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A market power model with price caps and compact DC power flow constraints
Electrical Power and Energy Systems 25 (2003) 301–307 www.elsevier.com/locate/ijepes
A market power model with price caps and compact DC power flow constraints Zuwei Yu* State Utility Forecasting Group, School of Industrial Engineering, Purdue University, Room 334, 1293 A.A. Potter, West Lafayette, IN 47907, USA
Abstract This paper presents a spatial gaming model with price caps for deregulated electricity markets. There has been heated debate on price caps that have been enforced in deregulated electricity markets. Opponents argue that price caps may send wrong economic signals while advocates argue that price caps are good for damping market power. This paper does not intend to take a stand in the argument. Given the fact that price caps are enforced in several deregulated regional electricity markets in the US, a logical step is to reflect this reality in gaming modeling. However, current gaming models have not included any price cap formulation. This paper is the first one to address the issue. DC power flow equations are used for representing the spatial nature of an electrical network. An algorithm is proposed to find a generalized Nash equilibrium under the enforcement of price caps based on the Kuhn– Tucker Vector Optimization Theorem. Case studies show the successful application of the model. The conclusion is that market power impact can be reduced under appropriate price caps. q 2003 Elsevier Science Ltd. All rights reserved. Keywords: Nash equilibrium; Cournot strategy; Kuhn–Tucker vector optimization theorem; Spatial gaming; Price caps; DC power flow
1. Introduction Market power refers to producer’s capability of pricing much higher than marginal cost. This can happen when a producer has a large market share. It has been a major concern since the start of electricity industry deregulation. As a result, there has been increased interest in market power or gaming modeling. These studies are either based on the assumption of supply function competition, or the Cournot competition, or the Stackelberg competition. A combination of different methods can also be found. These studies can also be spatial or non-spatial (i.e. supply and demand are assumed to be at one point). A typical example of non-spatial model, with the supply function competition, is from Green and Newbery [1]. The non-spatial formulation of gaming for electricity markets ignores transmission network capacity limits and wheeling charges. The second class of the studies is based on transportation formulation, e.g. [2 – 4] and a simplified version in this class is based on radial systems [5]. The third class of such studies uses DC power flow equations to represent the spatial nature of the problem. This paper focuses on the discussion of the third class of studies due to the increased attention paid to the gaming modeling with DC power flow constraints in recent years. For example, * Tel.: þ1-765-494-4224; fax: þ 1-765-494-2351. E-mail address: [email protected] (Z. Yu).
Hogan in Ref. [6] presented a spatial model for simulating gaming in electricity markets. He adopted a DC power flow formulation to represent the transmission network. Hobbs in Ref. [7] presented a linear complementary programming (LCP) model with an alternative DC power flow formulation. Hobbs et al. also conducted a spatial gaming study for electricity markets to include possible supply function bidding strategy [8]. Berry et al. studied the effect of transmission system on market power with an application to a system of identical line reactances [9]. The authors in Refs. [10 – 12] developed mixed complementarity programming (MCP) gaming models using DC power flow models based on Wood and Wollenberg [13]. In Ref. [10] we present a gaming model considering transmission line losses while in Ref. [11] we compare the models with and without DC power flow constraints, and present a procedure for resolving multiple equilibria problem. And in Ref. [12] we propose a gaming model for markets with a few oligopoly producers and a group of fringe producers who produce according to their marginal costs. This paper differs from other papers in that it uses the concept of proper solutions rather than the concept of optimal solutions because a gaming problem may only have a proper solution. In addition, this paper is intended to establish a mathematical relationship between vector optimization and the Nash– Cournot gaming. All of the earlier mentioned studies have made
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Z. Yu / Electrical Power and Energy Systems 25 (2003) 301–307
contributions to the area. However, none of these studies have taken into consideration price caps. Price caps have been enforced in a number of deregulated electricity markets, such as the Pennsylvania – Jersey – Maryland (PJM) Power Pool, the New York Power Pool, the New England Power Pool, etc. In California, the price caps have been moving back and forth several times in the past few years [14]. While price caps in these markets have been under heated debate, they are in place and will unlikely be abandoned in the foreseeable future. Therefore, gaming models should reflect this reality. This paper is the first attempt to resolve the problem. In this paper, a spatial gaming model is proposed to take into consideration nodal price caps in regional electricity markets. If there is only one price cap for a power pool, the nodal price caps are then equal. A long- term expansion model (LTEM) is used for setting a price cap (or price caps). The LTEM is a modification to our earlier work [19], with the addition of revenue requirements for the producers. The objective of the LTEM is welfare maximization rather than cost minimization. The price cap is calculated to reflect the ‘long-term marginal cost’ as described in standard economic literature. In addition, compact DC power flow equations are used to approximate the physical laws of transmission networks. The spatial gaming problem is first formulated as a generic, vector profits maximization model, subject to a common set of constraints, including the Cournot conjectural variations. Each component of the vector represents the profit of a specific producer. The problem is then solved using the Kuhn – Tucker (KT) Vector Optimization Theorem (KTVOT) [15]. The mathematical proof of using the KTVOT is trivial and is thus omitted. The algorithm for solving this gaming problem is to convert the vector profit maximization into individual profit maximization using the KTVOT. The KT first order necessary conditions for an efficient point (or, a proper solution) are then developed. The KT necessary conditions for the vector optimization can be solved using an MCP solver. The solution is then checked to see if it is a generalized Nash equilibrium (we will omit the word generalized later on). The whole solution process is similar to the one that is proposed in Ref. [4]. The remainder of the paper is arranged as follows: Section 2 presents the mathematical formulation for the gaming model with price caps. Section 3 describes the solution procedure of the gaming problem. Section 4 illustrates case studies and analysis. Section 5 concludes the paper with summary and topics for further investigation.
