Solvng Envronmental/Economc Power Dspatch Problem by a Trust Regon Based Augmented Lagrangan Method Downloaded from eee.ust.ac.r at :07 IRST on Saturday December 9th 08 H. Mohammadan Bsheh*, A. Rahm Kan** and M. M. Seyyed Esfahan*** Abstract: Ths paper proposes a Trust-Regon Based Augmented Method (TRALM) to solve a combned Envronmental and Economc Power Dspatch (EEPD) problem. The EEPD problem s a mult-obectve problem wth competng and non-commensurable obectves. The TRALM produces a set of non-domnated Pareto optmal solutons for the problem. Fuzzy set theory s employed to extract a compromse non-domnated soluton. The proposed algorthm s appled to the standard IEEE 30 bus sx-generator test system. Comparson of TRALM results wth the varous algorthms, reported n the lterature shows that the solutons of the proposed algorthm are very accurate for the EEPD problem. Keywords: Envronmental and economc power dspatch, fuzzy set theory, trust-regon augmented Lagrangan method. Introducton The man obectve of Economc Power Dspatch (EPD) s to mnmze the operatng cost, whle satsfyng the load demand, and all unt and system equalty and nequalty constrants. In addton, the ncreasng publc awareness of the envronmental protecton gudelnes and the passage of the Clean Ar Act Amendment of 990 have mpelled the utltes to modfy ther desgn or operatonal strateges n order to reduce polluton and atmospherc emssons of thermal power plants [, ]. Several strateges have been proposed to reduce the atmospherc emssons [3, 4], some of whch are:. Plannng to reduce the power use of power plants wth hgher polluton rates and use the power statons wth lower emsson rates.. Installaton flters on power plants to purfy the pollutant gases. 3. Swtchng to low emsson fuels from hgh emsson ones (e.g., usng natural gas nstead of mazut). 4. Replacement aged and low effcent fuel-burners and generator unts by hgh effcent ones. The second to fourth optons requre nstallaton of new equpments, and need consderable captal Iranan Journal of Electrcal & Electronc Engneerng, 0. Paper frst receved July 0 and n revsed form 8 Jan. 0. * The author s wth the Department of Industral Engneerng of Mazandaran Unversty of Scence and Technology, Babol, Iran. ** The author s wth the Department of Electrcal Engneerng of Tehran Unversty, Tehran, Iran. *** The author s wth the Department of Industral Engneerng of Amrabr Unversty, Tehran, Iran. E-mals: hbmohammadan@yahoo.com, aran@ut.ac.r and msesfahan@aut.ac.r. nvestments, and normally are consdered as long-term plannng. Hence, the frst opton, that s plannng the power dspatch n such a manner that optmzes the fuel cost obectve, as well as emsson cost obectve, ndvdually, and especally smultaneously, s our concern for study. After deregulaton of electrcty marets, serous competton has arsen among generatng companes [5-7]. In ths stuaton, generatng companes try to reduce the cost of energy, to enable them compete n the compettve electrcty marets. One of the effectve methods of reducng the cost of electrc energy s envronmental-economc power dspatch (EEPD). In recent years, the EEPD category has consderably been nvestgated n dfferent ways. In [8-0], the emsson was consdered as a constrant wth a permssble lmt, and the problem was reduced to a sngle obectve optmzaton problem. The problem n ths method s that a compromse optmal soluton cannot be found between emsson and fuel costs. In [], a lnear programmng based optmzaton procedure was proposed n whch the obectves were consdered one at a tme. A compromse optmal soluton s mpossble n ths method ether. In [], a fuzzy mult-obectve optmzaton approach for the EEPD problem was proposed. The solutons produced by ths technque were suboptmal and the algorthm dd not provde a systematc framewor to drect the search towards the Pareto optmal set. Over the past decade, the EEPD problem has receved much nterest due to the development of a number of mult-obectve search strateges. Strength Iranan Journal of Electrcal & Electronc Engneerng, Vol. 8, No., June 0 77
