Electricity Market Modelling Using Dynamic LQ Games

Transcription

1 XII International PhD Workshop OWD 1, 3-6 October 1 Electricity Market Modelling Using Dynamic LQ Games Michał Ganobis, Institute of Automatics, AGH ( , prof. Wojciech Mitkowski, University of Science and Technology AGH) Abstract This paper presents possibilities of modelling of electricity market using linear-quadratic (LQ) game theory. First section is a short introduction. In the second section, the main characteristic of electricity market is presented - its legal base and assumptions of consumers behaviour. Third and fourth sections provide an overview of main results of LQ game theory, and exact algorithm for solving them. Fifth section contains an example and its simulations in MATLAB. The last part are the conclusions, where pros and cons of such modelling are analysed. 1. Introduction Dynamic game theory takes into consideration the situation, when two or more controls (called players) are trying to optimise their performance indexes with respect to some dynamical system. Applications of this theory are very wide, such as robotics [1], military [1], or biology. The most natural area for applications for dynamic game theory appears to be the area of economy sciences - many macro- and microeconomic problems can be described and solved as a game. In this paper, linear-quadratic games are taken into consideration. It means that we deal with the situation with linear dynamical system, and quadratic cost functional. Linear systems are well-known and very deeply analysed (see e.g. [9], [11]). This is because of the possibility of linearization for many nonlinear systems, which is very efficient method for many problems. The quadratic performance index is typical for minimal-cost and minimal-energy problems, and for this reason used very frequently. In optimization theory (i.e. with only one control and performance index), linear-quadratic problems are the subject of many publications (e.g. [8]).. The main characteristic of electricity market The legal base of electricity market in Poland is an act "Energetic Law" (see Polish Journal of Laws, 1th April 1997). According to this law, there is a possibility of free choose of electric energy for companies and, since 1st July 7, for private subjects. It is distinguished between producer and provider of energy. The delivery services are monopolised, i.e. there is no possibility to connect a consumer to more then one electric network. As a result, energy delivering is not taken into account in the model - the only competition on the market takes place between producers of energy. Energy market is an area where interests of producers should be balanced with interest of the country and its citizens. Tendency to profit maximisation is natural for producers, but it should come with stability and long term development of the market, and be aligned with energetic policy of the country. A simple model of the electricity market will be based on the following assumptions: 75

2 1. Consumer can change the producer of energy, but it requires some effort from both sides. As a result, electricity market seems to be characterized with relatively big inertia. The price of energy is partly regulated by Urząd Regulacji Energetyki (URE). In particular, price can not exceed some fixed value, further called maximal price 3. Business model for energy producers is based on the long-term strategies, and, besides of profits, must take into account the trust and opinion of the consumer Basing on above assumptions, we can create linear model as follows: x i (state) represents the level of aversion of a client to the particular producer. Greater level of aversion causes with lower amount of energy purchased by clients u i (control) - the level of discount. As mentioned in assumptions, the price of energy has its upper bound, defined by URE. Nevertheless, producers can sell their energy cheaper. The value of u i defines how much the price is lower in compare to maximal price, i.e. u i = U C i, where U -maximal price, C i - the price of energy for producer i J i - performance index for i-th producer. The goal of producer is some kind of compromise between maximization of profit and gain of clients trust. We will try to get linear differential equation of (1), representing the dynamic of clients attitude to particular producer. Such model takes into account the inertia of customers trust, which, as mentioned, appears on the real markets (see e.g. [4]). It can be expected, than elements of matrix A will be negative on its diagonal and also besides it. The justification of signs depends on the following observations of customers behaviour: 1. Elements on the diagonal represent the influence of current opinion of the producer on the future opinion. This elements should be negative, because the natural tendency of consumer is to assure them in their opinions (phenomenon of such called after decision discord, see e.g. []). The choice of some producer make consumer "convincing themselves", even without any particular actions from the side of the producer.. Elements which are not on the diagonal represent the influence of other produrers opinion on opinion of particular producer. When the opinion is bad in comparing to others, it will cause with further making it worse. Matrix B decides of the influence of discount on the produrer s opinion. It is obvious, that more expensive energy causes with worse opinion. This is because b i element of the vector B i (it means the element linking x i with B i ) should be negative. As mentioned, in the characteristic of energy market is favoring long-term planning, with taking into account the opinion and trust of the consumer. Because trust can not be explicitly converted to money, it becomes the separate factor in the performance index which is optimized by producers of energy. The relationship between matrices Q and R (see next section) tells us if the particular operator more counts on profits and ignore consumers opinion, or rather prefers building of trust and relations, dealing with partial reduction of profits. Of course, the situation when producer completely ignores its profits may not be taken into account. 3. Basics of -person LQ game theory In the linear-quadratic game theory, linear invariant systems (1) are taken into account ẋ(t) = Ax(t) + B 1 u 1 (t) + + B N u N (t) x = x() (1) whre x(t) R n, u i (t) R, and A, B - constant matrices. Additionally, players optimise quadratic payoff functions in form (). J = T [x T (t)qx(t) + u T 1 (t)r 1 ut 1 + u T (t)r ut ]d t + xt (T )Q T x(t ) () where x(t) is a state vector, and u 1, u are controls of players. When we assume the horizon T, thus () will take a form 76

