Mixed Integer Programming to Globally Minimize the Economic Load Dispatch Problem With Valve-Point Effect

Transcription

1 1 Mxed Integer Programmng to Globally Mnmze the Economc Load Dspatch Problem Wth Valve-Pont Effect Mchael Azzam, S. Easter Selvan, Augustn Lefèvre, and P.-A. Absl arxv: v1 [math.oc] 16 Jul 2014 Abstract Optmal dstrbuton of power among generatng unts to meet a specfc demand subject to system constrants s an ongong research topc n the power system communty. The problem, even n a statc settng, turns out to be hard to solve wth conventonal optmzaton methods owng to the consderaton of valve-pont effects whch make the cost functon nonsmooth and nonconvex. Ths dffculty gave rse to the prolferaton of populaton-based global heurstcs n order to address the multextremal and nonsmooth problem. In ths paper, we address the economc load dspatch problem (ELDP) wth valve-pont effect n ts classc formulaton where the cost functon for each generator s expressed as the sum of a quadratc term and a rectfed sne term. We propose two methods that resort to pecewse-quadratc surrogate cost functons, yeldng surrogate problems that can be handled by mxed-nteger quadratc programmng (MIQP) solvers. The frst method shows that the global soluton of the ELDP can often be found by usng a fxed and very lmted number of quadratc peces n the surrogate cost functon. The second method adaptvely bulds pecewsequadratc surrogate under-estmatons of the ELDP cost functon, yeldng a sequence of surrogate MIQP problems. It s shown that any lmt pont of the sequence of MIQP solutons s a global soluton of the ELDP. Moreover, numercal experments ndcate that the proposed methods outclass the state-of-the-art algorthms n terms of mnmzaton value and computaton tme on practcal nstances. Index Terms Economc load dspatch, Global convergence, Mxed nteger quadratc programmng, Valve-pont effect. I. INTRODUCTION ECONOMIC LOAD DISPATCH PROBLEM (ELDP) attempts to mnmze the cost assocated wth the power generaton by optmally schedulng the load across generatng unts to meet a certan demand subject to system constrants [1]. It s not uncommon to notce that the cost functon for a generator s approxmated by a quadratc functon for the sake of smplcty. Nevertheless, when the cost functon also takes nto account hghly nonlnear nput-output characterstcs due to valve-pont loadngs, even the statc ELDP problem that gnores the ramp-rate constrants turns out to be dffcult to solve. The challenges faced by the solvers stem from () the nonsmooth cost functon and () the mult-extremal nature of the problem. A popular strategy to address the ELDP s to rely on a (populaton-based) stochastc search algorthm. Indeed, a myrad of such algorthms have been proposed durng recent years, ncludng genetc algorthms [2], evolutonary programmng [1], partcle swarm optmzaton [3], ant swarm optmzaton [4], dfferental evoluton [5], frefly algorthm [6], bacteral foragng algorthm [7], and bogeography-based optmzaton [8]. Snce heurstcs enable the global exploraton, and a local method ads to converge to a local optmum, a focussed effort has been made by many to ntegrate a global heurstcs wth a local optmzer, resultng n hybrd algorthms. Interested readers may refer to [9] for an exhaustve lst of such algorthms n the ELDP context. A few well-known local optmzers ntegrated wth a global scheme to tackle the ELDP are the Nelder-Mead method [7], generalzed pattern search [10], sequental quadratc programmng [11], and a Remannan subgradent steepest descent [12]. Note that the equalty constrant must be handled by way of a slack varable or a barrer approach, and the nequalty constrants wth the help of a penalty approach n global heurstcs. Furthermore, even though the global heurstcs favor fndng the global mnmum, these hybrd algorthms are guaranteed at best to fnd a local mnmum. In ths paper, we ntroduce new technques to effcently buld surrogates of the ELDP cost functon n order to take advantage of powerful modern mxed-nteger programmng (MIP) solvers. In partcular, the adaptve technque ntroduced n Secton V bulds a sequence of surrogate pecewsequadratc cost