Influence of feasibility constrains on t (1)

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Electric Power Systems Research

j o u r n a l h o m e p a g e :w w w . e l s e v i e r . c o m / l o c a t e / e p s r

Influence of feasibility constrains on the bidding strategy selection in a

day-ahead electricity market session

Alberto Borghetti

a,∗

, Stefano Massucco

b

, Federico Silvestro

b a_{Dept. of Electrical Engineering, University of Bologna, Viale risorgimento 2, 40136 Bologna, Italy}

b_{Dept. of Electrical Engineering, University of Genova, via all’Opera Pia 11a, 16145 Genova, Italy}

a r t i c l e

i n f o

Article history:

Received 18 December 2008 Received in revised form 10 June 2009 Accepted 26 July 2009

Available online 5 September 2009

Keywords: Electricity market Bidding strategies Feasibility constrains Game theory Unit commitment

a b s t r a c t

Large part of liberalized electricity markets, including the Italian one, features an auction mechanism, called day-ahead energy market, which matches producers’ and buyers’ simple bids, consisting of energy quantity and price pairs. The match is achieved by a merit-order economic dispatch procedure indepen-dently applied for each of the hours of the following day. Power plants operation should, however, take into account several technical constraints, such as maximum and minimum production bounds, ramp constraints and minimum up and downs times, as well as no-load and startup costs. The presence of these constraints forces to adjust the scheduling provided by the market in order to obtain a feasible scheduling. The paper presents an analysis of the possibility and the limits of taking into account the power plants technical constraints in the bidding strategy selection procedure of generating companies (Gencos). The analysis is carried out by using a computer procedure based both on a simple static game-theory approach and on a cost-minimization unit-commitment algorithm. For illustrative purposes, we present the results obtained for a system with three Gencos, each owning several power plants, trying to model the bidding behaviour of every generator in the system. This approach, although complex from the computational point of view, allows an analysis of both price and quantity bidding strategies and appears to be applicable to markets having different rules and features.

© 2009 Elsevier B.V. All rights reserved.

1. Introduction

Large part of liberalized electricity markets, including the Ital-ian one, features an auction mechanism, called day-ahead energy market, which matches producers’ and buyers’ simple bids, consist-ing of energy quantity and price pairs.1_{The match is achieved by a} merit-order economic dispatch procedure independently applied for each of the hours of the following day. This type of public day-ahead energy market is referred in the literature as a power exchange (e.g.[1]) or MinISO[2]in order to distinguish it from more centralized market architectures defined as power pools or MaxIso. In the latter architecture, generators provide extensive data other than price–quantity pairs by means of complex bids, such as startup costs, ramp rate limit, etc. With these extensive data, the unit commitment (UC) and dispatch that maximize social welfare, taking into account all important aspects of generator’s

operat-∗ _{Corresponding author. Tel.: +39 051 2093475; fax: +39 051 2093470.}

E-mail addresses:alberto.borghetti@unibo.it(A. Borghetti),

stefano.massucco@unige.it(S. Massucco),fsilvestro@epsl.die.unige.it(F. Silvestro). 1_{The producers’ offers state the aim to sell a certain amount of energy at a given}

price or higher. The buyers’ offers state the aim to buy a certain amount of energy at a given price or lower.

ing costs and physical constraints, are obtained by the auctioneer’s optimization computer program; side payments to some genera-tors are therefore needed in order to cover all the costs declared in their bids[1,3]. The presence of transmission constraints justifies the wide interest in the development of approaches/methods able to solve security constrain unit commitment (SCUC) problems (e.g. [4]).

The paper focuses on the former market architecture with sim-ple bids, which does not use side payments. The rational operator is expected to present bids to this market with the purpose of maxi-mizing its benefits. For the particular case of a generating company (Genco), the optimal bidding strategy is the one that allows the attainment of good profits and, at the same time, results in feasible schedules of its power plants, taking into account all their technical characteristics and constraints. For this latter purpose, in general, Gencos have at their disposal detailed software tools which can solve the so-called cost-based UC problem, i.e., they are able to cal-culate the optimal scheduling and power dispatching in order to feasibly satisfy an assigned load profile with the minimum variable production costs (e.g.[5,6]).

Many optimization approaches have been applied to address the optimal bidding strategy selection, e.g.[7–16]. In particular, for both the case of Gencos without market power and for the case of a single company with market power in the system – i.e., for

the case of a single company able to influence the market clearing price (MCP) by means of its own bidding strategy – the optimal bidding strategy selection problem can be suitably transformed in cost-minimization problems and, therefore, can be addressed by using a traditional cost-based UC algorithm (e.g.[15]). If MCP is assumed as an exogenous (random) variable, each generating unit can be considered separately[17].

Many methods have been proposed to solve the strategic bid-ding problem under the assumption of an exogenous MCP by using dynamic programming (e.g. [14]), stochastic linear pro-gramming[18], mixed integer programming (MIP) (e.g.[13]), and population-based search methods (e.g.[19,20]). Two-level opti-mization approaches are often applied in order to represent the strategic interaction among suppliers (e.g. [21]), also in hybrid markets where electrical energy and spinning reserve are simul-taneously traded (e.g.[22]) or in the presence of future contracts (e.g.[23]) and bilateral contracts (e.g.[24]). Also the influence of extra objectives, such the minimization of supplier emission of pol-lutants (e.g.[25]), or the influence of unit reliability (e.g.[26]) has been analyzed. The competition process can also be represented as a dynamic feedback system (e.g. as in[27]).

