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2006 International Conference on Power System Technology A Nucleolus-Based Profit Allocation Method for Determine Individual Power System Stabilizer's Contribution to System Stability Wei Pan, Wenying Liu, Yihan Yang, Yangnan Li, Lin Cheng, and Yunlong Tang Abstract-- The issue of pricing the services provided by Power System Stabilizers (PSS) to damp out low-frequency oscillations in large interconnected electrical power grids is proposed. Taking the dominant eigenvalue method for analytical tool to evaluate the increased transfer capability when PSS-control services are utilized, a method based on nucleolus in the cooperative game theory to determine the individual PSS services provider's contribution to power systems stable operation, and allocate the profit obtained by total transaction level increment among all PSS services providers is presented. The calculation results show that the proposed method can achieve rational allocation, and have incentive effect on popularizing PSS application. These are helpful for the stable and economical operation of electricity markets. Index Terms-- Electricity market; Cooperative game theory; Power system stabilizer; Nucleolus I. INTRODUCTION power systems are under increasing stress as market policies introduce new economic objectives for operation. To achieve those economic objectives, power systems are being operated closer to their limits. As a result, any one of a large number of factors (such as weak connections, unexpected events, hidden failures in protection system, and human errors) may cause a system to lose stability, possibly leading to catastrophic failure. With the interconnection of large electric power systems, low frequency oscillations have become the main problem for power system small signal stability. They restrict the steady- state power transfer limits, which therefore affects operational system economics and security. Considerable effort has been placed on the application of Power System Stabilizer (PSS) to damp low frequency oscillations and thereby improve the small signal stability of power systems [1]-[4]. A. Template Use of a supplementary control signal in the excitation Wei Pan, Wenying Liu , Yihan Yang, Yangnan Li, Lin Cheng are with The Key Laboratory of Power System Protection and Dynamic Security Monitoring and Control in Ministry of Education, North China Electric Power University, Beijing, 102206, China. (e-mail: panway2004q163.com) Yunlong Tang is with the Xuji Hitachi Company, Xuchang, 461000, China. (email: [email protected]) system and/or the governor system of a generating unit can provide extra damping for the system and thus improve the unit's dynamic performance. Power system stabilizers (PSSs) aid in maintaining power system stability and improving dynamic performance by providing a supplementary signal to the excitation system. This is an easy, economical and flexible way to improve power system stability. Over the past few decades, PSSs have been extensively studied and successfully used in the industry. To date, PSS's have proved to be very effective and economical tools and therefore have been widely used by utilities. Game theory, in which the preciseness mathematical model is used to solve the conflict of interest of actual society, can be divided into uncooperative game theory and cooperative game theory. The former can be applied to analyze the agents' interaction between individuals or groups in a competition environment and the later can be applied to analyze their interaction in a cooperation environment. Cooperative game theory is widely used in electricity markets, such as allocating transmission loss, transmission cost [5], [6] and analyzing the economic effect of interconnected power system [7], identifying noncompetitive situations in energy marketplaces, providing supports for minimizing risks involved in price decisions in energy marketplaces [8], making bid strategies of the generation companies [8], estimating the pricing of transmission in power markets [10], etc. In this paper, the issue of pricing the services provided by Power System Stabilizers (PSS) to damp out low-frequency oscillations in large interconnected electrical power grids is proposed. Taking the dominant eigenvalue method for analytical tool to evaluate the increased transfer capability when PSS-control services are utilized, a method based on nucleolus in the cooperative game theory to determine the individual PSS services provider's contribution to power systems stable operation, and allocate the profit obtained by total transaction level increment among all PSS services providers is presented. The calculation results show that the proposed method can achieve rational allocation, and have incentive effect on popularizing PSS application. These are helpful for the stable and economical operation of electricity markets.. 1-4244-0111-9/06/$20.00c2006 IEEE. I

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The issue of pricing the services provided by PowerSystem Stabilizers (PSS) to damp out low-frequency oscillationsin large interconnected electrical power grids is proposed.Taking the dominant eigenvalue method for analytical tool toevaluate the increased transfer capability when PSS-controlservices are utilized, a method based on nucleolus in thecooperative game theory to determine the individual PSS servicesprovider's contribution to power systems stable operation, andallocate the profit obtained by total transaction level incrementamong all PSS services providers is presented. The calculationresults show that the proposed method can achieve rationalallocation, and have incentive effect on popularizing PSSapplication. These are helpful for the stable and economicaloperation of electricity markets.

