assumed to be characterized as the fuzzy numbers. The real fuzzy numbers a~ form a convex continuous fuzzy subset of the real line whose membership function μa~ ðaÞ is defined by: 1) 2) 3) 4) 5) 6)

a continuous mapping from R1 to the closed interval [0,1]; μa~ ðaÞ ¼ 0 for all a A ½  1; a1 ; strictly increasing on ½a1 ; a2 ; μa~ ðaÞ ¼ 1 for all a A ½a2 ; a3 ; strictly decreasing on ½a3 ; a4 ; μa~ ðaÞ ¼ 0 for all a A ½a4 ; þ 1;

Assume that a~ in the FM-RAP are fuzzy numbers whose membership functions are μa~ ðaÞ. Definition 1. (α-level set). The α-level set or α-cut of the fuzzy ~ for which the degree numbers a~ is defined as the ordinary set Lα ðaÞ of their membership functions exceeds the level α A ½0; 1: ~ ¼ fajμa~ ðaÞ Zαg: Lα ðaÞ For a certain degree α, the (FM-RAP) can be represented as a nonfuzzy α-VMP as follows: 9 Min ff 1 ðX; aÞ; f 2 ðX; aÞ; :::::; f m ðX; aÞg > > > > = subject to gðX; aÞ r 0 ð2Þ X ¼ ðx1 ; x2 ; :::xn Þ; a ¼ ða1 ; a2 ; ::::; an Þ > > > > ; Lαi r ai r U αi where constraint Lαi r ai rU αi gives the lower and upper bound for the parameters ai Definition 2. (α–Pareto optimal solution). xn A X is said to be an α–Pareto optimal solution to the (α-VMP), if and only if there does ~ such that f i ðx; aÞ Z f i ðxn ; an Þ; not exist another x A X, a A Lα ðaÞ i ¼ 1; 2; ::; k; with strictly inequality holding for at least one i, where the corresponding values of parameters ani are called αlevel optimal parameters. In real world application problems, input data or related parameters are frequently imprecise/fuzzy owing to incomplete or unobtainable information, so the concept of Pareto stability is introduced for the Pareto optimal solutions of a vector valued problem of the allocation of resources to activities. Definition 3. (Stable Pareto optimality) A Pareto- optimal solution x of the problem FVMP is said to be stable if and only if there exists a real number α A ½0; 1 such that x is still Pareto-optimal if ais replaced by any a0 satisfying the following requirement: ~ ¼ fajμa~ ðaÞ Z αg a0 A Lα ðaÞ ð3Þ Such a solution x is said to be a stable Pareto-optimal solution. In practical sense only Pareto optimal solutions that are stable are of interest since there are always uncertainties associated with the efficiency data (input data).

2. Fuzzy multiobjective optimization

3. Multiobjective formulation of EELD problem

A Multi-objective Optimization Problem (MOP) can be defined as determining a vector of design variables within a feasible region

The economic emission load dispatch involves the simultaneous optimization of fuel cost and emission objectives which are

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13

conflicting ones. The deterministic problem is formulated as described below.

3.2.2. Maximum and minimum Limits of power generation The power generatedP Gi by each generator is constrained between its minimum and maximum limits, i.e.,

3.1. Objectives

P Gi min r P Gi r P Gi max ; Q Gi min r Q Gi r Q Gi max ;

3.1.1. Fuel cost objective The classical economic dispatch problem of finding the optimal combination of power generation, which minimizes the total fuel cost while satisfying the total required demand, can be mathematically stated as follows: n

n

i¼1

i¼1

f 1 ð U Þ ¼ C t ¼ ∑ C i ðP Gi Þ ¼ ∑ ðai þ bi P Gi þ ci P 2Gi Þ$=hr

ð4Þ

where C t : Total fuel cost ($/hr), C i : Fuel cost of generator i, ai ; bi ; ci : Fuel cost coefficients of generator i, and P Gi : power generated (p. u.) by generator i, n: Number of generators. 3.1.2. Emission objective The emission function can be presented as the sum of all types of emission considered, such as NOx ,SO2 , thermal emission, etc., with suitable pricing or weighting on each pollutant emitted. In the present study, only one type of emission NOx is taken into account without loss of generality. The amount of NOx emission is given as a function of generator output, that is, the sum of a quadratic and exponential function: n

f 2 ð U Þ ¼ ENO ¼ ∑ ½10  2 ðαi þ βi P Gi þ γ i P 2Gi Þ þ ξi expðλi P Gi Þton=h

ð5Þ

i¼1

where, αi ; βi ; γ i ; ξi ; λi : coefficients of the ith generator's NOx emission characteristic.

