Transcript of "Bid based economic electrical load dispatch using improved genetic algorithm"
Bid-Based Economic Electrical Load DispatchUsing Improved Genetic AlgorithmGwo-Ching Liao IEEE MemberAbstract This paper presented an improved algorithm-IsolationNiche Immune Genetic Algorithm for solving Bid-based EconomicDispatch (INIGA-BED) in a power system. Economic Dispatch de-termines the electrical power to be generated by the committedgenerating units in a power system so that the generation cost canminimized, while satisfying various load demands simultaneously.The model of Bid-Based Economic Dispatch is proposed in order tomaximize the social profit under the competitive environment ofelectricity market. This model synthetically considers various con-straints on ramp rates, transmission line capacity and polluting gasemission constraints etc. The Isolation Niche Immune Genetic Al-gorithm was induced as a new solution for this model. With the in-troduction of niche technology, the capability of the immune ge-netic algorithm in dealing with the optimization of multi-peakmodel function was enhanced. This paper proposed the Niche basedon Isolation mechanism which possesses with biological basis. Itnot only effectively ensures the diversity solutions of the group, butalso has a strong ability to guide evolution.Index Terms Isolation Niche, Immune Genetic Algorithm, Eco-nomic Dispatch, Bid-Based.I. INTRODUCTIONThe Economic Dispatch (ED) plays an important role inthe operation of power system as well as real-time control,and is of great significance in improving the economy andthe reliability of the operation of power system. With the re-form of power industry system, the traditional economicdispatch of power system has been given a lot of new mean-ing. In order to improve the competitiveness of the partici-pating parties as well as the most optimal allocation of re-sources in the power system, the economic dispatch inpower market has been converted from the traditional goalof cost minimization to the goal based on Bid-Based mecha-nism and maximization of social profit. In the market, theparticipants present their own one-day or one-hour in ad-vance bid curve (including bid 1ding power for eachtime-period, electricity price and the corresponding operat-ing parameters) to the Independent System Operator (ISO).The ISO, in meeting the context of systemic economy andsecurity as well as the objective of maximizing the socialbenefits, conducts the economic dispatch of each tradingtime-period. The Bid-based Economic Dispatch Model(BED) - proposed by this paper not only took into ac-count of the bidding perspectives of generation and users,but also incorporated the users’ flexible effect to make elec-tricity as a commodity to adjust the supply and demand aswell as the stability of electricity price through its own lev-Gwo-Ching Liao is with Department of Electrical Engineering, Fortune In-stitute of Technology, Kaohsiung Taiwan.(E-mail:email@example.com)erage. The solution of the economic dispatch model pro-posed in this research, that took into the consideration ofconstrains of ramp rate of unit, the security of grid, envi-ronmental pollution, was to calculate the operational trajec-tory of each unit in continuous region of time and space, be-ing more complicated in nature with features of high order,nonlinear, and multiple constraints. This research used a newevolutionary computation method, Isolation Niche ImmuneGenetic Algorithm [8-10] to solve the Bid-based EconomicDispatch. There are many ways in the past were used tosolve the economic dispatch, such as: dynamic programmingmethod [11-13]. The advantage of this method is that thecost function can be for discontinuity or monotonically in-creasing; while the disadvantage of it is that when the num-ber of units increases, the demand for its computer memoryand computing time required increases exponentially, andsince then, owing to multiple local minimums it has, thismethod often can only find a suboptimal solution. On theother hand, the recent development of artificial intelligenceoptimization methods has been