Firefly Algorithm for Economic Power Dispatching With Pollutants Emission Latifa Dekhici, Pierre Borne, Belkadi Khaled
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Latifa Dekhici, Pierre Borne, Belkadi Khaled. Firefly Algorithm for Economic Power Dispatching With Pollutants Emission. Informatica Economică vol. 16, no 2/2012, 2012, 16 (2), pp.45-57. hal- 00719362
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Firefly Algorithm for Economic Power Dispatching With Pollutants Emission
Latifa DEKHICI1, Khaled BELKADI1, Pierre BORNE2 1 LAMOSI, University of Sciences and the Technology (USTOMB) 31000 Oran, Algeria 2LAGIS, Ecole Centrale de Lille 59651 Villeneuve d'Ascq, Lille, France [email protected] , [email protected], [email protected].
Bio-inspired algorithms become among the most powerful algorithms for optimization. In this paper, we intend to provide one of the recent bio-inspired metaheuristic which is the Firefly Algorithm (FF) to optimize power dispatching. For evaluation, we adapt the particle swarm optimization to the problem in the same way as the firefly algorithm. The application is done in an IEEE-14 and on two thermal plant networks. In one of the examples, we neglect power loss and pollutant emissions. The comparison with the particle swarm optimization (PSO), demonstrate the efficiency of firefly algorithm to reach the best cost in less than one second. Keywords: Firefly Algorithm, Economic Power Dispatching, Particle Swarm Optimization, Pollutant Emissions
Introduction meet the required load demand at minimum 1 Currently, a set of nature-inspired cost fulfilling all system operational con- metaheuristics based on the natural behavior straints. Improvement can lead to significant of birds, ants, swarms, and bees, have cost saving (from 0, 03 $ to 0,20 $ per kWh) emerged as an alternative to rise above the [4]. difficulties presented by conventional meth- Methods in the literature: ods in optimizations problems. Various conventional methods like nonlinear Economic power dispatching is one of the programming [32] [39], Bundle method [33], difficult optimization problems. Resolution dynamic programming [45], mixed integer by metaheuristics can avoid significant fi- linear programming [6] [17] [18] [31] [39], nancial loss. quadratic programming [12], Lagrange relax- In this paper, we adapt the firefly algorithm ation method [12] [49], network flow method to economic dispatching problem. For this, [14], direct search method [56] reported in we describe in the second section the eco- the literature are used to resolve such prob- nomic problem and its formulation. In the lems. third section, we present the used Practically, economic dispatching problem is metaheuristic, its origin and its parameters. nonlinear, no convex with multiple local op- In the fourth section, we describe our adapta- timal points due to the inclusion of valve tion of the particle swarm optimization point loading effect, multiple fuel options (PSO) and the Firefly algorithm (FF) to the with diverse equality and inequality con- problem. In the last section, we discuss the straints. results of the metaheuristics on two net- Conventional methods have failed to solve works. Finally, we give a conclusion. such problems as they are sensitive to initial estimates and converge into local optimal so- 2 Power Economic Dispatching lution and computational complexity. Description: Heuristic optimization techniques based on Economic dispatch problem has become a artificial intelligence concepts operational re- decisive task in the operation and planning of search, such as Tabu search [29] [31], neural power system. The aim is to schedule the network [26] [28] [33] [36] committed generating units output so as to [58],evolutionary, programming [36] [45]