V i min r V i r V i max ; i ¼ 1; ::::::; n whereP Gi min : minimum power generated, andP Gi max : maximum power generated.

3.2.3. Security constraints A mathematical formulation of the security constrained EELD problem would require a very large number of constraints to be considered. However, for typical systems the large proportion of lines has a rather small possibility of becoming overloaded. The EELD problem should consider only the small proportion of lines in violation, or near violation of their respective security limits which are identified as the critical lines. We consider only the critical lines that are binding in the optimal solution. The detection of the critical lines is assumed done by the experiences of the DM. An improvement in the security can be obtained by minimizing the following objective function. k

S ¼ f ðP Gi Þ ¼ ∑ ðjT j ðP G Þj=T max Þ j

ð7Þ

j¼1

is the maximum limit of where, T j ðP G Þ is the real power flow T max j the real power flow of the j th line and k is the number of monitored lines. The line flow of the j th line is expressed in terms of the control variables PGs, by utilizing the generalized generation distribution factors (GGDF) [38] and is given below. n

T J ðP G Þ ¼ ∑ ðDji P Gi Þ

ð8Þ

i¼1

3.2. Constraints On the other hand, The economic emission load dispatch involves the simultaneous optimization of fuel cost and emission objectives subjected to the following constraints:

where, Dji is the generalized GGDF for line j, due to generator i For secure operation, the transmission line loading Sl is restricted by its upper limit as Sℓ r Sℓ max ; ℓ ¼ 1; ::::; nℓ Where nℓ is the number of transmission line.

3.2.1. Power balance constraint The total power generated must supply the total load demand and the transmission losses [37]. n

∑ P Gi  P D  P Loss ¼ 0

i¼1

where P i ¼ P Gi  P Di ; Q i ¼ Q Gi  Q Di ; Rij Rij cos ðδi  δj Þ; Bij ¼ sin ðδi  δj Þ Aij ¼ V iV j V iV j n

n

P Loss ¼ ∑ ∑ ½Aij ðP i P j þ Q i Q j Þ þ Bij ðQ i P j þP i Q j Þ i¼1i¼1

PD: total load demand (p.u.) n: number of buses Rij: series resistance connecting buses i and j Vi: voltage magnitude at bus i Ploss: transmission losses (p.u.). δi: voltage angle at bus i Pi: real power injection at bus i Qi: reactive power injection at bus i

ð6Þ

4. ACO metaheuristic ACO makes use of agents, called ants, which mimic the behavior of real ants in how they manage to establish shortest-route paths from their colony to feeding sources and back [39]. Ants communicate information through pheromone trails, which influence which routes the ants follow, and eventually lead to a solution route. ACO was initially designed to solve the Traveling Salesman Problem (TSP) as indicate in Fig. 1 and works as follows. In the TSP, a given set of n cities has to be visited exactly once and the tour ends in the initial city. We call dij ði; j ¼ 1; 2; :::; nÞ the length of the path between cities i and j. In the case of Euclidean TSP, dij is the Euclidean distance between i and j (i.e., dij ¼ jjxi  xj jj2 ). The cities and routes between them can be represented as a connected graph ðn; EÞ, where n the set of towns and E is the set of edges between towns (a fully connected graph in the Euclidean TSP). The ants move from one city to another following the pheromone trails on the edges. Let τij ðtÞ be the trail intensity on edge ði; jÞ at iteration t. Then, each ant k; k ¼ 1; 2; :::m chooses the next city to visit depending on the intensity of the associated trail. When the ants have completed their city tours, the trail intensity is updated according to: τij ðt þ 1Þ ¼ ρτij ðtÞ þ Δτij ðtÞ;

t ¼ 1; 2; :::::T

ð9Þ

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Fig. 1. Example of a TSP problem, (a) TSP of 10-cities problem and (b) shortest route path.