markets with multiple-price zones (e.g. the California spot electricity markets). However, the model can also be applied to markets with a mechanism that is based on nodal pricing. To facilitate the presentation of the model, we first list our notations below. i,j,m,n,l zone or node indices N total number of nodes S,k producer/supplier indices, with S ¼ 1; 2; … Sm total number of producers xði; jÞ line ði; jÞ’s reactance (per unit) Fmax ði; jÞ line ði; jÞ’s power flow limit (e-10) (MW) Pmaxði; SÞ S0 maximum production limit at node i (MW) wh(i,j ) fixed wheeling cost from i to j($/MWh) PG (i,S ) power production by S at i (MWh) C(i,S ) production cost of S at i ($/MWh) F(i,j,S ) (contractual) power flow on (i,j ) (MW) attributed to S P(i ) price at i ($/MWh) q(i,S ) power sale or supply to i by S (MW) p(S ) profit of S ($) a0(i ) price-axis intercept for i’s demand function (people also call it the inverse demand function) b0(i ) slope of demand function at node i Pnet(i ) net power injection at node i X(i,j ) element of the inverse of the B matrix [13] cap(i ) price cap at node i for real power or energy ($/ MWh) The mathematical formulation of the gaming model can then be expressed as: Min½2pð1Þ; 2pð2Þ; …; 2pðSm Þ; Subject to PGði; SÞ 2 Pmax ði; SÞ # 0; X X qðm; SÞ ¼ PGðm; SÞ; m
m
qði; SÞ 2 PGði; SÞ þ
The gaming model developed in this paper is only for real power markets and the time span is for 1 h (MW and MWh are equal in this case). No arbitrage is assumed. Losses are ignored. The model is proposed for electricity
X
;i; S:
ð2:2Þ
;S:
ð2:3Þ
½Fði; n; SÞ 2 Fðn; i; SÞ # 0;
n7 !i
ð2:4Þ
;i; S: X Fði; j; kÞ 2 Fmax ði; jÞ # 0;
;i; j:
ð2:5Þ
k
2PGði; SÞ # 0;
;i; S:
ð2:6Þ
2Fði; j; SÞ # 0;
;i; j; S:
ð2:7Þ
2qði; SÞ # 0; PðiÞ 2 capðiÞ # 0;
2. The gaming model with price caps
ð2:1Þ
X
ð2:9Þ
;i:
½Fði; j; kÞ 2 Fðj; i; kÞ ¼
N X n¼1
k
Pnet ðiÞ ¼
ð2:8Þ
;i; S:
X k
½Xði; nÞ 2 Xðj; nÞ Pnet ðnÞ: xði; jÞ ð2:10Þ
½PGði; kÞ 2 qði; kÞ;
;i:
ð2:11Þ
Z. Yu / Electrical Power and Energy Systems 25 (2003) 301–307
The Cournot strategy for each S:
ð2:11aÞ
P Where producer S0 profit is defined as pðSÞ ¼ m P P P PðmÞqðm; SÞ 2 m Cðm; SÞ 2 m n7!m Fðm; n; SÞwhðm; nÞ and the Cournot strategy as ›qðm; SÞ=›qði; kÞ ¼ 0 for i – m and k – S k – S; as described in microeconomics. PðiÞ is a function of the qði; SÞs and has a general P P functional form of PðiÞ ¼ fn½ k qði; kÞ; with k qði; kÞ as demand at node i and fn(·) representing a function relationship. Linear demand functions are used in our case studies and this would not degrade the generality of the model. In our formulation, n 7 ! m means that all of the nodes n’s are connected to node m. Here the power flow is expressed as the difference of two positive parts: Fði; j; SÞ 2 Fðj; i; SÞ: This will guarantee that wheeling charges are always positive (see Ref. [4] for more details). The production cost function is defined as: Cði; SÞ ¼ c0 ði; SÞ þ c1 ði; SÞPGði; SÞ þ c2 ði; SÞ
½PGði; SÞ2 : Eq. (2.3) states that S0 total supply equals its total production. This constraint is only for illustrative purpose since the meaning of Eq. (2.3) is implied in constraint (2.4). Eq. (2.4) represents an equilibrium supply for S: its sale to i is no more than its production at i minus its net export. Eq. (2.4) is necessary to capture gaming and accounting. This is because each producer’s profit is determined by its sales to all of the individual nodes minus its operating costs, including its wheeling cost. Note that inequality sign is used in Eq. (2.4) and this will limit the search space for its LaGrange multipliers. The reason for the way Eq. (2.5) is formulated is to avoid the situation that Fði; j; SÞ $ Fmax ði; jÞ (see [4] for more details). Eq. (2.9) represents the nodal (or zonal) price cap formulation. There is a possibility that a single cap is enforced for a power pool. In this case, cap(i )s are equal in the formulation. Eqs. (2.10) and (2.11) represent DC power flow equations (it is easy to deduce these equations from Ref. [13] by eliminating power flow angles). Eq. (2.11a) is an indicative constraint indicating that the Cournot conjectural variations will be used in finding a Nash equilibrium. Notice that the Cournot conjectural variations in quantity are also implied in the KT first order necessary conditions for vector optimization (There are no conjectural variations in the KTVOT while the Cournot conjectural variations are zero in the Cournot strategy. This means that both are equivalent). Table 1 lists the constraints and the corresponding LaGrange multipliers. Constraint (2.11) is combined into constraint (2.10) so that there is no need to assign a LaGrange multiplier for it. According to the KTVOT, the KT function (a partial LaGrange function), f(S ), can then be formulated below for each producer. The basic procedure for constructing the KT conditions for a proper solution model is explained in Section 3.
303
Table 1 The list of Lagrange multipliers Constraint
LaGrange multiplier
(2.1) (2.2) (2.4) (2.5) (2.9) (2.10)
m(S ) . 0 v(I,S ) $ 0 b(I,S ) $ 0 v(i,j ) $ 0 d(i ) $ 0 a(i,j ) unrestricted
fðSÞ ¼ mðSÞ
X
"
X
Cðm; SÞ þ
Fðm; n; SÞwhðm; nÞ
n7 !m
m
#
2 PðmÞqðm; SÞ þ
X
vðm; SÞ½PGðm; SÞ 2 Pmax ðm; SÞ
m
þ
X X
"
vðm; nÞ
X
m n7 !m
þ
X
# Fðm; n; kÞ 2 Fmax ðm; nÞ
k
"
bðm; SÞ qðm; SÞ 2 PGðm; SÞ þ
X
ðFðm; n; SÞ
n7 !m
m
# 2 Fðn; m; SÞÞ þ
X
dðmÞ½PðmÞ 2 capðmÞ
m
þ
X X
(
aðm; nÞ
X ½Fðm; n; kÞ 2 Fðn; m; kÞ
m n7 !m N X Xðm; lÞ 2 Xðn; lÞ 2 xðm; nÞ l¼1
"
X
#) ðPGðl; kÞ 2 qðl; kÞÞ
;
k
ð2:12Þ where vðSÞ . 0; as is required by the KTVOT.
3. A solution procedure As indicated in our earlier paper [4], there are several techniques that can be used for solving a multiple-player, non-cooperative gaming problem. The goal is to find Nash equilibrium (or Nash equilibria). However, according to our experience, the MCP technique often outperforms other methods in speed and accuracy. Besides, standard MCP solvers are readily available. The idea is to construct the KT necessary conditions for vector optimization and solve these conditions using an MCP solver to obtain Nash equilibrium. The following is a brief description of how the conversion is carried out. According to the KTVOT, from Eq. (2.12), the first order KT conditions, i.e. the derivatives of f(S ) with respect to qði; SÞ are represented by the following equation.
