Optimal Dispatch in Electricity Markets

Transcription

1 QUANTITATIVE FINANCE RESEARCH CENTRE QUANTITATIVE FINANCE RESEARCH CENTRE Research Paper 06 October 007 Optmal Dspatch n Electrcty Markets Vladmr Kazakov and Anatoly M Tsrln ISSN

2 Optmal dspatch n electrcty markets Vladmr Kazakov and Anatoly M. Tsrln School of Fnance and Economcs, Unversty of Technology, Sydney, PO Box 13 Broadway, NSW 007 Australa, vladmr.kazakov@uts.edu.au Program System Insttute, Russan Academy of Scences, set Botk, Pereaslavl-Zalesky, Russa 1500 September 1, 007 Abstract The problem of calculatng the optmal dspatch and prces n a sngle-perod electrcty aucton n a wholesale electrcty market s consdered here. The novel necessary and suffcent condtons of optmalty for ths problem are derved and computatonal algorthms for solvng these condtons are constructed. Keywords: Optmal dspatch; Electrcty market; Nonlnear programmng; Non-convex problems; Dynamc programmng. Introducton Economc dspatch problem for electrcty generaton has been the focus of numerous studes eg [1], [5]) for a long tme. Recent market deregulaton led to ts reformulaton. Calculatng market-based economc dspatch stll requres solvng non-lnear programmng problem but ths problem became qualtatvely dfferent. Frstly, ts obectve functon became the cost of generaton, defned as supplers prce-volume bd functons ntegrated over ther dspatched volumes. In most markets these bds are step-wse functons. Ths yelds dscontnutes of gradents and reduces the effcency of standard numercal methods. Secondly, because negatve prce steps are commonly used n bds the market-based dspatch problem s non-convex and may have multple solutons. These are the dstngushng characterstcs of dspatch problem n a deregulated electrcty market whch make t mportant to develop algorthms especally talored to solve them effcently. 1

3 In ths paper we develop novel necessary and suffcent condtons for dspatch and prcng problem n a deregulated electrcty market and construct computatonal methods, based on these condtons, capable of effcently calculatng dspatch and prces. Dspatch n Australan electrcty market We consder normal tradng only, dd not consder any tradng n auxlary markets, reserves etc. Electrcty market s descrbed as a network of scalar flows, whch s standard n modellng of economc electrcty dspatch. The model consdered n ths paper s used n practce to dspatch the Australan market, NEM see [3], [4]. Thus, we consder electrcty market that ncludes n regonal markets connected by a network of nterregonal connectors, whch transfer electrcty between them, Fg. ). Each partcpant generator or consumer) s located n one of the regonal markets. All partcpants n one regonal market are pad or pay the same regonal prce for electrcty they sell or consume. That s, each regonal aucton s a sngle-prce aucton. As a rule, regonal prces n dfferent regons are dfferent. Every tradng day s dvded nto a sequence of dentcal sngle-perod auctons. A sngle perod aucton s n fact a number of lnked regonal auctons that take place n all regons smultaneously. The results of a sngle perod aucton depends on generators prce-volume bds, the state of the market regonal generatons and nter-regonal flows) before the aucton, and the regonal demands. These aucton results determne regonal dspatches - the amounts of electrcty sold by every generators n the regon, power flows between regons and the regonal prces. Generators submt prce-volume bds to the market operator. These bds are step-wse functons that show how much power they are prepared to supply for a partcular prce Fg. 1). Prce steps P k n prce bds could be and ndeed often are n practce) negatve. That s, the producer offers to pay a consumer who agrees to buy ts electrcty. Tradng day s dvded nto sequence of equal-tme perods. Market operators calculates dspatch for each of ths perod by runnng sngle perod electrcty aucton, usng the followng nformaton: Combned regonal demands by all consumers d ; Combned regonal prce-bds P q ) by all generators, Combned regonal generatons before the aucton q 0), Inter-regonal flows before the aucton g 0).

