Impacts of suppliers’ learning behaviour on market
equilibrium under repeated linear supply-function
bidding
Y.F. Liu, Y.X. Ni and F.F. Wu
Abstract: The paper studies the impacts of learning behaviour of electrical-power suppliers on
electricity-spot-market equilibrium under repeated linear supply-function bidding. In the markets,
the supplier will conduct ‘learning’ to improve his strategic bidding in order to obtain greater profit.
Therefore, it is significant to explore the impacts of such learning behaviour on market equilibria
and market-clearing price (MCP). First the mathematical model for supplier’s optimal bidding is
established. This is then used to solve for market equilibrium. It is shown that supply-function
equilibrium is a Nash equilibrium; and that under certain conditions the overall learning behaviour
will reduce the MCP, which in turn increases consumers’ surplus and decreases suppliers’ profits,
while in some other conditions the results are just the contrary. In either case, the MCP at
equilibrium induced by the overall learning behaviour reflects the true relationship of supply and
demand. Numerical results support the analytical conclusions very well.
1
Introduction
In recent years, competition has been introduced to the
power industry in order to increase social welfare and
improve market efficiency. A variety of restructuring modes
have been proposed in different countries. Among these
modes, the market structure of the power pool (‘poolco’
type) is the most popular one. In poolco, electricity spot
markets, the suppliers (gencos) bid for the next-day hourly
price and generation; then the independent system operator
(ISO) creates the next-day hourly generation schedules to
meet the demand at a minimum cost, and establishes the
corresponding market-clearing price (MCP) for each hour
[1]. Theoretically, in a perfectly competitive market,
suppliers will bid their true marginal production costs to
maximise their profits. However, it is well known that the
power industry is more akin to an oligopoly, where
suppliers can influence the market-clearing price by their
strategic bidding [2–4]. Therefore, each supplier faces the
problem of how to improve its strategic bidding to
maximise its own profit based on its production cost, its
expectation about rivals’ behaviour, and some other
available market information.
Numerous papers [5–17] have addressed the strategicbidding issue in competitive power markets. An optimal
bidding strategy to maximise a bidder’s profit in the
England & Wales power market is presented in [7] under the
assumption of perfect competition. In [8], the optimal
bidding strategy is derived as a function of one’s own cost
and the rivals’ cost distributions. In [9–12], game theory is
r IEE, 2006
IEE Proceedings online no. 20050137
doi:10.1049/ip-gtd:20050137
Paper first received 19th April and in final revised form 5th September 2005
The authors are with the Department of Electrical and Electronic Engineering,
Hong Kong University, Pokfulam Road, Hong Kong
E-mail: [email protected]
44
applied to find Nash equilibria of power markets. Other
methods, such as genetic algorithm [13], Markov decision
process [14], Lagrangian relaxation [15], stochastic optimisation [16] and ordinal optimisation [17], are also used to
solve the optimal-bidding problem.
Recently, supply-function models have been used widely
to investigate the optimal-bidding problem [4, 18–24], since
the power-market rules require the individual supplier to
submit its supply function for representing the willingness of
generating at various electricity prices. The concept of
supply-function equilibrium (SFE) was originally developed
by Klemperer and Meyer [25] as a way to model market
players’ behaviour in a market with uncertain demand. In
[4], the Klemperer–Meyer equation is first used to compute
SFE in the England &Wales spot market. In [18], a linear
version of the supply-function-based model is developed. In
[19], some basic issues about learning through the SFE
approach are discussed. In [20], a two-point boundary
problem is defined using the SEF concept. A generalisation
of Green’s model [18] is presented in [21]. Then, Baldick and
Hogan [22] show that only the affine supply function can be
stable. A conjectural supply-function model is introduced in
[23]. In [24], the existence and uniqueness of linear SFE are
studied.
