Electric Power Systems Research 70 (2004) 187–193
A price competition model for power and reserve market auctions
Deqiang Gan a,∗ , Chen Shen b
a
College of Electrical Engineering, Yuquan Campus, Zhejiang University, Hangzhou, Zhejiang, PR China
b Department of Electrical Engineering, Tsinghua University, Beijing, PR China
Received 25 August 2003; received in revised form 23 November 2003; accepted 9 December 2003
Abstract
In our recent work, we formulated a price competition game model for modeling oligopolistic competition in a single-period electricity
market auction. An assumption made in the study is that generator marginal costs are un-identical. In this paper, we continue to study
single-period auction games, with the assumption that generator marginal costs are identical. We found that the later assumption facilitates a
sharper result, which states that the market clearing price (CP) at equilibrium under tight capacity constraint is unique. This assumption also
permits us to obtain simpler equilibrium characterization and mathematical proofs. Furthermore, we extend the model to study the auction
game in which power and reserve are auctioned simultaneously. As an application, we derive market power indices out of the suggested
game-theoretic models.
© 2004 Elsevier B.V. All rights reserved.
Keywords: Power system; Electricity market; Reserve; Game theory; Optimization
1. Introduction
As electricity markets are rapidly emerging around the
world, a timely topic arises, that is, how to predict or analyze
the performance of a given market design. The answer of
the question can significantly influence policy-making. The
mainstream approach is that of game theory. The idea of this
approach is to predict market performance assuming that
market players will take Nash equilibrium strategies.
The micro-structure of an electricity market is typically
complicated. On the other hand, to enhance the predicting
power of a game model, one favors analytic, rather than,
computational model. A tradeoff has to be carefully made
between the modeling capability and predicting capability
of a game model. These considerations have motivated our
recent development of a price competition game model for
modeling oligopolistic competition in electricity market auctions [1].
In a nutshell, we consider in [1] a single-period electricity market auction as a simultaneous-move, price competition, continuous game. This model works rather well if one
∗
Corresponding author. Tel.: +86-571-8795-1831;
fax: +86-571-8795-2591.
E-mail addresses: [email protected], [email protected] (D. Gan).
0378-7796/$ – see front matter © 2004 Elsevier B.V. All rights reserved.
doi:10.1016/j.epsr.2003.12.007
is interested in analyzing the market behavior for a given
load level. Related work can be found in, say [4–14]. Results of laboratory experimental studies can also be found
in, for example, reference [15]. As is well-known, capacity
constraints is one of the most important factors that affect
bidding behavior. A key feature of the proposed game model
is in its flexibility of modeling capacity constraints.
An assumption required in [1] is that generator marginal
costs are mutually un-identical. In this paper, we provide
results under the assumption that generator marginal costs
are identical. We show that equilibrium characterization
with this assumption is simpler, and the main results are
sharper. Furthermore, the suggested model is extended to
study single-period auction games where power and reserve
are auctioned simultaneously. Finally, as an application, we
describe how the introduced results can be used to derive
numerical market power indices.
2. Equilibrium points in energy-only markets
The model is intended for single-period auction game.
Unit commitment is not considered in this paper. In this
section we will first describe the auction game and then
provide an equilibrium characterization.
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D. Gan, C. Shen / Electric Power Systems Research 70 (2004) 187–193
2.1. The game model
The power dispatch and pricing is obtained from the following optimization problem [1]:
pT P
min
P
(1.1)
πk (pk ) = (ρe − c)T P k
s.t. e P − DP = 0,
(1.2)
0 ≤ P ≤ P̄,
(1.3)
T
where P̄ is the high operating limit vector, e is a vector
with all ones, and DP is the total load. To solve the above
problem, we form Lagrangian function as follows:
T
T
Γ = pT P + ρ(DP − eT P) − τ P + τ (P − P̄)
(2)
where ρ, τ , τ are the Lagrange Multipliers. As is
well-understood, the price of the last accepted generator is
the market clearing price (CP), which is equal to ρ in the
above setting.
