356 IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 19, NO. 1, FEBRUARY 2004
Abstract— This paper presents a methodology for the development of bidding strategies for electricity producers in a competitive electricity marketplace. Initially, the problem is modeled as a two level optimization problem where, at the first level, a market participant tries to maximize his expected profit under the constraint that, at the second level, an independent system operator dispatches power solving an optimal power flow problem that minimizes total system cost. It is assumed that each supplier bids a linear supply function and chooses his bidding strategy based on probabilistic estimates of demand and rival behavior. Monte Carlo simulation is used to calculate the expected profit and Genetic Algorithms are employed to find the optimal strategy. Subsequently, the formulation is expanded to account for different market participants’ risk profiles. It is shown that risk aversion may influence the optimal bidding strategy of an individual.
Index Terms— Bidding strategies, electricity market, genetic algorithms, Monte Carlo simulation, risk aversion, utility theory.
network admittance matrix (size ) bus to generating unit incidence matrix (size
) bus to consumer incidence matrix (size ) branch to bus incidence matrix (size ) diagonal branch primitive reactance matrix
(size ) bus on which unit is connected generating unit bid vector (size ) generating unit actual production cost function generating unit submitted cost function branch rating vector (size ) total profit of an individual utility of an individual’s total profit
,
,
,
,
,
I. N OMENCLATURE bus index number (set) of buses branch index number (set) of branches generating unit index number (set) of generating units number (set) of generating units controlled by an individual set of generating units controlled by an individual’s competitors consumer (demand) index number (set) of consumers bus active power injection vector (size ) generating unit output vector (size ) generating unit maximum output vector (size
) generating unit minimum output vector (size
) consumer demand vector (size ) expected consumer demand vector (size ) consumer demand covariance matrix (size
) bus voltage phase angle vector (size ) system reference bus voltage phase angle nodal price vector (size )
Manuscript received July 8, 2003. This work was supported by the National
Fellowship Foundation of Greece.
The authors are with the Department of Electrical Engineering, Aristotle University of Thessaloniki, 54006 Greece.
Digital Object Identifier 10.1109/TPWRS.2003.821474
II. I NTRODUCTION
I N the last years, the power industry is undergoing massive changes around the world. The electricity supply, once regarded as public service provided by state owned or regulated utilities, is now being opened to competition and is subject to market rules. What brought about this major shift, from the intensely regulated monopoly to a deregulated electricity market, was mainly the notion that individual initiative could result in a better allocation of resources, thus supplying the end customer with a lower cost but high reliability energy supply. Following the early attempts of Britain, California, Australia, Argentina and the Scandinavian countries, most countries have embarked on the task of creating an effective electricity market.
In this new framework, new mathematical tools have to be devised that facilitate the analysis of the new market structure.
Such tools are essential both to the power producers in order to define their optimal bidding strategy, and the regulatory authorities in order to prevent anticompetitive acts by market participants. A significant amount of research has been conducted the past several years concerning the market structure and the development of efficient bidding strategies for power producers.
In this effort, a variety of different approaches was used. An effective way is to develop a forecasting model for electricity markets. A short term price forecasting model based on artifi-
cial neural networks is presented in [1]. In [2], time series anal-
ysis is used to create price forecasting tools based on dynamic
regression and transfer functions. In [3], a combined time se-
ries and stochastic forecasting method, which incorporates “rational” strategic behavior modeling of the market participants, is presented. Having developed a forecast of the electricity prices, bidding strategies for power suppliers can be created. A comprehensive tool for a supplier’s simultaneous participation in the
0885-8950/04$20.00 © 2004 IEEE

GOUNTIS AND BAKIRTZIS: BIDDING STRATEGIES FOR ELECTRICITY PRODUCERS IN A COMPETITIVE ELECTRICITY MARKETPLACE 357
energy, AGC and reserve markets can be found in [4]. Although
price forecasting techniques have small average errors (3% and
5% are reported in [2]), the major drawback of this approach is
that it can neither examine the actual mechanism of electricity price fluctuation, nor determine the way electricity prices can be influenced by individual suppliers since market participants are viewed upon as price takers. Key issues like potential monopolistic behavior, regional market power and network constraint exploitation can not be addressed.
Game theory has also been used as a means of examining the
emerging electricity markets [5]. This approach takes into con-
sideration the fact that market participants react to competitor
strategies in order to maximize their pay-off. In [6], the en-
ergy market is modeled as a noncooperative game of incomplete information and a Cournot game is solved to determine the market equilibrium state. In this game, individuals establish rival behavior based on an estimate of rival cost functions.
Although modeling the game as a noncooperative game of incomplete information is realistic, this method cannot be easily extended to real markets since many simplifying assumptions have to be made and technical constraints are not incorporated.
In [7], a game of complete but imperfect information is formu-
lated where technical constraints are ignored and the Nash Equilibrium solution is determined via the creation of the expected
payoff matrix. In [8], more technical constraints are incorpo-
rated, but the game is regarded as a noncooperative game of
complete information which is unrealistic. In [9], necessary and
sufficient conditions of Nash Equilibrium bidding strategies are derived and Nash equilibrium bidding strategies are constructed via a network optimization technique. System losses and transmission charges are also taken into consideration but complete information is assumed and significant simplifying assumptions concerning the power allocation have been made that prohibit the application of the method on real markets.