46 Informatica Economică vol. 16, no. 2/2012
[50] [64] and genetic algorithms [2] [7] [39] ( ) = + + + [51] [54] [57], simulated annealing [4] [54] 푛 <2 (5) ∑푘=1 푘 푔푘 푘 푔푘 푘 [55],ant colony optimization [20] [47], parti- where:퐸 푝 ME is 훿푘훼the푝푔푘푝 maximum훽 푝 allowable푐 푘 cle swarm optimization [9] [10] [22] [23] amount of pollutant휂 푒 during푀퐸 the dispatch peri- [24] [25] [27] [39] [40] [41] [48] [50] [52] od which is the EPA’s hourly emission target [53] [64] [65] provide the better solution. [65] and , , , , are emission co- PSO has gained popularity as the best suita- efficients. 푘 푘 푘 푘 푘 ble solution algorithm for such problems. 훼 훽 푐 훿 휂 Formulation : 3 Firefly Algorithm Having a network with N generators nodes Inspiration: where: Fireflies, belong with family of Lampyridae, : Power of generator k. are small winged beetles capable of produc-
푔 푘: Lost power. ing a cold light flashes in order to attract ma- 푝 : Requested power. tes. They are believed to have a capacitor- 퐿 Economic푝 dispatch problem can be repre- like mechanism, that slowly charges until a 퐷 sented푝 as a quadratic fuel cost objective func- certain threshold is reached, at which they re- tion as described in (1). lease the energy in the form of light, after which the cycle repeats [11] = ( ) 푛 Firefly algorithm, developed by [60] is in- 푔 (1) 푘 푔푘 spired by the light attenuation over the dis- 푓�푝 � �푘=1 푓 푝 : Total cost tance and fireflies’ mutual attraction, rather than by the phenomenon of the fireflies’ light : Cost of node k 푓 flashing. Algorithm considers what each fire- with considering equality constraint (2) to 푘 fly observes at the point of its position, when demand푓 and inequality constraint (3). trying to move to a greater light-source, than = (2) its own. Cold light is a light producing little 푛 or no heat. , 푔푘 퐿 퐷, (3) �푘=1 푝 − 푝 푝 Algorithm: 푘 The Firefly Algorithm is one of the newest 푃푔 min ≤ 푝푔 ≤ 푃푔 푚푎푥 A cost of a power generator can be for- meta-heuristics developed by Yang [59] [60] mulated by (4). [61] [62]. One can find few articles concern- 푝푔푘 = + + + ing the continuous firefly algorithm [1] [13] [15] [19] [30]. A validation of continuous sin ( , )(4) 2 푓푘 �푝푔푘� 푐푘 푏푘푝푔푘 푎푘푝푔푘 firefly algorithm on stochastic functions is So that: 푑푘 푒푘 푝푔 푚푖푛 − 푝푔푘 given in [59]. , , :Power Cost coefficients Sayadi et al. [42] proposed the first discrete , : Thermal power Cost coefficients. 푘 푘 푘 version for permutation flow shop problem 푎 푏 푐 푘 푘 using a binary coding of solution and a prob- The푑 푒4th term is for thermal power but can be ability formula for discretization. We can al- neglected. so find other discretization for economic problem such as [3][11][21]. Thermal Emission Constraint: Pseudo-code of the Firefly Algorithm (FF) In the case of thermal power, the atmospheric may look as follows: pollutants such as sulphur oxides (SOx) and nitrogen oxides (NOx) caused by conven- Procedure FF Metaheuristic (Nbr_gen: the tional thermal units can be modeled separate- max. number of generations) Begin ly. However the total emission of these pollu- γ: the light absorption coefficient tants which is the sum of a quadratic and an Define the objective function of f(x), exponential function can be expressed as (5) : where x=(x1,...... ,xd) in domain d Generate the initial population of fire- flies or xi (i=1, 2 ,..., nb)
Informatica Economică vol. 16, no 2/2012 47