where ρ is a coefficient such that ð1  ρÞ represents the evaporation of trail between iteration t and tþ1; T is the total is the number of iterations (generation cycles), m

Δτij ðtÞ ¼ ∑ Δτkij

ð10Þ

k¼1

where Δτkij is the quantity per unit of length of trail substance (pheromone in real ants) laid on edge ði; jÞ by the kth ant between iteration t and tþ1, it is given by the following equation: ( ) C if the kth ant uses edge ði; jÞ in its tour Wk k Δτij ¼ ð11Þ 0 otherwise where C is a constant and W k is the tour length of the kth ant. An ant k at city i chooses the city j to go to with a probability pkij ðtÞ, which is a function of the town distance and of the amount of pheromone trail present on the connecting edge. 8 α β > < ½τij ðtÞ ½ηα ij  β ; 8 jA U k k ∑ ½τij ðtÞ ½ηij  pij ðtÞ ¼ k A U ð12Þ > : 0 otherwise where U k is a set of the cities can be chosen by the kth ant at city i for the next step ηij is a heuristic function which is defined as the visibility of the path between cities i and j, ηij ¼ ð1=dij Þ; parameters α, and β determine the relative influence of the trail information and the visibility [40].

5. The proposed approach Dealing with several objectives in ant colonies that use the principles of MACO necessitates to answer three questions: (1) how to globally update pheromone according to the performance of each solution based on each objective, where each colony having its own pheromone structure (2) how does a given ant locally selects a path, according to the visibility and the desirability, at a given step of the approach (3) how to build the Pareto front. In this section, a framework for the proposed approach that involves two phases was presented. The first one employs the heuristic search by enhancing ant colony optimization through steady state genetic algorithm SSGA to obtain Pareto solution, while the other phase employs Topsis approach to identify the best compromise solution from a finite set of alternative. The proposed approach differs from the traditional ones in its design of a multipheromone ant colony optimization (MACO). 5.1. Stage 1: enhanced ant colony optimization system Implementing the multipheromone ant colony optimization for a certain problem requires a representation of n variables for each ant, with each variable i has a set of ni options (nodes) with their

Fig. 2. Ant representation.

values lij which we generate (a fully connected graph ðn; ni Þ, and their associated pheromone concentrations {τij } (see Fig. 2); where i ¼ 1; 2; :::; n, and j ¼ 1; 2; :::; ni . The process starts by generating m ants' position (solutions) from the population which is generated randomly, thus each ant k; k A f1; 2; :::; mg has a position with a selected value for each variable (lij ji ¼ 1; 2; :::::; n 4 j ¼ 1; 2; ::::; ni) according to the associated pheromone with this value. This process continues for each objective. Consequently, path of each ant was consisted of n nodes with a value lij for each node. The main steps of the MACO are summarized as follows: Step 1: Construct Q colonies In a multiobjective optimization problem, multiobjective functions F ¼ ff 1; f 2 ; :::f Q g need to be optimized simultaneously, there does not necessarily existence a solution that is best with respect to all objectives because of incommensurability and confliction among objectives. For this step, the number of colonies is set to Q with its own pheromone structure, where Q ¼ jF j is the number of objectives to optimize. Step 2: Initialization First, pheromones trails are initialized to a given value τq0 ; ðq ¼ 1; 2; ::::Q Þ where τq0 is the pheromone information in the current iteration. Also, Pareto set are initialized to an empty set. Step 3: Evaluation The MACO parameterized by the number of ant colonies Q and the number of associated pheromone structures. All the colonies have the same number of ants. Each colony k; k A f1; 2; :::; Q g tries to optimize an objective considering the pheromone information associated for each colony, where each colony is determined knowing only the relevant part of a solution. This methodology enforces all colonies to search in different regions of the nondominated front, e.g. MACO for two objective functions in Fig. 3, enforces two colonies to search in different regions of the nondominated front.