used by some scholars tosearch for optimal methods in economic dispatch, such asSimulated Annealing (SA) [14-16], Genetic Algorithm (GA)[17-21], Evolutionary Programming (EP) [22-24], and Evo-lutionary Strategy (ES) , these are the best optimizationtechniques with the ability to find the global optimal solution.We use the Immune Genetic Algorithm to discriminate bothantibody affinity and concentration, which allows its optimalsolution effectively to escape the trap of optimal solutions inthe region as well as to maintain the diversity of populationfor avoiding the solution prone to be premature. Therefore,we integrated these two methods; capturing individual ad-vantages to become a superior method for solving the eco-nomic dispatch problem.II. THE MATHEMATICAL MODEL BASES ON BID-BASEDECONOMIC DISPATCH IN POWER SYSTEMA hypothesis was set while constructing the mathematicalmodel for Dynamic Economic Dispatch in a market envi-ronment. In a competitive of electricity market all bid curvesare summated by participants (power producers and users)equal to the individual generation cost functions and usersbenefit functions in the market environment. Based on thishypothesis, the social profit is the total user benefit minus thetotal generation cost. To simplify the calculation, the bidcurves submitted by participants are expressed with quad-ratic function approximation.2.1 Objective functionA mathematical model based on the power systemProceedings of 2011 8th Asian Control Conference (ASCC) WeB2.5Kaohsiung, Taiwan, May 15-18, 2011- 1387 -
Bid-Based Dynamic Economic Dispatch under the condi-tion of meeting system operation constraints, is used to op-timize the generator output and user load to maximize thesocial profit. The mathematical model for this system is asfollows:( ) ( )L gT L dj j ,t i i ,tt 1 j 1 i 1T gd CBM ax F(1)and( ) ( )2j j,tj,t dj dj j,t dj dd dB c b a (2)( ) ( )2i gi gi i,tgii,t i,t gg gC c b a(3)i = 1, 2, ..., Lg, j =1, 2,…, L d, t =1, 2……,Twhere T is the total number of time slots; LL gdarethe set of generator system putting into operation and the setof users, respectively; g ti,is for the power output of unit iat time slot of t; d tj,for the user load in time slot of t;( )i i,tgC for bid function of the unit i in the electricity market,gi gi gic b a for constant coefficients of the bidding func-tion ; )( ,dB tjifor the bid function of user j in the electric-ity market, abc djdjdjfor the constant coefficients oftender curve .2.2 Constraints(1). The balance between load and generationLg Ldtj,ti,ti 1 j 1g d Loss(4)t =1, 2……,Twhere Losst is the loss of power of the system distribu-tion network in the time slot t; due to the consideration of theconstraint of branch flow, Losst is obtained by the flow al-gorithm.(2). Maximum(minimum) units bid generation poweri,tmin i,t i,tmaxg g g (5)i = 1, 2, ..., L gwheregg titi max,min,,is for the minimum(ormaximum) bid generation of power unit i at time slot t, re-spectively.(3). Minimum (or maximum) users bid loadj,tmin j,t j,tmaxd d d(6)j = 1, 2, ..., Ldwhere ,, ,j tmin j tmaxd dis for the minimum(or maximum)users bid load of user j at time slot of t, respectively.(4). The constraints of unit ramp rate( )Ri Rii,t 1 i,tt tg g UD (7)where U Ri and D Ri are the maximum rate of up anddown of unit per unit; t is for the duration of each timeslot.(5). The constraint of transmission line capacitylmin l,t lmaxF F F (8)where,l tFis transmission power of time slot t, branch l;lmaxFfor the maximum limits of transmission power ofbranch l.(6). The constraint of emission( )LgTii,tt 1 i 1APMgH (9)where ( )ii,tH g is the linear function of emissions of gen-erator i, APM is the index of allowable maximum amount ofemissions.III. THE PRINCIPLE OF IMMUNE GENETIC ALGORITHM BASEDON ISOLATION NICHE3.1. The immune genetic algorithm based on Isolation Niche1). Dividing into several small antibody groups by IsolationNiche method and codingWe mimicked the lymphocytes and antibody of humanbody immune system to solve economic dispatch. The codein the immune system is somewhat similar to genetic algo-rithms, its data structures of the genetic factors - antibodycoding was shown in Figure 1. First we use Isolation Nichemethod to divide a large population into separated smallpopulations. And every one small population there is a totalof Z antibodies coding as the demonstration in Figure 1 ,each antibody is with M number of genetic factors, whichmeans that there are Z number of antibodies in one genera-tion, each gene locus in each antibody is marked as k1, k2, ...,k, ..., ks.- 1388 -
