A Secure-Coordinated Expansion Planning of Generation and Transmission Using Game Theory and Minimum Singular Value

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1 Internatonal Journal of Electrcal and Computer Engneerng (IJECE) Vol. 4, No. 6, December 2014, pp. 868~881 IN: A ecure-coordnated Expanson Plannng of eneraton and ransmsson Usng ame heory and Mnmum ngular Value Mehd Zarean Jahrom, Mohsen ajdnan, Mojtaba Jalalpour Amrkabr Unversty of ecnology (ehran Polytechnc) No. 424, Hafez Avenue, ehran , Iran Artcle Info Artcle hstory: Receved Jul 15, 2014 Revsed ep 15, 2014 Accepted Oct 10, 2014 Keyword: ame theory eneraton expanson plannng Mnmum sngular value ransmsson expanson plannng Voltage securty margn ABRAC Correspondng Author: Mojtaba Jalalpour, Amrkabr Unversty of ecnology (ehran Polytechnc) No. 424, Hafez Avenue, ehran , Iran Emal: Jalalpur@aut.ac.r A In ths paper a novel method have been proposed for expanson plannng of generaton and transmsson that consdered statc securty of the system such as voltage securty margn And loadablty lmt. In the same study of expanson plannng ecurty constrants of the system are neglected. In ths study at the frst step mnmum sngular value technque s used to evaluate voltage securty margn and loadabolty lmt, n order to select best bus for load ncrmnaton. After t, n order to upply the load, coordnated expanson plannng of generaton and transmsson s needed, therefor the strategc nteracton between transmsson company (ransco) and generaton company (enco) for ransmsson expanson plannng (EP) and generaton expanson plannng (EP) n a compettve electrcty market s proposed usng ame heory (). Copyrght 2014 Insttute of Advanced Engneerng and cence. All rghts reserved. 1. INRODUCION Power system restructurng and deregulaton, ntroduce new defntons and methods to the power system plannng. In the monopoly power market, the decson maker s just one organzaton whch decdes for the eneraton Expanson Plannng (EP) and ransmsson Expanson Plannng (EP) altogether. Due to emergence of competton n the power market, the decson makers of EP and EP become separated such that the transmsson company (ransco) decdes for EP and the generaton company (enco) decdes for EP. In such an envronment, the coordnaton between these two organzatons becomes more crucal as capacty expanson of each organzaton affects the other sde capacty expanson and as a consequence the proft of each company s affected. In a compettve power market wth open access to the transmsson system, the generaton company s expected to supply the load wthout any congeston n the transmsson lnes. ransmsson companes are oblged to provde a congeston-free, relable and non-dscrmnatve path for the generaton companes to the consumers of electrcty. herefore, the transmsson networks must be regulated so that optmal operaton of power system s performed. In a restructured power market, enco decdes on capacty, place and tme of constructon of new power plants at ts own dscreton. he capacty expanson strategy used by enco nvolves ransco and produces uncertanty and challenges to EP. On the other hand, there s no absolute certanty that a transmsson network can provde suffcent capacty for new generaton capacty constructed by a enco [1-5]. hs nteracton between enco and ranco leads to a new method of EP and EP that consders both enttes proft. In such an envronment, ame theory seems to be a useful method to predct the strateges of enco and ransco for generaton expanson capacty (EC) and transmsson expanson Journal homepage:

2 869 IN: capacty (EC) and evaluate the power market equlbrum [6-10]. EP and EP have been studed separately n many artcles but the nteracton between them n a deregulated market s studed n a few papers. In [11] n order to study the relatonshp between generaton and transmsson nvestment, a threestage model s presented. A study n [12] shows that n a deregulated power market, the degree of competton between dfferent generators s dependent on the capacty of transmsson lnes. he nteracton between transmsson and generaton expanson plannng usng game theory, s studed n a three-bus system n [13]. o study the strategc nteracton between enco and ransco, a sngle-stage determnstc model s proposed n [14]. he expanson behavors of both enco and ransco are smulated usng Cournot model. he equlbrum n the power market n [14] s obtaned usng Mxed Complementarty Problem approach and the proposed model s appled to a three-bus system and the IEEE 14-bus system. On the other hand, power system securty whch s the ablty of the power system to wthstand dsturbances aganst any volaton n system operatng condtons should be consdered n ths new method of capacty expanson. he aforementoned studes n the feld of generaton and transmsson expanson plannng haven t consdered power system securty. However n ths paper, frst the load pattern of a sx-bus power system s mproved and then the best bus for load ncrement s determned usng a senstvty characterstc of ANN. Afterwards the strategc nteracton between transmsson company (ransco) and generaton company (enco) for EP and EP n a compettve electrcty market s proposed usng ame heory. It should be taken nto consderaton that the load dscussed n ths paper s a manageable load whch can be ncreased or decreased usng reward or penalty. he paper s organzed as follows. In secton II, the loadablty lmt as a securty ndex s ntroduced. he neural network used for the mprovement of load pattern s presented n secton III. Applcaton of Cournot model of duopoly for EP and EP s dscussed n secton IV. ecton V proposes ame heory for solvng EP and EP problem. A case study s presented n secton VI. Fnally, conclusons are presented n secton VII. 2. LOADABILIY LIMI AND VOLAE ECURIY MARIN A A ECURIY INDEX In order to study the statc voltage stablty of the power system, loadablty lmt of system s proposed as voltage stablty ndex for system securty evaluaton. Loadablty lmt of a power system s defned as the maxmum load, whch can be mposed on buses of a system wthout loss of voltage stablty. the mnmum sngular value technque s used to evaluate voltage securty margn for each state of loadng. he mnmum sngular value of the load flow Jacoban matrx s obtaned from solvng the load flow equatons that are shown n (1) and (2): n j j j j j 1 P V V Y cos (1) n j j j j j 1 sn (2) Q V V Y Where: P : Actve power transfer from bus I; Q : Reactve power transfer from bus ; V : Voltages of bus and bus j;, V j Y : Admttances from bus to bus j; j j : angle of mpedances from bus to bus j; Y j : Admttances from bus to bus j;, j : angle of voltage at bus and bus j. And: P J1 J2 Q J3 J 4 V (3) P : mall devaton n actve power; Q : mall devaton n reactve power; IJECE Vol. 4, No. 6, December 2014 :

3 IJECE IN: J1 J : Jacoban matrx elements of load flow equatons; 4 : mall devatons n voltage angle; V : mall devatons n voltage magntude. 3. MINIMUM INULAR VALUE One of the effectve methods of determnaton of voltage collapse pont s mnmum sngular value of the load flow Jacoban matrx. In ths paper, mnmum sngular value technque s used to selectve best bus for loadng. On hs way load ncreases n desred bus when load s fxed at another bus then calculate voltage securty margn wth approach technque. When voltage securty margn s calculated for all of the state, the state wth maxmum VM s the best state for loadng. Flowchart of ths process has been shown n fgure 1. Equaton (3) can be wrtten by equaton (4). Under normal operatng condton, applcaton of sngular value decomposton s appled to Jacoban matrx that has been shown n equaton n (5) to (9): P J P (4) 2( n 1) J u v (5) j 1 j j j 2( n 1) 1P 1 P J j V ju j V j 1 (6) 1 P 2( n1) V2( n1) u2( n1) V P U Let 2( n 1) (7) (8) hen V V 2( n 1) 2( n 1) (9) Where: n: number of buses n the power network, u j,v j : sngular vectors that u j and v j are the j th columns of untary matrx, : Postve real sngular values. Above analyss clearly shows where the mnmum sngular value of the load flow Jacoban matrx s almost zero, ths sutable ndcator detects the closeness of power system operatng condton to the voltage collapse pont. A ecure-coordnated Expanson Plannng of eneraton and ransmsson Usng (Mojtaba Jalalpour)