A.A.A. Mousa / Swarm and Evolutionary Computation 18 (2014) 11–21

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Fig. 4. The model for steady state for genetic algorithms.

Fig. 3. Search regions of MACO for two objectives.

Step 4: Trail update and reward solutions When updating pheromone trails, one has to decide on which of the constructed solutions laying pheromone. The quantity of pheromone laying on a component represents the past experience of the colony with respect to choosing this component. Then, at each cycle every ant constructs a solution, and pheromone trails are updated. Once all ants have constructed their solutions, pheromone trails are updated as usually in eq. 9: first, pheromone trails are reduced by a constant factor to simulate evaporation to prevent premature convergence; then, some pheromone is laid on components  of the best solution. Accordingly, pheromoneconcentration τqij ðtÞ; q ¼ 1; 2; :::; Q & i ¼ 1; 2; :::n & J ¼ 1; 2; :::; ni associated with each possible route (variable value) is changed in a way to reinforce good solutions, and the change in pheromone concentration Δτqij is expressed as follows: ) 8 C=f q f or Min f q ; > < if lij is chosen by ant k Δτqij ¼ C*f q f or Max f q ; ð13Þ > : 0 otherwise A possibility is to reward every nondominated solution of the current cycle as follows Δτqib ðtÞ ¼ ð1  ρÞτqib ðt  1Þ þ Δτqib ; b ¼ 1; 2; ::::::B & B D ni where τqib ðtÞ the revised concentration of pheromone is associated with option b A ni at iteration t, τqib ðt  1Þ is the concentration of pheromone at the previous iteration ðt  1Þ; Δτqib is change in pheromone concentration, and B is the size of reward solutions. Step 5: Solution construction Once the pheromone is updated after an iteration, the next iteration starts by changing the ants’ paths (i.e. associated variable values) in a manner that respects pheromone concentration and also some heuristic preference. For each ant and for each dimension construct a new candidate group to replace the old one. As such, an ant k will change the value for each variable according to the transition probability. The transition probability is done for each colony ðpqij ðtÞ; q ¼ 1; 2; :::; Q Þ as expressed in the following equation. 8 α β τq ðtÞ η > < ½ ijq  ½α ij q β ; 8 j A ni ∑ ½τ ij ðtÞ ½η ij  k pij ðtÞ ¼ j A ni ð14Þ > : 0 otherwise where pqij ðtÞ is Probability that option lij is chosen by ant k for variable i at iteration t. Step 6: Nondominated solutions The set of nondominated solutions is stored in an archive. During the optimization search, this set, which represents the Pareto front, is updated. At each iteration, the current solutions obtained are compared to those stored in the Pareto archive;

the dominated ones are removed and the nondominated ones are added to the set. Step 7: Steady state genetic algorithm Steady state genetic algorithm was implemented in such way that, two offspring are produced in each generation. Parents are selected to produce offspring and then a decision is made as to which individuals in the population to select for deletion to make room for the new offspring (Fig. 4). A replacement/ deletion strategy defines which member of the population will be replaced by the new offspring. Steady state genetic algorithms overlapping systems, since parents and offspring compete for survival. (i) Selections Selection determines which individuals of the population will have all or some of their genetic material passed on to the next generation of individuals. The mechanism for selecting the parents is based on a tournament selection. Tournament selection operates by choosing some individuals randomly from a population and selecting the best from this group to survive into the next generation. For example, pairs of parents ðx; yÞ are randomly chosen from the initial population. Their fitness values are compared and a copy of the better performing individual becomes part of the mating pool. The tournament will be performed repeatedly until the mating pool is filled. That way, the worst performing patent in the population will never be selected for inclusion in the mating pool. Tournaments are held between pairs of individuals are the most common. In this way all parents necessary for a reproduction operator are selected. (ii) Recombination through crossover and mutation After selection has been carried out, then the mechanisms of crossover and mutation are applied to produce an offspring, the following subsection outlines these genetic operators.  Crossover: Once the parents are created, the crossover step is carried out by replacing the current value with a new one which produced stochastically with a probability proportional to the crossover probability. Suppose the crossover probability set by the system is pc . Generating a random number r A ½0; 1, the crossover operation could be carried out only if r o pc . Suppose x and y are two parents and α is a random number (i.e. α A ½0; 1 ). The result of crossover operation x0 and y0 can be obtained by the following linear combination of x and y: x0 ¼ αx þ ð1  αÞy  y0 ¼ ð1  αÞx þ αy ð15Þ