itFigure 1 The code of antibody2). The calculation of diversity and affinity:Suppose there were Z numbers of antibodies in antibodypool, each antibody is with M number of genetic factors, inhere, we proposed a concept of "information entropy" ex-pressed as follows:( )1,g 2 ,g n ,gn ,gi,t i,t i,t i,t, , ,x x xX (10)whereX gnti,,represents as any antibody in human antibodies.N represents as the number of antibody (n = 1, 2, ..., u,v, ..., Z),g is the number of evolution generations,i is the number of genes of antibody n (i = 1, 2, ...., M).Assuming that each antibody has i units of electricity gen-eration, i is determined by the generation unit of power sta-tion,( )t 1Sn,gi i,t i,t i,tlogIE X P P (11)where ),,(XIEgntiirepresents the i-th "information en-tropy" of the n-th antibody, P ti, is the probability of geneticfactor t coming from the gene i, and the value of P ti, isset between [0.1,0.9], its calculation of the genetic character-istics uses the equation (12), while the "average informationentropy" is expressed asM( )M( )n ,g n ,g1i ,ti ,t ii 1IE XX IE (12)where the value of IEiis set from 0.1 to 0.4, as shown inequation (12) which indicates the entropy can be expressedas the diversity of antibody population.This research used two calculation formula of affinity;the first one expresses the affinity between the two antibod-ies:( )( )u ,gi,t1v,gbi,t u ,g v,gi,t i,tIE ,X X,X XA (13)where the ( )i,u,g v,gi,t t,b X XA indicates the affinity be-tween two antibodies, u,gi,tXand v,gi,tXjust represents theinformation entropy between two antibodies v and u.In equation (13), when the affinity between the two antibod-ies is small, it represents a greater diversity.The second calculation of affinity is based on the rela-tionship between antibody and antigen, which is expressed asequation (14) as follows:( )( )u ,gi ,t u ,gi ,t1gs1A XF Xe(14)where the )(,,XFsgutiis a adaptive value, which is usedto show the similarity between antibody u,gi,tXand antigenand is written as follows:min max min,,1( )) ( )(u gi tT T TTs XFF F F F(15)TF is as mentioned in equation (1),maxTF is the maximum value ofTF ,minTF is the minimum value ofTF .3.2 The implementation of new immune genetic algorithmStep 1: definition of antigen and antibodyAntigen 1: The values of objective function ( )u ,gi,tFs Xof equation (15).Antigen 2: Constraints.Antibody: reasonable solution.Step 2: The generation of initial antibody populationThe production of different sub-group of antibody, eachgroup is consisted of different adaptive value of antibodies.Step 3: The computation of antibody evolutionary1). )(,,XFs gutiis the adaptive value calculated from equa-tion (15).2). Use equation (16) to calculate the concentration( )u ,gi,tn XC of antibod u ,gi,tX( ) ( )Dv ,gu ,g u ,gi ,t i ,t i ,tv 1C1n ,DC AX X X (16)where- 1389 -
( ),otherwise10( ) OL OHu,g v,gu,g v,g i,t i,ti,t i,t,C C Cb, X XX X,AA (17)D represents the number of antibody; such as when u = 1and v = 4, then D = 4-1 = 3.,OL OHC Crepresent predetermined thresholds, where, thevalue ofOHCis set to 0.96 while the value ofOLCis set to0.9.Step 4: Select the better solution (antibody), and store it inthe memory cells.Those antibodies having higher affinities with the antigensare selected and store into memory cells. Since the numberof memory cells is limited, the antibodies with lower affini-ties will be replaced when the cells are full.Step 5: use equation (18) to calculate the expected reproduc-tion rate of the u-th solution.