A Genetic Algorithm Solution to the Unit Commitment Problem Based on Real-Coded Chromosomes and Fuzzy Optimization

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1 A Geneic Algorihm Soluion o he Uni Commimen Problem Based on Real-Coded Chromosomes and Fuzzy Opimizaion Alma Ademovic #1, Smajo Bisanovic #, Mensur Hajro #3 Faculy for Elecrical Engineering Universiy of Sarajevo Sarajevo, Bosnia and Herzegovina 1 aademovic@ef.unsa.ba s.bisanovic@elekroprivreda.ba 3 mensur.hajro@ef.unsa.ba Absrac This paper presens a combined Geneic Algorihm Fuzzy Opimizaion approach o he Uni Commimen problem. The Uni Commimen problem is a high complex combinaorial opimizaion ask, nonlinear and large-scale. In order o obain a near opimal soluion in low compuaional ime and sorage requiremens, wih respec o all specified consrains, a Geneic Algorihm using real-coded chromosomes is proposed in opposie o he more commonly used binary coded scheme. Gahering daa from a lis of sric prioriy order he Geneic Algorihm generaes differen candidae soluions o he problem, as Fuzzy Opimizaion guides he whole search process under an uncerain environmen (varying load demand, renewable energy sources). A sysem consising of 10 generaing unis is presened o demonsrae applicaion of he proposed algorihm o he Uni Commimen problem. The obained resuls show saisfacory oucome in oal cos, compared o Dynamic Programming based applicaions and he sole Geneic Algorihm based soluion o he Uni Commimen problem. I. INTRODUCTION The Uni Commimen (UC) problem refers o he process of opimal power generaing unis sar-up and shu-down schedule deerminaion, subjec o forecas load demand over a shor-erm planning horizon (4-168h). The objecive of a generaion scheduling opimizaion problem is o minimize oal operaing coss, while meeing a large se of sysem operaing consrains. The problem iself is a complex mahemaical opimizaion ask, which includes boh ineger and coninuous variables. An exac soluion could be obained by complee enumeraion, sill his canno be applied o realisic power sysems, due o excessive compuaional ime and sorage requiremens [1]. Since improvemen of opimizaion echniques also means decreasing coss of elecric power uiliies, grea effor has been spen on finding a beer opimizaion soluion. The complexiy of he UC problem and he benefis of is soluion improvemen keep he research coninuously aracive. Various opimizaion echniques have been proposed and implemened up o now. A nice survey is presened in []. The mos commonly used are Branch and Bound mehod (BB), Prioriy Lising mehod (PL), Dynamic Programming (DP) and Lagrange Relaxaion mehod (LR). Moreover, some heurisic and sof compuing echniques such as Simulaed Annealing (SA), Neural Neworks (NN), Geneic Algorihms (GA) and Fuzzy Logic (FL) have been recenly examined. Geneic Algorihm was inroduced by J.H. Holland [3] and more recenly reaed and enhanced by D.E. Goldberg [4], L. Davis [5] and many ohers. I is a general-purpose random parallel search opimizaion echnique, inspired by naural selecion and geneic recombinaion. A very nice overview and possible implemenaion of GAs on he UC problem was given in [6] and [7]. Fuzzy Logic emerged from he proposal of fuzzy ses inroduced by L.A. Zadeh in [8]. I represens a mahemaical heory based on he principle of vagueness, or he basic idea ha he ruh of a saemen is a maer of degree. The implemenaion of fuzzy ses and Fuzzy Logic in he UC problem was nicely illusraed in [9]-[11]. This paper gives a combined Geneic Algorihm Fuzzy Opimizaion (FO) soluion o he UC problem and is mainly inspired by work given in [9]. The proposed approach performs guided sochasic search in a varying, large-scale and complex sae space, wih low compuaional ime and sorage requiremens. The paper illusraes a possible use of he realcoded chromosome GA combined wih he FO approach o UC problem solving. Inspired by Prioriy Lising, he work is using he idea of sric prioriy order liss generaion, o reduce he size of he problem search space. The economic dispach problem is being solved wih help of a LR based mehod [1]. Tesing was being conduced on a 10 generaing unis es sysem [13]. This