The two-level model of interaction between power producers (GC) and System Operator (SO) is considered. The upper level corresponds to GC which try to increase their pro t by distortion information about the characteristics operating costs of their power plants. The lower level corresponds to SO which solves the generation scheduling problem on the basis of technical parameters of the power plants provided by the producers. SO is aimed at minimizing total production costs in Electric Power System (EPS). Two formulations of this model are considered. In the rst case the bounds on generation and capacities of lines are considered insigni cant. For this case we study properties of the objective function of the upper level and investigate the existence of the Nash equilibrium. In the second case we will take into account additional bounds on the power generation and power ows. In this case the standard replacement of the lower level problem by the Karush-Kuhn-Tucker (KKT) optimality conditions is proposed. In this paper, we present a simple numerical example to validate the proposed approach and demonstrate its main features.
A lot of investigations are devoted to research of the electricity market. Today an actual task is study of interaction between System Operator and power producers [1]. Producers in the electricity market are Generation Companies (GC).
Power producers provide their costs to the System Operator in the market conditions. SO solves a generation scheduling problem, minimizing total costs of electricity generation and calculating the nodal prices (dual variables) on the basis of technical parameters of the power plants provided by the producers. To increase their pro t power producers deliberately distort real values of some technical parameters of the power plants thereby implicitly in uencing the prices. This interaction between SO and GC can be modeled using a two-level mathematical programming problems [2, 3]. Earlier the bilevel programming technique was used for modeling the heat energy market [4]. Copyright c by the paper's authors. Copying permitted for private and academic purposes. In: A. Kononov et al. (eds.): DOOR 2016, Vladivostok, Russia, published at http://ceur-ws.org
The two-level model of interaction between System Operator and producers is considered in this paper. The upper level corresponds to the pro t maximization of Generation Companies with true costs functions. The lower level of the problem corresponds to SO e orts to schedule generation and calculate local marginal prices (LMP) on the basis of total production costs minimization. On the upper level there are several GC, each seeks to maximize its own pro ts, therefore on the upper level there is the equilibrium search problem.
In the considered model the limits on generation and capacities of lines are considered insigni cant. For this case we research properties of the objective function of the upper level and prove the existence of the Nash equilibrium.
This paper presents a numerical example demonstrating the applicability of the proposed approach to the solution of the considered problem. 2
The Electric Power System (EPS), consisting of n nodes and m lines is considered [1, 5]. We assume that the rst k (k < n) nodes are power producers and the others (n k) are consumers. Each power producer operates one power plant.
The network topology is given by the incidence n m matrix A with entries aij = >8 1; if line j enters node i, <
0; if line j is not connected to node i, > : 1; if line j leaves node i.
The power ows must satisfy Kirchho 's rst law
Ax = b; where xj is power ow on the line j, bi 0 is power generation for i = 1; k and power demand (in this case bi 0) for i = k + 1; n. Note that matrix A has the following property: if T A = 0, then 1 = 2 = : : : = n.
In this paper the demand is inelastic, i.e. bi = const, i = k + 1; n.
Generation of power producers is determined by quadratic costs function ci(bi) =
ibi2 + ibi + i; i = 1; k; where i > 0, i > 0, i > 0.
System Operator solves the convex quadratic programming problem k k X ci(bi) = X i=1 i=1 ibi2 + ibi + i ! (x;bm1;:i:n:;bk);
Ax = b:
Constraints (
In (
Such pricing system is aimed only to meet the demand with minimum costs and completely ignores the interests of producers. GC are aimed at maximizing their pro ts in market conditions. The minimum total production costs in the EPS is not the objective for them.
In this situation power producers deliberately distort real values of some technical
parameters of the power plants. This means that parameter values i, i, i in (
In this paper it is assumed that i = ^i; = ^i, i.e. we study the impact of parameters i on GC pro t. The problem of existence and nding the Nash equilibrium between GC is investigated. Therefore variable parameters i are used in problem formulation of SO, and for calculating producer's costs we use true values ^i. 3
Search for the Nash equilibrium
To solve the problem (
L(b; x; ) = k X ^ibi2 + ibi + ^i + i=1
T (Ax b): Calculate the partial derivatives and equal them to zero: Lx = AT = 0;
Lx = :
Using the property of matrix incidence A, from (
1 = 2 = : : : =
n = p: 2 ^ibi + i = p:
Note that the value of 2 ^ibi + i is the value of marginal costs of power producer
i, because c0i(bi) = 2 ^ibi + i. Therefore in optimal solution all power producers facing
the same level of marginal costs, which is equal to the established price.
(
Thus in the model all calculations are performed assuming a uniform price determined by GC's current costs.
Now we describe the pro t for power producer i as a function of the zero marginal costs i = c0i(0). We assume, that in practice the impact of parameter i on pro t is more signi cant.
As noted above the main aim of power producer is to maximize pro t
p ( )bi ( )
^ibi ( )2
^ibi ( )
^i ! max;
i
where p ( ) and bi ( ) are the corresponding to a given parameter vector optimal
values of the dual and primal variables, obtained after solving the problem (
Pro t functions of power producers
i( ) = p ( )bi ( )
^ibi ( )2 + ^ibi ( ) + ^i ; i = 1; k;
(
From (
Since A is matrix incidence, then, summing all the equations of the system (
From (
Analytical expression of pro t functions as functions of parameters
substitution of expressions (
Let us make some remarks. We obtained analytical problem solution for SO (the lower level problem) and substituted this solution in objective functions of the upper level problems thus excluded the lower level problem from further consideration. As a result, we received non-cooperative k-person game with pro t functions (12). It should be noted that the pro t of power producer q depends not only on its own zero marginal costs, but also zero marginal costs other power producers. Besides, the increase of its own costs (parameter q) will reduce its own production, and thus, will lead to production increase of other power producers.