( )( )( )u ,gu ,g i,ti,t u ,gi,tAg Xr XCn XE (18)As the equation (18) shown, when a solution is with lowerconcentration and i higher affinity, its regeneration ratio isgreater.Step 6: The determination of crossover rate and mutation ra-tio by fuzzy systemIn the Immune Genetic Algorithm (IGA), the Fuzzy Sys-tem (FS) method is used to adjust the crossover and mutationrates to achieve a proper population size. The FS comprisesfour basic elements: a fuzzifier, a fuzzy rule base, an infer-ence engine and a defuzzifie. The crossover and mutationrates can be adjusted based on statistics (such as best, aver-age fitness and fitness variance).Step 7: The conduction of crossoverThe so-called “selected-window” method is used to selecta stated number of genes, as a window, for the two parentlists. The window genes for Lists A and B are then altered.The method introduced by F. Zhuang and F. D. Galiana isadopted to safeguard the list structure, and the crossover ratiois adjusted by the FS method.Step 8: The progress of mutationMutation is the primary GA operation that promotes a newregion for exploration in the search space. According to amutation ratio, pre-determined by the FS, mutation is a ran-dom change in value for a selected position, such as from“1” to “0” or from “0” to “1” .Step 9: Annealing Immune Operator (AIO)The annealing immune operator consists of two parts,namely (a).Vaccine inoculation, and (b).Annealing immunealgorithm.(a). Vaccine inoculation1). Calculate and select the best adaptive value of the indi-vidual parent, that is the largest value of Fs of the solutionof X gnti,,.2). And then comparing the solutions of the rest of the anti-bodies one by one to choose the one with the best adaptivevalue.(b). The conduction of annealing immune algorithm.Decisions and choices of the next-generation were madeby comparing the above stated inoculation of vaccine (solu-tion). If the inoculated adaptive value of next-generation ishigher than the value of parent, it will be accepted to the so-lution of next-generation as shown in equation (22); if not,use the discrimination equations from (23) to (25) will beused to decide whether to include it into next-generation so-lution.1). If ( ) ( ))n,g 1 n,gi,t i,ts s ( sF F FX X > 0 (22)2). If ) ( ))n,g 1 n,gi,t i,ts s ( sF F FX X 0 (23)The acceptance of the solution is determined by the prob-ability obtained from equation (24).( )( )1s rs/ r1,TTexpP FF(24)1rr 0T T (25)where the s( )F represents the adaptive value of anti-body with the value set between [0,1],T rrepresents the current temperature,T0represents the initial temperature, such as 1,000 ,r represents the stacked algebra, such as 0 to 1,000,represents the cooling rate value, 0 < <1.Step 10: Antibody updatedThe affinities of antibodies were compared using the“Roulette Wheel Selection” to determine its turnover rate, inother words, when a solution is with greater affinity, thewheel will possess large areas thus determining the solutionof this antibody in the new pool is with more number of itsnew generation, and the average affinity of antibody pool ofnew generation calculated by step 3 is higher than the solu-tion of the older generation.The updated rate can be obtained by equation (26) as fol-lows:- 1390 -
( )( )( )u ,gg i ,tu ,gi ,tp n,gg i ,tnA XC XA X(26)where the ( )u,gg i,tA X represents the affinity between an-tibody u and antigen.Step 11: Stopping criterionIf the values of generation evolution meet the set, the cal-culation steps can be stopped then output the result, other-wise go to Step 3.IV. SIMULATION AND RESULTS(1). Basic informationThe simulation system used in this research includednine transmission lines, 7 Buses, 4 generators and 3 users.Figure 2 demonstrates this system. The node for power gen-erator 1 was set as a balance node. A three hour time slot wasset to solve the model based on the Bid-Based Dynamic Eco-nomic Dispatch. This research also set the power generator’semission as a linear function. The power generator 1 emis-sion constant coefficient value was 0.7lb/MBTU, that forpower generators 2 and 3 were 0.5lb/MBTU, and that forpower generator 4 was 0.2lb/MBTU. The parameters and ba-sic information (biding data) for 4 power generators areshown in Table 1.Table 1 4 power generators’ bidding dataNo. ofGenerator cgi b