Short-term scheduling of thermal units: emission constraints and trade-off curves

Transcription

1 Short-term schedulng o thermal unts: emsson constrants and trade-o curves J. P. S. Catalão 1, *,, S. J. P. S. Marano 1, V. M. F. Mendes 2 and L. A. F. M. Ferrera 3 1 Department o Electromechancal Engneerng, Unversty o Bera Interor, Covlha, Portugal 2 Department o Electrcal Engneerng and Automaton, Insttuto Superor de Engenhara de Lsboa, Lsbon, Portugal 3 Department o Electrcal Engneerng and Computers, Insttuto Superor Técnco, Lsbon, Portugal SUMMARY Ths paper provdes an approach to short-term schedulng o thermal unts, desgned to smultaneously address the economc ssue o the uel cost ncurred on the commtment o the unts and the envronmental consderaton due to emsson allowance tradng. The smultaneous address o the uel cost wth the emsson s modelled by a mult-objectve optmzaton problem, whch s solved by a combnaton o the weghted sum method wth the -constranng method. A numercal example or derent values o a scalng actor s consdered n order to obtan the non-domnated solutons o the trade-o curve between uel cost and emsson. Our approach presents a new parameter, rato o change, and the correspondng gradent angle, to enable the proper selecton o a compromse commtment or the unts. Copyrght 2006 John Wley & Sons, Ltd. KEY WORDS: unt commtment; emsson constrants; mult-objectve optmzaton 1. INTRODUCTION Fossl uels represent a relable and aordable source o energy, necessary to satsy the demand or electrc energy. The wdespread adopton o energy economes based on ossl uels has brought wth t the potental harmul problem o the emsson o gaseous and partculate products o combuston. When the concentraton o emssons reaches a pre-speced threshold, the phenomenon s termed polluton [1]. *Correspondence to: J.P.S. Catalão, Dep. Electromec., Unv. Bera Interor, Covlha, Portugal. E-mal: catalao@ub.pt Copyrght 2006 John Wley & Sons, Ltd.

2 2 Fossl uelled power plants are classed as statonary sources o emsson polluton. The amount o emsson polluton as a result o burnng ossl uels to convert nto electrc energy depends on the uel used, the level o power and the ecency o the technology used n the power plant operaton. For nstance, electrc companes wth old coal-red power plants technology have hgher emsson levels. As a comparson, the emsson level or a pulverzed coal-red power plant s about two tmes and a hal hgher than the emsson level or a natural gas-red power plant n combned cycle conguraton. Although wth the actual technology coal-red power plants are pollutng less, they are n the vcnty o urban or rural zones then concentraton o polluton due to weather condtons or the usual nght temperature nverson eect [2] can orgnate envronmental mpact, even when compared wth old coalred power plants not n the vcnty o urban or rural zones. Envronmental concerns are becomng ncreasngly relevant or companes as regulatons on pollutants become more strngent and customer awareness o envronmental mpacts ncreases. Frstly, t s now recognzed that the greenhouse eect can be slowed down only emssons o carbon doxde and other harmul gases are reduced drastcally. A major step n ths drecton s the Kyoto Protocol [3], whch establshes a 5% drop n carbon doxde emssons compared to emssons n 1990 or the ndustralzed countres. Recently, the European Parlament adopted the world s rst mult-natonal Emssons Tradng Scheme coverng greenhouse gases n the Unon [4]. The sx man greenhouse gases are: carbon doxde, methane, ntrous oxde, hydro luorocarbons, per luorocarbons and sulphur hexaluorde. The Emssons Tradng Scheme, entered nto orce on January 2005, caps emssons rom plants n the ol renng, smeltng, steel, cement, ceramcs, glass and paper sectors and allows tradng o emsson allowances. The envronmental ssues mposed by the Kyoto Protocol mply new emsson constrants regardng producton decson n thermal unts burnng ossl uels. The poston o a power plant n the mert order lst has been based tradtonally on mnmzed total cost. Hence, old coal-red power plants acheved a superor mert order, producng at the lowest cost electrc energy, although wth consderable mpact on the envronment. The letme o these plants was extended, causng a delay n development o ecent and clean technologes. An unprecedented change s bound to occur n the new carbon constraned world and wth the new envronmental regulatons mplemented worldwde the role o the old coal-red power plant s lely to change. In the presence o emsson allowances, old coal-red power plants may move down n the mert order, due to hgher carbon emsson ntensty. They wll run less than t was normal n the old carbon unconstraned world. Hence, natural gas-red power plants n combned cycle conguraton, or even the new promsng technology or coal power plants wth zero emssons, wll go up n the mert order. Gas plants wll need less emsson allowances than coal plants, resultng n a tendency or a sht n the mert order o power plants. Also, the utlty manager has to consder that the emsson rghts can be reely traded, havng the opton to sell the allowances not needed. Market prces to buy more emsson allowances wll add up a cost to the margnal cost o electrc energy producton or ossl-uelled power plants [5]. Instead, hydro plants whch can be regarded as renewable energy sources [6] wll not have to buy or own allowances or ther electrc energy produced and thereore wll ace a compettve advantage.