It is known that some games allow the reduction to optimization problem [6]. Such games are called potential and nding Nash equilibrium for them is signi cantly facilitated. The majority of practically interesting games are not potential and this is an obstacle to develop computational procedures, guaranteed determine the equilibrium point.
Lemma 1 ([7]). The non-cooperative k-person game with pro t functions (12) is not potential.
Proof. The proof of this lemma can be found in [7].
We prove existence of the Nash equilibrium in this game.
Lemma 2 ([7]). The values Kq < 0 8q = 1; k.
Proof. By simple mathematical reformulation we get
Kq = bq + q (13)
1 " 1 2 ^q ^q # 1 :
Substitute the found partial derivatives in (13) and equate the received expression to zero.
As a result, nding the equilibrium is reduced to the solution system of linear equations of the following form:
q ^q 2 = ^q ^q 2
In vector-matrix form equation (14) can be written as follows: Denote
k 1 = P i=1 ^i
, then where k X i=1 ^i
i Lemma 3 ([7]). If k > 1 then matrix D 1 exists and all its elements are negative. Proof. The complete proof can be found in [7].
Lemma 4 ([7]). The elements of the vector r are negative.
Theorem 1. If k > 1 then the solution of the system (15) exists and is unique, the components of the vector are positive and = 12 6666BBBB ^...1 . . . ... CCC +
C 0
1 1 AC ^k
1 2 ^q ^q Proof. The proof follows from the two last lemmas.
The existence of the Nash equilibrium under assumption of bounded parameters i variation was proved above. As follows from this theorem, there is no need to set upper bounds on i. It is possible even to admit also the negative values of these parameters, still the equilibrium is achieved at the interior point of non-negative orthant, which corresponds to the practical interpretation of i. 4
The two-level problem formulation with constraints on the power generation and power ows In this section we will take into account additional bounds on x and b. Consider rst the problem with one power producer. Without loss of generality, assume that the power producer is in node 1. The problem of the System Operator ^1b12 + 1b1 + ^1 ! (mx;ibn);
A1x = d1
b1; Aix = di; i = 2; n; b1 = bT = n X di: i=1 where A is the incidence matrix, Ai are the rows of matrix A, d is demand. It's obvious that Thus the optimization is carried out only in the variable x. From the optimality conditions obtain 1 = 2 ^1bT + 1 = 2 = 3 = : : : = n: The power producer's pro t subject to (16)-(17) 1b1 ( ^1(b1)2 + ^1b1 + ^1) = (2 ^1bT + 1)bT ( ^1b2T + ^1bT + ^1) is linear, increasing relative to 1 function. Therefore under the constraints pro t reaches its maximum at the right end of the segment: 1 = 1. Let us pass now to the general statement. The problem of the System Operator 0 6 1 6
1 n X( ^ibi2 + ibi + ^i) ! (mx;ibn); i=1
Ax = d
b; x 6 x 6 x; 0 6 b 6 b: (16) (17) (18) (19) (20)
The optimality conditions represent the stationarity conditions, conditions of complementary slackness, constraints on the sign of the dual variables corresponding to inequality constraints: iais s1 +
s2 = 0; s = 1; m; xj ) = 0; j2(xj
xj ) = 0; j = 1; m; i bi = 0; i2(bi 1
bi) = 0; i = 1; n; j1 > 0; j2 > 0; j = 1; m; i1 > 0; i2 > 0; i = 1; n; (22) (23) (24) (25) (26) (27) supplemented by the conditions (19)-(21).
If in some nodes there are no power producers then we set i = i = 0 and bi = 0 for these nodes.
Now interaction between GC and SO is formulated as the bilevel problem with GC at the rst level and SO at the second. We replace the lower level by the system of necessary optimality conditions (19)-(27) obtaining the one-level nonconvex problem.
It is known [9, 10] that the nonlinear constraints of the form
wv = 0; w > 0; v > 0 can be rewritten in the form of linear constraints by introducing an additional Boolean variable # as follows: w 6 M #; v 6 M (1
#); w > 0; v > 0: Let M be a su ciently large constant. In the example below M = 106. Introduce the following Boolean variables: j1, j2, i3, i4 . Then the constraints (24)-(27) are equivalent
The results of numerical example demonstrate the e ect achieved by power producers due to distortion information about the characteristics operating costs of their power plants. Producers increase the level of market prices by changing i. The prices increase provide producers the opportunity to obtain additional pro t. 6
The electricity market model of interaction between power producers (GC) and System Operator (SO) was considered. This model is formulated as two-level mathematical programming problem. Thus on the upper level GC maximizes its own pro ts due to the distortion of technical parameters of the power plants, transmitted SO. Two formulations of this model are considered. In the rst one the bounds on generation and capacities of lines are considered insigni cant. For this case we prove the existence of the Nash equilibrium. The equilibrium is unique and is achieved at the interior point of non-negative orthant. In the second one the model takes into account additional bounds on the power generation and power ows. For this case the equivalent one-level mixed-integer quadratic programming problem is proposed.