4 871 IN: Fgure 1. Flowchart of best bus electon for loadng 4. APPLICAION OF COURNO DUOPOLY MODEL O EP AND EP PROBLEM In the Cournot duopoly model, two companes produce a homogenous product and wthout knowng the decson of the other one, they must decde how much product they should produce to obtan the maxmum proft [18]. As the Cournot model s somehow smlar to the EP and EP behavors, t can be appled to the problem of EP and EP n the restructured power market. he frst smlarty between the cournot model and EP and EP problem s the expanson capacty of EP and EP, whch s quantty n Cournot model. he second smlarty s that n a pool market the product of each generaton unt s a homogenous product whch s offered by enco at each supply bd. he Cournot model maxmzes the proft of each company and defnes the amount of ther outputs. In terms of mathematcal formulas, the optmal * * quantty par ( q, q ) s the Coumot equlbrum, f for frm 1, q solves (10): 1 2 * 1 max ( q, q ) (10) * Where: : Proft for frm ; =1, 2; q : Quanttes produced by frm ; q : Optmal quanttes produced by frm. * he proft functon for frm 1 can be represented by (11): IJECE Vol. 4, No. 6, December 2014 :

5 IJECE IN: ( q, q ) p( q q ) q c ( q ) (11) Where: p (.) : Market prce for aggregate quantty; C (.) :cost functon for frm. 5. APPLYIN AME HEORY O EP AND EP PROBLEM 5.1. Assumptons In order to formulate the strategc nteracton between enco and ransco n the expanson plannng game, some assumptons are consdered. ransco s owner of all transmsson lnes. Power market structure s Poolco-type n whch enco offers the prce and quantty bds to a Independent ystem Operator (IO). hen, the IO dspatches the cheapest power consderng operatonal constrants lke transfer capacty lmtatons and energy balance constrants. Demand s consdered as a constant load. Expanson strateges of ransco and enco are dscrete. hs means that ransco can ether expand ts lne capacty or mantan the ntal transfer capacty of the lne. Expanson behavors of ransco and enco are based on Cournot model Problem Formulaton As proft maxmzaton s the man purpose of each sde of the game theory and proft s the dfference between revenues and costs, the revenue of the ransco for each lne s gven by a congeston charge of that lne [19] whch s the dfference between local margnal prces (LMPs) [20-21]. he sum of congeston charges of each lne s total revenue of ransco. In order to make the problem more smple, ransco total cost s not consdered. o, proft of the ransco can be expressed as (12): (12) ( ) P j j j Where: : proft of the ransco n $/hr; : LMP at node n $/MWhr; j : LMP at node j n $/MWhr; P j : Actve power flow from node to node j n MW. As EP s performed at node, he enco proft from EP s obtaned usng (13): P c( P ) ( ) (13) Where: : proft of enco n $/hr; : LMP at node n $/MWhr; P : actve power generaton n MW; cp ( ): quadratc cost functon of actve power generaton n $/hr; ( ): nvestment cost n terms of generaton expanson capacty n $/hr oluton Methodology he followng notatons are used n the soluton methodology. : th expanson strategy of EP; = 1,2,...,n; j : j th expanson strategy of EP; j=1,2,..,n; :transmsson expanson capacty (EC) n MW; A ecure-coordnated Expanson Plannng of eneraton and ransmsson Usng (Mojtaba Jalalpour)

6 873 IN: : eneraton expanson capacty (EC) n MW; 0 : ntal EC n MW; 0 : ntal EC n MW; : ncrement n EC n MW; : ncrement n EC n MW; : maxmum EC n MW; : maxmum EC n MW. Algorthm of fndng the Nash equlbrum among all possble combnatons of expanson strateges s shown n flowchart of Fgure 7 and the Cournot equlbrum s found by usng an teratve search procedure [22] as shown n Fgure CAE UDY he proposed algorthm s appled to a sx-bus system shown n Fgure 9. he Data of the sx-bus system s gven n ables 1 to 3. In the proposed method, frst the most approprate bus for load ncrement s selected usng mnmum sngular value technque. he mnmum sngular value technque shows bus 2 s the best and the most sutable bus for load ncremental because ths state has maxmum VM, ths ssue s clearly shown n Fgure 2. Afterwards, n order to supply the ncremented load n the selected bus, ame heory s used to study the strategc nteracton between the EP and EP. Fgure 2. VM and Mn ngular Value of system when load ncrease at bus2 IJECE Vol. 4, No. 6, December 2014 :