If the problem (1) has multiple optimal solutions, it implies that the bid prices of some units are the same and these
generators set clearing price. Under such circumstances,
these generators are dispatched in proportion to their capabilities. In a two-generator system, this widely accepted
protocol can be described mathematically as
P1 =
P̄1
L̂,
P̄1 + P̄2
As usual the payoff function of the game is assumed to
be the profit of the player. The problem the kth player faces
is to choose a price vector, pk , so as to maximize his profit,
assuming that the price vectors of the other players p∗−k are
given. The profit of kth player πk (pk ) is given by
P2 =
P̄2
L̂
P̄1 + P̄2
(3)
where L̂ is the residual load that the two generators supply (Fig. 1). In a multi-generator system, the same concept
applies.
Assumptions.
1. eT P̄ − DP ≥ 0. This ensures that there are feasible solutions to the auction problem.
2. Each unit has a constant marginal cost which is publicly
known.
3. Bid prices are subject to a price cap c̄, that is, p ≤ c̄. Here
c̄ represent the price cap which is a scalar, c̄ represents
price cap vector with suitable dimension.
4. Price caps are strictly greater than costs, that is, c̄ > c.
5. A generator is allowed to bid one block only. This assumption is not essential but is helpful for explanation.
6. There are at least two players, each of which owns generators with non-zero capability.
Fig. 1. Residual load when two generators bid the same price.
(4)
Notice that P and ρ are single-valued functions of pk
Let us denote this function as P(·) and ρ(·), respectively.
Now the profit maximization problem can be formulated as
follows:
max
pk
s.t.
πk (pk ) = [ρ(pk )e − c]T P k (pk )
c ≤ pk ≤ c̄k ,
(5.1)
(5.2)
It is worth stating that the above problem is a bi-level optimization problem because computing P(·) and ρ(·) involves
solving an optimization problem defined in Eq. (1) [16].
Apparently, the kth player’s best response, pk , is in general a function of its rival’s strategies, p∗−k . The reason is that
if p∗−k changes, then the optimization solution pk changes.
Let rk (·) denote this function, then pk = r k (p∗−k ). It is
convenient to re-write this relationship into compact matrix
form as s = r(s). The intersection of the graph of functions
rk (·), k = 1, 2, . . . , Ns is Nash equilibrium, here Ns is the
number of players. Mathematically, the solution of the simultaneous equations s = r(s) is Nash equilibrium.
Given a strategy p∗−k , there could be as many as infinite
number of best responses pk . Therefore r k (·) is in general
a point to set map. It follows that pk ∈ rk (p∗−k ). The intersection(s) of the graphs of these point-to-set maps are Nash
equilibrium points [1,17]. A fundamental property of Nash
equilibrium is that no player has in incentive to deviate from
Nash equilibrium strategy.
2.2. Equilibrium analysis
When the capacity constraint is very tight, the clearing
price at equilibrium is likely to be high because large players
could manipulate the clearing price. This well-known wisdom must be proved. The following result serve the purpose.
Lemma 1 (existence of critical load level). There exists a
critical load level DP∗ such that DP > DP∗ then the auction
game possesses an equilibrium at which ρ∗ = c̄.
Proof. Since the objective is to prove the existence of DP∗ ,
it is reasonable to assume that, if one stacks up generators
bids randomly, then all generators would be fully accepted
except one. Let the index of this partially accepted generator
be m. Consider the following configuration:
(a) The mth generator bids price cap, the others bid
marginal costs.
Then apparently generators, i (i = m), do not have incentives to deviate from bidding their marginal cost because
D. Gan, C. Shen / Electric Power Systems Research 70 (2004) 187–193
these generators are fully accepted and the clearing price
reaches maximum.
The mth generator’s profit
above configuration
is
in the
given by πm (DP ) = (c̄ − c) DP − j=m P̄j . Apparently,
one can always find a DP∗ such that πm (DP∗ ) > 0.
If the mth generator bids a price that is between c and c̄, it
would profit less than πm (DP∗ ). If this generator bids a price
equal to c, all generators would be dispatched in proportion
to their capacity, the market clearing price would be c, the
profit of the mth generator would become zero.
