Evolutonary Mult-Objectve Envronmental/Economc Dspatch: Stochastc vs. Determnstc Approaches Robert T. F. Ah Kng, Harry C. S. Rughooputh and Kalyanmoy Deb 2 Department of Electrcal and Electronc Engneerng, Faculty of Engneerng, Unversty of Maurtus, Redut, Maurtus {r.ahkng, r.rughooputh}@uom.ac.mu 2 Kanpur Genetc Algorthms Laboratory, Department of Mechancal Engneerng, Indan Insttute of Technology, Kanpur, PIN 208 06, Inda deb@tk.ac.n KanGAL Report Number 200409 Abstract. Due to the envronmental concerns that arse from the emssons produced by fossl-fueled electrc power plants, the classcal economc dspatch, whch operates electrc power systems so as to mnmze only the total fuel cost, can no longer be consdered alone. Thus, by envronmental dspatch, emssons can be reduced by dspatch of power generaton to mnmze emssons. The envronmental/economc dspatch problem has been most commonly solved usng a determnstc approach. However, power generated, system loads, fuel cost and emsson coeffcents are subjected to naccuraces and uncertantes n real-world stuatons. In ths paper, the problem s tackled usng both determnstc and stochastc approaches of dfferent complextes. The Nondomnated Sortng Genetc Algorthm II (NSGA-II), an eltst multobjectve evolutonary algorthm capable of fndng multple Pareto-optmal solutons wth good dversty n one sngle run s used for solvng the envronmental/economc dspatch problem. Smulaton results are presented for the standard IEEE 30-bus system. Introducton The classcal economc dspatch problem s to operate electrc power systems so as to mnmze the total fuel cost. Ths sngle objectve can no longer be consdered alone due to the envronmental concerns that arse from the emssons produced by fosslfueled electrc power plants. In fact, the Clean Ar Act Amendments have been appled to reduce SO 2 and NO x emssons from such power plants. Accordngly, emssons can be reduced by dspatch of power generaton to mnmze emssons nstead of or as a supplement to the usual cost objectve of economc dspatch. Envronmental/economc dspatch s a mult-objectve problem wth conflctng objectves because polluton s conflctng wth mnmum cost of generaton. Varous technques have been proposed to solve ths mult-objectve problem whereby most researchers have concentrated on the determnstc problem. Economc dspatch calculates the cost of generaton based on data relatng fuel cost and power output. Ths cost functon s approxmated by a quadratc equaton wth
2 Ah Kng, Rughooputh and Deb cost coeffcents. In conventonal economc dspatch the coeffcents are assumed to be determnstc, but n real-world stuatons, these data are subjected to naccuraces and uncertantes. These devatons are attrbuted to () naccuraces n the process of measurng and forecastng of nput data and () changes of unt performance durng the perod between measurng and operaton []. Thus, the operatng pont n practce wll dffer from the planned operatng pont and wll thus affect the actual fuel cost. Smlarly, emsson coeffcents may also be subjected to some devatons resultng n defnte dfferences n practcal systems. There has been much research usng the determnstc approach to solve the envronmental/economc dspatch problem. Gent and Lamont [2] ntroduced the mnmum-emsson dspatch concept where they developed a program for on-lne steam unt dspatch that results n the mnmzng of NO x emsson. These authors ntroduced the mathematcal representaton of NO x emsson of steam generatng unts and used a Newton-Raphson convergence technque to obtan base ponts and partcpaton factors. Zahav and Esenberg [3] proposed a dspatch procedure for power that meets the demand for energy whle accountng for both cost and emsson consderatons. A tradeoff curve whch present the decson maker wth all possble courses of acton (dspatch polces) for a gven demand was ntroduced. Nanda et al. [4] presented an mproved Box complex method for economc dspatch and mnmum emsson dspatch problems. Dhllon et al. [5] formulated the multobjectve thermal power dspatch usng noncommensurable objectves such as operatng costs and mnmal emsson. The epslon-constrant method was used to generate non-nferor solutons to the multobjectve problem consderng the operatng cost as the objectve and replacng emsson objectve as a constrant. More recently, mult-objectve evolutonary algorthms have been appled to the problem at hand. Abdo has poneered ths research by applyng NSGA [6], NPGA [7] and SPEA [8] to the standard IEEE 30-bus system. In fact, t has been shown that NSGA-II can obtan mnmum cost and mnmum emsson solutons comparable to Tabu search [9]. Not long after the ntroducton of the envronmental consderaton n the economc dspatch problem, researchers started consderng stochastc approaches bearng n mnd the uncertantes that are nherent n real-world stuatons. Vvan and Heydt [0] ncorporated the effects of uncertan system parameters nto optmal power dspatch. Ther method employed the multvarate Gram-Charler seres as means of modelng the probablty densty functon (p.d.f.) whch characterzes the uncertan parameters. Part et al. [] extended the Lagrange multpler soluton method to solve the economc thermal power dspatch problem usng an objectve functon consstng of the sum of the expected producton costs and expected cost of devatons (a penalty term proportonal to the expectaton of the square of the unsatsfed load because of possble varance of the generator actve power). Bunn and Paschents [] developed a stochastc model for the economc dspatch of the electrc power. These authors used a form of stochastc lnear programmng method for onlne schedulng of power generaton at 5 mnute ntervals takng nto account the msmatch between dspatched generaton and actual load demanded. Expermental results on real data demonstrated the effcency of the approach compared to conventonal determnstc lnear programmng model. Dhllon et al. [2] have used the weghted mnmax technque to obtan trade-off relaton between the conflctng objectves and fuzzy set theory s subsequently used to help the operator choose an optmal operatng pont. In another
Evolutonary Mult-Objectve Envronmental/Economc Dspatch 3 attempt, Dhllon et al. [3] solved the multobjectve stochastc economc dspatch problem whereby the weghted sum technque and Newton-Raphson algorthm are used to generate the non-nferor solutons consderng expected operatng cost and expected rsk assocated wth the possble devaton of the random varables from ther expected values. In ther study, the random varables are assumed to be normally dstrbuted and statstcally dependent on each other, hence the determnstc objectve functons have both varance and covarance terms. Recently, Bath et al. [4] presented an nteractve fuzzy satsfyng method for mult-objectve generaton schedulng wth explct recognton of statstcal uncertantes n system producton cost data. However, the mult-objectve problem s converted nto a scalar optmzaton problem and solved usng weghted sum method. Hooke-Jeeves pattern search, evolutonary optmzaton and weght smulaton methods are used to fnd the optmal weght combnatons and fuzzy sets are used to obtan the 'best' soluton from the non-nferor solutons set. In ths paper, both the determnstc and stochastc approaches are addressed. More precsely, the stochastc problem s consdered n a unque way due to the nature of the problem when the load flow calculatons determne the power generated by the slack bus. Thus, a relablty measure s used to test the power system under dfferent stochastc consderatons. The paper s organzed as follows. The envronmental/economc dspatch problem s defned n Secton 2. Secton 3 outlnes the system parameters consdered n ths study. The smulaton results of the determnstc approach are gven n Secton 4 whle those of the stochastc approach are presented n Secton 5. Based on these results, the man fndngs and some conclusons are outlned n Secton 6. 2 Envronmental/Economc Dspatch The envronmental/economc dspatch nvolves the smultaneous optmzaton of fuel cost and emsson objectves whch are conflctng ones. The determnstc problem s formulated as descrbed below. 2. Objectve Functons Fuel Cost Objectve. The classcal economc dspatch problem of fndng the optmal combnaton of power generaton, whch mnmzes the total fuel cost whle satsfyng the total requred demand can be mathematcally stated as follows [5]: where n ( + ) 2 C = a b P + c P $/hr = C: total fuel cost ($/hr), a, b, c : fuel cost coeffcents of generator, P G : power generated (p.u.) by generator, and G G ()
4 Ah Kng, Rughooputh and Deb n: number of generators. NO x Emsson Objectve. The mnmum emsson dspatch optmzes the above classcal economc dspatch ncludng NO x emsson objectve, whch can be modeled usng second order polynomal functons [5]: n 2 E NO x = ( an + bn PG + cn PG + d = N sn( e N P G )) ton/hr (2) 2.2 Constrants The optmzaton problem s bounded by the followng constrants: Power balance constrant. The total power generated must supply the total load demand and the transmsson losses. where where n P G = P P D : total load demand (p.u.), and P L : transmsson losses (p.u.). The transmsson losses s gven by D P L = 0 N N ( rj / VV j )cos( δ δ j )( PP j + QQ j ) + = = (4) PL = j ( rj / VV j )sn( δ δ j )( Q Pj PQ j ) N : number of buses r : seres resstance connectng buses and j j V : voltage magntude at bus δ : voltage angle at bus P : real power njecton at bus Q : reactve power njecton at bus (3) Maxmum and mnmum lmts of power generaton. The power generated P G by each generator s constraned between ts mnmum and maxmum lmts,.e., P Gmn P G P Gmax (5) where P Gmn : mnmum power generated, and P Gmax : maxmum power generated.
