Abstract—This paper studies the maintenance decisions of
generating companies (GENCOs) which are fully engaged in
oligopolistic electricity market. Maintenance decisions in an
oligopolistic electricity market have a strategic function, because
GENCOs usually have impacts on market prices through
capacity outages. The main contribution of this paper is modeling
a game theoretic framework to analyze strategic behaviors of
GENCOs. Each GENCO tries to maximize its payoff by
strategically making decisions, taking into account its rival
GENCOs’ decisions. Cournot-Nash equilibrium is used for
decision making on maintenance problem in Oligopolistic
electricity market. The analytic framework presented in this
paper enables joint assessment of maintenance and generation
strategies; it also considers the regulation of ISO on GENCOs'
desired maintenance plan.
Index Terms—Cournot-Nash equilibrium, game theory,
maintenance scheduling of generating units, Oligopolistic market.
Power generated by unit j of GENCOi in period
Total energy supply in period t (MWh).
Binary maintenance decision variable for unit j
of GENCOi in period t (1 if unit j is on
maintenance in period t and 0 otherwise).
Time periods (week).
Number of time periods (52).
Intercept of the linear price/demand curve
Slope of the linear price/demand curve
Power demanded in period t (MW).
M. A. Fotouhi is with K. N. Toosi University of Technology, Tehran, Iran
S. M. Moghaddas Tafreshi is with K. N. Toosi University of Technology,
Tehran, Iran (e-mail: firstname.lastname@example.org).
Forced outage rate of unit j of GENCOi.
Set of indices of generating units owned by
Hours of period t (168 hours).
Number of GENCOs.
Maintenance cost of unit j of GENCOi ($/MW).
Maximum numbers of units in maintenance for
Duration of the maintenance outage of unit j of
Capacity of unit j of GENCOi.
Production cost of unit j of GENCOi ($/MWh).
Reserve requirement in period t.
REVENTIVE maintenance scheduling of generating units
is an important mission in power system and plays vital
role in operation and planning of the system. Maintenance
strategy is a complicated optimization problem for mid-term
power systems operations planning. Several methods have
been proposed recently to solve the maintenance scheduling of
generating units. It is in fact determining the maintenance
period of time for individual generators subject to several
constraints over a given horizon.
In a centralized electric power system, an appropriate
generation maintenance scheduling is derived by the system
operator and imposed to producers . The reliability
evaluation of maintenance scheduling and least-cost
optimization algorithms had been one of the main concerns for
the last few decades . In most of the previous works,
however, generators are maintained or not for system-wide
reliability or least-cost rather than for their own profitability.
In the new competitive environment, customers request for
high reliability services with lower electricity prices, while
GENCOs have to make their own profit.
In a competitive market environment which the
management of GENCO and ISO is separated, unit
maintenance scheduling is determined through multiple
interactions between ISO and GENCOs each maximizing its
A Game Theoretic Framework for Generation
Maintenance Scheduling in Oligopolistic
M. A. Fotouhi and S. M. Moghaddas Tafreshi
own benefit. GENCOs will try to schedule their units for
maintenance in order to maximize their benefit. The ISO seeks
a generation maintenance annual plan that ensures similar
reliability through the weeks of the year, prior to the ISO’s
coordination process, individual GENCOs should have their
own maintenance strategies in advance .
In deregulated power systems usually GENCOs have
independence to maintain their generators in a decentralized
manner . Strategic behaviors of GENCOs can be modeled
in a game theoretic framework, and players of the game
correspond to GENCOs.
Maintenance of generating units may cause outage of a
significant amount of its capacity. Withdrawing generation
capacity is a strategic decision GENCOs adopt in order to
increase electricity prices. Maintenance decisions is a way of
withdrawing some parts of generation capacity for a period of
time, therefore, it has potentially major impacts on spot prices
In this study, a game theoretic framework is suggested to
solve maintenance scheduling of generating units under
competitive market environment. Gaming considerations can
earn GENCOs higher profit because they can effectively
exercise market power when their competitors’ capacity is on
Most of the research done on the electricity markets was
based on the Cournot model. The Cournot oligopoly model
assumes that each strategic firm decides its quantity to
produce, while treating the output level of its competitors as a
constant. The optimal strategy profile is defined by Cournot-
Nash equilibrium of the game.
