Mult-Objectve Evolutonary Programmng for Economc Emsson Dspatch Problem P. Venkatesh and Kwang. Y.Lee, Fellow, IEEE Abstract--Ths paper descrbes a new Mult-Objectve Evolutonary Programmng (MOEP) method to solve the Combned Economc Emsson Dspatch (CEED) and Economc Emsson Dspatch (EED) problems. The CEED s a b-objectve optmzaton problem that consders two objectves such as fuel cost and O x emsson. It s converted nto a sngle objectve optmzaton problem usng weghted sum method. The EED s a three-objectve optmzaton problem that consders the fuel cost, O x and SO 2 emssons as objectves. on-domnated soluton rankng s employed as selecton mechansm n the proposed MOEP for the CEED and EED problems. The developed algorthm s tested for a three-unt and a sx-unt systems, and sx and 30 bus systems. The results demonstrate the capabltes of the proposed approach to generate well-dstrbuted Pareto optmal solutons of the mult-objectve problems n a sngle run. Index Terms-- Combned economc emsson dspatch, economc emsson dspatch, evolutonary programmng, nondomnated soluton, pareto optmal solutons. E I. ITRODUCTIO COOMIC load dspatch s one of the man functons of energy management system whch determnes the optmal real power settng of generatng unts wth an objectve of mnmzng the total fuel cost []. Artfcal ntellgence technques such as Genetc Algorthm (GA) [2] and Artfcal eural etwork [3] are appled to solve economc load dspatch problems. Modern electrc utlty companes can no longer dspatch electrc power wth mnmum cost as the only objectve. The ncreasng envronmental concern pressures the utlty towards mult-objectve dspatch of electrc power n meetng the load demand. Generaton of electrcty from fossl fuel releases several contamnants, such as sulphur doxdes, ntrogen oxdes and carbon doxde, nto the atmosphere. In addton, ncreasng publc awareness of envronmental protecton and the passage of the clean ar act amendments of 990 [4] have also forced utltes to modfy the objectves of ther electrc power dspatch problems. Reducton of emsson of gases such as SO 2 and O x, are consdered as an objectve and a soluton procedure based on The work s supported n part by BOYSCAST Fellowshp Scheme from Government of Inda (SR/BY/E-08/06). P. Venkatesh s wth the Department of Electrcal Engneerng, The Pennsylvana State Unversty, Unversty Park, PA 6802, USA. (e-mal: vup@psu.edu) Kwang Y. Lee s wth the Department of Electrcal and Computer Engneerng, Baylor Unversty, One Bear Place #97356, Waco, TX 76798-7356, USA (e-mal: Kwang_Y_Lee@baylor.edu) the LaGrange multpler technque for envronmentally constraned economc dspatch problem s also proposed [4]. Varous methods and emsson model that reduce emsson nto the atmosphere have been summarzed n [5]. In the recent years, combned emsson economc dspatch receved much attracton [6-0]. Dfferent technques have been reported n the lterature pertanng to envronmental economc dspatch problem. Dhllon et al. [7] have solved a problem on CEED by a stochastc approach that takes nto account uncertantes n the system producton cost and nature of load demand. In a weghted sum method [7] a set of tradeoff soluton s obtaned by varyng the weghts. Ths method requres multple runs. In Song et al. [8], Fuzzy logc controlled genetc algorthm s used to solve a CEED problem. Wong et al. [9] have proposed Evolutonary Programmng (EP) based algorthm for envronmental constraned economc dspatch problem. The comparson of EP technques such as GA, mcro GA and EP to CEED problems was presented n [0]. Each of these methods [6-0] was based on optmzaton of the most preferred objectve and consderng the remanng objectve as constrants. These methods do not deal wth the Pareto optmal solutons. In recent years, researchers have been showng a great deal of attenton to the development of effcent, real, multobjectve optmzaton technques for solvng varous real world mult-objectve problems. A fuzzy mult-objectve optmzaton technque for CEED problem was proposed n []. A mult-objectve stochastc search method for solvng the economc dspatch problem was presented n [2]. However, ths technque has ts own dsadvantages such as suboptmal soluton and hgh computatonal tme. On the other sde, the researchers have been explorng many multobjectve algorthms, to elmnate most of the abovementoned dffcultes n dfferent applcatons. Most of the methods are tested n the test examples to show the ablty n solvng dfferent mult-objectve problems [3-5]. A new Strength Pareto Evolutonary Algorthm (SPEA) based approach s proposed to solve CEED problem [6]. In ths SPEA method, Pareto optmal solutons are found n a sngle run. The comparatve study proves that the SPEA s hghly capable of fndng Pareto optmal solutons over the other method lke ched Pareto GA (PGA). Eltst multobjectve evolutonary algorthm based on on-domnated Sortng GA II (SGA-II) for solvng CEED problem was proposed [7]. The three objectves (fuel cost, O x and SO 2 2008 IEEE.