The above demonstrates that configuration (a) is at equilibrium and ρ∗ = c̄. The proof is complete.
䊐
As an example, consider a system with three players and
three generators, assume that P̄1 = 100, P̄2 = 200, P̄3 =
500. Then DP∗ = 300.
The above result is of qualitative nature. Proposition 1
below provides a more specific characterization of equilibrium points of the game. Before stating the proposition, we
need to prove a useful result. The idea of this result under
slightly different assumptions has actually been utilized but
not explicitly stated in our recent work [1].
Lemma 2 (price competition lemma). Consider a dispatch
configuration (P ∗ , ρ∗ ). If the clearing price ρ∗ is set by
generators owned by several players and eT P̄ − DP > 0,
then the configuration is not at equilibrium.
Proof. See Appendix A.
䊐
The idea of the above result is as follows. Suppose a player
has a generator which ties with his rival’s generator. The
player should undercut his rival’s generator. By doing so,
he would increase his output though the market CP would
become lower, he would gain more than he lose.
Now let us state one of the main results of this paper.
Proposition 1 (equilibrium under strong
constraints). Suppose
there exists a player A such that j∈A
/ P̄j < DP , then the
game possesses an equilibrium at which ρ∗ = c̄. Furthermore, ρ∗ = c̄ uniquely.
Proof. Consider the following configuration:
Player A bids c̄ for all his generator(s), player A’s rival(s)
bid c.
The above configuration is an equilibrium point. The reason is as follows.
First, A’s rivals do not have incentives to deviate from
bidding c because by hypothesis their generators are all fully
accepted and the CP is highest possible, so all of them obtain
maximum payoff.
Second, A does not have incentive to deviate because if A
bids a price between c and c̄, A would profit less. If A bids
c, A would profit zero which is lessthan what he canobtain
in the above configuration, (c̄ − c) DP − j∈A
/ P̄j > 0.
189
Now let us prove the uniqueness part of the proposition.
Apparently there does not exist equilibrium at which ρ = c.
Now suppose c < ρ < c̄, consider the following possible
configurations.
(a) The clearing price is set by generator(s) owned by a
single player.
(b) The clearing price is set by generators owned by several
players.
Configuration (a) is not at equilibrium because no matter
which generator sets clearing price, he has incentive to raise
the price. To prove that configuration (b) is not at equilib䊐
rium, the readers are referred to Lemma 2.
Remark. The statement “ρ∗ = c̄ uniquely” does not necessarily
means that the equilibrium is unique. For example,
when j P̄j = DP , there can be multiple equilibriums.
In [1], we studied market equilibrium under weak capacity
constraints when marginal costs are unequal. The next result
characterizes equilibrium when marginal costs are identical.
Proposition 2 (equilibrium under weak constraints). Let I
be the indexset of all players. Suppose that for every player
A, A ∈ I, j∈A
/ P̄ j > Dp , then there does not exist equilibrium at which ρ > c. In other words, if an equilibrium
exists, then at the equilibrium ρ = c.
Proof. Suppose there exists an equilibrium at which ρ > c,
the following are the possible configurations.
(a) The clearing price is set by generator(s) owned by a
single player.
(b) The clearing price is set by generators owned by several
players.
To show that configuration (a) is not at equilibrium, recall
the hypothesis j∈A
/ P̄j > Dp , for every player A, A ∈ I,
therefore there must be unaccepted generator(s) owned by
the price-setting player’s rival. The unaccepted generator(s)
have incentive to undercut the price-setting generator because ρ > c.
To prove that configuration (b) is not at equilibrium, the
readers are referred to Lemma 2.
䊐
The above result includes the classic Bertrand Paradigm
as a special case. Consider a market with two sufficiently
large generators, at the equilibrium point of the market the
CP is equal to marginal cost. As a more general example,
consider a market with A, B, C three generators, each is
independently owned, none of the generators can serve the
load independently, but two of them can always serve the
load, the clearing price of the market at equilibrium is equal
to marginal cost c.
