ECONOMIC AND MINIMUM EMISSION DISPATCH

Transcription

1 ECOOMIC AD MIIMUM EMISSIO DISACH Y. Demrel A. Demroren İstanbul echncal Unversty, Electrcal Engneerng Department Maslak / İstanbul / urkey ABSRAC In ths paper, Hopfeld eural etwork (H and Lagrange Multpler (LM solutons to economc dspatch (ED, O x emsson dspatch (EmD, and economc-emsson dspatch (EED of a sample system consstng of sx thermal generators are presented. ransmsson losses are ncluded. he results of H are compared wth the results of LM. Keywords : Economc - Emsson Dspatch, Hopfeld eural etwork, Lagrange Multpler,. OMECLAURE F :otal producton cost F :roducton cost of the th plant E :otal O x emsson E :O x emsson of the th plant :Real power output of the th generator :otal number of unts on system a,b,c :Cost coeffcents of the th generator d,e,f :O x emsson coeffcents of the th generator Ф :otal ectve functon w,w :Weght factors h :Rate coeffcent,mn :Mnmum generaton lmt of the th generator,max :Maxmum generaton lmt of the th generator D :otal demand L :otal losses B j :ransmsson loss coeffcents L(p, λ :Lagrange functon λ :Lagrange multpler g(p :Equalty ant E :Hopfeld network s energy functon E :Optmzaton ectve functon x :n dmenson varable vector of ectve functon :n n symmetrcal matrx of ectve functon coeffcents :n dmenson vector of ectve functon A :Equalty ant matrx b :m dmenson ualty ant vector m :Equalty ants A n :Inualty ant matrx b n :m n dmenson ınualty ant vector m n :Inualty ants :n+m n dmenson feasble subspace projecton matrx s :n+m n dmenson feasble subspace offset vector A :Extended ant matrx b :(m +m n dmenson extended ant vector I :Identty matrx X :Extended varable vector ncluded slack varables :enalty factor c y p :Slack varable of the pth nualty ant f(x :he actvaton functon of the varables ρ :Momentum term s coeffcent :n+m n dmenson matrx of Hopfeld dfferental uaton I :n+m n dmenson vector of Hopfeld dfferental uaton η :Coeffcent whch belongs to varables η :Coeffcent whch belongs to slack varables :otal power except for transmsson loss net I. IRODUCIO he economc dspatch (ED problem s to determne the optmal combnaton of power outputs for all generatng unts whch mnmzes the total fuel cost whle satsfyng load demand and operatonal ants. A number of studes have been presented to solve ED problems such as ark et al.[], and Yalcnoz and Short[]. Under the strct governmental regulatons on envronmental protecton, the conventonal operaton at mnmum fuel cost can no longer be the only bass for dspatchng electrc power. he contrbutons of the electrc energy ndustry to envronmental polluton rase questons concernng envronmental protecton and methods of reducng polluton from power plants ether by desgn or by operatonal strateges. Especally, emssons contrbuton of fossl-fred electrc power plants whch use coal, ol, gas or combnatons as the prmary energy resource cannot be neglected. hese emssons are CO, CO, SO, O x, partculates, and thermal emsson. Emssons may be reduced through these methods : swtchng to fuels wth low emsson potental, nstallng post-combuston cleanng system, and dspatchng of generaton to each generator unt wth the ectve of mnmum emsson dspatch [3-4]. Selectng the thrd method s aduate because t s easy to mplement and rures mnmal addtonal costs, so, n ths study t s used. Several researchers have consdered emssons ether n the ectve functon or treated emssons as addtonal ants. Kulkarn et al. [3], Song et al. [4], Dhllon et al. [5], and Kng et al. [6] presented EED dspatch.