gi agiminMWG maxMWG1PG 650 8.35 0.0017 100 7802PG 610 8.64 0.0033 80 6203PG 420 7.22 0.0038 50 5404PG 150 6.37 0.0052 10 410Figure 2 4 Units 7 Bus System Structure(2). The model bases on Bid-based Economic Dispatchwithout the consideration of some constraintsTo obtain the effective isolation niche immune geneticalgorithm (INIGA) application to the dynamic economicdispatch model, and simplify the calculation, this researchdid not consider the transmission line capacity Ramp Rateand emissions constraints. Table 2 shows the final powerproducer and user bid amount obtained from INIGA.Table 2 The optimization results of power generators and users’ bidding ofcase4PG3PG2PG1PGThis research used dynamic programming (DP) methodand genetic algorithms (GA) together, under the same condi-tions, to conduct the optimal system dispatch. Table 3 showsthe optimal results obtained from the three algorithms. Thetable data shows that the GA and INIGA methods are clearlybetter than the DP method, as the required time for INIGAmethod calculation was 8.35 sec. Calculation for the GAwas 17.22 sec., while the DP method was 21.89 sec., whichdemonstrates that the calculation speed of the INIGAmethod was faster than both GA and DP. From the perspec-tive of algebraic stack convergence, the INIGA converged tothe optimal solution after the 18th generation while the GAand DP calculated to the 45th and 65th generations. Thisdemonstrated that the convergence ratio of the INIGAmethod was significantly more superior to that of the GAand DP methods.Table 3 The results comparison of different calculation method under thesame conditions.(3). The consideration of the constraint of ramp rates of unitIn here the upper and lower ramp rate limits for systempower generator 1 and 2 are 60 MW/h and 20 MW/h, re-spectively, and the limits for power generators 3 and 4 are40 MW/h and 10 MW/h, respectively, taking the unit con-straints into account, the optimal dispatch results using theINIGA method are shown in Table 5. Adding the ramp rateconstraint, the total social profit dropped form U.S.$ 72,115to U.S.$ 71,256.- 1391 -
Table 4 The optimization results of power generators and users’ biddingunder the constraints of ramp rate.1PG3PG2PG4PGV. CONCLUSION(1). This research proposed the model based on Bid-basedEconomic Dispatch. This model could optimize the powergenerators’ economic dispatch and operation and controlproblems of users’ electric power system and demonstratedthat the Isolation Niche Immune Genetic Algorithm was aneffective tool for solution.(2). As the economic dispatch is with features of high di-mensional, nonlinear, constraints, this research proposed aIsolation Niche Immune Genetic Algorithm, through a cal-culation analysis to a 7 Buses to compared with the dynamicprogramming genetic algorithm Simulated AnnealingEvolutionary Programming, to verify the correctness and ef-fectiveness of the algorithm.(3). The best advantages of Isolation Niche Immune GeneticAlgorithm is its fast convergence, usually finding the optimalsolution after the implementation for 30-50 generations. Forthe operation and control of power system with large-scale,high order, nonlinear, multi-constrained range, the IsolationNiche Immune Genetic Algorithm seemed to be an effectiveand practical tool for solution.VI. REFERENCES F. N. Lee and A. M. Breipohl, “Reserve Constrained Economic Dis-patch with Prohibited Operating Zones”, IEEE Trans. On Power Sys-tem, Vol. 8, No. 1. pp. 246-254, Feb. 1993. D. C. Walters and G. B. Sheble, “Genetic Algorithm Solution of Eco-nomic Dispatch with Valve Point Loading”, IEEE Trans. On PowerSystem, Vol. 8, No. 3, pp. 1325-1332, Aug. 1993. F. Li, R. Morgan and D. Williams, “Hybrid Genetic Approach toRamping Rate Constrained Dynamic Economic Dispatch”, EPSR,Vol.43, pp. 97-103, Nov. 1997. H. T. Yang, P. C. Yang and C. C. Huang, “Evolutionary ProgrammingBase Economic Dispatch for Units with Non-Smooth Fuel Cost Func-tions”, IEEE Trans. On Power System, Vol. 11, No. 1, pp. 112-118,Feb. 1996. D. Attaviriyanupap, H. Kita, E. Tanaka and J. Hasegawa, “A HybridEP and SQP for Dynamic Economic Dispatch