 Mutation: Once the, the crossover is performed, the mutation step is carried out by replacing the current value with a new one which produced stochastically with a probability proportional to the mutation probability pm . Generating a random numberr A ½0; 1, the

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A.A.A. Mousa / Swarm and Evolutionary Computation 18 (2014) 11–21

(6) Develop a distance measure over each criterion to both ideal (D þ ) and nadir (D  ). rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Diþ ¼ ∑ ðvij  vjþ Þ2 ; Di ¼ ∑ ðvij  vj Þ2 j

j

(7) For each alternative, determine a ratio R equal to the distance to the nadir divided by the sum of the distance to the nadir and the distance to the ideal, R¼

Fig. 5. Algorithm 1: the strategy of deletion.

mutation operation is implemented only if r o pm . Suppose xðjÞ will be transformed into x0 ðjÞ after mutation as follows: x0 ðjÞ ¼ LðjÞ þ λðUðjÞ  LðjÞÞ; j ¼ 1; 2; ::::; n

ð16Þ

where λ is a random number (i.e. λ A ½0; 1 ). Here L and U are the lower and upper bounds respectively. (iii) Replacement/deletion strategy A widely used combination is to replace the worst individual only if the new individual is better. In the paper, this strategy will be suggested that the individual will be deleted if it was dominated by the new offspring as in algorithm 1 (Fig. 5). 5.2. Stage 2: identifying the best compromise solution Optimization of the above-formulated objective functions using multiobjective genetic algorithms yields not a single optimal solution, but a set of Pareto optimal solutions, in which one objective cannot be improved without sacrificing other objectives. For practical applications, however, we need to select one solution (operating point), which will satisfy the different goals to some extent. Such a solution is called best compromise solution. TOPSIS method given by Yoon and Hwang [41,42] has the ability to identify the best alternative from a finite set of alternatives quickly. It stands for “Technique for Order Preference by Similarity to the Ideal Solution” which based upon the concept that the chosen alternative should have the shortest distance from the positive ideal solution and the farthest from the negative ideal solution. TOPSIS can incorporate relative weights of criterion importance. The idea of TOPSIS can be expressed in a series of steps. (1) Obtain performance data for n alternatives over M criteria xij (i¼1,…., n, j¼1,…., M). (2) Calculate normalized rating (vector normalization is used) r ij . (3) Develop a set of importance weights W j , for each of the criteria. The basis for these weights can be anything, but, usually, is adhoc reflective of relative importance. V ij ¼ wj :r ij (4) Identify the ideal alternative (extreme performance on each criterion) S þ .   þ g ¼ max vij jj A J 1 ; S þ ¼ fv1þ ; v2þ ; ::; vjþ ; ::; vm    min vij jj A J 2 ; i ¼ 1; ::::; n Where J 1 is a set of benefit attributes and J 2 is a set of cost attributes. (5) Identify the nadir alternative (reverse extreme performance on each criterion) S  .  S  ¼ fv1 ; v2 ; ::; vj ; ::vm g ¼ fðmin vij jj A J 1 Þ;

ðmax vij jj A J 2 Þ; i ¼ 1; ::::; ng

D D þDþ 

ð17Þ

(8) Rank alternative according to ratio R (in Step 7) in descending order. (9) Recommend the alternative with the maximum ratio A relative advantage of TOPSIS is the ability to identify the best alternative from a finite set of alternatives quickly. TOPSIS is attractive in that limited subjective input is needed from decision makers. The only subjective input needed is weights which reflect the degree of satisfactory of each objective. The flowchart of the proposed algorithm is shown in Fig. 6.