Determine the light intensity Ii at xi where the first and second term is due to the via f(xi) attraction while the third term is randomiza- While (iter 48 Informatica Economică vol. 16, no. 2/2012 • Position. The new position is the calcu- the particle swarm optimization, the con- late by the formula 8: striction of velocity is based on the same pi =pi+vi (8) principle of the first corrector. where pi is the current position of a particle In the two algorithms we don’t admit any vi- i. olation of loss and demand constraint (for- • Velocity. We keep the equation of ve- mula (3)). locity vi of a solution i as shown in for- mula 9. 5 Computational Results vi = vi rand(besti- Data: The implementation is done in C++ builder -xi) (9) 32 bits in a personal computer with a Intel ω +φp xi)+φg rand(gbest CPU of 22.67 Ghz and a RAM of 3,12 GO. where φg, φp, are coefficients initialized de- In a first example, we consider a IEEE net- pending to sample . work of 14 nodes [38] with two generators G1 and G2.There costs are: The ω parameter decrease following the for- mula (10)[43]: 2 f1(Pg1) = 0.006 Pg1 + 1.5 Pg1 + 100 ( )× = (10) 2 _ f2(Pg2) = 0.009 Pg2 + 2.1 Pg2 + 130 ωmax−ωmin iter under equality constraint: ω ωmax − max gen We initialize , depending to the Pg1+Pg2 – PD– PL = 0 sample in the section V. And inequality constraint : max min ω ω 135 ≤ Pg1 ≤ 195 (MW) Firefly algorithm to economic dispatching: 70 ≤ Pg2 ≤ 145 (MW) To accelerate the FF algorithm, we suggest Requested Power is fixed to: that alpha parameter increase following the PD=259 (MW) formula (11): Lost power is constant and equal to: ( )× PL=16.2 (MW) = (11) _ αmax−αmin iter Correction and Constriction: In a second example, we consider a four unit α max gen − αmax Fireflies’ attractiveness, randomly move- thermal plant system (CS4)[43]. It has 4 gen- ments and Particle shifting provide float var- erators where the total power losses PL are iables which sometimes do not respect con- considered 0. The data for the 4 generators straints. We add before the intensity or the (cost coefficients and limits of generated fitness update two functions. The first cor- powers) are presented in Table 1. The total rects position so that it stays in the domain power demanded in the system is PD=520 [pmin, pmax]. The second corrects it so that the MW. total power will be equal to request power. In Table 1. Data of a CS4 thermal plant power System (2nd example) 2 Generator a($/MW ) b($/MW) c($) pmin pmax (MW) (MW) 1 0.00875 18.24 750 30 120 2 0.00754 18.87 680 50 160 3 0.0031 19.05 650 50 200 4 0.00423 17.9 900 100 300 In the third example, we consider a network Requested Power is fixed to: PD=518(MW) inspired from [34] with 5 generators. The We consider a constant Loss power of emission is considered and it should be up to PL=7.34 (MW) ME where ME=1700. under equality constraint: Pg – PD– PL = 0. Informatica Economică vol. 16, no 2/2012 49 The data for the generators (cost coefficients in Table 2. and limits of generated powers) are presented Table 2. Data of a 5 generators wind power System (3rd example) Gene- c($) b($/ a($/ e($/ d($ α β γ η δ pmin pmax rator MW) MW2) MW2) /MW2) (MW) (MW) 1 786.79 38.53 0.152 0.041 450 103.39 -2.444 0.031 0.503 0.0207 470 135 2 451.32 46.16 0.105 0.036 600 103.39 -2.444 0.031 0.503 0.0207 470 150 3 1049.99 40.39 0.028 0.028 320 300.39 -4.069 0.051 0.496 0.0202 340 73 4 1243.53 38.30 0.035 0.052 260 300.39 -4.069 0.051 0.496 0.0202 300 60 5 1658.57 36.30 0.021 0.063 280 320 -3.813 0.034 0.497 0.0200 243 73 In the three examples, PSO parameters are We optimize the first system with 10 fireflies φg,=1.75, φp=2.75, = , = in 50 iterations. We Remarque as shown in 0.4 while FF parameters are = Figure 1 that the firefly algorithm reach a max min , = 0.02 ,ω = 0.51, ω= 0. . good cost of 790,48 $/h since the 16th itera- max Results: ∗ α tion and its reach an optimum 781,95 $/h at 10 αmin 훽0 γ 1 the 21th iteration . f($) Improvement of cost 794 792 790 788 786 784 782 780 778 776 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 Iterations Fig. 1. Improvement of total cost with 10 fireflies in 50 iteration. 