emssons) of quadratc nature are consdered n the envronmental/economc dspatch problem. Pareto optmal front s also gven for the three objectves to the three unts test system. Recently, (SGA-II) s proposed to solve the CEED problem, and compared wth weghted sum method, PGA, SGA & SPEA [8]. Abdo has compared the varous mult-objectve evolutonary algorthms to the electrc power dspatch problem [9]. The two objectves such as fuel cost n quadratc functons and emsson n exponental term n addton to the quadratc functons are consdered n the electrc power dspatch problem for the varous test systems. Ths paper presents the applcaton of MOEP to the CEED problem havng two objectves such as fuel cost and O x emsson. EED problem havng three objectves such as fuel cost, O x and SO 2 emsson s solved usng MOEP algorthm. The MOEP method s developed to fnd the exact Pareto optmal solutons to CEED and EED problems. The proposed MOEP algorthm s demonstrated to solve the CEED problem for a three-unt and a sx-unt system. The results of the MOEP are compared wth SGA-II method and weghted sum method. The proposed MOEP algorthm s demonstrated to solve the EED problem for a sx and the IEEE 30 bus systems. The effectveness and potental of the proposed approach to solve the CEED and EED problems are demonstrated. II. PROBLEM FORMULATIO The objectve of CEED problem s to mnmze two conflctng objectve functons, fuel cost and O x emsson, whereas the objectve of EED problem s to mnmze the three conflctng objectve functons, fuel cost and O x and SO 2 emssons, whle satsfyng several equalty and nequalty constrants. The problem formulaton s as follows: A. Objectve Functons. ) Fuel cost objectve. Total fuel cost of generaton s formulated as F = F PG = a PG + b PG + c PG + d hr () ( 3 2 ) $/ = where s the number of generators, a, b, c and d are the cost coeffcent of generator and P G s the power generated by the th unt. 2) O x emsson objectve. It s expressed as 2 3 2 x( ) = F = EO PG = na PG + nb PG + nc PG + nd Kg/hr (2) where na j, nb j, nc j and nd j are the coeffcents of emsson for unt and F 3 s the total O x emsson of pollutants of the generatng unts. 3) SO 2 emsson objectve. Lke the fuel cost gven (), the SO 2 emsson s also expressed as 3 2 3 2 = F = ESO ( PG ) = sa PG + sb PG + sc PG + sd Kg/hr (3) where sa, sb, sc and sd are the coeffcents of emsson for unt and F 2 s the total SO 2 emsson of pollutants of the generatng unts. B. Constrants The problem constrants are ) Generatng capacty constrant. where PGmn PG PGmax for =, 2,... (4) PG mn and max power output of the th unt. 2) Power balance constrants. PG are the mnmum and maxmum (5) = PD = PG PL where PD s the system load demand and PL s the transmsson loss, whch s determned usng the B-matrx loss coeffcents. The calculaton of PL for the bus systems s determned by solvng the load flow problem, whch has equalty constrants on real and reactve power at each bus as follows: 3) Real and Reactve Power constrants. B j j j j j j = PG PD = V V G cos( δ δ ) + B sn( δ δ ) B = j δ δ j j δ δ j j= QG QD V V G sn( ) B cos( ) (7) where =,2,,B; B s the number of buses; QG s the reactve power generated at the th bus; PD and QD are the th bus load real and reactve power, respectvely; G j and B j are the transfer conductance and susceptance between bus and bus j, respectvely; and δ and δ j are the voltage angles at bus and j, respectvely. The equalty constrants n (6) and (7) are nonlnear equatons that can be solved usng ewton- Raphson method to generate a soluton of the load flow problem. Durng the course of the soluton, the real power output of one generator, called slack generator, s left to cover the real power loss and satsfy the equalty constrant n (5). C. Optmzaton Problem The EED problem s formulated by ncludng the reducton of O x and SO 2 emsson objectves along wth cost mnmzaton. The CEED problem s formulated by ncludng the reducton of O x emsson objectve along wth cost mnmzaton. (6)