However, most of the prior work is based on the static
analysis of market equilibrium; little attention has been paid
to market evolution in the situation of repeated bidding. It
is clear that, during repeated bidding, suppliers will conduct
‘learning’ so as to make full use of available information
and then improve their bidding strategies based on knowledge acquired. Apparently, it is significant to explore
whether the suppliers have incentives to learn and the
subsequent impacts of suppliers’ rational learning behaviour
on the evolution of market equilibrium and MCP. The two
issues will be addressed in the paper, with proof. In
addition, [2] and [10] mention a ‘prisoner-dilemma’ issue
induced by supplier’s strategic bidding in power markets,
but the mechanism on why and how it may happen has not
IEE Proc.-Gener. Transm. Distrib., Vol. 153, No. 1, January 2006
been illustrated. This issue will also be explained in this
paper. Computer test results of a six-genco power market
show clearly that analytical results are correct, i.e.
(i) Suppliers have incentive to learn;
(ii) In certain condition, the overall learning behaviour will
reduce the MCP which, in turn, increases consumers’
surplus and decreases suppliers’ profits, while in some other
conditions the results are just the contrary; and
(iii) In either case, the MCP at equilibrium induced by the
overall learning behaviour reflects the true relationship of
supply and demand.
2
Bidding with optimal linear supply function
2.1 Assumptions on demand curve and
production-cost function
Assume that each electrical-power supplier has a cluster of
generators and hence the ISO has no need to consider
suppliers’ operation constraints, such as ramp rate, minimum up and minimum down time, generation capacity
etc., when we explore supplier’s strategic-bidding behaviour.
Also, for simplicity, the network impacts (congestion
and losses) are not considered at this stage. Also, suppose
that the overall demand can be described by a linear
inverse demand curve, and the demand is stationary
for the same hour of any day. Hence for the studied hour
we have:
n
X
qi
ð1Þ
p ¼ e fD ¼ e f
i¼1
where D ¼ total demand; qi ¼ electrical-power output of
supplier i; e, f ¼ positive coefficients publicly known from
historical records, constant here; n ¼ number of suppliers
(generator); and p ¼ MCP. As
Pelectrical-power energy can
not be stored, we have Q ¼ ni¼ 1 qi ¼ D.
Suppose that each supplier has a quadratic productioncost function which takes the form
1
Ci ðqi Þ ¼ ai þ bi qi þ ci q2i ; i ¼ 1; . . . ; n
ð2Þ
2
where ai , bi , ci are parameters of the cost function, generally
nonnegative.
2.2
Determination of MCP
For the studied hour, if all suppliers bid their supply
functions as qi ¼ qi ðpÞ ði ¼ 1; . . . ; nÞ, the ISO will
aggregate these bids, determine the MCP (p) and
individuals’ outputs to satisfy the demand at lowest costs.
It is clear that supplier i’s profit function is given by [see
(2) for Ci ðqi Þ]:
ð3Þ
pi ¼ pqi Ci ðqi Þ
Assume that each supplier is asked to submit a linear supply
function:
p ¼ b0i þ xi qi
ð4Þ
It is proven in [21] that optimal supply-function coefficient
b0i is equal to the true cost-function coefficient bi in (2).
Therefore, in our formulation, b0i is replaced by bi , and only
slope xi in (4) should be optimised. When ISO receives all
the bids, he will solve the following supply–demand-balance
equation for MCP (p) based on (1) and (4):
n
n
X
X
p bi
ep
¼ D
ð5Þ
qi ¼
¼
Q ¼
f
x
i
i¼1
i¼1
Here we assume that p4bi ði ¼ 1; . . . ; nÞ; otherwise the
corresponding supplier will not be dispatched. Solving (5),
IEE Proc.-Gener. Transm. Distrib., Vol. 153, No. 1, January 2006
yields MCP:
n b
P
i
i ¼ 1 xi
p ¼
n 1
P
1þf
i ¼ 1 xi
eþf
ð6Þ
Given the parameters e, f and fbi ; xi gi ¼ 1;...;n , the MCP can
be calculated according to (6). The individual generation qi
and the total supply and demand can be calculated
thereafter without difficulty. It is clear that parameters e,
f, bi are basically stationary and easily estimated from
historical data, while xi is versatile and time varying in the
repeated bidding. It is the decision of xi by an individual
supplier for maximising its profit that exercises market
power and causes spikes of MCP.
