Congestion Management in Restructured Power Systems Using an Optimal Power Flow Framework

Transcription

1 PSERC Congeston Management n Restructured Power Systems Usng an Optmal Power Flow Framework Masters Thess and Proect Report Power Systems Engneerng Research Center A Natonal Scence Foundaton Industry/Unversty Cooperatve Research Center snce 1996

2 Power Systems Engneerng Research Center Congeston Management n Restructured Power Systems Usng an Optmal Power Flow Framework Masters Thess and Proect Report A.S. Nayak and M.A. Pa Unversty of Illnos at Urbana-Champagn PSERC Publcaton 0-3 May 00

3 Informaton about ths Report For nformaton about ths report contact: M.A. Pa Professor Electrcal and Computer Engneerng Unversty of Illnos at Urbana-Champagn 345 Evertt Lab 1406 W. Green Street Urbana, IL Phone: Fax: Emal: Power Systems Engneerng Research Center Ths s a proect report from the Power Systems Engneerng Research Center (PSERC). It s avalable on the PSERC webste, The PSERC publcaton number s 0-3. For addtonal nformaton, contact: Power Systems Engneerng Research Center Cornell Unversty 48 Phllps Hall Ithaca, New York Phone: Fax: PSERC s a mult-unversty center for research and educaton on challenges facng the electrc power ndustry. Informaton about PSERC can be found at the above webste address. Notce Concernng Copyrght Materal Permsson to copy wthout fee all or part of ths publcaton s granted f approprate attrbuton s gven to ths document as the source materal. 00 Unversty of Illnos. All rghts reserved.

4 Acknowledgements The work descrbed n ths report was sponsored by the Power Systems Engneerng Research Center (PSERC). It s one of the research products from the PSERC proect New System Control Methodologes. The proect leader s Chrs DeMarco from the Unversty of Wsconsn-Madson. Proect team members nclude Ian Dobson (Unversty of Wsconsn- Madson), M. A. Pa (Unversty of Illnos at Urbana/Champagn) and Ian Hskens (Unversty of Illnos at Urbana-Champagn). We express our apprecaton for the support provded by PSERC s ndustral members and by the Natonal Scence Foundaton under grant NSF EEC receved under the Industry/Unversty Cooperatve Research Center program. We would also lke to acknowledge the support of the Granger Foundaton.

5 Executve Summary The restructurng of the electrc power ndustry has nvolved paradgm shfts n the real-tme control actvtes of the power grds. Managng dspatch s one of the mportant control actvtes n a power system. Optmal power flow (OPF) has perhaps been the most sgnfcant technque for obtanng mnmum cost generaton patterns n a power system wth exstng transmsson and operatonal constrants. In ths report we look at a modfed OPF whose obectve s to mnmze the absolute MW of reschedulng. In ths framework, we also consder dspatchng blateral contracts n case of serous congeston, wth the knowledge that any change n a blateral contract s equvalent to modfyng the power nectons at both the buyer and the seller buses. Ths hghlghts the fact that, n a restructured scenaro, contracts between tradng enttes must be consdered as system decson varables (n addton to the usual generaton, loads and flows). The dspatch problem has been formulated wth two dfferent obectve functons: cost mnmzaton and mnmzaton of transacton devatons. Congeston charges can be computed n both the cases. In a pool market mode, the sellers (compettve generators) may submt ther ncremental and decremental bd prces n a real-tme balancng market. These can then be ncorporated n the OPF problem to yeld the ncremental/decremental change n the generator outputs. Smlarly, n the case of the blateral market mode, every transacton contract may nclude a compensaton prce that the buyer-seller par s wllng to accept should ts transacton be curtaled. Ths can then be modeled as a prortzaton of the transactons based on the latter s senstvtes to the volated constrant n case congeston occurs. In ths report, we also seek to develop an OPF soluton ncorporatng FACTS devces n a gven market mode (pool or blateral dspatch). FACTS devces assume mportance n the context of power system restructurng snce they can expand the usage potental of transmsson systems by controllng power flows n the network. FACTS devces are operated n a manner so as to ensure that the contractual requrements are fulflled as far as possble by mnmzng lne congeston. Varous optmzaton technques avalable n the lterature have been used to solve OPF problem.

6 Table of Contents 1 INTRODUCTION. 1 CONGESTION MANAGEMENT METHODOLOGIES. 5.1 Introducton.. 5. Vertcally Integrated Operaton 5.3 Unbundled Operaton Congeston Management Methodologes 7.5 Example of Congeston Management n an Economc Dspatch Framework 7.6 Congeston Management Usng Prcng Tools OPTIMAL DISPATCH METHODOLOGIES IN DIFFERENT MARKET STRUCTURES Introducton Pool Dspatch Pool structure Pool dspatch formulaton Example of correctve reschedulng n pool dspatch Blateral Dspatch Blateral market structure Blateral dspatch formulaton Test results Treatment of Transacton-Based Groups Dspatch formulatons Test case... 4

7 3.5 Conclusons. 9 4 OPTIMAL DISPATCH USING FACTS DEVICES IN DEREGULATED MARKET STRUCTURES Introducton Statc Modelng of FACTS Devces Thyrstor-controlled seres compensator (TCSC) Thyrstor-controlled phase angle regulator (TCPAR) Statc VAr compensator (SVC) Problem Formulaton for OPF wth FACTS Devces FACTS Devces Locatons Reducton of total system VAr power loss Selecton of optmal placement of FACTS devces Test Cases Sx-Bus system Fourteen-bus system Conclusons CONCLUSIONS AND FUTURE WORK 44 REFERENCES.. 45 v

