Estimating conjectural variations for electricity market models
S. Lo´pez de Haro
a,*, P. Sa´nchez Martı´n
a,
J.E. de la Hoz Ardiz
b, J. Ferna´ndez Caro
ba
Instituto de Investigacio´n Tecnolo´gica, Comillas University, CnAlberto Aguilera 25, 28015 Madrid, Spain
bSubdireccio´n de Planificacio´n y Mercados Direccio´n de Gestio´n de la Energı´a Viesgo Generacio´n S.L., Plaza Pablo Ruiz Picasso 1, Torre Picasso, pl. 19, 28020 Madrid, Spain
Received 1 December 2004; accepted 1 December 2005 Available online 31 July 2006
Abstract
Agents’ behavior in oligopolistic markets has traditionally been represented by equilibrium models. Recently, several approaches based on conjectural variations equilibrium models have been proposed for representing agents’ behavior in electrical power markets. These models provide insight of market equilibrium sensitivity to agents’ strategies and external variables, and therefore, they are widely applied. Unfortunately, not enough analysis has been done in how these user-supplied parameters, the conjectural variations, should be estimated. This paper proposes a parameter inference procedure based on two stages. The first stage infers historical values of the parameter by fitting the models’ results to historical market data. The second stage is based on a statistical time-series model whose objective is to forecast parameter values in future scenarios. Additionally, results of this procedure’s application to a real-size case are presented.
2006 Elsevier B.V. All rights reserved.
Keywords: Economics; Game theory; Generation scheduling; Equilibrium models
1. Introduction
In liberalized electricity markets, utilities are completely responsible for their generation schedule and, therefore, they assume new operation strategies and investment decision risks. Centralized decision-support models such as those applied for medium and long-term planification should be modified to take into account competitive behavior at liberalized markets.
In this liberalized context, specific market models are used to analyze the influence of agents’ bidding strat-egies in a competitive auction market such as the generation electricity market. A common approach in several
0377-2217/$ - see front matter 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.ejor.2005.12.039
* Corresponding author.
E-mail addresses:[email protected](S. Lo´pez de Haro),[email protected](P. Sa´nchez Martı´n),[email protected](J.E. de la Hoz Ardiz),[email protected](J. Ferna´ndez Caro).
proposed models consists of forecasting firms’ strategies as a multilateral equilibrium among agents [1–5]. Market equilibrium, as it is explained by game theory, is the strategy for each agent such that its unilateral variation would be less profitable for itself.
Commonly used approaches for the estimation of agents’ behavior in power market models can be classi-fied into three main groups:
• Supply function equilibria (SFE): This kind of equilibrium is achieved by finding each firm’s optimal supply function to be matched to the aggregated supply and demand functions. This approach is only suitable for short-term planification tools due to its computational complexity.
• Cournot-based equilibria (CE): This kind of equilibrium is achieved by finding each firm’s optimal output subject to the satisfaction of a certain demand function. It has been traditionally used[5–7]in oligopolistic markets, particularly in electricity markets. Unfortunately, it provides barely credible prices due to the fact that market agents neglect their competitors’ supply functions.
• Stackelberg equilibria: This kind of equilibrium defines a ‘‘leader’’ whose decisions take into account reac-tions of ‘‘followers’’, who do not recognize how their reacreac-tions affect the leader’s decisions.
• Some conjectural variations-based equilibria (CVE): This kind of equilibrium is based on user-supplied parameters, conjectural variations (CV), that represent the firms’ behavior in a competitive market. CV’s usually stand for the residual demand slope in electricity industry models. This parameter measures the influence each firm’s additional output has on market equilibrium. It stands for firms’ expectancy of how competitors will react to unilateral output changes. CVE allows a more flexible representation of firms’ behavior and an accurate price generation process since Cournot and perfect competition equilibria can be achieved for particular values of the CV.