2 In ths paper, H and LM are used to solve economcemsson dspatch problem. As an llustraton, only O x emsson reducton s consdered. he ualty ant of power balance and nualty generator capacty ants are taken nto consderaton. Also, transmsson loss s consdered. hese methods have been demonstrated through a sample system consstng of sx thermal generators. In smulaton secton, the results of H are compared wth LM as the classcal method. II. FORMULAIO OF ECOOMIC- EMISSIO DISACH In ths paper a system consstng of thermal generatng unts connected to a transmsson network servng a receved electrcal load D [MW] wll be studed. he total cost rate of ths system s, the sum of the cost rate of the ndvdual unts. he fuel cost curve s assumed to be approxmated by a quadratc functon of [MW] [-7] : F ( a + b + c...[rs/h] ( = = In Eq.(, a [Rs/MW -h], b [Rs/MWh], and c [Rs/h] are cost coeffcents. For emsson dspatch problem, the amount of O x emsson s expressed as a quadratc functon lke the cost functon [3-6] : E ( d + e + f...[kg/h] ( = = In Eq.(, d [kg/mw -h], e [kg/mwh], and f [kg/h] are O x emsson coeffcents. Emsson functon as an ectve s added to Eq.( as follows to obtan the ectve functon of the economc-emsson dspatch problem [3-6] : Φ Φ subject to = w F + hw. E...[Rs/h] (3 = = = = = w ( a + b+ c + hw. ( d + e+ f (4,mn,max (=... (5 = D + L (6 = = L = j= B [MW] (7 j j ow, the problem s to fnd the rate coeffcent, h. A practcal way of determnng h s dscussed by Kulkarn et al. [3]. It s necessary to obtan the rate coeffcents of each generator at ts maxmum output: F ( E (,max,max,max,max = h (=... [Rs/kg] (8 h (=,..., s then arranged n ascendng order; the maxmum capacty of each unt,,max, one at a tme, startng from the smallest h unt, untl D.At ths stage, h assocated wth the last unt n the process n the rate coeffcent h [Rs/kg] for the gven load. III. LAGRAGE MULILIER It s well known as the Lagrange functon and s shown n Eq.(9 [7]. L p, λ = Φ( p + λ. g( p (=,..., (9 ( g( p = D + L = ( = IV. HOFIELD EURAL EWORK eural networks are hghly smplfed models of the human nervous systems, exhbtng abltes such as learnng, generalzaton, and abstracton. It s well known that the H converges very slowly and normally takes several thousand teratons. In ths study momentum term s used to speed up convergence for the H. Also, Improved Euler Method and (RK-4 Method are used to solve dfferental uatons n H. he H method uses a mappng technque, whch has been descrbed n reference [], to solve quadratc programmng problems. For the mappng of quadratc programmng problems, nualty ants have been combned wth a slack varable technque. A. Mappng echnque Mappng technque refers to plannng technque of energy functon whch s used to adapt economc-envronmental dspatch problem to H form. he dfferental uatons of Hopfeld s contnuous model [,6,8] are defned as follows : dx dt η = Ο X + F( x + I η ( hs model s based on contnuous varables and responses. Hopfeld s energy functon [,6,8,9] s defned as follows : E( x = X X X I ( he energy functon s a quadratc functon that s assocated wth the cost functon and the emsson functon to mnmze the optmzaton problem. he quadratc problem can be wrtten as Mn E ( x = X X X (3

3 Under the ualty and the nualty ants : A X = b (4 n n A X b or and the sde ants may be gven as X n, mn,max n A X b (5 X X (6 X=[x,...,x n ] s the vector of varables and X,mn are X,max are lower and upper bounds, respectvely. he feasble soluton can be descrbed as : X = X + s (7 = I A ( A A A (8 s = A ( A A b (9 Energy functon can be wrtten accordng to penalty factor (c as E = E + c X ( X + s ( he network s weghts ( and nput bases (I are set as follows to satsfy the energy functon Eq.( : I [ I ] [ I ] s = + c ( = + c ( In these uatons, because of converted nualty ants to ualty ants by ntroducng slack varables, varables X are set as X =[x y ]. y s the vector of slack varables [y, y,..., y n ]. F(x m functon s explaned n next secton. B. Mappng of Economc-Emsson Dspatch Frst, we have to set weghts and nput bases for the EED problems. We use n neurons for generators and m n neurons for nualty ants. he ectve functon of the aned EED problem gven n Eq.