functons that ams at keepng low the number of peces for the sake of effcency whle nevertheless offerng the guarantee that the sequence of surrogate solutons converge to the global soluton of the ELDP. It s nterestng to note that the MIP method vested wth theoretcal guarantees as appled to a statc ELDP wth the valve-pont effect may be extended to a dynamc settng, provded the number of generatng unts s not too large. Fnally, we demonstrate that the mnmzaton results from a 3-, two nstances of 13-, and a 40-generator settngs are lower than the results reported thus far n the lterature wth the same datasets. Whle the rectfed-sne model (1b) of the valve-pont effect was proposed more than 20 years ago [2], t s only n the past few months that MIP technques appeared n the lterature to handle ths problem formulaton [13], [14], [15]. The methods ntroduced n ths paper contrbute beyond ths recent lterature n two ways. () The approach proposed n Secton III shows that t s often possble to obtan the exact global soluton wth a fxed and very lmted number of lnear peces n the surrogate cost functon. () The adaptve approach proposed n Secton V, whle sharng several aspects wth the recent report [14], ntroduces fewer breakponts n the surrogate cost functon, allowng for a reduced complexty. Moreover, t takes

2 2 nto account that t s only the rectfed sne term that s pecewse concave n (1b); ths leads a pecewse-quadratc under-approxmaton of (1b) that s handled by mxed-nteger quadratc programmng (MIQP). The remander of the artcle s organzed as follows. In Secton II, the ELDP wth the valve-pont effect s brefly presented. The prncple behnd our approach s frst expounded n Secton III usng a smple lnear approxmaton of the term that accounts for the nonconvexty and nonsmoothness. Secton IV revews MIP formulatons for general pecewselnear objectve functons and shows how ths technque can be ntegrated n the ELDP. The adaptve algorthm s ntroduced and ts convergence analyss carred out n Secton V. Numercal results are reported n Secton VI, and conclusons are drawn n Secton VII. f(p) 2,500 2,000 1,500 1, p II. PROBLEM STATEMENT In ths secton, we recall the formulaton of the wdely nvestgated ELDP wth valve-pont effect, as descrbed, e.g., n [9]. In the ELDP, the man component of the cost that needs to be taken nto account, s the cost of the nput, that s to say the fuel. The objectve functon used n the problem s thus defned by how we represent the nput-output relatonshp of each generator. The total cost s then naturally the sum of each contrbuton. The objectve functon s thus wrtten f(p) = mn p R n n f (p ), =1 (1a) where f s the total cost functon n $/h, equal to the sum of the n f unvarate functons that gve the ndvdual contrbuton n the total cost of the th generator, dependng on the p amount of power, n MW, assgned to ths unt. A classcal, smple and straghtforward approach to construct the cost functons s to use a quadratc functon for each generator,.e., f (p ) = a p 2 + b p + c, where a, b and c scalar coeffcents. However, n realty, performance curves do not behave so smoothly. In the case of generatng unts wth mult-valve steam turbnes, rpples wll typcally be seen n the curve. Large steam turbne generators usually have a number of steam admsson valves that are opened n sequence to meet an ncreasng demand from a unt. And as each steam admsson valve starts to open, a sharp ncrease n losses due to wre drawng effects occur [9], [16]. Ths s the so-called valvepont effect. To try to capture ths effect n the model, a rectfed sne term s usually added to the fuel cost functons, so that they become f (p ) = a p 2 + b p + c + d sn(e (p p mn )) (1b) where d and e are addtonal postve coeffcents needed to take the valve-pont effect nto account [2]. We can see how the new term affects the cost functon n the example of Fg. 1. Unfortunately, ths addton brngs two detrmental propertes n the problem: non-convexty and non-dfferentablty. These are two major hndrances that prevent the drect applcaton of usual optmzaton algorthms. Fg. 1. Examples of cost functons for a generator, wthout (sold) or wth (dashed) valve-pont effect