In order to explicitly represent individual market power, i.e., its ability to manipulate market price via its strategic bidding behaviour, Gencos bidding in an oligopolistic electricity market can be modelled as a supplier game. In particular, recent paper [28]thoroughly analyzes Nash equilibria (NE) and the conditions for such equilibria to exist when Gencos game through their sup-ply functions. However, as mentioned in[28], it is not rigorously defined the link between market spot prices and on/off variables of the UC problem, with fixed and startup costs, as well as with non-zero lower production bounds. Therefore, in game-theory based methods, UC is, in general, not included in the Gencos gaming strategy, i.e., it is assumed to be known (e.g.[28,29]). However, it is a general belief that UC will remain an important support to the hourly bidding strategy builder tool, if accurate forecast of the Genco loads and hourly prices will be used as inputs for solving the UC problem (e.g.[2,30–32]).

As mentioned, in several electricity markets, day-ahead energy auctions adopt relative simple procedures in which they initially neglect transmission line capability constraints and network losses, as well as detailed power plant inter-temporal constraints, which are accounted for by ex-post procedures, i.e., adjustment market sessions. As shown in[33]for the case of transmission constraints and network losses, these simple auction procedures may result in some loss of economic efficiency and cross-subsidies between market participants with respect to the solution of a detailed opti-mization model.

In this paper, we do not consider improved energy auction procedures that may be implemented in order to avoid these inef-ficiencies and we assume that the electricity market is based on simple day-ahead energy auctions that clear the market at every hour of the following day without consideration of inter-temporal constraints, i.e., it does not allow a bidder to explicitly specify some technical constraints, such as ramp rate and minimum up and down times. Therefore, the differences between hourly energy programs and feasible generation schedules must be compensated by each Genco through its participation to the following adjust-ment and balancing market sessions (as described for example in [34]).

The present paper aims at investigating how a Genco bidding strategy selection procedure may take into account the power plant operational constraints and, also, try to guide market auc-tion results toward almost feasible generaauc-tion schedules in order to limit the costly participation to the adjustment market sessions. Among the various game-theory based approaches proposed for the analysis of oligopolistic electricity markets, the one based on

the payoff matrix of the game represented in normal-form allows the direct treatment of nondifferentiable and nonconvex functions (e.g.[35–40]). Therefore, the proposed analysis is carried out by using a computer procedure coupling the use of a multiperiod and multiplayer payoff matrix – calculated by applying simple auction rules to the combinations of the operators’ pure strategies for all the hours of the following day – with a cost-based UC algorithm, which allows the refined calculation of the production costs for each of the power plants of a specific Genco and to take into account all the relevant constraints.

The analysis is carried out by choosing a preliminary limited number of pure strategies for each Genco in order to limit the com-putational effort. The discretization of pure bidding strategies may be improved in a following refining stage[38,39]. The use of the payoff matrix results in a procedure almost independent of the spe-cific market rules, although we have implemented it with reference to a uniform-price day-ahead energy auction. The uniform price can be differentiated by taking into account the network constraints between different zones of the system.

The structure of the paper is the following. Section2describes the scheme proposed for the analysis of the bidding decisions. Then, the assumptions and the details of a simplified implementation into a computer code are presented in Section3, namely the main char-acteristics of the various strategies, the selection criteria of the most convenient bidding strategy based on the game theory and the inte-gration with a typical cost-based UC code. The inteinte-gration with the UC code permits to analyze heuristics procedures conceived with the aim to select bidding strategies that are expected to result in fea-sible schedules of the power plants. Section4presents the results of the analysis carried out for a system with three Gencos, each own-ing several power plants. The results show the biddown-ing behaviour of every generator in the system. This approach, although com-plex from the computational point of view, illustrates the effects of the considered power plant costs and constraints on the bidding strategy selection in a typical day-ahead electricity market session. Section4concludes the paper.

2. Structure of the procedure adopted for the analysis

As already mentioned, the procedure adopted for the proposed analysis is based on the coordinated use of a normal-form game-theory representation of the day-ahead electricity market and of an algorithm for the solution of UC problems.

The scheme of the procedure is shown inFig. 1. As described below, the procedure is based on the calculation of the payoff matrix, which represents the normal-form of the game for each of the periodstof the considered horizonT (e.g. the 24 h of the following days), and, then, by a feasibility enforcement procedure that applies a cost-based UC program in order to update the payoff matrix.

2.1. Normal-form game-theory approach

A finite number of pure strategies are defined for each market participant or player, i.e., for each energy demand and generation bidder. For each combination of strategies and for each period, the market results (i.e., the MCP and the accepted bids) are deter-mined through a procedure that implements the market specific rules.

Fig. 1.Structure of the adopted feasibility-constrained bidding selection procedure.

pairs. Both the aggregated curves of all the generation and demand bids are computed and the market clears at the matching point of these two curves.

Genco bids are expected to internalize all the operating costs, including startup and shutdown costs. Therefore, also these costs appear to deserve to be taken into account in the bidding strategy selection procedure.

Moreover, specific market rules can incorporate the possibility that the market is split into several regional markets or zones due to the presence of tie-line constraints. The revenues for each Genco can be calculated with respect to the specific zonal prices relevant to the regions where the production units are located.