2.3
Optimal decision on xi
The slope of supply function xi should be nonnegative. In
this paper we assume xi 40 (if xi ¼ 0, we can assume it is a
very small positive value x). According to microeconomics,
we know that a rational supplier will not submit a supply
i
function below its marginal-cost function MCi ¼ @C
@qi ¼
bi þ ci qi ; therefore we have xi maxðci ; xÞ ði ¼ 1; . . . ; nÞ
and hence (because f 40; xj 40):
!
!
n
n
X
X
1
1
1þf
40; and
1þf
40 ð7Þ
x
x
j¼1 j
j ¼ 1;j6¼i j
Since p4bi ði ¼ 1; . . . ; nÞ, then with (6) and (7), we can
see that
n
X
bj bi
p bi 40 ) e bi þ f
40;
xj
ð8Þ
j ¼ 1;j6¼i
i ¼ 1; . . . ; n
According to (2), (3) and (4), the supplier’s profit function
can be rewritten as [eliminate p with (6)]:
p bi
p bi
1
p bi 2
pi ¼ p
ai bi
ci
2
xi
xi
xi
32
n
P
bj bi
7 6 e bi þ f
xj 7
6
1
j ¼ 1;j6¼i
7
6
!
¼ 6
7 x i 2 c i ai
n
P
5
4
1
xi 1 þ f
þf
x
j ¼ 1;j6¼i j
2
ð9Þ
It is clear from (9) that, if bj6¼i is fixed and xj6¼i can be
estimated from historical data; then, in the nonco-operative
situation, each supplier can determine its optimal xi based
on (9) to maximise its profit. Differentiating pi in (9) with
respect to xi , yields
!2
!
n
n
P
P
bj bi
1
e bi þ f
1þf
xj
xj
j ¼ 1;j6¼i
j ¼ 1;j6¼i
9
>
>
=
8
>
>
<
@pi
¼
@xi
f
þ ðci xi Þ
n
P
>
>
>
>
1
;
: 1þf
xj
j ¼ 1;j6¼i
(
xi 1 þ f
n
P
j¼1
!)3
1
xj
ð10Þ
45
For (10), with conditions (7) and (8), we can conclude that
0
1
B
C
@pi
f
B
C
þ
c
x
sign
¼ signB
C
i
i
n
P
1
@
A
@xi
1þf
j ¼ 1;j6¼i xj
and the supplier i’s optimal decision should be
f
xoipt ¼
þ ci
n
P
1
1þf
j ¼ 1;j6¼i xj
ð11Þ
ð12Þ
Eqns. (11) and (12) are reasonable from microeconomics,
i.e. when the number of supplier
P is infinite ðn ! 1Þ (or the
competition is perfect), then nj¼ 1;j6¼i x1j ! 0 and we have
xoipt ! ci ; when there is only one market supplier (monopoly) or the suppliers conduct a Cournot game (e.g.
duopoly of Cournot game), it is easy to prove [26] that
xoipt ¼ ci þ f . It is well known that a real market will be
less competitive than ‘perfect competition’, but more
competitive than ‘monopoly’; thus a supplier in an
oligopoly market should submit a supply function with
xoipt 2 ½ci ; ci þ f .
2.4
Features of SFE
When all suppliers learn, [24] has proven that there exists
the unique equilibrium, called supply-function equilibrium
(SFE).
Definition 1: A set of linear supply-function coefficients x
¼ ðxi Þi ¼ 1;...;n with xi 4 maxðci ; 0Þ is a Nash-supply-function equilibrium (SFE) if:
xi 2 arg maxfpi ðxi ; xi Þg; 8i ¼ 1; . . . ; n
ð13Þ
xi
where xi ¼ fxj jj ¼ 1; . . . ; n; j ¼
6 ig.