) Mult-objectve problem. The objectve of the EED problem s to mnmze the three objectve functons, fuel cost, and O x and SO 2 emssons, smultaneously, whle satsfyng all equalty and nequalty constrants. Ths problem s formulated as a non-lnear constraned mult-objectve optmzaton problem as follows: [,, ] Mn f = F F F 3 subject to g( PG ) = 0 (8) hpg ( ) 0 Whereas the CEED problem s to mnmze the two objectve functons, fuel cost and O x emsson, smultaneously, whle satsfyng all equalty and nequalty constrants. Ths problem s formulated as a non-lnear constraned mult-objectve optmzaton problem: [, ] Mn f = F F (9) subject to g( PG ) = 0 hpg ( ) 0 where g and h are the problem equalty and nequalty constrants. 2) Weghted sum method. In ths method, the CEED problem,.e., the cost functon gven n () and O x emsson functon n (2), are weghted accordng to ther relatve mportance and than two weghted functons are added together to produce the objectve functon as Mn f = ωf + ( ω) F (0) where F s the fuel cost functon, F 2 s the emsson functon and ω s the weghtng coeffcent n the range 0 and. When ω = 0, only the emsson objectve s consdered and when ω =, only the economc objectve s accounted. By varyng the values of ω, the tradeoff between the fuel cost and emsson cost can be determned over the range of values of ω. Actually n ths method the b-objectve CEED problem s converted to a sngle objectve usng the lnear combnaton of all objectves. It s dffcult to solve the EED problem usng weghted sum method because three weghts have to assume n such a way that the sum of all the weghts s equal to one. Hence the results of MOEP to CEED problem s possble to compare wth weghted sum method for the test systems. III. PRICIPLE OF MULTI-OBJECTIVE OPTIMIZATIO Many real world problems nvolve smultaneous optmzaton of multple objectves that often are competng. In a Mult-objectve Optmzaton Problem (MOP), there may not exst, one soluton that s best wth respect to all objectves. Usually the am s to determne the trade-off surface, whch s a set of non-domnated soluton ponts, known as Pareto optmal solutons. Every ndvdual n ths set s an acceptable soluton. Mathematcally a MOP can be formulated as follows: Mnmze f ( x) =, 2,..., () obj Subject to g j ( x) = 0 j =, 2,..., M h ( x) 0 k =, 2,..., K k where f s the th objectve functons, x s a decson vector that represents a soluton, obj s the number of objectves. For a MOP problem, any two solutons x and x can have one of two possbltes: one domnates the other or none domnates the other. In a mnmzaton problem, wthout loss of generalty, a soluton covers x, or domnates x 2, f the followng two condtons are satsfed: { obj} { obj} j j.,2,... f ( x ) f ( x ) (2) 2. j,2,... f ( x ) < f ( x ) If any of the above condtons s volated, the soluton x does not domnate the soluton x 2. If x domnates the 2 soluton x, x s called the non-domnated soluton. The solutons that are non-domnated wthn the entre search space are denoted as Pareto optmal soluton and consttute the Pareto optmal set. Ths set s also known as Pareto optmal front. The concept of Pareto optmal front for a MOP s as shown n Fg.. Fg.. Pareto optmal front for a MOP. Recently, the studes on evolutonary algorthms have shown that these algorthms can be effcently used to elmnate most of the dffcultes of classcal methods, such as multple runs n order to get the Pareto optmal front [9]. The goal of a mult-objectve optmzaton algorthm s not only to gude the search toward the Pareto optmal front, but also to mantan populaton dversty n the set of nondomnated solutons. Varous researchers have proposed the mult-objectve evolutonary algorthms such as PGA, SGA-II, and SPEA [2]. IV. MULTI-OBJECTIVE EVOLUTIOARY PROGRAMMIG (MOEP) The proposed MOEP method s effcently used to fnd Pareto optmal front n a sngle run for the CEED & EED problems. In ths algorthm, non-domnated sortng s ncorporated wth EP for selecton purpose.
