C-GRASP APPLICATION TO THE ECONOMIC DISPATCH PROBLEM

Transcription

1 C-GRASP APPLICATION TO THE ECONOMIC DISPATCH PROBLEM By INGRIDA RADZIUKYNIENE A THESIS PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE UNIVERSITY OF FLORIDA 2010

2 c 2010 Ingrda Radzukynene 2

3 I dedcate ths to my wonderful son, Matas 3

4 ACKNOWLEDGMENTS I am grateful to many people for supportng me throughout my graduate study n Unted States. Frst of all, I would lke to express my earnest grattude to my advsor, Dr. Panos M. Pardalos, for drectng ths study and readng prevous drafts of ths work. Wthout hs gudance, nspraton, and support throughout the course of my research, ths work would not be complete. Many thanks to Arturas who has been there for me, lstenng to me and supportng me. I am also thankful to my frends at the Center for Appled Optmzaton who mentally supported and made my student lfe more colorful. 4

5 TABLE OF CONTENTS page ACKNOWLEDGMENTS LIST OF TABLES LIST OF FIGURES ABSTRACT CHAPTER 1 INTRODUCTION Motvaton Lterature Overvew ECONOMIC DISPATCH (ED) PROBLEM ED Constrants Load-Generaton Balance Generaton Capacty Constrant Generatng Unt Ramp Rate Lmts Reserve Contrbuton System Spnnng Reserve Requrement Te-lne Lmts Prohbted Zone Objectve Functons Smooth Cost Functon Non-smooth Cost Functons wth Valve-pont Effects Non-smooth Cost Functons wth Multple Fuels Non-smooth Cost Functons wth Valve-Pont Effects and Multple Fuels Emsson Functon SOLUTION METHODS Contnuous Greedy Randomzed Adaptve Search Procedure (C-GRASP) Genetc Algorthms (GA) Smulated Annealng (SA) Constrants Handlng Penalty-Based Approach Heurstc Strategy EXPERIMENTS AND RESULTS Experments System

6 4.1.2 System System System System Results Case Case Case Case CONCLUSION REFERENCES BIOGRAPHICAL SKETCH

7 Table LIST OF TABLES page 4-1 Generatng unts characterstcs of fve-unt system Load demand Generatng unts characterstcs of sx-unt system Rump-up lmts and prohbted zones of sx-unt system Generatng unts characterstcs of 13-unt system Generatng unts characterstcs of 40-unt system Generatng unts characterstcs of 10-unt system Load demand for 24 hours Generaton costs for case Best soluton for case Best solutons for case Best results, when demand s1263 MW Generaton costs for 13-unt system wth demand 1800 MW Generaton costs for 40-unt system wth demand MW Best soluton for case

8 Fgure LIST OF FIGURES page 2-1 Example of cost functon wth two prohbted operatng zones Cost functon wth valve-pont effects Cost functon wth multple fuels

9 Abstract of Thess Presented to the Graduate School of the Unversty of Florda n Partal Fulfllment of the Requrements for the Degree of Master of Scence C-GRASP APPLICATION TO THE ECONOMIC DISPATCH PROBLEM Char: Panos M. Pardalos Major: Industral and Systems Engneerng By Ingrda Radzukynene August 2010 Economc dspatch plays an mportant role n power system operatons, whch s a complcated nonlnear constraned optmzaton problem. It has non-smooth and non-convex characterstc when generaton unt valve-pont effects are taken nto account. Ths work adopts the C-GRASP algorthm to solve dfferently formulated economc dspatch problems. The comparson of the feasblty and effectveness of the C-GRASP, SA and GA s gven as well. 9

10 CHAPTER 1 INTRODUCTION 1.1 Motvaton The economc dspatch (ED) optmzaton problem s one of the fundamental ssues n power systems to obtan optmal benefts wth the stablty, relablty and securty [52]. Essentally, the ED problem s a constraned optmzaton problem n power systems that have the objectve of dvdng the total power demand among the on-lne partcpatng generators economcally whle satsfyng the varous constrants. ED problem have complex and nonlnear nonconvex characterstcs wth equalty and nequalty constrants. Therefore, good solutons of the ED problem would result n great economcal benefts. Over the years, many efforts have been made to solve ths problem, ncorporatng dfferent knds of constrants or multple objectves, through varous mathematcal programmng and optmzaton technques [42]. In the conventonal methods such as the lambda-teraton method, the base pont and partcpaton factors, and the gradent methods, an essental assumpton s that the ncremental cost curves of the unts are monotoncally ncreasng pece wse lnear functons, but the practcal systems are nonlnear [52]. Hence, global optmzaton technques, such as the genetc algorthms (GAs), smulated annealng (SA), and partcle swarm optmzaton (PSO) have been studed n the past decade and have been successfully used to solve the ED. However, the references wth contnuous greedy randomzed adaptve search procedure (C-GRASP) applcaton to such type of problems hadn t appear yet. The am of ths work s to apply the C-GRASP to the ED problem and compare ts effectveness and produced soluton feasblty wth ones of other heurstc methods lke the GAs and SA. 10