Definition 1 is correct since it is based on the definition of
Nash equilibrium. Thus at linear-supply-function equilibrium, we have [see (11)]:
f
; i ¼ 1; . . . ; n
ð14Þ
xi ¼ ci þ
n
P
1
1þf
j ¼ 1;j6¼i xj
Since (14) satisfies Definition 1, so the set of xi ði ¼
1; . . . ; nÞ constitutes a Nash SFE. This is an important
feature of SFE. From (14), we can see that to find the Nash
equilibrium of a power market under linear-supply-function
bidding is to solve a set of coupled nonlinear equations.
Also we can see that the equilibrium is independent of the
demand characteristic e and the marginal-cost-function
intercepts bi .
3 Impacts of overall learning on the market
equilibrium and market price
3.1
Concept of learning
In power markets, certain market information ðp; qj ; j ¼
1; . . . ; nÞ will be publicised. Supplier i can make estimates of
rivals’ bidding coefficients xj , based on the market
information (history data). Then, in the next round of
46
xi ½k ¼
1þf
Eqn. (11) shows that each supplier i has an incentive
to learn about xj of its rivals so as to get maximum
profit pi .
6 iÞ, we can conclude from (11) that a
Since xj 40 ðj ¼
rational supplier will take
ci xoipt f þ ci
bidding, it will improve its supply-function coefficient xi to
be the best response to the estimated coefficients xj;j6¼i . It is
clear that according to (11), supplier i can make the
following rational adjustment:
f
n
P
1
j ¼ 1;j6¼i xj ½k 1
þ ci
ð15Þ
where k is the time series and [k1] and [k] represented
the same hour of two subsequent days of k 1 and k.
Apparently, each supplier will have an incentive to make
the optimal decision through learning in the market in
order to obtain maximum profit. Here, learning in
the market means that supplier i adjusts its supply-function
coefficient xi according to (15) based on the estimation of
supply-function coefficients xj;j6¼i of others.
3.2
Impacts of rivals’ behaviour
From (15), we can see that each supplier’s profit pi is
also dependent on all the supply-function coefficients.
We can see that, during the repeated bidding, each supplier
will adjust its supply-function coefficient through
learning, which will in turn influence not only its own
profit, but also the others according to (9). Thus we need to
explore the impacts of one’s learning behaviour on
others’ profits and the impacts of overall learning on
6 iÞ
MCP. Differentiating pi in (9) with respect to xj ðj ¼
yields
!
n
X
@pi
1
bk bi
¼ 2 xi ci
e bi þ f
2
@xj
xk
k¼1
n
! e bj þ f P bk bj
ð16Þ
xk
fxi
k¼1
3
n 1
x2j
P
xi 1 þ f
k ¼ 1 xk
It is clear that we have [see (7), (8) and, with the constraint
xj maxðcj ; xÞ ðj ¼ 1; . . . ; nÞ]:
1
xi ci 40
2
!
n
X
bk bi
e bi þ f
40
xk
k¼1
!
fxi
40
x2j
!
n
X
bk bj
e bj þ f
40
xk
k¼1
(
xi
n
X
1
1þf
x
k¼1 k
!
)
þ f 40
Thus a very important observation can be made, i.e.
@pi
40
@xj
j¼
6 i
ð17Þ
Inequality (17) means that, if supplier j increases its
supply-function coefficient xj , its rivals can obtain more
profit, which lays a foundation for the study of interaction
of suppliers’ learning. To show the underlying economic
IEE Proc.-Gener. Transm. Distrib., Vol. 153, No. 1, January 2006
meaning, define supplier i’s residual demand function as
n
n
X
X
p bj
ep
ep
qi ð p Þ ¼
Qdi ¼
f
f
xj
j ¼ 1;j6¼i
j ¼ 1;j6¼i
!
!
n
n
X
X
bj
e
1
1
þ
þ
¼
p
f j ¼ 1;j6¼i xj
f j ¼ 1;j6¼i xj
ð18Þ
Thus, it can be seen that, if any other supplier increases its
6 iÞ, supplier i will face a
supply function coefficient xj ðj ¼
less elastic residual-demand curve, which is apparently
beneficial to supplier i.