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Page 1: Nucleolus-Based Profit Allocation Method for Determine Individual Power System Stabilizer's Contribution to System Stability

2006 International Conference on Power System Technology

A Nucleolus-Based Profit Allocation Method for

Determine Individual Power System Stabilizer's

Contribution to System StabilityWei Pan, Wenying Liu, Yihan Yang, Yangnan Li, Lin Cheng, and Yunlong Tang

Abstract-- The issue of pricing the services provided by PowerSystem Stabilizers (PSS) to damp out low-frequency oscillationsin large interconnected electrical power grids is proposed.Taking the dominant eigenvalue method for analytical tool toevaluate the increased transfer capability when PSS-controlservices are utilized, a method based on nucleolus in thecooperative game theory to determine the individual PSS servicesprovider's contribution to power systems stable operation, andallocate the profit obtained by total transaction level incrementamong all PSS services providers is presented. The calculationresults show that the proposed method can achieve rationalallocation, and have incentive effect on popularizing PSSapplication. These are helpful for the stable and economicaloperation of electricity markets.

Index Terms-- Electricity market; Cooperative game theory;Power system stabilizer; Nucleolus

I. INTRODUCTION

power systems are under increasing stress as marketpolicies introduce new economic objectives for operation.

To achieve those economic objectives, power systems arebeing operated closer to their limits. As a result, any one of alarge number of factors (such as weak connections,unexpected events, hidden failures in protection system, andhuman errors) may cause a system to lose stability, possiblyleading to catastrophic failure.

With the interconnection of large electric power systems,low frequency oscillations have become the main problem forpower system small signal stability. They restrict the steady-state power transfer limits, which therefore affects operationalsystem economics and security. Considerable effort has beenplaced on the application of Power System Stabilizer (PSS) todamp low frequency oscillations and thereby improve thesmall signal stability of power systems [1]-[4].

A. TemplateUse of a supplementary control signal in the excitation

Wei Pan, Wenying Liu , Yihan Yang, Yangnan Li, Lin Cheng are withThe Key Laboratory of Power System Protection and Dynamic SecurityMonitoring and Control in Ministry of Education, North China Electric PowerUniversity, Beijing, 102206, China. (e-mail: panway2004q163.com)

Yunlong Tang is with the Xuji Hitachi Company, Xuchang, 461000,China. (email: [email protected])

system and/or the governor system of a generating unit canprovide extra damping for the system and thus improve theunit's dynamic performance. Power system stabilizers (PSSs)aid in maintaining power system stability and improvingdynamic performance by providing a supplementary signal tothe excitation system. This is an easy, economical and flexibleway to improve power system stability. Over the past fewdecades, PSSs have been extensively studied and successfullyused in the industry. To date, PSS's have proved to be veryeffective and economical tools and therefore have been widelyused by utilities.

Game theory, in which the preciseness mathematical modelis used to solve the conflict of interest of actual society, can bedivided into uncooperative game theory and cooperative gametheory. The former can be applied to analyze the agents'interaction between individuals or groups in a competitionenvironment and the later can be applied to analyze theirinteraction in a cooperation environment.

Cooperative game theory is widely used in electricitymarkets, such as allocating transmission loss, transmissioncost [5], [6] and analyzing the economic effect ofinterconnected power system [7], identifying noncompetitivesituations in energy marketplaces, providing supports forminimizing risks involved in price decisions in energymarketplaces [8], making bid strategies of the generationcompanies [8], estimating the pricing of transmission in powermarkets [10], etc.