4 3 P P P P q ) P 1 P 0 0 Q 1 Q Q3 q Fgure 1: Prce-volume bds by producers. g 1 P q d 1 1 ) 1 P q ) d g 14 g 13 g3 P q 4 d 4 ) P q d ) 3 g 43 Fgure : Fragment of regonal markets network. Functons P q ) are contnuous from the rght and have left lmts lm q Q P q ) =. Regonal generatons are range-constraned qmn regonal energy flows g P q q max. Inter- from the -th to -th regon are also range constraned, g mn g g max dynamcal reasons. Note that g = g.. These constrants are due to technologcal and We assume that when the power g s transferred from -th to the -th regonal markets some of t s lost and ths loss s descrbed by the functon L g ). Ths functon s a gven contnuous functon of g such that L g ) = L g ) = L g ), L x) 0, L 0) = 0, L = L, L > 0 for g 0 and d L d g > 0 for g. Characterstc dependence L g ) on g s shown n Fg. 3. The loss L s dvded between correspondng markets n a fxed proporton gvng addtonal demand α L n the -th regon and α L n the -th regon, Fg. 4. α are constants, 0 α 1, α = 1 α, α = 0. The net energy balance for the whole network of regonal markets s n q = =1 n d + 1 n L g ). 1) =1,=1, Snce g = g and α + α = 1,, = 1,..., n ths balance holds f the 3

5 Fgure 3: Characterstc dependence of power loss L on the power flow g. Fgure 4: Two markets lnked by nter-connector. regonal energy balances q = d + n =1, g + α L g ) ), = 1,..., n ) hold. The cost of generaton n the -th market s defned as C q ) = q 0 P x)dx. 3) The optmal dspatch s determned by mnmzng the combned cost of generaton of all generators n all regonal markets, I, on q, g subect to range constrants Id 1,..., d n, q,..., q n, g ) = q mn n =1 C q ) mn q,g 4) q q max, = 1,..., n 5) 4

6 g mn g g max = 1,..., n, = 1,..., n 6) and regonal markets energy balances ). We denote ts soluton as q, g and the value of the obectve functon the mnmal cost of supply) as I d 1,..., d n ) = mn q,g Id 1,..., d n, q,..., q n, g ) = = Id 1,..., d n, q,..., q n, g ). 7) The cost of generaton I and the mnmal cost of generaton I both depend not only on d but also on the bds P q ) and pre-aucton state of the market q 0), g 0) va range constrants on q and g ). The unknown varables n the optmal dspatch problem 4), 5), 5), ) are the regonal generatons and nter-regonal flows after the aucton. If network contans connectons between all regonal markets then the number of unknown varables here s n n +n = nn+1), n 0. The frst term s the number of unknown exchange flows g and the second term s the number of generated regonal powers q. Snce these varables obey n balance equatons ) the number of free unknowns n the dspatch problem s equal to the number of nter-regnal flows, nn 1) for a network where all regonal markets are connected. For n = 1 there s one unknown, for n = 3 - three, etc. Note that the problem 4-5) s not easy to solve, t has both range constrants on ts ndependent varables and constrants on ts dependent varables q. Necessary condtons of optmalty The optmal dspatch problem 4), 5), ) can be rewrtten as a non-lnear programmng problem n [ C d + =1 subect to constrants and range constrants q mn d + g mn n =1, n =1, g + α L g ) )] max g 8) g + α L g ) ) q max. 9) g g max, = 1,..., n. 10) 5

7 The ndependent unknown varables now are the flows g. Regonal generatons are calculated from balances ). We defne reduced prce of the regon wth respect to regon as P g ) = P d + g + α L g ) ))[ 1 + α dl g ) dg ], 11) and for -th wth respect to -th t s P g ) = P d + g + α L g ) ))[ 1 α dl g ) dg ]. We also defne ther left and rght lmts P + ab = lm ɛ 0 P ab g + ɛ), P ab = lm ɛ 0 P ab g ɛ). The necessary condtons of optmalty for dspatch problem s then gven by the followng Theorem. If {g } s an optmal soluton of the problem 8) 10) then one of the followng condtons holds Non-bndng range constrants q mn Bndng flow constrants g mn < g < g max, q mn < d + g + α L g) ) < q max, 1) < d + g + 1 α )L g) ) < q max, P + g ) P g ), P g ) P + g ) or q mn g = g max, < d + g + α L g) ) < q max, 13) < d + g + 1 α )L g) ) < q max, P g ) P + g ), q mn q mn g = g mn < d + g + α L g) ) < q max, 14) < d + g + 1 α )L g) ) < q max, P + g ) P g ) q mn 6,