3.3
Two dominant scenarios of learning
Two significant issues arising from learning in the markets
are how one’s decision will influence one’s rivals’ profits and
how the overall learning will affect MCP and total power
exchange. Two dominant scenarios are used as examples,
followed by further discussion. In the first case, all suppliers
have an initial guess of ‘highly competitive’ on market
conditions. In this case, supplier i revises its xi through
learning, and the MCP will increase with respect to time
and all suppliers will obtain more profits until market
equilibrium is reached. In the second case, all suppliers have
an initial guess of ‘highly oligopolistic’ on market conditions; then the MCP in equilibrium will decrease with
respect to time and at least some suppliers will obtain less
profit when the market equilibrium is reached, although
their incentives are to increase their profits. The detailed
proof is presented below.
Case 1: ‘Highly competitive’ initial guess
For this case, assume that the supplier’s initial guess is
that the market is ‘highly competitive’; therefore each
supplier will submit a supply function the same as its true
marginal cost function, i.e. xi ½0 ¼ ci ði ¼ 1; . . . ; nÞ.
According to (10), obviously we have
@pi ðx ½0 ¼ ci Þ 40 ði ¼ 1; . . . ; nÞ
@xi i
Since each supplier has the incentive to learn in the market
for maximum profit, we can expect that each supplier will
update its bid according to the rational learning process
(15).
Differentiating xi ½k in (15) with respect to xj ½k 1,
yields
f2
@xi ½k
¼
@xj ½k 1
ðxj ½k 1Þ
j ¼ 1; . . . ; n;
2
n
P
1
1þf
x
½k
j ¼ 1;j6¼i j 1
!2 40;
We know that the upper-bounded, increasing time series
fxi ½kg will finally converge [Here, it will converge to the
point xi ði ¼ 1; . . . ; nÞ, the unique solution of coupled
nonlinear equation (14)]. In the meantime, from (17), we
know that this adjustment will also increase others’ profits
(the profit function moves in the same direction as supply
function coefficients). Therefore, from the initial guess of a
‘highly competitive’ market, the overall learning behaviour
will benefit to all suppliers, i.e.
pi ðx1 ; x2 ; . . . ; xn Þ4pi ðx1 ½0; x2 ½0; . . . ; xn ½0Þ;
xi ½0 ¼ ci ; i ¼ 1; . . . ; n
Without loss of generality, it is easy to prove that, for any
xi ½0 2 fci xi ½0oxi g, inequality (19) still holds.
Case 2: ‘Highly oligopolistic’ initial guess
In this case, we assume that the supplier’s initial guess is
that the market P
is ‘highly oligopolistic’. Differentiating the
industry profit ( nj ¼ 1 pj , the total profit of all suppliers)
with respect to the variable xi at the equilibrium point fxi g,
i
we have [see (10), (14) and (17) and note that @p
@xi ¼ 0 at
xi ¼ xi ]:
n
n
X
X
@pj @pj ðxi ¼ xi Þ ¼
ðx ¼ x Þ 40; i ¼ 1; . . . ; n
@xi
@xi i i
j¼1
j ¼ 1;j6¼i
ð21Þ
Equation (21) implies, that at the equilibrium point fxi g,
the industry profit is not maximised ( from game theory; we
know that the industry profit is maximised at the cooperative-equilibrium point. Here we study the noncooperative equilibrium, and it is reasonable that the industry
profit is not maximised at the equilibrium). Also, from (9),
we know that the profit function pi ði ¼ 1; . . . ; nÞ is twice
continuously differentiable with respect to all coefficients xj
ðj ¼ 1; . . . ; nÞ, which means that the derivative given by
(21) is also continuous. Without loss of generality, we can
always find an upper-bound positive x, such that:
n
X
@pk
0; xi 2 xi oxi ^xi ; i ¼ 1; . . . ; n ð22Þ
@x
i
k¼1
where ^xi ¼ xi þ x ði ¼ 1; . . . ; nÞ.