The proposed strategy selection procedure of each Genco is based on the following optimization problem solved independently for each of the hourtof the following day in order to obtain each element of the payoff matrix that represents the game in normal-form.

max ,Pg,Pd

Nz

z=1

⎡

⎣

i∈˝

d∈˝dz,i

Nh

h=1

d,h(si)·Pd,h(si)

−

i∈˝

g∈˝gz,i

Nh

h=1

g,h(si)·Pg,h(si)

⎤

⎦

(1)

so that

Pm

d ≤Pd≤PdMorPd=0

∀

d (2)

Pm

g ≤Pg≤PgMorPg=0

∀

g (3)

Pℓ= Nz−1

z=1

⎡

⎣ptdf

ℓ,z·

⎛

⎝

i∈˝

g∈˝gz,i Pg−

i∈˝

d∈˝dz,i Pd

⎞

⎠

⎤

⎦

≤P_ℓM

∀

ℓ∈˝l (4)

Nz

z=1

⎛

⎝

i∈˝

g∈˝gz,i Pg−

i∈˝

d∈˝dz,i Pd

⎞

⎠

=0 (5)

where

-Nzis the number of zones;

-˝,˝l,˝dz,i,˝gz,i are the set of players, of network links between different zones, of demand sites and generation units in zonezthat belongs to playeri, respectively;

-d,h,Pd,hare the price and quantity pairs of each stephof demand

bidd, whilstg,h,Pg,hare the price and quantity pairs of each step

hof generation bidg; -Pm

d, P M d, P

m

g, PgMare the lower and upper limits of the consumption Pdof demand sitedand of the outputPgof generation unitg,

respectively;

-Pℓis the power flow absolute value in network linkℓandP_ℓMthe maximum limit;

-ptdfℓ,zis theℓ, zelement of the power transfer distribution factors (ptdf) matrix, i.e., the sensitivity of the power flow in network link ℓto an injection at zonez(and equivalent extraction at zoneNz); -␳is the vector of the MCPs, one for each zonez, whilstPgand P_dare the vectors of all the accepted demand and generation quantities, i.e., the vector of all the selectedPdandPgvalues, one

for each demand sitedand generation unitg, respectively

The price and quantity pairs of each bid depend on the particular bidding strategysi∈Siof each playeri, beingSithe relevant set of

strategies (also called strategy space).

We focus the analysis to the Gencos. Each element of the Genco payoff matrix, for each periodt, is composed by a vector of values Vi(si,s−i), one for each Gencoi, beings−i(∈S−i) the vector (and the

relevant set) of the strategies corresponding to every player with the exclusion of Gencoi. Each valueVi(si,s−i) represents the profit

obtained by Gencoiduring the considered periodt, i.e., the differ-ence between the revenues and costs. The revenues are calculated by the summation of the all the products between pricez and

selected outputPgof each generating unitgowned by Gencoiin

every zonez. The costs are the summation of the operating costs associated to all the generating units owned by the Genco.

Each Genco is considered as an intelligent agent that chooses its bidding strategy in order to maximize its profits, under the assumption that each Genco could estimate also the cost or bene-fit functions of the others market participants and their possible strategies, but, obviously, does not know their final choice. The model is therefore conceived under the complete (but imperfect) knowledge assumption.

The problem solved by each Gencoito find theT-elements vector of the optimal strategiess∗

ifor all periodstcould be represented as an optimization problem

s∗

i =argmax_s

i,s−i

T

t=1

Vi,t(si,t,s−i,t)

=argmax si,s−i

T

t=1

Nz

z=1

g∈˝gz,i

zPg,t−ug,tcg(Pg,t)−g(ug,t, ug,t−1)

(6)

without violating any physical and operating constraint of the generation units, being binary variableug,tthe commitment state

during periodtof each generation unitgof zonezowned by Genco i,cg(pg,t) the variable operating cost of generation unitgworking

at production levelpg,tduring periodtandg(ug,t, ug,t−1) the

tran-sition cost incurred at every change of the commitment state of generation unitgbetween two consecutive periodst−1 andt.

In order to calculate these optimal bidding strategies, the algo-rithm carries out a preliminary elimination of the strategies that are dominated by the others, for each periodtand Gencoi. A strategy sd

i is said to be dominated when it provides profitsVi,td lower than those provided by every other strategysi, for every combination of

strategiess−ithat could be chosen by the competitors: Vd

i,t(sdi,s−i)< Vi,t(si,s−i)

∀

si∈Si:si=/sid,

∀

s−i∈S−i (7) The elimination of the dominated strategies significantly reduces the computational efforts and it implies that the

so-called non-credible threats are disregarded. This is justified by the assumption of participants’ rational behaviour, which prevents the adoption of the strategies against the profits maximization goal.