11 1.2 Lterature Overvew Snce Carpenter ntroduced a network constraned economc dspatch problem n 1962 [9] and the frst paper n the area of dynamc dspatchng was publshed by Bechert and Kwatny n 1972 [6], a lot of researches have employed varous mathematcal programmng optmzaton methods for solvng ED problems [30]. These optmzaton technques can be classfed nto three man categores. The frst category contans determnstc methods that nclude the lnear programmng algorthm [26, 57, 69], quadratc programmng algorthm [18, 37], non-lnear programmng algorthm [39], etc. The LP method applcaton to the power-system reschedulng problem wth securty-constraned economc dspatch/control for multple-valved-turbne unts was gven by Stott and Marnho [57]. Rosehart et al. [48] dscovered that for the economc dspatch problem, SLP appears to be a better tool than SQP. An approach based on effcent SLP technques to solve the mult-objectve envronmental/economc load dspatch problem was presented by Zehar and Sayah [69]. Granell et al. [18] solved a securty constraned economc dspatch problem usng modfed SQP technques. A dual feasble startng pont s found by relaxng transmsson lmts and then constrant volatons are enforced applyng the dual quadratc algorthm. In [59] and [35], a securty constraned economc dspatch problem was solved by SLP and the nteror pont dual-affne scalng algorthm. Momoh et al. [37] proposed an IPM for ED problem formulated as lnear and convex QP. However each of tradtonal methods has some defects: t would generate large errors to use the lnear programmng algorthm to lnearze the ED model; for the quadratc programmng and nonlnear programmng algorthms, the objectve functon should be contnuous and dfferentable [30]. The second category contans the methods based on artfcal ntellgence. Artfcal ntellgence technology has been successfully used to solve the ED problem. A chaos optmzaton algorthm (CAO) has been proposed by Jang et al. [29] to deal wth the economc dspatch problem of a hydro power plant. Zhjang et al. [71] also appled a 11

12 COA and the smulaton results verfed that the proposed approach s effectve and precse. A mutatve scale COA was appled by Xu et al. [65] to the economc operaton of power plants. However, the results showed that the method s tme-consumng. An mproved mutatve scale COA hes been developed by Han and Lu [19]. Accordng to the authors, ther algorthm s hghly effcent and can be appled not only to ED but to many power system problems, such as economc operaton, OPF, system dentfcaton and optmal contro, as welll. In [36], Mahdad et al. proposed an effcent decomposed parallel GA to solve the mult-objectve envronmental/economc dspatch problem. In the frst stage, the orgnal network s decomposed nto mult sub-systems and the problem s transformed to optmze the actve power demand assocated wth each parttoned network. In the second stage, an actve power dspatch strategy s proposed to enhance the fnal soluton of the optmal power flow of the orgnal network. The proposed approach was tested on the Algeran 59-bus test system. The computatonal results showed the convergence at the near soluton and obtan a compettve soluton at a reduced tme. GAs wth fuzzy logc controllers to adjust ts crossover and mutaton probabltes was appled by Song et al. [56] to solve a combned envronmental economc dspatch problem. SA technques were used by Roa-Sepulveda and Pavez-Lazo [47], however, long computatonal tme to obtan an optmal soluton was reported. Tabu search was appled by Altun and Yalcnoz [2]. Smulaton results on power systems consstng of 6 and 20 generatng unts exhbted good performance. In [38], an applcaton of TS for solvng securty constraned ED problem was gven by Muthuselvan and Somasundaram. Base case and contngency case lne flow constrants were consdered. Tests on 66-bus and 191-bus Indan utlty systems revealed the relablty, effcency and sutablty of the proposed algorthm for practcal applcatons. The thrd category conssts the hybrd methods, whch combne two or more technques n order to get best features n each algorthm. Typcally, sgnffcant mprovement wth hybrd methods can be acheved over each of the ndvdual methods. 12

13 Hybrd methods ganed ncreasng popularty n the last 10 years. For the ED problem, Wong and Wong [63] combned an ncremental GA wth SA technques. Coelho and Maran [12] proposed a method combnng a DE algorthm wth self-adaptve mutaton factor n the global search stage and chaotc local search technques n the local search to solve an ED problem assocated wth the valve-pont effect. The same authors report another successful applcaton of chaotc PSO n combnaton wth an mplct flterng local search method to solve economc dspatch problems [13]. The chaotc PSO approach s used to produce good potental solutons, whle the mplct flterng s used to fne-tune the fnal soluton of the PSO. The hybrd methodology s valdated for a test system consstng of 13 thermal unts whose ncremental fuel cost functon takes nto account the valve-pont loadng effects. In [11], Coelho and Lee mproved PSO approaches for solvng an ED problem takng nto account non-lnear generator features such as ramp-rate lmts. Prohbted operatng zones n the power system operaton are developed as well. Ther algorthm combnes the PSO, Gaussan probablty dstrbuton functons and/or chaotc sequences. The PSO and ts varants are valdated for two test systems consstng of 15 and 20 thermal generaton unts, respectvely. A combnaton of chaotc and self-organzaton behavor of ants n the foragng process was presented by Ca et al. [8]. Ths algorthm was appled to ED problems wth thermal generators. The thess s organzed as follows: In Secton 2, we brefly dscuss a general ED problem formulaton. The methods appled to solve ED are shortly dscussed n Secton 3. Secton 4 descrbes expermental cases and presents calculaton results. We conclude wth Secton 5. 13