6. Implementation of the proposed approach The described methodology has been described for Q-objective function, but it is applied to the standard IEEE 30-bus 6-generator test system with two objectives (Q¼2). The single-line diagram of this system is shown in Fig. 7 and the detailed data are given in [13,43]. For comparison purposes with the reported results, the system is considered as losses and the security constraint is released. The values of fuel cost ($/h) and emission (ton/h) coefficients are given in Table 1. Naturally, these data (fuel cost and emission coefficients) involve many controlled parameters whose possible values are vague and uncertain. Consequently each numerical value in the domain can be assigned a specific “grade of membership” where 0 represents the smallest possible grade of membership, and 1 is the largest possible grade of membership. Thus fuzzy parameters can be represented by its membership grade ranging between 0 and 1. The fuzzy numbers shown in Fig. 8 have been obtained from interviewing DMs or from observing the instabilities in the global market and rate of prices fluctuations. The idea is to transform a problem with these fuzzy parameters to a crisp version using α -cut level. This membership function can rewrite as follows: 8 1; a ¼ aij > > > > >  19 0:95a < 20a jk ra raij aij μðaij Þ ¼ ð18Þ 20a > 21  aij aij r a r 1:05aij > > > > :0 a o 0:95aij or a 41:05aij So, every fuzzy parameter a~ ij can be represented using the membership function. By using α-cut level, these fuzzy parameters can be transformed to a crisp one having upper and lower bounds ½aLij ; aUij , which declared in Fig. 8. Consequently, each α-cut level can be represented by the two end points of the alpha level. Consequently, each α-cut level can be represented by the two end points of the alpha level. For example given: let the parameter a has a value of 150, by taking α ¼ 1, its value remain as it is as in Fig. 9(A). But for α ¼ 0 its value is changed to become a A ½142:5; 157:5 as in Fig. 9(B). Also for α ¼ 0:6 we get another bounds for the parameter a A ½147; 153 as in Fig. 9(C).

A.A.A. Mousa / Swarm and Evolutionary Computation 18 (2014) 11–21

17

Fig. 6. A flowchart of the proposed approach.

7. Results and discussion Here, the problem is how to determine the optimal power flow for considering the minimum cost and the minimum emission objectives simultaneously. In order to efficiently and effectively obtain the solution, the search for the optimal solution is carried out in two steps. Firstly, a set of nondominated solutions is obtained by exploring the optimal Pareto frontier using different α cut levels. To study the influence of fuzzy parameters on the obtained Pareto optimal solutions, all the range of the parameter fluctuation were scanned, two bounds of α value have been considered α ¼ 0; α ¼ 1, and also we take

some values between these bounds α ¼ 0:2; 0:4; 0:6; 0:8. Based on the definition of Pareto stability, the Pareto frontier may be reduced to a manageable size (i.e., Stable Pareto optimal solutions). The proposed optimization system used in this study were developed and implemented on dual-core processor PC using MATLAB environment. We empirically determined the parameter setting used in the algorithm. Table 2 lists the parameter setting used in the algorithm for all runs. Graphical presentations of the experimental results are presented in Figs. 10–15 for six instances problems. For comparison purposes with the reported results, Tables 3 and 4 show the best fuel cost and best NOx emission obtained by proposed

18

A.A.A. Mousa / Swarm and Evolutionary Computation 18 (2014) 11–21

Fig. 7. Single line diagram of IEEE 30-bus 6-generator test system.

Table 1 Generator cost and emission coefficients. G1

G2

G3

G4

G5

G6

Cost($/h)

a b c

10 200 100

10 150 120

20 180 40

10 100 60

20 180 40

10 150 100

Emission(ton/h)