Downloaded from eee.ust.ac.r at :07 IRST on Saturday December 9th 08 Pareto Evolutonary Algorthm (SPEA) [], Nched Pareto enetc Algorthm (NPA) [3], Non-domnated Sortng enetc Algorthm (NSA) [4], Multobectve Stochastc Search Technque (MOSST) [5], Fuzzy Clusterng-based Partcle Swarm Optmzaton (FCPSO) [6], Mult-obectve Partcle Swarm Optmzaton (MOPSO) [7], Epslon Constrant (EC) approach [8], etc., consttute the poneerng multobectve approaches that have been appled to solve the mult-obectve EEPD problem. In the above approaches, the EEPD problem were converted to a sngle obectve problem by usng a lnear combnaton of the obectves as a weghted sum wth a long range plannng le swtchng to low emsson fuels. The postve characterstc of these methods s that a set of Pareto optmal solutons can be obtaned by changng the weghts. In general, whle there are more than one obectve functon n a problem, especally when these obectve functons are noncommensurable or even conflctng, nstead of havng one optmal soluton, a set of optmal solutons are of nterest. The reason for the optmalty of many solutons s that no one can be consdered to be better than any other wth respect to all the obectve functons. These optmal solutons are nown as Pareto optmal solutons. There are some problems assocated wth tang a lnear combnaton of dfferent obectves as a weghted sum:. The combned obectve functon may lose sgnfcance due to the ncorporaton of multple non-commensurable factors nto a sngle functon.. The lac of suffcent nformaton regardng the operaton condtons mae t dffcult for the decson maer to decde on the preferences of obectve n gvng the weghtng factors. The frst problem can be addressed by a proper selecton of the scalng factor λ and multplyng the emsson obectve by ths factor. To deal wth the second problem, fuzzy set theory has been used to effcently derve a canddate Pareto optmal soluton for the decson maer [9]. Ths approach wll be explaned later. Problem Statement The EEPD problem s to mnmze two noncommensurable and competng obectve functons, fuel cost and emsson, whle satsfyng several equalty and nequalty constrants []. The problem s generally formulated n the followng subsectons.. Problem Varables The varables of the problem are the quanttes of real power of commtted power plants, that s, P, =,,..., N and N s the number of commtted power plants n the nterconnected networ.. Problem Obectves There are two obectves whch are mnmzaton of fuel cost and mnmzaton of emsson amount. Mnmzaton of fuel cost The generators cost curves are represented by quadratc functons [,]. The total $/h fuel cost F( P ) can be expressed as N ( P ) = a + b P c P = F + where N s the number of generators, a, b, and c are the cost coeffcents of the th generator, and P s the real power output of the th generator. P s the vector of real power outputs of generators whch s defned as () P = P, P,..., P ] () [ N Mnmzaton of emsson amount The total emsson E ( P ) n atmospherc pollutants such as sulphur oxdes (SO ) and ntrogen oxdes (NOx) caused by the operaton of fossl fueled thermal generaton can be expressed as []: E( P ) = N = 0 ( α + β P + ζ exp( λ P ) + γ P ) (3) where α, β, γ, ζ and λ are coeffcents of the th generator emsson characterstcs..3 Problem Constrants.3. Power Balance Constrant The total power generaton must cover the total power demand P D and the real power loss n transmsson lnes P loss. Hence, N = P P P = 0. (4) D loss The real power loss P loss n Eq. (4) s represented by calculaton of the AC load flow problem, whch has equalty constrants on real and reactve power at each bus as follows []: P Q NB = PD + V V = [ cos( δ δ ) B sn( δ δ) ] (5) = QD + V V [ sn( δ δ ) = (6) + B cos( δ δ )] NB where NB s the number of buses; P and Q are the real and reactve power generated at the th bus respectvely, P D and Q D are the th bus load real and 78 Iranan Journal of Electrcal & Electronc Engneerng, Vol. 8, No., June 0
Downloaded from eee.ust.ac.r at :07 IRST on Saturday December 9th 08 reactve power, respectvely, and B are the transfer conductance and susceptance between bus and bus, respectvely, V and V are the voltage magntudes at bus and bus, respectvely, δ and δ are the voltage angles at bus and bus, respectvely. There are several methods of solvng the resultng nonlnear system of Eqs. (5) and (6) whch the most popular s nown as the Newton-Raphson Method. The load flow soluton gves all bus voltage magntudes and angles that can be used to calculate the transmsson losses as follows: P loss NL = = g [ V + V V V cos( δ δ )] where, NL s the number of transmsson lnes and g s the conductance of the th lne that connects bus to bus. (7).3. eneraton Capacty Constrant For stable operaton, the real power output of each generator s lmted by lower and upper lmts as follows []: mn = P P P,,,..., N. (8) mn where, P and P are the lower lmt