7 IJECE IN: Fgure 3. VM and Mn ngular Value of system when load ncrease at bus3 Fgure 4. VM and Mn ngular Value of system when load ncrease at bus4 A ecure-coordnated Expanson Plannng of eneraton and ransmsson Usng (Mojtaba Jalalpour)

8 875 IN: Fgure 5. VM and Mn ngular Value of system when load ncrease at bus5 Fgure 6. VM and Mn ngular Value of system when load ncrease at bus6 IJECE Vol. 4, No. 6, December 2014 :

9 IJECE IN: (, ) Fgure 7. Flowchart of the strategc nteracton between ransco and enco Fgure 8. Flowchart of the Cournot soluton algorthm A ecure-coordnated Expanson Plannng of eneraton and ransmsson Usng (Mojtaba Jalalpour)

10 877 IN: Fgure 9. x-bus system enerator able 1. enerator Data Mn eneraton Power Mn eneraton Power able 2. Lne Data From Bus o Bus R (pu) X (pu) ransmsson Capacty able 3. Bus Data Bus Voltage P Bus ype gen P load λ Number (pu V) (pu MW) (pu MW) ( $/MWhr) 1 wng en en Load Load Load A. electon of the best bus for load ncrement As shown n Fgure 2 the best loadablty lmt and the most approprate bus for load ncrement, s bus 2 because n ths state loadablty lmt ( P max ) and mnmum egenvalue of Jacobean matrx ( mn ) of power flow equatons that s presented n (14) are the most secure state among other states. IJECE Vol. 4, No. 6, December 2014 :

11 IJECE IN: Pmax 850MW mn 0.3 (14) B. ransmsson and eneraton Expanson Plannng After selecton of the best bus for load ncement, n order to supply the load, ame heory studes the strategc nteracton between transmsson expanson plannng (EP) and generaton expanson plannng (EP). Under normal operaton condton, generator 1 output s 76.5 MW and generator 2 and 3 generate ther maxmum allowable output power. upposng the 40 MW load ( =40 MW) nstalled at bus 2 and as the capacty of lne1-2 s 15.28MW, a lne congeston occurs. herefore, n order to supply ths load, transmsson capacty expanson between buses 1 and 2 or generaton capacty expanson of generator at bus 2, s requred. As a consequence, a competton occurs between ransco for the capacty expanson of lne 1-2 and enco for the generaton expanson capacty of generator at bus 2. able 4 shows the expansons strateges of the two players (ransco & enco) n ths case. Equaton (12) represents the proft of ransco due to congeston n transmsson lnes, and (13) represents the proft of enco due to expanson of generator capacty at bus 2 and cp ( 2 ) for generator 2 s presented n (15). cp ( ) P P (15) , , 200 and ( ) 200$ / hr In the next step, each of these strateges s appled to the system and at each step, the maxmum proft of each company s obtaned. Fnally, the most proftable strategy for both player usng ame heory and Nash equlbrum s obtaned. able 4. Expanson trateges of ransco and enco ymbol Expanson trategy 1 ransco wll not expand lne ransco wll expand lne enco wll not expand lne enco wll expand lne 1-2 1) 1 st 1 1 Expanson scenaro (, ) In ths scenaro, nether ransco nor enco expands ts capacty, so there s no revenue or cost for enco and ransco gans proft from the congeston charges. profts of the two players n ths strategy are gven n able 5. 2) 2 nd 2 1, Expanson scenaro ( ) ransco wll expand lne 1-2 capacty from MW to MW n steps of 5 MW. he EC wth maxmum proft for ransco s MW wth proft of $/hr. Once agan there s no proft for enco n ths scenaro. he profts of ransco and enco n ths scenaro are gven n able 6. able 5.Profts of wo Players n the 1 st Expanson cenaro Lne 1-2 capacty otal proft of ransco ($/hr) EC n bus 2 Proft of enco ($/hr) A ecure-coordnated Expanson Plannng of eneraton and ransmsson Usng (Mojtaba Jalalpour)