(4 s consdered as the energy functon of the H. herefore weghts and nput bases of the ectve functon are set as follows : j = ( w a = = ( w b + h. w d + h. w e (3 he ants of the EED problems can be handled by addng correspondng terms to the energy functon. We can convert nualty ants to ualty ants, then A and b can be wrtten as n n [ A ; A ] and b = [ b b ] A = ; (4 A = ( n generators n m slack varables b = + (6 D L A n and b n are defned from nualty ant uatons gven n Eq.(5. Generaton lmts are taken as nualty ants. Inualty ants can be converted to ualty ants by usng slack varables. For example, the upper lmt of the th generator may be converted to :, max,max y p = y p (y p s a slack varable of the p-th nualty ant and we can defne A p n and b p n as : [ ] n =, max A p - th generator (n + p- th column (7 and b p n =. Smlarly the lower lmts of generators can be fxed as n the above example. hen A s created as A = Ο,mn,max Ο 6,mn 6,max 3 8 (8 F(x functon n Eq.( s chosen as a symmetrc ramp functon whch can be shown n Fg.. he actvaton functon of each neuron s modfed to lmt the output value between lower and upper bounds. It s descrbed as,mn,,mn > = f ( p,,mn,max (9,max, >,max f(p,mn,max Fgure. Input-output functon of the varable. After fndng A and b, and s can be determned usng Eq.(8 and (9. hen we can set weghts and nput bases usng Eq.( and (. Fnally, H s created for solvng the aned EED problem. 3

4 V. ES SYSEM he test system [3,5], whch has sx thermal generators, s chosen. he fuel cost and O x emsson uatons are gven n able and able [5]. able. Fuel cost [Rs/h] uatons. F = F = F 3= F 4= F 5= F 6= able. O x emsson [kg/h] uatons. E = E = E 3= E 4= E 5= E 6= able 3. Operatng lmts [MW]. Generator o. Lower Lmt [MW] Upper Lmt [MW] ransmsson loss coeffcents are taken from Ref.5. he operatng lmts of the generators are gven n able 3 [5]. In ths paper, Improved Euler Method and RK-4 Method are used to solve dfferental uatons n H. he tme step (or step length t s gven values from. to.3. Also, to speed up convergence to optmum pont n H, momentum term s used as follows : X ρ (3 n+ = X n + X n +. X n In Eq.(3, momentum term s coeffcent s obtaned from trals [6]. In ths paper, ρ s selected to.95 by tralerror. VI. SIMULAIO RESULS In ths secton, smulaton results of pure ED, pure EmD, and EED for the two condtons wth transmsson loss and wthout transmsson loss are demonstrated. he results obtaned from H Method are compared wth the results of LM Method. he programs for these two optmzaton technques were wrtten n Matlab. hese programs are executed on a entum III 733 MHz C wth 64 MB RAM. In ths paper, these optmzaton technques are appled to a test system whch has sx generatng unts, for 5, 6, and 7 MW loads and for four study modes as follows.. w =, w = ure ED. w =.8, w =. EED 3. w =.5, w =.5 EED 4. w =, w = ure EmD wth transmsson loss and wthout transmsson loss. For the condton neglectng transmsson loss,. Improved Euler Method. Method 3. momentum term and Open Euler M. 4. momentum term and RK-4 Method 5. Lagrange Multpler Wth the above technques, smulaton s mplemented. Wth a system load of 6 MW, smulaton s performed for the whole study modes and the results are demonstrated n able 4 and able 5. able 4. Results of the condton neglectng transmsson loss for D =6 MW. MEHODS Improved Euler ure SUDY MODES Results of able 4 and able 5 ndcate how a reducton n O x emsson could be acheved by a change n generaton dspatch schedules. hs s obtaned at the expense of fuel cost. he results of these methods do not volate the ndvdual generator capacty lmts, and the transmsson losses are also nearly the same as LM method. From the results of pure ED and pure EmD dspatches, t s observed that there s an ncrease n fuel cost of 3Rs/h and a reducton n O x emsson of.99 kg/h for H used momentum term and RK-4 Method together neglectng L. hus, for a reducton of kg of O x /h, there s an ncrease n cost of 3.3 Rs. For the case ncludng L, ths value s 3.64 Rs. Accordng to LM Method, ths value s 6.48 Rs for the condton neglectng L, and ths value s.37 Rs for the condton ncludng L. ED EED EED ure EmD [MW] F [Rs/h] Method E [kg/h] [MW] F [Rs/h] Method E [kg/h] Momentum term [MW] F [Rs/h] Open Euler M. E [kg/h] Momentum term [MW] F [Rs/h] E [kg/h] Lagrange Multpler [MW] F [Rs/h] E [kg/h]