MOEP algorthm s descrbed n the followng steps: Step ): Randomly generate P number of ntal tral solutons as parent soluton and ntate the teraton, =. Step 2): Create P number of offsprng solutons from the parent solutons. Step 3): Create 2 P number of solutons n a present populaton, by combnng the parent and offsprng solutons. Step 4): Identfy the non-domnated solutons and assgn a rank to each soluton by countng the number of solutons that domnates each soluton n the current populaton. Step 5): Sort the total 2 P soluton n the ascendng order wth respect to the rank assgned to each solutons n Step 4. Step 6): Select the frst P solutons as parent soluton to the next hgher teraton. Step 7): Check, f the stoppng crteron (Maxmum o. of Iteratons) s met, then the present parent solutons are Pareto optmal front solutons. Otherwse, the procedure s repeated from Step 2. V. IMPLEMETATIO OF MOEP TO CEED AD EED PROBLEMS Ths proposed MOEP approach s a probablstc, global search technque. Ths starts wth a populaton of randomly generated canddate solutons and evolves towards better set of solutons (Pareto optmal front solutons) over number of generaton or teraton. The man stages of ths technque nclude ntalzaton, mutaton, evaluaton, non-domnated soluton rankng, and selecton. Its mplementaton for CEED and EED problems s as gven below. Step (Intalzaton): The ntal populaton comprses combnatons of only the canddate of dspatch solutons whch satsfy all the constrants. It conssts of PS,. =, 2,, I, tral parent ndvduals. The elements of a parent are the power outputs of generatng unts randomly chosen by a random number rangng over [PG mn, PG max ]. In order to satsfy the power balance constrant, a generator s arbtrarly selected as a dependent generator d. Its output s gven by PGd = PD + PL PG (3) =, d Step 2 (Mutaton): It s performed on each vector element by addng a normally dstrbuted random number wth mean zero and standard devaton σ denoted as (0,σ 2 ). Ths results n PS+ k= [ PG', PG2',..., PG '], for =, 2,..., I (4) PG ' = PG + (0,σ 2 ) for =, 2,..., (5) where s the number of generators. In ths the standard devaton σ s gven by the expresson. f σ = β ( PG PG ) (6) max mn fmax where β s a scalng factor whch has to be tuned durng the process of each search for the optmum around the ntal ponts, f s the ftness value of the th ndvdual, and f max s the maxmum ftness among the I parents. Mutaton results n creaton of I offsprng ndvduals. The parent ndvduals are canddate dspatch solutons, whch satsfy all constrants. However after mutaton, the elements of offsprng PG ' may volate constrant n (3). Ths volaton s corrected as follows: PG mn, f PG ' < PG mn PG ' = (7) PG max, f PG ' > PG max Step 3 (Evaluaton): For the EED problem, the combnaton of parent and offsprng solutons are evaluated by three ftness functons F, F 2 and F 3 separately and stored n F, F 2, and F 3, for =,2,, 2*I. In the case of CEED problem, the combnaton of parent and offsprng solutons are evaluated by two ftness functons F and F 2 separately and stored n F and F 2 for =, 2,, 2*I. Step 4 (on-domnated Rankng): For the EED problem, the rankng procedures conssts of fndng the non-domnated soluton n the current populaton 2*I usng ther ftness F, F 2, and F 3,. Assgn a rank to each soluton by countng the number of solutons that domnates each soluton. In the case of CEED