14 CHAPTER 2 ECONOMIC DISPATCH (ED) PROBLEM ED s one of the mportant optmzaton problems n power system operatons, whch s used to determne the optmal combnaton of power outputs of all generatng unts to mnmze the total fuel cost whle satsfyng varous constrants over the entre dspatch perods [67]. The tradtonal or statc ED problem assumes constant power to be suppled by a gven set of unts for a gven tme nterval and attempts to mnmze the cost of supplyng ths energy subject to constrants on the statc behavor of the generatng unts lke system load demand. Shortly, statc ED determnes the loads of generators n a system that wll meet a power demand durng a sngle schedulng perod for the least cost. Therefore, t mght fal to capture large varatons of the load demand due to the ramp rate lmts of the generators. Due to large varaton of the customers load demand and the dynamc nature of the power systems, t became necessary to schedule the load beforehand so that the system can antcpate sudden changes n demand n the near future. Dynamc ED s an extenson of statc ED to determne the generaton schedule of the commtted unts so that to meet the predcted load demand over the entre dspatch perods at mnmum operatng cost under ramp rate and other constrants [64]. The ramp rate constrant s a dynamc constrant whch used to mantan the lfe of the generators,.e. plant operators, to avod shortenng the lfe of the generator, try to keep thermal stress wthn the turbnes safe lmts [20]. Snce the volatons of the ramp rate constrants are assessed by examnng the generators output over a gven tme nterval, ths problem cannot be solved for a sngle value of MW generaton [20]. The objectve functon of dynamc ED s formulated as follows mnc (P) = T t=1 N =1 C (P t ) (2 1) 14

15 where N s the set of commtted unts; P s the generaton of unt ; C (P ) s the cost of producng P from unt ; T s the number of ntervals n the study perod. The fuel cost functons C ( ) s derved from the fuel consumpton functon that can be measured and are dscussed n Secton 2.2. The dynamc ED s not only the most accurate formulaton of the economc dspatch problem but also the most dffcult to solve because of ts large dmensonalty [3]. The DED problem s normally solved by dscretzaton of the entre dspatch perod nto a number of small tme ntervals, over whch the load demand s assumed to be constant and the system s consdered to be n a temporal steady state. Over each tme nterval a statc ED problem s solved under statc constrants and the ramp rate constrants are enforced between the consecutve ntervals [34]. In the DED problem the optmzaton s done wth respect to the dspatchable powers of the unts. Some researchers have consdered the ramp rate constrants by solvng SED problem nterval by nterval and enforcng the ramp rate constrants from one nterval to the next. However, ths approach can lead to suboptmal solutons [23]; moreover, t does not have the look-ahead capablty. Snce dynamc ED was ntroduced, varuos methods have been used to solve ths problem. However, all of those methods may not be able to provde an optmal soluton and usually gettng stuck at a local optmal. 2.1 ED Constrants The constraned ED problem s subjected to a varety of constrants dependng upon assumptons and practcal mplcatons. Usually, formulaton of ED problem ncludes such constrants as load generaton balance, mnmum and maxmum capacty constrants. To mantan system relablty and securty, spnnng reserve constrants and securty constrants can be added to the dynamc ED problem. The ncluson of the prohbted zones, ramp-rate lmts and other practcal constrants results n nonconvex ED of generatng unts. All these constrants are dscussed bellow. 15

16 2.1.1 Load-Generaton Balance The generated power from all the runnng unts must satsfy the load demand and the system losses gven by (2 2) N =1 P t = D t + Loss t, t = 1, 2,..., T (2 2) where D t s the demand and Loss t s the system transmsson loss. Ther sum represents the effectve load to be satsfed at the tth nterval. The transmsson lne losses can be expressed n terms of the unt outputs: Loss t = N =1 N j=1 P t B jp t j + N =1 B 0 P t + B 00 where B j s the jth element of the loss coeffcent square matrx, B 0 s the th element of the loss coeffcent, and B 00 s the constant loss coeffcent. Sometmes the last two terms are omtted. In a compettve envronment, the load-generaton balance constrant s relaxed and each generatng company schedules ts producton to maxmze ts profts gven a forecast of electrcty prces for the schedulng perod. As a frst approxmaton, each generatng unt could be optmzed separately n ths problem because of the decouplng made possble by the avalablty of prces at each perod. Dynamc constrants (such as ramp rates and mnmum up and down tme constrants) complcate the problem because a generatng company that owns a portfolo of unts must then decde whether to buy flexblty on the market or meet the dynamc constrants wth ts own resources [21] Generaton Capacty Constrant For normal system operatons, real power output of each generator s restrcted by lower and upper bounds as follows: P t + S t P max = 1, 2,..., N, t = 1, 2,..., T (2 3) 16

17 where P mn and P max P mn P t = 1, 2,..., N, t = 1, 2,..., T (2 4) are the mnmum and maxmum power produced by generator, S t s the reserve contrbuton of unt durng tme nterval t Generatng Unt Ramp Rate Lmts One of unpractcal assumpton that prevaled for smplfyng the problem n many of the earler research s that the adjustments of the power output are nstantaneous [43]. Therefore, the power output of a practcal generator cannot be adjusted nstantaneously wthout lmts. The operatng range of all onlne unts s restrcted by ther ramp-rate lmts durng each dspatch perod. So, the subsequent dspatch output of a generator should be lmted between the constrants of up and down ramp-rates [66] as follows P t+1 P t UR t (2 5) P t P t+1 DR t = 1, 2,..., N, t = 1, 2,..., T 1 (2 6) where UR and DR are the maxmum ramp up/down rates for unt and t s the duraton of the tme ntervals nto whch the study perod s dvded. The ncluson of ramp rate lmts modfes the generator operaton constrants (2 3, 2 4) as follows max(p mn, P t Reserve Contrbuton DR ) P mn(p max, P t 1 + UR ) (2 7) The maxmum reserve contrbuton has to satsfy followng constrants: 0 S t S max = 1, 2,..., N, t = 1, 2,..., T (2 8) where S max s the maxmum contrbuton of unt to the reserve capacty. Maxmum-ramp spnnng reserve contrbuton s defned as n (2 9) 0 S t UR t = 1, 2,..., N, t = 1, 2,..., T (2 9) where S t s the spnnng reserve of unt. 17