8 1 INTRODUCTION The restructurng of the electrc power ndustry has nvolved paradgm shfts n the realtme control actvtes of the power grds. Managng dspatch s one of the mportant control actvtes n a power system. Optmal power flow (OPF) has perhaps been the most sgnfcant technque for obtanng mnmum cost generaton patterns n a power system wth exstng transmsson and operatonal constrants. The role of an ndependent system operator n a compettve market envronment would be to facltate the complete dspatch of the power that gets contracted among the market players. Wth the trend of an ncreasng number of blateral contracts beng sgned for electrcty market trades, the possblty of nsuffcent resources leadng to network congeston may be unavodable. In ths scenaro, congeston management (wthn an OPF framework) becomes an mportant ssue. Real-tme transmsson congeston can be defned as the operatng condton n whch there s not enough transmsson capablty to mplement all the traded transactons smultaneously due to some unexpected contngences. It may be allevated by ncorporatng lne capacty constrants n the dspatch and schedulng process. Ths may nvolve redspatch of generaton or load curtalment. Other possble means for relevng congeston are operaton of phase-shfters or FACTS devces. In ths report we look at a modfed OPF whose obectve s to mnmze the absolute MW of reschedulng. In ths framework, we consder dspatchng the blateral contracts too n case of serous congeston, wth the knowledge that any change n a blateral contract s equvalent to modfyng the power nectons at both the buyer and the seller buses. Ths hghlghts the fact that, n a restructured scenaro, contracts between tradng enttes must be consdered as system decson varables (n addton to the usual generaton, loads and flows). Fgure 1.1 shows a transacton network [1] n a typcal deregulated electrcty system. It dsplays lnks of data and cash flow between varous market players. In the fgure, G stands for generator-servng enttes (or gencos), D for load or demand-servng enttes (LSEs or dscos), E for marketers, and ISO for the ndependent system operator.

9 G D Informaton flow Money flow ISO E Fgure 1.1 Transacton network The dspatch problem has been formulated wth two dfferent obectve functons: cost mnmzaton and mnmzaton of transacton devatons. Congeston charges can be computed n both the cases. In a pool market mode, the sellers (compettve generators) may submt ther ncremental and decremental bddng prces n a real-tme balancng market. These can then be ncorporated n the OPF problem to yeld the ncremental/decremental change n the generator outputs. Smlarly, n case of a blateral market mode, every transacton contract may nclude a compensaton prce that the buyer-seller par s wllng to accept should ts transacton be curtaled. Ths can then be modeled as a prortzaton of the transactons based on the latter s senstvtes to the volated constrant n case congeston occurs. In ths report, we also seek to develop an OPF soluton ncorporatng FACTS devces n a gven market mode (pool or blateral dspatch). FACTS devces assume mportance n the context of power system restructurng snce they can expand the usage potental of transmsson systems by controllng power flows n the network. FACTS devces are operated n a manner so as to ensure that the contractual requrements are fulflled as far as possble by mnmzng lne congeston. Varous optmzaton technques have been used to solve OPF problems. These may be classfed as sequental, quadratc, lnear, nonlnear, nteger and dynamc programmng

10 methods, Newton-based methods, nteror pont methods, etc. Nonlnear programmng methods nvolve nonlnear obectve and constrant equatons. These make up the earlest category of OPF technques as they can closely model electrc power systems. The benchmark paper by Dommel and Tnney [] dscusses a method to mnmze fuel costs and actve power loss usng the penalty functon optmzaton approach. Dv and Kesavan [3] use an adapted Fletcher s quas-newton technque for optmzaton of shfted penalty functons. Lnear programmng deals wth problems wth constrants and obectve functon formulated n lnear forms. Sterlng and Irvng [4] solved an economc dspatch of actve power wth constrants relaxaton usng a lnear programmng approach. Chen et al. [5] developed a successve lnear programmng (SLP) based method for a loss mnmzaton obectve n an ac-dc system. In the SLP approach, the nonlnear OPF problem s approxmated to a lnear programmng problem by lnearzng both the obectve functon as well as the constrants about an operatng state. At every teraton, a suboptmal soluton s found and the varables are updated to get a new operatng state. The process s then repeated untl the obectve functon converges to an optmal level. Megahed et al. [6] have dscussed the treatment of the nonlnearly constraned dspatch problem to a seres of constraned lnear programmng problems. Smlarly, Waght et al. [7] have used the Dantzg-Wolfe decomposton method to break the dspatch problem nto one master problem and several smaller lnear programmng subproblems. Combnatons of lnear programmng methods wth the Newton approach have been dscussed n the lterature [8]. In [9], Burchett and Happ apply an optmzaton method based on transformng the orgnal problem to that of solvng a seres of lnearly constraned subproblems usng an augmented Lagrangan type obectve functon. The subproblems are optmzed usng quas-newton, conugate drectons, and steepest descent methods. Quadratc programmng s another form of nonlnear programmng where the obectve functon s approxmated by a quadratc functon and the constrants are lnearzed. Nanda et al. [10] dscuss an OPF algorthm developed usng the Fletcher s quadratc programmng method. Burchett et al. [11] dscuss a successve quadratc programmng (SQP) method where the approxmaton-soluton-update process s repeated to convergence ust as n the SLP method. In ths method, a sequence of quadratc programs s created from the exact analytcal frst and second dervatves of the 3

11 power flow equatons and the nonlnear obectve functon. Interor pont methods are farly new entrants n the feld of power system optmzaton problems. Vargas et al. [1] dscussed an nteror pont method for a securty-constraned economc dspatch problem. In [13], Momoh et al. present a quadratc nteror pont method for OPF problems, economc dspatch, and reactve power plannng. The report s organzed as follows. In Chapter we look at congeston management methodologes and how they get modfed n the new compettve framework of electrcty power markets. A smple example s gven for the calculaton of congeston charges n a scenaro where the obectve of optmzaton s to maxmze socetal beneft. In Chapter 3, we work out dfferent OPF formulatons. Obectve functons that are treated nclude cost mnmzaton and transacton curtalment mnmzaton. Market models nvolvng pool and blateral dspatches are consdered. The possblty of usng these formulatons n an open access system dspatch module and n real-tme balancng markets s dscussed. In Chapter 4, we treat the subect of ncludng FACTS devces n the OPF framework. Varous devce models are consdered and then appled n the problem formulaton. The mpact of these devces on mnmzng congeston and transacton devatons s studed. In Chapter 5, the OPF results are dsplayed on two test systems and nferences are drawn from the same. Further areas of research n ths feld are then explored n the concludng chapter. 4