Naturally, the problem has also some constrants that must be satsfed and restrct the search space. Frst, the producer must meet the demand, even though some power wll be lost n the network. Ths s the power balance constrant, formulated through an equalty constrant n p = p D + p L (p) (2) =1 wth scalar p D and functon p L beng respectvely the demand on the system and the transmsson loss n the network, both n MW. The transmsson loss s computed usng the so-called B-coeffcents as p L (p) = p t Bp + p t b 0 + b 00 (3) where B s a symmetrc postve-semdefnte matrx, b 0 a vector of sze n and b 00 a scalar. The other type of constrants are generator capacty constrants, whch take the form of box constrants mposng that each unt has ts own range of possble power generaton, from p mn to p max. Ths s easly transcrbed as nequalty constrants p mn p p max, (4) where p mn and p max are obvously the mnmum and maxmum power output of the th generator, n MW. The set of ponts that satsfy constrants (2) and (4) s termed the feasble set of the ELDP. As s often done, we wll gnore the transmsson loss n the network,.e., we set p L = 0 n (2). The statc ELDP wthout losses s then classcally wrtten as mn p R n subject to n =1 a p 2 + b p + c + d sn(e (p p mn )) n p = p D =1 p mn p p max {1,..., n} (5)

3 3 However, as we pont out n the concludng secton, consderng the loss does not add much complexty. 1.5 III. A FIRST SIMPLE MIQP APPROACH We now ntroduce our frst, crude but effectve way of buldng a pecewse-quadratc surrogate of the ELDP cost functon (1), and we show how to solve the resultng surrogate problem wth an MIQP solver. As dscussed n the prevous secton, the term y sn(e (p p mn )) (6) of the objectve functon makes the optmzaton problem challengng because t breaks ts smoothness and convexty. In ths paper, we overcome the dffculty by approxmatng (6) wth functons that are more manageable, namely pecewse-lnear functons. Even though they do not restore smoothness nor convexty of the problem, they are convenently handled by mxed-nteger programmng. A frst smple pecewse-lnear approxmaton conssts of replacng sn x by x over [ π/2, π/2] and completng the approxmaton over the whole doman by perodc extenson, observng that sn x s perodc of perod π. The underlyng motvaton s to keep low the number of lnear peces whle capturng accurately the behavor of (6) around ts knk ponts, as the optmum tends to be located at those knk ponts. The resultng functon can be compactly wrtten as arcsn(sn x), whch has the sawtooth shape shown on Fg. 2. Ths functon can be nterpreted as the dstance between x and the closest multple of π. Ths can be wrtten as a small mxed nteger optmzaton problem such as arcsn(sn x) = mn x kπ k Z whch can be reformulated to become lnear, as follows : mn t (k,t) Z R s.t. x kπ t x kπ t. The problem s thus very smple, even though we ntroduce an nteger varable. By ntroducng ths n the orgnal problem, we get mn p,k,t a p 2 + b p + c + d t s.t. p mn p = D p p max t e (p p mn ) πk t p R t R k N whch s an MIQP problem. Ths class of problems can be solved exactly by solvers, for nstance, CPLEX, Gurob or Mosek. In Secton VI, we see that ths model gves results that are compettve wth other methods suggested n the lterature. (7) π π π x Fg. 2. Plot of sn(x) on the nterval [-4,4] and the suggested surrogate : arcsn(sn(x)). Sne s n dashed lne and the surrogate s sold. IV. FINER APPROXIMATION WITH PIECEWISE LINEAR FUNCTIONS In ths secton, we show how to handle general pecewselnear surrogate objectve functons, then we consder specfcally an over-approxmaton obtaned from tangents to the rectfed sne functon. Let us say we want to mnmze on an nterval [v, w], a pecewse lnear functon g descrbed by the slopes α and ntercepts β of ts m lne segments components as well as ts breakponts X 1 = v, X 2,..., X m+1 = w. Ths objectve functon can be expressed as a mxed nteger lnear problem such as mn s.t. m α χ + β η =1 m χ = x =1 m η = 1 =1 X η χ X +1 η {1,..., m} η {0, 1} {1,..., m}. Ths system of constrants ensures that for a gven x, only one of the bnary varables η wll be equal to one, ndcatng whch segment s actve, whle one of the real varables χ wll hold the value of x and the others wll be equal to zero. Thus, only one term of the objectve wll be non zero. It wll be equal to the value of the lnear functon of slope α and ntercept β, at ths abscssa. We now ntegrate ths technque n our model. Instead of replacng sn x by mn k Z x kπ, we use the output of ths as the abscssa for fner approxmaton through pecewse lnear functons. For example, wth m segments for the cost functon of unt, π (8)