At the point of f^xi gi ¼ 1;...;n , we have [see (7)]:
f
1þf
We know that xi ½k is a monotonically increasing function
6 iÞ.
of xj ½k 1 ðj ¼
Apparently, from the ‘highly competitive’ initial condition with xj ½0 ¼ cj ðj ¼ 1; . . . ; nÞ and (15), we have
f
f
þ ci ¼
xi ½1 ¼
n
n
P
P
1
1
1þf
1þf
x
c
½0
j ¼ 1;j6¼i j
j ¼ 1;j6¼i j
ði ¼ 1; . . . ; nÞ
It will in turn induce others’ responses with (19), so we
obtain an increasing time series fxi ½kg for any
i ¼ 1; . . . ; n. The series has its boundary condition (12).
IEE Proc.-Gener. Transm. Distrib., Vol. 153, No. 1, January 2006
1
^
x
j ¼ 1;j6¼i j
¼
j¼
6 i
þ ci 4xi ½0 ¼ ci
n
P
1þf
ð19Þ
ð20Þ
f
n
P
1
j ¼ 1;j6¼i xj þ x
x 1 þ f
¼
þ ci ^xi
n
P
j ¼ 1;j6¼i xj
n
P
f
1
1þf
þx
x
j ¼ 1;j6¼i j
1þf
n
P
1
j ¼ 1;j6¼i xj
1
þf
þx
!
n
P
x
1
x
j ¼ 1;j6¼i j
n
P
!
1
1þf
x
j ¼ 1;j6¼i j
ð23Þ
! o0
Therefore, with (14) and (23), it is not difficult to show that
@pi i ¼ 1; . . . ; n
ð24Þ
ðx ¼ ^x ;j ¼ 1;...;nÞ o0;
@xi j j
This is an important feature of case 2.
Now, under the ‘highly oligopolistic’ initial condition
ði ¼ 1; . . . ; nÞ,
we
have
with
xi ½0 ¼ xi þ x
@pi @xi ðxj ¼ xj þx;j ¼ 1;...;nÞ o0. Thus each supplier has the incentive
to decrease its supply-function coefficient, i.e., xi ½1oxi ½0
47
ði ¼ 1; . . . ; nÞ. Furthermore from (19), we know that the
time series fxi ½kgi ¼ 1;...;n is monotonically decreasing. The
series also has its boundary condition defined in (12).
We know that the low-bounded, decreasing time series
fxi ½kgi ¼ 1;...;n will finally converge [here, it will converge to
the equilibrium point xi ði ¼ 1; . . . ; nÞ, the unique solution
of coupled nonlinear (14)].
For the decreasing series fxi ½kgi ¼ 1;...;n , with condition
(22),
nP we knowothat the corresponding total-profit time series
n
j ¼ 1 pj ½k is decreasing too, i.e.
n
X
pj ðx1 ½k; x2 ½k; . . . ; xn ½kÞ
j¼1
o
n
X
pj ðx1 ½k 1; x2 ½k 1; . . . ; xn ½k 1Þ;
ð25Þ
(c) Moreover, the overall learning behaviour will lead to
reasonable market equilibrium. For example, if all suppliers
guess that the market initial competition is more likely
‘highly oligopolistic’ and bid with a large xi , then the
learning behaviour will guide suppliers to reduce the
bidding coefficients xi rationally. At the new equilibrium
point, the MCP will decrease, which reflects the true
relationship between market demand and supply. The
opposite is also true.
(d) The fact of (c) shows that the MCP at the market
equilibrium is actually an important economic signal to
reflect the true relationship of market demand and supply.
That is to say, suppliers’ rational learning and optimal
decisions are not a bad thing, since they can help to reach
an appropriate power-market equilibrium, which shows
clearly the advantages of free markets as ‘an invisible hand’.
j¼1
k ¼ 1; 2; . . .