18 2.1.5 System Spnnng Reserve Requrement Suffcent spnnng reserve s requred from all runnng unts to maxmze and mantan system relablty [14]. There are many ways to determne the system spnnng reserve requrement. It can be calculated as the sze of the largest unt n operaton or as a percentage of forecast load demand or even as a functon of the probablty of not havng suffcent generaton to meet the load [64]. The spnnng rezerve can be defned by (2 10) N =1 S t SR t t = 1, 2,..., T (2 10) where SR t s the system spnnng reserve requrement for tme nterval t. Also, the system spnnng reserve requrement for nterval t can sometmes be gven by the followng equaton [20]: SR t = α d D t + α g max(p max scheduled at tme t, = 1, 2,... N) (2 11) where α d and α g are constants whch depend on the system requred relablty level [55]. Besdes the determnaton of the system spnnng reserve requrement, the ssue of allocaton the spnnng reserve among the commtted unts s very mportant; however, t has receved very lttle attenton n the dynamc ED lterature Te-lne Lmts The economc dspatch problem can be extended by mportng addtonal constrant lke transmsson lne capacty lmt gven by (2 12) P Tjk,mn P T jk + S jk P Tjk,max (2 12) where P Tjk,mn and P T jk,max specfy the te-lne trasnmsson capablty,.e. the transfer from area j to area k should not exceed the te-lne transfer capactes for securty consderaton [28]. Each area has own specal load and ts spnnng reserve [68]. 18

19 2.1.7 Prohbted Zone The generatng unts may have certan ranges where operaton s restrcted on the grounds of physcal lmtatons of machne components or nstablty, e.g. due to steam valve or vbraton n shaft bearngs. So, there s a quest to avod operaton n these zones n order to economze the producton [43]. These ranges are prohbted from operaton and a generator wth prohbted regons (zones) has dscontnuous fuel-cost characterstcs (Fg ) [53]. The acceptable operatng zones of a generatng unt can be formulated as follows P mn P t P l,1 (2 13) P u,j 1 P t P l,j, θ, j = 2, 3,..., n, t = 1, 2,..., T (2 14) P u,n P t P max (2 15) where n s the number of the prohbted zones n unt, θ s the set of unts that have prohbted zones, P l,j, P u,j are the lower and upper bounds of the jth prohbted zone. Fgure 2-1. Example of cost functon wth two prohbted operatng zones 2.2 Objectve Functons The dynamc ED problem has been solved wth many dfferent forms of the cost functon, such as the smooth quadratc cost functon (2 16) or the nonsmooth cost 19

20 functon due to the valve-pont effects (2 17). Also, a lnear cost functon [20] and pecewse lnear cost functon [27, 41] have been employed. For smooth cost functon t s usually assumed that ts ncremental cost functon. In some power systems combned cycle unts are used to supply the base load. For these unts the cost functon can be gven as lnear, pecewse or quadratc wth decreasng ncremental cost functon [41]. For unts wth prohbted zones, the fuel cost functon s dscontnuous and nonconvex. An nterestng departure from ths standard formulaton s the approach proposed by Wang and Shahdehpour [61] who nclude n the objectve functon a term representng the reducton n the lfe of the turbne caused by excessve rampng rates. Ths flexble technque makes possble a tradeoff between the system operatng cost and the lfe cycle cost of the generatng unts [21] Smooth Cost Functon The most smplfed cost functon of each generator can be represented as a quadratc functon as gven n (2 16) whose soluton can be obtaned by the conventonal mathematcal methods where a, b,c are cost coeffcents of generator. C (P t ) = a + b P t + c (P t )2 (2 16) Non-smooth Cost Functons wth Valve-pont Effects The generatng unts wth mult-valve steam turbnes exhbt a greater varaton n the fuel cost functons because n order to meet the ncreased demand a generator wth mult-valve steam turbnes ncrease ts output and varous steam valves are to be opened [67]. Ths valve-openng process produces rpple lke effect n the heat-rate curve of the generator. The ncluson of valve-pont loadng effects makes the modelng of the ncremental fuel cost functon of the generators more practcal [60]. Therefore, n realty, the objectve functon of ED problem has non-dfferentable property. Consequently, the objectve functon should be composed of a set of non-smooth cost functons. Consderng non-smooth cost functons of generaton unts wth valve-pont 20

21 Fgure 2-2. Cost functon wth valve-pont effects effects, the objectve functon s generally descrbed as the superposton of snusodal functons and quadratc functons [52] C (P t ) = a + b P t + c (P t )2 + e sn(h (P mn P t )) (2 17) where e and h are the coeffcents of generator reflectng valvepont effects. As shown n Fg , ths ncreases the non-lnearty of curve as well as number of local optma n the soluton space [60] compared wth the smooth cost functon due to the valvepont effects. Also the soluton procedure can easly trap n the local optma n the vcnty of optmal value Non-smooth Cost Functons wth Multple Fuels Snce the dspatchng unts are practcally suppled wth mult-fuel sources [49], each unt should be represented wth several pecewse quadratc functons reflectng the effects of fuel type changes, and the generator must dentfy the most economc fuel to burn. The resultng cost functon s called a hybrd cost functon. Each segment of the hybrd cost functon mples some nformaton about the fuel beng burned or the 21