4 4 mn p,k,t,χ,η s.t. m a p 2 + b p + c + d (α j χ j + β j η j ) p = D p mn p p max j=1 t e (p p mn ) πk t X j η,j χ,j X j+1 η,j m t = j=1 m j=1 p R χ,j η,j = 1 t, χ,j R + k N η,j {0, 1} where t are the dstances to the nearest root, χ,j and η,j are auxlary varables that help us get the rght value n the objectve functon dependng on whch segment we are located n and α j and β j are constants that parametrze the lne segments. A course of acton we could opt for s followng the logc of the prevous secton and buld an over-approxmaton wth frst-order Taylor approxmaton to the sne at dfferent ponts. Wth three segments, t would mean we want to approxmate sn x by mn(x, T 1 (x), T 2 (x)), where T 1 (x) and T 2 (x) are the tangents to sn x at respectvely θ 1 and θ 2. A bt of algebra gves us the parameters α j = cos θ j β j = sn θ j θ j cos θ j and the endponts of the ntervals X 1 = sn θ 1 θ 1 cos θ 1 1 cos θ 1 X 2 = θ 2 cos θ 2 θ 1 cos θ 1 sn θ 2 + sn θ 1 cos θ 2 cos θ 1 Ths s llustrated n Fg. 3. Numercal results are reported n Secton VI. V. GLOBAL METHOD BASED ON ADAPTIVE UNDER-APPROXIMATION In ths secton, we wll descrbe an algorthm that provably converges toward a global mnmum of the ELDP. The way to acheve ths s to change our approach to use underapproxmatons nstead of over-approxmatons. Over-approxmatons wth tangents seem natural because we can then nfer propertes from Taylor s theorem, but underapproxmatons have the useful property that f the underapproxmaton and the true functon concde at a global mnmzer x of the under-approxmaton, then x s also a global mnmzer of the true functon. (9) y π π 0 θ π 2 1 θ 2 x Fg. 3. Plot of sn(x) on the nterval [-4,4] and a pecewse lnear approxmaton, bult from tangents. Sne s n dashed lne and the approxmaton s sold. θ 1 and θ 2 are the tangency ponts. Proposton 1: Let f : X R and g : X R be two functons such that g(x) f(x) x X and x X a global mnmzer of g. If g(x ) = f(x ), then x s a global mnmzer of f. Proof: Snce x s a global mnmzer of g, g(x ) g(x) for every x X. And because g f, we have g(x ) g(x) f(x) for every x X. Fnally, snce f(x ) = g(x ), we can conclude. So f we were able to fnd such an under-approxmaton, then we would have found the global mnmum. What we suggest s to buld a sequence of under-approxmatons that acheve ths goal at the lmt. As before, we wll make use of pecewse lnear functons to approach the sne part of the real cost functon. We start off wth a smple chord of sn x that lnks ts extreme ponts. We then solve problem (9). We want the next approxmaton n the sequence to be equal to the true objectve functon at the soluton p 0 we found for the frst approxmaton. Therefore, we use the values of t as breakng ponts for the new pecewse lnear approxmaton and compute the coeffcents of the lne segments so that g 1 (p 0 ) = f(p 0 ). We can now smply repeat ths procedure agan, fnd the new soluton p 1, buld a new approxmaton g 2, so on, tll convergence s reached. At each teraton, the approxmaton becomes closer to the true functon. Of course, t s very possble that for a generator, ts assgned power does not change from one teraton to another and so, the approxmaton wll stay the same. Ths s actually desred because t means we add less complexty than we mght have expected. Ths algorthm can be formalzed for our specfc needs as follows, where X s the set of break ponts for cost functon, m the number of segments for ts approxmaton, X,j the jth element of X n ascendng order, α j and β j the coeffcents of lne segment j of cost functon, δ the change n optmal value as a proxy to measure convergence, and ɛ a gven postve parameter. We now analyze the convergence of Algorthm 1.