2.2. Criterion for the selection of the optimal bidding strategy

After the calculation of the payoff matrix, for the solution of problem(6), i.e., for the selection of optimal bidding strategys∗

iof each Gencoi, the definition of a criterion is needed. Such a crite-rion should provide the profits maximization, taking into account that the competitors’ choice is not known and, therefore, taking into account the associated risks. We have implemented a typical minimum-risk criterion that consists in the choice of the so-called maxmin strategy, i.e., the optimal strategy is the one that ensures the maximum profit at the end of the following day by assuming that the competitors will choose, in every periodt, the combination of strategies that results in the minimum profit levelV_- i,t(si,t) for each strategysi,tof the operator of interest (Gencoi):

s∗

i =argmax si

T

t=1

V_{- i,t}(s_i,t) (8)

where

V_- i,t(si,t)=min s−i,t

Vi,t(si,t,s−i,t)

∀

s−i,t. (9)

The literature on the subject often refers to another criterion based on Nash equilibria (NE)[41], under the assumption that NEs may provide a coherent set of offers (e.g.[28,42]). If all the play-ers choose to follow NEs, each Genco not only has the potential to obtain a satisfactory profit but also has nothing to gain by being the only one to modify its own offer. Indeed, NE, if it exists, is a combi-nation of strategiess∗_{= {(s}∗

i,s∗−i)}so that each operator could not obtain any benefit by unilaterally deviating from it:

V∗

i,t(s∗i,t,s∗−i,t)≥Vi,t(si,t,s∗−i,t)

∀

i,

∀

si (10) This calculation may also allow inferring how a repeated game will be played, in the sense that if all players predict that a particular NE will occur, then no player has an incentive to play differently.

2.3. Feasibility enforcement by using a cost-based UC program

A Genco is expected to select the optimal strategy also taking into account that the load profile attributed by the market-clearing solution must be feasibly covered by its power plants. This is of importance even for the markets in which, as the Italian one, the clearing mechanism does not take into account all the power plants constraints, in particular those that couple the decisions in dif-ferent periods, such as the typical minimum up and down time constraints, ramp constraints and the optimal use of an assigned water quantity for the reservoirs of the hydro power plants.

For this purpose, the proposed procedure includes an itera-tive process, in which, for each Genco, a feasible solution of the corresponding cost-based UC problem is calculated for the entire load profile of the following day defined by the selected strate-gies sequence and the payoff matrix is updated on the basis of the production costs provided by the UC solutions.

For each Gencoi, the cost-based UC problem may be written as the problem of minimizing the sum of operating costs and transi-tion costs of committed units between consecutive periods

min T

t=1

Nz

z=1

g∈˝gz,i

ug,tcg(Pg,t)+g(ug,t, ug,t−1)

(11)

the payoff matrix elements associated to the last-selected optimal strategys∗

i, namely

g∈˝gz,i

Pg,t=Dz,t(V_- i,t(s∗i,t))

∀

t,

∀

z (12)

if maxmin criterion (8) is adopted, or

g∈˝gz,i

Pg,t=Dz,t(V_i,t∗)

∀

t,

∀

z (13)

if the NEs defined by(10)are followed (as illustrated byFig. 1). The UC solution should not violate physical and operating constraints of the generation units, both of thermal and hydro ones, such as the minimum and maximum output limits, the minimum up and down times, the ramp constraints and reservoir storage constraints (e.g.[5]). As the UC calculation is carried out independently for each Genco and each zone, by assuming the corresponding load profile as defined by the payoff matrix, the UC solution also satisfies the network constraints between zones.

2.4. Construction of the bidding offer curves

The last step of the procedure is the translation of the selected most convenient strategies in bidding offer curves for each hour and generating unit or group of units. Such a procedure should take into account all the specific electricity market administrative rules and regulations.

The UC solution provides both a more refined calculation of the payoff matrix elements and also a feasible scheduling with respect to inter-temporal constraints. As we assume that these constraints are not explicitly enforced in the electricity market auctions, it appears useful that, in every period, each Genco may differenti-ate the offers of the power plants that the bidding procedure has selected as not in operation from the offers relevant to the power plants that are expected to be dispatched, in order to limit the costly participation to the adjustment market sessions. For such a purpose, a specific heuristic procedure is here introduced, able to reduce the probability the on/off commitment of some units selected by the bidding procedure will be changed by the market auction results in several periods.

The following Section 3 describes the assumptions and the details of the computer procedure developed for the proposed anal-ysis of the influence of feasibility constrains on the bidding strategy selection.

3. Details of the implemented procedure

The various blocks illustrated inFig. 1have been implemented in Matlab scripts. In the implemented procedure, we do not con-sider network constraints and we concon-sider only Gencos as market participants (i.e., the level of the demand is fixed and known for each hour of the following day). Their objective function is given by(6), i.e., the maximization of expected profits (expected revenues minus the estimated operating costs) in the day-head market with-out cooperation with other market participants and withwith-out taking into consideration the existence of other electricity market sessions (e.g. balancing and spinning reserve sessions).

3.1. Payoff matrix calculation

Each producer offers a step-wise offer curve for each of his generating units depending on the selected bidding strategy, con-sidered to be the same for all the units of the same zone. The list of implemented strategies refers to both price strategies and pro-duction repro-duction strategies. The strategies are formulated with

Fig. 2.Example of payoff matrix for the case of three market participants.

reference to an offer which will be called “at the marginal costs”, calculated for each step as the ratio between the operating cost vari-ation and the corresponding output varivari-ation. We consider offer curves composed by a number of equal power steps with the first steps grouped in order to fulfil the minimum power output of the station.