12 CONGESTION MANAGEMENT METHODOLOGIES.1 Introducton In ths chapter, we look at congeston management methodologes and how they get modfed n the new compettve framework of electrcty power markets. A smple example s gven for the calculaton of congeston charges n a scenaro where the obectve of optmzaton s to maxmze socetal beneft.. Vertcally Integrated Operaton The unbundlng of the electrc power market has led to the evoluton of new organzatonal structures. Unbundlng mples openng to competton those tasks that are, n a vertcally ntegrated structure, coordnated ontly wth the obectve of mnmzng the total costs of operatng the utlty. In such a tradtonal organzatonal structure, all the control functons, lke automatc generaton control (AGC), state estmaton, generaton dspatch, unt commtment, etc., are carred out by an energy management system. Generaton s dspatched n a manner that realzes the most economc overall soluton. In such an envronment, an optmal power flow can perform the dual functon of mnmzng producton costs and of avodng congeston n a least-cost manner. Congeston management thus nvolves determnng a generaton pattern that does not volate the lne flow lmts. Lne flow capacty constrants, when ncorporated n the schedulng program, lead to ncreased margnal costs. Ths may then be used as an economc sgnal for reschedulng generaton or, n the case of recurrng congeston, for nstallaton of new generaton/transmsson facltes..3 Unbundled Operaton In a compettve power market scenaro, besdes generaton, loads, and lne flows, contracts between tradng enttes also comprse the system decson varables. The followng pool and blateral compettve structures for the electrcty market have evolved/are evolvng: 5

13 (1) Sngle aucton power pools, where wholesale sellers (compettve generators) bd to supply power n to a sngle pool. Load servng enttes (LSEs or buyers) then buy wholesale power from that pool at a regulated prce and resell t to the retal loads. () Double aucton power pools, where the sellers put n ther bds n a sngle pool and the buyers then compete wth ther offers to buy wholesale power from the pool and then resell t to the retal loads. (3) In addton to combnatons of (1) and (), blateral wholesale contracts between the wholesale generators and the LSEs wthout thrd-party nterventon. (4) Multlateral contracts,.e., purchase and sale agreements between several sellers and buyers, possbly wth the nterventon of thrd partes such as forward contractors or brokers. In both (3) and (4) the prce-quantty trades are up to the market partcpants to decde, and not the ISO. The role of the ISO n such a scenaro s to mantan system securty and carry out congeston management. The contracts, thus determned by the market condtons, are among the system nputs that drve the power system. The transactons resultng from such contracts may be treated as sets of power nectons and extractons at the seller and buyer buses, respectvely. For example, n a system of n buses, wth the generator buses numbered from 1 to m, the nodal actve powers may be represented as [14] where P D K P, k K = Ppo, + PT + loss compensaton, =1,, m (.1) K D = D po + DT K, k K = actve nected power at generator bus = actve extracted power at load bus = set of blateral / multlateral transactons,, = m+1, n (.) P po,i = pool power nected at bus D po, = pool power extracted at bus P Tk,I = power nected at bus n accordance wth transacton T K D Tk, = power extracted at bus n accordance wth transacton T K Loss compensaton = power suppled at bus by all transacton partcpants to make good the transmsson losses. 6

14 .4 Congeston Management Methodologes There are two broad paradgms that may be employed for congeston management. These are the cost-free means and the not-cost-free means [15]. The former nclude actons lke outagng of congested lnes or operaton of transformer taps, phase shfters, or FACTS devces. These means are termed as cost-free only because the margnal costs (and not the captal costs) nvolved n ther usage are nomnal. The not-cost-free means nclude: (1) Reschedulng generaton. Ths leads to generaton operaton at an equlbrum pont away from the one determned by equal ncremental costs. Mathematcal models of prcng tools may be ncorporated n the dspatch framework and the correspondng cost sgnals obtaned. These cost sgnals may be used for congeston prcng and as ndcators to the market partcpants to rearrange ther power nectons/extractons such that congeston s avoded. () Prortzaton and curtalment of loads/transactons. A parameter termed as wllngness-to-pay-to-avod-curtalment was ntroduced n [14]. Ths can be an effectve nstrument n settng the transacton curtalment strateges whch may then be ncorporated n the optmal power flow framework. In the next chapter we look at OPF formulatons ncorporatng both (1) and () above. These models can be used as part of a real-tme open access system dspatch module [16]. The functon of ths module s to modfy system dspatch to ensure secure and effcent system operaton based on the exstng operatng condton. It would use the dspatchable resources and controls subect to ther lmts and determne the requred curtalment of transactons to ensure uncongested operaton of the power system..5 Example of Congeston Management n an Economc Dspatch Framework We now look at an example of calculatng optmal bus prces and congeston costs for a power system, wheren an ndependent company (ISO) controls the transmsson network and sets nodal prces that are computed as part of a centralzed dspatch. A smple power system s consdered here for the calculaton of congeston charges. A three-bus system s shown n Fgure.1 wth generator cost/margnal cost and load beneft/margnal beneft 7