Evolutonary Mult-Objectve Envronmental/Economc Dspatch 5 2.3 Multobjectve Formulaton The multobjectve determnstc envronmental/economc dspatch optmzaton problem s therefore formulated as: Mnmze [ C, E NOx ] (6) n subject to: P P P = 0 (power balance), and G = D L P Gmn P G P Gmax (generaton lmts) 3 System Parameters Smulatons were performed on the standard IEEE 30-bus 6-generator test system (Fg. ) usng the Eltst Nondomnated Sortng Genetc Algorthm (NSGA-II) for both determnstc and stochastc approaches. Detals of the algorthm of NSGA-II can be found n [6]. The power system s nterconnected by 4 transmsson lnes and the total system demand for the 2 load buses s 2.834 p.u. Fuel cost and NO x emsson coeffcents for ths system are gven n Tables and 2 respectvely. 2 4 3 5 6 8 7 6 4 3 2 0 7 5 9 25 8 9 G 6 26 20 G 5 30 24 23 22 27 2 G 29 28 G 4 G 2 G 3 Fg.. Sngle-lne dagram of IEEE 30-bus test system [8]
6 Ah Kng, Rughooputh and Deb Table. Fuel Cost coeffcents Unt a b c P Gmn P Gmax 0 200 00 0.05 0.50 2 0 50 20 0.05 0.60 3 20 80 40 0.05.00 4 0 00 60 0.05.20 5 20 80 40 0.05.00 6 0 50 00 0.05 0.60 Table 2. NO x Emsson coeffcents Unt a N b N c N d N e N 4.09e-2-5.554e-2 6.490e-2 2.0e-4 2.857 2 2.543e-2-6.047e-2 5.638e-2 5.0e-4 3.333 3 4.258e-2-5.094e-2 4.586e-2.0e-6 8.000 4 5.326e-2-3.550e-2 3.380e-2 2.0e-3 2.000 5 4.258e-2-5.094e-2 4.586e-2.0e-6 8.000 6 6.3e-2-5.555e-2 5.5e-2.0e-5 6.667 In all smulatons, the followng parameters were used: populaton sze = 50 crossover probablty = 0.9 mutaton probablty = 0.2 dstrbuton ndex for crossover = 0 dstrbuton ndex for mutaton = 20 The smulatons were run for fve dfferent cases: Case D: Determnstc - System s consdered as lossless Case D2: Determnstc - Transmsson losses are consdered Case S: Stochastc power generated Case S2: Stochastc power generated and system loads Case S3: Stochastc power generated, system loads, fuel cost and emsson coeffcents 4 Determnstc approach Usng the determnstc parameters as gven n Tables and 2, the smulaton results obtaned are presented.
Evolutonary Mult-Objectve Envronmental/Economc Dspatch 7 4. Case D: Determnstc wthout Transmsson Losses Fg. 2 shows a good dversty n the nondomnated solutons obtaned by NSGA-II after 200 generatons. 0.225 0.22 NSGA-II epslon-constrant 0.25 NOx Emsson (ton/hr) 0.2 0.205 0.2 0.95 0.9 600 605 60 65 620 625 630 635 640 Fuel Cost ($/hr) Fg. 2. Nondomnated solutons for Case D Table 3 and 4 show the best fuel cost and best NO x emsson obtaned by NSGA-II as compared to Lnear Programmng (LP) [5], Mult-Objectve Stochastc Search Technque (MOSST) [7], Nondomnated Sortng Genetc Algorthm (NSGA) [6], Nched Pareto Genetc Algorthm (NPGA) [7] and Strength Pareto Evolutonary Algorthm (SPEA) [8]. It can be deduced that NSGA-II fnds comparable mnmum fuel cost and comparable mnmum NO x emsson to the last three evolutonary algorthms. To confrm that NSGA-II s able to obtan the Pareto front for the problem, the epslon-constrant method [8] has been used as shown on the plot of Fg. 2. Genetc algorthm was used to solve the resultng sngle-objectve problem. Table 3. Best fuel cost LP [5] MOSST NSGA NPGA SPEA NSGA-II [7] [6] [7] [8] P G 0.500 0.25 0.567 0.080 0.062 0.059 P G2 0.3000 0.3020 0.2870 0.3284 0.2897 0.377 P G3 0.5500 0.53 0.467 0.5386 0.5289 0.526 P G4.0500.0208.0467.0067.0025.046 P G5 0.4600 0.53 0.5037 0.4949 0.5402 0.559 P G6 0.3500 0.3625 0.3729 0.3574 0.3664 0.3583 Best cost 606.34 605.889 600.572 600.259 600.5 600.55 Corresp. emsson 0.22330 0.22220 0.22282 0.226 0.225 0.2288