22 Fgure 2-3. Cost functon wth multple fuels unts operaton. Thus, generally, the fuel cost functon s a pecewse quadratc functon descrbed as follows a 1 + b 1 P t + c 1 (P t )2 f P,mn t P t P t,1 a 2 + b 2 P t c (P ) =. a n + b n P t + c 2 (P t )2 + c n (P t )2 f P t P t,1 P t,2. f P t P t,n 1 P,max t (2 18) where are a p, b p, c p the cost coeffcents of generator for the pth power level. The ncremental cost functons are llustrated n Fg. (2.2.3) Non-smooth Cost Functons wth Valve-Pont Effects and Multple Fuels To obtan an accurate and practcal economc dspatch soluton, the realstc operaton of the ED problem should consder both valve-pont effects and multple fuels. The reference [10] proposed an ncorporated cost model, whch combnes the valve-pont loadngs and the fuel changes nto one frame. Therefore, the cost functon should combne (2 17) wth (2 18), and can be realstcally represented as shown n 22

23 (2 19) a 1 + b 1 P t + c 1 (P t )2 + e,1 sn(h,1 (P mn P t t,1,1 )) f P,mn P t P t,1 a 2 + b 2 P t c (P ) =. + c 2 (P t )2 + e,2 sn(h,2 (P mn P t t,2,2 )) f P P t,1 P t,2. a n + b n P t + c n (P t )2 + e,n sn(h,n (P mn,n P t )),n f P t P t,n 1 P,max t (2 19) Emsson Functon Due to ncreasng concern over the envronmental consderatons, socety demands adequate and secure electrcty,.e. not only at the cheapest possble prce, but also at mnmum level of polluton. In ths case, two conflctng objectves,.e., operatonal costs and pollutant emssons, should be mnmzed smultaneously [4, 5, 7, 62]. The atmospherc pollutants such as sulphur oxdes (SO x ) and ntrogen oxdes (NO x ) caused by fossl-fueled generatng unts can be modeled separately or as the total emsson of them whch s the sum of a quadratc [4] and an exponental functon and can be expressed as T t=1 N =1 α + β P t + γ (P t )2 + η exp(δ P t ) (2 20) where α, β, γ, η, and δ are emsson coeffcents of th generatng unt. 23

24 CHAPTER 3 SOLUTION METHODS 3.1 Contnuous Greedy Randomzed Adaptve Search Procedure (C-GRASP) Contnuous-GRASP (C-GRASP) extends the greedy randomzed adaptve search procedure (GRASP) that was ntroduced by Feo and Resende [16, 17] from the doman of dscrete optmzaton to that of contnuous global optmzaton n [24, 25]. It s descrbed as a mult-start local search procedure, where each C-GRASP teraton conssts of two phases, namely, a constructon phase and a local search phase [24]. Constructon combnes greedness and randomzaton to produce a dverse set of good-qualty startng solutons for local search. The local search phase attempts to mprove the solutons found by constructon. The best soluton over all teratons s kept as the ntal soluton. The advantages of ths method s smplcty to mplement and no requrement for dervatve nformaton Pseudo-code for C-GRASP s shown n (3.1). C-GRASP works by dscretzng the doman nto a unform grd. Both the constructon (see the hgh level pseudo-code 3.2) and local mprovement phases (see the hgh level pseudo-code 3.3) move along ponts on the grd. As the algorthm progresses, the grd adaptvely becomes more dense. The man dfference between GRASP and C-GRASP s that an teraton of C-GRASP does not consst of a sngle greedy randomzed constructon followed by local mprovement, but rather a seres of constructon-local mprovement cycles wth the output of constructon servng as the nput of the local mprovement, as n GRASP, but unlke GRASP, the output of the local mprovement serves as the nput of the constructon procedure [25]. Snce C-GRASP s essentally an unconstraned optmzaton algorthm, the constrants handlng strategy needs to be ncorporated nto t n order to deal wth the constraned ED problem. Approaches to manage these constrants are dscussed n secton

25 pseudo-code 3.1 C-GRASP (n, l, u, f ( ),MaxIters, MaxNumIterNoImprov, NumTmesToRun, MaxDrToTry,α) 1: f 2: for j 1,..., NumTmesToRun do 3: x UnfRand(l, u); h 1; NumIterNoImprov 0; 4: for Iter 1,..., MaxIters do 5: x ConstructGreedyRandomzed(x, f ( ), n, h, l, u, α); 6: x LocalSearch(x, f ( ), n, h, l, u, MaxDrToTry ); 7: f f (x ) < f then 8: x x ; f f (x ); NumIterNoImprov 0; 9: else 10: NumIterNoImprov NumIterNoImprov+1 11: end f 12: f NumIterNoImprov MaxNumIterNoImprov then 13: h h/2; NumIterNoImprov 0; {/}*make grd more dense*/ 14: end f 15: end for 16: end for 17: return x pseudo-code 3.2 ConstructGreedyRandomzedSoluton (Problem Instance) 1: Soluton ; 2: whle Soluton constructon not done do 3: MakeRCL(RCL); 4: S SelectRandomElement(RCL); 5: Soluton Soluton S; 6: AdaptGreedyFuncton(S); 7: end whle 8: return (Soluton); pseudo-code 3.3 LocalSearch(Soluton,Neghborhood) 1: Soluton* Soluton 2: whle Soluton* not locally optmal do 3: Soluton* SelectRandomElement(Neghborhood(Soluton*)); 4: f Soluton better than Soluton* then 5: Soluton* Soluton; 6: end f 7: end whle 8: return (Soluton*) 25