5 able 5. Results of the condton wth transmsson loss for D =6 MW. MEHODS Improved Euler Method Method Momentum erm+ Open Euler ure ED SUDY MODES EED EED ure EmD net [MW] F [Rs/h] E [kg/h] L [MW] net [MW] F [Rs/h] E [kg/h] L [MW] net [MW] F [Rs/h] E [kg/h] Method L [MW] Momentum erm+ net [MW] F [Rs/h] E [kg/h] Method. L [MW] Lagrange Multpler net [MW] F [Rs/h] E [kg/h] L [MW] o obtan the performance of H method, the error s calculated as the percentage dfference between the values of H method and LM method. he error s formulated as H' s cost - LM's cost Err = % (3 LM's cost he maxmum error s 6.53% for the H method. he mnmum error s (-6.% for the H method. he negatve sgn refers to the advantage of the H method. In addton, whle the error of F decreases from pure ED to pure EmD, the error of E ncreases. he error values are almost the same for all dfferental uaton technques. Also, the error s calculated as the percentage dfference between Improved Euler Method and H used other technques to fnd out whch dfferental uaton soluton technque s the best. Although, n general the results of the momentum term and RK-4 method together are very good, the whole error values are very lttle and can be neglected. he teratons of the H methods are (4-78 for 5 MW load, (56-73 for 6 MW load, and ( for 7 MW load. he momentum term and RK-4 method together has the mnmum teratons. he LM method has no teratons except some exceptons. he executon (CU tmes are (-.6s for 5 MW load, (.5-.7s for 6 MW load, and ( s for 7 MW load. Although the H method used momentum term and RK-4 method together has the maxmum executon tmes, t can be neglected. he LM method takes almost no tmes. he mnmum memores are 34 Bytes for the condton neglectng L and 85 Bytes for the condton ncludng L for LM method. he maxmum memores are 74 Bytes for the condton neglectng L and 879 Bytes for the condton ncludng L for momentum term and RK-4 method together. VII. COCLUSIOS H and LM solutons to the economc-emsson dspatch problem have been presented. Although t s well known that the H converges very slowly and t takes several thousand teratons, ths paper has presented an analyss of the performance of the H methods whch have acheved effcent and accurate solutons for test system sx generatng unts for 5, 6, and 7 MW loads. A comparson of H method wth LM method has been presented. he errors of H method are neglgble even H has an advantage of O x emsson. he H method has acheved very fast solutons accordng to a lot of studes n lterature. he paper demonstrated that the H method can be appled easly to the economc-emsson dspatch problems. VIII. REFERECES [] ARK, J.H., KIM, Y.S., and LEE, K.Y., 993. Economc Load Dspatch for ecewse Quadratc Cost Functon Usng Hopfeld eural etwork, IEEE ransactons on ower Systems, 8/3,pp.3-38,August. [] YALCIOZ,., CORY, B.J., and SHOR, M.J.,. Hopfeld eural etwork Approaches to Economc Dspatch roblems, Electrcal ower and Energy Systems, 3,pp [3] KULKARI,.S., KOHARI,A.G., and KOHARI, D..,. Combned Economc and Emsson Dspatch Usng Improved Backpropagaton eural etwork, Electrc Machnes and ower Systems, 8, pp [4] SOG, H., WAG, S., WAG, Y., and JOHS,.,997. Envronmental / Economc Dspatch Usng Fuzzy Logc Controlled Genetc Algorthms, IEEE roceedngs, Generaton, ransmsson and Dstrbuton, 44/4, pp , July. [5] DHILLO, J.S., ARI, S.C., and KOHARI, D.., 993. Stochastc Economc Emsson Load Dspatch, Electrc ower Systems Research, 6, pp [6] KIG,.D., EL-HAWARY, M.E., and EL-HAWARY, F., 995. Optmal Envronmental Dspatchng of Electrc ower Systems va an Improved Hopfeld eural etwork Model, IEEE ransactons on ower Systems, /3, pp , August. [7] WOOD, A.J. and WOLLEBERG, B.F., 984. ower Generaton Operaton & Control, John Wley & Sons. [8] MEHROA, K., MOHA C.K., and RAKA, S., 997. Elements of Artfcal eural etworks, MI ress, Cambrdge. [9] LVEBERGER, D.G., 984. Introducton to Lnear and onlnear rogrammng, Addson-Wesley, nd Edton. 5