15 functons as shown. Also shown n the fgure are the maxmum lne flow lmts and lne susceptances. C 1 = P 1 $/hr MC 1 = P 1 $/MWhr G 1 G 1 P max = 5 MW b 1 = - p.u. C = 3P $/hr MC = 3.34P $/MWhr P max = 15 MW b 13 = -1.0 p.u. P max = 15 MW b 3 = -1.5 p.u. 3 B 3 = -55P 3 $/hr MB 3 = -55 $/MWhr Fgure.1 Sample power system For smplcty we make the followng approxmatons: (1) Each transmsson lne s represented by ts susceptance b. () A lossless DC power flow model s assumed;.e., the bus voltage angular dfferences are assumed to be small and the voltage magntudes approxmately 1.00 p.u. The real power flow on each lne s gven by P = δ δ ) (.3) b ( where δ and δ represent the voltage angles at buses and, respectvely. The total power necton at bus s gven by P = P (.4) As mentoned above, we solve ths problem n a centralzed dspatch framework where the obectve s to maxmze socal beneft. Ths optmzaton problem thus seeks to mnmze the system operatng costs mnus the consumer beneft, subect to the bndng 8

16 lne flow nequalty constrants and the power flow equalty constrants. The problem nvolves solvng a quadratc Lagrangan (quadratc n the decson varables and multplers). The varables are gven by z = [ P, δ, λ, µ ] (.5) where P denotes the net power nectons at all the buses δ denotes the voltage angles λ denotes the Lagrangan multplers for the equalty constrants µ denotes the multplers for the nequalty constrants. The problem may be thus stated as mn{ C1( P1 ) + C ( P ) B3 ( P3 )} P, δ (.6) subect to P = δ δ (.7) 1 3 P = 3.5δ 1.5δ (.8) 3 P = 1.5δ +.5δ (.9) 3 3 P P, P P, P P (.10) max max max In ths example, the nequalty constrant lmtng the flow on lne 1- s taken as bndng. The Lagrangan functon for ths problem may be gven as l = P + 3P + 55 P + λ ( δ δ P) + λ (3.5δ 1.5 δ P) + λ ( 1.5δ +.5 δ P) µ ( δ 5) (.11) 1 The optmalty condton s gven by l = 0 (.1) z and 9

17 T 1 T l l l ( z) = z z + z (.13) z z z=0 From equatons (.1) and (.13), t can be seen that the optmal value of z may be obtaned by solvng l l z = z z z = 0 Solvng the problem n the above example yelds the followng optmal values: (.14) z = [ ] T (.15) The Lagrange multplers λ = [ ] T can be nterpreted as the optmal nodal prces at each of the three buses n $/MWhr. In other words, f these had been used as the bus prces, the generator and load responses to these prces would have been the same as what was obtaned n the above optmal dspatch. We now compute the congeston charges (for the flow on each transmsson lne). The congeston charge may be looked upon as the nherent cost of transmttng power across the lne. A smple way to compute ths s gven here. The congeston charge c for lne s the dfference n the congeston costs c and c at buses and, respectvely;.e., c = c - c, (.16) Now, each bus nodal prce λ s made up of three components, vz., the margnal cost of generaton at the slack bus, the margnal cost of losses, and the congeston cost. Hence, C1 ( P1 ) P1 λ = + c (.17) P P where C 1 (P 1 ) s the cost functon at bus 1, whch has been consdered as the slack bus n ths example. 1 We have consdered the lossless case n ths example. Hence we have, c = λ - λ, (.18) 10

18 Thus the congeston charge for any lne may be computed as the dfference n the nodal prces between buses and. The values obtaned n ths problem are c 1 = $/MWhr, c 3 = 6.58 $/MWhr, c 13 = $/MWhr..6 Congeston Management Usng Prcng Tools In [15], Glavtsch and Alvarado dscuss congeston prcng as may be done by an ISO n the absence of nformaton on the margnal costs of the generators. The methodology suggested nvolves observng the behavor of generators under a varety of condtons, based on whch quadratc coeffcents for all generators may be nferred. In [17], Bhattacharya et al. dscuss the method of market splttng to allevate transmsson congeston. The basc prncple of ths method les n sendng prce sgnals that ether exceed or are less than the margnal costs to generators and thereby affectng a change n the generaton pattern. The market s splt nto dfferent bd areas and the area-prces are calculated for each bd area usng a capacty fee. In the next chapter we work out dfferent OPF formulatons n the varous market modes dscussed earler. 11

19 3 OPTIMAL DISPATCH METHODOLOGIES IN DIFFERENT MARKET STRUCTURES 3.1 Introducton In ths chapter, we look at ways of managng the power dspatch problem n the emergng electrcty market structures. The operatng strateges that may be used by the ISO n dfferent market modes have been explored and test cases have been studed to determne the compatblty of the strateges wth the market envronment. Emphass s placed on dealng wth congeston management. The conventonal OPF problem comprses schedulng the power system controls to optmze a gven obectve functon under a set of nonlnear nequalty constrants and equalty constrants. Under a deregulated envronment, mechansms for competton and tradng are created for the market players. Ths leads to the ntroducton of new OPF controls. In ths chapter we look at how to deal wth these controls. The fundamental entty n all compettve market structures s an ISO. Successful tradng requres that the ISO match the power bds from the supply sde (gencos) wth the offers from the demand sde (dscos). Ths s true for all market structures. The mportant way n whch market structures dffer s n the manner of the man contractual system that s followed by the market players on both the supply and demand sdes. We look at two dfferent market modes, vz., pool dspatch and blateral dspatch. 3. Pool Dspatch 3..1 Pool structure Interconnected system operaton becomes sgnfcant n a deregulated envronment. Ths s because the market players are expected to treat power transactons as commercal busness nstruments and seek to maxmze ther economc profts. Now when several gencos decde to nterchange power, complcatons may arse. An economc dspatch of the nterconnected system can be obtaned only f all the relevant nformaton, vz., generator curves, cost curves, generator lmts, commtment status, etc., s exchanged 1