For each hour of the following day the payoff matrix is indeed a multi-dimensional array, obtained by the market-clearing solu-tion that, at each hour, matches the Gencos step-wise bids with the forecasted demand level. For each hour, the array dimensionality is equal to the number of Gencos and the bound on each dimension is equal to the number of the strategies assumed for the correspond-ing Genco. Each element of the matrix is given by the vector of the expected profits for all the Gencos for a particular combination of pure strategy. The market-clearing calculation is carried out for each strategy combination and for each period.Fig. 2illustrates the structure of the payoff matrix for the particular case of three Gen-cos, in which Genco 1 operates with 3 strategies, Genco 2 with 4 strategies and Genco 3 with 2 strategies.

3.2. Implementation of two strategy selection criteria

As mentioned before, two bidding selection criteria have been implemented in order to select strategys∗

i,tby each Gencoiin each hourt: namely, the maxmin criterion and the NE-based one. Both criteria use the calculated payoff matrix.

Fig. 3illustrates the procedure adopted for the implementation of the maxmin criterion, for the case of the 4 strategies of Genco 2 of Fig. 2. First of all, the minimum profit valuesV_- i,t(si,t) are calculated, for each strategysi,tof the considered Gencoiand for every hour

t, as specified by (9). On the basis of these minimum profit values, a forward dynamic algorithm is implemented over the entire 24-h optimization 24-horizon, in order solve problem (8), i.e., to find t24-he final maximum profit. By using the forward dynamic algorithm, not only the summation of variable operating costscg(pg,t), associated

to each generation unitgowned by the considered Genco working at production levelpg,tduring periodt, but also the summation of

all transition costsg(ug,t, ug,t−1), incurred at every change of the

commitment stateug,tof a generation unit between two

consecu-tive periodst−1 andt, could be efficiently taken into account as requested by(6).

The implemented computer tool also determines the NEs by means a complete enumeration algorithm. As we consider only pure strategies, in some periods more than one NE may be obtained, whilst, in others, none is found. In case of multiple NEs, the program discards those that are dominated by others, i.e., those that are char-acterized by inferior profit valuesV∗

i,tfor all the participants. Being Kthe set of multiple NEs that fulfil condition(10), equilibriumhis discarded if there exists, at the same hourt, another NEkable to assure better profits for each Gencoi

V_i,t∗,h(s∗,h t )< V

∗,k i,t(s

∗,k

t )

∀

i,

∀

k∈K, k=/h (14)

Moreover, we have often found (see Section4) similar profits and market prices corresponding to different non-dominated NEs relevant to the same hour. For this reason, the computer tool also calculates, for each hour, the average value of the profits of each producer and the average value of the market prices relevant to all the calculated non-dominated NEs.

3.3. The adopted cost-based UC program

As already mentioned, our intention is that the proposed bidding strategy tool will also guide the market to a feasible dispatch-ing of the power plants, startdispatch-ing from the deeper knowledge that each Genco is able to obtain relevant to costs and con-straints of his own generation units with respect to the data used in the payoff matrix calculation. This justifies the use of a UC computer program in order to refine the solution. The UC pro-gram provides a self-dispatching solution for each Genco and for each zone, taking into account both the most convenient strategies defined by the game-theory based model and the most detailed information available on the operating costs and con-straints.

Several approaches have been proposed for the solution of the UC problem. For an exhaustive overview we refer the reader to the recent survey[43]. Refined UC codes have been presented in the literature and are commercially available, which allow a detailed description of the various units, taking into account also the pres-ence of hydro stations. The optimal scheduling of hydrostations should take into account various peculiar constraints that may require the development of specific optimization procedures, as shown, for example in[44,45].

The main points of the proposed analysis appears to be ade-quately illustrated also by using a simple UC code that takes into account only the presence of thermal units. The implemented UC program minimizes the variable operating costs, described by quadratic functions, and the startup and shutdown costs of the Genco’s units in order to satisfy the load profile, over the 24-h horizon of the following day, that has been defined by the current optimal strategy selected by applying one of the two implemented selection criteria. The main operating constraints and the physical characteristics of the power generation system, usually considered for the problem of interest (e.g.[5]), are enforced in the UC calcu-lation.

The adopted UC solution algorithm is based on the Lagrangian relaxation of the load balancing constraint. The Lagrangian relax-ation approach is often preferred due to its ability to include more detailed system representation than would be possible with other techniques (e.g. [46]). As known [5], the Lagrangian relaxation technique allows to decouple the problem into a minimization sub-problem for each generating unit. Then the solution is guided by the dual maximization problem. Each of the minimization sub-problem is solved by means of a dynamic programming algorithm. The solution of the dual problem is carried out through an itera-tive procedure where the Lagrangian multipliers are updated by using the so-called sub-gradient method with adequate heuristic

procedures in order to obtain a feasible solution of the primal prob-lem from the solution of the dual. Various other techniques for the solution of the dual problem have been proposed in the literature on the subject, such as Bundle methods (e.g.[47,48]). The quality of the obtained UC solution is indicated estimated by the duality gap value, i.e., by the difference between the optimal value of the objective function of the primal problem and the solution of the Lagrangian dual[5].

As illustrated inFig. 1, the UC results are used to update the relevant values of the payoff matrix. The optimal strategy selection criterion is then applied again to the updated payoff matrix and the iterative process is repeated if the resulting optimal strategy differs from the one selected before the application of the payoff matrix refinement by using the UC algorithm. We have not encountered convergence problems in this iterative process for the examined cases and the feasibility-constrained optimal sequence of strategies is usually found after few iterations.