3 3 The majorty o the studes concernng emsson constrants are on the economc dspatch problem [7 13], decdng only the power contrbuton o each thermal unt, but not decdng on whch unts should be commtted or generaton at each hour. The tradtonal short-term schedulng problem o thermal unts, mnmzng uel cost durng a tme horzon o one day up to one week, does not nclude concerns due to emsson polluton comng rom the operaton o power plants. Ths schedule or a thermal unt comprses both decdng the commtment status, a dscrete value, and the power contrbuton, a contnuous value. Ths problem s also known as unt commtment. The account o envronmental actors n the unt commtment problem [14 16], dd not receve as much attenton as n the economc dspatch problem. However, the recent advent o carbon doxde tradng n the European Unon has renewed nterest n the envronmentally constraned unt commtment problem [17 18]. Emsson allowances are yearly allocated. Hence, a short-term schedulng o thermal unts ruled by an emsson allowance market requres a medum-term schedulng [5]. An estmaton o the daly allowances o each unt s obtaned by means o annual allowances. The schedulng o the thermal unts ruled by an emsson allowance market can be organzed through a herarchcal structure le the one descrbed n [19]. The schedule o thermal unts s dvded nto three herarchcal levels: the year level, the month level and the day level, wth tme perods respectvely o one month, one day and one hour. At each level, an average load dspatch s computed. Year level average load dspatch s computed, establshng a monthly emsson allowance or each unt. Month level average load dspatch s computed, establshng a daly emsson allowance or each unt. These daly allowances wll comply wth the yearly allocated emsson allowances. At the year and month levels, all unts are supposed to be commtted or generaton. At the day level, the commtment o the unts s decded. It s possble to use a week level nstead o a day level or the unt commtment. Snce mnmzng uel cost and mnmzng emsson tend to be conlctng objectves, an approach based on mult-objectve optmzaton s proposed n ths paper to obtan the best compromse soluton rom the non-domnated or Pareto-optmal soluton set. The trade-o curve between uel cost and emsson s presented, graphcally llustratng ths non-domnated soluton set. Ths paper s structured as ollows. In Secton 2, we present a mathematcal ormulaton or schedulng o thermal unts wth emsson constrants, modelled as a dynamc, mxed-nteger non-lnear constraned optmzaton problem. In Secton 3, an approach desgned to smultaneously address the economc ssue o the uel cost ncurred on the commtment o the unts and the envronmental consderaton due to emsson allowance tradng s shown. In Secton 4, we present a case study wth 11 thermal unts and a schedulng tme horzon o 168 hours. Fnally, Secton 5 outlnes the man conclusons. 2. PROBLEM FORMULATION The tradtonal problem o short-term schedulng o thermal unts s dened as the task o establshng the mnmum uel cost or the hourly generaton schedule o the thermal unts durng a tme horzon o one day up to one week, satsyng the demand o electrcal energy and the consdered constrants.