paper is organized as follows. In Secion II he UC problem formulaion is presened. Secion III gives a shor overview of possible opimizaion echniques used in UC problem solving. The proposed algorihmic scheme is demonsraed wihin Secion IV, as simulaion resuls are presened in Secion V. Conclusion is given in Secion VI. II. PROBLEM FORMULATION The main goal of he Uni Commimen problem is o minimize he oal operaing cos of elecric power uiliies over he scheduling ime horizon, while meeing a variey of consrains. The overall objecive funcion of he UC problem for N hermal power unis over a scheduling ime horizon T is: /10/$ IEEE 1476

2 T N min TC = { Fi ( Pi ) vi + SUiyi + SDizi } (1) = 1i= 1 N number of hermal power unis; T number of considered ime inervals under sudy; P i power oupu of hermal uni i a hour ; TC oal coss of hermal unis under sudy ($); F i fuel cos of uni i a hour ($/h); SU i sarup cos of uni i a hour ($/h); SD i shu-down cos of uni i a hour ($/h); v i 0/1 variable ha is equal o 1 if uni i is on-line a hour ; y i 0/1 variable ha is equal o 1 if uni i is sared-up a he beginning of hour ; z i 0/1 variable ha is equal o 1 if uni i is shu-down a he beginning of hour. The major componen of he operaing cos funcion is he fuel cos F i ( P i ), which is usually modeled in a quadraic form as given in (): a i, b i, c i i ( i ) i i i i i F P = a P + b P + c () are cos coefficiens of uni i given in ($/MW h), ($/MWh) and ($/h), respecively. The sarup cos SU i is described as follows: X = + i SU i TSi BSi 1 exp (3) τ i TS i urbine sarup cos for uni i a hour ($/h); BS i boiler sarup cos for uni i a hour ($/h); X number of hours down for uni i a hour ; i τ i ime consan characerizing uni i cooling speed. The consrains for he problem are: Sysem power balance: N Pi = Pd (4.a) i = 1 Capaciy limis of hermal power unis: min max Pi vi Pi Pi vi (4.b) Minimum up ime consrains: on on ( X i, 1 Ti ) ( ν i, 1 ν i, ) 0 (4.c) Minimum down ime consrains: ( X i, 1 Ti i, i, 1 ) ( ν ν ) 0 (4.d) The minimum up ime and minimum down ime consrains sae ha a uni ha is running mus be up for a leas T on hours and a uni ha is down mus say down for a leas T hours. The minimum up ime consrains arise from physical consideraions associaed wih hermal sress on he unis and are designed o preven equipmen faigue. The minimum down ime consrains, on he oher hand, are based on economic consideraions inended o preven excessive mainenance and repair coss due o frequen uni cycling. Ramp-rae limis: Pi Pi, 1 RUi (4.e) Pi, 1 Pi RDi P d forecas load demand during ime inerval ; min P i minimum oupu limi on uni i ; max P i maximum oupu limi on uni i ; on T i minimum up ime ; T i minimum down ime ; on X i duraion a which uni i has been ON a ime ; X i duraion a which uni i has been OFF a ime ; RU i ramp-up rae limi of power uni i (MW/h); RD ramp-down rae limi of power uni i (MW/h). i III. PREVIEW OF UC OPTIMIZATION TECHNIQUES In his chaper a shor preview of he mos commonly used mehods o solving Uni Commimen calculaions is given. A. Prioriy Lising Mehod The Prioriy Lising mehod is based on sric prioriy order Uni Commimen scheduling, designed in order o mee he hourly load demand on a lowes operaing cos. Unis are being sored in ascending order, wih he lowes cos operaing uni a he op of he lis, so he mos economic base load unis are commied firs. The PL mehod is very fas and raher simple o implemen, bu is main disadvanages are ha i is no considered o be very accurae [14] and i gives schedules wih relaively high producion coss [7]. In his paper he idea of sric prioriy order lis creaion is used o reduce he size of he search space, as he economic dispach problem is being solved wih help of a Lagrange Relaxaion based mehod. B. Dynamic Programming Dynamic Programming [15] is an opimizaion echnique based on wo-direcional problem processing, performed over an adequae decomposiion in ime. Saring a he firs hour of he scheduling ime horizon, generaing unis are commied one hour a a ime. Obained combinaions and informaion needed for furher problem reamen are sored in he one which is called he forward pah. By backracking from he 1477