3.4. Bidding offer construction

As illustrated inFig. 1, each Genco bidding selection procedure ends with the bidding offer construction, usually based on refined procedures able to meet all the specific electricity market admin-istrative rules and regulations. As justified in Section2.3, we here examine the effects of a simple heuristic procedure that differenti-ates the offers of the power plants that the bidding procedure has selected as not in operation from the offers of the power plants that are expected to be dispatched. The latter offers are chosen on the basis of the selected optimal strategy; the former ones are increased in order to reduce the probability that those power plants will be forced to operate by the market results. The considered heuristic is therefore not a part of the market-clearing mechanism but is assumed to be introduced in the Genco bidding offer construction in order to limit the costly participation to the adjustment market sessions.

4. Simulation results

4.1. Test system

The considered test system is composed by three Gencos with only thermal units: the first (Genco 1) has 20 units (corresponding to 54.7% of the system capacity), the second (Genco 2) has 10 units (27.3% of the total capacity), and the third (Genco 3) has 4 units (18% of the total capacity).

The power plant characteristics, the parameter values of the variable-cost function, assumed to be quadratic, and the values of the startup costs have been adapted from[49].Table 1shows the values of the parameters of all the considered power plants, wherea,b, andcare the coefficients of the quadratic cost function, and the minimum up and down timesTare considered equal for all.

Table 1

Data of the power plants of the three Gencos in the considered test system (m.u. indicates a generic monetary unit).

Genco no. Unit no. Min outputPm(MW) Max outputPM(MW) Quadratic cost function coefficients Startup costs

(m.u.)/10

Minimum up and down timesT(h)

a(m.u./h) b(m.u./MWh) c(10−2_{) (m.u./MW}2_h)

1 1 150 455 1000 23 0.048 900 8

1 2 150 455 970 17.26 0.031 1000 8

1 3 20 130 700 13 0.2 110 5

1 4 20 130 680 11 0.211 112 5

1 5 25 162 450 19.07 0.398 180 6

1 6 20 80 370 22.26 0.712 34 3

1 7 25 85 480 27.74 0.079 52 3

1 8 150 455 1000 13 0.048 900 8

1 9 150 455 970 17.26 0.031 1000 8

1 10 20 130 700 20 0.2 110 5

1 11 20 130 680 18 0.211 112 5

1 12 25 162 450 19.07 0.398 180 6

1 13 20 80 370 22.26 0.712 34 3

1 14 25 85 480 27.74 0.079 52 3

2 1 150 455 1000 12 0.048 900 8

2 2 150 455 970 17.26 0.031 1000 8

2 3 20 130 700 21 0.2 110 5

2 4 20 130 680 16.05 0.211 112 5

2 5 25 162 450 19.07 0.398 180 6

2 6 20 80 370 22.26 0.712 34 3

2 7 25 85 480 27.74 0.079 52 3

3 1 150 455 1000 12 0.048 900 8

3 2 150 455 970 16 0.031 1000 8

3 3 20 130 700 16.06 0.2 110 5

1 15 20 55 660 25.92 0.413 6 1

1 16 20 55 665 27.27 0.222 6 1

1 17 20 55 670 27.79 0.173 6 1

1 18 20 55 660 25.92 0.413 6 1

1 19 20 55 665 27.27 0.222 6 1

1 20 20 55 670 27.79 0.173 6 1

2 8 20 55 660 25.92 0.413 6 1

2 9 20 55 665 27.27 0.222 6 1

2 10 20 55 670 27.79 0.173 6 1

3 4 20 55 660 25.92 0.413 6 1

plants larger than 500 MW (strategy 4), and, finally, bidding at price values 50% lower than marginal costs (strategy 5). There-fore, Gencos can choose both price and quantity strategies. For example, strategy 3 is typical of price-taker participant, whilst strategy 4 may be adopted by dominant players to drive market price.

A twenty-four 1-h horizon is assumed, with a predefined typical demand profile characterized by a first peak at 12 a.m. and a second peak at 8 p.m., as reported inTable 2.

Fig. 4.Example of the bidding offer at marginal costs and quadratic cost function of the first power plant belonging to Genco 1.

4.2. Bid strategy selected for the base test system

The adopted procedure is applied to the test system by consid-ering both the maxmin criterion and NEs.

4.2.1. Maxmin criterion

The strategies selected by the three Gencos by adopting the maxmin criterion are listed inTable 3. The table presents both the results obtained before and after the application of the UC-based feasibility enforcement.

The solution of bidding strategy selection procedure obtained before the application of the UC-based feasibility enforcement corresponds to a sequence of power plants startups and shut-downs with various violations of the minimum up and down times constraints. The list of units whose scheduling violates these

con-Table 2

Demand profile.