4 4 The problem o short-term schedulng o thermal unts wth emsson constrants can be solved by consderng that there s a medum-term schedulng whch provdes daly or weekly allowances or the unts. Some o the data nvolved n the unt commtment problem are stochastc n nature, but or the short-term tme horzon consdered the correspondng orecasted values are assumed as determnstc data. Thereore, the problem s vewed as a determnstc one [20]. The problem nvolves nteger varables assocated wth dscrete states, contnuous varables and also equalty, nequalty and logcal constrants. The problem s a dynamc, mxed-nteger non-lnear constraned mathematcal programmng problem. The economc consequences o unt commtment are recognzed as very mportant; savngs o a small percent value represent a sgncant reducton n the uel consumpton [21]. The problem s wrtten as a mathematcal programmng problem o the type: mnmze ( x, subject to ( x, F (1) where the objectve uncton s gven by: ( x, C ( x, u, p ) (2) kk I K s the set o hours n the schedulng tme horzon, I s the set o thermal unts n the power system, s the total uel cost ncurred by a thermal unt n hour k, and x, u, C p are respectvely the state, the dscrete decson and the power producton varables assocated wth the thermal unt n hour k. nto: The constrants may be dvded nto global and local constrants. Global constrants may be dvded (a) Hourly generaton constrants. For nstance: the power produced by the thermal unts equals the demand D k n each hour k, gnorng transmsson losses I p ; k K (3) D k (b) Cumulatve constrants. For nstance: the maxmum emsson o a group o unts over the schedulng tme horzon cannot exceed a pre-speced value kk B n H n ( x, u, p ) H req n ; n N (4) where B n s the set o thermal unts on the nth cumulatve constrant, H n s the uncton whch descrbes

5 5 a contrbuton o thermal unt to nth cumulatve constrant, constrant and N s the set o cumulatve constrants. The local constrants may be dvded nto: (a) State equatons or the thermal unts req H n s the upper bound on nth cumulatve x 1 A ( x, u ); u U, I, k K (5), k yeldng the state varable n hour k+1 or the state varable n hour k and or the dscrete decson varable n hour k belongng to the set o easble dscrete decson varables (b) Power producton admssble set U. p P u ); I, k K (6) ( or nstance, the unt s on, the power producton s between the mnmum value and the maxmum value o the power or the unt n hour k; the unt s o, the power producton s null. (c) Intal state x o and nal state x x X 0 0 x X I (7) belongng respectvely to the ntal state set Constrants (3) to (7) dene the set o easble varables: F 0 X and the nal state set X. ( x, : constrants (3),(4) (7) are satsed The total uel cost ncurred by a thermal unt s gven by the sum o the start up cost wth the operaton cost. We consder the start up cost gven as a constant, and the operaton cost mathematcally modelled as a second order Taylor expanson. Hence, the operaton cost s gven by: C op ( u, p 2 ) u ( p p ) (8) where, and are the cost coecents or thermal unt. The objectve uncton may be consdered as the total emsson nstead o the total uel cost. We consder the emsson due to ossl-uelled unts also mathematcally modelled as a second order Taylor expanson, gven by: E em ( u, p 2 ) u ( a b p c p ) (9) where a, b and c are the emsson coecents or thermal unt.