3 sae wih he leas operaing cos a he final hour, using he opimal link-back pah o he iniial sae, he schedule wih he mos economic soluion is obained. The main disadvanage of DP is he dimensionaliy of he Uni Commimen problem, since all possible combinaions for N hermal power unis and T ime inervals need o be sored (T ( N 1) calculaed values), which is considered large even for moderae size generaing sysems [7]. Eliminaion of hese limis could be conduced hrough relaxaion of some of he imposed consrains. One possibiliy is he inroducion of new variables: K for he number of saes in every ime inerval o be considered and M for he number of pahs (sraegies) o be sored in every sep. Based on he lis of sric prioriy order, he upper bound for K would become N (reduced from N 1) which is he number of power unis observed. Also, by rejecing combinaions wih high operaing cos i.e., soring only currenly bes candidae soluions for every of he considered T ime inervals, he M value or he number of pahs o be sored in every sep could also be reduced [1]. As DP always resuls in an opimal soluion, o disinguish boh menioned mehods, he laer approach will be called Dynamic Programming relaxed form (DPr). C. Geneic Algorihm Geneic Algorihm is a general-purpose random parallel search opimizaion echnique inspired by naural selecion and geneic recombinaion. GA usually sars wih a random iniialized populaion consising of P members called chromosomes. Chromosomes are binary or coninuous encoded srings, represening poenial soluions o he opimizaion problem. Each member becomes evaluaed on he finess funcion (objecive funcion), giving a measure of he soluion qualiy called he finess value. Upon candidae soluion selecion, recombinaion (crossover) is being performed, ending in a new candidae soluion populaion. In comparison o oher opimizaion echniques, GA operaes on a populaion of feasible soluions complying wih consrains (i.e., increasing he reliabiliy of finding a beer soluion) and generaes he nex populaion of poenial soluions combining currenly bes individuals. The laer propery provides he inheriance of he informaion on currenly bes individuals. Moreover, a muaion operaor makes i possible o deal wih local minima, even improving sampling of a complex search space. D. Fuzzy Logic Fuzzy Logic is a mahemaical heory ha provides convenien ways o represen uncerainies or vague conceps. Fuzzy Logic resembles human reasoning while making decisions in an uncerain environmen [10]-[11] and enables problem inerpreaion in a way more comprehensible o human beings, as well as easy problem processing and resolving, especially in lack of adequae mahemaical models or in case he soluion is no easy o find. The idea emerged from he concep of a fuzzy se, a heory based on he noion of parial membership, meaning each elemen belongs parially or gradually o a se of elemens, depending on is degree of membership. This propery makes he FL a naural ool when uncerainies are included wihin he problem. IV. PROBLEM SOLVING USING PROPOSED APPROACH Basic calculaions referring o condiional decision making when using he proposed approach are performed by he: PL mehod creaing a lis of sric prioriy order while making decisions on unis o be commied and Lagrange Relaxaion based mehod solving he economic dispach problem. A. GA approach The main feaure of he proposed GA approach for solving he UC problem is he coninuous encoded sring chromosome, consising of 4 sae variables (one for each hour of he 4h ime horizon) for N hermal power unis. Each variable k ( = 1,,...,4) represens he number of power generaing unis o be commied a hour of he 4h ime horizon, according o he lis of sric prioriy order (k є Z, 0 < k N, = 1,,,4). The proposed chromosome is shown in Fig. 1. k 1 k k 3... k k 3 k 4 k є Z, k > 0 ( = 1,,...,4) Fig. 1. Example of he proposed real-coded chromosome During he search process, consrains are being observed in following order: minimum up ime and down ime (4.c and 4.d), oupu limis (4.b) and ramp-rae limis (4.e), respecively. If any of he proposed variables k does no mee he given consrains, a penaly facor is being added o corresponding value of he performance