1. INTRODUCTION
Currently, most electricity oligopoly markets are regulated on the
basis of competition among the companies or agents seeking, from the point
of view of the regulators, to establish an autoregulated price-fixing
mechanism. In this context, one important challenge for research has been
to obtain proper market behavior models to compute the agent energy
productions, using, among other approaches, the so-called Cournot
equilibrium approach (see [1] for
a review of its application in the electricity market). The Cournot
productions {Pe, e =
1,…,E} of each producer (i.e., the strategies) result from
maximizing the individual agent profit
Be(Pe) =
λPe −
Ce(Pe) when the
productions of the other agents are supposed to be fixed (λ is the
market price and
Ce(Pe) are the
generation costs) and when they do not respond to changes in the market
price (i.e., Pe(λ) =
Pe). If the demand curve D is a
linear function of λ, the Cournot equilibrium conditions are
formulated as
where the demand at price λ0 is D0
and the slope demand is 1/μ [i.e., D = d
− (1/μ)λ with d = D0 +
(1/μ)λ0]. Note that in the equilibrium, the
balance equation (demand D equal to the total generation
P1 + ··· +
PE) must also be satisfied.
Nevertheless, Cournot equilibrium presents an important weakness due
to its high sensitivity with respect to the residual demand curves (RDCs)
of each agent (see [6]). If they are
supposed to be linear, then it is (see [16])
where λ is the market price if the considered agent sells the
amount of production Pe and if
PE−{e}0 is sold by
the other agents.
In the literature, the market models found that take into account the
RDCs' sensitivity are not formulated as equilibrium problems and
assume that a probabilistic estimation for these curves is always
available (see [1] for a review).
Nevertheless, some drawbacks can lead to inappropriate uses of this type
of uncertainty modeling since (1) very often, different probability
distributions can be fitted for the same set of statistical information
(see [14]); (2) sufficient historical
information may not be available (see [11]); (3) to obtain a satisfactory computational
efficiency with probabilistic models, several simplifying hypotheses are
needed (normality, independence) reducing the rigor of the probabilistic
approach; and (4) it is very convenient to represent the subjective
(usually linguistic) information provided by the experts about the
RDCs' slope behavior. In these cases, possibility theory
(see [19]) emerges as an alternative
tool to model not only the uncertainty but also the imprecision and
vagueness (see [15]), and, therefore,
it can be more flexible than probability theory, although, to some extent,
less informative (see [10]). That is
why in this paper, possibility distributions have been chosen to
model the uncertainty in the RDC slope, which, according to (2), coincides
for each agent with μ (from now on denoted by
), which is the inverse of the demand slope. A common point
(Pe0, λ0) to all
possible RDCs of each agent is also assumed (see Fig.
1), which could be established by considering an approximated
equilibrium solution.
Additionally, when modeling real electricity markets, it becomes
convenient to obtain sensible energy agent productions when they face
different unfavorable, but possible to a certain degree, RDC slope
scenarios. Otherwise, seemingly good strategies could lead to a general
loss of producers profit and to a general inefficiency of the power
system, which can be avoided by seeking robust equilibrium solutions. In
this paper, robust market equilibria have been computed using two dual
approaches introduced in [4],
developed for possibilistic objective optimization problems (for a review,
see [12]).
The next section describes the fundamental theoretical aspects of
possibility theory to model the uncertainty of the RDC slope. Applying the
two dual approaches introduced in [4],
Section 3 proposes two alternative and complementary methods to compute a
robust Cournot equilibrium. Both approaches have also been combined into a
biobjective linear fractional optimization problem that provides better
compromise solutions. Section 3.3 presents an application to a real power
system with a medium-term horizon and discusses the numerical results.
Finally, the conclusion remarks are given.
2. PRELIMINARIES
Let Ω be the variability range of the RDC slope
,
and let
be the possibility measure defined on the power set of Ω,
with Π(A) being the degree of possibility that A
occurs. The fundamental axioms of Π are (see [19])
A possibility distribution π: Ω →
[0,1] can be defined from Π as π(μ) =
Π({μ}), ∀μ ∈ Ω, such that π(μ)
quantifies the possibility that the variable
takes the value μ. In this paper, only continuous possibility
distributions are considered. In particular, left and right
(LR)-possibility distributions (see Fig. 2 and
[8])
are used. Functions
are differentiable and strictly decreasing in (0,1),
L(s) = R(s) = 1 for s ≤ 0
and L(s) = R(s) = 0 for s
≥ 1.
LR-possibility distribution.