5 5 y π π π x Fg. 4. Plot of the two frst approxmatons of the sne term for a gven generator. We assume that the soluton for ths unt n the frst teraton was 1. Algorthm 1 Adaptve pecewse-quadratc underapproxmaton for = 1,..., n do X {0, π/2} m 1 end for 5: repeat for = 1,..., n do for j = 1,..., m do α,j (sn(x,j+1 ) sn(x,j ))/(X,j+1 X,j ) β,j sn(x,j ) α,j X,j 10: end for end for (ˆp, ĝ) solve problem (9), where ˆp denotes the optmal p and ĝ the optmal value of the surrogate cost functon; δ f(ˆp) ĝ for = 1,..., n do 15: X X {t } m #X 1 end for untl δ < ɛ Theorem 2 (convergence to the global mnmum): Let denote the optmal value of the cost functon f (1) of the ELDP (5). For m = 0, 1,..., let g m denote the pecewsequadratc surrogate cost functon used by Algorthm 1 at teraton m, and let p m R n denote the power outputs produced by Algorthm 1 at teraton m by solvng problem (9). Recall that g m (p m ) = mn g m and denote t by gm. Then lm m gm = lm m f(p m ) = f, and every lmt pont of (p m ) m N s a global soluton of the ELDP. Proof: We frst show that f and g m, m = 0, 1,..., are Lpschtz contnuous on the ELDP feasble set wth a common Lpschtz constant K. Indeed, for every, the cost functon π f f (1b) satsfes the Lpschtz property f (p + ) f (p ) (2a p max + b + d e ) for all p and p + that satsfy the generator capacty constrants (4). The ELDP cost functon f (1), beng the sum of Lpschtz contnuous functons, s thus Lpschtz contnuous on the ELDP feasble set, wth a constant K = n =1 2a p max +b +d e. Snce g m s obtaned by replacng the rectfed snes of f by chords, t follows that 2a p max + b + d e s stll a Lpschtz constant for the contrbuton of generator to g m, and hence that K s also a Lpschtz constant for g m. Snce buldng g m+1 from g m conssts of nsertng, for each generator, one new breakpont for the pecewse-lnear under-approxmaton of the pecewse-concave rectfed sne, t follows that, for every m, g m g m+1 f. (10) Therefore (g m) m N s a nondecreasng sequence bounded by f. Thus (g m) m N converges and lm m g m f. We show that lm m g m = f. By contradcton, assume that lm m g m = f ε wth ε > 0. Snce (p m ) m N s bounded n vew of the generator capacty constrants (4), there exsts a subsequence (p mk ) k N that converges. We then have the followng nequaltes whch we justfy hereafter: p mk p mk+1 1 K [g m k+1 (p mk ) g mk+1 (p mk+1 )] (11) 1 K [f g mk+1 (p mk+1 )] (12) 1 K [f (f ε)] (13) ε K, a contradcton snce (p mk ) k N converges. Inequalty (11) states Lpschtz contnuty of g mk+1 wth constant K. Inequalty (12) follows from g mk+1 (p mk ) = f(p mk ) f ; ndeed, by constructon, p mk s a breakpont of g mk +1 and all subsequent surrogate cost functons. Fnally, (13) follows from g mk+1 (p mk+1 ) = gm k+1 f ε. We now show that lm m f(p m ) = f. By contradcton, suppose not. Then there s an nfnte subsequence (p mk ) k N and ε > 0 such that f(p mk ) f + ɛ. We assume w.l.o.g. that (p mk ) k N converges; f not, we extract a sub-subsequence that does. By the trangle nequalty, we have f(p mk ) f f(p mk ) g mk+1 (p mk ) + g mk+1 (p mk ) g mk+1 (p mk+1 ) + g mk+1 (p mk+1 ) f. The frst term of the bound s zero by constructon of the surrogate functons. The second term goes to zero as k n vew of the common Lpschtz constant K and the convergence of (p mk ) k N. The thrd term goes to zero as k snce g mk+1 (p mk+1 ) = gm k+1 and lm m gm = f. Hence f(p mk ) f goes to zero, a contradcton. Fnally, let (p mk ) k N be a convergent subsequence and let p m denote ts lmt. By contnuty of f, we have that f(p m ) = lm k f(p mk ) = f. Note that, n vew of the breakpont nserton procedure (step 15 of Algorthm 1), t generally does not hold that g m