20 among all the gencos. To overcome ths complex data exchange and the resultng nonoptmalty, the gencos may form a power pool regulated by a central dspatcher. The latter sets up the nterchange schedules based on the nformaton submtted to t by the gencos. Whle ths arrangement mnmzes operatng costs and facltates system-wde unt commtment, t also leads to several complextes and costs nvolved n the nteracton wth the central dspatcher. Conventonally, the optmal operaton of a power system has been based on the economc crteron of loss mnmzaton,.e., maxmzaton of socetal beneft. Pool dspatch follows the same crteron but wth certan modfcatons necesstated by the coexstence of the pool market wth a short-term electrcty spot market. Namely, these effects are demand elastctes and the varaton n the spot prce wth the purchaser s locaton on the grd. The exstence of the spot market or blateral market behnd the scene does not explctly affect the operaton of the ISO. 3.. Pool dspatch formulaton Neglectng the effects of prce elastctes and locaton, the dspatch formulaton may be stated as mn C ( PG ) B ( PD ) (3.1) P, P G D subect to g( x, u) = 0 h( x, u) 0 (3.) where g and h are the sets of system operatng constrants, ncludng system power flow equatons and lne flow lmts u s the set of control varables, vz., actve powers at the generator and load buses x s the set of dependent varables and are the set of gencos and dscos, respectvely Ths OPF uses the bds and offers submtted by the partcpants and sets the nodal prces (that are obtaned as the Lagrangan multplers), whch are n turn used to charge for the power consumpton at every node. The vectors of generaton and load are denoted as P G 13

21 and P D, respectvely. The nodal prces appled to the generaton and load controlled by players and are obtaned as a byproduct of the OPF and are represented as λ and λ, respectvely. The cost and beneft functons of each generator and load are denoted by C and B, respectvely. The cost and beneft functons are assumed to be well descrbed by quadratc functons. C ( P ) = a P + b P + c, G (3.3) D G G, G G, G G, B ( P ) = a P + b P + c, D (3.4) D, D D, D D, where G represents the set of all gencos and D represents the set of dscos. The equalty constrant may be wrtten as P D where L s the transmsson loss functon. P G + L = 0 (3.5) The capacty constrant (nequalty) may be gven as P P 0 (3.6) G G, max Problem (3.1) leads to the soluton and Kuhn-Tucker condtons gven as B P D p L hk λ ( 1+ ) + π k = 0 P P D k D C P G L hk λ ( 1 ) µ π k = 0 P P G k G µ P P ) 0 and µ 0 ( G G, max = π = 0 and π 0 (3.7) k h k where λ represents the system ncremental cost (dual multpler on the equalty constrant) and µ and π represent the sets of Kuhn-Tucker dual varables on the capacty and operatng constrants, respectvely. k 14

22 3..3 Example of correctve reschedulng n pool dspatch When the system s nsecure and there are volatons n the system, the obectve of the pool central dspatcher s to elmnate the system overload and come up wth the correctve reschedulng to elmnate the volatons as fast as possble. Mnmum operatng cost, mnmum number of controls, or mnmum shft from the optmum operaton may be used as the obectve functon. We now look at an OPF example where the obectve functon s to mnmze the reschedulng of generaton. 70 MW 0.06 G 3 1 G MW 30 MVar MW 80 MW 0 MVar 10 MW 60 MVar G 100 MW 10 MW 30 MVar 5 Fgure 3.1 Three-generator fve-bus system Consder a fve-bus system as shown n Fgure 3.1. The system data s gven n Table 3.1 Table 3.1 Bus data for Fgure 3.1 Bus number Load MW MVar Gen MW Gen mn MW Gen max MW Voltage setpont Cost ($/MWhr) 1 (slack)

23 Table 3. Lne data for Fgure 3.1 From bus To bus p.u. mpedance MVA ratng Base case power flow(mw) The base case power flow for the system shows (Table 3.) that congeston occurs on lne 1-. The am s to reschedule generaton to remove ths congeston and any other nduced congeston. We frst compute the senstvtes of lne flow P k to changes n generaton P G1, P G, P G4. For that we use the chan rule: P P k G P Pk f = θ θ T 1 f P P G (3.8) where p f represents the power flow equaton at bus, whch s gven as 1 ( θ θ ) ( PG PD ) = 0 (3.9) x In matrx formulaton the power flow equaton s θ = B 1 P, where B s the bus susceptance matrx computed from the lne mpedance data. Fxng bus 1 as the slack, we can then get the equatons for lne flows and the lne flow senstvtes to generaton. The sum of all the products of lne flow senstvtes wth changes n generaton (reschedulng) gves the overload n that partcular lne. 16

24 In ths partcular example, the obectve s to mnmze the reschedulng of generaton requred to lmt the flow on lne 1- to 150 MVA. The OPF problem can then be gven as subect to mn( P + P + P + P + P + P ) (3.10) G1 G1 G G P P + P P + P P = 0 (3.11) G1 G1 G G G4 G4 G4 G4 and P P 1 G + T P1 + T [ P P ] + [ P P ] = G G P G 4 G 4 G 4 (3.1) where 0.47 s the overload on lne 1-. Ths OPF problem can be solved to mnmze the reschedulng of generaton. We get the result that bus 1 must drop ts generaton by 56. MW, bus must rase ts generaton by 5.37 MW, and bus 4 must rase ts generaton by 3.88 MW; G 1 P = 56. MW + G + G 4 P = 5.37 MW P = 3.88 MW (3.13) 3.3 Blateral Dspatch Blateral market structure The conceptual model of a blateral market structure s that gencos and dscos enter nto transacton contracts where the quanttes traded and the prces are at ther own dscreton and not a matter for the ISO;.e., a blateral transacton s made between a genco and a dsco wthout thrd party nterventon. These transactons are then submtted to the ISO. In the absence of any congeston on the system, the ISO smply dspatches all the transactons that are requested, makng an mpartal charge for the servce. 17