26 3.2 Genetc Algorthms (GA) Ths secton engages nto the concept of genetc algorthms that reflects the nature of chromosomes n genetc engneerng. GAs are a class of stochastc search algorthms that start wth the generaton of an ntal populaton or set of random solutons for the problem at hand. Each ndvdual soluton n the populaton called a chromosome or strng represents a feasble soluton. The objectve functon s then evaluated for these ndvduals. If the best strng (or strngs) satsfes the termnaton crtera, the process termnates, assumng that ths best strng s the soluton of the problem. If the termnaton crtera are not met, the creaton of new generaton starts, pars, or ndvduals are selected randomly and subjected to crossover and mutaton operatons. The resultng ndvduals are selected accordng to ther ftness for the producton of the new offsprng. Genetc algorthms combne the elements of drected and stochastc search whle explotng and explorng the search space [31]. More detals about GA can be found n [22, 46, 58]. pseudocode 3.4 Genetc algorthm 1: ntalze populaton() 2: whle not converge do 3: assgn populaton ftness() 4: for 1,..., npopsz do 5: select parents(p1,p2) 6: reproducton(p1,p2,chld) 7: end for 8: select next generaton() 9: end whle The advantages of GA over other tradtonal optmzaton technques can be summarzed as follows: GA searches from a populaton of ponts, not a sngle pont. The populaton can move over hlls and across valleys. GA can therefore dscover a globally optmal pont, because the computaton for each ndvdual n the populaton s ndependent of others. GA has nherent parallel computaton ablty. 26

27 GA uses payoff (ftness or objectve functons) nformaton drectly for the search drecton, not dervatves or other auxlary knowledge. GA therefore can deal wth non-smooth, non-contnuous and non-dfferentable functons that are the real-lfe optmzaton problems. Ths property also releves GA of the approxmate assumptons for a lot of practcal optmzaton problems, whch are qute often requred n tradtonal optmzaton methods. GA uses probablstc transton rules to select generatons, not determnstc rules. They can search a complcated and uncertan area to fnd the global optmum. GA s more flexble and robust than the conventonal methods [33]. The frst attempt of the applcaton of genetc algorthms n power systems s n the load flow problem [70]. It has been found that the smple genetc algorthm quckly fnds the normal load flow soluton for small-sze networks by specfyng an addtonal term n the objectve functon. A number of approaches to mprovng convergence and global performance of GAs have been nvestgated [70]. 3.3 Smulated Annealng (SA) The SA s a generc probablstc meta-heurstc for the global optmzaton problem that was proposed by Krkpatrc et al. [32]. In the SA method, each pont s of the search space s analogous to a state of some physcal system, and the functon E(s) to be mnmzed s analogous to the nternal energy of the system n that state. The goal s to brng the system, from an arbtrary ntal state, to a state wth the mnmum possble energy. In each step of the SA algorthm the current soluton s replaced by a random nearby soluton, chosen wth a probablty that depends on the dfference between the correspondng functon values and on a global parameter T (called the temperature), that s gradually decreased durng the process. The dependency s such that the current soluton changes almost randomly when T s large, but ncreasngly downhll as T goes to zero. The allowance for uphll moves saves the method from becomng stuck at local mnma whch are the bane of greeder methods. For certan problems, SA may be more effectve than exhaustve enumeraton. It has been shown that ths technque converges asymptotcally to the global optmal soluton wth probablty one [1]. 27

28 [50]: SA s an effectve global optmzaton algorthm because of the followng advantages sutablty to problem n wde area, no restrcton on the form of cost functon, hgh probablty to fnd global optmzaton, easy mplementaton by programmng. The pseudocode mplementng SA s gven bellow. It starts from state s0 and contnue for kmax of steps or untl a state wth energy emax or less s found. The call neghbour(s) should generate a randomly chosen neghbour of a gven state s; the call random() should return a random value n the range [0,1]. The annealng schedule s defned by the temp(r), whch should yeld the temperature to use, gven the fracton r of the tme budget that has been expended so far. pseudocode 3.5 Smmulated Annealng 1: s s 0 ; e E (s) 2: s best s; e best e; 3: k 0; 4: whle k < k max and e > e max do 5: s new neghbour(s) 6: e new E (s new ) 7: f e new e best then 8: s best s new ; e best e new 9: end f 10: f P(e, e new, temp(k/max )) > random() then 11: s s new ; e e new 12: k k : end f 14: end whle 15: return s best Actually, the pure SA algorthm does not keep track of the best soluton found so far: t does not use the varables s best and e best, t lacks the frst f nsde the loop, and, at the end, t returns the current state s nstead of s best. Whle savng the best state s a 28