funcion, making he soluion candidae exremely unfavorable, hus reducing he probabiliy of ha individual's conribuion o new generaion soluions. An evaluaion of he finess funcion akes place during he process of economic dispach problem decision making, and is performed over a se of operaing unis proposed by he GA. The whole procedure is being repeaed in a number of loops, he algorihm is always given a new iniial chromosome (he soluion reained during he former cycle), if ha soluion is beer hen he one obained before. The main advanages of his approach are low compuaional ime and sorage requiremens. B. GA FO approach The combined GA FO approach relaes o he use of fuzzy se conceps in he segmen of GA finess funcion evaluaion. Considering he fac ha characerisic values like sysem load demand mus be known in advance o UC problem solving, shor-erm forecasing mus be performed. Since hese values depend on behavior of consumers and oher variables, here are always errors in he forecas. Fuzzy se noaion is used o ake ino accoun errors, ending o find a beer soluion o he opimizaion problem under an uncerain environmen. To solve he UC problem using he proposed approach, firs he membership funcions for operaing cos (OC) and demand (L) are esablished [9]. 1478

4 The membership funcions are described as follows: 1) Membership funcion for operaing cos (OC): The membership funcion relaed o he operaing cos is defined in a way ha a high cos is given a low membership value. By keeping he membership funcion as high a possible, a desirable soluion wih low operaing cos can be obained. In Fig. he membership funcion for he fuzzy variable ΔC is illusraed. A membership value of 1 is assigned o any ΔC ha is less han zero. As ΔC becomes larger han zero, he degree of saisfacion will decrease. C L a, L f η L L +, L - 1, ΔL 0 ΔL 1 + η L + μ = L L (7) 1, ΔL < 0 ΔL 1 + η L L La Lf Δ L = 100% (8) Lf acual and forecas load; weighing facor; average percenage error when he acual load is greaer/lower han he forecas load. Fig.. Membership funcion for operaing cos (OC) The membership funcion μ C is expressed as exp ( ωδc), ΔC 0 μ C = (5) 1, ΔC < 0 TC σtc Δ C = max (6) σtcmax ω weighing facor; σ cos olerance facor ( σ < 1); TC max highes operaing cos. ) Membership funcion for load demand (L) L Fig. 3. Membership funcion for load demand (L) In he UC problem, L denoes he fuure hourly load demand, and can be obained hrough load forecasing. In Fig. 3 ΔL represens he percenage error in he load forecas. The membership value will be 1 for ΔL = 0, meaning ha no forecas error is observed. For oher ΔL values, he membership funcion decreases wih increasing forecas error. The membership funcion μ L is expressed as C L Membership funcion μ L is used o esimae he qualiy of a candidae soluion, i.e. GA chromosome, wih respec o load demand consrains. The shape of funcion μ L direcly influences he course of he search process. This work is using he funcion shape as he one given in [9], wih an excepion ha average percenage error L + and L - are being calculaed as a muliplied value of an assumed forecas maximum load demand percenage error ΔL on a 4 hour ime horizon. The opimizaion mehod implemened for solving he UC problem is based on he GA approach described in Secion A wih he finess funcion formed as follows [9]. Firs he percenage errors of he operaing cos and load demand are calculaed, o obain membership values μ C and μ L ( = 1,,...,T; T number of considered ime inervals under sudy). Evaluaion of he finess funcion of he curren candidae soluion (chromosome) consiss of deermining he characerisic value μ L* as he highes percenage error in load demand forecas over he considered ime horizon T, formally given by following expression: { } μ = min μ, μ,, μ (9) L* L1 L LT Using he compued values for μ C and μ L*, he value of he finess funcion can be obained as he membership value μ Dj ( j = 1,,,m; m populaion size), : { } μ = min μ, μ (10) Dj C L * The bes ou of m soluions is he candidae soluion wih he highes finess value, formally wrien as: { } μ = max μ, μ,, μ (11) D D1 D Dm The soluion obained is sored and he seps described above are repeaed unil a cerain crierion is me (maximum number of generaions, ime limi, ec). V. TEST RESULTS A sysem consising of 10 hermal generaing unis [13] has been used o demonsrae applicaion and efficiency of he 1479