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Z. Yu / Electrical Power and Energy Systems 25 (2003) 301–307
8 > > <
9 > > =
›PðiÞ dqði; SÞ ¼ mðSÞ 2PðiÞ 2 ½qði; SÞ 2 dðiÞ X > › qði; kÞ > > > : ; k
þ bði; SÞ þ
X X m n7 !m
aðm; nÞ
Xðm; iÞ 2 Xðn; iÞ ; xðm; nÞ
;i; S: P P In this equation, the term, m n7!m aðm; nÞ½Xðm; iÞ 2 Xðn; iÞ=xðm; nÞ; represents the effect of the DC power flow constraints. The KT necessary conditions with respect to qði; SÞ for a proper solution can then be expressed below. dqði; SÞ $ 0;
dqði; SÞ £ qði; SÞ ¼ 0;
;i; S:
ð3:1Þ
Eq. (3.1) can also be regarded as the mixed complementarity conditions. Other KT conditions for a proper solution can be derived similarly and are omitted for brevity. An MCP solver, MILES [16], is used for obtaining the Nash equilibrium in the paper. It is a program for solving non-linear complementarity problems. The solution procedure is a generalized Newton method with a backtracking line search, coupled with the Lemke’s almost complementarity pivoting scheme in which lower and upper bounds are represented implicitly. The solution from MILES must be verified to make sure that Nash equilibrium is obtained. This can be done by remaximizing one producer’s profit while keeping other producers’ production, sales and flows fixed. This procedure must be repeated for each and every producer. If no producer’s profit is changed, the solution is proved to be a Nash equilibrium. The solution process can be shown in Fig. 1. Note that when no Nash equilibrium can be found,
further analysis of the problem is needed, including reexamining the MCP conditions. According to our experience, the MCP conditions can be complicated and the mishandling of the conditions can cause problems in finding Nash equilibrium. This is why we emphasize the reexamination of the MCP conditions. We otherwise have not encountered any serious problem in obtaining Nash equilibria for different problems. Moreover, the market power model (1) in this paper has linear constraints that are well known to be convex and differentiable, and the vector objective functions are all convex and differentiable too, according to Theorem 3.5.2 in Ref. [20], it is sufficient for the model to have a proper efficient solution. Note that each equality constraint can be represented by two inequality constraints to establish the above argument. In addition, as proved in a proposition in the appendix, the KTVOT necessary conditions characterize a Nash equilibrium.
4. Case studies In this section, two case studies are presented to illustrate the application of the proposed model. In Case I, with the use of an econometric model the long-term coal price are projected to be $1.1/MMBTU and the gas prices are $4.5/MMBTU for 2001, $4.1/MMBTU for 2002 and $3.7/MMBTU for 2003 and the next 13 years (these prices are similar to the projection done by the Energy Information Administration of the US). A capital recovery factor of 20% is used for plant capital recovery, based on our joint study with several utilities. The capital costs are $400 kW21 for gas peaking plants, $580 kW21 for combine cycle gas plants and $900 kW21 for base load coal plants. A long-term marginal cost of $148.75 MWh21 is obtained from the LTEM model. This is why the price cap is set at $150 MWh21. As for Case II, the long run marginal cost is assumed because the system data is dated back in the 1960s. Therefore, Case II is for illustrating the effect of price caps on the system outcome of gaming without considering applicability. 4.1. Case 1. A six-node system
Fig. 1. Solution flowchart.
This six-node system is the same one as introduced in Ref. [4] except that there are no emission costs and limits. The detailed data of this system can be found in Ref. [4]. There are three Cournot-type producers in this case. Price caps for this case are $150 MWh21 for all nodes (This price cap coincides with the price cap that has been suggested for the California electricity markets [17]). The profits and nodal prices with and without price caps are illustrated in Tables 2 and 3. Both solutions are proved to be Nash equilibria. We can see that with price cap enforcement, the producers’ profits are reduced somewhat (about 5%) but the reduction in profit is not uniform across the producers. The percentage reduction of profit from producer one is the
Z. Yu / Electrical Power and Energy Systems 25 (2003) 301–307
305
Table 2 Profits without and with price caps ($)
Table 4 Nodal demand and price data for case 2
Profits
Node
Producer
W/o caps With caps Change
1
2
3
Total
134358.15 133069.56 21288.59
81830.67 74466.71 27363.96
69231.07 63088.17 26142.90
285419.89 270624.44 214795.45
least. This may be caused by the fact that producer one’s generators are located strategically better than the other two producers’ generators. Due to the enforcement of price caps, all of the nodal prices are no more than 150 MWh21. As a result, all consumers benefit due to the decline of the nodal prices. The total social welfare, by intuition, is increased but detailed calculation is trivial and is thus omitted. According to microeconomics, market power of the producers is reduced in this case. Since the model is a short-term model of one hour, the market power reduction is also short-term in nature. 4.2. Case 2. An IEEE 30-node system In Case 2, an IEEE 30-node system is used for testing the gaming model. The electrical network and generation unit data of this system can be found in Ref. [18]. Three producers are assumed to compete on the Cournot terms. Assume that producer one owns the plant at node 1, producer two owns the plants at nodes 2 and 5, and producer three owns plants at nodes 8, 11 and 13. The demand function for each node is also assumed linear with a slope of one. The intercepts of the nodal demand functions on the vertical axis (price axis) are listed in Table 4. These intercepts are generated arbitrarily without degrading the generality of the model. The wheeling cost for each line is derived using the following formula: whði; jÞ ¼ 1:0 þ uniformð20:1; þ0:1Þ where uniform( ) is a uniform random number generator. The results are summarized in Tables 4– 6. The highest price without a price cap is $21.426 MWh21 at node 5, as is illustrated in Table 4. Now suppose a price cap of $16 MWh21 is enforced. As a result, the price at node 5 is reduced to $16 MWh21. It is interesting to notice that the nodal prices at nodes other than node 5 are all increased somewhat after the price cap enforcement. However, as is
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30
Price w/o caps
Price with caps
25.00 30.85 25.00 35.00 67.10 25.00 31.50 23.80 25.00 22.90 21.20 25.00 25.00 23.10 24.10 21.75 24.50 21.60 24.75 21.10 28.75 25.00 21.60 24.35 25.00 21.75 25.00 25.00 21.20 25.30