4
Thus, at the equilibrium point fxi g there must be
n
X
pj ðx1 ; x2 ; . . . ; xn Þo
j¼1
n
X
pj ðx1 ½0; x2 ½0; . . . ; xn ½0Þ
j¼1
ð26Þ
If all suppliers have the same cost function (symmetrical
situation), then, at the equilibrium point fxi g, all supplies
have the same profit ½p1 ðx1 ; x2 , . . . , xn Þ ¼ p2 ðx1 , x2 , . . . ,
xn Þ ¼ . . . ¼ pn ðx1 , x2 , . . . , xn Þ. According to (26), we know
that individual suppliers’ profit will also decrease with
respect to time, i.e.
pi ðx1 ; x2 ; . . . ; xi ; xiþ1 ; . . . ; xn Þopi ðx1 ½0; x2 ½0; . . . ; xn ½0Þ;
xi ½0 ¼ xi þ x;
i ¼ 1; . . . ; n
ð27Þ
It is, indeed, a scenario of ‘prisoner dilemma’, in which each
supplier conducts ‘learning’ for more profit, but the
outcome is just the contrary.
In the unsymmetrical situation (i.e. suppliers’ cost
functions are different), according to (26), we know that
at least one supplier (say supplier l) will obtain less profit,
i.e.
pl x1 ; x2 ; . . . ; xi ; xiþ1 ; . . . ; xn opl ðx1 ½0; x2 ½0; . . . ; xn ½0Þ;
xi ½0 ¼ xi þ x;
i ¼ 1; . . . ; n;
9l 2 f1; . . . ; ng
Numerical test results
The IEEE six-generator 30-bus system is used for testing.
The suppliers’ cost-function coefficients are listed in Table 1.
Suppose that the (hourly-based) inverse demand function in
a spot market is known as (p in $/MWh, D and qi in MW):
p ¼ 50 0:02D ¼ 50 0:02
6
X
ð29Þ
qi
i¼1
Suppose that the market information is complete and the
supplier conducts learning based on (15). ISO receives all
the submitted linear supply functions [as (4)], determines the
MCP according to (5) and the individual generation. We
have simulated and analysed eight cases, where case 1 is a
pure Cournot competition, where all suppliers adopt the
Cournot competition strategy and the corresponding supply
function is p ¼ bi þ ðci þ f Þqi ; and cases 2–7 corresponding to different numbers of learners ( from one to six) with
the remaining suppliers adopting the Cournot strategy;
while in case 8, all suppliers adopt the perfect competition
strategy (being a price taker, the supply function is
p ¼ bi þ ci qi ). Table 2 shows the MCP, each supplier’s
generation output qi and profit pi at the equilibrium for the
studiedPhour as well as MCP and industry output Q and
profit i pi .
ð28Þ
For supplier l, the ‘prisoner dilemma’ still exists. (Moreover,
if the unsymmetry is not too serious, we can still expect that
most suppliers will obtain less profit.)
In both situations of case 2, the series fxi ½kgi ¼ 1;...;n are
all decreasing; thus the total supply will increase and the
MCP will decrease, which is beneficial to the consumers.
We can make further comments on the learning impacts
as follows:
(a) As an extension of cases 1and 2, we can expect that, if
the initial condition is mixed, i.e. some suppliers’ initial
actions are based on some ‘more competitive’ market
estimation, while others are just the contrary, then the
outcome of overall learning is uncertain. Generally, the
former will increase its profit, while the latter will experience
profit loss during the learning.
(b) In any situation, a rational supplier will always have the
willingness to learn, as learning means making optimal
decisions and bring about more profit compared with
nonlearning, no matter whether others are learning or not.