Hour Demand (MW) Hour Demand (MW)

1 2296 13 5409

2 2518 14 4964

3 2963 15 4519

4 3408 16 3852

5 3630 17 3630

6 4075 18 4075

7 4297 19 4519

8 4519 20 5409

9 4964 21 4964

10 5409 22 4075

11 5631 23 3185

Table 3

Strategies selected through the maxmin criterion. Genco Period

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

Before the application of the UC-based feasibility enforcement

1 3 3 3 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

2 1 1 1 1 1 1 2 2 1 1 1 1 1 1 2 2 3 2 2 1 1 1 1 1

3 3 3 3 2 2 2 1 1 1 2 2 1 2 1 1 2 2 2 1 2 1 2 1 1

After the UC-based feasibility enforcement

1 3 3 3 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

2 3 3 3 3 3 2 2 2 1 1 1 1 1 1 2 2 3 2 2 1 1 2 1 1

3 3 3 3 2 2 3 3 1 1 5 5 1 5 1 2 3 2 3 1 3 1 1 2 2

Table 4

List of the units that violate min. up and down time constraints before the application of the UC procedure (maxmin criterion).

Units that violate the constraints

Min. up time Min. down time

Genco 1 1, 5, 6, 10, 13 11, 12

Genco 2 5

Genco 3 3 3

straints is reported inTable 4. For the considered case, the UC-based feasibility enforcement converges and solves all the constraints vio-lations ofTable 2after 3 iterations when it is applied to Genco 1 and Genco 2 (with a reduction of the minimum expected profit equal to 4.4% and 2.7%, respectively) and after 6 iterations when it is applied to Genco 3 (with a profit reduction equal to 8.7%).

The feasibility enforcement does not change the strategies of Genco 1 and change the strategy of Genco 2 only in low load peri-ods (periperi-ods 1–6 and 22). The strategies of Genco 3 are the most affected.

4.2.2. NEs

Table 5shows the strategies corresponding for each period to the calculated single dominant NE, when it exists, without the application of the UC-based feasibility enforcement.

For the case of NE calculations, the results of periods 1 and 4 are not shown inTable 5: in fact, in period 4 there is no equilibrium, whilst in period 1 there are two equilibriums for the two triples of strategies{1,1,3}and{1,2,1}(for the three Gencos respectively), with market prices equal to 24.3 m.u. and 23.6 m.u.

Table 5shows that, as expected, the equilibrium is obtained for strategies characterized by offers with high prices, because of the absence of a direct participation from the buyers. Only for the smallest Genco (namely, Genco 3) the NE includes strategy 3 (at the marginal costs) in two low load periods (2 and 24).

Fig. 5.Results obtained by adopting the strategies selected by the maxmin criterion and by the NEs: (a) market prices and load demand and (b) profits for the generation companies.

4.3. Market results

The adoption of the strategies provided by the proposed bid-ding selection procedure accorbid-ding to the maxmin criterion and of those obtained by adopting the strategies relevant to the

cal-Table 5

Strategies corresponding to the NEs.

Genco Period

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

1 – 1 2 – 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

2 – 2 1 – 1 1 1 1 2 1 1 1 1 2 1 1 1 1 1 1 2 1 2 2

3 – 3 2 – 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 3

Table 6

Genco owner of the power plant that sets the price in the various periods for the two criteria.

Period 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

Maxmin 1 1 1 1 1 2 1 2 1 1 2 2 3 1 2 2 1 2 2 1 1 2 1 1

Table 7

Marginal units that set the price in the various periods for the two criteria.

Period 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

Maxmin 2 2 9 9 2 3 6 6 1 1 1 8 1 1 6 5 2 3 6 1 1 3 9 2

NE 2 9 11 9 11 10 12 5 6 13 8 8 13 6 5 5 11 10 5 13 6 10 2 2

Table 8

List of the units that violate minimum up and down time constraints after the market-clearing simulation.

Units that violate the constraints

Min. up time Min. down time Maxmin criterion

Genco 1 1, 6, 10, 13 1, 5, 10, 11, 12

Genco 2 6 3, 6

Genco 3 – –

NE

Genco 1 1, 2, 5, 6, 11, 12, 13 1, 5, 10, 11, 12

Genco 2 2, 3, 4, 5, 6 5

Genco 3 – –

Table 9

List of the units that violate min. up and down time constraints after the market-clearing simulation with the application of the heuristic procedure.

Heuristic parameters Units that violate the constraints Min. up time Min. down time Maxmin criterion

Genco 1 ˛1= 1.1,ˇ1= 0.9 1, 5, 6, 10, 12 1, 5, 6, 10, 11, 12 Genco 2 ˛2= 1.1,ˇ2= 0.9 3 3

Genco 3 ˛3= 1,ˇ3= 1 – –

NE

Genco 1 ˛1= 1.1,ˇ1= 0.9 1, 7, 11, 14 2, 7, 10, 11 Genco 2 ˛2= 1.1,ˇ2= 0.9 2, 3, 5, 7 3, 5

Genco 3 ˛3= 1,ˇ3= 1 – 3

culated NEs produces, as expected, different market solutions. To compare them, we here present the results obtained by the sim-ulation of the day-ahead market by concurrently adopting, for the considered three Gencos, the strategies ofTable 3and we repeat the simulation by adopting the strategies ofTable 5. In every period in which there is not a unique NE, all the Gencos bid at the marginal costs (strategy 3).

Fig. 5shows the comparison between the market clearing results obtained, both for the case of maxmin and NE selection criterion. Fig. 5(a) shows the given load profile and the market prices, whilst Fig. 5(b) shows the corresponding profits of the companies.

As expected, higher market prices result by adopting the strate-gies relevant to NEs, with respect to those obtained by using the strategies selected by maxmin criterion, which is based on the concept of adversity to risk, above all in low load hours.