10.985 12.161 12.794 14.798 21.426 11.894 13.335 11.165 12.520 12.454 10.721 11.820 12.096 11.970 12.961 11.648 12.981 13.017 14.182 12.787 14.735 13.652 13.119 14.009 14.258 14.179 13.607 12.940 13.489 14.389
11.071 12.247 12.841 14.882 16.000 11.978 13.421 11.246 12.567 12.490 10.802 11.901 12.183 12.051 13.042 11.729 13.023 13.098 14.219 12.824 14.771 13.689 13.200 14.045 14.306 14.227 13.693 13.026 13.575 14.475
indicated in Table 5, the profits of the producers are all reduced after the price cap enforcement. This again indicates that consumers, on average, benefit from the price cap. To be precise, only consumers at node 5 actually benefit from the price cap and the consumers at nodes other than node 5 see minor price increases as a result of the price cap enforcement. A caution is offered here that the result for this case is just a result of the way the price cap is set and different results can be obtained with different price capping strategies. It is worthwhile to make a further comment on the profit reduction in Case 2. Remember that producer two has a generator at node 5 where demand is high (see a0(5) in Table 4). Therefore, without a price cap, producer two has
Caps
W/o caps W/ caps
a0(i )
Table 5 Comparison of profits with and without price caps
Table 3 Nodal prices without and with price caps for case 1 ($ MWh21) Price
Parameter
Profit
Node 1
2
3
4
5
6
171.87 150
166.44 150
167.20 150
154.08 150
164.36 150
152.18 150
W/o caps With caps Change
Producer 1
Producer 2
Producer 3
Total profit
1275.791 1206.687 269.104
1206.545 983.316 2223.138
904.004 848.898 255.106
3386.340 3038.901 2347.439
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Table 6 Comparison of production quantities Producer Cases
1
2
3
Total
W/o caps With caps Change
161.87 165.16 3.29
123.81 124.01 0.20
105.00 105.00 0.00
390.68 394.17 3.49
Then, f ðxÞ ¼ ½f1 ðxÞ; …fSm ðxÞ: Similarly, the constraints in model (2.1) can be written as a row vector G(x) # 0, with the provision that each equality constraint is converted to two inequalities. It follows that model (2.1) can be expressed as: Minf ðxÞ Subject to GðxÞ # 0;
an advantage over its competitors. It is then understandable that this producer would reduce its profit most after the price cap enforcement as compared with the other two producers. Even though producer two’s total production quantity is not reduced very much (see Table 6), the price cap forces it to reduce its sale to node 5. It is also possible that producer two would increase its sale to other nodes that have lower prices.
5. Conclusions This paper is the first to model spatial gaming with price caps for deregulated electricity markets. The logic for this effort is that price caps are enforced in several electricity markets in the US and existing electricity gaming models have not reflected this fact. The DC power flow equations are included to represent the physical characteristics of electrical networks. The generic gaming problem is formulated as a vector maximization model. The KTVOT is used to find Nash equilibria. The paper claims that the KTVOT includes the concept of the Cournot strategy in gaming. The MCP conditions are then formulated and solved by our using MILES. Two case studies are used to illustrate the application of the proposed model with satisfactory results. Price caps can reduce market power in the short-term as shown by the case studies. The paper by no means has concluded the gaming modeling effort in this area and more work needs to be done, for example, to include losses in the formulation.
Appendix A
x $ 0:
Define a positive vector m and a non-negative vector l of LaGrange multipliers for the constraints, the KT necessary conditions for a proper solution can then be expressed below. (i) km; 7f ðxp Þl þ kl; 7Gðxp Þl ¼ 0; (ii) kl; Gðxp Þl ¼ 0; (iii) m . 0; l $ 0: where k l represents dot product, 7 gradient and x p the proper solution point. Note that the above KT necessary conditions are modified by the Cournot assumptions (These KT conditions may be called the diagonal conditions. See J.B. Rosen (1965) for more details).We next show that the above KT necessary conditions characterize Nash equilibrium. From the KT definition of a proper efficient solution, (see Definition 3.1.10 in Ref. [20]), there is no h [ Rn such that k7fS ðxp Þ; hl # 0 for all S with at least one S resulting in k7fS ðxp Þ; hl , 0 and that kh; 7gj ðxp Þl # 0;
for any j [ Jðxp Þ ¼ {j : gj ðxp Þ ¼ 0}:
Notice that the above definition says that for the KT proper solution, there is no h that can result in k7fS ðxp Þ; hl , 0 for a single S and satisfy the constraints at the same time. (Remember that k7fS ðxp Þ; hl , 0 indicates that producer S can improve its objective and also note that the equivalence of KT necessary conditions and the KT definition for a proper solution is proved in Theorem 3.5.2 in Ref. [20].) This means that no producer can improve its objective unilaterally, which is a characterization of a generalized Nash equilibrium. The proof is complete. A
On the equivalence of the KTVOT and a Nash equilibrium Proposition. The KTVOT necessary conditions for a proper solution is equivalent to a Nash equilibrium for the gaming model (2.1) considering the Cournot assumptions. Proof. Define a row vector, x, for representing all of the variables qði; SÞ; Fði; j; SÞ; PGði; SÞ (Note that PðiÞ and Cði; SÞ are functions of these variables). Define fS ðxÞ ¼ 2pðSÞ þ 0ðSÞx1; where 0(S ) is a null row vector and x1 is a column vector representing those variables in x that are not in p(S ).
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with CO2 emission tax and limits. Int J Electric Power Energy Syst 2001;23(6):451–7. Bushnell J, Day C, Green R, Halseth A, Read E, Rogers J, Rudkevich A, Scott T, Smeers Y, Huntington H. An international comparison of models for measuring market power in electricity. Energy Modeling Forum, Stanford University; 2000. Hogan W. A market power model with strategic interaction in electricity networks. Energy J 1997;18(4):107–41. Hobbs B. LCP models of Nash–Cournot competition in bilateral and POOLCO-based power markets. Proceedings of the IEEE PES Winter Meeting; 1999. Hobbs B, Metzler C, Pang J-S. Strategic gaming analysis for electric power systems: an MPEC approach. IEEE Trans Power Syst 2000; 15(2):638–45. Berry C, Hobbs B, Meroney W, O’Neill R, Stewart Jr. W. Understanding how market power can arise in network competition: a game theoretic approach. Utility Policy 1999;8:139–58. Yu Z. A proposed CLP method for allocating real power losses of multiple flows. IEEE PES Lett, IEEE Power Engng Rev 2001; November:45–7. Yu Z, Morin T. Comparing transportation-constrained gaming to DC power flow-constrained gaming. Under review, IEEE Trans Power Syst.