25 3.3. Blateral dspatch formulaton In a blateral market mode, the purpose of the optmal transmsson dspatch problem s to mnmze devatons from transacton requests made by the market players. The goal s to make possble all transactons wthout curtalments arsng from operatng constrants. The new set of rescheduled transactons thus obtaned wll be closest to the set of desred transactons, whle smultaneously satsfyng the power flow equatons and operatng constrants. One of the most logcal ways of reschedulng transactons s to do t on the bass of ratonng of transmsson access. Ths may be modeled as a user-pay scheme wth wllngness-to-pay surcharges to avod transmsson curtalment. The mathematcal formulaton of the dspatch problem may then be gven as mn f ( x, u) where f ( u, x) ] o T o T T = [( u u ) A] W [( u u ) A (3.14) subect to g( x, u) = 0 h( x, u) 0 where W s a dagonal matrx wth the surcharges as elements A s a constant matrx reflectng the curtalment strateges of the market partcpants u and u o are the set of control varables, actual and desred x s the set of dependent varables g s the set of equalty constrants, vz., the power flow equatons and the contracted transacton relatonshps, h s the set of system operatng constrants ncludng transmsson capacty lmts The blateral case can be modeled n detal. We consder transactons n the form of ndvdual contracts where a seller nects an amount of power T at one generator bus and the buyer extracts the same amount at a load bus. Let the power system consst of n buses wth the frst m assumed to be seller buses and the remanng n-m as buyer buses. One partcular bus (bus 1) may be desgnated as the slack to take nto account 18

26 transmsson losses. The total power nected/extracted at every bus may be gven by the summaton of all ndvdual transactons carred out at those buses. Thus, for = to m, P = T, and for = m+1 to n, P = (3.15) T The transactons T also appear n the power flow equalty constrants snce they act as the control varables along wth the usual generator bus voltages. The set of control T varables can thus be represented as u = { T, V}, where V s the vector of generator bus voltages. The real and reactve power flow equatons can be wrtten n the usual form represented by g ( x, u) = 0 The transacton curtalment strategy s mplemented by the ISO n collaboraton wth the market partcpants. In the case of blateral dspatch, ths strategy concerns the ndvdual power contracts. One such strategy s such that, n case of an ndvdual contract, the curtalment of the transacted power nected at the genco bus must equal the curtalment of the transacted power extracted at the dsco bus. In ths case, we may rewrte the dspatch formulaton as mn f ( x, u) where f m n 0 ( x, u) = w ( T T = = m+ 1 ) (3.16) where w = the wllngness to pay factor to avod curtalment of transacton 0 T = the desred value of transacton T 19

27 3.3.3 Test results We consder a sx-bus system representng a deregulated market wth blateral transactons. An OPF wll be solved for ths system to determne the optmal generaton schedule that satsfes the obectve of mnmzng devatons from the desred transactons. Table 3.3 provdes the system data pertanng to generaton and load. Table 3.4 provdes the system network data. Fgure 3. shows the system network confguraton. Buses 1 and are genco buses and, beng PV buses, the voltages here are specfed exactly. At the other buses, the allowable upper and lower lmts of voltage are specfed. The losses are assumed to be suppled only by the generator at bus 1. G G Fgure 3. Two-generator sx-bus system Table 3.3 System data Bus Generaton capacty, MW Generator cost characterstc, $/hr Voltage, pu P P +.5P P 00.4P + 5.5P V V V V

28 Table 3.4 System network data From bus to bus Resstance, pu Reactance, pu Lne chargng admttance, pu In ths case, blateral contracts have been consdered between each genco and each dsco. Table 3.5 shows the desred power transactons. Table 3.5 Desred transactons before curtalment Bus # Desred transactons, MW Three strateges for the curtalment of transactons are adopted for congeston management: (1) The curtalment on the dsco loads s assumed to be lnear. In ths case, all the wllngness to pay factors are taken to be equal. () Same as case (1), except that the wllngness to pay prce premum of loads on buses 1 to 3 s assumed to be twce that of loads on buses 4 to 6. 1

29 (3) In ths case, the prce premum of loads on buses 4 to 6 s assumed to be twce that of loads on buses 1 to 3. The OPF problem s solved usng the MINOS-5.0 nonlnear programmng solver n the Generalzed Algebrac Modelng Systems (GAMS) programmng envronment [18]. Table 3.6 shows the constraned generaton and load data obtaned from the OPF soluton. It can be seen that the wllngness to pay and the partcpants curtalment strategy are two factors that sgnfcantly affect the constraned dspatch. The hgher the wllngness to pay, the less s the curtalment of that partcular transacton. The curtalment strateges mplemented have complex effects. These factors not only affect the curtalment of ts own transacton, but wll also mpact that of other transactons. Table 3.6 Constraned generaton and load data after runnng OPF Bus # Constraned generaton and load, MW Case (1) Case () Case (3) Treatment of Transacton-Based Groups In a compettve market scenaro, relatonshps among market players may develop over tme and may lead to the formaton of electrcty supply and consumpton groups. The concept of a group as a collecton of buyers, sellers, and market brokers functonng together n a cohesve manner has to be dealt wth. The formaton of such transactonbased groups n a power system necesstates changes n power dspatch. In the followng sectons we look at dspatch formulatons takng nto account the group concept.