3 J = [x T (t)q x(t) + u T 1 (t)r 1 ut 1 + ut (t)r ut ]d t (3) Moreover, the unique equilibrium actions are given by u i (t) = R 1 i i BT P i i Φ(t) (8) where Φ(t) satisfies the transition equation It should be noticed, that, in general case, each player has his own performance index () or (3) being minimized. So, in -person game with infinite time horizon, we can have two indexes with Φ = A c l Φ, Φ() = x (9) A c l = A S 1 P 1 S P and J 1 = J = [x T (t)q 1 x(t) + u T 1 (t)r 11 ut 1 + ut (t)r 1 ut ]d t (4) [x T (t)q x(t) + u T 1 (t)r 1 ut 1 + ut (t)r ut ]d t (5) We also assume, that R i i >, i = 1,. Now we will introduce a concept of a Nash equilibrium. We say that a pair of strategies (u 1, u ) is in the Nash equilibrium, when they satisfies simultaneously J 1 (u 1, u ) J 1 (u 1, u ) and J (u 1, u ) J 1 (u 1, u ) for every admissible u 1, u. In other words, each strategy in the Nash equilibrium is a best response to all other strategies in that equilibrium. In case of a game given by linear equation (1) and index (3), it can be proven (see [7]) that the equilibrium exists and is unique for every initial state if the following conditions are satisfied The set of coupled Riccati equations = A T P 1 P 1 A Q 1 + P 1 S 1 P 1 + P 1 S P = A T P P A Q + P S P + P S 1 P 1 (6) where S i = B i R 1 i i BT, has a stabilising solution i the two Riccati equations = A T K 1 + K 1 A K 1 S 1 K 1 + Q 1 = A T K + K A K S K + Q (7) has symmetric stabilising solution The costs of using this actions is (for player i = 1,) J i = x T Φ i xt (1) where Φ i is the matrix defined as an unique solution of the Lyapunov equation A T c l Φ i + Φ i A c l + Q i + P T i S i P i = (11) There are several ways to solve the problem given by Riccati equations (6), (7). Below we present an algorithm introduced by J.Engwerda [6]. Its advantage is that it can be relatively easy implemented in numerical environments such as MATLAB. 4. Algorithm for solving LQ game In general, there are two groups of algorithms for solving LQ games - accurate (see e.g. [6]) and iterative ([3]). Accurate algorithms can provide an exact solution (if we can obtain the exact eigenvalues of some matrices), but can not be applied to big problems because of its complexity. Nevertheless, ranks of presented problems are small - so this methodology can be used. Presented algorithm comes from [5]. Let s define an invariant subspace as Definition 4.1 Subspace S R n of the matrix A is called inwariant, if Ax S for any x S Invariant subspace can be obtained using eigenvectors of matrix A. Definition 4. Invariant subspace S of matrix A is called stable, if eigenvalues of matrix A related with subspace S have negative real parts In case of invariant subspaces composed with eigenvalues of A, stable invariant subspaces are created by using combinations of eigenvectors related with stable eigenvalues of matrix A 77