and upper lmt power outputs of th generator, respectvely, and N s the number of generators..3.3 Securty Constrant For secure operaton, the transmsson lne loadng S s restrcted by ts upper lmt as follows: l S S, =,... NL (9) l l, where S l and Sl are respectvely the transmsson loadng and upper lmt transmsson loadng of th transmsson lne. It should be noted that the transmsson lne flow th connectng bus to bus can be calculated as S l = (0) * ( V δ ) I where, I s the current flow from bus to bus and can be calculated as I ( V δ V δ )( y ) = ( V ) δ y + ( V )( ) () δ where y s the lne admttance, whle y s the shunt susceptance of the lne []..4 Problem Formulaton By summng up the aforementoned obectves and constrants, the problem can mathematcally be formulated as a nonlnear constraned mult obectve optmzaton problem as follows: Mnmze F ( P ), E( P )] () [ Subect to g ( ) = 0 (3) P h ( ) 0 (4) P where, g s the equalty constrant representng the power balance, and h s the nequalty constrant representng the power system securty and the generator capacty constrants. The power balance constrant s as follows: d b + ( Bbδ ) Pb = 0, b B (5) where b and are number of the buses (nodes) n the electrc networ, d b s the demand at bus b, B b s networ susceptance matrx, δ s phase angle at bus, P b s the generatng unt actve power at bus b, and B s the set of all buses. The forward and bacward power system securty constrants are: l l cl cl ( H δ ) 0, l NL (6) l + ( H δ ) 0, l NL (7) l where, l s number of the lne between bus b and bus, l cl s the capacty lmt of lne l, H l s the networ transfer matrx, and NL s the set of all lnes. The generator capacty constrant s as follows: P + P 0, b B b b (8) where, P b s the mum generaton capacty at bus b. In order to solve OPF problem t s necessary to select one of the buses as swng (slac) bus wth the followng relaton: sw δ = 0, B (9) where, sw s the swng bus vector. There are two nonnegatve decson varables, wth the followng relatons: d b P b 0, b B (0) 0, b B () Mohammadan Bsheh et al: Solvng Envronmental/Economc Power Dspatch Problem Problem... 79
Downloaded from eee.ust.ac.r at :07 IRST on Saturday December 9th 08 3 Prncples of Mult Obectve Optmzaton enerally, nonlnear constraned mult obectve optmzaton problems can be shown as follows [0]: Mnmze f ( x) =,..., () Subect to N ob g ( x) = 0 =,..., ME (3) h ( x) = 0 =,..., MI (4) where f s the th obectve functon, x s a decson varable vector whch representng a soluton, N ob s the number of obectves, ME s the number of equalty constrants, and MI s the number of nequalty constrants. As stated already, the obectve functons often do not have a common scale, and normally compete wth each other. For such competng obectves, nstead of loong for one optmal soluton, a set of optmal solutons s of nterest. The reason for the nterest n these several optmal solutons s the fact that no soluton can be consdered to be better than any other one wth respect to all obectve functons. These optmal solutons are nown as Pareto optmal solutons. In ths stuaton, any two solutons x and x for a mult obectve optmzaton problem can have one of the two followng possbltes: The frst soluton x domnates or covers the other soluton. In ths case x s called non domnated (or domnatng) soluton or vce versa. In a mnmzaton problem, a soluton x covers or domnates x f and only f the followng two condtons are satsfed:. {,,..., N }: f ( x ) f ( x ) (5) ob. {,,..., N } : f ( x ) f ( x ) (6) ob < The solutons that are non-domnated wthn the entre search space consttute the Pareto optmal set [] and [0]. 4 Trust Regon Based Augmented Lagrangan Method (TRALM) The Augmented Lagrangan Method (ALM) [] solves a generc optmzaton problem mn f ( ) (7) Subect to H ( ) = 0 (8) ( ) 0 (9) 0 (30) By convertng t nto a sequence of unconstraned optmzaton problems wth penalty terms as follows: mn T T L ( ) = f ( ) + ( λ ) H ( ) + H ( ) [ W ] H ( ).