The regulation of ISO is also considered, and the difference
between their scheduling is compared. The objective in using
such a method is to attain an applicable blend between
maximum reliability of the system and maximum benefit of
generating firms. Note that maintenance outages decrease
reliability and increase operation cost. Although the
coordination process how ISO adjusts the individual
GENCOs' maintenance schedules and how each GENCO
responds to the ISO's proposed schedule is important, it is
beyond the scope of this paper.
A hypothetical test system is considered to show the
applicability of the proposed model. The results obtained point
out that maintenance scheduling can be one of the important
strategic behaviors whereby GENCOs maximize their profit in
a competitive market environment.
III. COURNOT BEHAVIOR OF GENCOS
In order to analyze real markets, economists have developed
models between two extreme cases, perfect competition and
pure monopoly. Oligopoly competition refers to a market
structure where a few players coexist. Taking perfect
competition and monopoly models as the end points, there is
an infinite number of theoretical possibilities for oligopoly
models, all of which differ mainly in the assumptions used to
characterize market structure and firm interdependencies.
The Cournot model has become a classic in microeconomic
oligopoly theory. Cournot games have the following
characteristics in common:
competition occurs only in quantities
product is homogeneous
market price is determined by auction
Players schedule for their production simultaneously.
In the Cournot model, each firm chooses an output quantity
to maximize profit. It is assumed that quantities produced are
immediately sold. Market price in the model is determined
through an auction process that equates industry supply with
aggregate demand. The model also assumes that all firms in
the industry can be identified at the start of the game, and that
decision-making by firms occurs simultaneously .
Each firm is sufficiently large to influence market price
received by all, and the quantity produced by other firms.
Each firm maximizes its own profit given the quantity chosen
by other firms expressed as .
other players i′ ;
Price as a function of all q’s i.e., firms i, i', etc.
This is the main characteristic of an oligopolistic
market that distinguishes such markets from a
perfectly competitive one—players can influence
market price by changing their production as
opposed to a “price taker” behavior exhibited in a
Cost as a function of production strategy
Number of players.
The solution of the game is obtained by solving a set of
simultaneous equations representing the first order optimality
conditions for each firm i .
The Nash equilibrium formation of Cournot’s duopoly
model is shown in Fig. 1. The two axes define the output of
the firms. In this model the reaction curve represents how
much each firm would produce given the production decision
from the other firm. The Nash equilibrium defines where each
firm has maximized profit, given the output of the other.
The Cournot model is often used to describe the behavior of
generating companies (GENCOs) in electricity markets. We
consider n GENCOs, assume that each GENCO uses a
Cournot model to derive its generation and maintenance
scheduling in the market, and obtain some relevant analytical
results characterizing the market. GENCOs are assumed to
maximize profit according to the Cournot assumption using
production quantities as the decision variable. It is considered
that the transmission network does not influence the
equilibrium, i.e., that no network constraint is binding.
Cournot models are often encountered in the technical
literature as they adequately represent producer behavior in
real-world markets .
A Cournot-Nash equilibrium outcome is characterized by
the familiar Nash equilibrium concept that no player can gain
any additional profit by changing its own generation strategy
while every other player keeps its own generation unchanged
Profit of player given the production strategy of all
Production strategy of player i;
Fig. 1. Nash-Cournot Equilibrium in duopoly model.
IV. LINEAR DEMAND EQUATION
In characterizing market participants, producers are usually
represented by their cost curves and consumers by their
demand curves. It is relatively easier to quantify cost curves,
for example producers with thermal generators are often
modeled by their heat rate and fuel cost. In comparison,
demand curves are more difficult to empirically quantify as
they are derived from the subjective utility of consuming
volumes electricity . If the GENCOs do not know the linear
demand function, they must estimate the demand. Different
methods have been proposed for estimating linear demand
function which is not a main concern of this paper.