The maximization of individual profit is an approach where the market is examined from a market participant’s point of view. The search of optimal bidding strategies for an individual assumes the knowledge or the estimate of competitor behavior.
Although this approach can neither capture the dynamics of competitor reaction nor pinpoint equilibrium market points, it can present a more realistic model of the electricity marketplace, incorporating constraints that are present in real markets and provide market participants with a tool to define their
market tactics. In [10], a simple framework for the develop-
ment of bidding strategies in an energy brokerage is presented.
In [11], evolving trading agents, whose evolution is based on a
genetic algorithm (GA), are used to simulate the electricity auction. The drawback of the method is that all agents are adapting their strategies at each GA generation making it very difficult to
identify if and why a particular strategy is a good one. In [12],
the problem is formulated as a two-level optimization procedure with a centralized economic dispatch that determines market clearing prices at the top level, and a self-unit commitment simulator at the second level. Time coupling constraints are also taken into consideration, but complete information about com-
petitors is assumed. An extension of the work presented in [11],
which uses finite state automata and investigates adaptive strate-
gies, is presented in [13]. In [14], the effects of electricity market
rules are investigated, using a simple market of three players
where one is not bidding strategically. In [15], a framework is
suggested that uses Monte Carlo simulation in order to create efficient bidding strategies based on estimates of rival behavior. In
[16], Monte Carlo simulation and Genetic Algorithms are em-
ployed in order to coordinate the bidding strategies of a supplier participating in energy and spinning reserve markets. Both
models, [15] and [16], assume incomplete information but disre-
gard the load uncertainty. Moreover, the effects of transmission
constraints are ignored. In [17], optimal multi-period bidding
strategies are developed with the application of a discrete-state and discrete-time Markov decision process. The model incorporates load uncertainty and rival behavior uncertainty but ignores transmission constraints. Moreover, the computational requirements explode with the number of bidding options examined.
In [18], a model is proposed where a supplier’s bidding and
self-scheduling problem is solved using Lagrangian relaxation.
In [19], two methods are presented to create optimal bidding
strategies. One of them is based on Monte Carlo simulation as
in [15], the other is optimization based and more efficient computationally. In [20], ordinal optimization is used to isolate a set
of “good” strategies. Subsequently, hydrothermal scheduling or unit commitment is used to select the best strategy from the ones
selected. In [21] and [22] the development of bidding strate-
gies is modeled as a two-level optimization problem, where market participants, suppliers and consumers, try to maximize their profits under the constraint that an independent system operator (ISO) determines market prices and dispatches using an optimal power flow program that maximizes social welfare.
This model allows for the investigation of transmission constraint issues at the deregulated market. It is assumed that market participants know, or have a single, non probabilistic estimate, of their competitors’ behavior.
This paper proposes a methodology for the development of optimal bidding strategies for electricity producers. Initially a
two-level optimization model, similar to that in [21] and [22], is
defined, where an ISO solves an OPF to determine dispatches and nodal prices. This model however assumes incomplete knowledge of competitors’ behavior. This behavior is modeled probabilistically and is allowed to have an arbitrary probability density function. Moreover, the formulation accounts for load uncertainty and the fact that demands of different nodes are correlated. The expected profit is calculated using a Monte
Carlo simulation and the optimal bidding strategy is derived either by means of exhaustive search of the search space, for small size problems, or with the utilization of Genetic algorithms for larger problems. The proposed approaches are then extended to incorporate an individual’s risk profile. The need
for such an investigation is stated in [17] where risk neutral
market participants are considered. In this paper, the economic notion of the utility function is used and bidding strategies that maximize expected utility are derived. It is shown that, due to uncertainty, the optimal bidding strategy changes with the degree of risk aversion and the type of utility function used.
The approaches developed, (Monte Carlo/exhaustive search and Monte Carlo/Genetic Algorithms) have been tested on sample systems and their results are completely consistent with each other. Moreover, they provide a significant insight on how

358 IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 19, NO. 1, FEBRUARY 2004 certain constraints can be exploited by market participants, the effect of a certain participant’s risk profile on his bidding strategy and the effects of his choice on the market.
In competitive electricity markets, the suppliers are required to submit MW outputs, along with associated prices. The general rule that applies is that the price at which the power is offered is a non-decreasing function of the amount of power. Two bidding schemes have been implemented, the block bidding and the continuous bid curve (Fig. 1).
In this paper, a continuous bid curve model for suppliers is assumed where a supplier is requested to submit his marginal cost function. Consumer bidding will not be addressed in this paper. However, the methodology presented can easily incorpo-
rate consumer bidding with minor modifications [21].
The bids of all the suppliers (and consumers where applicable) are collected by the Independent System Operator (ISO) who determines the power output of every unit. To do so, the
ISO solves the social welfare maximization problem. In case of no consumer bidding, this is the problem of total system cost minimization. This involves running an OPF algorithm that determines supplies and prices. In this paper a nodal pricing model is assumed, in which every supplier is paid the price of electricity at the node he is connected to. The Lagrange multipliers of the OPF solution define the nodal price vector, .