Fig. 5(b) shows that, in the first three periods, the profits obtained through the maxmin-selected strategies are particularly low (even slightly negative for Genco 1 and Genco 2). This is due to the fact that minimum up and down constrains are not considered binding at period 0 and startup costs are not assigned to period 1. Therefore, in order to save the startup costs, it results conve-nient to have some large plants in service already since period 1 at low load, thus supporting their high production costs, instead

Fig. 6. Results obtained by adopting the strategies selected by the maxmin criterion and by the NEs, modified by the heuristic procedure: (a) market prices and load demand and (b) profits for the generation companies.

of putting them in operation during the day. Anyhow, the total profits for the three companies obtained from the market by using the maxmin-selected strategies are larger than the minimum ones evaluated in the bidding selection procedure: namely 49.6%, 165.7% and 211.3% larger for Gencos 1, Genco 2 and Genco 3, respec-tively.

Table 6reports, for each period, the Genco which owns the so-called marginal unit, i.e., the power plant which establishes the market price. The list of the marginal units is shown inTable 7. It can be noticed that Genco 1 results the owner of the marginal units in 15 periods when maxmin-selected strategies are adopted and in 22 periods when those relevant to NEs are considered.

As already mentioned, the market-clearing procedure is assumed not to take into account several power plants constraints. Table 8shows the violations of the minimum up and down time constraints that results both when maxmin and NE selected strate-gies are used.

Table 8shows that, although the strategies selected according to the maxmin criterion have been modified by the UC-based feasibil-ity enforcement procedure, there are several constraints violations.

Table 10

Genco owner of the power plant that sets the price in the various periods for the two criteria, after the application of the heuristic procedure with the˛andˇfactors values ofTable 9.

Periods 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

Maxmin 1 1 3 1 2 2 1 2 1 1 2 2 2 1 1 1 1 2 1 2 1 1 2 2

Table 11

Marginal unit that sets the price in the various periods for the two criteria, after the application of the heuristic procedure with the˛andˇfactors values ofTable 9.

Periods 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

Maxmin 2 9 2 2 5 3 11 3 1 13 6 8 3 1 6 10 9 3 1 3 12 6 2 2

NE 2 9 4 3 5 5 12 1 1 13 7 14 13 1 5 5 5 5 6 9 6 5 2 2

This is due to the fact that by concurrently adopting the maxmin strategies for all the Gencos, each one of these sees competitors’ behaviours different from those conjectured by applying the pes-simistic minimum profit condition.

However, a heuristic procedure can be implemented in order to limit the violations ofTable 8, by using the feasible UC calculated by the proposed bidding selection procedure. Such a procedure dif-ferentiates, in every period, the offers of the power plants that are set not in operation in the feasible UC from the offers of the power plants that are expected to be dispatched in the same feasible UC. The aim of the heuristic is to try to force the market to follow the feasible scheduling that was devised by the UC-based feasibility enforcement.

In order to apply the heuristic procedure also to the case of strategies related to NEs, a UC self-dispatching calculation is per-formed for each Genco, taking into account the relevant load profile allocated by the calculated NEs.

Table 9reports the violations still present after the application of the simple heuristic procedure that multiplies by a factor˛i(equal

or greater than 1) the offers of all the power plants of Gencoithat are not considered in operation by the bidding selection proce-dure and by a factorˇi (equal or lower than 1) the offers of all

the power plants of Gencoithat are considered in operation. This simple heuristic procedure is able to reduce the number of units that violate the constraints from 9 to 7 and from 13 to 11, for the case of maxmin-selected strategies and NEs, respectively.

Fig. 6 shows the comparison between the market-clearing

results obtained for the case of maxmin and NE selection criterion, by applying the offers modified by the heuristic procedure with the ˛andˇfactors values ofTable 9.Fig. 6(a) shows the given load pro-file and the market prices, whilstFig. 6(b) shows the corresponding profits of the companies.

Fig. 6shows that the application of a heuristic procedure may substantially affect the final results, particularly when the units that violate the constraints are also those that set the price in some periods.

The results ofTable 6andTable 7are modified by the appli-cation of the heuristic procedure as shown inTables 10 and 11. Genco 1 results the owner of the marginal units in 13 periods when maxmin-selected strategies are adopted and in 19 periods when those relevant to NEs are considered. Also Genco 3 is the owner of the marginal units in 1 and 2 periods, for the case of maxmin-selected strategies and NEs, respectively.

5. Conclusions

The paper has presented an analysis of the influence of feasi-bility constrains on the bidding strategy selection based on the use of a computer procedure that integrates a normal-form game-theory representation of the day-ahead electricity market with a cost-based UC program.

The integration with the UC program allows the selection of the most convenient bidding strategy and at the same time takes into account both an accurate estimation of the production costs and the various power plants technical constraints, which are often disregarded in day-ahead auctions.

The results of the analysis indicates that a procedure allow-ing the definition of convenient biddallow-ing strategies on the basis of adequate selection criteria, which would also facilitate the

achieve-ment of a feasible power scheduling, is needed as a decision support tool for Gencos in oligopolistic markets.

Acknowledgements

The authors would like to warmly thank Prof. C.A. Nucci for his helpful comments and R. Ferrante for his collaboration in perform-ing the calculations. This work was supported in part by the Italian Ministry of University and Scientific Research and in part by the University of Bologna under Project Decisopelet 2006.

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