6 6 The objectve uncton or the problem becomes the total emsson, gven by: g ( x, E ( x, u, p ) (10) kk I The smultaneous address o the uel cost wth the emsson s modelled n ths paper by a multobjectve optmzaton problem, gven by: mnmze ( x,, g ( x, subject to ( x, F (11) 3. PROPOSED APPROACH In the mult-objectve problem ormulaton the two objectve unctons consdered, total uel cost and total emsson, tend to be conlctng. Hence, t s mpossble to obtan the mnmum at the same pont when the objectve unctons are ndependently optmzed,.e., or the mnmum total uel cost and or the mnmum total emsson. A gan n one objectve uncton s due to a sacrce n the other objectve uncton. Our approach ams to get the best compromse soluton rom the non-domnated soluton set, consderng the two objectve unctons smultaneously. The two objectve unctons must be traded o n some way. We treated them by a convex combnaton, a weghted sum gven by: h ( x, w ( x, (1 w) g ( x, (12) where w s a weghtng actor varyng between 0 and 1 to generate the non-domnated solutons: w 0 corresponds to the best emsson commtment (BEC), and w 1 corresponds to the best cost commtment (BCC); s a scalng actor, gven or nstance by the carbon market prce, whch s uncertan but assumed constant over the schedulng tme horzon. Nevertheless, each scalng actor consdered s used to dene a scenaro. Hence, the uncertanty regardng the scalng actor s dvded nto a nte number o possbltes. The weghted sum method obtans the set o non-domnated solutons, M, by varyng the weghtng actor. Our approach combnes the weghted sum method, usng a convex combnaton o the objectve unctons, wth the -constranng method, constranng the objectves by some allowable levels : kk I C req C (13) E kk I ε req E (14)

7 7 n order to overcome the dculty on ndng the non-convex Pareto-optmal soluton set or the multobjectve optmzaton problem. A non-domnated soluton m n the Pareto-optmal soluton set, gven by a 168 hours schedule, s characterzed by a total uel cost and a total emsson n the space o crterons. The percentage ncrease n the total uel cost over the total uel cost obtaned or the BCC, computed or each soluton m as ollows: BCC, s % m m m BCC m m m ( x, u, p ) ( x, u, p ) 100 % (15) BCC The percentage decrease n the total emsson over the total emsson obtaned or the BCC, computed or each soluton m as ollows: BCC g, s g % BCC m m m m m m g g ( x, u, p ) ( x, u, p ) 100 % (16) BCC g The proposed approach upon havng the set o non-domnated solutons, graphcally llustrated by the trade-o curve between uel cost and emsson, extracts a compromse soluton. Ths soluton s dened by the amount o percentage ncrease n the total uel cost that the decson maker s wllng to accept n exchange or a certan amount o percentage decrease n the total emsson. We obtan the rato o change or each non-domnated soluton m wth respect to the prevous non-domnated soluton m-1, comparatvely to the maxmum rato o change, gven by: g ( x, u, p ) g ( x, u, p m m m m-1 m-1 m-1 BEC m % % % (17) m m m m-1 m-1 m-1 BCC % ( x, u, p ) % ( x, u, p ) g% ) We also obtan the correspondng gradent angle, gven by: m tan 1 ( m ) (18) The value o the gradent angle ncreases rom 0 to 90º as the weghtng actor vares between 0 and 1. On the one hand, the gradent angle assumes small values, the percentage decrease n the total emsson would be small or a sgncant percentage ncrease n the total uel cost. On the other hand, the gradent angle assumes large values, the decson maker may decde n avour o a urther percentage decrease n the total emsson at the expense o some percentage ncrease n the total uel cost. In our approach, the best compromse commtment s chosen or a rato o change equal to 1, correspondng to a gradent angle o 45º, snce a rato o change less than 1 means that the percentage decrease n the total emsson s less than the correspondng percentage ncrease n the total uel cost.