In a perfectly competitive market a supplier would maximize his profits by bidding his true marginal cost function. However, mainly due to the limited number of power companies, the fact that investments are capital intensive and the constraints of the transmission system, the electricity market is more akin to an oligopoly. Therefore, market power can be exercised to ensure profits well above the ones that would be realized in a perfectly competitive market.
It is assumed that the units have an actual production cost function true marginal cost function, that corresponds to a
, which would have been the unit’s linear bid curve under perfect competition. Fixed costs are ignored since this methodology is con-
cerned with one-period auctions [15]. It is also assumed that
suppliers have the ability to price away from their true marginal productions costs, thus developing a bidding strategy. The variation in bidding will be modeled as the variation of a single parameter,
III. E LECTRICITY M ARKET S ETUP U SING OPF
, for each supplier unit as shown in Fig. (1b). This parameter will vary the bid around the true marginal cost curve of the generating unit:
Although this may seem limiting as far as potential market
behavior of a supplier is concerned, it has been shown [22] that
from a supplier’s viewpoint, is an adequate variation parameter. Note that modifying a bid in this manner is the same as multiplying the cost function used in the OPF by , i.e.
(1)
Fig. 1.
Supplier block bid and continuous bid curve with bidding variation.
IV. I NDIVIDUAL E XPECTED P ROFIT
The total profit of an individual is:
M AXIMIZATION
In power markets, a particular power company controls a number of generators. In order to accommodate the fact that a set of units may decide on a strategy under the same goal, the notion of the individual will be used. An individual is the economic entity that controls a combination of several generators.
If consumer bidding is allowed, the definition of the individual can be expanded to include consumers as well.
It is assumed that each individual seeks to maximize his personal profit. A single unit’s profit is the amount of revenue received from selling the power, minus the cost of supplying the power. An individual wants to maximize the total profit of all the units that he controls.
Let denote the set of all generating units, the subset of units controlled by the individual and the subset of units controlled by the individual’s competitors. Let be the vector of supply allocated to the units in the set and the corresponding vector of bidding parameters. Let be the vector of the nodal prices paid to each unit , depending on the node it is connected, .
Define and in a similar fashion. Then:
.
..
.
..
(2)
Note that an individual’s profit is not an explicit function of his bid vector . However, is an implicit function of since and are determined by an OPF solution, which is a function of . Thus, and are implicit functions of the bids vector . Assuming the individual knows what other individuals in the market are going to bid, the individual’s goal is to maximize his welfare by choosing a bid that is a best response to the other individuals’ bids. As a result, the maximization of an individual’s profit forms a two-level optimization problem where the individual maximizes his welfare subject to an OPF solution, which minimizes total system cost, based on all bids
in the market [21].
However, maximizing the profit assuming that competitors’ bids are known is an unrealistic assumption for electricity markets. In the deregulated environment, an individual has complete information concerning the production cost functions and technical constraints of the units under his control, and the rules under which the system administrator allocates the production.

GOUNTIS AND BAKIRTZIS: BIDDING STRATEGIES FOR ELECTRICITY PRODUCERS IN A COMPETITIVE ELECTRICITY MARKETPLACE 359
On the other hand, an individual is uncertain of the production cost functions and technical constraints of the competitors’ units. Moreover, he is unaware of the bidding strategy his competitors will follow. Furthermore, the load vector is not known when the bidding occurs.
Load forecasting is an issue that has been treated for decades and is beyond the scope of this paper. It will be assumed that a load forecasting technique is used to estimate the actual load.
Therefore, the bus load vector will be treated as a random vector with a multinormal distribution. The correlation between the energy demands of different buses is mainly due to regional phenomena, such as weather conditions.
Let denote the mean vector (size ) and the covariance matrix (size ) of . The probability density function (pdf) of the random vector is given by the model significantly. However, these correlations will be ignored, since they are difficult to estimate in practice.
Under these assumptions, the Individual Expected Profit
Maximization Problem forms a two level optimization problem where, at the first level, the individual is required to determine his bid vector, , which maximizes his expected profit under the constraint that, at the second level, the ISO solves an OPF problem to determine the value of the supply vector, , that minimizes total system cost. This problem can be stated as:
(4)
(5) where is the determinant of
(3)
, which is positive definite and symmetric. In compressed form, the former can be stated as
.
Since neither the competitors’ true marginal cost curve nor their bidding strategies are known with certainty, an individual must treat the bid curve submitted to the system administrator by a competitor as a random quantity. Its estimate will be based on the type of unit and fuel used and its
past strategic behavior [15]. Past bidding data are often publicly
available several (e.g. six) months after market clearing and are rather extensive, including all bids submitted, the unit that submitted the bid and the economic entity that controls the unit.
Although PJM, for example, masks firm and unit identities, the consistency of the generator and company codes makes them easy to decipher. An individual can thus create an estimate of rival bidding behavior. From his standpoint, the bid curve contains all the necessary information for the development of his strategy. In order to simplify modeling, it will be assumed that the true production cost coefficients are known, implying that the uncertainty about them has been incorporated in the uncertainty of the bidding strategy estimate. Therefore the model will incorporate only an estimate of the vector. As far as the competitor units’ technical constraints are concerned ( , ), it will be assumed that an individual can have a good estimate about them (e.g from past bidding data, unit type, relevant press releases etc) so they will be treated as known. The proposed method can be easily extended to treat competitor unit technical constraints as random variables at the expense of higher execution time.