CVE-based models have been widely applied in electricity markets fundamental modeling[1,6,8,9]. Unfor-tunately, no such concern has been shown in how the CV should be estimated. In this paper a new approach for CV estimation is proposed. It is based on an iterative algorithm that finds the parameters that fit the CVE to historical results and a statistical time-series model that forecasts parameters’ evolution over time. This approach has both the advantages of fundamental and statistical modeling. A fundamental model considers most of the elements involved in firms’ decision taking and a statistical model is used to cover up its inade-quacy to represent every aspect of the electricity market.
The sequel of this paper is organized as follows. A symbol reference is provided in Section 2. Section3
shows the market equilibrium conditions in an oligopolistic context and points out two main procedures for solving conjectural variations-based equilibria. Section 4 describes two different conjectural variations inference procedures. Section 5 shows a small case implementation of these procedures. Section 6 shows the results provided by one of these procedures in a real-size medium-term-horizon scenario. Finally, conclu-sions are stated in Section 7. Some examples of take-or-pay contracts and capacity payments modeling are presented in Appendix A.
2. Notation
Capital letters have been used to differentiate variables from parameters in linear programming models.
Index Description
f Generation firms involved in the market
f All generation firms butfÆf’s competitors
u Generation units involved in the market
s Time scope periods
k Iterations
3. Conjectural variations-based models
3.1. Discussion
CVE have been widely applied in firms’ behavior forecasting[1,6,8,9]and market power measurement[10– 13]. Firms involved in electricity industry usually apply them to simulate oligopolistic markets behavior, to
Parameters Description Units
Monoperiod formulation
Pf Firmf’s profit €
qf Firmf’s production MW h
hf Firmf’s production contracts MW h
p Market price €/MW h
d System’s demand MW h
Cf Firmf’s total costs €
MCf Firmf‘s marginal costs €/MW h
MRf Firmf’s marginal revenues €/MW h
RDf Firmf’s residual demand function
D System’s demand function in period
Sf Firmf’s supply function
hf Negative value of firmf’s residual demand function’s slope.
The user-supplied parameter to be inferred
€/MW h2
^
hf Estimator forhf €/MW h
Multiperiod formulation
qf,s Firmf’s total production in periods MW h
Qf,s
^
qf;s Expected value ofQf,s(based on a historical scenario) MW h
QUu,s Unitu’s production in periods MW h
hf,s Firmf’s production contracts in periods MW h
ts Duration of periods h
ps Market price in period s €/MW h
^
ps Expected value ofps. It can be obtained
from a real past scenario
€/MW h
ds System’s demand of periods MW h
MCf,s Firmf‘s marginal costs of period s €/MW h
hf,s Negative value of firmf’s residual demand
function’s slope in periods(to be inferred)
€/MW h2
^
hf;s Estimator forhf,s €/MW h2
Dhf,s Increment ofhf,s €/MW h2
es Price estimation error of periods €/MW h2
a Updating computational coefficient p.u.
pmaxu Maximum power generation of unitu MW
gcu Variable generation cost of unitu €/MW h
capu Available hydraulic energy of unitu MW h
cpu Capacity payment €
ccpu Amount of energy of the condition of capacity payment MW h
CPSu Capacity payment satisfaction indicator of unit u [0, 1]
tpeu Take-or-pay contract equivalent amount of energy MW h
design their medium and long term strategies and to measure the influence that some external variables, such as fuel prices or hydrology inflows, have on their expected benefit. Another application of CVE is market power measurement[14]. It has been used by several studies that follow the ‘‘New Empirical Industrial Orga-nization’’ methodology. This technique measures market power by estimating the firms’ price-cost margins within a conjectural variations framework. Its main advantage is that it infers price-cost margins without actual cost data.
Several drawbacks of CVE application to market power measurement have been identified in the theoret-ical literature. First, it has been shown [15]that unless very specific information assumptions are provided, only Cournot model can be considered consistent1. Additionally,[10]states that if firms are competing in a dynamic setting then CV estimation can be biased since the influence of collusion should also be considered. Another argument was provided by[11], where it is shown that CV do not accurately reflect the market’s com-petitiveness since they depend on the demand’s functional form.