8 Ah Kng, Rughooputh and Deb Table 4. Best NO x emsson LP [5] MOSST NSGA NPGA SPEA NSGA-II [7] [6] [7] [8] P G 0.4000 0.4095 0.4394 0.4002 0.46 0.4074 P G2 0.4500 0.4626 0.45 0.4474 0.4532 0.4577 P G3 0.5500 0.5426 0.505 0.566 0.5329 0.5389 P G4 0.4000 0.3884 0.387 0.3688 0.3832 0.3837 P G5 0.5500 0.5427 0.5553 0.575 0.5383 0.5352 P G6 0.5000 0.542 0.4905 0.5259 0.548 0.50 Best 0.9424 0.948 0.9436 0.9433 0.942 0.9420 emsson Corresp. cost 639.600 644.2 639.23 639.82 638.5 638.269 4.2 Case D2: Determnstc wth Transmsson Losses Consdered In ths case, the transmsson losses are consdered and the NSGA-II algorthm was run for 200 generatons. Fg. 3 shows the nondomnated solutons obtaned by NSGA-II for Case D2 where a good dstrbuton of the solutons s observed. 0.22 0.25 NSGA-II epslon-constrant NOx Emsson (ton/hr) 0.2 0.205 0.2 0.95 0.9 605 60 65 620 625 630 635 640 645 Fuel Cost ($/hr) Fg. 3. Nondomnated solutons for Case D2 The best fuel cost and best NO x emsson obtaned by NSGA-II as compared to NSGA, NPGA and SPEA are gven n Table 5 and 6. It s observed that NSGA-II agan fnds better mnmum fuel cost and emsson level than the other evolutonary algorthms.
Evolutonary Mult-Objectve Envronmental/Economc Dspatch 9 Table 5. Best fuel cost NSGA [6] NPGA [7] SPEA [8] NSGA-II P G 0.68 0.245 0.086 0.82 P G2 0.365 0.2792 0.3056 0.348 P G3 0.544 0.6284 0.588 0.590 P G4 0.9447.0264 0.9846 0.970 P G5 0.5498 0.4693 0.5288 0.572 P G6 0.3964 0.3993 0.3584 0.3548 Best cost 608.245 608.47 607.807 607.80 Corresp. emsson 0.2664 0.22364 0.2205 0.289 Table 6. Best NO X emsson NSGA [6] NPGA [7] SPEA [8] NSGA-II P G 0.43 0.3923 0.4043 0.44 P G2 0.459 0.4700 0.4525 0.4602 P G3 0.57 0.5565 0.5525 0.5429 P G4 0.3724 0.3695 0.4079 0.40 P G5 0.580 0.5599 0.5468 0.5422 P G6 0.5304 0.563 0.5005 0.5045 Best 0.9432 0.9424 0.9422 0.949 emsson Corresp. cost 647.25 645.984 642.603 644.33 Agan, t can be deduced that the algorthm s capable of obtanng the Pareto front for the gven problem as verfed by the mnmum of each objectve and ponts obtaned by the epslon-constrant method n Fg. 3. It has been shown that NSGA-II can obtan the Pareto front of the problem and t s therefore deal for solvng the multobjectve envronmental/economc dspatch optmzaton problem whch has conflctng objectves from the fact that the multobjectve approach yelds multple Pareto-optmal solutons n a sngle smulaton run whereas multple runs are requred for the sngle objectve approach wth weghted objectves. 5 Stochastc approach Prevous stochastc approaches nvolved the ncluson of devatonal (recourse) costs to account for msmatch between scheduled output and actual demand n the formulaton of the objectve functon [], and converson of stochastc models nto ther determnstc equvalents by takng ther expected values and formulatng the problem as the mnmzaton of cost and emsson plus addtonal objectve for the expected devaton between generator outputs and load demand (unsatsfed load
0 Ah Kng, Rughooputh and Deb demand) [2, 3, 4]. The approach adopted n ths paper s based on the relablty concept and smulatons are performed to test the relablty of the stochastc system under dfferent problem formulatons. Decson varables P G ( =,,6) are assumed to be normally dstrbuted wth Mean P G and Standard Devaton (SD) σ = 0.P G. For each soluton P G ( = 2,,6), 00 random nstantates havng Mean P G and SD σ are created wthn 2σ. A good measure of system performance n the case of stochastc systems s ts relablty [9]. We defne relablty R as: R = and an addtonal constrant s ncluded n the optmzaton problem: n m (7) cr R R (8) cr where R s the