30 3.4.1 Dspatch formulatons Here the concern s to make possble a group transfer wthout curtalment, even f the ndvdual generators wthn the group or utlty have to be rescheduled. The obectve functon s mn f ( x, u) where K m 0 f ( u, x) = [ w ( T T ) ] (3.17) k k = 1 = k m = k where w k = the wllngness to pay factor to avod curtalment of the kth group transacton 0 T k = the desred value of transacton T In ths group curtalment dspatch formulaton, there s the need to develop a strategy to allocate the total group power curtalment among all the group partcpants. That s, f the genco powers wthn a group need to be curtaled, the resultng shortfall has to be allocated to all the group dscos n accordance wth some predetermned strategy. Another way of mplementng curtalment of group transactons s by mnmzng the change to every nected or extracted power transacton at the generator bus and load bus of a group based on the wllngness to pay factors. In ths case, the obectve functon may be expressed as where mn f ( x, u) K m 0 (, ) = [ k ( k k ) ] k= 1 = f u x w T T (3.18) where w = the wllngness to pay factor to avod curtalment of the nected power block T k. k In ths optmal transmsson dspatch problem, all power transactons are requred to be as close as possble to the ntal desred power transfers, and the curtalment decsons are 3

31 based on the market players wllngness to pay to avod curtalment, ther preferred curtalment strateges, and on the system securty condtons. The dspatch procedure starts wth the market partcpants submttng ther multlateral transactons to the ISO. If the operatng and capacty constrants are satsfed whle all the desred transactons are dspatched, there s no need to go through the curtalment routne. Otherwse the optmal dspatch models descrbed above (Sectons 3.., 3.3., 3.4.1) are used to curtal the requested power transfers. Fnally, the orgnal/curtaled power transfers are dspatched and the ISO buys the requred regulatng power at bus 1 to compensate for transmsson losses Test case We now look at an optmal transmsson dspatch problem n a deregulated market havng transacton-based groups. We consder the IEEE 14-bus system here (Fgure 3.3) G G TR TR- 9 6 TR-3 5 G 1 G G Fgure 3.3 IEEE fve-generator fourteen-bus system Some slght modfcatons are made. Bus 4 s renumbered as bus 1 and t s assumed that ths bus s contracted by the system ISO to provde for the transmsson losses;.e., bus 1 4

32 s the system slack bus. Ths bus, n addton to bus 5, s usually shown connected to a synchronous condenser. But n ths problem, we treat bus 1 as a generator bus owned by a genco. Smlarly, bus 5 s treated as a PV-bus n the problem. Table 3.7 provdes the generaton bus data. Table 3.8 provdes the system network data. The voltages at the genco buses are specfed snce they are P-V buses, whereas at the dsco buses, the allowable upper and lower lmts of voltage are specfed. Table 3.7 Generaton bus data Bus Generaton capacty, Generator cost Voltage, pu MW characterstc, $/hr P P P P P P P P +.45P Table 3.8 System network data From bus to bus Resstance, pu Reactance, pu Lne chargng admttance, pu

33 Table 3.8 (cont.) We now assume that there are two groups n ths power system: Group 1 conssts of buses and 3 and makes transfers to dsco buses 7, 9, 11, and 14. Group conssts of the sngle genco bus 4 and makes tranfers to dsco buses 8, 10, 1, and 13. Table 3.9 shows the desred power generaton and load for both groups. Table 3.9 Desred generaton and load before curtalment Bus # Pre-curtalment MW

34 It s seen from the power flow soluton that the dspatch of the contracted transactons wthout any curtalment leads to overloadng of the lnes between buses 3 and 11, and buses 7 and 9. Therefore, to remove ths congeston and to ensure that the system securty lmts are not volated, the ISO needs to curtal the power transactons The followng four strateges for the curtalment of transactons are adopted for congeston management. The results are shown n Table (1) Both groups 1 and employ the group curtalment formulaton as descrbed by (3.17). The curtalment on the dsco loads s assumed to be lnear. The total group power curtalment s taken as a lnear combnaton of the ndvdual dsco curtalments. In ths case, all the wllngness to pay factors are taken to be equal to unty. () Same as case (1), except that the wllngness to pay prce premum of the players n group s assumed to be twce that of the players n group. (3) In ths case, group 1 employs the curtalment strategy gven n (3.17), whereas group adopts the curtalment formulaton descrbed n (3.16). Wllngness to pay premums are mantaned at unty. (4) Same as case (3), except that the wllngness to pay premums on the transactons between buses 4 and 10, and buses 4 and 1, are doubled. Table 3.10 shows the constraned generaton and load data obtaned from the OPF solutons usng the four curtalment strateges. 7

35 Table 3.10 Constraned generaton and load data after runnng OPF Bus # Constraned generaton and load, MW Case (1) Case () Case (3) Case (4) (group #1) (genco) (genco) (group #) 4 (genco) (loss compensator) 1 (genco) The optmal dspatch gves an uncongested system soluton (Table 3.10);.e., all the lne overloads are removed. In case (1), both the groups use the same curtalment strateges wth dentcal wllngness-to-pay factors, and ths results n all power transactons gettng curtaled n varyng degrees. In case (), the wllngness to pay of group 1 s ncreased. Ths does not lead to a proportonate reducton n the curtalment of the transactons n group 1 or a proportonate ncrease n the curtalment of transactons n group. In case (3), the use of two dfferent curtalment strateges for the two groups seems to affect some transactons more than others. For nstance, the transacton between buses 4 and 10, and buses 4 and 1, get relatvely heavly curtaled. Ths s remeded n case (4) where the wllngness to pay for both these pars of players s doubled. 8

36 3.5 Conclusons Ths chapter has focused on the dspatch curtalment problem n a compettve market scenaro. A framework for prce-based operaton under these condtons s explored and an optmal transmsson dspatch methodology s developed. The case studes show the complex nteractons between the market partcpants. 9