6 6 TABLE I FOUND SOLUTION FOR THE 3-UNITS STUDY CASE (I) USING THE SIMPLE MODEL (7) Unt Power (MW) p p p Total cost ($/h) Best n lt. ($/h) Real tme (s) CPU tme (s) TABLE II FOUND SOLUTION FOR THE 13-UNITS STUDY CASE (IIA) USING THE SIMPLE MODEL (7) Unt Power (MW) p p p p p p p p p p p p p Total cost ($/h) Best n lt. ($/h) Real tme (s) CPU tme (s) converges to f pontwse; otherwse the proof above could have been more drect. VI. NUMERICAL RESULTS After havng bult such a model, we can hand t over to a solver that can handle MIQP. Here we wll use the Gurob solver [17]. To study the effcency of our method, we wll test t on the most common test cases n the lterature : a 3-unts settng wth a demand of 850 MW [2] (I), a 13-unts settng wth a demand of 1800 MW [1] (IIa) and 2520 MW [18] (IIb), and a 40-unts settng wth a demand of 10,500 MW [1] (III). Usng model (7) and these datasets, and feedng them to Gurob, we get the solutons gven n tables I, II, IV. The hardware used s a PC wth a Intel Core 2 Duo P8600 CPU (two cores at 2,4 Ghz) and 3 GB of RAM, on GNU/Lnux. If we compare these results to those found n the lterature [19], we can see that they are close to the best solutons found to date. In fact, for I and IIa, we do get the best results, whle for IIb and III, t s less than 0.005% worse. Furthermore, t s acheved on a determnstc bass, wthout the uncertanty and rreproducblty of the commonly used heurstcs. For the model of Secton IV, table V shows the soluton found for a choce of parameter θ 1 = 0.35π and θ 2 = 0.47π. TABLE III FOUND SOLUTION FOR THE 13-UNIT CASE (IIB) USING THE SIMPLE MODEL (7) Unt Power (MW) p p p p p p p p p p p p p Total cost ($/h) Best n lt. ($/h) Real tme (s) CPU tme (s) TABLE IV FOUND SOLUTION FOR THE 40-UNITS STUDY CASE (III) USING THE SIMPLE MODEL (7) Unt Power (MW) Unt Power (MW) p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p Total cost ($/h) Best n lt. ($/h) Real tme (s) CPU tme (s) The resultng optmal value s slghtly better than what we found earler. The real tme needed s s for a CPU tme of s. We also tested the method of Secton V on the dfferent study cases. It so happens that t fnds exactly the same solutons (except for a few swaps n equvalent generators) as the method of Secton IV for case I, IIa and III thereby provng ther optmalty. The excepton s case IIb, for whch a slghtly better soluton s found (table VI). Note that the algorthm of Secton V always needs more tme snce t takes at least two teratons to stop: one to get a soluton and another

7 7 TABLE V SOLUTION FOR 40-UNITS CASE STUDY (III) USING THE MODEL OF SECTION IV. Unt Power (MW) Unt Power (MW) p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p Total cost ($/h) Best n lt. ($/h) Real tme (s) CPU tme (s) TABLE VI GLOBAL SOLUTION FOR CASE IIB FOUND WITH THE ALGORITHM OF SECTION V to prove the optmalty. Unt Power (MW) p p p p p p p p p p p p p Total cost ($/h) Best n lt. ($/h) Real tme (s) CPU tme (s) VII. CONCLUSION Ths artcle concerns a pecewse quadratc underapproxmaton of the ELDP cost functon that takes nto account the valve-pont effect, thereby provdng a means to solve t globally wth the MIQP method, despte the nonsmoothness and the nonconvexty of the orgnal cost functon. Furthermore, a convergence analyss s presented to show that, under mld assumptons, ths strategy guarantees