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[12] Yu Z, Sparrow FT, Morin TL, Nderitu G. A Stackelberg price leadership model with applications to deregulated electricity markets. Proceedings of the IEEE PES 2000 Winter Meeting, Singapore; January 27–February 2000. [13] Wood AJ, Wollenberg BF. Power generation, operation and control. New York: Wiley; 1984. [14] EEnergy Informer, What’s The Right Price Caps, If Any? August 2000. p. 1–2. [15] Kuhn HW, Tucker AW. Nonlinear programming. Proceedings of the Second Berkley Symposium on Mathematical Statistics and Probability; 1951. p. 481– 92. [16] GAMS—the solver manuals, GAMS Development Corporation. [17] Electric Utility Week, California Drama Rises: State Keeps Pressing FERC, Talking of Rebellion; November 20, 2000. p. 1 –3. [18] Freris LL, et al. Investigation of the load flow problem. Proc IEE 1968;115(10):1459–69. [19] Sparrow FT, Bowen BH, Yu Z. Modeling long-term capacity expansion options for the Southern African power pool. Proceedings of the IASTED International Conference on Power and Energy Systems; November 1999. [20] Sawaragi Y, Nakayama H, Tanino T. Theory of multiobjective optimization. London: Academic Press; 1985. p. 90.
A market power model with price caps and compact DC power flow constraints Zuwei Yu* State Utility Forecasting Group, School of Industrial Engineering, Purdue University, Room 334, 1293 A.A. Potter, West Lafayette, IN 47907, USA
Abstract This paper presents a spatial gaming model with price caps for deregulated electricity markets. There has been heated debate on price caps that have been enforced in deregulated electricity markets. Opponents argue that price caps may send wrong economic signals while advocates argue that price caps are good for damping market power. This paper does not intend to take a stand in the argument. Given the fact that price caps are enforced in several deregulated regional electricity markets in the US, a logical step is to reflect this reality in gaming modeling. However, current gaming models have not included any price cap formulation. This paper is the first one to address the issue. DC power flow equations are used for representing the spatial nature of an electrical network. An algorithm is proposed to find a generalized Nash equilibrium under the enforcement of price caps based on the Kuhn– Tucker Vector Optimization Theorem. Case studies show the successful application of the model. The conclusion is that market power impact can be reduced under appropriate price caps. q 2003 Elsevier Science Ltd. All rights reserved. Keywords: Nash equilibrium; Cournot strategy; Kuhn–Tucker vector optimization theorem; Spatial gaming; Price caps; DC power flow
1. Introduction Market power refers to producer’s capability of pricing much higher than marginal cost. This can happen when a producer has a large market share. It has been a major concern since the start of electricity industry deregulation. As a result, there has been increased interest in market power or gaming modeling. These studies are either based on the assumption of supply function competition, or the Cournot competition, or the Stackelberg competition. A combination of different methods can also be found. These studies can also be spatial or non-spatial (i.e. supply and demand are assumed to be at one point). A typical example of non-spatial model, with the supply function competition, is from Green and Newbery [1]. The non-spatial formulation of gaming for electricity markets ignores transmission network capacity limits and wheeling charges. The second class of the studies is based on transportation formulation, e.g. [2 – 4] and a simplified version in this class is based on radial systems [5]. The third class of such studies uses DC power flow equations to represent the spatial nature of the problem. This paper focuses on the discussion of the third class of studies due to the increased attention paid to the gaming modeling with DC power flow constraints in recent years. For example, * Tel.: þ1-765-494-4224; fax: þ 1-765-494-2351. E-mail address: [email protected] (Z. Yu).
Hogan in Ref. [6] presented a spatial model for simulating gaming in electricity markets. He adopted a DC power flow formulation to represent the transmission network. Hobbs in Ref. [7] presented a linear complementary programming (LCP) model with an alternative DC power flow formulation. Hobbs et al. also conducted a spatial gaming study for electricity markets to include possible supply function bidding strategy [8]. Berry et al. studied the effect of transmission system on market power with an application to a system of identical line reactances [9]. The authors in Refs. [10 – 12] developed mixed complementarity programming (MCP) gaming models using DC power flow models based on Wood and Wollenberg [13]. In Ref. [10] we present a gaming model considering transmission line losses while in Ref. [11] we compare the models with and without DC power flow constraints, and present a procedure for resolving multiple equilibria problem. And in Ref. [12] we propose a gaming model for markets with a few oligopoly producers and a group of fringe producers who produce according to their marginal costs. This paper differs from other papers in that it uses the concept of proper solutions rather than the concept of optimal solutions because a gaming problem may only have a proper solution. In addition, this paper is intended to establish a mathematical relationship between vector optimization and the Nash– Cournot gaming. All of the earlier mentioned studies have made
0142-0615/03/$ - see front matter q 2003 Elsevier Science Ltd. All rights reserved. PII: S 0 1 4 2 - 0 6 1 5 ( 0 2 ) 0 0 0 4 0 - 6
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contributions to the area. However, none of these studies have taken into consideration price caps. Price caps have been enforced in a number of deregulated electricity markets, such as the Pennsylvania – Jersey – Maryland (PJM) Power Pool, the New York Power Pool, the New England Power Pool, etc. In California, the price caps have been moving back and forth several times in the past few years [14]. While price caps in these markets have been under heated debate, they are in place and will unlikely be abandoned in the foreseeable future. Therefore, gaming models should reflect this reality. This paper is the first attempt to resolve the problem. In this paper, a spatial gaming model is proposed to take into consideration nodal price caps in regional electricity markets. If there is only one price cap for a power pool, the nodal price caps are then equal. A long- term expansion model (LTEM) is used for setting a price cap (or price caps). The LTEM is a modification to our earlier work [19], with the addition of revenue requirements for the producers. The objective of the LTEM is welfare maximization rather than cost minimization. The price cap is calculated to reflect the ‘long-term marginal cost’ as described in standard economic literature. In addition, compact DC power flow equations are used to approximate the physical laws of transmission networks. The spatial gaming problem is first formulated as a generic, vector profits maximization model, subject to a common set of constraints, including the Cournot conjectural variations. Each component of the vector represents the profit of a specific producer. The problem is then solved using the Kuhn – Tucker (KT) Vector Optimization Theorem (KTVOT) [15]. The mathematical proof of using the KTVOT is trivial and is thus omitted. The algorithm for solving this gaming problem is to convert the vector profit maximization into individual profit maximization using the KTVOT. The KT first order necessary conditions for an efficient point (or, a proper solution) are then developed. The KT necessary conditions for the vector optimization can be solved using an MCP solver. The solution is then checked to see if it is a generalized Nash equilibrium (we will omit the word generalized later on). The whole solution process is similar to the one that is proposed in Ref. [4]. The remainder of the paper is arranged as follows: Section 2 presents the mathematical formulation for the gaming model with price caps. Section 3 describes the solution procedure of the gaming problem. Section 4 illustrates case studies and analysis. Section 5 concludes the paper with summary and topics for further investigation.