with Non-Smooth FuelCost Functions”, IEEE Trans. On Power Systems, Vol. 17, No. 2, pp.411-416, May, 2002. X. S. Han, H. B. Gooi and D. S. Kirschen, “Dynamic Economic Dis-patch: Feasible and Optimal Solutions”, IEEE Trans. On Power Sys-tems, Vol. 16, No. 1, pp. 22-28, Feb. 2001. W. R. Barcedo and P. Rastgonfard, “Dynamic Economic Dispatch Us-ing the Extended Security Constrained Economic Dispatch Algorithm”,IEEE Trans. On Power Systems, Vol. 12, No. 2, pp. 961-967, May1997. T. K. A. Rahman, Z. M. Yasin and W. N. W. Abdullab, “Artificial Im-mune-Based for Solving Economic Dispatch In Power System”, Na-tion Power & Energy Conference, pp. 31-35, Aug. 2004. G. C. Liao and T. P. Tsao, “Application Evolving Immune Algorithmfor Short-Term Thermal Generating Unit Commitment Problem”, IEEProc. Generation, Transmission & Distribution, Vol. 153, No. 3, pp.309-320, May 15 2006. G. C. Liao and T. P. Tsao, “Using Chaos Search Immune Genetic Algo-rithm and Fuzzy System for Short-Term Unit Commitment”, Interna-tional Journal of Electrical Power & Energy Systems, Vol. 28, No. 1,pp. 1-12, January 27, 2006. D. W. Ross and S. Kim, “Dynamic Economic Dispatch of Generation”,IEEE Trans. On Power Apparatus and Systems, Vol. 99, No. 6, pp.2060-2088, 1980. R. R. Shoults, S. V. Venkatesh, S. D. Helmick and M. J. Lolla, “A Dy-namic Programming Based Method for Developing Dispatch Curveswhen Incremental Heat Rate Curves Are Non-Monotonically Increas-ing”, IEEE Trans. On Power Systems, Vol. PWRS-1, No. 1, pp. 10-16,1986. Z. X. Liang and J. D. Glover, “A Zoom Feature for a Dynamic Pro-gramming Solution to Economic Dispatch Including TransmissionLosses”, IEEE Trans. On Power Systems, Vol. 7, No. 2, pp. 544-549,1992. A. H. Mantawy, Y. L. Abdel-Magid and S. Z. Selim, “A SimulatedAnnealing Algorithm for Unit Commitment”, IEEE Trans. On PowerSystems, Vol. 13, No. 1, pp. 197-204, 1998. K. P. Wong and C. C. Fung, “Simulated Annealing Based EconomicDispatch Algorithm”, IEE Proceeding-C, Vol. 140, No. 6, pp. 509-515,1993. F. Zhuang and F. D. Glaiana, “Unit Commitment by Simulated An-nealing”, IEEE Trans. On Power Systems, Vol. 5, No. 1, pp. 311-318,1990. G. C. Liao and T. P. Tsao, “Application of a Fuzzy Neural NetworkCombined with a Chaos Genetic Algorithm and Simulated Annealingto Short-Term Load Forecasting”, IEEE Transactions on Evolution-ary Computation, Vol. 10, No. 3, pp. 330-340, June 12 2006. P. H. Chen and H. C. Chang, “Large Scale Economic Dispatch by Ge-netic Algorithm”, IEEE Trans. On Power Systems, Vol. 10, No. 4, pp.1919-1926, 1995. G. B. Shebel and K. Brittiy, “Refined Genetic Algorithm EconomicDispatch Example”, IEEE Trans. On Power Systems, Vol. 10, No. 1,pp.117-124, 1995. C. W. Richter and G. B. Sheble, “Genetic Algorithm Evolutionary ofUtility Bidding Strategies for the Competitive Marketplace”, IEEETrans. On Power Systems, Vol. 13, No. 1, pp. 256-261, 1998. T. T. Maifeld and G. B. Sheble, “Genetic Based Unit Commitment Al-gorithm”, IEEE Trans. On Power Systems, Vol. 11, No. 3, pp.1359-1370, 1996. S. A. Kazarlis, A. G. Bakirtzis and V. Petridis, “A Genetic AlgorithmSolution to the Unit Commitment Problem”, IEEE Trans. On PowerSystems, Vol. 11, No. 1, pp. 83-92, 1996. K. P. Wong and J. Yuryevich, “Evolutionary Programming Based Al-gorithm for Environmentally-Constrained Economic Dispatch”, IEEETrans. On Power Systems, Vol. 13, No. 2, pp. 301-306, 1998. J. Yuryevich and K. P. Wong, “Evolutionary Programming Based Op-timal Power Flow Algorithm”, IEEE Trans. On Power Systems, Vol.14, No. 4, pp. 1245-1250, 1999. A. J. Gaul, E. Handschin and C. Lehmkoster, “Establishing a RuleBase for a Hybrid ES/XPS Approach to load Management”, IEEETrans. On Power Systems, Vol. 13, No. 1, pp. 86-93, 1998.VIII. BIOGRAPHIESGwo-Ching Liao (M’04) He received hisMS-EE from the National Cheng KungUniversity, Tainan, Taiwan in 1991 and thePh.D degree from National Sun Yat-SenUniversity, Kaohsiung. He works onDepartment of Electrical Engineering, FortuneInstitute of Technology, Kaohsiung County,Taiwan. His interests are deregulation, powersystem operations and AI application in powersystem.- 1392 -