8 Bndng generaton constrant A) q mn or B) q max = d + = d + g mn < g < g max, P P +, 15) g + α L g) ), or q max = d + g + 1 α )L g) ). g mn < g < g max +, P P, 16) g + α L g) ), or q mn = d + g + 1 α )L g) ). Proof. We derve the proof by examnng the frst varaton of the functon I. The partcular form of problem s constrants allows us to smplfy the problem by solvng some of ts constrants explctly. Frst we consder the case were both range constrants 9), 10) are not bndng and P and P are contnuous at g. Dfferentaton leads to the followng expresson for the varaton of the obectve functon I = P P ) g. Snce here both postve and negatve varatons of g are feasble, the + current arguments are where the bd functons are contnuous, P = P = P and P + = P = P, the necessary condton of optmalty condton of non-mprovement of I by a feasble nfntesmal varaton of g ) concdes wth 1). Let us consder the general case of possble dscontnuous ponts of P and/or P. Snce P s a non decreasng and P s a non ncreasng functons of the same flow g, from 8) t follows that I can be wrtten as I = [ P + P ) P + P )] δ g )Θ + g ) Θ g )) + + P + P )Θ + g ) + P P + )Θ g ) 17) where δx) s the Drac functon, Θ + x) = 1, f x > 0 and 0 otherwse. The frst term here descrbes the net effect of two umps n bds. Suppose all range constrants are not bndng. Then g are not constraned and can have arbtrary sgn. Therefore, the condton I g 0 that guarantees that g s not mprovable by nfntesmal varaton of g become 1). Fnally, f at least one of the followng condtons g = g max or q max = d + g + α L g ) ) hold, then only negatve varatons g are feasble. In ths case the condton I g 0 takes the form 16). The same dervatons for the case, where only postve g are feasble conclude the proof. 7

9 Long-chan condtons of optmalty Consder a lnear subnetwork of three regonal markets. Suppose that P < P, Pk < P k, q = q max. The flows s drected from -th to the -th and then -th to the k-th markets. The range constrant on the generaton n the -th market s actve. Ths prohbts postve varaton of g. However, two smultaneous postve varatons δg = δg k > 0 are feasble. Analyss dentcal to a sngle connector analyss above yelds the followng condtons of optmalty for non-bndng range constrants for q and q k on the ntervals where P.) and P k.) are contnuous P 1 α L ) 1 αk L k) = Pk 1 + α L ) 1 + αk L k), 18) and the general condton of optmalty for dscontnuous ponts ) ) ) P + 1 α L 1 αk L k) P k 1 + α L 1 + αk L k, 19) ) ) ) P 1 α L 1 αk L k) P + k 1 + α L 1 + αk L k, 0) For bndng range constrants these condton of long-range optmalty s exactly the same as for short-range optmalty above - but for long-range reduced prces lhs and rhs of 18)) nstead of short-range reduced prces 11). They state that for the optmal dspatch two regonal markets connected by a network of generaton-constraned ntermedate markets are dfferent, can have dfferent long-range reduced prces f and only f at least one of the flows between them s constrant or the generaton n one these two markets s range constraned. P q ) g P q ) P g k q k ) q =q max Fgure 5: 3-market lnear fragment. k Numercal methods In practce dspatch and prces n most of the markets now s solved by lnearzng dspatch problem and then solvng t numercally usng classcal lnear programmng algorthms see [3], [4]). 8