(3) n {([ ( ),0]) + μ ( ) + U μ } U = In Eq. (3), s the number of nequalty constrants, λ and μ are tral Lagrange multplers, and W and U are penalty parameters. In the so-called multpler method, λ, μ, W, and U are updated after each round of unconstraned optmzaton λ + = λ + [ W ] H( ) (3) { μ U } μ + = + ( ),0 (33) W U + βww f H ( ) γw H ( ) f = W f H ( ) γ W H ( ) = β ( ) ( ) UU f γ U + f U f ( ) γ U( ) (34) (35) where, s the soluton of Eq. (3). Convergence s acheved provded that γw f 0, γu p, βw f, βu f, and the followng relatons are satsfed: ( L ) ε (36) + λ λ /( λ ) ε λ + (37) + μ μ /( μ ) ε μ + (38) In Eqs. (36-38), ε s are the tolerance parameters and ε decreases to a near-zero value ε as the suboptmzaton ncreases. Combned wth a sutable unconstraned optmzaton algorthm, the augmented Lagrangan method can solve large-scale nonlnear constraned optmzaton problems very relably and generate accurate Lagrangan multplers. In the TRALM algorthm, we use a trust regon method to solve Eq. (3). enerally trust-regon methods are used to solve unconstraned optmzaton problems []. Hence the constraned problems are converted to unconstraned ones by Lagrangan multplers and are solved by trust-regon algorthms. Branch et al. proposed a two-dmensonal trust-regon method for solvng large-scale optmzaton problems [3]. The pseudo code for the trust-regon method adopted n TRALM s shown n Fg.. 5 Implementaton of the Proposed Algorthm For mplementng the proposed algorthm (TRALM), the parameters have been selected as follows: 80 Iranan Journal of Electrcal & Electronc Engneerng, Vol. 8, No., June 0
Downloaded from eee.ust.ac.r at :07 IRST on Saturday December 9th 08 Let τ η γ γ Δ0 0p p p,0 p p, f 0, and 0 be gven, 0 whle L ( ) f ε do T T ψ ( S) L( ) S + S L( ) S S = arg mn ψ ( S) S Δ L( + S) L( ) ρ = ψ ( S ) f ρ f τ, + + S else + end f f ρ τ, Δ+ γ S else f ρ f η and S = Δ, Δ+ γ Δ else Δ+ Δ end f + end do Fg. Pseudo code for the trust-regon method adopted n TRALM ε = 5e 3, ε λ = e, ε = e0, ε η = 0.75, γ = 0., γ =.0, β μ 0 w, u = 3, γ = e, τ = 0.5 w, u = 0.33 Then the TRALM has been mplemented n MATLAB on a Pentum 33 MHz PC and was tested on the standard IEEE 30 bus sx-generator test system. The sngle lne dagram and the generator fuel cost and emsson coeffcents are shown n Fg. and Tables and, respectvely []. The detaled data could be obtaned from []. To compute the dfferent Pareto optmal solutons, obectve functons are lnearly combned to consttute a sngle obectve functon as follows: Mnmze wf ( P ) + ( w) λ E( P ) (39) where, the scalng factor λ was selected to be 3000 n our study and w s a weghtng factor []. As can be seen from Eq. (39), when, w = 0, the sngle obectve functon calculates only the emsson amount, and when w =, t calculates only the fuel cost. When w changes from 0 to, for each w, there s a Pareto optmal soluton. In order to generate evenlydstrbuted Pareto optmal soluton set, w s ncreased evenly by a fxed amount Δ w n each step from 0 to. The number of w counts the number of Pareto optmal solutons. As a matter of fact there s not a defnte rule to choose the number of w, but we generated the number of Pareto optmal sets, from to 0 and computed the best compromse solutons. Comparng these best compromse solutons showed that, there s not much dfference between the best compromse solutons when the number of w changes from to 0. So, was selected for w due to the advantage of less computaton tme. Fg. Sngle-lne dagram of IEEE 30 bus test system Table enerator fuel cost coeffcents en F = a+ bp + cp $/h P P mn No a b c Per 00 Per 00 MW MW 0 00 00.5 0.05 0 50 0.5 0.05 3 0 80 40.5 0.05 4 0 00 60.5 0.05 5 0 80 40.5 0.05 6 0 50 00.5 0.05 Table enerator emsson coeffcents en E = 0 ( α + βp + γp ) + ξexp( λp)( ton/ h) No α β γ ξ λ 4.09-5.554 6.490.0 E-4.857.543-6.047 5.638 5.0 E-4 3.333 3 4.58-5.094 4.586.0 E-6 8.000 4 5.36-3.550 3.380.0 E-3.000 5 4.58-5.094 4.586.0 E-6 8.000 6 6.3-5.55 5.5.0 E-5 6.667 By varyng w from 0 to wth the abovementoned procedure, the compromse soluton set has been computed usng TRALM, and the results are shown n Fgs. 3, 4 and 5. In Fg. 3, the system s consdered as lossless and the securty s released (Case). In Fg. 4, Mohammadan Bsheh et al: Solvng Envronmental/Economc Power Dspatch Problem Problem... 8
Downloaded from eee.ust.ac.r at :07 IRST on Saturday December 9th 08 the transmsson power loss has been taen nto account and the securty s released (Case). At last n Fg. 5 all three constrants have been taen nto account (Case3). Emsson 0.5 0. 0.5 0. 0.05 0. 0.95 0.9 600 605 60 65 60 65 630 635 640 Fuel cost($/h) Fg. 3 The soluton of TRALM approach for Case Emsson 0.5 0. 0.5 0. 0.05 0. 0.95 0.9 605 60 65 60 65 630 635 640 645 Fuel cost($/h) Fg. 4 The soluton of TRALM approach for Case Emsson 0.35 0.3 0.5 0. 0.5 0. 0.05 0. 