THE ROLE OF ENERGY IMBALANCE MANAGEMENT ON POWER MARKET STABILITY

Transcription

1 Proceedings of HICSS-31, Big Isand of Hawaii, January 6-9, 1998, Voume III, pp THE ROLE OF ENERGY IMBALANCE MANAGEMENT ON POWER MARKET STABILITY Fernando L. Avarado Department of Eectrica and C. Engineering The University of Wisconsin - Madison avarado@engr.wisc.edu Weington S. Mota Departamento de Engenharia Eetrica UFPB - Campina Grande PB - BRASIL wsmota@dee.ufpb.br Abstract Power System Markets represented by dynamic equations provide insights into the market behavior which are not avaiabe from static modes. In this paper, the dynamic couping between energy imbaance and markets is studied by modeing the market dynamics together with the eectromechanica power system dynamics. Matab-based software is used to evauate the eigenvaues of a inearized version of an interconnected power system incuding dynamic market equations. The effect of network congestion on a market with severa suppiers and severa consumers is considered. Numerica resuts are presented to iustrate the proposed mode and to show how improper handing of energy imbaance in a system can ead to an unstabe market. Keywords: Market dynamics, power system dynamics, eigenvaues. Introduction The response of markets to changes in prices is not instantaneous. Neither is the response of a power system to an order to increase the power output of a particuar unit or units. These deays in the abiity to respond to changing conditions can give rise to instabiities, periods of excess suppy at cheap prices are foowed by periods of insufficient suppy. This paper studies the dynamics of markets and their interaction with the eectromechanica inertias and other dynamic effects of an actua system. Severa assumptions are made [1: Margina production costs λ g are inear functions of power generation order P g. Margina benefit function λ d are negativey-soping inear functions of power consumption P d. Response of suppiers and consumers to observed prices is not instantaneous. It is governed by first order singe time constant differentia equations. If power is not baanced precisey at a times, an energy imbaance resuts. An energy imbaance eads to the need to contro such imbaance to prevent system damage or unwanted reay action. The synchronous generators can be represented by differentia equation modes. Some of these modes incude the effect of votage reguator and power system stabiizing signas (PSSS). The action of the governor/turbine system is aso representabe by a differentia equation mode.. Production is a function of margina cost and price. Consumption is a function of margina benefit, price, and system votage. The network is representabe either as a set of inear agebraic equations, or, in some cases, just by a modified power baance condition. The two ast assumptions above provide coupe the market to the network. 1 Market Dynamics 1.1 The energy imbaance case In a rea power system, energy imbaance cannot be sustained indefinitey [1. It must be reduced or driven to zero. In a traditiona utiity environment, such an objective is attained by automatic generation contro. In a market-driven environment, it can be assumed that prices wi refect the degree of energy imbaance. That is, an excess of the quantity suppied (the power generation order P g ) in the grid wi sighty depress the vaue of the power, and thus wi decrease the price of power. Such a situation can be represented by adjusting prices depending on the degree of energy imbaance. The changing of price depending on excess or shortfa of rea time energy is referred as frequency reguation pricing or ACE (Area Contro Error) pricing. The equations representing the dynamics of ACE pricing for m suppiers and n consumers are:

2 τ gj P gj b gj c gj P gj + λ K j E, j 1,m(1) τ di P di b di + c di P di λ, i 1,n (2) Ė m n P gj P di (3) j1 i1 τ λ λ E (4) b gj + c gj P gj b di + c di P di τ gj τ di λ τ λ E K j is the margina cost of suppier j is the margina benefit of consumer i is the power generation order time constant of suppier j is the demand time constant of consumer i is the market price is the market price time constant is the power system stored energy is a market stabiizer gain sent to suppier j The interpretation of these equations is as foows: generators act in a way that tends to increase production when prices exceed production margina costs. Consumers act in a way that tends to increase consumption when margina benefits exceed price. Since it may be impossibe to perfecty baance suppy and consumption at a times, any discrepancy accumuates as an energy error. In practica system, this resuts in either an increase in frequency or an increase in the ACE. The resut of an excess of energy is a reduction in the system vaue of eectric energy. The consequence of this is a probabe reduction in price, which takes pace according to some time constant τ λ. This reduction in system price increases consumption and decreases production, thereby eading to a decrease in the excess energy. Stabiity requirements aso necessitate the presence of a suppementary stabiizing price signa to be sent to either the suppiers or the consumers [1. Here, the signa is sent to the suppiers. The stabiizing signa is a constant gain times the accumuated energy error. This can be interpreted as a bias that is added to prices whenever the energy error is nonzero. This suppementary signa is essentia to asymptotic stabe market behavior. 1.2 The network congestion case Consideration of the dynamics of congestion itsef is beyond the scope of this paper. Consideration of energy dynamics with a singe differentia equation for the energy imbaance can ead to unsovabe modes. Thus, for the network congestion case, the market is modeed as in equation (1) and (2), except that a requirement for strict power baance at a times repaces equations (3) and (4): τ gj P gj b gj c gj P gj + λ, j 1,m (5) τ di P di b di + c di P di λ, i 1,n (6) m n P gj P di (7) j1 i Congestion constraints Congestion means that the fow in one or more ines must be reguated to avoid exceeding a rating and damaging the component, or other such reason. Introducing conditions for congestion avoidance within the previous dynamic formuation (incuding energy imbaance) presents severa chaenges, since congestion can resut in a functiona decouping of the system. Congestion can be expressed in terms of the network injections (both generation and oad). For n s 1congestion conditions: P g S g21 S g2m S d21 S d2n. P gm s P d1. S gns1 S gnsm S dns1 S gnsn. s ns P dn The compete dynamic equations for the congested m-suppier n-consumer case with n s active agebraic congestion conditions in matrix notation are: [ T [. P. Λ [ [ C S t P S Λ [ b + s T is a diagona matrix with the time constants for power generation as we as demand. P is a vector of a power (m generation powers and n demand powers). Λ is a vector with a Lagrange mutipiers (the price λ is the first of these). C is the diagona matrix of a quadratic cost coefficients c gi as we as c di S is a matrix of sensitivity of constraints to injections. Its first row is the power baance condition. s is a vector with a zero in its first position and the vaues of the right hand sides in the the constraint equations in the remaining positions. The dynamic equations for this probem are of reduced order. Reduction of the dynamic agebraic equation probem to a purey differentia equation can be done by first eiminating the n s redundant state variabes in terms of a reduced set of n + m n s of non-redundant variabes, obtaining in the end a set of n + m n s purey differentia equations, organized as foows:

3 T 1 S1 t T 2 S2 t S 1 S 2 Λ P 1 P 2 C 1 S1 t C 2 S2 t P 1 P 2 S 1 S 2 Λ S 1 corresponds to a subset of n s by n s of S, and P 1 corresponds to a subset n s of P denoting the redundant power variabes. Reduction and eimination of Λ and P 1 yieds the foowing reduced purey differentia equations: [ T2 + S2S t 1 t T 1S1 1 S 2 P 2 [ C 2 + S2S t 1 t C 1S1 1 S 2 P2 (8) Thus, the mode for a congested power system is of ower order than the mode for the uncongested system. 2 Eectromechanica Dynamics It is of interest to investigate whether it is possibe to put together a mode that incudes both the dynamics of a market as we as the detaied eectromechanica dynamics of a system. In order to do so, the generation and transmission eectromechanica dynamics are modeed according to [ Turbine/Governor dynamic mode The Turbine/Governor is modeed according the bock diagram figure 1, P g corresponds to the Power Generation Order associated with the market mode. ω is the machine veocity, Y is the vave position, and P m the mechanica power output. Fig. 2: AVR mode bock diagram [ Ė q 1 E T fd + x d x d d x d x E q x d x x d x E q + (xd x d )(x d x ) x d x I d (1) Ė d 1 [ E T d + E d +(x q x q )I q+ x q x x q q x Ė d (11) Ė q 1 [ E T q + E q (x d x d )I d+ x d x x d d x Ė q (12) ω 1 2H (P m P e D(ω 1)) (13) δ ω s (ω 1) (14) With the corresponding agebraic equations: Fig. 1: Governor Turbine bock diagram 2.2 Automatic Votage Reguator mode The Automatic Votage Reguator is modeed through a genera mode according the bock diagram figure 2, the AVR type is seected by appropriate choice of parameters. The incuded PSSS type derived by veocity or machine termina power is aso seected by appropriate choice of parameters. 2.3 Synchronous Machine dynamic mode The synchronous machine is modeed by a sixth order mode, according to dynamic equations: Termina Votages Interna Votages Termina Active Power Air-gap Power Vt 2 Vd 2 + V q 2 V d E d x q I q R a I d V q E q x d I d R a I q P t V d I d + V q I q Ė d 1 T q [ xq x q x q x E d x q x x q x E d + (x q x q)(x q x ) x q x I q (9) P e P t + R a (I 2 d + I2 q ) To represent the system by a purey agebraic mode, repace the derivative terms above with zero. The resuting mode wi represent the agebraic behavior of the system.