10 The derved condton of optmalty requre that reduce prces between lnked regonal markets be equal unless flow constrant or generaton constrant becomes bndng. They can be used to fnd the dspatch optmzaton numercally. As an ntal pont ths algorthm requres a feasble set of flows g such that range constrants5) and 9) hold. Ths soluton s then mproved teratvely by applyng the followng elementary computatonal operaton: 1) calculate the reduced prces dfference P = P P for every par of regons connected by nter-connector; ) mark all par where one of the optmalty condtons holds; 3) for each unmarked par f P > P + then g mn < g < g max, q max < q g ) < q max, q max < q g ) < q max then g s ncreased, untl one of these nequaltes becomes equalty. Ths ncrease contnues across umps of the prce-volume bds, provded that the reduced -th prce at these new steps s stll lower that reduced prce P. Smlarly, f P < P + then g s reduced untl ether reduced prces equalze or one of range constrants becomes bndng. The mproved soluton s agan submtted as an nput to step ). The teratons wll converge because varaton of g ncrease one of the prces P, P and decreases another. After teratons converge, one can use ts output as an ntal soluton for another teratve algorthm whch uses the mprovement operaton for 3-market substrngs nstead of -market ones, then for 4-market etc. The condtons of maxmal equalzaton of reduced prces between all regons across the network can be mantaned by an automatc feedback control system. Ths would allow to replace the sequence of sngle-auctons wth a sngle contnuous real-tme aucton and wll approve dspatch dramatcally. Non-convexty of the dspatch problem when bds have negatve prce steps We llustrate non-convexty by plottng the cost of generaton Ig) as a functon of g n Fgure 7 for the market wth two regonal markets, regonal bds P 1 q) and P q) shown n Fgure 6, and losses Lg) = 0.14g equally dvded between regonal markets. The graph clearly shows that Ig) has two mnma. The left mnmum corresponds to I = , g = 11.1 and dspatches q1 = and q = 60. The lnk between non-convexty of dspatch problem and negatve prces n bds can be traced usng the followng analyss. Each regonal cost of generaton ntegrated prce-volume bds) C q ) s a convex, pece-wse lnear functon. Therefore, the total cost of generaton - the sum of regonal costs 9

11 P q) P q) q Fgure 6: Bds by generators n two regonal markets. 0 Ig) g 16 Fgure 7: Cost of supply for two regonal markets n the example as the functon of nter-regonal flow. of generaton - s also convex on q. But after we express these varables n terms of nter-regonal flows g usng energy balances ) the obectve functon of the sngle-perod aucton problem becomes non-convex. Indeed, the second dervatve of C g ) wth respect to g s Snce dl dg d dg C = d dg dc dq dl dg ) = P dl dg 1) > 0 the sgn of d dg C concdes wth the sgn of the prcevolume bd current prce step). Thus, the use of negatve prce steps n bds s the causes of non-convexty. Thus, f one solves the necessary condtons of optmalty derved above or uses a drect search to mnmze the cost of supply, then there s no guarantee the the soluton found s one of the local mnma. That s why t s mportant 10

12 to verfy f the soluton found s local or global mnmum and t s local then how close the local mnmum s from the global one, how much mprovement n terms of the cost of supply can be acheved f search for optmal dspatch contnues from another ntal pont. Globally optmal dspatch. Consder relaxaton of the optmal dspatch problem 4), 5), 5), ) by deletng range constrants on the flows 5) and regonal energy balances ). The generalzed optmal dspatch problem then becomes subect to Id 1,..., d n, q,..., q n, g )) = q mn n =1 C q ) mn q ) q = M 3) q q max, = 1,..., n 4) M here s the parameter that descrbes nter-connectors losses and whch must obey the range constrants Q mn d M q max. 5) Bellman functon [6]) s defned usng the followng recurrent equaton [ φ 1 x 1 ) = C 1 x 1 ), φ x ) = mn q mn q q max C q ) + φ 1 x q ) ], [ φ ν x ν ) = mn q mn ν q ν qν max Cν q ν ) + φ ν x ν q ν ) ] 6) By constructon φ n M) = mn q 1, q,..., q n C q M)) 7) gves global mnmum and correspondng network losses are gven by the network energy balance d + 1 L g ) = M 8) Suppose we used numercal algorthm descrbed n the secton above and obtaned g, q, I and M = q. φ h M ) then gves the lower bound on 11