60 65 60 65 630 635 640 645 650 655 Fuel cost($/h) Fg. 5 The soluton of TRALM approach for Case 3 6 Best Compromse Soluton Optmzaton of the formulated obectve functons Eqs. () and (3) usng TRALM yelds not a sngle optmal soluton, but a set of Pareto optmal solutons, n whch one obectve cannot be mproved wthout sacrfcng another obectve. For practcal applcatons, however, we need to select one soluton, satsfyng the dfferent goals to some extent. Such a soluton s called best compromse soluton. One of the challengng factors for the tradeoff decson s the mprecse nature of the decson maer's udgment. For ths consderaton fuzzy set theory s employed [9]. The th obectve value, F correspondng to a soluton s represented by a membershp functon μ F μ = F F F 0 mn F mn F F < F F F mn < F (40) mn where, F s the value of an orgnal obectve functon whch s supposed to be completely satsfactory, and, s the value of the obectve F functon whch s clearly unsatsfactory to the decson maer. For each non-domnated soluton, the normalzed membershp functon μ s calculated as follows: μ Nob μ = = M Nob μ = = (4) where, M s the number of non-domnated solutons, and N ob s the number of obectve functons. The functon μ n equaton Eq. (4) represents a fuzzy cardnal prorty ranng of the non-domnated solutons. The soluton whch attans the mum membershp μ n the fuzzy set can be chosen as the best compromse soluton or that havng the hghest cardnal prorty ranng. The values of μ for non-domnated solutons of the proposed algorthm have been calculated by a MATLAB program. 7 Results and Dscussons To demonstrate the effectveness of the proposed algorthm TRALM, our obtaned results are compared wth the results obtaned by the seven other algorthms reported as SPEA [], LP [], NPA[3], NSA[4], MOSST[5], FCPSO[6], and EC[8] for three Cases as follows: 8 Iranan Journal of Electrcal & Electronc Engneerng, Vol. 8, No., June 0
Downloaded from eee.ust.ac.r at :07 IRST on Saturday December 9th 08 7. Case At ths level the system s consdered as lossless and only the capacty constrants are consdered. The results obtaned from the proposed algorthm on the test system, have been shown n Fg. 3. Table 3 s provded to compare the best fuel costs of varous algorthms. As can be seen from ths table the fuel cost calculated by the proposed algorthm (TRALM) s less than or at least equal to the other concerned algorthms. Table 4 has been provded to compare the best emssons of dfferent algorthms. Ths table shows that the emsson amount obtaned from the proposed algorthm s less or at least equal to the other algorthms. It s worth to be noted that when the emsson of TRALM s equal to other algorthms ts fuel cost s less than the ones of the others. Table 5 has gathered the exstng best compromse solutons of the concerned algorthms. Snce n compromse soluton, decreasng of one obectve functon occurs n compensaton of ncreasng the other one. So t s not a good measure for comparson, but Table 5 has been provded to show that the compromse soluton of the proposed algorthm s qute reasonable. The run tme of TRALM, for Case s seconds, whch s greater than the run tme reported n [8], and less than the one reported n []. 7. Case In ths case the transmsson losses and power balance constrants are consdered, but the securty constrants are released. The numercal results obtaned from the proposed algorthm on the test system are shown n Fg. 4. Table 3 Comparson of best fuel costs of varous algorthms for Case Table 6 s provded to compare the best fuel costs of varous algorthms. Ths table shows that the fuel cost obtaned by the proposed algorthm except n EC approach s less than the ones of the other concerned algorthms. Table 7 has been provded to compare the best emssons of the varous algorthms. It shows that the emsson amount obtaned from the proposed algorthm s less than or at least equal to the other algorthms. It s worth to be noted that when the emsson of TRALM s equal to other algorthms ts fuel cost s less than the ones of the others. Table 8 has gathered the exstng best compromse solutons of the concerned algorthms. Ths table represents a reasonable compromse soluton for the proposed algorthm. 