Assume that an individual has developed a probabilistic estimate concerning competitors’ strategies and expected system load. Let , denote the probability density functions of the strategies of the units in the set. Correlation among them is introduced mainly due to the following reasons: (a) rival individuals may also control a group of units whose bids are correlated through the individual’s profit maximizing goal, (b) there exists some correlation through the load vector estimate. The incorporation of these correlations to the model would not alter
(7)
(8) where follows the probability density function components of
, the follow the probability density functions
, and operator means expectation.
The individual profit maximization problem (4)–(8) is a nonconvex, stochastic optimization problem, which is solved using
Monte Carlo Simulation and Genetic Algorithms as described in the following.
(6)
A. Individual Expected Profit Calculation by Monte Carlo
Simulation
The problem of the exact calculation of the individual expected profit, given the individual bid vector, and considering the random variation of the nodal active power demands,
, and the competitors’ bid strategies, , is a computationally demanding combinatorial problem. It requires the discretization of all associated random variables, namely the nodal demands and the competitors’ bids, and the enumeration of all the possible combinations of the discrete random variables. Under any such combination the individual profit is calculated (note that given specific values for .
..
and the individual profit can be calculated solving (5)–(8)) and multiplied by the joint probability of the random variables attaining the specific discrete values in the combination. It is evident that the computation time of the above process explodes with the number of the associated random variables. Alternatively, the individual expected profit, given the individual bid vector, can be estimated using Monte Carlo (MC) simulation by drawing random samples of the and vectors based on their probability distributions as described in the following.
During every trial of the MC simulation, a random sample of each associated random variable is drawn, according to its probability distribution. Since the competitors’ strategies are modeled as independent random variables the samples of the individual bids are drawn independently to form the competitor bid

360 IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 19, NO. 1, FEBRUARY 2004 vector, , based on samples from the corresponding
. The inverse transform method [23] is used to create
random samples that follow a specific pdf.
The generation of the random load vector, derivation of the lower triangular matrix
Such a matrix exists and is unique since
, requires the that satisfies: is positive definite.
Its derivation can be achieved through the “square root method”
[23]. Once
is derived, random load vectors, generated according to the following formula:
, can easily be
, 6) Execute a Monte Carlo simulation for every member of the population to calculate the Expected Profit as follows:
A.
Specify the number of MC trials allowed,
.
.
B.
Set simulation trial counter
C.
Sample the , and determine the and vectors.
functions to
D.
Solve 2nd level of the Expected Profit maximization problem (5)–(8) using sparse quadratic programming.
E.
F.
Calculate
. If
.
go to 6.C.
G.
Calculate
7) Use tion members.
8) Perform standard genetic algorithm operators, i.e. parent selection, crossover, mutation, etc.
9)
10) Extract
. If as the fitness function of the populago to 6.
.
for the fittest member of the genetic algorithm. Stop.
where is a normal vector of size with zero mean and covariance matrix equal to the identity matrix. Such a vector can be constructed by means of sampling of a standard normal distribution Norm (0,1).
Having formed the and vectors, can be calculated by solving the 2nd level OPF problem. The expected profit is derived from averaging the
Carlo simulation trials.
values of the Monte
When the number of the units controlled by the individual is very small, Monte Carlo simulation can be used to calculate the different expected individual profits for the whole range of
. The continuous variables , values each. This creates a set of
, are discretized into possible vectors to be examined. For every vector of this set, the expected profit is calculated via a separate Monte Carlo simulation. This approach delivers the whole surface of expected profit as a function of .
However, since the computational requirements are proportional to , the approach is not suited for individuals with many units under their control.
V. I NDIVIDUAL E XPECTED U TILITY M AXIMIZATION
Up to now the discussion focused on finding methods to maximize the expected profit of an individual under uncertainty.
However, risk was not taken into consideration. Economic theory indicates that most individuals do prefer more to less, but have aversion against risk. The risk is often represented as the variance in return.
In general, if the utility, , of expected wealth, , is greater, less than, or equal to the expected utility of wealth, an individual is said to be risk averse, risk lover or risk neutral respectively.
Alternatively, depending on whether is linear, strictly concave or convex, an individual is respectively risk neutral, risk averse
or risk lover [24].
In order to model the utility that an individual receives from a certain amount of wealth, typical utility functions are used. Two commonly used utility functions are the following:
B. Individual Expected Profit Maximization by Monte Carlo
Simulation and Genetic Algorithms
In an electricity market, an individual may control a significant number of units. In order to overcome the problem of excessive computational requirements of examining all possible strategies, genetic algorithms can be employed.
As mentioned, the vector is regarded as input for a specific
MC simulation. Maximizing the individual Expected Profit requires a genetic algorithm whose population members represent the vector of strategies for the units owned by the individual.
The fitness function is the Expected Profit and is calculated using a separate MC simulation for every population member.
Executing the algorithm involves the following actions:
1) Specify the
2) Specify the
, of competitor strategies.
of the system load vector.
3) Create a genetic algorithm whose population members represent the vector, i.e. each member’s chromosome consists of genes, each gene representing the bidding parameter of a different unit.
4) Initialize the GA population and specify the number of generations allowed, .
5) Set genetic algorithm generation counter .