13 the cost of generaton I. If t turns out that I = φ n M ) then g, q s the global mnmum. If I φ n M ) s small enough then one may reasonably decde to stop search and use the current local mnmum nstead of global one. Entrepreneural nter-connectors Entrepreneural lnes represent another new feature of modern electrcty markets. Physcally they are no dfferent to the standard nter-regonal connectors that are controlled by market operator. But entrepreneural lnes operate dfferently. Before tradng starts they submt prce-volume bds, that are dentcal to generators prce-volume bds. They submts these bds for two regonal markets connected by ther lne. These bds are the offers to flow energy, whch are taken nto account when sngle-perod aucton s calculated. Let us consder for smplcty two regonal market -th and -th lnked by an entrepreneural lne, see Fg. 8. We denote ther prce-volume bds as E x) and E x), the flow and losses n the entrepreneural lne as e and LEe), the coeffcent that shows how these losses are apportoned to the -th and -th regons as α e and the contrbuton of the lne to the net energy balances n the -th and -th regons amount flown n/out of the regon) as qe and qe. The dspatch s then found by solvng the extended optmal dspatch problem 4), 5), 5), ), Id 1,..., d n ) = C q ) + C x ) + sgne) + 1 qe 0 E x)dx sgne) 1 qe 0 E x)dx mn q,g,qe,qe,e 9) subect to regonal energy balances q = d g α L g )) + e + α e LEe), 30) range constrants 5), and e mn e e max 31) 1

14 furthermore balances for the entrepreneural lne qe = e + α e LEe), qe = e + 1 α e )LEe), qe + qe = LEe) 3) In 9) sgns) = 1 f s 0 and sgnx) = 1 f x < 0. Ths reflects the nature of operatons by entrepreneural lne - t flows energy from one regonal market to another. The cost of generaton now ncludes the cost of supply for entrepreneural lne tself. If flow s drected from to then the extra term correspondng to ths cost scqe ) = qe E 0 x)dx, f the energy s transferred from to then ths added term os Cqe ) = qe E 0 x)dx. For the postve e the varaton of I s δi = [ P 1+α e d de LE))+P 1+1 α e ) d de LE))+E e)1+α e d de LE))] δe 33) If e belongs to an nterval where functon E.)s contnuous, then the followng necessary condton of optmalty take the form d E = P P de ) + LE) d 1 + α e LE)P 34) de E qe ) E qe ) LE e) e α LE E e ) 1 α ) LE e ) E E x) E x) qe qe q x x Fgure 8: Entrepreneural nter-connectors. 13

15 That s, the hghest prce band dspatched va the entrepreneural lne s determned by the dfference between spot prces n these two markets plus correctonal term that depends on the network losses n entrepreneural lne. In algorthmc terms f P 1 + α e LE ) + P α e )LE ) < E e)1 + α e LE ) 35) then one needs to ncrease e untl ths condton s no longer true. Conclusons The dspatch and prcng problem for sngle-perod electrcty aucton n a network of regonal markets s consdered. Its condtons of optmalty are derved and solved. It s shown these condtons requre that the values of some well-defned functon, called reduce prce, n connected regonal markets be as close to each other as possble. Computatonal algorthms for solvng these condtons numercally are constructed. It s shown that dspatch problem s non-convex f negatve prces are used n bds. The dynamc programmng based algorthm for calculaton of the lower bound on the cost of generaton s constructed. It s shown how t can be used to verfy f the obtaned soluton s global. References [1] Schweppe, F. C., Caramans, M. C., Tabors, R. D., and Bohn, R.E., Spot Prcng of Electrcty, Kluwer Academc Publshers, Norwell, MA, 1988). [] Bohn, R, Caramans, M and Schweppe,F 1984) Optmal Prcng n Electrcal Networks Over Space and Tme. Rand J. on Economcs, 183). [3] Gllett, R 1998). Pre-dspatch process descrpton, NEMMCO, Australa. [4] Chong-Whte, C C 001) Investgaton nto aspects of Australan natonal electrcty market central dspatch algorth, MS Thess, UTS. [5] Chowdhury, B H Rahman, S 1990) A revew of recent advances n economc dspatch, IEEE Trans. Pow. Syst., 5, 4, 148. [6] Bellman, R 1957) Dynamc programmng, Prnceton Unv. Press. 14