In this context the validity of CVE to firms’ behavior forecasting could be questioned. It is true that a CV estimate that fits the static CVE2to historical results of a real dynamic game where agents are using collusive strategies would be hardly consistent. Nevertheless, this CV estimate could be observed as a ‘static equivalent’ parameter that provides information about agents’ behavior in the real market. And, therefore, future values of this parameter could be forecasted by time-series models. Additionally, it can be assumed that firms are not using collusive strategies since agents’ behavior is under surveillance in most liberalized electricity markets.
And second, it has been discussed that there are difficulties in CV empirical estimation when marginal cost data are absent. While it may be true that, due to confidentiality reasons, marginal cost estimates are hardly available in other industries, it is not so in electrical industry. Usually hourly units’ output and market clearing price are publicly known in most liberalized markets[11,12]. Fossil–fuel prices can be obtained from various public energy agencies and technological efficiency rates from historical data or from plant builders’ catalogue. In Spanish electricity market, other variables, such as planned unavailability ratios, are only at the disposal of firms involved in the market.
3.2. Market equilibrium
From a mathematical point of view, game theory equilibria are represented by a set of equality constraints whose solution provides the strategy that each agent should use so that any unilateral variation from it would be less profitable for itself.
The profitPf of firmfafter selling an amount of energyqfto the market at a unitary pricepand taking into
account production costsCf(qf) is given by
PfðqfÞ ¼pðqfÞ qf CfðqfÞ 8f. ð1:1Þ
The optimality condition can be obtained differentiating and zeroing Eq.(1.1)
oPf
oqf ¼pðqfÞ þ
op
oqfqfMCfðqfÞ ¼0. ð1:2Þ
Note that marginal costs MCfare defined as the first derivative of production costs when firmf’s output isqf.
Actually a cost minimization problem must be solved in order to compute this variable. Also, it must be no-ticed that market pricep has been considered as dependent on the firms’ outputqfgiven that agents are
as-sumed to have market power. The first two terms of the equation make up firms’ marginal revenue MRf.
Marginal revenue measures how firms’ revenues change when they incrementally modify their production. As seen in Eq. (1.2), firm’s production in the equilibrium is achieved when its marginal revenue is equal to its marginal cost. Therefore, marginal costs and marginal revenues are assumed to be equal in any real market context.
1
‘Consistent equilibria’ are those which consider the conjectural variation model literally, i.e., the parameter is supposed to represent a firm’s beliefs regarding how its competitors will react if it changes its output.
2
Then, the optimality second order condition, assuming that the relationship between price and production is linealðd2p=dq2
f ¼0Þ, as follows:
o2Pf
oq2
f
¼2 op
oqf
oMCfðqfÞ
oqf 60. ð1:3Þ
The following condition stands for the demand satisfaction constraint, which establishes the relationship be-tween firms’ production scheduling problems:
d¼X
f
qf. ð1:4Þ
Fig. 1summarizes agents’ production scheduling problems and their relationship.
3.3. Conjectural variations-based equilibrium
In an oligopolistic competition context, the influence of agents’ output on market price is usually repre-sented by the residual demand RD(p) function[1], which provides the amount of energyqf= RD(p) that firm fwould sell for a certain pricep. RD represents the influence of competitors’ bidding from firmf’s point of view. It can be obtained by substracting the competitors’ supply functionSffrom the aggregated demand
functionD(p)
RDfðpÞ ¼DðpÞ SfðpÞ. ð1:5Þ
Models[8,9]are both based on the same conjectural variation-based equilibrium. They characterize the resid-ual demand in terms of a user-supplied parameterhfwhich stands for the negative value of firmf’s RD’s slope
at the equilibrium. This way, firmf’s marginal revenue is expressed in terms ofhfas follows:
MRfðqfÞ ¼pþoqf
op qf ¼phf qf. ð1:6Þ
Market equilibrium can be represented as the solution of the set of equations resulting from(1.4)and the sub-stitution of Eq.(1.2)in(1.6). It is expressed as
0¼MRðqfÞ MCðqfÞ ¼phf qfMCðqfÞ 8f;
d¼X
f
qf :p. ð1:7Þ
Note that only a unique period equilibrium has been modeled (the multiperiod generalization can be found in next section). The solution of this problem is not straightforward since market pricep, the demand satisfaction constraint’s dual variable[6], is shared by the set of equations that relates marginal revenues and costs.