markets with multiple-price zones (e.g. the California spot electricity markets). However, the model can also be applied to markets with a mechanism that is based on nodal pricing. To facilitate the presentation of the model, we first list our notations below. i,j,m,n,l zone or node indices N total number of nodes S,k producer/supplier indices, with S ¼ 1; 2; … Sm total number of producers xði; jÞ line ði; jÞ’s reactance (per unit) Fmax ði; jÞ line ði; jÞ’s power flow limit (e-10) (MW) Pmaxði; SÞ S0 maximum production limit at node i (MW) wh(i,j ) fixed wheeling cost from i to j($/MWh) PG (i,S ) power production by S at i (MWh) C(i,S ) production cost of S at i ($/MWh) F(i,j,S ) (contractual) power flow on (i,j ) (MW) attributed to S P(i ) price at i ($/MWh) q(i,S ) power sale or supply to i by S (MW) p(S ) profit of S ($) a0(i ) price-axis intercept for i’s demand function (people also call it the inverse demand function) b0(i ) slope of demand function at node i Pnet(i ) net power injection at node i X(i,j ) element of the inverse of the B matrix [13] cap(i ) price cap at node i for real power or energy ($/ MWh) The mathematical formulation of the gaming model can then be expressed as: Min½2pð1Þ; 2pð2Þ; …; 2pðSm Þ; Subject to PGði; SÞ 2 Pmax ði; SÞ # 0; X X qðm; SÞ ¼ PGðm; SÞ; m
m
qði; SÞ 2 PGði; SÞ þ
The gaming model developed in this paper is only for real power markets and the time span is for 1 h (MW and MWh are equal in this case). No arbitrage is assumed. Losses are ignored. The model is proposed for electricity
X
;i; S:
ð2:2Þ
;S:
ð2:3Þ
½Fði; n; SÞ 2 Fðn; i; SÞ # 0;
n7 !i
ð2:4Þ
;i; S: X Fði; j; kÞ 2 Fmax ði; jÞ # 0;
;i; j:
ð2:5Þ
k
2PGði; SÞ # 0;
;i; S:
ð2:6Þ
2Fði; j; SÞ # 0;
;i; j; S:
ð2:7Þ
2qði; SÞ # 0; PðiÞ 2 capðiÞ # 0;
2. The gaming model with price caps
ð2:1Þ
X
ð2:9Þ
;i:
½Fði; j; kÞ 2 Fðj; i; kÞ ¼
N X n¼1
k
Pnet ðiÞ ¼
ð2:8Þ
;i; S:
X k
½Xði; nÞ 2 Xðj; nÞ Pnet ðnÞ: xði; jÞ ð2:10Þ
½PGði; kÞ 2 qði; kÞ;
;i:
ð2:11Þ
Z. Yu / Electrical Power and Energy Systems 25 (2003) 301–307
The Cournot strategy for each S:
ð2:11aÞ
P Where producer S0 profit is defined as pðSÞ ¼ m P P P PðmÞqðm; SÞ 2 m Cðm; SÞ 2 m n7!m Fðm; n; SÞwhðm; nÞ and the Cournot strategy as ›qðm; SÞ=›qði; kÞ ¼ 0 for i – m and k – S k – S; as described in microeconomics. PðiÞ is a function of the qði; SÞs and has a general P P functional form of PðiÞ ¼ fn½ k qði; kÞ; with k qði; kÞ as demand at node i and fn(·) representing a function relationship. Linear demand functions are used in our case studies and this would not degrade the generality of the model. In our formulation, n 7 ! m means that all of the nodes n’s are connected to node m. Here the power flow is expressed as the difference of two positive parts: Fði; j; SÞ 2 Fðj; i; SÞ: This will guarantee that wheeling charges are always positive (see Ref. [4] for more details). The production cost function is defined as: Cði; SÞ ¼ c0 ði; SÞ þ c1 ði; SÞPGði; SÞ þ c2 ði; SÞ
½PGði; SÞ2 : Eq. (2.3) states that S0 total supply equals its total production. This constraint is only for illustrative purpose since the meaning of Eq. (2.3) is implied in constraint (2.4). Eq. (2.4) represents an equilibrium supply for S: its sale to i is no more than its production at i minus its net export. Eq. (2.4) is necessary to capture gaming and accounting. This is because each producer’s profit is determined by its sales to all of the individual nodes minus its operating costs, including its wheeling cost. Note that inequality sign is used in Eq. (2.4) and this will limit the search space for its LaGrange multipliers. The reason for the way Eq. (2.5) is formulated is to avoid the situation that Fði; j; SÞ $ Fmax ði; jÞ (see [4] for more details). Eq. (2.9) represents the nodal (or zonal) price cap formulation. There is a possibility that a single cap is enforced for a power pool. In this case, cap(i )s are equal in the formulation. Eqs. (2.10) and (2.11) represent DC power flow equations (it is easy to deduce these equations from Ref. [13] by eliminating power flow angles). Eq. (2.11a) is an indicative constraint indicating that the Cournot conjectural variations will be used in finding a Nash equilibrium. Notice that the Cournot conjectural variations in quantity are also implied in the KT first order necessary conditions for vector optimization (There are no conjectural variations in the KTVOT while the Cournot conjectural variations are zero in the Cournot strategy. This means that both are equivalent). Table 1 lists the constraints and the corresponding LaGrange multipliers. Constraint (2.11) is combined into constraint (2.10) so that there is no need to assign a LaGrange multiplier for it. According to the KTVOT, the KT function (a partial LaGrange function), f(S ), can then be formulated below for each producer. The basic procedure for constructing the KT conditions for a proper solution model is explained in Section 3.
303
Table 1 The list of Lagrange multipliers Constraint
LaGrange multiplier
(2.1) (2.2) (2.4) (2.5) (2.9) (2.10)
m(S ) . 0 v(I,S ) $ 0 b(I,S ) $ 0 v(i,j ) $ 0 d(i ) $ 0 a(i,j ) unrestricted
fðSÞ ¼ mðSÞ
X
"
X
Cðm; SÞ þ
Fðm; n; SÞwhðm; nÞ
n7 !m
m
#
2 PðmÞqðm; SÞ þ
X
vðm; SÞ½PGðm; SÞ 2 Pmax ðm; SÞ
m
þ
X X
"
vðm; nÞ
X
m n7 !m
þ
X
# Fðm; n; kÞ 2 Fmax ðm; nÞ
k
"
bðm; SÞ qðm; SÞ 2 PGðm; SÞ þ
X
ðFðm; n; SÞ
n7 !m
m
# 2 Fðn; m; SÞÞ þ
X
dðmÞ½PðmÞ 2 capðmÞ
m
þ
X X
(
aðm; nÞ
X ½Fðm; n; kÞ 2 Fðn; m; kÞ
m n7 !m N X Xðm; lÞ 2 Xðn; lÞ 2 xðm; nÞ l¼1
"
X
#) ðPGðl; kÞ 2 qðl; kÞÞ
;
k
ð2:12Þ where vðSÞ . 0; as is required by the KTVOT.