α β γ ζ λ

4.091  5.554 6.490 2.0E-4 2.857

2.543  6.047 4.638 5.0E-4 3.333

4.258  5.094 4.586 1.0E-6 8.000

5.426  3.550 3.380 2.0E-3 2.000

4.258  5.094 4.586 1.0E-6 8.000

6.131  5.555 5.151 1.0E-5 6.667

Further we need to determine stable Pareto set solution, which is a Pareto optimal for all runs (different α cut levels), there was 7 Pareto solutions was detected as a stable Pareto solution. Table 5 lists the set of the stable set of optimal solution. To select the best compromise solution, TOPSIS method is used. To show the effect of changing the weights on the best compromise solution, 6 cases are considered. In each case one weight is changed linearly taking 6 different values. The other one are obtained using the relation w1 þw2 ¼ 1. Table 6 show the values of the weights in three cases. The objective functions obtained from the six solutions corresponding to the six weights are drawn versus weights for the six cases. The drawings are shown in Fig. 16. Therefore it can be said that TOPSIS method is attractive since limited subjective input (namely the weight values) is needed from the DM to get a satisfactory results from the Pareto set quickly. Also, this method can be classified as interactive approach, where the DM specifies input values according his needs. In this subsection, a comparative study has been carried out to assess the proposed approach concerning Pareto optimal solutions, decision maker preference, and computational time. On the one hand, evolutionary techniques suffer from the large-size of the Pareto optimal solution. Therefore the proposed approach has been used to reduce the Pareto optimal front by detecting the stable Pareto optimal solution under uncertainty of the efficiency data matrix, which enable the DM to take correct decision by visualizing the Pareto front which is validate under the instabilities in the global market, also it maintains the diversity of the Pareto set and good distribution over the front range. On the other hand, the proposed approach can be considered posteriori method, where after the Pareto optimal set (or a part of it) has been generated; it is presented to the decision maker, who selects the most preferred among the alternatives according to relative weights of criterion importance. In such away our algorithm aims to find a single point (solution) depending on the decision maker preference. On the next hand the proposed algorithm takes the effect of impreciseness of the mathematical modeling (occurring due to environmental fluctuations or due to instabilities in the global market and the rapid fluctuations of prices) into consideration by choosing appreciate α cut level. On the contrary, the proposed approach generates a set of solutions (Pareto set with a manageable size) at each iteration according to designer preference. Accordingly, it provides the facility to save computing time. Finally, the feasibility of using the proposed approach to handle multiobjective EELD has been empirically approved.

8. Conclusions

α−

→ ↓

↓

Fig. 8. Fuzzy numbers of the effectiveness of resource.

algorithm (α ¼ 0:0) as compared to the nondominated sorting genetic algorithm (NSGA) [44], niched pareto genetic algorithm (NPGA) [45], strength pareto evolutionary algorithm (SPEA) [46] and Epsilon dominance approach [13]. It can be deduced that the proposed algorithm finds comparable minimum fuel cost and comparable minimum NOx emission to the three evolutionary algorithms.

On the basis of the application, we can conclude that the proposed method can provide a sound optimal power flow by simultaneously considering multiobjective problem. Ant colony optimization has been and continues to be a fruitful paradigm for designing effective combinatorial optimization solution algorithms, in this paper; we proposed a new hybrid optimization system for solving multiobjective optimization with an application in optimal power flow considering two objectives (cost and emission). Our approach has two characteristic features. Firstly, a fuzzy representation of the optimal power flow problem has been defined. Secondly, by enhancing ant colony optimization through SSGA, a strong robustness and more effectively algorithm was created. Also, stable Pareto set of solutions has been detected, where in a practical sense only Pareto optimal solutions that are stable are of interest since there are always uncertainties associated with efficiency data. Moreover to help the decision maker DM to extract the best compromise solution from a finite set of alternatives a TOPSIS method is adopted. The main

A.A.A. Mousa / Swarm and Evolutionary Computation 18 (2014) 11–21

19

Fig. 9. Alpha level representation. 0.23

Emission (Ton/h)

Table 2 Parameters for the proposed approach. Parameters Number of objective Number of colonies Number of iteration m p α β C τo pr pm

2 2 200 100 0.5 1 0 100 10 0.85 0.02

0.19

600

610

620

630

640

Fig. 12. Pareto optimal set for α cut level¼0.4.

0.21

Alpha=0

0.2 0.19 0.18

0.205 Alpha=0.6

0.2 0.195 0.19

580

590

600

610

620

630

0.185 590

640

Cost($/h)

600

610

620

630

640

650

Cost($/h)

Fig. 10. Pareto optimal set for α cut level¼0.