In this paper, the issue of pricing the services provided byPower System Stabilizers (PSS) to damp out low-frequencyoscillations in large interconnected electrical power grids isproposed. Taking the dominant eigenvalue method foranalytical tool to evaluate the increased transfer capabilitywhen PSS-control services are utilized, a method based onnucleolus in the cooperative game theory to determine theindividual PSS services provider's contribution to powersystems stable operation, and allocate the profit obtained bytotal transaction level increment among all PSS servicesproviders is presented. The calculation results show that theproposed method can achieve rational allocation, and haveincentive effect on popularizing PSS application. These arehelpful for the stable and economical operation of electricitymarkets..

1-4244-0111-9/06/$20.00c2006 IEEE.

I

Page 2: Nucleolus-Based Profit Allocation Method for Determine Individual Power System Stabilizer's Contribution to System Stability

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II. TECHNICAL METHOD OF Low FREQUENCY OSCILLATIO,PSS AND COOPERATIVE GAME

A. Small signal analysis method and lowfrequencyoscillation

Small signal stability is defined as the ability of the powersystem to remain stable in the presence of small disturbances[11]. These disturbances could be minor variations in load orgeneration on the system. If sufficient damping torque doesn'texist, the result can be rotor angle oscillations of increasingamplitude. Generators connected to the grid utilizing highgain automatic voltage regulators can experience insufficientdamping to system oscillations

A power system can be represented by the following set ofdeferential and algebraic equations:

L]0 g(X, Y)] ( ) 1

Where x is the vector of the state variables and y thevector of algebraic variables.

In a small signal stability analysis, (1) can be linearizedaround an equilibrium or operating point (x0, y0) for the

given values of the parameters p0. Thus,

[A<] [gVx 7Vyf7 [x]= J[ J2][ X] [][zx](2)Where J is the complete system Jacobian matrix. If it is

assumed that J4 is nonsingular, which is a requirement forequations (1) to appropriately represent the system, the systemeigenvalues can be readily computed eliminating the vector ofthe algebraic variable Jy in (2), as follows:

Ax = (J1 - J2J4-1J3)Ax = Aix (3)The system state matrix eigenvalues and eigenvectors are

defined by{Av2 (4)

Where pu is the eigenvalues, and v and w are thecorresponding right and left eigenvectors respectively.

The stability of system is determined by the eignevalues asfollows:

A real eigenvalue corresponds to a non-oscillatory mode.A negative real eigenvalue represents a decaying mode, Thelarger its magnitude, the faster the decay. A positive realeigenvalue represents a periodic instability. Complexeigenvalues occur in conjugate pairs, and each paircorresponds to an oscillatory mode. The real component of theeigenvalues gives the damping, and the imaginary componentgives the frequency of oscillation. A negative real partrepresents a damped oscillation whereas a positive real partrepresents oscillation of increasing amplitude. Thus, for acomplex pair of eigenvalues:

X = C ±j2 o (5)The frequency of oscillation in Hz is given by

(6)2fTThis represents the actual of damped frequency.The damping ratio is given by

-C

cO2 +w

(7)

The damping ratio j determines the rate of decay of theamplitude of the oscillation.

When J, < 0.03, it indicates that system is lightly damped.If J, < 0, it means that system damp is changing to negativeand will emerge the oscillation with increasing amplitude. Atypical damping factor of J, = 0.05

The participation factor pj of the ith state variable to the

jth eigenvalue can be defined as:

PjWij V i

(8)WjVII

In case of complex eigenvalues, the amplitude of eachelement of the eigenvectors is used:

WiCVjiVPi n

|Wjk |Vkjk=l

(2)

Participation factor pij is actually equal to the sensitivity

of eigenvalue Ai to the diagonal element akk of the state

matrix A . The participation factors are generally indicative ofthe relative participations of respective states in thecorresponding modes.