the global mnmzer n the statc ELDP context. In order to support our clam, the mnmzaton results are furnshed for the datasets correspondng to a 3-, two nstances of 13-, and a 40-generator settng, wheren the transmsson losses are omtted. Interestngly enough, one the one hand, n accordance wth the global convergence guarantee, the outcome of the cost mnmzaton by the method of Secton V s never surpassed by the hybrd methods, and on the other hand, the computatonal tmes are qute mpressve. Whle these wdely used datasets n the ELDP lterature enable us to nvestgate how our approach compares wth the state-of-the-art methods, the same framework s applcable for scenaros that do not gnore losses. The only modfcaton needed s to add the loss term n the constrant to take t nto account and to relax the equalty nto an nequalty. The reason behnd the relaxaton s to make the constrant convex and the model easy enough to solve. Ths does not change the soluton of the problem as long as the objectve functon s monotoncally ncreasng. Moreover, these methods can be used for other cost functons, where the valve-pont effect would be modeled dfferently. As long as the valve-pont effect s represented by a perodc pecewse-concave functon, only mnor adaptaton would be needed. REFERENCES [1] N. Snha, R. Chakrabart, and P. K. Chattopadhyay, Evolutonary programmng technques for economc load dspatch, IEEE Trans. Evol. Comput., vol. 7, no. 1, pp , Feb [2] D. C. Walters and G. B. Sheblé, Genetc algorthm soluton of economc dspatch wth valve pont loadng, IEEE Trans. Power Syst., vol. 8, no. 3, pp , Aug [3] J. B. Park, K. S. Lee, J. R. Shn, and K. Y. Lee, A partcle swarm optmzaton for economc dspatch wth nonsmooth cost functons, IEEE Trans. Power Syst., vol. 20, no. 1, pp , Feb [4] J. Ca, X. Ma, L. L, Y. Yang, H. Peng, and X. Wang, Chaotc ant swarm optmzaton to economc dspatch, Electrc Power Systems Research, vol. 77, no. 10, pp , Aug [5] L. S. Coelho and V. C. Maran, Combnng of chaotc dfferental evoluton and quadratc programmng for economc dspatch optmzaton wth valve-pont effect, IEEE Trans. Power Syst., vol. 21, no. 2, pp , May [6] X. S. Yang, S. S. S. Hossen, and A. H. Gandom, Frefly algorthm for solvng non-convex economc dspatch problems wth valve loadng effect, Appled Soft Computng, vol. 12, no. 3, pp , Mar [7] B. K. Pangrah and V. R. Pand, Bacteral foragng optmsaton: Nelder-Mead hybrd algorthm for economc load dspatch, IET Generaton, Transmsson & Dstrbuton, vol. 2, no. 4, pp , Jul [8] A. Bhattacharya and P. K. Chattopadhyay, Hybrd dfferental evoluton wth bogeography-based optmzaton for soluton of economc load dspatch, IEEE Trans. Power Syst., vol. 25, no. 4, pp , Nov [9] M. S. P. Subathra, S. E. Selvan, T. A. A. Vctore, A. H. Chrstnal, and U. Amato, A hybrd wth cross-entropy method and sequental quadratc programmng to solve economc load dspatch problem, IEEE Syst. J., 2014, n press. [10] J. S. Alsumat, J. K. Sykulsk, and A. K. Al-Othman, A hybrd GA PS SQP method to solve power system valve-pont economc dspatch problems, Appled Energy, vol. 87, no. 5, pp , May [11] T. A. A. Vctore and A. E. Jeyakumar, Hybrd PSO SQP for economc dspatch wth valve-pont effect, Electrc Power Systems Research, vol. 71, no. 1, pp , Sep [12] P. B. Borckmans, S. E. Selvan, N. Boumal, and P.-A. Absl, A Remannan subgradent algorthm for economc dspatch wth valvepont effect, Journal of Computatonal and Appled Mathematcs, vol. 255, pp , Jan

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