problem, ths rankng procedures conssts of fndng the non-domnated soluton n the current populaton 2*I, usng ther ftnesses F and F 2. Assgn a rank to each soluton by countng the number of solutons that domnates each soluton. Step 5 (Selecton): Dependng on the EED or CEED problems then sort the 2*I solutons n the ascendng order wth respect to the rank assgned to each solutons. ow the frst I solutons are selected as parents wthout ther rank values for the next generaton. Steps 2 to 5 are repeated untl ther stoppng crteron (Maxmum number of teratons) s met, and then the present parent solutons are the optmal Pareto optmal front solutons. VI. SIMULATIO RESULTS AD DISCUSSIO To test the effectveness of the proposed MOEP algorthm to the CEED and EED problems, the solutons were obtaned for a three-unt and a sx-unt system wth quadratc fuel cost functons and emsson functons. Whereas the fuel cost, O x and SO 2 emsson coeffcents for the test systems, such as sx bus and IEEE 30 bus systems, are of cubc functons by neglectng the quadratc term. To determne the effectveness of MOEP method, the test results of the MOEP method s compared wth SGA-II [7] for three-unt system and wth weghted sum method for sx-unt system. MOEP algorthm based EED problem for the three objectves such as fuel cost, and O x and SO 2 emsson s also solved for the three-unt system. MOEP based CEED and EED problem s solved for sxbus system. Practcal standard test system such as IEEE 30 bus system s used for the effectveness of the MOEP based
EED algorthm. The losses are calculated by performng load flow usng ewton-raphson method for the sx bus and IEEE 30 bus systems. The scalng factor s consdered as 0. n the EP algorthm for the three test systems. The smulaton studes were carred out n Matlab envronment. A Three-Unt System The cost coeffcents, generaton lmts, and smplfed loss expresson for the three-unt system are taken from [7]. The proposed MOEP method s appled to ths system to fnd the Pareto optmal solutons. In the three-unt system the proposed MOEP s compared wth SGA-II. In ths smulaton the populaton sze was chosen as 500, number of teratons and scalng factor (β) are 00 and 0.0, respectvely. The obtaned global Pareto optmal front usng MOEP method n a sngle run s shown n Fg. 2. The best cost and O x emsson solutons obtaned n a sngle run are also compared wth those obtaned usng SGA-II from [7] are shown n Table I. As compared wth SGA-II the proposed MOEP gves good results. The proposed MOEP to EED problem by consderng the three objectves such as fuel cost, O x, and SO 2 for the same system s carred out. The best soluton to EED problem wth SGA-II comparson s as gven n Table II. The obtaned Pareto- optmal front s as shown n the Fg. 3 and t s smlar to the Pareto front as gven n [7]. Ox Emsson (ton/hr) 0. 0.099 0.098 0.097 0.096 0.095 9.05 9 SO2 Emsson (ton/hr) 8.95 8340 8350 8360 8370 Fuel cost ($/hr) 8380 8390 Fg.3. Pareto-optmal solutons of MOEP to EED problem of three-unt system. B Sx-Unt System The cost coeffcents, generaton lmts, and transmsson loss co-effcents for ths system are taken from [8]. Varous lnearly dstrbuted weghts (ω) from 0 to n the nterval of 0.025 are used n the weghted sum method. The trade-off optmal solutons are obtaned by runnng the weghted sum method program n 8 separate runs for sx-unt system s shown n Fg. 4. Ths