4 2.4 Transmission System The transmission system can be represented by means of the noda admittance matrix expanded into its rea and imaginary components. I DQ YV DQ 2.5 Load Representation The reactive oad can be represented by a constant admittance and the active oad by a function of termina bus votage. P L V D I D V Q I Q 2.6 Axis Transformation In order to transform votage and current from system reference to machine reference frame, the foowing axis transformation is required: [ [ [ VD sin δ cos δ Vd V Q cos δ sin δ V q 3 Market/Power System Linearized Mode The inearized market and the inearized power system are represented by a set of differentia equations together with a set of agebraic equations [2: [ [ [ Ẋ J1 J 2 X J 3 J 4 Z (15) J 3 J 4 [ J1 J 2 is the Jacobian matrix. The symbo signifies an incrementa change from a steady-state vaue and wi be omitted in the remainder of this paper. Equation (15) can be organized as foows: Ė λ P d P g ẋ gov ẋ avr ẋ syn x mkt x gen z {}}{{}}{{}}{ J 1 J 2 J 3 J State Variabes The state vector is formed by the foowing: [ Xmkt X,k 1,ng X genk E λ P d P g x gov x avr x syn P t V t V DQ I DQ V dq I dq For m suppiers and n consumers: E X mkt λ P di, i 1,n; j 1,m P gj is the market state vector. For n g generation units: X genk X gov X avr X syn X gov X avr X syn,k 1,n g State vector for the turbine/governor. State vector for the AVR incuding PSSS. State vector for the generator. 3.2 Agebraic variabes The agebraic variabes vector is: P tk V tk Z V DQ I DQ V dqk I dqk P t V t V DQ I DQ V dq I dq Synchronous generator termina active power. Synchronous generator termina votage magnitude. Rea and imaginary components of bus votage (system reference). Rea and imaginary components of bus injected current. Rea and imaginary components of the generator votage (machine reference). Rea and imaginary components of the generator termina current (machine reference). 3.3 Market/Power System state matrix The Market/Power System state matrix can be obtained by eiminating the vector of agebraic variabes Z in equation (15). Ẋ (J 1 J 2 J 1 4 J 3) X A X 4 Energy imbaance numerica exampe A three generator/infinite bus 7-ine 6-bus system as shown in figure 3 is used to iustrate the proposed ideas. For this system, three suppiers and two consumers are considered. Numerica vaues for the market parameters are presented in the appendix.

5 Tabe 1: Eigenvaues for the market/power system Stabe (K g.1) Unstabe (K g.) i i i i i i i i i i i i i i i i i i The dynamic market and the dynamic eectromechanica power system are modeed together, using equation (15). The interconnected dynamic mode market/power system produces an instabiity of the market when a market stabiizer gains are not considered. See the second coumn of tabe 1. Assuming a market stabiizer gains equa to.1, the system becomes stabe. See the first coumn of tabe 1. Aso, the market dynamic equations can be interconnected to the network through the network agebraic equations as previousy mentioned. The market eigenvaues are shown in the first coumn of tabe 2. This provides the same number of eigenvaues when the isoated market is considered. See the second coumn of tabe 2. Tabe 2: Eigenvaues of the dynamic market mode connected to the network agebraic equations. Market/Network Isoated Market i i Fig. 3: Singe Line Diagram of the Three Generators One Infinite Bus System. The main difference from the market connected to the power system through the network agebraic equations and the isoated market is due to the consideration of the network osses. 5 Concusions A method to evauate the stabiity and dynamic properties of eectric power system markets using eigenvaues is deveoped and impemented using Matab. A inearized power system mode aows eigenvaue computation for severa situations: the power system aone, the uncouped market and combined market/power system. The market dynamics study was modeed by considering the eectromechanica power system dynamics. A method to incorporate the effect of congestion was aso demonstrated. Resuts show that the parameters of a Power System Stabiizer can have an effect on the stabiity of a market. Appendix 1 Data Three-suppier two-consumer market parameters Suppiers Data Consumers Data τ g c g b g K g τ d c d b d Transmission system eectromechanica data (pu) From To R X Y Power in Mw and MVar

6 Bus V θ P g Q g P Q Generator data Bus H X d X d X q R a T do Excitation system Bus K a T a Turbine/Governor system Bus R T s T m REFERENCES [1 Fernando L. Avarado. The dynamic of power system markets technica report pserc The University of Wisconsin - Madison, March [2 Weington S. Mota and Fernando L. Avarado. Eigenvaue cacuation of a inearized mode of an interconnected power system incuding market dynamics and using the matab program. technica report dce. The University of Wisconsin - Madison, Juy [3 H. J. C. Pinto N. Martins and L. T. G. Lima. Sma signa stabiity of arge scae power system - pacdyn user s manua version 4.. CEPEL, Rio de Janeiro - Brazi, Apri 1996.