4 Definition 4.3 Let s assume that invariant subspace V of matrix A is created as follows X1 V = X If exists X 1, then V is called a graph subspace Using above definitions, the following algorithm can be used (see [5]). 1. Find eigenvalues and eigenvectors of matrices A S1 H i = Q i A T (1) If both of this matrices have stable invariant subspaces, go to step ). Otherwise there are no Nash equilibrium in this game.. Build a matrix M = A S 1 S Q i A T Q A T (13) and find its eigenvalues. If the number of stable eigenvalues (including multiplies) is less then n, there is no Nash equilibrium in this game 3. Calculate all stable invariant subspaces Θ of M. 4. Decompose Θ on three n n matrices as below Then calculate Θ = X Y Z (14) P 1 = Y X 1 (15) P = ZX 1 (16) Obtained matrices P 1, P are solutions of Riccati equation (6) The first step of presented algorithm checks if equations (7) have stabilizing solutions. In steps ) and 3) we verify, if matrix M has stable graph subspaces (and, as a result, if there exist P 1, P ), and step 4) allows to obtain the solution of (6). In case when M has more then one stable n n graph subspaces, we have more than one pair of strategies in Nash equilibrium (the solution is not unique). Performing calculations from step 4) for all of them, we can otbain all of Nash equilibria. It can be easily verified, that maximal number of n strategies in equilibrium is - it is because n of limited number of eigenvectors of matrix M. 5. Simulations Let s take into consideration the following example A = B 1 =.5, B = where weight matrices are 1 Q 1 =, Q =.5 3 and R 1 =.4, R =.. It is easy to notice, that in this model Producer emphasises trust and customer s opinion, and Producer 1 counts on bigger profit. In first step, we obtain two matrices H 1, H. It can be easily proven, that stable graph subspaces exist for both of them. Then, we are calculating M matrix. It has the following eigenvalues λ 1 =.895 λ =.861 λ 3 =.9 λ 4 =.9983 λ 5 =.7111 λ 6 =.84 As we can see, two of them have negative real parts - the condition from step 3) is fulfilled, and the solution is unique. Stable graph subspace exists. Then, using calculations from step 4) we have the following unique solution of Riccati equation (6) P 1 =

5 data which can be helpful in such modelling, is confidential and not available. Nevertheless, on short horizon, such model could probably be enough for many applications. Bibliography Fig. 1. Trajectories in Nash equilibrium P = which leads us to the following closed loop matrix A c l = and trajectories of the system will be as on the figure (1) It can be noticed that, as expected, optimal strategy for Player (the "greedy" one) is to offer a low discount - it will make his profit greater. Player 1 offers much bigger discount, which results with much better opinion after a short while (to make it more visible, in the example the initial aversion to both producers is identical). 6. Conclusions In paper, linear-quadratic model of electricity market has been introduced, which allows to plan and predict some strategies on electricity market. The justification of its parameters has been presented. Then, an algorithm for finding a Nash equilibrium for such model is provided. Thanks to its simplicity, proposed model can be analysed both using numerical and more algebraic methods. The main disadvantage of presented model is the problem with model parameters. There is no easy way to obtain entries of the matrices A, B, Q, R, because they represent such volatile things as trust, attitude, influence of price and so on. Additionally, in most cases, the kind of [1] A.Gałuszka and A.Świerniak. Non-cooperative game approach to multi-robot planning. International Journal of Applied Mathematic and Computer Science, 15(3):359 67, 5. [] J. Brehm. Postdecision changes in the desirability of alternatives. Journal of Abnormal and Social Psychology, 5: , [3] Tobias Damm, Vasile Dragan, and Gerhard Freiling. Lyapunov iterations for coupled riccati differential equations arising in connection with nash differetial games. Mathematical Reports, 9(59):35 46, 7. [4] Alan S. Dick and Kunal Basu. Customer loyalty: Toward an integrated conceptual framework. Journal of the Academy of Marketing Science, (): 113, [5] Jacob Engwerda. LQ Dynamic Optimization and Differential Games. Chichester, Wiley, 5. [6] Jacob Engwerda. Algorithms for computing nash equilibria in deterministic lq games. CMS, Springer-Verlag, 6. [7] Jacob Engwerda. The open-loop linear quadratic differential game revisited. Tilburg University, discussion paper, 6. [8] Henryk Górecki, Stanisław Fuksa, Adam Korytowski, and Wojciech Mitkowski. Sterowanie optymalne w systemach liniowych z kwadratowym wskaźnikiem jakości. Biblioteka Naukowa Inżyniera. Państwowe Wydawnictwa Naukowe, [9] Tadeusz Kaczorek. Teoria Sterowania, volume I. Wydawnictwa Naukowe PWN, [1] P. K. Menon. Short-range nonlinear feedback strategies for aircraft pursuit-evasion. Joumal of Guidance, Control, and Dynamics, 1(1):7 3, [11] Wojciech Mitkowski. Stabilizacja Systemów Dynamicznych. Wydawnictwo AGH, Kraków,

6 Authors: M.Sc.Michał Ganobis University of Science and Technology AGH al.mickiewicza Kraków ganobis@agh.edu.pl This work was supported by Ministry of Science and Higher Education in Poland in the years 8-11 as a research project No N N