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D. Gan, C. Shen / Electric Power Systems Research 70 (2004) 187–193
generators). If the auction problem (6) has one unique solution, there can be only two marginal generators. The problem may have multiple solutions which can appear in the
following forms:
3. Equilibrium points in energy and reserve markets
3.1. The game model
The power/reserve dispatch and pricing are obtained from
the following optimization problem [2]:
pT P + r T R
min
P,R
s.t.
(6.1)
eT P = DP ,
(6.2)
eT R = DR ,
0≤P
and
(6.3)
P + R ≤ Γ = pT P + ρ(DP − eT P)
T
T
− τ P + τ (P − P̄), (6.4)
T
T
0 ≤ R ≤ Γ = pT P + ρ(DP − eT P) − τ P + τ (P − R̄)
(6.5)
The details of the above market structure, which is essentially an abstraction of the market model of major North
America electricity markets such as New England and New
York, are further described in [2,3]. It is worth noting that
the above reserve market structure is not necessarily the best.
One piece of evidence is that these markets are often suspended when the load and reserve demands exceed or approach the available generating capacity. The reason is that
under such circumstances, generators have too much market
power to disturb the market. Eliminating the requirement on
reserve capabilities can (greatly) help to obviate this problem, this is the innovating idea of treating reserve capabilities as insurance product, as suggested in [21].
To solve the above optimization problem, form the Lagrangian function as follows:
T
Γ = pT P + r T R + ρ(DP − eT P) + γ(DR − eT R) − τ P
T
T
+ τ (P + R − Γ = pT P + ρ(DP − eT P) − τ P
T
T
+ τ (P − P̄)) − τ R + τ (R − Γ
T
= pT P + ρ(DP − eT P) − τ P
T
To deal with the above solution multiplicity problem,
again, we apply the proportion rule described in Eq. (3). If
generators tie in reserve market, we dispatch the reserve in
proportion to their reserve capability R̄. If generators tie in
both power and reserve markets, we dispatch power and reserve in proportion to their power capability.
Assumptions.
1. There are feasible solutions to the auction problem.
2. Each unit has constant marginal cost which is publicly
known. Reserve costs are zero.
3. Bid prices are subject to a price cap c̄, that is, p ≤ c̄,
r ≤ c̄.
4. c̄ > c.
5. A generator is allowed to bid one block only. Again, this
assumption is not essential but is helpful for explanation.
6. There are at least two players, each of which owns generators with non-zero power and reserve capability.
The payoff function of the game is again assumed to be
the profit of the player. The problem that the kth player faces
is to choose a price vector, (pk , rk ), so as to maximize his
profit, assuming that the price vectors of the other players
(p∗−k , r ∗−k ) are given. The profit of kth player πk (pk , r k ) is
given by
πk (pk , rk ) = (ρe − ck)T P k + γeT Rk
T
+ τ (P − R̄)) [0, 1] [0, 1]
• Multiple generators set power CP jointly.
• Multiple generators set reserve CP jointly.
• Multiple generators set power and reserve CPs jointly,
ρ = γ.
• Multiple generators set power and reserve CPs jointly,
ρ = γ.
(7)
where ρ, γ, τ , τ , σ , σ are the Lagrangian multipliers. At
the optimum, by the Kuhn–Tucker optimality condition, we
have

∂Γ


= pi − ρ − τ i + τ i = 0

∂Pi
,
(8)
∂Γ



= ri − γi + τ i − σ i + σ i = 0
∂Ri
As in power-only market, ρ, γ are set as clearing prices for
power and reserve, respectively. They equal to the bid prices
of power and reserve of price-setting generators (marginal
(9)
Notice that P, R, ρ, and γ are single-valued functions of
(pk , r k ). Let us denote this function as P(·), ρ(·), R(·) and,
γ(·), respectively. Now the profit maximization problem can
be formulated as follows:
max
pk ,rk
πk = [ρ(pk , rk ) − ck ]T P k (pk , rk )
+ γeT (pk , rk )Rk (pk , rk )
s.t.
ck ≤ pk ≤ c̄
and
0 ≤ rk ≤ c̄
(10.1)
(10.2)
Again kth player’s best response, (pk , rk ), is in general
a point-to-set map of its rival’s strategies, (p−k , r −k ). The
intersection(s) of the graphs of these point-to-set maps are
Nash equilibrium points [17].