8 8 4. CASE STUDY We consder a case study consstng o 11 thermal unts and a schedulng tme horzon o 168 hours. Table I shows the coecents or cost and emsson. "See Table I at the end o the manuscrpt". Thermal unts are avalable or producton durng the entre tme horzon o optmzaton. Note that thermal unts 1 to 6 have neror uel cost but hgher emsson n comparson wth thermal unts 7 to 11. The demand to be satsed durng the tme horzon s shown n Fgure 1. Fgure 1. Hourly demand. The computatonal approach was developed and mplemented on a 1.6-GHz-based processor wth 512 MB o RAM usng FORTRAN language. We carred out the ollowng computaton strategy: at rst, uel cost and emsson are ndependently optmzed to determne the extreme ponts o the trade-o curve: the BCC and the BEC; then, uel cost and emsson are merged n the weghted sum method, as mentoned n our approach. The BCC and BEC results or unts 1 to 6 are shown n Fgure 2. Fgure 2. Hourly total generaton or thermal unts 1 to 6. The sold lne represents best cost commtment results whle the dashed lne represents best emsson commtment results. The BCC and BEC results or unts 7 to 11 are shown n Fgure 3. Fgure 3. Hourly total generaton or thermal unts 7 to 11. The sold lne represents best cost commtment results whle the dashed lne represents best emsson commtment results. Unts wth neror uel cost are commtted typcally at ull power regardless o emsson, n the BCC. Thus, t was expected that the lesser pollutant unts were not needed to be commtted n order to satsy the demand, because these unts have hgher uel cost. The commtment status o thermal unts acheved or the BCC ollows the demand prole, as shown n Fgure 4. Fgure 4. Matrx structure representng commtment status o thermal unts or best cost commtment results: a lled spot means that the unt s commtted, whle a blank spot means that the unt s not commtted. In the BEC all unts are commtted, as shown n Fgure 5, and the power o unts 1 to 6 s reduced n order to acheve the mnmum emsson, mplyng a hgher total uel cost. Fgure 5. Matrx structure representng commtment status o thermal unts or best emsson commtment results: a lled spot means that the unt s commtted, whle a blank spot means that the unt s not commtted. Fgure 6 to Fgure 9 show 100 non-domnated solutons o the trade-o curve consderng a scalng actor respectvely o 7, 14, 21 and 28, thus posng our derent scenaros or the scalng actor.

9 9 Fgure 6. Non-domnated solutons o the trade-o curve wth a scalng actor o 7. Fgure 7. Non-domnated solutons o the trade-o curve wth a scalng actor o 14. Fgure 8. Non-domnated solutons o the trade-o curve wth a scalng actor o 21. Fgure 9. Non-domnated solutons o the trade-o curve wth a scalng actor o 28. The trade-o curve has a sharp slope at the BCC neghbourhood. The gradent angle s close to 90º, meanng that a sgncant percentage decrease n the total emsson, about 16.5%, s obtaned wth a small percentage ncrease n the total uel cost, about 2.0%. It should be noted that at the end o the curve the opposte occurs, snce or the same ncrease o 2.0% n the total uel cost only a 0.9% decrease n the total emsson s obtaned, because at ths pont the gradent angle s close to 0º. An overall decrease n the total emsson o about 42.1% s obtaned by a total uel cost ncrease o about 12.4%. The new parameter, rato o change, and the correspondng gradent angle, enable the proper selecton o a compromse commtment or the unts between the BEC and the BCC. The ncrease o the scalng actor avours the predomnance o the total emsson objectve uncton over the total uel cost objectve uncton, thus shtng the best compromse commtment nto the neghbourhood o the BEC, as shown n Table II. "See Table II at the end o the manuscrpt". The total CPU-tme or the computaton o the trade-o curve was about 270 s, wth an average 2.7 s or each soluton correspondng to a 168 hours schedule. Ths demonstrates that the proposed approach has an acceptable CPU-tme n handlng ths problem. 5. CONCLUSION Ths paper provdes an approach or the short-term schedulng o thermal unts wth emsson constrants. A compromse between the uel cost ncurred on the commtment o the unts and the emsson mples the consderaton o a mult-objectve optmzaton problem or developng an normaton management system adng the decson maker. The non-domnated solutons o the trade-o curve between uel cost and emsson are presented or derent values o a scalng actor, assumed constant over the schedulng tme horzon. Our approach presents a new parameter, rato o change, and the correspondng gradent angle, whch can be used by the decson maker to choose a compromse commtment or the unts. A lmtaton may result when the carbon market prces are hghly volatle over the schedulng tme horzon. In ths case, the decson maker has to readjust hs schedulng contnuously. Numercal results show that the proposed approach s ecent or obtanng the trade-o curve and the best compromse commtment or the unts wth an acceptable CPU-tme requrement.