37 4 OPTIMAL DISPATCH USING FACTS DEVICES IN DEREGULATED MARKET STRUCTURES 4.1 Introducton In the prevous chapters we have looked at congeston management n deregulated power systems usng models that nclude prcng tools such as prortzaton and curtalment of transactons. In ths chapter we look at treatng congeston management wth the help of flexble AC transmsson (FACTS) devces. We consder an ntegrated approach to ncorporate the power flow control needs of FACTS n the OPF problem for allevatng congeston. Two man types of devces are consdered here, namely, thyrstor controlled seres compensators (TCSC) and thyrstor controlled phase angle regulators (TCPAR). The concept of flexble AC transmsson systems (FACTS) was frst proposed by Hngoran [19]. FACTS devces have the ablty to allow power systems to operate n a more flexble, secure, economc, and sophstcated way. Generaton patterns that lead to heavy lne flows result n hgher losses, and weakened securty and stablty. Such patterns are economcally undesrable. Further, transmsson constrants make certan combnatons of generaton and demand unvable due to the potental of outages. In such stuatons, FACTS devces may be used to mprove system performance by controllng the power flows n the grd. Studes on FACTS so far have manly focused on devce developments and ther mpacts on the power system aspects such as control, transent and small sgnal stablty enhancement, and dampng of oscllatons [0]-[3]. Here we look at solvng the OPF problem n a power system ncorporatng FACTS devces. As we have seen n the earler chapters, dfferent soluton approaches are possble to solve the OPF problem. The man conventonal control varables are the generaton MWs when the DC power flow model s used. Wth the ncreased presence of ndependent gencos n the deregulated scenaro, the operaton of power systems would requre more sophstcated means of power control. FACTS devces can meet that need. 30

38 4. Statc Modelng of FACTS Devces For the optmal power dspatch formulaton usng FACTS controllers, only the statc models of these controllers have been consdered here [4]. It s assumed that the tme constants n FACTS devces are very small and hence ths approxmaton s ustfed Thyrstor-controlled seres compensator (TCSC) Thyrstor-controlled seres compensators (TCSC) are connected n seres wth the lnes. The effect of a TCSC on the network can be seen as a controllable reactance nserted n the related transmsson lne that compensates for the nductve reactance of the lne. Ths reduces the transfer reactance between the buses to whch the lne s connected. Ths leads to an ncrease n the maxmum power that can be transferred on that lne n addton to a reducton n the effectve reactve power losses. The seres capactors also contrbute to an mprovement n the voltage profles. Fgure 4.1 shows a model of a transmsson lne wth a TCSC connected between buses and. The transmsson lne s represented by ts lumped π-equvalent parameters connected between the two buses. Durng the steady state, the TCSC can be consdered as a statc reactance -x c. Ths controllable reactance, x c, s drectly used as the control varable to be mplemented n the power flow equaton. Bus S R +X S Bus -x c B c B c Fgure 4.1 Model of a TCSC Let the complex voltages at bus and bus be denoted as V δ and V δ, respectvely. The complex power flowng from bus to bus can be expressed as S * = P Q = V * * = V [( V V ) Y + V ( B I c )] 31

39 * = V [ G + ( B + B )] V V ( G + B ) (4.1) c where G + B = ( R + X X ) (4.) 1 L L C Equatng the real and magnary parts of the above equatons, the expressons for real and reactve power flows can be wrtten as P = V G V V G cos( δ δ ) V V B sn( δ δ ) (4.3) Q = V ( B + B ) V V G sn( δ δ ) + V V B cos( δ δ ) (4.4) c Smlarly, the real and reactve power flows from bus to bus can be expressed as P = V G V V G cos( δ δ ) + V V B sn( δ δ ) (4.5) Q = V ( B + B ) + V V G sn( δ δ ) + V V B cos( δ δ ) (4.6) c The actve and reactve power loss n the lne can be calculated as P = P + P L = V G + V G V V G cos( δ δ ) (4.7) L Q = Q + Q = V ( B + B ) V ( B + B ) + V V B cos( δ δ ) (4.8) c These equatons are used to model the TCSC n the OPF formulatons. c 4.. Thyrstor-controlled phase angle regulator (TCPAR) In a thyrstor-controlled phase angle regulator, the phase shft s acheved by ntroducng a varable voltage component n perpendcular to the phase voltage of the lne. The statc model of a TCPAR havng a complex tap rato of 1:a α and a transmsson lne between bus and bus s shown n Fgure 4.. Bus S 1:a α R +X Fgure 4. Model of TCPAR S Bus 3

40 The real and reactve power flows from bus to bus can be expressed as and P V a V av Y * * = Re{ [( ) ]} = a V G avv G cos( δ δ + α) avv B sn( δ δ + α) (4.9) Q V a V av Y * * = Im{ [( ) ]} = a V G avv B cos( δ δ + α) avv G sn( δ δ + α) (4.10) Smlarly, real and reactve power flows from bus to bus can be wrtten as and P V V av Y * = Re{ [( ) ]} = V G avv G cos( δ δ + α) + avv B sn( δ δ + α) (4.11) Q V V av Y * = Im{ [( ) ]} = V B + avv B cos( δ δ + α) + avv G sn( δ δ + α) (4.1) The real and reactve power loss n the lne havng a TCPAR can be expressed as Pl = P + P = av G + V G VVG cos( δ δ + α) (4.13) Ql = Q + Q = av B V B + VVB cos( δ δ + α) (4.14) Ths mathematcal model makes the Y-bus asymmetrcal. In order to make the Y-bus symmetrcal, the TCPAR can be smulated by augmentng the exstng lne wth addtonal power nectons at the two buses. The nected actve and reactve powers at bus ( P, Q ) and bus ( P, Q ) are gven as P = a V G av V [ G sn( δ δ ) B cos( δ δ )] (4.15) P = av V G sn( δ δ ) + B cos( δ δ )] (4.16) [ Q = a V B + av V [ G cos( δ δ ) + B sn( δ δ )] (4.17) Q = av V G cos( δ δ ) B sn( δ δ )] (4.18) [ These equatons wll be used to model the TCPAR n the OPF formulaton. 33