5 proposed opimizaion echnique o he UC problem, along a 4 hour ime horizon. Parameers for each hermal uni are given in Table I, as sysem operaing consrains are given in Table II. TABLE I PARAMETERS OF TEN THERMAL UNITS uni Pmin Pmax a b c ($/MW h) ($/MWh) ($/h) # # # # # # # # # # TABLE II SYSTEM OPERATING CONSTRAINTS uni BS TS RU RD T on T ($/h) ($/h) # # # # # # # # # # TABLE III LOAD DEMAND FORECAST (P F) P F () P F () P F () P F () TABLE IV LOAD DEMAND ACTUAL (P A) P A () P A () P A () P A () The hourly forecas load demand P F is presened in Table III and he acual load demand P A in Table IV. Table V gives he specific GA parameers used in he experimen wihin he proposed algorihms. For he sake of comparison, hree differen approaches for UC problem solving were implemened and displayed, DPr, GA and GA FO, wihin wo differen cases. Firs case assuming P A = P F, while in he second one uncerainies are included, assuming P A P F. TABLE V PARAMETERS GA Number of Generaions 100 (wih increase of 50) Populaion Size 100 Crossover one poin (middle of he chromosome) Selecion ournamen selecion ( members) Loop Coun 5 The values for average percenage error parameers were calculaed as a hree imes greaer value han he assumed forecas maximum load demand percenage error ΔL on a 4 hour ime horizon. A. Case P A = P F This case assumes ha acual and forecas load demand mach. Unlike es example in Secion B, hese wo quaniies differ, he described example does no correspond o real case siuaions and is used for comparison purpose only. Resuls, wih respec o maching calculaion ime, are given in Table VI for each of he hree approaches individually. TABLE VI TEST RESULTS FOR CASE P A = P F Cos Comparison Approach DPr GA GA FO OC ($) OC OC DPr ($) OC OC DPr (%) compuaional ime (s) OC operaing cos ($); OC OC DPr comparison of operaing cos o leas saisfacory resul (mehod DPr) ($). The main characerisic of DPr is ha compuaion is being performed successively for every ime inerval on he observed ime horizon T, as he soluion obained is direcly affeced by he choice made in he sep prior o i. Uni on/ saus is being deermined upon he lis of sric prioriy order, obained by he PL mehod, only hose combinaions meeing he given consrains are aken ino accoun. For economic dispach, a LR based mehod is being used o allocae sysem demand among operaing unis during each hour of operaion. The combinaion resuling in lowes operaing cos is being aken, as all oher soluions are being omied. The obained resul is mainly far away from he opimal one because good resuls in individual ime inervals do no necessarily mean a good overall soluion. This could be seen from Table VI. Ignorance of resuls from oher poenial seps does no leave he possibiliy o make decisions upon heir 1480

6 qualiy. Considering he fac ha uni on/ saus grealy depends on previous sysem demand allocaion, he algorihm can easily converge o local opima, missing he overall good resul. From hese resuls i is possible o conclude ha he GA-FO approach provides a beer soluion (lowes operaing cos), which could be explained by he fac ha he GAs sochasic parallel search resuls in higher probabiliy o locae a beer resul, guided by FO under an uncerain environmen. The indicaed GA-FO approach was on he oher hand more compuaionally expensive, which is mosly affeced by he number of proposed repeaing loops. This can be reduced wih parial impac on he qualiy of resuls obained. B. Case P A P F If he more realisic case is considered, acual and forecas load demand do no mach, quie subsanial deviaion in calculaed resuls appears, ha is demonsraed in Table VII. TABLE VII TEST RESULTS FOR CASE PA PF Cos Comparison Approach DPr GA GA FO OC ($) OC OC DPr ($) OC OC DPr (%) compuaional ime (s) Considering he fac ha UC also includes uncerainies, he exended GA approach using FO may beer deal wih uncerain evens and herefore improve he soluion obained by he GA. The GA-FO approach differs only in he