Fig. 13. Pareto optimal set for α cut level¼0.6.

0.23

0.225

0.22

0.22

0.21

Emission(ton/h)

Emission (ton/h)

590

0.215

0.21

Alpha=0.2 0.2 0.19 0.18 580

Alpha=0.4

0.2

Cost($lh)

Emission (ton/h)

Emission (ton/h)

0.21

0.18 580

0.22

0.17 570

0.22

590

600

610

620

630

640

Cost($/h) Fig. 11. Pareto optimal set for α cut level¼ 0.2.

0.215

Alpha=0.8

0.21 0.205 0.2 0.195 0.19 595

600

605

610

615

620

625

630

635

640

Cost($/h)

features of the proposed algorithm could be summarized as follows:

Fig. 14. Pareto optimal set for α cut level¼0.8.

(a) The proposed technique has been effectively applied to solve the EELD problem considering two objectives simultaneously, with the facility in handing more than two objectives. (b) The non-dominated solutions in the obtained Pareto-optimal set are well distributed and have satisfactory diversity characteristics. This is useful in giving a reasonable freedom in choosing operating point from the available finite alternative.

(c) The proposed approach is efficient for solving nonconvex multiobjective optimization problems where multiple Pareto-optimal solutions can be found in one simulation run. (d) This approach seems to be an interactive approach where the DM specifies the relative weights of criterion importance (e) The proposed approach is capable of determining the stable Pareto optimal solution, where in practical sense only Pareto

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A.A.A. Mousa / Swarm and Evolutionary Computation 18 (2014) 11–21 630

0.225 0.22 0.215 Alpha=1.0

0.21

Cost($/h)

Emission(ton/h)

0.21

f (Emission) f (Cost)

0.205 0.2

620

0.2

0.195 0.19 595

600

605

610

615

620

625

630

635

640

645

Cost($lh)

610

1

1.5

2

2.5

3

3.5

4

4.5

5

5.5

6

0.19

Runs

Fig. 15. Pareto optimal set for α cut level¼1.0.

Fig. 16. Best compromise solution for different weights in six runs of Table 6. Table 3 Best fuel cost. NSGA Best cost Corresponding Emission

NPGA

SPEA

ε  dominance Proposed

608.245 608.147 607.807 606.4533 0.21664 0.22364 0.22015 0.2028

600.1185 0.2210

Table 4 Best NOx Emission. NSGA Best emission. Corresponding cost

NPGA

SPEA

ε  dominance Proposed

0.19432 0.19424 0.19422 0.1882 647.251 645.984 642.603 642.8976

0.1928 641.2515

Table 5 The stable optimal pareto solutions by the proposed hybrid optimization system. Pareto index

PG1

PG2

PG3

PG4

PG5

PG5

Emission (ton/h)

Cost ($/h)

1 2 3 4 5 6 7

0.2556 0.2510 0.2523 0.2563 0.2616 0.2642 0.2990

0.4041 0.4035 0.4023 0.4055 0.4079 0.4080 0.4336

0.5315 0.5291 0.5307 0.5285 0.5293 0.5299 0.5298

0.6798 0.6827 0.6848 0.6743 0.6632 0.6594 0.5914

0.5350 0.5394 0.5359 0.5374 0.5355 0.5355 0.5319

0.4271 0.4274 0.4272 0.4312 0.4355 0.4360 0.4476

0.1996 0.1998 0.1998 0.1994 0.1989 0.1988 0.1963

610.6031 610.3254 610.2686 610.9785 611.6744 611.9128 617.2925

Table 6 Different weights (W1 is changed linearly). Run

W1

W2

1 2 3 4 5 6

0 0.2 0.4 0.6 0.8 1

1 0.8 0.6 0.4 0.2 0

optimal solutions that are stable are of interest since there are always uncertainties associated with the efficiency data (input data). (f) On the basis of the application, we can conclude that the proposed method can provide a sound optimal power flow by simultaneously considering multiobjective problem. The performance improvement of ACO algorithm still remain in the experimental stage for lack of solid theoretical support; thus,

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