(9)
(10)
These are special cases of the Hyperbolic Absolute Risk
Aversion (HARA) function [25].
Given the uncertainties in the electricity market, it is preferable to use the method presented in order to find bidding strategies that maximize the individual expected utility rather than the expected profit. Moreover, modern economic theory promotes the concept of the “corporate utility function” as an approach
capturing a company’s risk and return preferences [26].
In order to find the optimal bidding strategy that maximizes expected utility the method presented in Section IV-B will be used with the only modification that instead of using as the fitness function, the function

GOUNTIS AND BAKIRTZIS: BIDDING STRATEGIES FOR ELECTRICITY PRODUCERS IN A COMPETITIVE ELECTRICITY MARKETPLACE 361 will be used. In real applications, the individual’s utility function can be determined by the individual’s preferences. Note that maximizing expected utility is an effective approach since expected utility summarizes all the
preference information of an individual [27].
A. Utility Function Properties
Two popular measures of risk are the absolute risk aversion
(ARA) and the relative risk aversion (RRA) defined as:
Fig. 2.
Case A sample system.
TABLE I
2-N ODE S AMPLE S YSTEM U NIT
D ATA
(11)
(12)
ARA measures risk aversion for a given level of wealth. RRA measures risk aversion towards a proportional loss of wealth.
The Quadratic utility function is one of the most commonly used since it possesses the attractive feature that individual preferences may be represented as mean-variance preferences, meaning that individuals choose strategies based on expected profit and variance in return alone. Unfortunately, the quadratic utility function exhibits increasing absolute risk aversion
(ARA) and increasing Relative Risk Aversion (RRA). Both these properties do not make sense intuitively when modeling
individual behavior. Friend and Blume [28] have used Internal
Revenue Service data to estimate changes in ARA and RRA as a function of wealth. The results were consistent with decreasing ARA and constant RRA equal to 2.0, which can be modeled as an iso elastic utility function with equal to 2.0.
It is not suggested that this is the case with power producers as well, since there are no relevant data. Besides, an individual’s preferences determine his utility function and not vice-versa (a
thorough axiomatic treatment of utility can be found in [27]).
However, practitioners in the field of energy trading [26] seem
to be content with the properties of iso elastic utility functions.
Fig. 3.
Case A competitor strategy.
, while keeping constant, constitutes an increase in both the
ARA and RRA since:
VI. C ASE S TUDIES
The developed methods are applied to two, small size test systems: a 2-node and a 9-node system. The test system size was kept small so that test results can be thoroughly analyzed. Coalitions of up to two units are presented because in these cases multiple Monte Carlo simulations can yield the full payoff surface of an individual. The combined Monte Carlo-Genetic Algorithm (MCGA) approach can deal with problems of greater size. However, it will be used on the same sample systems so that its results can be verified with the ones of the exhaustive search.
The analysis that follows will demonstrate the optimal bidding strategy for different degrees of risk aversion. When the iso elastic function is considered, an increase of the value of
, indicates an individual with higher RRA and ARA. Therefore, results will be demonstrated as a function of . When the quadratic utility function is considered, will be held constant and results will be demonstrated as a function of . Increasing
A. Case A: 2-Node System
Consider the simple 2-node system of Fig. 2. The unit data are shown in Table I and the estimate of competitor strategy in
Fig. 3. The transmission line rating is 135 MW. Units 1 and 2 are controlled by the individuals 1 and 2 respectively. In order to simplify the presentation of input parameters, it is assumed that both individuals estimate that the load at bus 2 is normally distributed with a mean of 235 MW and a variance of 500
For the same reason, the estimate of the competitor strategy in
.
Fig. 3 applies symmetrically to both individuals, i.e. when computing the bidding strategy of individual 1, Fig. 3 represents the estimate of individual 1 for the strategy followed by individual
2 and vice-versa.
Different Monte Carlo simulations were executed for different . Values of ranged from 1 to 2.98 with a 0.03 step and every simulation consisted of 2000 trials. The expected profit and the profit variance of both individuals as functions of their bidding strategy are shown in Figs. 4 and 5 respectively.
Using the same parameters the MCGA method was used. The results, presented in Table II, were completely consistent with the exhaustive search results of Fig. 4. Due to the simplicity of

362 IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 19, NO. 1, FEBRUARY 2004
Fig. 4.
Expected profit chart for individuals 1 and 2 (Case A).
Fig. 6.
Individual 1 expected utility chart for quadratic and iso elastic utility functions (Case A).
Fig. 5.
Profit variance chart for individuals 1 and 2 (Case A).
Fig. 7.
Individual 1 best strategy chart for quadratic and iso elastic utility functions (Case A).
TABLE II
C ASE A MCGA E XPECTED P ROFIT M AXIMIZATION R ESULTS the 2-node system, the GA algorithm had a population of 20 members and was allowed to evolve for 20 generations. When coding the phenotype, 7 bits were used to allow for sufficient accuracy (less than 0.01). Every member’s objective function calculation involved a Monte Carlo simulation with 2000 trials.
It is clear that individual 1 is better off exploiting individual’s
2 200 MW constraint. With the load estimate of Fig. 2, the expected power to be served with a very high bid is 35 MW (Table II) so it is more profitable for individual 1 to bid very high in order to raise the expected price of electricity to around 31.4 Euro/MWh instead of trying to increase his market share.