In today's practical power systems, small-signal stability islargely a problem of insufficient damping of oscillations. Thestability of the following two types of low -frequencyoscillations is of concern:

Local modes or machine-system modes oscillations aregenerally involve one or more synchronous machines at apower station swinging together against a comparatively largepower system or load center. The frequency of oscillation is inthe range of 0.7 Hz to 2 Hz.

Interarea modes oscillations are usually involvecombinations of many machines on one part of a powersystem swinging against machines on another part of thepower system. Inter-area oscillations are normally in thefrequency range of less than 0.5 Hz.

Low-Frequency oscillations are a quite common problemin most interconnected power systems today. Theseoscillations are due to dynamic interactions between thevarious generators of a system, through its transmissionnetwork, and are usually associated to the presence ofautomatic voltage regulators (AVRs) with high gain in thegenerators and long transmission lines establishing weakconnections between distinct areas of a system. Due to thisphenomenon, limits often have to be placed on the maximumamount of power that can be transferred over these strategiclines, to ensure a safe operation of the system. Therefore,

Page 3: Nucleolus-Based Profit Allocation Method for Determine Individual Power System Stabilizer's Contribution to System Stability

3

equipment and procedures to enhance the damping ofoscillations become mandatory for the system operati(order to allow a better use of the existent transmnetwork.

B. Power system stabilizerFrom a single generator viewpoint, the basic nature

oscillation problem was presented in [12]. The reductithe damping component of the electrical torque caushigh gain AVRs was determined as the root of the prcThe solution proposed was to add a damping signal (inwith the oscillation) to the AVR reference input, thro-phase lead compensator, so the component of the ele(torque generated by this signal could enhance the dampithe oscillation. This approach for damping controller deswidely used nowadays and the controllers designed wittphilosophy are known as power system stabilizer (PSS).

The basic function of a power system stabilizer (PSSadd damping to the generator rotor oscillations by contrits excitation using auxiliary stabilizing signals. To proidamping, the stabilizer must produce a componeielectrical torque in phase with the rotor speed deviations.

A typical PSS mathematical model are shown in Fig

I) )t3 A.,2.

)t I.- K+ I -+-S'TI Pf I+T1>l5I'

.... .... _.3....0~~~~~~~I I + fT

Fig. 1 Block diagram representation of PSS

Model in Fig.1 accept as input signals the rotordifference t - o, the active power difference P - P,o ai

bus voltage magnitude difference V, - Voof generatwhich the PSS is connected through the automatic viregulator. Kql, is the stabilizer gain used for speed diffe

Kq2 and Kq3 are the gain for active power and gain fc

voltage magnitude.The model consists of three blocks: a gain block, a

washout block and a phase compensation block.The stabilizer gains Kql , Kq2 and Kq3 determin

amount of damping introduced by the PSS. Ideally, theshould be set at a value corresponding to maximum danhowever, it is often limited by other considerations.

The signal wash-out block serves as a high-passwith the time constant Tq high enough to allow s

associated with oscillations in - co to pass unchaWhen K = 1, the signal wash-out block is the usual irdiff block, whereas if K = 0 is the phase-shift block.

The phase compensation block provides the approphase-lead characteristic to compensate for the phasbetween the exciter input and the generator electrical t(Tl, T2, T3 and T4 are the time constant of the

compensation block.

theseon, iniission

of theion in

ed byblem.phase,ugh actricaling of;ign isl such

Vmax and V, mjn is the output limit of PSS.

III. NUCLEOLUS BASED PROFIT ALLOCTION OF PSSSERVICE

The model of cooperative game has two basic elements:the set of players and the eigenfunction. The set of players iscomposed of all independent benefit entities affecting theissue of the problem. If the problem deals with n(n > 1)benefit entities, the set of players is expressed asN = {1, 2, ..., n}, V1S c N. The meaning of eigenfunction v(s)is a real function of set N; any possible set S of players (acoalition) would produce the benefit v(s) . Apparently, theeigenfunction should have the following characters.