3. A solution procedure As indicated in our earlier paper [4], there are several techniques that can be used for solving a multiple-player, non-cooperative gaming problem. The goal is to find Nash equilibrium (or Nash equilibria). However, according to our experience, the MCP technique often outperforms other methods in speed and accuracy. Besides, standard MCP solvers are readily available. The idea is to construct the KT necessary conditions for vector optimization and solve these conditions using an MCP solver to obtain Nash equilibrium. The following is a brief description of how the conversion is carried out. According to the KTVOT, from Eq. (2.12), the first order KT conditions, i.e. the derivatives of f(S ) with respect to qði; SÞ are represented by the following equation.
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Z. Yu / Electrical Power and Energy Systems 25 (2003) 301–307
8 > > <
9 > > =
›PðiÞ dqði; SÞ ¼ mðSÞ 2PðiÞ 2 ½qði; SÞ 2 dðiÞ X > › qði; kÞ > > > : ; k
þ bði; SÞ þ
X X m n7 !m
aðm; nÞ
Xðm; iÞ 2 Xðn; iÞ ; xðm; nÞ
;i; S: P P In this equation, the term, m n7!m aðm; nÞ½Xðm; iÞ 2 Xðn; iÞ=xðm; nÞ; represents the effect of the DC power flow constraints. The KT necessary conditions with respect to qði; SÞ for a proper solution can then be expressed below. dqði; SÞ $ 0;
dqði; SÞ £ qði; SÞ ¼ 0;
;i; S:
ð3:1Þ
Eq. (3.1) can also be regarded as the mixed complementarity conditions. Other KT conditions for a proper solution can be derived similarly and are omitted for brevity. An MCP solver, MILES [16], is used for obtaining the Nash equilibrium in the paper. It is a program for solving non-linear complementarity problems. The solution procedure is a generalized Newton method with a backtracking line search, coupled with the Lemke’s almost complementarity pivoting scheme in which lower and upper bounds are represented implicitly. The solution from MILES must be verified to make sure that Nash equilibrium is obtained. This can be done by remaximizing one producer’s profit while keeping other producers’ production, sales and flows fixed. This procedure must be repeated for each and every producer. If no producer’s profit is changed, the solution is proved to be a Nash equilibrium. The solution process can be shown in Fig. 1. Note that when no Nash equilibrium can be found,
further analysis of the problem is needed, including reexamining the MCP conditions. According to our experience, the MCP conditions can be complicated and the mishandling of the conditions can cause problems in finding Nash equilibrium. This is why we emphasize the reexamination of the MCP conditions. We otherwise have not encountered any serious problem in obtaining Nash equilibria for different problems. Moreover, the market power model (1) in this paper has linear constraints that are well known to be convex and differentiable, and the vector objective functions are all convex and differentiable too, according to Theorem 3.5.2 in Ref. [20], it is sufficient for the model to have a proper efficient solution. Note that each equality constraint can be represented by two inequality constraints to establish the above argument. In addition, as proved in a proposition in the appendix, the KTVOT necessary conditions characterize a Nash equilibrium.
4. Case studies In this section, two case studies are presented to illustrate the application of the proposed model. In Case I, with the use of an econometric model the long-term coal price are projected to be $1.1/MMBTU and the gas prices are $4.5/MMBTU for 2001, $4.1/MMBTU for 2002 and $3.7/MMBTU for 2003 and the next 13 years (these prices are similar to the projection done by the Energy Information Administration of the US). A capital recovery factor of 20% is used for plant capital recovery, based on our joint study with several utilities. The capital costs are $400 kW21 for gas peaking plants, $580 kW21 for combine cycle gas plants and $900 kW21 for base load coal plants. A long-term marginal cost of $148.75 MWh21 is obtained from the LTEM model. This is why the price cap is set at $150 MWh21. As for Case II, the long run marginal cost is assumed because the system data is dated back in the 1960s. Therefore, Case II is for illustrating the effect of price caps on the system outcome of gaming without considering applicability. 4.1. Case 1. A six-node system
Fig. 1. Solution flowchart.
This six-node system is the same one as introduced in Ref. [4] except that there are no emission costs and limits. The detailed data of this system can be found in Ref. [4]. There are three Cournot-type producers in this case. Price caps for this case are $150 MWh21 for all nodes (This price cap coincides with the price cap that has been suggested for the California electricity markets [17]). The profits and nodal prices with and without price caps are illustrated in Tables 2 and 3. Both solutions are proved to be Nash equilibria. We can see that with price cap enforcement, the producers’ profits are reduced somewhat (about 5%) but the reduction in profit is not uniform across the producers. The percentage reduction of profit from producer one is the
Z. Yu / Electrical Power and Energy Systems 25 (2003) 301–307
305
Table 2 Profits without and with price caps ($)
Table 4 Nodal demand and price data for case 2
Profits
Node
Producer
W/o caps With caps Change
1
2
3
Total
134358.15 133069.56 21288.59
81830.67 74466.71 27363.96
69231.07 63088.17 26142.90
285419.89 270624.44 214795.45
least. This may be caused by the fact that producer one’s generators are located strategically better than the other two producers’ generators. Due to the enforcement of price caps, all of the nodal prices are no more than 150 MWh21. As a result, all consumers benefit due to the decline of the nodal prices. The total social welfare, by intuition, is increased but detailed calculation is trivial and is thus omitted. According to microeconomics, market power of the producers is reduced in this case. Since the model is a short-term model of one hour, the market power reduction is also short-term in nature. 4.2. Case 2. An IEEE 30-node system In Case 2, an IEEE 30-node system is used for testing the gaming model. The electrical network and generation unit data of this system can be found in Ref. [18]. Three producers are assumed to compete on the Cournot terms. Assume that producer one owns the plant at node 1, producer two owns the plants at nodes 2 and 5, and producer three owns plants at nodes 8, 11 and 13. The demand function for each node is also assumed linear with a slope of one. The intercepts of the nodal demand functions on the vertical axis (price axis) are listed in Table 4. These intercepts are generated arbitrarily without degrading the generality of the model. The wheeling cost for each line is derived using the following formula: whði; jÞ ¼ 1:0 þ uniformð20:1; þ0:1Þ where uniform( ) is a uniform random number generator. The results are summarized in Tables 4– 6. The highest price without a price cap is $21.426 MWh21 at node 5, as is illustrated in Table 4. Now suppose a price cap of $16 MWh21 is enforced. As a result, the price at node 5 is reduced to $16 MWh21. It is interesting to notice that the nodal prices at nodes other than node 5 are all increased somewhat after the price cap enforcement. However, as is