weghted sum method gves the tradeoff solutons for the two objectves. Fg.2.Pareto optmal soluton of MOEP to CEED problem of three-unt system TABLE I. BEST SOLTUIO TO CEED PROBLEM FOR THREE UIT SYSTEM. Best fuel cost ($/hr) Best O x emsson (ton/hr) MOEP SGA-II MOEP SGA-II PG 437.6 437.45 502.56 505.00 PG 2 298.56 299.455.24 253.9 PG 3 29.6 28.94 07.02 06.59 Losses 5.78 5.80 4.82 4.78 Fuel Cost 8344.59 8344.62 8364.04 8363.7 O x Emsson 0.09874 0.09846 0.09593 0.09593 TABLE II. BEST SOLTUIO TO EED PROBLEM FOR THREE UIT SYSTEM. Best fuel cost ($/hr) Best SO 2 emsson (ton/hr) Best O X emsson (ton/hr) MOEP SGA-II MOEP SGA- SGA- MOEP II II PG 435.25 43.680 54.32 538.527 506.84 508.367 PG 2 299.85 302.925 223.09 227.87 25.54 250.444 PG 3 30.73 3.34 00.06 98.85 06.38 05.934 Loss 5.83 5.99 4.47 4.528 4.76 4.745 Fuel Cost 8344.6 8344.65 839.76 8385.7 8364.75 8364.99 SO 2 9.0226 9.0254 8.96620 8.96670 8.97389 8.97374 O x 0.0987 0.09892 0.09653 0.09632 0.09592 0.09592 Fg. 4. Pareto optmal front of lnear combnaton n 8 Separate runs for the sxunt system. The proposed MOEP method s appled to the CEED problem for sx-unt system to fnd the Pareto optmal solutons. The obtaned global Pareto optmal front usng MOEP method n a sngle run s compared wth weghted sum method and t s as shown n Fg. 5. It s exactly matchng wth the 8 optmal soluton ponts those obtaned usng weghted sum method. Furthermore the other ntermedate solutons are also avalable whch s qute useful.
To verfy the effectveness and the convergence capablty of the proposed MOEP method, the solutons status at ntermedate teraton level such as at the end of 20 th teratons s shown n Fg. 6. Ths gves the better understandng of the proposed MOEP method. The parameters used n smulatons of the three-unt and sx-unt systems are gven n Table III. The drawbacks of the weghted sum method are the dffculty of selectng the weghts properly and huge computaton tme needed for several runs. Smlarly computaton tme for SGA-II s more compared wth MOEP algorthm. Table IV shows the smulaton results of the MOEP method to CEED problem along wth SGA II comparson. It proves the ablty of the MOEP method to obtan the Pareto optmal soluton for CEED problem n a sngle run wth the reduced computaton tme. TABLE III. PARAMAETERS USED I WEIGHTED SUM, MOEP & SGA-II METHODS ALOG WITH COMPUTATIO TIME FOR THREE AD SIX UIT SYSTEMS. Method Weghted MOEP SGA-II sum Populaton Sze 20 500, 500 500,500 o of Iteraton 00 00,300 000,2000 Scalng factor (β) 0.0 0.0,0.055 - Computaton Tme n Sec *66.0 5.6, 2.92 36, 83 *Tme requred for 8 runs, umbers n bold font represents the sx- unt system TABLE IV. BEST SOLTUIO TO CEED PROBLEM FOR SIX- UIT SYSTEM. Best fuel cost ($/hr) Best emsson (Kg/hr) MOEP SGA-II MOEP SGA-II PG 75.90 79.48 06.2 04.8 PG 2 48.98 5.23 75.68 75.40 PG 3 45.23 49.7 93.00 9.05 PG 4 03.55 02.48 09.40 09.96 PG 5 265.08 258.33 83.4 84.8 PG 6 92.39 88.76 70.09 70.93 Losses 3.3 29.99 37.52 36.96 Fuel Cost 3822.6 38223.3 39436.89 39368.39 Emsson (kg/hr) 535.9 523.29 462.73 462.78 C Sx-bus System The cost, and emssons (O x and SO 2 ) coeffcents of the three generatng unts wth the generaton lmts n the sx-bus system are gven n Tables V and VI. The bus data and lne data of the system are taken from [4,22]. The system demand s 20 MW. Slack Generator s consdered as the frst generatng unt n the st bus. The number of