7.3 Case 3 In ths case, all constrants ncludng transmsson losses, power balance, and securty constrants are consdered. The numercal results obtaned from the proposed algorthm on the test system are shown n Fg. 5. Table 9 s provded to compare the best fuel costs of the varous algorthms. Ths table shows that the fuel cost obtaned by the proposed algorthm except n EC approach s less than the ones of the other concerned algorthms. Table 0 has been provded to compare the best emssons of the varous algorthms. It shows that the emsson amount obtaned from the proposed algorthm except n EC approach s less than the ones of other algorthms. Table has gathered the exstng best compromse solutons of the concerned algorthms. Ths table represents a reasonable compromse soluton for the proposed algorthm. LP [] MOSST [5] FCPSO [6] NPA [3] NSA [4] SPEA [] EC [8] TRALM [Proposed] P 0.500 0.097 0.070 0.6 0.038 0.009 0.097 0.097 P 0.3000 0.998 0.897 0.353 0.38 0.386 0.998 0.998 P 3 0.5500 0.543 0.550 0.549 0.53 0.5400 0.543 0.543.0500.06.050.045.0387 0.9903.06.06 P 5 0.4600 0.543 0.5300 0.476 0.534 0.5336 0.543 0.543 P 6 0.3500 0.3597 0.3673 0.35 0.34 0.3507 0.3597 0.3597 Cost ($/h) 606.34 605.8890 600.35 600.3 600.34 600. 600.4 600.4 Emsson 0.33 0. 0.3 0.38 0.4 0.3 0. 0. Table 4 Comparson of best emssons of varous algorthms for Case LP [] MOSST [5] FCPSO [6] NPA [3] NSA [4] SPEA [] EC [8] TRALM [Proposed] P 0.4000 0.4095 0.4097 0.40584 0.407 0.440 0.4060 0.4054 P 0.4500 0.466 0.4550 0.4595 0.4538 0.4577 0.4590 0.459 P 3 0.5500 0.546 0.5363 0.53797 0.4888 0.530 0.5379 0.538 0.4000 0.3884 0.384 0.38300 0.430 0.37 0.3830 0.383 P 5 0.5500 0.547 0.5348 0.5379 0.5836 0.53 0.5380 0.538 P 6 0.5000 0.54 0.540 0.50 0.4707 0.590 0.500 0.5099 Emsson 0.943 0.948 0.94 0.943 0.946 0.94 0.94 0.94 Cost ($/h) 639.600 644.8 638.3577 636.04 633.83 640.4 638.703 638.387 Mohammadan Bsheh et al: Solvng Envronmental/Economc Power Dspatch Problem Problem... 83
Downloaded from eee.ust.ac.r at :07 IRST on Saturday December 9th 08 Table 5 Comparson of best compromse solutons of varous algorthms for Case LP [] MOSST [5] FCPSO [6] NPA [3] NSA [4] SPEA [] EC [8] TRALM [Proposed] P 0.663 0.5 0.63 Comprom 0.50 P Comprom Compromse Comproms 0.3700 0.36 0.3765 se 0.3700 P 3 se soluton soluton was e soluton 0.5 0.5 0.548 soluton 0.5394 was not not was not 0.70 0.7660 0.6838 was not 0.7080 P 5 consdered calculated consdered 0.556 0.5397 0.538 calculated 0.5394 P 6 n the n detals n the paper 0.496 0.487 0.4305 n detals 0.496 Cost($/h) paper 6.758 608.90 606.03 60.977 60.9634 608.834 Emsson 0.968 0.05 0.04 0.005 0.000 0.05 Table 6 Comparson of best fuel costs of varous algorthms for Case L [] MOSST [5] FCPSO [6] NPA [3] NSA [4] SPEA [] EC [8] TRALM [Proposed] P 0.30 0.45 0.447 0.79 0.076 0.9 P Case was Case was 0.345 0.693 0.3066 0.363 0.30 0.304 P 3 not not 0.586 0.5908 0.5493 0.5803 0.5970 0.53 consdered consdered 0.9860 0.9944 0.9894 0.9580 0.9897.05 P 5 n the paper n the paper 0.564 0.535 0.544 0.558 0.50 0.53 P 6 0.3450 0.339 0.354 0.3589 0.35 0.369 Cost($/h) 607.786 608.06 607.98 607.86 605.8363 606.805 Emsson 0.0 0.07 0.9 0.76 0.08 0. Table 7 Comparson of best emssons of varous algorthms for Case LP [] MOSST [5] FCPSO [6] NPA [3] NSA [4] SPEA [] EC [8] TRALM [Proposed] P 0.4063 0.4064 0.399 0.445 0.40 0.406 P Case was Case was 0.4586 0.4876 0.3937 0.4450 0.4633 0.46 P 3 not not 0.550 0.55 0.588 0.5799 0.5447 0.5438 consdered consdered 0.4084 0.4085 0.436 0.3847 0.39 0.3966 P 5 n the paper n the paper 0.543 0.5386 0.5445 0.5384 0.5447 0.5438 P 6 0.494 0.499 0.59 0.505 0.55 0.56 Emsson 0.94 0.943 0.947 0.943 0.94 0.94 Cost($/h) 64.8964 644.3 638.98 644.77 646.03 644.079 Table 8 Comparson of best compromse solutons of varous algorthms for Case LP [] MOSST [5] FCPSO [6] NPA [3] NSA [4] SPEA [] EC [8] TRALM [Proposed] P 0.976 0.935 0.75 Comproms 0.694 P Comprom Comproms Comproms 0.3956 0.3645 0.375 e soluton 0.387 P 3 se soluton e soluton e soluton 0.5673 0.5833 0.5796 were not 0.5463 were not were not were not 0.698 0.6763 0.6770 consdered 0.685 P 5 consdered consdered consdered 0.50 0.5383 0.583 n the paper 0.5463 P 6 n the n the paper n the paper 0.3904 0.4076 0.48 for Case 0.4388 Cost($/h) paper for Case 67.79 67.80 67.57 67.506 Emsson 0.004 0.00 0.00 0.00 Table 9 Comparson of best fuel costs of varous algorthms for Case 3 LP [] MOSST [5] FCPSO [6] NPA [3] NSA[ 4] SPEA [] EC [8] TRALM [Proposed] P 0.596 0.7 0.358 0.39 0.83 0.648 P Case3 was Case3 was 0.3535 0.3747 0.35 0.3654 0.3554 0.3456 P 3 not not 0.7974 0.8057 0.848 0.779 0.5776 0.669 consdered consdered 0.979 0.903.043 0.98 0.7590.079 P 5 n the paper n the paper 0.0864 0.347 0.063 0.308 0.5393 0.69 P 6 0.49609 0.533 0.4664 0.59 0.4080 0.447 Cost($/h) 60.8 60.46 60.87 69.60 6.98 63.9360 Emsson 0.83 0.43 0.368 0.44 0.043 0.3 84 Iranan Journal of Electrcal & Electronc Engneerng, Vol. 8, No., June 0