D. Gan, C. Shen / Electric Power Systems Research 70 (2004) 187–193
3.2. Equilibrium analysis
The following result, which states that the market cannot
be competitive if demand level is higher than a critical value,
is an immediate extension of Lemma 1.
Lemma 3 (existence of critical load/reserve level). There
∗ ) such that if
exists a critical load/reserve level (DP∗ , DR
∗
∗
DP > DP , DR > DR , then the auction game possesses an
equilibrium at which ρ∗ = c̄, γ ∗ = c̄.
The proof of the above result is omitted. Similarly, the
following result is an extension of Proposition 1.
Proposition 3 (equilibrium under strong constraints). Suppose there exists player A such that
j ∈A
/ P̄j > Dp ,
j ∈A
/ R̄j > DR , then the game possesses an equilibrium
at which ρ∗ = c̄, γ ∗ = c̄.
Proof. Consider the following configuration:
Player A bids c̄ for energy and reserve for all his generator(s), player A’s rival(s) bid c for energy, and 0 for reserve,
so ρ∗ = s̄, γ ∗ = c̄.
The above configuration is at equilibrium, to prove the
assertion, notice that by hypothesis generators of A’s rivals
are all fully accepted in power and reserve auctions, the
power and reserve CPs are highest possible, so A’s rivals
already secure maximal payoff in the above configuration.
On the other hand, A does not have incentive to change his
power price either, because if he bids a price between c and
c̄ for power, he would profit less. If he bids c, he would profit
zero in power market which is less than
whathe already
has
in the above configuration, (c̄ − c) DP − j∈A
/ P̄j > 0.
By the same token, A does not have incentive to change his
reserve price.
䊐
Naturally, one questions if the result of Proposition 2 can
be extended to the situation in power/reserve combined market. The answer to this question is satisfactorily yes, as stated
in Proposition 4 below.
Proposition 4 (equilibrium
under weak
constraints). If for
every A, A ∈ I, j∈A
/ P̄j > DP +DR ,
j ∈A
/ R̄j > DR , then
there does not exist equilibrium at which ρ > c or γ > 0.
Proof. Suppose there exists an equilibrium at which ρ > c
or γ > 0. The following are the possible configurations.
(a) Power CP is set by generator(s) owned by a single player,
reserve CP is set by generator(s) owned by a single
player.
(b) Power CP is set by generator(s) owned by a single player,
reserve CP is set by generators owned by several players.
(c) Power CP is set by generators owned by several players,
reserve CP is set by generator(s) owned by a single
player.
191
(d) Power CP is set by generators owned by several players,
reserve CP is set by generators owned by several players.
We need to show that none of the above configurations is
at equilibrium (sketch). To show that configuration (a) is not
at equilibrium, notice that, by hypothesis there must exist a
generator (say generator x owned by player X) that is not accepted in power market. Suppose X is not the price-setting
player, then if x lower power bid price to undercut the
price-setting generator, X could improve his profit; suppose
X is the price-setting player, then if x raise its power bid
price, X could improve X’s profit.
To show that configuration (b) is not at equilibrium, notice
that if one of the price-setting generators in the reserve market lower its reserve bid price, he could improve his profit in
reserve market. This assertion is true following the argument
of Lemma 2. So configuration (b) is not at equilibrium.
Using the same argument, one can easily show that con䊐
figurations (c) and (d) are not at equilibrium.
The above result asserts that, if equilibrium exists, then
at the equilibrium ρ > c and γ > 0.
As an example, consider
a three-player
(player X, Y,
Z) market. Suppose
P̄
>
D
,
R̄j > DR ,
j
P
j∈X
j∈Y
P̄j > DP ,
j∈Z R̄j > DR ,
j∈Z P̄j > DP ,
j∈Y
R̄
>
D
,
then
Proposition
4
asserts
that, in this
R
j∈X j
market, there does not exist equilibrium at which ρ > c and
γ > 0.