10 10 6. LIST OF SYMBOLS x u p K I C x Array o all state varables Array o all dscrete decson varables Array o all power producton varables Set o hours n the schedulng tme horzon Set o thermal unts n the power system Total uel cost ncurred by a thermal unt n hour k State varable assocated wth a thermal unt n hour k u Dscrete decson varable assocated wth a thermal unt n hour k p D k B n Power producton varable assocated wth a thermal unt n hour k Demand o electrcal energy n each hour k Set o thermal unts on the nth cumulatve constrant H n Functon whch descrbes a contrbuton o thermal unt to nth cumulatve constrant req H n Upper bound on nth cumulatve constrant N Set o cumulatve constrants U Set o easble dscrete decson varables or thermal unt n hour k P 0 X Set o admssble power producton varables or unt at stage k Set o ntal states or unt X Set o nal states or unt E w M Total emsson caused by a thermal unt n hour k Weghtng actor Scalng actor Set o non-domnated solutons Allowable levels

11 11 REFERENCES 1. Bellhouse GM, Whttngton HW. Smulaton o gaseous emssons rom electrcty generatng plant. Electrcal Power and Energy Systems 1996; 18: Palanchamy C, Babu NS. Day-nght weather-based economc power dspatch. IEEE Transactons on Power Systems 2002; 17: Unted Natons Framework Conventon on Clmate Change, UNFCCC, Kyoto Protocol [Onlne]. Avalable: 4. Drectve 2003/87/EC o the European Parlament and o the Councl o 13 October 2003 establshng a scheme or greenhouse gas emsson allowance tradng wthn the Communty and amendng Councl Drectve 96/61/EC [Onlne]. Avalable: 5. Pulgar-Panemal HA. Short-term generaton schedulng under a SO 2 emssons allowances market. Electrc Power Systems Research 2005; 74: Catalão JPS, Marano SJPS, Mendes VMF, Ferrera LAFM. Parametersaton eect on the behavour o a headdependent hydro chan usng a nonlnear model. Electrc Power Systems Research 2006; 76: Talaq JH, EI-Hawary F, EI-Hawary ME. A summary o envronmental/economc dspatch algorthms. IEEE Transactons on Power Systems 1994; 9: Wong KP, Yuryevch J. Evolutonary-programmng-based algorthm or envronmentally-constraned economc dspatch. IEEE Transactons on Power Systems 1998; 13: Dhllon JS, Kothar DP. The surrogate worth trade-o approach or multobjectve thermal power dspatch problem. Electrc Power Systems Research 2000; 56: Huang CM, Huang YC. A novel approach to real-tme economc emsson power dspatch. IEEE Transactons on Power Systems 2003; 18: Abdo MA. Envronmental/economc power dspatch usng multobjectve evolutonary algorthms. IEEE Transactons on Power Systems 2003; 18: Muslu M. Economc dspatch wth envronmental consderatons: tradeo curves and emsson reducton rates. Electrc Power Systems Research 2004; 71: Chang CL, Law JH, Su CT. New approach wth a genetc algorthm ramework to mult-objectve generaton dspatch problems. European Transactons on Electrcal Power 2005; 15: Kuloor S, Hope GS, Mal OP. Envronmentally constraned unt commtment. IEE Proceedngs-Generaton Transmsson and Dstrbuton 1992; 139: Gjengedal T. Emsson constraned unt-commtment (ECUC). IEEE Transactons on Energy Converson 1996; 11: Abdul-Rahman KH, Shahdehpour SM, Aganagc M, Mokhtar S. A practcal resource schedulng wth OPF constrants. IEEE Transactons on Power Systems 1996; 11: Mendes VMF, Marano SJPS, Catalão JPS, Ferrera LAFM. Emsson constrants on short-term schedule o thermal unts. 39th Internatonal Unverstes Power Engneerng Conerence (UPEC 2004), Brstol, UK, Catalão J, Marano S, Mendes V, Ferrera L. Unt commtment wth envronmental consderatons: a practcal approach. 15th Power Systems Computaton Conerence (PSCC'05), Lège, Belgum, Hu YL, Wee WG. A herarchcal system or economc dspatch wth envronmental constrants. IEEE Transactons on Power Systems 1994; 9: Ferrera LAFM, Andersson T, Imparato CF, Mller TE, Pang CK, Svoboda A, Vojdan AF. Short-term resource schedulng n mult-area hydro-thermal power systems. Electrcal Power and Energy Systems 1989; 11: Wood AJ, Wollenberg BF. Power Generaton, Operaton and Control; 2nd edn, Wley: New York, 1996.