29 standard optmzaton, that can be used n any metaheurstc, t breaks the analogy wth physcal annealng snce a physcal system can store a sngle state only. In strct mathematcal terms, savng the best state s not necessarly an mprovement, snce one may have to specfy a smaller k max n order to compensate for the hgher cost per teraton. However, the step s best s new happens only on a small fracton of the moves. Therefore, the optmzaton s usually worthwhle, even when state-copyng s an expensve operaton. SA has the ablty to avod gettng local solutons; then t can generate global or near global optmal solutons for optmzaton problems wthout any restrcton on the shape of the objectve functons [44]. SA s not memory ntensve [45]. However, the settng of control parameters of the SA algorthm s a dffcult task and the computaton tme s hgh [3]. The computatonal burden can be reduced by means of parallel processng [44]. 3.4 Constrants Handlng Constrants le at the hear to fall constraned engneerng optmzaton applcatons. Practcal constrants, whch are often nonlnear and non-trval,confne the feasble solutons to a small subset of the entre search space. There are several approaches whch can be appled to handle constrants n heurstc approaches. These methods can be grouped nto four categores: methods that preserve the feasblty of solutons, penalty-based methods, methods that clearly dstngush between feasble and unfeasble solutons, and hybrd methods [15, 62] Penalty-Based Approach The penalty functon method s frequently appled to manage constrants n evolutonary algorthms. Such a technque converts the prmal constraned problem nto an unconstraned problem by penalzng constrant volatons. The penalty functon method s smple n concept and mplementaton. However, ts prmal lmtaton s the degree to whch each constrant s penalzed. These penalty terms have certan weaknesses that become fatal when penalty parameters are large. Such a penalty 29

30 functon tends to be ll condtoned near the boundary of the feasble doman where the optmum pont s usually located [10]. The penalzed fuel cost functon n ED problem was employed n [51]. In [40] the ED problem was transformed nto an unconstraned one by constructng an augmented objectve functon ncorporatng penalty factors for any value volatng the constrants: N eq N ueq H(X ) = J(X ) + k 1 (h j (X )) 2 + k 2 max[0, g j (X )] 2 (3 1) j=1 j=1 where J(X ) s the objectve functon value of the ED problem. N eq and N ueq are the number of equalty and nequalty constrants, respectvel; h j (X ) and g j (X ) are the equalty and nequalty constrants, respectvely; k 1 and k 2 are the penalty factors. Snce the constrants should be met, the value of thek 1 and k 2 parameters were chosen to have hgh value of 10,000. Ths approach was epmpoyed when applyng SA method. The heurstc startegy that s dscussed n nex secton was used to get a feasble soluton whle applyng C-GRASP method Heurstc Strategy When the C-GRASP s appled to solve ED problem, a key problem s how to handle constrants wth effcency. In ths secton we manly focus on handlng the real power lmts and generators ramp-up constrants. Other than penalty based way to satsfy the real power balance equalty constrants (2 2), s to specfy the output of (N 1) generatng unts and to fnd the Nth from the equalty constrant lke n [4, 67]. In reference [67], authors employed a dependent generaton power p t l of randomly selected unt l. The heurstc strategy appled n LocalSearch() procedure n C-GRASP algorthm can be formulated n a followng way: Step 1. Set the dspatch perod ndex t = 1 and teraton = 1. 30

31 Step 2. Calculate the volaton of power blance constrant P t err at dspatch tme t s calculated from 3 2 as follows P t err = D t + Loss t N =1 P t (3 2) If P t err = 0, then go to Step 5, otherwse to Step 3. Step 3. Randomly generate l the ndex of generatng unt and calculate the real power of selected dependent generatng unt p t l from (3 3). P t l = D t N P t t = 1, 2,..., T (3 3) =1 l However, consderng transmsson losses (2.1.1), these equalty constrants become nonlnear and the output of dependent generatng unt for every dspatch perod t can be found from by solvng a followng equaton B ll (P t l )2 +(2 N N N B l P t +B l0 1)P t l +(D t + =1 l =1 l j=1 j l P t B jp t j +B 00+ N =1 l B 0 P t N =1 l P t ) = 0 (3 4) If t doesn t volate the generator operatng lmts and ramp-up constrants (f they are present), go to Step 5. Otherwse, the value has to be modfed accordng to 3 5 P max l f P t l > P max l P t l = P mn l f P t l < P mn l (3 5) If ED ncorporates ramp-up lmts and dspatch perod t > 1, then dependent unt output has to be calculated as 3 6 max(p mn P t l = mn(p max l l, P t 1 l DR l ) f P t l, P t 1 + UR l ) f P t l > max(p mn l, P t 1 l DR l ) < mn(p max l, P t 1 + UR l ) (3 6) After adjustment, go to Step 4. 31

32 Step 4. Increase the teraton number by 1,.e. l = l + 1. If l < l max, go to Step 2, otherwse go to Step 5. Step 5. Increase the perod number by 1,.e. t = t + 1. If t T, go to Step 2, otherwse stop. The appled strategy for constrants handlng wll produce solutons satsfyng real power lmts constrant and generatng unt ramp rate lmts constrant, howerver not always the the real power balance constrant wll be satsfed n dynamc ED due to ramp-up lmts. The stuaton can be that n one dspatch perod demand wll meet generaton, however n the next perod the demand can be not because due to generatng unt power ncrease or reducton lmtaton. In order not to consder such nfeasble soluton a large penalty s added to objectve functon value. 32