When maximizing Unit 2 Expected profit, it is obvious that the line rating constraint can be exploited. Due to this constraint, individual 2 has potential monopoly power on the 2nd node and can increase his bid up to the maximum ating congestion and raising the price at node 2.
, thus cre-
The results presented in Table II indicate that the expected profit of Individual 1 with bidding strategy maximizing his own expected profit is lower than the one computed when the competitor’s expected profit is maximized. This contradictory, at first sight, result is due to the fact that the assumptions made by the individuals about the competitor’s bidding strategy probability distribution are inconsistent with each other. In this example, it is assumed that each individual believes that his competitor’s bidding strategy follows the probability distribution of
Fig. 3. Given the substantial power of both individuals to increase their bids exploiting the system constraints, the beliefs on rival behavior are obviously incorrect. Correcting these beliefs by incorporating the reaction of rivals and estimating market equilibrium points is an issue of game theory and beyond the scope of this paper.
If a utility function is introduced, the results vary significantly. Assuming a quadratic utility function, multiple Monte
Carlo simulations were used to create the surface of expected utility for different degrees of risk aversion. The results in
Fig. (6a) illustrate individual 1 expected utility for a quadratic utility function with equal to 60 and ranging from 0 to
0.015. The same calculations were made using an iso elastic utility function. Note that absolute values of utility have little meaning. The idea of relative more or less utility is significant
[27]. The resulting expected utility chart for the iso elastic
utility function is shown in Fig. (6b). This chart is considerably different from the previous one. To illustrate this, the best bidding strategies of individual 1 according to his degree of risk aversion for both utility functions are presented in Fig. 7.
Table III presents the results of the MCGA method which are completely consistent with the results of Fig. (7a). It is obvious that the best bidding strategy of an individual depends not only on the expected profit but on the utility function and the degree of risk aversion. As far as individual 1 is concerned, the high bid strategy presents the maximum expected profit. However, bidding very high is risky since it may in some cases yield very low revenues. Since the preferences of an individual with a

GOUNTIS AND BAKIRTZIS: BIDDING STRATEGIES FOR ELECTRICITY PRODUCERS IN A COMPETITIVE ELECTRICITY MARKETPLACE
TABLE III
C ASE A MCGA E XPECTED U TILITY M AXIMIZATION R ESULTS
363
Fig. 9.
Case B sample system.
TABLE IV
C ASE B U NIT D ATA
TABLE V
C ASE B L OAD D ATA
Fig. 8.
Individual 2 best strategy chart for quadratic and iso elastic utility functions (Case A).
quadratic utility function are based on mean and variance only, the variance chart, presented in Fig. (5a) illustrates the previous arguments.
The same methodology was applied to individual 2. His expected profit, profit variance and expected utility chart for both utility functions are presented in Fig. (4b), (5b) and 8 respectively. In all cases the optimal bidding strategy is to bid the maximum allowed bid. The reason behind this is that individual 2 has a greater advantage exploiting the transmission constraint which yields very high payoffs and is optimal even if the individual is considerably risk averse and although the variance in return increases significantly.
B. Case B: 9-Node System
Consider the 9-node system of Fig. 9. All lines have a 200
MW rating. The unit and load data are shown in Tables IV and V respectively. The nodal load covariance matrix is given in terms of the nodal load correlation matrix and the standard deviation vector . Note that a higher correlation was selected for neighboring nodes. The correlation decreases with distance (it has been assumed that the distance between the nodes is proportional to the difference of node indexes). The covariance matrix is calculated from .
Assume the existence of an individual, called “individual A”, that controls units 8 and 9, connected at nodes 8 and 9 respectively. The probability distributions of the competitor units’ strategies are assumed identical and are presented in Fig. 10.
Through the execution of multiple Monte Carlo simulations for different vectors, the pay off surface of Fig. (11a) was created. The expected profit surface, which is identical to the utility surface of a risk neutral individual , presents 2 local maxima. One of them is encountered when both units bid close to marginal cost and the other when both bid very high.
For the parameters examined, the global maximum is the latter.
In both cases, the controlled units must follow the same tactics.
Fig. (11a) indicates that if the individual decides to exploit the grid by congesting lines, both its units are required to adopt a high bid in order to impose congestion. In case one of the units deviates from the common strategy the total profit is reduced significantly.
Fig. (11b) shows the Expected utility chart assuming an iso elastic function and (risk aversion). This chart also presents 2 local maxima. However, it is the low bidding strategy that presents the global maximum since the high bids present a higher expected profit but also an increased risk as far as total revenue is concerned.
The results of Fig. 11 are completely consistent with the results of the MCGA method which are presented in Table VI.
The genetic algorithm had a population of 30 members and was allowed to evolve for 30 generations. This consistency in the results of the MCGA method and the exhaustive search of the search space (by means of multiple Monte Carlo simulations)

364 IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 19, NO. 1, FEBRUARY 2004
Fig. 10.
Case B competitor strategy.
Fig. 12.
Case B expected prices vs. individual A risk profile.
TABLE VII
C OMPUTATIONAL R EQUIREMENTS
Fig. 11.
Individual 1 expected utility chart (iso elastic utility function,
= 0
,
= 2
) (Case B).