It is generally difficult to determine demand curves from
fundamental utility considerations as utility is intrinsically an
ordinal quantity, also the consumer income constraints
required for the derivation are largely unknown. Consequently
with few exceptions, most market simulations model demand
by directly specifying a representative demand function which
relates the price a consumer is prepared to pay for a volume of
If it is assumed that there is negligible transmission loss,
then the aggregate demand Q will be equal to the total output
of all the Gencos in the market as
The market price P depends on Q and their relationship is
represented by the inverse linear demand function in (3)
Whereγ ,λ are the positive coefficients of the linear demand
V. FIRMS' SCHEDULING
GENCOs would try to extract maximum profit by
scheduling their maintenance in a period to incur least
opportunity costs, and to make the most of all other periods
when its competitors’ generators are scheduled for
maintenance . Note that all GENCOs considered are price-
maker, i.e., they have the capability of altering market clearing
prices. We may think of each GENCO trying to work out the
best generation strategy as well as the maintenance decisions
by looking at the mutual impact of maintenance on their base
reaction function . GENCOs' objective for maximization is
ij ij ij
For the purpose of simplicity, no uncertainty is considered,
which means that appropriate linear demand functions are
forecasted for each week.
The set of constraints of the maintenance scheduling
problem of the GENCOi are defined below.
1) Maintenance Outage Duration: The following constraint
guarantees for each unit that it is maintained the required
number of time periods 
2) Continuous Maintenance: The constraint below ensures that
the maintenance of any unit must be completed once it begins
,,) 1 () 1
maintenance: Constraint (10) limits the maximum number of
units that GENCOi can maintain at the same time 
4) Maintenance Exclusion: This constraint enforces the
impossibility of maintaining two prespecified unit of the same
GENCO at the same time 
It means parallel maintenance of unit j and j′ of GENCOi is
VI. REGULATION BY ISO
It should be noted that the role of ISO is to ensure system
security. Therefore, it must agree with GENCOs on a
generation maintenance plan that preserves system security.
The ISO solves a maintenance scheduling problem involving
all units, independently on which GENCO owns each unit,
with the target of maximizing the reliability throughout the
weeks of the year. Sufficiently accurate demand forecasts for
the whole year are considered known .
The ISO compares the maintenance outage timing
scheduled by GENCOs with its desired timing. If the
reliability index of two plans are closed to each other, the ISO
accepts the GENCO's decision, If not ISO encourages
GENCOs to alter their plan by proposing incentives. The
objective function of ISO can be formulated as follow
The ISO considers the constraints mentioned for generating
units. It also has a constraint which guarantees a net reserve
above a specified threshold for all periods 
VII. CASE STUDY
A hypothetical system formed by 11 generating units
belonging to 3 GENCOs is introduced (Table I). The specific
data of generating units is based on IEEE reliability test
system . GENCOs' individual limitations shown in Table
II. depend to there structure and desires.
u1 400 8 40 30
GENCO1 u2 197 6 44 27
u3 197 6 38 27
u4 76 3 31 21
u5 350 8 42 32
GENCO2 u6 155 5 28 25
u7 100 4 34 33
u8 20 1 35 18
u9 400 8 40 35
GENCO3 u10 350 8 42 32
u11 12 1 31 19
Maintenance exclusion z
Maximum number of
units on maintenance at t
In Fig. 2 the peak demand and average demand during 52
weeks of the year is shown. The weekly electricity price and
demand of Omel deregulated power system is used and they
are normalized to match our hypothetical system . Linear
demand function parameters are actually captured from loads,
but for the sake of simplicity they are estimated from demand
and price data.
Table III illustrates GENCOs initial maintenance plan
which should be checked by ISO. In time periods not shown in
the table no maintenance outage is anticipated by GENCOs.
The model is implemented using GAMS.
ISO has its own maintenance plan, hence GENCOs send
their initial plan to ISO and it will be compared with ISO's
desired plan. Table IV illustrates ISO's ideal plan.
GENCOS' INITIAL MAINTENANCE PLAN
147101316 192225 28 3134 37 4043 46 49 52
Fig. 2. Demand graph during 52 weeks.
In Fig. 3 the difference between maintenance scheduling of
GENCOs and ISO's ideal plan is shown. Now it needs
coordination between them to achieve the final plan.
ISO'S IDEAL MAINTENANCE PLAN
The proposed framework is simple to implement in practice,
and requires a reasonably small amount of computing time and
a small amount of data communication. It is presented based
on the fact that GENCOs will try to strategically place their
maintenance and dispatch their generators taking into
consideration their rivals’ strategies. This implicitly assumes
each GENCO is able to at least “guess” the capacity and cost
parameters of its competitor GENCOs—an assumption that is
not unreasonable or unrealistic.
The results emphasize the need to an advanced coordination
procedure between GENCOs and ISO in order to acquire final
maintenance scheduling plan.
0 200400 600800
Fig. 3. Different plans of GENCOs and ISO.
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