48
Table 1: Cost-function coefficients
Supplier
ai
bi
ci
1
0
2
0.02
2
0
1.75
0.0175
3
0
3
0.0175
4
0
3
0.025
5
0
1
0.0625
6
0
3.25
0.00834
Without loss of generality, let us examine supplier 2’s
profit fluctuation in different cases. In case 1 (all supplies
adopt Cournot strategy, i.e. the initial guess of market
competition is ‘highly oligopolistic’ and all suppliers do not
learn), supplier 2’s profit is $3461.7. In case 2 (only supplier
1 learns), supplier 2’s profit is reduced to $3031.4 for
nonlearning. In case 3, supplier 2 will also learn, and its
IEE Proc.-Gener. Transm. Distrib., Vol. 153, No. 1, January 2006
Table 2: Computation results (q in MW, p in $, p in $/Mwh)
Case 1
q1
319.06
p1
3054
q2
347
p2
3461.7
q3
261.39
2220.5
p3
261.39
q4
2220.5
p4
q5
166.82
1426.2
p5
406.23
q6
3988.5
p6
p
Q
P
14.76
1761.9
i
pi
16371
Case 2
459.89
3370
324.71
3031.4
242.82
1916.2
242.82
1916.2
156.69
1258.2
376.74
3430.5
13.927
1803.7
14923
Case 3
Case 4
432.11
2874.8
416.73
2617.6
487.63
3392.6
3099.5
1596.6
1737.3
1432.8
145.14
343.12
2845.6
13386
5
IEE Proc.-Gener. Transm. Distrib., Vol. 153, No. 1, January 2006
451.33
2759.6
298.19
1516.6
298.19
1516.6
353.4
1730.4
405.12
2080.6
258.44
1085.4
258.44
Case 8
348.43
1214
412.49
1488.8
238.74
712.47
238.74
1085.4
712.47
132.77
162.13
142.9
127.5
903.46
931.76
709.48
507.98
324.58
307.12
302.17
560.18
2279.7
11.954
1902.3
12420
In this paper, we study the impacts of power suppliers’
learning behaviour on electricity spot-market equilibrium
under repeated linear supply-function bidding. In such
markets, each individual supplier will conduct ‘learning’ to
improve its strategic bidding in order to earn more profit.
Therefore, it is significant to explore the impacts of such
learning behaviour on market equilibria and market
clearing prices (MCP). In this paper, we first establish the
mathematical model for a supplier’s optimal bidding, which
is used to solve for the market equilibrium. Then it is shown
that supply-function equilibrium is a Nash equilibrium; and
that, in certain conditions, the overall learning behaviour
will reduce the market clearing price, which in turn increases
consumers’ surplus and decreases suppliers’ profits; while in
some other conditions the results are just the opposite. In
either case, the MCP at equilibrium induced by the overall
learning behaviour reflects the true relationship of supply
397.32
2320.5
Case 7
138.77
12.448
Conclusions
Case 6
986.92
1877.6
profit will increase from $3031.4 to $3392.6, i.e. supplier 2
has the incentive to learn. From case 3 to case 6, when other
suppliers join the learning team one by one, supplier 2’s
profit will decrease successively. In case 7 (when all suppliers
learn), supplier 2’s profit is reduced to $2080.6, which is far
less than the profit of $3461.7 in case 1. Considering other
suppliers’ profit evolution, we can also make similar
observations. Therefore, there exists a ‘prisoner dilemma’,
i.e. although all supplies learn for more profits, the outcome
is that all suppliers obtain less profits if, at the very
beginning, a ‘highly oligopolistic’ guess about the market is
made by all suppliers. Moreover, we can see that the MCP
continuously decreases from case 1 to case 7, which means
that, when the initial estimate of market competition is
‘highly oligopolistic’, suppliers’ rational learning behaviours
will lead to a reduction of xi and hence a lower MCP. When
all suppliers behave as price takers from the guess of ‘perfect
competition’ in case 8, all suppliers’ profits are even less
than that in case 7. Therefore we can expect that, if starting
from case 8, and all suppliers learn rationally, the learning
will lead to more benefits to all suppliers. Finally, the
market will still converge to the equilibrium in case 7 with
MCP larger than case 8, but smaller than cases 1–6.
302.24
1564.3
2546.3
12.974
1851.6
302.24
1564.3
209.97
1079.6
456.11
2833.7
315.88
221.65
401.84
2385
471.66
221.65
1596.6
Case 5
11531
2206.9
11.814
1909.3
11252
2713.8
10.43
685.68
1960.5
8.97
1978.5
2051.6
9405.2
6596.3
and demand. Numerical results support the analytical
conclusions very well.
6
Acknowledgment
The authors acknowledge financial support from the RGC,
Hong Kong Government, Hong Kong.
7
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