TABLE VI
C ASE B MCGA E XPECTED U TILITY M AXIMIZATION R ESULTS indicates that the MCGA method can be used effectively for individuals with many controlled units, where the exhaustive search is ineffective.
The effects of the different risk profiles of individual A on the system are presented in Fig. 12 where the expected nodal prices are shown. If individual A is risk averse and decides to adopt the low bidding strategy, no line is congested in all 2000 simulations and the nodal prices are the same for all system nodes. If, however, individual A is risk neutral and adopts the high bidding strategy, the line connecting nodes
7 and 8 and the line connecting nodes 5 and 9 are congested with a 89.25% and 3.35% probability respectively. This results in an overall increase in all expected nodal prices and especially the ones at nodes 8 and 9.
OPF solutions are required for the computation of the optimal bidding strategy.
The computation time required for every OPF solution depends on the size of the system being examined. The required number of MC trials in each MC experiment, , depends on the number of loads and the number of units controlled by competitors. However, this dependence does not seem to be very strong: extensive experimentation on small systems (up to 24 buses with up to 21 competitor units) indicates that both the expected profit and its standard deviation converge after at most 500 MC trials. In all our MCGA tests we have selected ters, and
, which is on the safe side. The genetic parame-
, are selected according to problem difficulty, which depends on (a) the chromosome length and (b) the shape and characteristics of the fitness landscape. The chromosome length depends on the number of the units controlled by the individual. The fitness landscape is very difficult to visualize, except for trivial cases (e.g. Fig. 11). The genetic parameters in the results of Table VII are selected based on experimentation,
following the procedure of [29].
Table VII presents execution times (on a Pentium IV-1.8 GHz computer) for the two cases discussed in Section V as well as
for the one-area IEEE RTS-96 [30]. The results of Table VII, as
well as the above analysis reveal the very extensive computational requirements of the proposed method. Its application to power systems of practical size may only be achieved with the utilization of parallel computers.
VII. C OMPUTATIONAL R EQUIREMENTS
The combined use of GAs and MC simulation raises the issue of the computational requirements of the proposed method. The GA computational requirements are determined by the required number of fitness function evaluations (NFE).
The product gives the required NFE of a GA, in which a population of size is allowed to evolve for generations. Every fitness function evaluation requires
MC trials, in which the computationally dominant task is the
2nd level OPF solution. Thus, a total number of
VIII. C ONCLUSIONS
A methodology for the development of bidding strategies for electricity producers has been presented. When considering a risk neutral individual, the optimal bidding strategy is derived from the maximization of expected profit. The model proposed incorporates crucial characteristics of the electricity

GOUNTIS AND BAKIRTZIS: BIDDING STRATEGIES FOR ELECTRICITY PRODUCERS IN A COMPETITIVE ELECTRICITY MARKETPLACE 365 markets such as nodal load uncertainty/correlation, incomplete knowledge of rival behavior and network constraints, thus providing a fairly realistic formulation of the market. In order to better model individual decision-making, risk aversion is taken into consideration through the use of utility functions. In this framework, the optimal bidding strategy is derived from the maximization of expected utility. The methodology has been tested on sample systems and has proved effective in finding optimal strategies and identifying the underlying causes of their optimality.
The main drawbacks of the presented method are the fact that genetic algorithms cannot guarantee convergence to the global optimum and the extensive computational requirements.
In order for it to be applied to practical problems, the utilization of parallel computers is required.
R EFERENCES
[1] B. R. Szkuta, L. A. Sanabria, and T. S. Dillon, “Electricity price short-term forecasting using artificial neural networks,” IEEE Trans.
Power Syst.
, vol. 14, pp. 851–857, Aug. 1999.
[2] F. J. Nogales, J. Contreras, A. J. Conejo, and R. Espinola, “Forecasting next-day electricity prices by time series models,” IEEE Trans. Power
Syst.
, vol. 17, pp. 342–348, May 2002.
[3] A. Kian and A. Keyhani, “Stochastic price modeling of electricity in deregulated energy markets,” in Proc. Hawaii Int. Conf. Syst. Sci.
, Jan.
4–7, 2000.
[4] J. M. Arroyo and A. J. Conejo, “Optimal response of a power generator to energy, AGC, and reserve pool-based markets,” IEEE Trans. Power
Syst.
, vol. 17, pp. 404–410, May 2002.
[5] M. Shahidehpour, H. Yamin, and Z. Li, Market Operations in Electric
Power Systems: Forecasting, Scheduling, and Risk Management .
New
York: Wiley, 2002.
[6] F. S. Wen and A. K. David, “Oligopoly electricity market production under incomplete information,” IEEE Power Eng. Rev.
, vol. 21, pp.
58–61, Apr. 2001.
[7] R. W. Ferrero, J. F. Rivera, and S. M. Shahidehpour, “Application of games with incomplete information for pricing electricity in deregulated power pools,” IEEE Trans. Power Syst.
, vol. 13, pp. 184–189, Feb. 1998.
[8] J.-B. Park, B. H. Kim, J.-H. Kim, M.-H. Jung, and J.-K. Park, “A continuous strategy game for power transactions analysis in competitive electricity markets,” IEEE Trans. Power Syst.
, vol. 16, pp. 847–855, Nov.
2001.