Downloaded from eee.ust.ac.r at :07 IRST on Saturday December 9th 08 Table 0 Comparson of best emssons of varous algorthms for Case 3 LP [] MOSST [5] FCPSO [6] NPA [3] NSA [4] SPEA [] EC [8] TRALM [Proposed] P 0.47969 0.4753 0.4403 0.449 0.4 0.4649 P Case3 was not Case was not 0.587 0.56 0.4940 0.4598 0.4667 0.564 P 3 consdered consdered 0.676 0.653 0.7509 0.6944 0.554 0.60 n the paper n the paper 0.538 0.4363 0.5060 0.466 0.4059 0.4764 P 5 0.57 0.896 0.375 0.95 0.573 0.09 P 6 0.599 0.5988 0.55364 0.63 0.4550 0.577 Emsson 0.047 0.07 0.084 0.09 0.944 0.008 Cost($/h) 65.6 657.57 649.4 65.7 64.70 65.5708 Table Comparson of best compromse solutons of varous algorthms for Case 3 LP [] MOSST [5] FCPSO [6] NPA [3] NSA [4] SPEA [] EC [8] TRALM [Proposed] P 0.998 0.7 0.305 Compromse 0.36 P Compromse Compromse Compromse 0.435 0.3670 0.4389 soluton was 0.46 P 3 soluton was soluton was soluton was 0.74 0.8099 0.763 not 0.6508 not not not 0.685 0.7550 0.6978 consdered 0.7994 P 5 consdered consdered consdered 0.560 0.357 0.55 n the paper 0.809 P 6 n the paper n the paper n the paper 0.556 0.539 0.5507 for Case 3 0.494 Cost for Case3 for Case 3 for Case 3 630.06 65.7 69.59 6.7865 ($/h) Emsson 0.079 0.36 0.079 0.094 8 Conclusons Most papers reported n the lterature revew, used evolutonary algorthms to solve mult-obectve envronmental-economc power dspatch problem. The trust regon based augmented Lagrangan method (TRALM) s a nown and powerful technque for solvng constraned nonlnear programmng problems. Therefore n ths paper the TRALM, was presented and appled to combned envronmental / economc power dspatch optmzaton problem. The problem was formulated as a mult obectve optmzaton problem wth competng fuel cost and envronmental mpact obectves. The two obectve functons were lnearly combned by weghtng factors to consttute a sngle obectve functon. By varyng the weghtng factor the Pareto optmal sets are acheved. A fuzzy based mechansm was employed to extract the best compromse soluton among the Pareto optmal soluton set. To demonstrate the effectveness of the proposed algorthm, we compared the results obtaned by an mplementaton of our algorthm wth the ones obtaned by seven dfferent algorthms reported n the lterature. The results of the comparsons showed our proposed approach s very compettve n the sense of beng accurate. In the real world EEDP s very mportant for generatng companes to acheve an optmal soluton for ther nstalled generatng unts. Ths approach can help them fnd an optmal soluton for ther generaton schedule. References [] Wood A. J. and Woolenburg B. F., Power eneraton, Operaton and Control, New Yor, Wley, pp.9-90, 996. [] Abdo M. A., Multobectve Evolutonary Algorthms for Electrc Power Dspatch Problem, IEEE Trans. on Evolutonary Computaton, Vol. 0, No. 3, pp. 35-39, 006. [3] Talaq J. H., El-Hawary F. and. El-Hawary M. E., A summary of envronmental/economc dspatch algorthms, IEEE Trans. Power syst., Vol. 9, No. 3, pp. 508-56, 994. [4] El-Keb A. A., Ma H., and Hart J. L., Economc dspatch n vew of the clean ar act of 990, IEEE Trans. Power Syst. Vol. 9, No. 9, pp. 97-978, 994. [5] Yousef S., Moghaddam M. P. and Mad V. J., Agent-Based Modelng of Day-Ahead Real Tme Prcng n a Pool-Based Electrcty Maret, Iranan Journal of Electrcal and Electronc Engneerng, Vol. 7, No. 3, pp. 03-, 0. [6] Barforosh T., Moghaddam M. P., Javd M. H. and Sheh-El-Eslam M. K., A New Model Consderng Uncertantes for Power Maret, Iranan Journal of Electrcal and Electronc Engneerng, Vol., No., pp.7-8, 006. [7] Monsef H. and Mohamad N. T., eneraton Schedulng n a Compettve Envronment, Iranan Journal of Electrcal and Electronc Engneerng, Vol., No., pp.68-73, 005. Mohammadan Bsheh et al: Solvng Envronmental/Economc Power Dspatch Problem Problem... 85
Downloaded from eee.ust.ac.r at :07 IRST on Saturday December 9th 08 [8] ranell. P., Montagna M., Pasn. L. and Marannno P., Emsson Constraned dynamc dspatch, Electrc Power syst. Res., Vol. 4, pp. 56-64. 