If coalition S and T are independent, namely, S rn T=0,then v(S u T) > v(S) + v(T) . In fact, it is the prerequisite to

,) is to realize larger coalition.'olling It is well known that proper PSS tuning and operation canviding eliminate low frequency oscillation under small disturbance.

nt of From the point of this view, it is easy to understand that PSSoperation can improve system transfer limit. The improvement

.1: of system transfer capability will increase the benefit ofelectricity market. How to allocate the increased profit amongthe market participants can be modeled as a cooperative game,having as its characteristic function a benefit formulationwhich is based on the objective function used in the tuningprocess, and where one or more generators are consideredtogether (in all possible coalitions) to obtain a fair revenueallocation. Thus the analysis of the worth of PSS-controlancillary service is based on two important issues: how the

speed coalitions are formed amongst the PSSs, and consequentlynd the how the benefit from a PSS service is allocated in these two

tor to interrelated issues.

torlta The nucleolus is a solution concept introduced byDltage Schmeidler in [13]. Two important characters of nucleolus are,rence. respectively, firstly, every game has one and only onewr bus nucleolus, and secondly, unless the core is empty, the

nucleolus is in the core.

signal The goal of using the nucleolus concept is to find a way tofairly allocate the profit that is jointly created by PSS services.

e the Suppose X {xl, x2, I... X, } is the set of each PSS service

gains provide profit, Y ={y, y2,1...Y,y } is the set of the profitnping; allocation imputation, and is the loss of the transaction

coalition S. Nucleolus is based on the minimum core and isfilter, represented byignals

inged.iertia-

priate,e lagorque.phase

C (£) = {y E y / (y) < E}

{((y) = max e(S, y)ScN

(10)

Where c represents an arbitrary small real number;e(S, y) is the coalition S 's excess value of imputation y E Y,

namely, e(S, y) = V(S) - , Y, ; q(y) is the maximum oficS

excess value; and V(S) is the profit, which is created by thealternation of the member of PSS service coalition , namely

'I + S-Ap-

I+S

Page 4: Nucleolus-Based Profit Allocation Method for Determine Individual Power System Stabilizer's Contribution to System Stability

4

V(S) = v(S) - Zv(i) (11)

Use linear programming to solve (10), namelymin £

s.t. V(S) = (12)

V(S) ZYi <

Where SI is the coalition of all PSS services, and S2 is allnonempty subcoalitions of PSS services.

The total benefit allocation for profit provided by eachPSS service should be the summation of the benefit created byall PSS services and one created by the individual PSS service,namely

xi = v(i) + y(i), i = 1,2, ..., n. (13)

IV. NUMERICAL EXAMPLE

In order to be able to accurately allocate payoffs to PSSs,it is important to have a good understanding of theircontribution to the system. In the present approach, thecontribution is measured in terms of the enhanced transfercapability due to addition of each PSS.

In this section, the New England System 39-bus system,composed of 10 generators, 39 buses and 46 lines, as definedin [14], was depicted in Fig. 2 to illustrate the proposed profitallocation method of PSS service.

the effect of PSS on the system transfer capability anddamping. The dominant eigenvalue of the system shiftstoward instability as the system loading is increased, or, inother words, the damping factor decreases monotonically asthe load scaling factor increases.

When the system loading increased to 108% of base casecondition, means than the total system loading reached6704MW and 1536Mvar and generator' outputs 6755MW and1596Mvar. At this time, the damping factor corresponding tothe dominant eigenvalue decreased to 0.031.

Under this condition, if a three-phase-to-ground faultapplied line from bus21 to 22, the generator relative anglesamong bus 31, 35 and bus 36 is show as follows:

- 31-32

207 -20

¢e -35.1

31-35 31-34

0 2 4 6 8 10 12 14 16 18 20Times [s]

Fig.3 Rotor relative angles at when a fault applied at line 21-22 under 1 12%base case condition

From Fig.3, it can be seen that the system is poor dampedbut still can keep secure operation under contingency, thedamping factor equals to 0.031 also proved this fact. On theother word, the system is at the critical point.