12 879 IN: able 6. Profts of wo Players n the 2 nd Expanson cenaro Lne 1-2 EC otal proft of ransco ($/hr) EC n bus 2 Proft of enco ($/hr) ) 3 rd 1 2 Expanson scenaro (, ) enco wll perform EP n bus 2 and the EC range s from 68 MW to 108 MW n steps of 5 MW. he EC wth maxmum proft for the enco s 30 MW wth proft of $/hr. In ths scenaro, ransco gans proft from congeston charges. he profts of ransco and enco n ths scenaro are gven n able 7. able 7. Profts of wo Players n the 3 rd Expanson cenaro Lne 1-2 Capacty otal proft of ransco ($/hr) EC n bus 2 Proft of enco ($/hr) ) 4 th 2 2 Expanson scenaro (, ) Both ransco and enco wll perform EC and EC. he most proftable expanson capacty s 37 MW for EP wth proft of $/hr and 32 MW for EP wth proft of $/hr. o the Cournot capacty par wth maxmum proft s (32 MW, 37 MW) among all other Cournot capacty pars. he profts of ransco and enco are gven n able 8. able 8. Profts of wo Players n the 4 th Expanson cenaro otal Lne 1-2 EC n Proft of proft of EC bus 2 enco ransco ($/hr) ($/hr) After consderng all of the mentoned scenaros, a payoff matrx that comprses the payoffs of ransco and enco s constructed whch s shown n able 9. In able 9, the frst entry n the box s the proft of ransco and the second entry s the proft of enco. In ths case, the domnant strategy whch s 2 2 characterzed as the Nash equlbrum s the par of strateges (, ) whch mples that the most proftable strategy for both ransco and enco s to perform EP and EP. In ths strategy, the proft of ransco s $/hr whle enco proft s $/hr. IJECE Vol. 4, No. 6, December 2014 :

13 IJECE IN: able 9. Payoff Matrx of the ame heoretc EC and EC ransco trategy Nash eq. enco trategy (14.3,0) (6.39,290.44) 2 (31.03,0) (32.03,291.43) 7. CONCLUION In ths study t was shown a new method for expanson plannng of generaton and transmsson that consdered statc securty of the system such as voltage securty margn And loadablty lmt n a sx-bus system. At the frst step a secure bus was selected for load ncrmnaton that was consdered voltage securty margne by usng mnmum sngular value technque. As the transmsson lne capacty s not suffcent for the load ncrement, a congeston n the transmsson lne occurs whch can be an ncentve for enco and ransco to gan the maxmum proft from supplyng ths load. Consderng multple scenaros, the Cournot model defnes the most proftable EC and EC n each scenaro. After consttutng the payoff matrx whch ncludes all of the possble strateges combnaton, the Nash equlbrum s obtaned. he results show that the most proftable scenaro for both enco and ransco s to perform generaton expanson and transmsson expanson. REFERENCE [1] H. He, Z. Xu and. H. Cheng, Impacts of transmsson congeston on market power n electrcty market, 2004 IEEE PE Power ystems Conference and Exposton, vol. 1, pp , Oct [2] N. Yang and F.. Wen, A chance constraned programmng approach to transmsson system expanson plannng, Electrc Power ystems Research, vol. 75, pp , Aug [3] C. W. Lee,. K. K. Ng, J. Zhong and F. F. Wu, ransmsson expanson plannng from past to future, n Proc. PCE 2006, Oct 2006, pp [4] F. F. Wu, F. L. Zheng and F.. Wen, ransmsson nvestment and expanson plannng n a restructured electrcty market, Energy, vol. 31, pp , May-Jun [5] C. W. Lee,. K. K. Ng and J. Zhong, Value-based transmsson expanson plannng n deregulated market, Proceedngs of the 38 th North Amercan Power ymposum, pp , ept [6] H. ngh, Introducton to game theory and ts applcaton n electrc power markets, IEEE Comp. Applcat. n Power, vol. 12, pp , Oct [7] J. Contreras and F. F. Wu, Coalton formaton n transmsson expanson plannng, IEEE rans. Power ystems, vol. 14, pp , [8] H.. Ko,. Nmura and K. Ozawa, A day-ahead electrcty prce predcton based on a fuzzy-neuro autoregressve model n a deregulated electrcty market, n Proc. Int. Jont Conf on Neural Networks, May 2002, vol. 2, pp [9]. De la orre, J. W. Feltes,... Roman and H. M. Merrl, Deregulaton, prvatzaton, and competton: ransmsson plannng under uncertanty, IEEE rans. Power ystems., vol. 14, pp , May [10] A. K. Davd and F.. Wen, trategc bddng n compettve electrcty markets: a lterature survey, n Proc. IEEE Power Eng. oc. ransmsson and Dstrbuton Conf:, 2000, vol. 4, pp [11] E. E. auma and.. Oren, Proactve transmsson nvestment n compettve power systems, IEEE Power Engneerng ocety eneral Meetng, pp , Jun [12]. Borensten, J. Bushnell and. toft, he compettve effects of transmsson capacty n a deregulated electrcty ndustry, RAND Journal of Economcs, vol. 31, pp , Jun [13]. K. K. Ng, C. W. Lee and J. Zhong, A ame-heoretc Approach to tudy trategc Interacton Between ransmsson and eneraton Expanson Plannng, Proceedngs of the 38 th North Amercan Power ymposum, pp , ept [14]. K. K. Ng, J. Zhong and C. W. Lee, A ame-heoretc tudy of the trategc Interacton between eneraton and ransmsson Expanson Plannng, n Proc. PCE 2009, March 2009, pp [15] H.P chmdt, R, N. Adams, Assessment of statc voltage stablty usng artfcal neural Network, proc. Of 11th PCC, avgon, agust 30, 1993 [16] M.R. Aghamohammad, H. atoh, J, oyada enstvty characterstc of Neural Network as a tool for secure generaton schedulng n power system, n proc. ICIC, 96, nternatonal Conference on Intellgent and cogntve system, ehran, Iran, ept [17] C. Charalambous, Conjugate gradent algorthm for effcent tranng of artfcal neural networks, IEEE Proceedngs, vol. 139, no.3, 1992, pp [18] R. bbons, A prmer n game theory, New York, Harvester wheatsheaf, [19].Krause, Congeston management n lberalzed electrcty markets - theoretcal concepts and nternatonal applcaton, [Onlne]. Avalable: A ecure-coordnated Expanson Plannng of eneraton and ransmsson Usng (Mojtaba Jalalpour)