12 12 Table I. Cost and emsson coecents. Cost ($) p mn p max Emsson (Gg) unt (MW) (MW) a b c total Table II. Computatonal results or the proposed approach. Scalng actor Total uel cost ($) Total generaton (GW) Total emsson (Gg) Best cost commtment - 12,994, Best emsson commtment - 14,611, ,510, Best compromse commtment 14 13,555, ,568, ,589, AUTHORS BIOGRAPHIES J. P. S. Catalão receved the electromechancal engneerng degree rom the Unversty o Bera Interor, Covlha, Portugal, n 1998 and the M.Sc. degree n electrcal and computer engneerng rom the Insttuto Superor Técnco, Techncal Unversty o Lsbon, Portugal, n He s an IEEE Member. Snce 1999, he has been wth the Department o Electromechancal Engneerng, Unversty o Bera Interor, where he s currently a Teachng Assstant nshng hs Ph.D. n electrcal engneerng. S. J. P. S. Marano receved the electrcal and computer engneerng degree and the M.Sc. degree rom the Insttuto Superor Técnco, Techncal Unversty o Lsbon, Portugal, n 1990 and 1994, respectvely, and the Ph.D. degree n electrcal engneerng rom the Unversty o Bera Interor, Covlha, Portugal, n Snce 1992, he has been wth the Department o Electromechancal Engneerng, Unversty o Bera Interor, where he s currently an Assstant Proessor. V. M. F. Mendes receved the electrcal engneerng degree and the M.Sc. and Ph.D. degrees n electrcal and computer engneerng rom the Insttuto Superor Técnco, Techncal Unversty o Lsbon, Portugal, n 1977, 1987 and 1994, respectvely. Snce 1997, he has been wth the Department o Electrcal Engneerng and Automaton, Insttuto Superor de Engenhara de Lsboa, Lsbon, Portugal, where he s currently a Proessor n charge o the Economc and Management group o dscplnes. L. A. F. M. Ferrera receved the electrcal engneerng degree rom the Insttuto Superor Técnco, Techncal Unversty o Lsbon, Portugal, n 1977 and the M.S.E.E. and Ph.D. degrees rom Georga Insttute o Technology, Atlanta, n 1983 and 1986, respectvely. From 1986 to 1989, he was wth the Pacc Gas and Electrc Company, San Francsco, USA, where he was a major developer o the Hydro-Thermal Optmzaton program. Snce 1989, he has been wth the Department o Electrcal Engneerng and Computers, Insttuto Superor Técnco, where he s currently an Assocate Proessor.