lnes n the system s. The parameters consdered such as populaton sze/the number of teratons for MOEP-based EED and CEED s 500/300 and 500/500, respectvely. The populaton sze s same for both the problems. The calculated loss of the system s found to be 3.4%. The MOEP algorthm s executed for the CEED problem and EED problem and the obtaned best soluton of the ndvdual objectves are gven n the Table VII. The Pareto-optmal front for the MOEP based CEED problem s as shown n Fg. 7. It s possble to vsualze the best values and the extreme ponts of fuel cost and O x emsson n the graph. TABLE V. GE. LIMITS AD THE FUEL COST COEFFICIETS - SIX BUS SYSTEM. Fg. 5. Pareto optmal solutons for sx-unt system. PG mn, PG max (MW) QG mn, QG max (MW) 50, 200-4.758 55E- 06 37.5, 50-00, 00 4.40 90E-06 45, 80-00, 00.394 9E-05 Fuel Cost Coeff a c d 8.693 54E+ 00 9.526 9E+ 00 7.684 84E+00 7.592 E+ 0 5.324 98E+ 0.73 24E+ 02 TABLE VI. O X AD SO 2 EMISSIO COEFFICIETS - SIX BUS SYSTEM. O x emsson Coeff. SO 2 emsson Coeff. Fg. 6. MOEP soluton status at the end of the 20 th teraton. na nc nd sa sc sd.478 97E-08 4.3 09E-04 5.073 8E-02 2.855 4E-09 5.26 2E-03 4.555 26E-02.488 0E-09.302 98E-03.057 4E-0 2.205 46E-09 4.763 46E-03 2.662 50E-02.445 8E-08 4.066 70E-04 2.04 66E-02 8.369 46E-09 4.60 90E-03.038 74E-0
TABLE VII. BEST VALUES OF IDIVIDUAL OBJECTIVES - SIX BUS SYSTEM. Best Fuel Cost Best O x emsson Best SO 2 emsson PG (MW) 54.36 88.048 5.69 PG 2 (MW) 73.80 37.5 75 PG 3 (MW) 88.56 9.642 90 PL (MW) 6.72 7.9 6.669 Fuel cost 270.70 283.52 270.80 ($/hr) ox Emsson 345.48 334.57 347.32 (Kg/hr) So 2 emsson (Kg/hr) 226.6 253.96 224.94 consdered as the frst generatng unt n the IEEE 30 bus system and the calculated losses are found to be 4%. The parameters consdered such as populaton sze/the number of teratons for MOEP-based EED of the test system s 600/ 600 respectvely. The best soluton of the ndvdual objectves such as fuel cost, O x and SO 2 of the IEEE 30 bus system are gven n Table VIII. The dfference of the best values of fuel cost and O x emsson s hgher compared to SO 2 emsson. The Pareto-optmal front for three objectves s as shown n the Fg. 9. It s possble to vsualze that the large varatons of ndvdual objectves such as fuel cost, O x are n the lnes n ther regons, whereas only curve s seen n the SO 2 emsson n ther regon because of the less varaton n ther values. The executon tme of MOEP-based EED of IEEE 30 bus system s obtaned as 56.578 seconds. 284 282 TABLE VIII. BEST VALUES OF IDIVIDUAL OBJECTIVE -IEEE 30 BUS SYSTEM FUEL COST ($/hr) 280 278 276 274 272 270 334 336 338 340 342 344 346 Ox EMISSIO (Kg/hr) Fg. 7. Pareto-optmal front of MOEP based CEED problem- sx bus system. The Pareto-optmal front for the MOEP based EED problem s as shown n Fg. 8. The executon tme for MOEP based CEED and EED problem of sx-bus system s obtaned as 7.23 and 8.89 seconds, respectvely. Best Fuel Cost Best O x Emsson Best SO 2 Emsson PG (MW) 72.86 204.76 87.84 PG 2 (MW) 39.24 28.78 34.3 PG 3 (MW) 26.22 20.2 23.35 PG 4 (MW) 8.02 3.66 5.96 PG 5 (MW) 6.4 2.93 4.77 PG 6 (MW) 20.98 6.0 8.68 PL (MW) 0.33 2.95.5 Fuel cost 3068.6 3083.43 3.07.67 ($/hr) O x 668.57 640.3 647.85 Emsson (Kg/hr) SO 2 emsson (Kg/hr) 82.88 83.0 8.36 ox emsson (Kg/hr) 350 345 340 335 ox emsson (Kg/hr) 850 800 750 700 650 330 250 240 SO2 emsson (Kg/hr) 230 220 270 272 278 276 274 Fuel cost ($/hr) 280 600 840 830 SO2 emsson (Kg/hr) 820 80 3060 3070 3080 Fuel cost ($/hr) 3090 300 Fg. 8. Pareto-optmal front of MOEP-based EED problem - sx bus system. D IEEE 30- bus System The cost, emssons (O x and SO 2 ) coeffcents of the sx generatng unts wth the generaton lmts n the IEEE 30-bus system and the system data are taken from [4,9,22]. The system demand for the IEEE 30 bus system s 283.4 MW. The numbers of lnes n the test system s 4. Slack generator s Fg. 9. Pareto-optmal front for three objectves IEEE 30 bus system. VII. COCLUSIO The proposed Mult-Objectve Evolutonary Programmng algorthm has been developed and appled to CEED and EED problems. The performance of the proposed algorthm s demonstrated and the results were compared wth SGA-II
method for three-unt system and also wth weghted sum method for sx-unt system. The executon tme of MOEP based EED problem of three-unt, sx bus and IEEE 30 bus test systems are 5.6, 8.89 and 56.58 seconds, respectvely. Results showed that the MOEP method s well suted for obtanng very good Pareto optmal solutons n a sngle run for EED problem wth reduced computaton tme. VIII. REFERECES [] F. J. Trefny and K. Y. Lee, Economc Fuel Dspatch, IEEE Transactons on Power Apparatus and Systems, Vol. PAS-00, pp.3468-3477, July/August 98. [2] G. B. Sheble and K. Brttg, Refned Genetc Algorthm-Economc Dspatch Example, IEEE Trans. Power Syst., vol. 0, pp. 7-24, Feb. 995. [3] J. H. Park, Y. S. Km, I. K. Eom, and K. Y. Lee, Economc Fuel Dspatch for Pecewse Quadratc Functon usng Hopfeld eural etwork, IEEE Trans. Power Syst., vol. 8, o. 3, pp. 030-038, Aug 993. [4] J. W. Lamount and E. V. Obess, Emsson dspatch models and algorthms for the 990 s, IEEE Trans. Power Syst., vol 0, no. 2, pp. 94-947, May 995. 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Wood and B.F. Woolenburg, Power Generaton, Operaton and Control. ew York: Wley, 996. IX. BIOGRAPHIES P. Venkatesh receved hs Degree n Electrcal and Electroncs Engneerng and Masters n Power System Engneerng wth Dstncton and Ph.D n 99, 994 and 2003, respectvely, from Madura Kamaraj Unversty, Inda. Hs area of nterest s applcaton of evolutonary computaton technques to power system problems and power system restructurng. He has receved the Boyscast Fellowshp award n the year 2006 from Department of Scence and Technology, Inda for carryng out Post Doctoral Research Work at the Pennsylvana State Unversty, U.S.A. Currently he s Assstant Professor n the Department of Electrcal and Electroncs Engneerng, Thagarajar college of Engneerng, Madura, Inda. Kwang Y. Lee receved hs B.S. degree n Electrcal Engneerng from Seoul atonal Unversty, Korea, n 964, M.S. degree n Electrcal Engneerng from orth Dakota State Unversty, Fargo, n 968, and Ph.D degree n System Scence from Mchgan State Unversty, East Lansng, n 97. He has been wth Mchgan State, Oregon State, Unv. of Houston, and the Pennsylvana State Unversty and Baylor Unversty, where he s currently a Professor and Char of the Electrcal and Computer Engneerng Department and Drector of Power Systems Control Laboratory. Hs nterests nclude power system control, operaton, plannng, and ntellgent system applcatons to power systems. Dr.Lee s a Fellow of IEEE, Assocate Edtor of IEEE Transactons on eural etworks, and Edtor of IEEE Transactons on Energy Converson. He s also a regstered Professonal Engneer.