1st Exemple The Table 3 shows the average CPU time, average total cost and minimum cost found in a simulation of 20 replications. Table 3. The average and minimum output in 20 trials with the best generators powers. 1st Example Trials avg. CPU Min Pg1 Pg2 Nb. Cost time Cost (hs.) 10 783,26 0 781,958 195 80,2 In another simulation, we optimize IEEE-14 of the first population at iteration 0 has the dispatching using 20 fireflies. We can ob- cost of 784,1 $/h. At the 3rd iteration the op- serve from Figure 2 that since initial solution timizer can reach the optimum of 781,95 $/h. 50 Informatica Economică vol. 16, no. 2/2012 Cost improvement Cost(f $/h) 784,5 784 783,5 783 782,5 782 781,5 781 780,5 0 1 2 3 4 5 Fig. 2. Improvement of total cost with 20 fireflies in 50 iteration.1st example. The firefly algorithm can find a cost function system with the same initial population is equal to 781,958 $/h at 20th iteration environ given in Figure 3.We clearly observe that the with only 10 fireflies and in a CPU time be- firefly algorithm is better than PSO one for low 1 hundredth second. this example. An example of an optimization of the CS4 12953 12950 12947 12944 12941 12938 FF 12935 PSO 12932 12929 12926 12923 12920 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 Fig. 3. Improvement of cost with 15 fireflies and 15 particles in 50 iterations, 2nd example. The average , best and worst Costs as the av- where the best cost found by PSO is erage of CPU times in 100 trials are given in 12920.01172 $. Firefly algorithm average Figure 4.They can confirm that firefly algo- and worst costs are also better than particle rithm get the best cost of 12919.78711 $ swarm optimization costs. Informatica Economică vol. 16, no 2/2012 51 12960,00 12959,06 12950,00 12940,00 12928,99 12928,16 12930,00 12920,01 12922,33 12919,79 12920,00 FF 12910,00 PSO 12900,00 12890,00 Cost ($/h) Cost($/h) Cost($/h) Avg. Best Worst Fig. 4. Comparison of average, worst and best cost, 2nd example, number of tri- als=100,number of iteration=50, number of particles=15. The CPU time of PSO which is equal to gether so it has a complexity of O(N2) while 0.2619 hundredth seconds (Figure 5) seems the PSO algorithm is based only on the com- to be less than FF time (0.8339 hundredth se- parison of each particle with its best position conds). This is due that the firefly algorithm in history so it has a complexity of O(N). is based on the comparison of fireflies to- 0,900 0,834 0,800 0,700 0,600 0,500 FF 0,400 PSO 0,300 0,262 0,200 0,100 0,000 Avg. CPU time (hs.) Fig. 5. Comparison of average CPU time(hundredth seconds), 2nd example, number of tri- als=100,number of iteration=50, number of particles=15. The rounded best powers are 91.3108 and Figure 6 that the firefly algorithm reach a 64.8567 MW best cost of at the 6th iteration so it is un- We optimize the second system with 6 fire- doubtedly faster than PSO algorithm in term flies in 50 iterations. We detect as shown in of iterations. 52 Informatica Economică vol. 16, no. 2/2012 32700 32650 32600 FF 32550 PSO 32500 32450 32400 1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 Fig. 6. Improvement of total cost with 15 fireflies or 15 particles in 50 iterations, 3rd example One hundredth trials with 50 iterations give fly algorithm gives an optimum slightly litter the results in Figure 7. We view that the fire- than the PSO one. 32427,01 32427,0078 32427,008 32427,0059 32427,006 32427,0039 32427,004 32427,002 FF 32427,002 PSO 32427 32427 32427 32426,998 32426,996 Avg Cost($) Best Cost($) Worst Cost($) Fig. 7. Comparison of average, worst and best cost, 3rd example, number of tri- als=100,number of iteration=50, number of particles=15. The average CPU time and the best emission found are given in Table 4. Table 4. Comparison of average CPU time and best emission, 3rd example, number