4. Applications
In our previous work, we defined a numerical index called
must-run-generation [18]. This numerical index, defined for
power market only, can be used to measure the price manipulation capability of market p layers. For player A, we can
calculate the following indices which measure the market
power of the player in power and reserve market:
DP − j∈A
/ P̄j
1
MRRA =
, 0 ≤ MRR1A ≤ 1
(11)
P̄
j
j∈A
MRR2A
DR −
=
j ∈A
/ R̄j
j∈A R̄j
,
0 ≤ MRR2A ≤ 1
(12)
The above indices are natural extensions of must-run-ratio
in power-only market. They are closely related to Supply
Sufficiency Indices, which have been used in California for
studying market power issue in power-only market [19,20].
The contribution of this and our previous work is to extend
the idea to power and reserve combined markets, and more
importantly to provide an analytical foundation for these
indices which appears to be lacking in the open literature.
Further work could be towards developing optimal regulation rules out of the introduced numerical market power
indices. For example, the price caps could be designed as a
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D. Gan, C. Shen / Electric Power Systems Research 70 (2004) 187–193
function of the size of the generator with highest market run
ratio. Alternatively, one could argue that if must run ratio of
a generator exceeds certain threshold, the generator should
be required to sign long term supply contract.
Now consider the situation where L̂ > P̄1 the new CP
remains unchanged, the profit of C is as follows:
(16)
π̈ =
(ρ∗ − c)P̄j + (ρ∗ − c)P̄1
j∈C,pj <ρ∗
The difference between π̇ and π̈ is then given by
5. Conclusions
In this work, we studied the electricity market auction
game under the assumption that generators marginal costs
are identical. As described in the paper, characterizing
the equilibrium points of identical cost game appears to
be simpler. We show that must-run-generation is an even
more important factor affecting the behavior of players in
such markets, and price caps have little impact on equilibrium of such auction games. The results introduced are
extended to study power and reserve simultaneous auction
games as demonstrated in the paper. The numerical indices
introduced in the paper are easily calculated and provide
a meaningful way of estimating market performance for a
wide range of market structures. Our study is motivated
from the work of reference [7] but our results are different
and more applicable as is demonstrated in the paper.
Appendix A
Proof of Lemma 2. Without loss of generality, consider the
situation where two generators 1 (owned by player C) and
2 (owned by player D) set CP. Let L̂ be the residual load
(refer to Fig. 1). The profit of player C is
π̇ =
(ρ∗ − c)P̄j + (ρ∗ − c)L̂
j∈C,pj <ρ∗
P̄1
P̄1 + P̄2
(13)
The first term is the profit earned by C’s infra-marginal
generators. This term could be zero. Note that L̂ can be
either greater or smaller than P̄1
Now suppose the price of generator 1 is lowered to (ρ∗ −
η), here η is a small positive number. Consider first that L̂ ≤
P̄1 , so the new CP becomes ρ∗ − η, the profit of C becomes
π̈ =
(ρ∗ − η − c)P̄j + (ρ∗ − η − c)L̂
(14)
j∈C,pj <ρ∗
The difference between π̇ and π̈ is then given by


P̄2
π̈ − π̇ = −η L̂ +
P̄j  + (ρ∗ − c)L̂
P̄
1 + P̄2
j∈C,p <ρ∗
j
(15)
Since L̂ +
j∈C,pj <ρ∗
P̄j > 0, and (ρ∗ −c)L̂[P̄2 /(P̄1 +
P̄2 )] > 0, therefore player C can always find an (possibly
small) η > 0, such that π̈ − π̇ > 0.
π̈ − π̇ = (ρ∗ − c)P̄1
P̄1 + P̄2 − L
>0
P̄1 + P̄2
(17)
The above indicates that player C has incentive to deviate
from bidding ρ∗ for generator 1. If player C has more than
one generators that tie in the auction, apparently the same
conclusion can be drawn.
The same argument holds for player D.
䊐
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