[9] H. Song, C.-C. Liu, and J. Lawarree, “Nash equilibrium bidding strategies in a bilateral electricity market,” IEEE Trans. Power Syst.
, vol. 17, pp. 73–79, Feb. 2002.
[10] J. W. Lamont and S. Rajan, “Strategic bidding in an energy brokerage,”
IEEE Trans. Power Syst.
, vol. 12, pp. 1729–1733, Nov. 1997.
[11] C. W. Richter and G. B. Sheble, “Genetic algorithm evolution of utility bidding strategies for the competitive marketplace,” IEEE Trans. Power
Syst.
, vol. 13, pp. 256–261, Feb. 1998.
[12] C.-A. Li, A. J. Svoboda, X. Guan, and H. Singh, “Revenue adequate bidding strategies in competitive electricity markets,” IEEE Trans. Power
Syst.
, vol. 14, pp. 492–497, May 1999.
[13] C. W. Richter, G. B. Sheble, and D. Ashlock, “Comprehensive bidding strategies with genetic programming/finite state automata,” IEEE Trans.
Power Syst.
, vol. 14, pp. 1207–1212, Nov. 1999.
[14] K. Seeley, J. Lawarree, and C.-C. Liu, “Analysis of electricity market rules and their effects on strategic behavior in a noncongestive grid,”
IEEE Trans. Power Syst.
, vol. 15, pp. 157–162, Feb. 2000.
[15] F. S. Wen and A. K. David, “Optimal bidding strategies for competitive generators and large consumers,” Electr. Power Energy Syst.
, vol. 23,
[16] no. 1, pp. 37–43, Jan. 2001.
, “Coordination of bidding strategies in energy and spinning reserve markets for competitive suppliers using a genetic algorithm,” in Proc.
IEEE Power Eng. Soc. Summer Meeting , vol. 4, 2000, pp. 2174–2179.
[17] B. R. Song, C.-C. Liu, J. Lawarree, and R. W. Dahlgren, “Optimal electricity supply bidding by Markov decision process,” IEEE Trans. Power
Syst.
, vol. 15, pp. 618–624, May 2000.
[18] D. Zhang, Y. Wang, and P. B. Luh, “Optimization based bidding strategies in the deregulated market,” IEEE Trans. Power Syst.
, vol. 15, pp.
981–986, Aug. 2000.
[19] F. S. Wen and A. K. David, “Optimal bidding strategies and modeling of imperfect information among competitive generators,” IEEE Trans.
Power Syst.
, vol. 16, pp. 15–21, Feb. 2001.
[20] X. Guan, Y.-C. Ho, and F. Lai, “An ordinal optimization based bidding strategy for electric power suppliers in the daily energy market,” IEEE
Trans. Power Syst.
, vol. 16, pp. 788–797, Nov. 2001.
[21] J. D. Weber and T. J. Overbye, “A two-level optimization problem for analysis of market bidding strategies,” in Proc. IEEE Power Eng. Soc.
Summer Meeting , vol. 2, 1999, pp. 1845–1849.
[22] J. D. Weber, “Individual welfare maximization in electricity markets including consumer and full transmission system modeling,” Ph.D. thesis,
Dept. Elec. & Comp. Eng., Univ. Illinois, Urbana-Champaign, 1999.
[23] R. Y. Rubinstein, Simulation and the Monte Carlo Method .
New York:
Wiley, 1981.
[24] T. E. Copeland and J. F. Weston, Financial Theory and Corporate Policy ,
3rd ed.
Reading, MA: Addison-Wesley, 1992.
[25] J. E. Ingersoll, Theory of Financial Decision Making .
Savage, MD:
Rowman & Littlefield, 1987.
[26] D. Pilipovic, Energy Risk .
New York: McGraw-Hill, 1997.
[27] R. D. Luce and H. Raiffa, Games and Decisions .
Mineola, NY: Dover,
1989.
[28] I. Friend and M. E. Blume, “The demand for risky assets,” American
Economic Review , vol. 65, no. 5, pp. 900–922, Dec. 1975.
[29] A. Bakirtzis, P. Biskas, C. Zoumas, and V. Petridis, “Optimal power flow by enhanced genetic algorithm,” IEEE Trans. Power Syst.
, vol. 17, pp.
229–236, May 2002.
[30] “The IEEE reliability test system—1996,” IEEE Trans. Power Syst.
, vol.
14, pp. 1010–1020, Aug. 1999.
Vasileios P. Gountis (S’00) was born in Düsseldorf, Germany, in March 1976.
He received the Dipl. Eng. Degree from the Department of Electrical Engineering, Aristotle University, Thessaloniki, in 1999. He is currently pursuing the Ph.D. degree at Aristotle University.
His research interests are in power system operation and control.
Mr. Gountis is a member of the society of Professional Engineers of Greece.
Anastasios G. Bakirtzis (S’77-M’79-SM’95) received the Dipl. Eng. Degree from the Department of Electrical Engineering at the National Technical University, Athens, Greece, in 1979 and the M.S.E.E. and Ph.D. degrees from Georgia
Institute of Technology, Atlanta, in 1981 and 1984, respectively.
Since 1986, he has been with the Electrical Engineering Department, Aristotle University of Thessaloniki, Greece, where he is currently a Professor. His research interests are in power system operation and control, reliability analysis, and alternative energy sources.