of tri- als=100,number of iteration=50, number of particles=15. Algorithm Avg. CPU time(hs.) Best Emission FF 36,44201 1617.878 PSO 0,519999 1617.873 We decrease the number of iterations to 20 emission constraint no improvement could be and of particles and fireflies to 6 and we de- possible beyond the values found. tect the same results with a CPU time of 10 The best powers of the 5 generators are 150, hundredth seconds of FF and 0.1 hundredth 135, 73 ,60 and 107.33995056 MW. second of PSO. We conclude that due to the Informatica Economică vol. 16, no 2/2012 53 6 Conclusion search”, Acta electrotehnica, 2011;vol. In this paper we showed the efficiency and 53, no.1. the feasibility of the recent algorithm which [6] Chan GW, Aganagie M, Waight JG, Me- is the firefly metaheuristic to resolve eco- dina J, Burton T, Reeves S. “Experiences nomic power dispatching problem. This with mixed integer linear programming problem with nonlinear function has also the based approaches on short term hydro particularity to have hard constraints. 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Mathe- pp. 1902– 1908. matics and Computers in Simulation [65] Zhao B, Jia Yi Cao, Multiple objective 2002;pp. 319–27. particle swarm optimization technique for [58] Yalcinoz T, Short MJ. Neural network economic load dispatch, Journal of approach for solving economic dispatch Zhejiang University Science. 2005,vol 6, problem with transmission capacity con- no.5,pp 420–427 Latifa DEKHICI is a PhD student in the department of computers sciences, sciences faculty, university of sciences and technology of Oran, Mohamed Boudiaf. In 2002, she had an engineering diploma in computer sciences in software engineering. She had his magister diploma in artificial intelligence and forms recognition in 2005. She works in the modeling, simulating and scheduling of the operating theatres of Oran. She had already published 3 pa- pers about operating theatres modeling and scheduling in reviews and jour- nals and she has among 21 manuscripts in several conferences. L. Dekhici is attached to LAMOSI (Laboratory of Simulation of Systems of Industries). Informatica Economică vol. 16, no 2/2012 57 Khaled BELKADI is actually a Research-professor at the USTO (university of sciences and technology of Oran, Mohamed Boudiaf) since September 6, 1987 .He has a PhD in computer science in July 1986 in Clermont Ferrand (France). He obtained a Doctorate in computer science in December 2006 in Oran (Algeria). He is a Member of the Scientific Council of the Faculty of Science at the USTO since February 2009. K. Belkadi works in modelling, simulation, optimization and performance evaluation of production systems and hospital systems. Pierre BORNE was born in Corbeil, France in 1944, he received the Master degree of Physics in 1967, the Masters of Electronics, of Mechanics and of Applied Mathematics in 1968. The same year he obtained the Diploma of "Ingénieur IDN" (French "Grande Ecole"). He obtained the PhD in Automat- ic Control of the University of Lille in 1970 and the DSc of the same Univer- sity in 1976. He became Doctor Honoris Causa of the Moscow Institute of Electronics and Mathematics in 1999, of the University of Waterloo (Canada) in 2006 and of the polytechnic University of Bucharest (Romania) in 2007. He is author or co-author of more than 200 Journal Articles and Book Chapters, and has presented 31 plenary lectures and more than 300 Communications in International Conferences. He is author of 21 books and has been the supervisor of 82 PhD theses. He has been Director of the CNRS group of research in Automatic Control, Scientific Director of the Ecole Centrale of Lille (French “Grande Ecole”), and President of the IEEE-SMC society and is presently President of the IEEE France Section