requred relablty whch s 95.6% for whch Pr{ µ 2σ < P < µ + 2σ }. Thus, Relablty R s calculated accordng to the number of cases for whch P s found to be wthn 2σ. In the stochastc approach, the objectve functons are now reformulated as follows: Mn. Mn. subject to the followng constrants: where Cost, NO x, Cost + 2 σ NO x n P G = + 2 σ P Cost NOx D P L = 0 P Gmn P G P Gmax R R cr σ Cost and σ NOx are the Expected Cost and Expected NO x emsson and SD of Expected Fuel Cost and SD of Expected NO x emsson respectvely. Note that P G s calculated from the loadflow program and ths satsfes mplctly the power balance constrant (equaton 3). The procedure used n ths stochastc method s descrbed as follows: For each feasble soluton P j ( j = 2,...,n ) obtaned by NSGA-II, ( j 2 Create m nstantates P ) ( j =,...,n ) by perturbng (9) each P j as N( P j, σ j ) where m = 00 and σ j = 0. P j.
Evolutonary Mult-Objectve Envronmental/Economc Dspatch Count the number of nstantates n for whch ( ) P [ P mn max P, where ] P 2 j = P mn = P σ P 0. 8 P 2 j = P max = P + σ P. 2 and Calculate R = n m Calculate Expected Cost Cost and Expected NO x σ SD of Cost σ Cost and SD of NO x NOx NO x and 5. Case S: Stochastc power generated wth fxed system load The mult-objectve optmzaton problem s formulated as above wth fxed total system load PD = 2.834 p.u. Thus, power generated P G are random varables. Fg. 4 shows the nondomnated solutons for the stochastc case wth fxed system load obtaned as compared to the determnstc case. 0.225 0.22 Case D2 Case S 0.25 NOx Emsson (ton/hr) 0.2 0.205 0.2 0.95 0.9 605 60 65 620 625 630 635 640 645 650 Fuel Cost ($/hr) Fg. 4. Nondomnated solutons obtaned for stochastc power generated wth fxed demand (Case S) as compared to determnstc case (Case D2) It can be nferred that for solutons excludng the mnmum fuel cost and mnmum NO x emsson (.e. the two extreme ponts on the curve, optmum values of the two
2 Ah Kng, Rughooputh and Deb objectves would be generally worst than the determnstc case. In other words, for pseudo weghts excludng (, 0) and (0, ), determnstc solutons always domnate stochastc ones. 5.2 Case S2: Stochastc power generated and system loads The mult-objectve optmzaton problem s formulated as above but the ndvdual loads on the system are treated as stochastc varables. Thus, power generated and system loads are random varables. Each of 2 loads s normally dstrbuted wth mean P L and σ = 0.P L. Power factor for each load s mantaned as at the base load,.e. rato P L to Q L s constant. Fg. 5 shows the nondomnated solutons obtaned for the three cases: determnstc (Case D2), stochastc power generated wth fxed system load (Case S), stochastc power generated and system loads (Case S2). It can be observed that the determnstc case shows that the mnmum fuel cost obtaned s no longer optmal when the decson varables are taken as stochastc. An nterestng observaton reveals that the mnmum NO x emsson s not affected by stochastc consderatons. 0.225 0.22 Case D2 Case S Case S2 0.25 NOx Emsson (ton/hr) 0.2 0.205 0.2 0.95 0.9 605 60 65 620 625 630 635 640 645 650 655 Fuel Cost ($/hr) Fg. 5. Nondomnated solutons for Cases D2, S and S2 on same plot Fg. 6 shows the varaton of average relablty wth pseudo weghts (, 0), (0.5, 0.5) and (0, ) of the two objectves. The average relablty was calculated over 00 runs for each set of decson varables (defnng an operatng pont). Ths fgure clearly verfes the statement above regardng the non-dependablty of the NO x emsson on the nature of the decson varables for mnmum emsson level.