The principal advantage of LR-possibility distributions is that their
shape is preserved for many operations performed via the Extension
Principle (EP) (see [17,18]). For example, the addition and the
multiplication by a scalar are defined by
3. SOME APPROACHES TO OBTAIN ROBUST COURNOT
EQUILIBRIA
A robust Cournot equilibrium can be interpreted as a low-risk
equilibrium in the sense that it should be somehow insensible to
unexpected but possible inputs. Let us analyze the influence of the RDC
slope variability on the equilibrium solution:
- When the equilibrium is solved for a given RDC slope μ, but the
real RDC slope μr is less than expected, then
according to the equilibrium constraints (see (1)), if the agents keep the
same productions Pe, the following results
are obtained (see Fig. 3): (1) a lesser market
price (λr instead of λ); (2) a lesser profit
agent (λrPe −
Ce(Pe) instead
of λPe −
Ce(Pe)); and (3)
a loss of the demand utility due to a larger amount of unsatisfied demand
(D instead of Dr).
- On the other hand, when μ ≤ μr, the
opposite results are obtained.
In this sense, sensible agents should protect themselves against the
first case, trying to reduce the risk of unexpected low profits. This
suggests more sophisticated Cournot equilibria, where market agents take
risk minimization into account. In the sequel, three approaches to solve
this type of equilibrium are presented. For each solution, the possibility
distribution of the market price λ (denoted
) can be computed by applying (4) in the following fuzzy
equation:
3.1. The Primal Approach
In this approach, each agent looks for a set of productions that
maximizes a pessimistic (minimum) profit
Be,min(Pe,αmax0)
such that the possibility Π of having profits lesser than
Be,min(Pe,αmax0)
is equal to a same value αmax0 ∈
[0,1] for any agent; that is,
where the fuzzy firm profit in terms of
is (D is a function of Pe through
the balance equation)
Applying (4) and assuming in the sequel d ≥ D
(e.g., when price λ0 is near zero), each firm profit has
been modeled by a known LR-possibility distribution given by (if
d < D, different LR-possibility distributions are
obtained, and the best solution found assuming d ≥ D
and d < D should be chosen)
According to [4],
Be,min(Pe,αmax0)
is reached for μ = μL −
L−1(αmax0)αμL;
therefore, problem (6) is equivalent to
This means that the original fuzzy model (6) is simplified into a
crisp one that can be solved as in [2], by minimizing the following cost in the
sequel denoted
Cmax({Pe},D,αmax0)
(note that the balance equation is denoted by F):
To solve (10), a resolution procedure similar to the approach in
[2] could be used.
3.2. The Dual Approach
An alternative approach consists of looking for a set of productions
that minimizes, for each agent, the possibility of having profits less
than a minimum profit target
Be,min0, maintaining the
productions of the other agents fixed:
Asymmetric solutions could be obtained using different
αe,max for the agents, reflecting the existence of
agents with a different attitude against risk, although a same risk
attitude has been assumed (αe,max = α,
∀e = 1,…,E). According to [4], problem (11) is then equivalent to
which is at the same time equivalent (see the Appendix) to minimize
the following possibility degree, denoted by
αmax({Pe},D,Cmax0):
where if Pe,min0 and
μmin0 are the productions and the RDC slope,
respectively, giving the profits
Be,min0, then
Cmax0 is
To solve (13), a resolution procedure similar to the approach shown in
Section 3.3 could be used.
3.3. The Combined Approach
A more sensible approach to robust Cournot equilibrium is to determine
a compromise solution between primal and dual approaches, maximizing
simultaneously the profit
Be,min(Pe,αmax0)
and the necessity
β(Pe,Be,min0)
= 1 −
α(Pe,Be,min0):
that is, the following biobjective optimization model (obtained by
combining (10) and (13)) must be solved:
In order to compute a compromise solution, it is interesting to define
a preference function to prioritize objectives (see [20]). To be able to compare both objectives in a
same [0,1] scale, the following linear functions are
proposed:
where the parameters Cumax0 and
αumax0 are some undesirable objective
values for (10) and (13), respectively.
Then, to calculate a compromise solution of problem (16), the minimum
operator function has been chosen as the preference function to combine
p() and d() (see [13] for other compensatory operators):
which is equivalent to
where, for each set of productions, the auxiliary variable θ can
be interpreted as the minimum “distance” to the primal and
dual optimal solutions.
This new problem is a nonlinear programming one. To apply linear
optimization software, first the quadratic functions in (10) and (13) must
be linearized using, for example, the same piecewise approximation found
in [2]. Then, the new problem is
linear in Pe but not in θ, and,
therefore, the robust Cournot equilibrium is the solution of the linear
feasible region obtained when the higher value of θ is fixed. Thus,
this model can be solved using the iterative algorithm suggested in
[7].
4. CASE STUDY
This case study has been implemented in GAMS language (see [3]) and solved using CPLEX solver (see
[5]). It has been executed in a
1.8-GHz PC with 524 MB and a Pentium IV processor.
4.1. Case Description
The represented model corresponds to the Spanish hydrothermal electric
power system. A multiperiod case is considered; it has 12 periods
(January–December), 2 subperiods for each period (working days and
weekends), and 3 load levels for each subperiod (peak, off peak 1, and off
peak 2). There are 7 agents in the market that own 29 hydraulic groups and
101 thermal groups. Together with the balance equation, some additional
constraints have been incorporated in the model as, for example, the
technical constraints of each generation group. The resulting case study
then has approximately 84,000 constraints and 120,000 variables.