As the system loading increased to 115% of base casecondition, total system loading reached 7016MW and1606Mvar and generator' outputs 7068MW and1794Mvar.the damping factor corresponding to the dominanteigenvalue decreased to 0.025 respectively. If the same lineoutage contingency occurred, the generator angles are asfollows:

* 31-35 31-38 -31-36X 40

o

Fig.2 One-line diagram ofNew England 10-generator 39-bus system

In the simulation, Generators are modeled using the two-axis model with fast AVR and PSS as shown in Figl.

In base case power flow distribution, the system loadingare 6151MW and 1409Mvar. The total generators active andreactive power output are 6193MW and 1258Mvar.

In base case condition, the damping factor correspondingto the dominant eigenvalue is 0.0496.

With all PSSs out of service, the system loading is nowincreased gradually and uniformly at all buses, using a loadscaling factor that denotes the load increase with respect to thebase load. This load increase scenario is only used to illustrate

-40"I

. -80

4 120

X f +R;l 1 1 ftT + A 1 + 1

It AS i1 1; VL F4,i ; fi~ ,W0AA AAA A

0 2 4 6 8 10 12 14 16 18 20Times Es]

Fig.4 Rotor relative angles at when a fault applied at line 21-22 under 115%base case condition

It is evidently that the system can not keep stable operationunder disturbance and low frequency oscillation emerged.By small signal analysis, it is known that under this

operation mode, bus 35, 36 and 38 had the largestParticipation factors as 0.07184, 0.06574 and 0.05668. SoPSS service is chosen to provide by the generators at these busand some assumption are considered in the analysis.

cl.C)

I

b.01-

Page 5: Nucleolus-Based Profit Allocation Method for Determine Individual Power System Stabilizer's Contribution to System Stability

5

For convenience, the set of tuned PSS parameters whichwill be considered in the analysis are given as follows:Kq2 =5, Tq = 10, T = 0.48, T2 = 0.23, T3 = 0.48, T4 = 0.23.

It is well known that different generator coalitions toprovide PSS services have different effect on the systemtransfer capability. Consequently, the different effect on

system transfer capability can be determined by assuming thatthe worth of unit increase in system loading with respect tothe unit loading profit is UP = 10$ /MW . By setting the PSSsto off-line status in all possible combinations, all the feasiblecoalitions in which the three PSS may operate are obtained. Incomputation, A typical damping factor of j = 0.03 isconsidered as a cut-off value, beyond which the systemloading can not be increased to maintain a "reasonable"margin of system security. The difference of loading levelbetween the 108% of base case condition without PSS serviceand the loading level with PSS service multiply the UP is theprofit obtained by coalition. By this method, the profit ofdifferent PSS service coalitions and the nucleolus value are

shown in below:TABLE I

Allocation Results ofNew England 39-Bus System

PSS ServiceCoalition

35

36

38Bus 35,36No.

35,3836,38

35,36,38

System Loading(MW)

6810

6854679569927046

70827243

Coalition Profit($)

1060

150091028803420

37805390

Nucleolus Value($/MW)1700

21401550

It can be seen from Table I that if more PSS service are

provided by generators, the more profit can be obtained by thewhole system. By nucleolus method each PSS serviceprovider can get more profit than by itself. This means theallocation is rational and proves the validity of nucleolusmethod.

V. CONCLUSION

PSS service is an important service in electricity market tof economic behavior in the power market. Decision making ofeach participant in the market must follow the law ofeconomics.

Based on cooperative game theory, a novel nucleolustheory based method for the PSS service scaling is proposed.

With this method, the contribution of individual PSSservice can be taken into consideration.