99. [9] Brodsy S. F. and Hahn R. W., Assessng the nfluence of power pools on emsson constraned economc dspatch, IEEE Trans. Power Syst., Vol. PER-6, Issue, pp. 57-6, 986. [0] El-Keb A. A. and Dng H., Envronmentally constraned economc dspatch usng lnear programmng, Electrcal Power System Research, Vol. 9, pp.55-59, 994. [] Farag A., Al-Baat S. and Cheng T. C., "Economc load dspatch multobectve optmzaton procedures usng lnear programmng technques, IEEE Trans. Power Syst., Vol. 0, pp73-738, 995. [] Srnvasan D., Cheng C. S. and Lew A. C., Multobectve generaton schedule usng fuzzy optmal search technque, Proc. Inst. Elect. Eng., en. Transm. Dst., Vol. 4, pp. 3-4, 994. [3] Abdo M. A, A Nched Pareto enetc Algorthm for Mult-obectve Envronmental/Economc Dspatch, Int. J.Electr. Power Energy Syst., Vol. 5, No., pp. 79-05, Feb. 003. [4] Abdo M. A., A New Mult-obectve Evolutonary Algorthm for Envronmental /Economc power Dspatch, Proc. IEEE Power Eng. Soc. Summer Meetng, pp. 63-68, Vancouver, BC, Canada, Jul. 5-9, 00. [5] Das D. B., and Patvardhan C., New multobectve stochastc search technque for economc load dspatch, Proc. Inst. Elect. Eng., en. Trans. Dst., Vol. 45, No. 6, pp. 747-75, 998. [6] Agrawal S., Pangrah B. K. and Twar M. K., Mult-obectve Partcle Swarm Algorthm wth Fuzzy Clusterng for Electrcal Power Dspatch, IEEE Transacton on Evolutonary Computaton, Vol., No. 5, pp.59-54, Oct. 008. [7] Zhao B. and Cao Y. J., Multple obectve partcle swarm optmzaton technque for economc load dspatch, J.Zheang Unversty Scence, Vol. 6, No. 5, pp. 40-47, 005. [8] Vahdnasab V. and Jadd S., Jont economc and emsson dspatch n energy marets: A multobectve mathematcal programmng approach, Energy 35, pp. 497-504, 00. [9] Saawa M., Yano H. and Yumne T., An nteractve fuzzy satsfcng method for multobectve lnear programmng problems and ts applcaton, IEEE Trans. Syst., Man, Cybern.,Vol. 7, No. 4, pp. 654-66, 987. [0] Zeleny M., "Multple Crtera Decson Mang", Mcraw-Hll Boo Company, pp. 34-373, 98. [] Wang H., Sanchez C.E., Zmmerman Ray D. and Thomas R. J., On Computatonal Issues of Maret-Based Optmal Power Flow, IEEE Transacton On Power Systems, Vol., No.3, pp.85-93, August 007. [] Bertseas D. P., Nonlnear Programmng, nd ed. Nashua, NH: Athena Scentfc, pp. 397-46, 999. [3] Branch M. A., Coleman T. F. and L Y., A subspace, nteror, and conugate gradent method for large-scale bound-constraned mnmzaton problems, SIAM J. Sc. Comput. Vol. 48, pp. - 3, Aug. 999. Hossen Mohammadan Bsheh Correspondng Author, Postal Address: Department of Industral Engneerng, Mazandaran Unversty of Scence and Technology, Babol, Iran. He receved the B.Sc. degree n Electrcal Engneerng from Sharf Unversty of Technology, Tehran, Iran n 97, and the MSc degree n Industral engneerng from Mazandaran Unversty of Scence and Technology, Babol, Iran n996. He s currently pursung the PhD degree n Industral Engneerng n Mazandaran Unversty of Scence and Technology. Hs research nterests are power system plannng and control, energy management systems, and optmzaton technques. Ashan Rahm-Kan (SM IEEE 08) receved the B.Sc. degree n electrcal engneerng from the Unversty of Tehran, Tehran, Iran, n 99 and the M.S. and Ph.D. degrees n electrcal engneerng from Oho State Unversty, Columbus, n 998 and 00, respectvely. He was the Vce Presdent of Engneerng and Development wth enscape, Inc., Lousvlle, KY, from September 00 to October 00 and a Research Assocate wth the School of Electrcal and Computer Engneerng (ECE), Cornell Unversty, Ithaca, NY, from November 00 to December 003. He s currently an Assocate Professor of Electrcal Engneerng (Control and Intellgent Processng Center of Excellence) wth the School of ECE, College of Engneerng, Unversty of Tehran. He s also the founder and drector of the Smart Networs Research Laboratory (SNL) at the school of ECE, UT. Hs research nterests nclude bddng strateges n dynamc energy marets, game theory and learnng, ntellgent transportaton systems, decson mang n multagent stochastc systems, stochastc optmal control, dynamc stoc maret modelng and decson mang usng game theory, smart grd desgn, operaton and control, estmaton theory and applcatons n energy and fnancal systems, rs modelng, and management n energy and fnancal systems. 86 Iranan Journal of Electrcal & Electronc Engneerng, Vol. 8, No., June 0
Downloaded from eee.ust.ac.r at :07 IRST on Saturday December 9th 08 Mr Mahd Seyyed Esfahn receved the B.Sc. degree n Industral Engneerng from Sharf Unversty of Technology, Tehran, Iran n 97, and M.Sc. and Ph.D. n Operaton Research from Bradford Unversty, England n 973 and 977 respectvely. He has been wth the Industral Engneerng Department, Amr Kabr Unversty, Iran, snce 977. Now, he s an assocate Prof. n aforesad unversty. Hs research nterests are operaton research and optmzaton technques. Mohammadan Bsheh et al: Solvng Envronmental/Economc Power Dspatch Problem Problem... 87