33 CHAPTER 4 EXPERIMENTS AND RESULTS 4.1 Experments In order to verfy the feasblty and effectveness of adopted C-GRASP capabltes for solvng ED problems, dfferent ED problem formulatons,.e. statc and dynamc ED and dfferent systems were used. The C-GRASP algothm wth heurstc strategy to deal wth constrants was mplemented n Matlab 7.5. For GA and SA algorthms, the standard Matlab functons form Genetc Algorthm and Drect Search Toolbox were employed. In standard GA functon ga(), t s possble to nclude both lnear and nonlnear equalty and nequalty constrants. However, SA functon smulannealbnd() ncorporates only lower and upper bound constrants, other constrants as a penalty functon s added to objectve functon. Next, we wll provde descrptons of systems used for our experments System 1 The system conssts of fve generatng unts, whose the maxmum total output s 925 MW. On ths system dynamc ED problem was solved wth the dspatch horzon one day wth 12 ntervals of one hour each. The demand of the system and generatng unt data are gven n Tables (4-1) and (4-2), respectvely. Table 4-1. Generatng unts characterstcs of fve-unt system a, $ /h b, $ /MWh c, $ /(MW) 2 h P mn, MW P max, MW Unt Unt Unt Unt Unt System 2 The system contans sx thermal generatng unts. The total maxmum output of generatng unts s 1470 MW. Ths system was used to solve statc ED problem where load demand on the system s 1263 MW. Parameters of all the thermal unts are 33

34 Table 4-2. Load demand Tme, h Load, MW Tme, h Load, MW reported n [30] and are gven n Tables 4-3 and 4-4. In normal operaton of the system, the loss coeffcents B are as follows: B j = B 0 = [ ] 10 3 B 00 = Table 4-3. Generatng unts characterstcs of sx-unt system Unt P mn, MW P max, MW a, $ /h b, $ /MWh c, $ /(MW) 2 h P 0, MW

35 Table 4-4. Rump-up lmts and prohbted zones of sx-unt system Unt UR,MW DR,MW Prohbted zone [ ] [ ] [90 110] [ ] [ ] [ ] [80 90] [ ] [90 110] [ ] [75 85] [ ] System 3 Ths system conssts of 13 generatng unts wth valve-pont loadng as gven n Table (4-5). The parameters of ths system showed s taken from [54]. The expected demand s 1800 MW and 2520 MW. Table 4-5. Generatng unts characterstcs of 13-unt system Unt P mn, MW P max, MW a b c e f System 4 Ths system s composed of 40 generatng unts wth valve-pont loadng effects supplyng a total demand of MW. Therefore, ths system has nonconvex soluton spaces and there are many local mnma due to valve-pont effects and the global mnmum s very dffcult to determne. The parameters of ths system showed n the Table (4-6) are avalable n [54] as well. 35

36 Table 4-6. Generatng unts characterstcs of 40-unt system Unt P mn, MW P max, MW a b c e f

37 4.1.5 System 5 Ths system has 10 generatng unts wth valve-pont loadng effects. Therefore, ths system has nonconvex soluton spaces and there are many local mnma due to valve-pont effects. The parameters of ths system are gven n the Table 4-10 and are avalable n [4] as well. The forecasted demand wth the dspatch horzon one day wth 24 ntervals of one hour each s shown n Table 4-8. Table 4-7. Generatng unts characterstcs of 10-unt system Unt P mn, MW P max, MW a b c e f UR UR Table 4-8. Load demand for 24 hours Tme, h Load, MW Tme, h Load, MW Tme, h Load, MW Results One of the features that the heurstc algorthms possess s randomness. Therefore, ther performances cannot be judged by the result of a sngle run and many trals wth dfferent ntalzatons should be made to reach a vald concluson about the performance of the algorthms. An algorthm s robust, f t can guarantee an acceptable 37

38 performance level under dfferent condtons. In ths paper, 50 dfferent runs of C-GRASP have been carred out Case 1 In ths case, the dynamc ED problem on system 1 s solved. It can be seen from Table (4-9) that the C-GRASP provded the best soluton compared to SA and GA. Table 4-9. Generaton costs for case 1 Method Mn Avg Max St.Dev. C-GRASP GA SA The smallest total producton cost s obtaned by SA and t s $ Morever, we can notce that on the average C-GRASP algorthm perfoms better than SA and GA. The lowest maxmum value s provded by C-GRASP as well, whle the hghest maxmum value was produced by SA. shows that SA solutons are very senstve to startng ponts and are more volatle. The best found soluton satsfyng demand and power lmts s gven n Table Table Best soluton for case 1 Hour Unt 1 Unt 2 Unt 3 Unt 4 Unt Case 2 Here, the statc ED problem ncludes the nonlnear generaton-demand equalty constrants due to ncluded transmsson losses. The ramp up lmts and prohbted 38

39 zones of generators are ncorporated as well. The effcent of C-GRASP s tested on sx-unt systems that s dscrbed n Secton The same problem has been solved n [30] and ther best soluton and appled methods are presented n Table The losses and total generaton cost are gven n Table The best solutons among all solutons have been llustrated n the bold prnts. From these data we can see that ther provded objectve functon values are smaller that one obtaned by C-GRASP, but t should be noted that solutons ganed by CPSO 1 and CPSO 2 volate the ramp-up lmts of generator 3. When the soluton of PSO has been pluged, t has been found the volaton of generaton-demand balance equalty by MW, because accordng to gven soluton, the generaton s equal to MW and loss s MW. The mnmum generaton cost found by C-GRASP s $ , whle the average cost s $ wth standard devaton of value $ Accordng to these facts, t can be stated that C-GRASP approch wth appled heurstc strategy can produced feasble and good solutons. The results produced by SA and GA were not feasble or reasonably close to results presneted here, so they are not presneted here. Table Best solutons for case 2 P 1 P 2 P 3 P 4 P 5 P 6 PSO CPSO CPSO C-GRASP Table Best results, when demand s1263 MW Total output Loss Total generaton cost PSO CPSO CPSO C-GRASP