Evolutonary Mult-Objectve Envronmental/Economc Dspatch 3 00 95 Case S2 Case S Case D2 Average Relablty (%) 90 85 80 75 (, 0) (0.5, 0.5) (0, ) Mnmum Fuel Cost Mnmum NOx Emsson Fg. 6. Average relablty for dfferent pseudo weghts for Cases D2, S and S2 5.3 Case S3: Stochastc power generated, system loads, fuel cost and emsson coeffcents Ths case s smlar to Case S2 but n addton the fuel cost and NO x emsson coeffcents are consdered as stochastc varables wth mean as gven n Tables and 2 respectvely and standard devaton as 0. of ther respectve means. Fg. 7 shows the nondomnated solutons obtaned for the stochastc case consderng power generated, system loads, fuel cost and emsson coeffcents consdered as random varables compared to the determnstc case as n Case D2. 0.22 0.25 Case D2 Case S3 NOx Emsson (ton/hr) 0.2 0.205 0.2 0.95 0.9 605 60 65 620 625 630 635 640 645 650 655 Fuel Cost ($/hr) Fg. 7. Nondomnated solutons for stochastc power generated, system loads, fuel cost and NO x emsson coeffcents (Case S3) as compared to determnstc case (Case D2) It can be observed that n ths case the nondomnated solutons obtaned are shfted away from those of the determnstc case, that s, the solutons obtaned when power
4 Ah Kng, Rughooputh and Deb generated, system loads, cost and emsson coeffcents are stochastc varables, are all domnated by the determnstc solutons. Therefore, hgher cost and emsson are expected when the power system s operated n real-world stuaton. The expected ncrease n mnmum cost and mnmum emsson are about 6 $/hr and 0.002 ton/hr. Thus, a % ncrease has been obtaned n both objectves when the standard devaton was taken as 0% of the mean value of the varables. It s to be noted that these fgures are not neglgble when the power system s operated over a year, correspondng fgures would be $52,560 and 7.52 tons respectvely. 6 Conclusons In ths paper, the mult-objectve envronmental/economc dspatch problem has been solved usng the eltst Nondomnated Sortng Genetc Algorthm. The algorthm has been run on the standard IEEE 30-bus system. Both the determnstc and stochastc approaches have been addressed. In the determnstc problem, two cases have been studed: () the lossless system and () when transmsson losses are taken nto consderaton and gven by the load flow soluton. In the frst case, the mnmum cost and mnmum emsson solutons found by NSGA-II are better than those found by the conventonal Lnear Programmng method. Moreover, these solutons are comparable, f not better than MOSST, NSGA, NPGA and SPEA reported from earler studes. Consderng the transmsson losses, smlar results were obtaned thus confrmng the superorty of NSGA-II as a fast evolutonary mult-objectve algorthm. For the stochastc problem, three cases wth dfferent complextes have been analyzed, all takng transmsson losses nto consderaton wth the followng stochastc varables: () power generated, () power generated and system loads, and () power generated, system loads and cost and emsson coeffcents. The followng nterestng fndngs can be stated after comparson wth the determnstc nondomnated solutons obtaned. In the frst case, the stochastc solutons obtaned are domnated by the determnstc ones except for the two extreme solutons. Ths means that n practce, real-world operaton cost and emsson would be always hgher except f the power system s operated at ether ts mnmum fuel cost (economc dspatch) or mnmum emsson (envronmental dspatch). In the second case, the mnmum emsson soluton s not affected by stochastc consderatons but all other solutons have hgher cost for the same emsson level. The mnmum cost soluton beng hgher than the determnstc one by about 6 $/hr. The relablty measure used n ths study confrms the non-dependence of the emsson level from comparson of operatng ponts based on dfferent pseudo-weghts of the two objectves. The thrd case shows that nondomnated solutons obtaned are shfted away from those of the determnstc case, the mnmum cost and mnmum emsson solutons beng hgher by about 6 $/hr and 0.002 ton/hr, respectvely. Thus, n real-world stuatons, the power system would be operated at an operatng pont, whch would have hgher fuel cost and hgher emsson level than the calculated and planned operatng pont. In other words, the real-world (stochastc) operatng pont would always be domnated by the determnstc one.
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