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30
Price w/o caps
Price with caps
25.00 30.85 25.00 35.00 67.10 25.00 31.50 23.80 25.00 22.90 21.20 25.00 25.00 23.10 24.10 21.75 24.50 21.60 24.75 21.10 28.75 25.00 21.60 24.35 25.00 21.75 25.00 25.00 21.20 25.30
10.985 12.161 12.794 14.798 21.426 11.894 13.335 11.165 12.520 12.454 10.721 11.820 12.096 11.970 12.961 11.648 12.981 13.017 14.182 12.787 14.735 13.652 13.119 14.009 14.258 14.179 13.607 12.940 13.489 14.389
11.071 12.247 12.841 14.882 16.000 11.978 13.421 11.246 12.567 12.490 10.802 11.901 12.183 12.051 13.042 11.729 13.023 13.098 14.219 12.824 14.771 13.689 13.200 14.045 14.306 14.227 13.693 13.026 13.575 14.475
indicated in Table 5, the profits of the producers are all reduced after the price cap enforcement. This again indicates that consumers, on average, benefit from the price cap. To be precise, only consumers at node 5 actually benefit from the price cap and the consumers at nodes other than node 5 see minor price increases as a result of the price cap enforcement. A caution is offered here that the result for this case is just a result of the way the price cap is set and different results can be obtained with different price capping strategies. It is worthwhile to make a further comment on the profit reduction in Case 2. Remember that producer two has a generator at node 5 where demand is high (see a0(5) in Table 4). Therefore, without a price cap, producer two has
Caps
W/o caps W/ caps
a0(i )
Table 5 Comparison of profits with and without price caps
Table 3 Nodal prices without and with price caps for case 1 ($ MWh21) Price
Parameter
Profit
Node 1
2
3
4
5
6
171.87 150
166.44 150
167.20 150
154.08 150
164.36 150
152.18 150
W/o caps With caps Change
Producer 1
Producer 2
Producer 3
Total profit
1275.791 1206.687 269.104
1206.545 983.316 2223.138
904.004 848.898 255.106
3386.340 3038.901 2347.439
306
Z. Yu / Electrical Power and Energy Systems 25 (2003) 301–307
Table 6 Comparison of production quantities Producer Cases
1
2
3
Total
W/o caps With caps Change
161.87 165.16 3.29
123.81 124.01 0.20
105.00 105.00 0.00
390.68 394.17 3.49
Then, f ðxÞ ¼ ½f1 ðxÞ; …fSm ðxÞ: Similarly, the constraints in model (2.1) can be written as a row vector G(x) # 0, with the provision that each equality constraint is converted to two inequalities. It follows that model (2.1) can be expressed as: Minf ðxÞ Subject to GðxÞ # 0;
an advantage over its competitors. It is then understandable that this producer would reduce its profit most after the price cap enforcement as compared with the other two producers. Even though producer two’s total production quantity is not reduced very much (see Table 6), the price cap forces it to reduce its sale to node 5. It is also possible that producer two would increase its sale to other nodes that have lower prices.
5. Conclusions This paper is the first to model spatial gaming with price caps for deregulated electricity markets. The logic for this effort is that price caps are enforced in several electricity markets in the US and existing electricity gaming models have not reflected this fact. The DC power flow equations are included to represent the physical characteristics of electrical networks. The generic gaming problem is formulated as a vector maximization model. The KTVOT is used to find Nash equilibria. The paper claims that the KTVOT includes the concept of the Cournot strategy in gaming. The MCP conditions are then formulated and solved by our using MILES. Two case studies are used to illustrate the application of the proposed model with satisfactory results. Price caps can reduce market power in the short-term as shown by the case studies. The paper by no means has concluded the gaming modeling effort in this area and more work needs to be done, for example, to include losses in the formulation.
Appendix A
x $ 0:
Define a positive vector m and a non-negative vector l of LaGrange multipliers for the constraints, the KT necessary conditions for a proper solution can then be expressed below. (i) km; 7f ðxp Þl þ kl; 7Gðxp Þl ¼ 0; (ii) kl; Gðxp Þl ¼ 0; (iii) m . 0; l $ 0: where k l represents dot product, 7 gradient and x p the proper solution point. Note that the above KT necessary conditions are modified by the Cournot assumptions (These KT conditions may be called the diagonal conditions. See J.B. Rosen (1965) for more details).We next show that the above KT necessary conditions characterize Nash equilibrium. From the KT definition of a proper efficient solution, (see Definition 3.1.10 in Ref. [20]), there is no h [ Rn such that k7fS ðxp Þ; hl # 0 for all S with at least one S resulting in k7fS ðxp Þ; hl , 0 and that kh; 7gj ðxp Þl # 0;
for any j [ Jðxp Þ ¼ {j : gj ðxp Þ ¼ 0}:
Notice that the above definition says that for the KT proper solution, there is no h that can result in k7fS ðxp Þ; hl , 0 for a single S and satisfy the constraints at the same time. (Remember that k7fS ðxp Þ; hl , 0 indicates that producer S can improve its objective and also note that the equivalence of KT necessary conditions and the KT definition for a proper solution is proved in Theorem 3.5.2 in Ref. [20].) This means that no producer can improve its objective unilaterally, which is a characterization of a generalized Nash equilibrium. The proof is complete. A
On the equivalence of the KTVOT and a Nash equilibrium Proposition. The KTVOT necessary conditions for a proper solution is equivalent to a Nash equilibrium for the gaming model (2.1) considering the Cournot assumptions. Proof. Define a row vector, x, for representing all of the variables qði; SÞ; Fði; j; SÞ; PGði; SÞ (Note that PðiÞ and Cði; SÞ are functions of these variables). Define fS ðxÞ ¼ 2pðSÞ þ 0ðSÞx1; where 0(S ) is a null row vector and x1 is a column vector representing those variables in x that are not in p(S ).
References [1] Green R, Newbery D. Competition in the British electricity spot market. J Politic Econom 1992;100(5):929–53. [2] Schmalensee R, Golub BW. Estimating the effect of concentration in deregulated wholesale electricity market. Rand J Econom 1984;15(1): 12– 26. [3] Wei J-Y, Smeers Y. Spatial oligopolistic electricity models with Cournot generators and regulated transmission prices. Oper Res 1999; 47(1):102–12. [4] Yu Z, Preckel P, Nderitu O, Sparrow FT. A locational gaming model
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[5]
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