14 881 IN: [20] Calforna Independent ystem Operator, Locatonal Margnal Prcng (LMP): Bascs of Nodal Prce Calculaton, [Onlne]. Avalable: [21] D. Krschen and. trbac, Fundamentals of power system economcs, Hoboken, NJ: John Wley & ons, [22] A.. Chuang, F. Wu, P. Varaya, A game-theoretc model for generaton expanson plannng: problem formulaton and numercal comparsons, IEEE rans. Power yst., vol. 16, pp , Nov Appendx ACOPF Equatons: mn f ( Pg ) Bg Pg Qj mn Qj Qj max st. : P Y V ( V cos V cos( )) V mn V V max g j j j j j Q Y V ( V sn V sn( )) 0 Pg Pg max g j j j j j mn max b n Pj mn Pj Pj max P P P g j Load ga j m Where: f (.) : total operaton cost n $/hr : generaton offerng prce for each generaton offerng energy n $/MWhr B g P : actve power generaton for each generaton offerng energy n MW g P max j Y j j : maxmum actve power generaton for each generaton offerng energy n MW P : actve power flow from bus to bus j n MW : admttance of branch connectng bus and bus j n Ω : admttance angle of branch connectng bus and bus j n radan V : voltage at bus n volt : voltage angle at bus n radan P j mn P j max Q j mn Q j max V mn V max mn max P Load : mnmum actve power transmsson capacty at branch j : maxmum actve power transmsson capacty at branch j : mnmum reactve power transmsson capacty at branch j : maxmum reactve power transmsson capacty at branch j : mnmum voltage at bus n volt : maxmum voltage at bus n volt : mnmum voltage angle at bus n radan : maxmum voltage angle at bus n radan : load at bus IJECE Vol. 4, No. 6, December 2014 :