Test results show that the nucleolus theory-based method isopen, equal, and impartial. The allocation solutions would notbe affected by the sequence that each coalition is formed andbe active. To a large-scale system, the whole calculatingprocess will take up much time and how to solve this problemdeserves more efforts.

VI. REFERENCES[1] E. V. Larsen and D. A. Swann, "Applying Power System Stabilizers Part

I, II, III," IEEE Trans. Power Apparatus and Systems, vol. PAS-100, no.6,pp.3017-3045, 1981.

[2] P. Kundur, M. Klein, G. J. Rogers, and M. S. Zywno, "Application ofPower System Stabilizers For Enhancement of Overall SystemStability," IEEE Trans. Power Systems, vol. 4, no. 2, pp. 614-625, 1989.

[3] Xiao-ming, M.; Yao, Z.; Lin, G.; Xiao-chen, W.,"Coordinated Controlof Interarea Oscillation in the China Southern Power Grid" IEEE Trans.Power Systems, vol. 21, no. 2, pp. 845-852, 2006.

[4] Kamwa, I.; Grondin, R.; Trudel, G.; IEEE PSS2B versus PSS4B: thelimits of performance of modern power system stabilizers. IEEE Trans.Power Systems, vol. 20, no. 2, pp. 903-915, 2005.

[5] X. Tan and T. T. Lie, "Application of the Shapley value on transmissioncost allocation in the competitive power market environment," Proc. Inst.Elect. Eng., Gener., Transm., Distrib., vol. 149, no. 1, pp. 15-20, Jan.2002.

[6] X. H. Tan and T. T. Lie, "Allocation of transmission loss cost usingcooperative game theory in the context of open transmission access," inProc. IEEE Power Engineering Society Winter Meeting, vol. 3, 2001, pp.1215-1219.

[7] P. Wei, X. Kaigui, and Z. Jiaqi, "Economic effect analysis ofinterconnected power system based on optimal power flow andcooperative game theory," Power Syst. Technology., vol. 28, pp. 35-39,Aug. 2004.

[8] M. Shahidehpour, H. Yamin, and L. Zuyi, Market Operations in ElectricPower Systems. New York: Wiley, 2002, p. 191.

[9] A. Maiorano, Y. H. Song, and M. Trovato, "Dynamics of noncollusiveoligopolistic electricity markets," presented at the IEEE PowerEngineering Society Winter Meeting, Singapore, 2000.

[10] Geerli, R. Yokoyama, and L. Chen, "Negotiation models for electricitypricing in a partially deregulated electricity market," presented at theIEEE Power Engineering Summer Meeting, Seattle, WA, 2000.

[11] Padiyar K R, Power System Dynamics Stability and Control. John wiley& Sons(Asia), 1996, p. 314

[12] F. P. DeMello and C. Concordia, "Concepts of synchronous machinestability as affected by excitation control," IEEE Trans. Power App.Syst., vol. PAS-88, pp. 316-329, July/Aug. 1969.

[13] D. Schmeidler. "The nucleolus of a characteristic function game," inSIAM J Appl. Math, vol.17, 1969, pp.1163-1170.

[14] Graham Rogers. Power system Oscillations. Kluwer AcademicPublishers,2000, p. 235

VII. BIOGRAPHIES

Wei Pan received the B.S. and M.S. degrees in the electrical engineering fromKuming Polytechnic University and Chongqing University in 1995 and 2004,respectively.Currently, he is a Ph.D candidate of North China Electric Power University.His research interests include power system operation, control, optimization,and economics.

Wenying Liu received her Bachelor's degree in 1982 in Northeast University.She is a professor of Electrical Engineering Department at North ChinaElectric Power University. Her major research interests are on power systemoperation and control.

Yihan Yang received his Bachelor's degree in 1949 in Northeast University,and received Master's degree in 1952 in Harbin Institute of Technology. He isa professor of Electrical Engineering Department at North China ElectricPower University. His major research interests are on power system operationand control.