seleced finess funcion evaluaion, since i is based on fuzzy ses. The fuzzy se noaion of uncerain variables, such as load demand, serves o cover hourly forecas errors and herefore assures obaining an improved UC under an uncerain environmen. From hese resuls i is possible o conclude ha he GA-FO approach provides he bes soluion among hese hree approaches as in he previous case when P A = P F. The combined algorihm approach gives he higher reliabiliy under uncerain condiions, providing an improved soluion in more realisic cases when P A P F. VI. CONCLUSION The Uni Commimen problem is a nonlinear, combinaorial and large-scale opimizaion ask, which is mainly caused by specific consrains. Some of hem depend on generaing unis behavior and properies, like he ramp-rae limis, as ohers are affeced by unpredicable evens, like load demand, waer level, wind speed or solar irradiaion. This primarily affecs he dimensionaliy of he search space, he oucome of he search process in whole, as well as compuaional ime and sorage requiremens. From resuls obained i is possible o conclude ha he DPr approach can easily converge owards local minima missing he overall good resul. In conrary, he GA approach using sochasic parallel search covers a broader region of he observed search space, resuling in higher probabiliy o locae a good qualiy resul. Considering he fac ha he UC problem also includes uncerainies, by expanding he previous approach wih FO, he proposed algorihm may beer deal wih uncerain variables. FO has an impac on guiding he GA search and herefore assures finding a beer soluion. I could be concluded ha he combined approach gives higher reliabiliy for UC problem solving especially in he more realisic case when uncerainies occur. Considering ime complexiy, DPr resuled in lowes ime demand of s, while he GA and he GA FO approach ook s and s, respecively. Compuaional ime of he las wo cases is mainly affeced by he number of proposed repeaing loops. This can be addiionally reduced if needed, wih a parial impac on he final soluion. ACKNOWLEDGMENT The auhors would like o hank anonymous reviewers for heirs valuable suggesions and commens. REFERENCES [1] Allen J. Wood, Bruce F. Wollenberg, Power Generaion, Operaion and Conrol, New York: John Wiley and Sons, [] Narayana Prasad Padhy, Uni Commimen A Bibliographical Survey, IEEE Transacions on Power Sysems, Vol. 19, No., May 004. [3] John H. Holland, Adapaion in Naural and Arificial Sysems, Ann Arbor: The Universiy of Michigan Press, [4] David E. Goldberg, Geneic Algorihms in Search, Opimizaion and Machine Learning, Reading, MA: Addison-Wesley Pub. Co., [5] L. Davis, Handbook of Geneic Algorihms, New York: Van Nosrand Reinhold, [6] Randy L. Haup, Sue Ellen Haup, Pracical Geneic Algorihms, New Jersey: John Wiley & Sons Inc., Hoboken, 004. [7] A. Kazarlis, A.G. Bakirzis, V. Peridis, A Geneic Algorihm Soluion o he Uni Commimen Problem, IEEE Transacions on Power Sysems, Vol. 11, No. 1, February 006. [8] Lofi A. Zadeh, Fuzzy ses, Informaion and Conrol, Vol. 8, [9] Ruey-Hsun Liang, Jian-Hao Liao, A Fuzzy-Opimizaion Approach for Generaion Scheduling Wih Wind and Solar Energy Sysems, IEEE Transacions on Power Sysems, Vol., No. 4, November 007. [10] S. Chenhur Pandian, K. Duraiswamy, Fuzzy Logic Implemenaion for Solving he Uni Commimen Problem, Inernaional Conference on Power Sysem Technology - POWERCON: Singapore, November 004. [11] Seyedrasoul Saneifard, Nadipuram R. Prasad, Howard A. Smolleck, A Fuzzy Logic Approach o Uni Commimen, IEEE Transacions on Power Sysems, Vol. 1, No., May [1] Mensur M. Hajro, Mirza R. Kusljugic, Power Sysem Operaion and Conrol, Sarajevo: IP ''SVJETLOST'' Sarajevo, dd, [13] Sishaj P. Simon, Narayana Prasad Padhy, R.S. Anand, An An Colony Sysem Approach for Uni Commimen Problem, Elecrical Power & Energy Sysems 8, Elsevier Ld., 006. [14] Kris R. Voorspools, William D. D'haeseleer, Long-erm Uni Commimen opimizaion for large scale power sysems: uni decommimen versus advanced prioriy lising, Applied Energy, Elsevier Ld., 003. [15] Richard Bellman, Dynamic Programming, New Jersesy: Princeon Universiy Press. (1957.), Dover paperback ediion,