The considered fuzzy RDC slope is modeled by an LR-possibility
distribution for each load level with L(s) =
R(s) = max(1 − s,0). The width
αμL = αμR has been
chosen equal to 40% of μL =
μR (see Table 1).
The risk parameter αmax0 has been fixed at
0.4 pd and the maximum cost Cmax0 is
−3 × 106 K$ through previous solution of the crisp
model of [2] for μ =
μL − αμL and
μ = μL, to obtain an estimation of the smaller
and larger cost Cmax0, respectively.
Finally, the parameters of function p() and d() in (17)
are shown in Table 2.
Parameters of p() and d() Functions
4.2. Results Discussion
Two different types of equilibrium have been compared. The first one,
denoted by {Pme}, has been computed for the
most possible RDC slope μL and solved with the
crisp model presented in [2]. The
execution time in this case is about 3 min. The second one, denoted by
{Pre}, has been obtained with the combined
approach proposed in this paper. The execution time has been about 10 min,
larger than the nonfuzzy approach because of stronger nonlinear
conditions. The maximum satisfaction level θ* obtained has been
0.95.
To check the robustness of these two equilibria defined by their
solutions, {Pme} and
{Pre}, the crisp cost function of [2], simply denoted
C({Pe},μ), that measures the
system efficiency can be evaluated under different possible RDC scenarios.
Lesser RDC slope scenarios than the most possible one (corresponding to
μL) can be represented by the parametric function
μ(s) = μL −
sαμL with s ∈
[0,1]. For each scenario μ(s), the best
alternative between {Pme} and
{Pre} is the one that provides the minimum
system cost. By computing Dif(s) =
C({Pme},
μ(s)) −
C({Pre},μ(s)),
it is possible to determine which is the best alternative, depending on
the value of s.
As Figure 4 shows, in most cases the cost
obtained with {Pre} is less than with
{Pme}, meaning that
{Pre} is a low-risk equilibrium in the sense
defined in the previous section. Nevertheless, the cost obtained with
{Pme} in the most-possible scenario
(s = 1) is necessarily lower than with
{Pre}, following the premise that risk
protection always implies a cost increment.
When market prices are compared for each load level, again in most
cases {Pre} leads to lower prices than
{Pme}, corresponding to larger system
efficiency. Figure 5 shows the fuzzy average
market price when the robust equilibrium
{Pre} is considered.
Average market price possibility distribution.
5. CONCLUSIONS
This paper shows that the possibility theory can be effectively used
to model the uncertainty of the RDC for an electricity market, where each
agent maximizes its benefit based on the Cournot equilibrium conjecture.
Two dual criteria have been proposed in order to obtain a robust Cournot
equilibrium, protecting the agents against an overestimated RDC slope,
which could lead to a significant loss of profits and to a general
inefficiency of the power system. These two criteria, dual in the sense
shown in [4], have been combined to
obtain a better compromise solution, leading to a nonlinear programming
model that can easily be solved with an iterative algorithm.
The combined approach has been applied to a real large-scale
electricity market. It has been checked that the solution it provides is
more robust than the one computed for the most possible RDC.
Future works could extend the proposed approach to other power-related
markets for which the Cournot framework is not appropriate, such as
markets where agents offer a supply function (or a quantity–price
function). Other interesting improvements could be the inclusion of fuzzy
constraints, such as those modeling generation units availability, or the
power network physical constraints.
Acknowledgments
The authors thank their colleagues J. Reneses, E. Centeno, and F. J.
Santos for the development of the programmed model for the nonfuzzy case
and gratefully acknowledge the comments that led to the improvement of
this paper.
APPENDIX
Proofs of the equivalence between problems (12) and (13) as stated in
the dual approach are given.
- On the one hand, the defined robust Cournot equilibrium is obtained
solving (12); that is, differentiating
α(Pe,Be,min)
with respect to Pe and zeroing for all
e:
As the derivate of L() is strictly negative in the range
(0,1) (because it is strictly decreasing), then it is
It is not difficult to show that the above equations are equivalent to
the following:
- On the other hand, the Lagrange function of (13) is
As in (A.1), for all e, one can check that
where
If d ≥ D, then applying (4), the LR-possibility
distribution of the cost defined in [2] when the RDC slope is fuzzy is equal to (if
d < D, a similar LR-possibility distribution is
obtained; there is no existing relevant difference in the proof with
regard to the case d ≥ D)
As one can check, the left-hand sides of (A.3) and (A.5) are equal;
that is,
In addition, according to the definition of
Cmax0,
Then, in terms of possibility measure for all e, it holds
true that
Because the agents independently choose the minimum production level
Pe,min0, the equality in the above
equation is obtained (see [9]).
Therefore, (A.3) and (A.5) are equal, and the equivalence between problems
(12) and (13) is proved.
Note that ∂Λ/∂D = 0 defines the Lagrange
multiplier η for each feasible solution, and
∂Λ/∂η = 0 is the demand balance equation.