Refs.[6,8] use different methodologies to find the equilibrium. While the first one is based on the mixed complementarity problem (MCP), the second one is based on quadratic programming (QP). Their objective
function consists of firms’ production schedule optimization in an oligopolistic competition environment. Both approaches provide powerful and flexible methodologies to forecast the firms’ long-term behavior since they resemble classical optimization of hydro-thermal coordination. The influence of important factors of power systems such as hydro-thermal coordination, capacity payments, carbon dioxide emission costs, national coal combustion bonuses or fuel take-or-pay contracts can be included in both approaches. Some examples where these elements have been modeled are shown inAppendix A. Due to the fact that most of the concepts explained can be easily extended to the MCP formulation, the sequel of this paper is based on the QP formulation formulated by[8].
3.4. QP formulation
An equivalent QP formulation for the market equilibrium as formulated in(1.7)is proposed in[8]. Dem-onstration of this equivalence was provided by showing that Karush–Kuhn–Tucker equations for the optimal-ity problem were equal to (1.6) and that the second order sufficiency condition (1.3) holds whenever cost functions are convex.
Next minimization problem’s solution provides the equilibrium forSperiods, where demand is supposed to be inelastic
min pf;s
X
f
CfðQf;sÞ þ X
f;s
hf;s
2 ðQf;shf;sÞ
2
;
s.t.
ds¼P
f
Qf;s 8s2S:ps;
Additional constraints:
( ð1:8Þ
Note that the objective function is calculated as the sum of two terms:
• PfCfðQf;sÞ, which represents system’s total costs given that each firmfhas a production scheduleQf,s.
• Pf;shf;s
2 ðQf;shf;sÞ
2
, which stands for a penalty term related to firms’ production deviations from the con-tracted amount in periods,hf,s.
Notice that neglecting the second term of the objective function would result in the centralized production problem, whose solution corresponds to the perfect competition equilibrium. Definitely, it is the second term which represents the oligopolistic influence in the equilibrium. In fact, ifhf,swas neglected for a certain firmf
and periods, then this firm would be considered as a price-taker in this period, i.e., unable to market pricep. The first constraint guarantees demand satisfaction, i.e., the total output of all firms must be equal to the demand of each period. Note that market pricepsis this constraint’s dual variable for each period s.
Addi-tional constraints can be used for modeling hydro-thermal coordination, fuel take-or-pay contracts, capacity payments or any other factor involved in the firms’ profit maximization problem (seeAppendix A).
Energy contractshf,s do also influence market equilibrium. They are usually used to reduce firms’ market
risk since part of their generation is sold for a fixed price which is independent of the market clearing price. It has been observed[16]that oligopolistic agents reduce competition in the day-ahead market by committing to sales in the forward market. Moreover, if a firm has sold more energy in the forward market than its produc-tion capacity, addiproduc-tional energy may be bought in the day-ahead market. Therefore, this firm’s strategy is using its power to reduce clearing price in the day-ahead market. This effect has been modeled by substracting hf,sfromQf,sin the second term of the objective function. If energy contracts are being considered,(1.6)should
be updated as follows:
MRfðqfÞ ¼phf ðqfhfÞ. ð1:9Þ
Usually, a cost minimization problem for each firm fis completed to compute its marginal costs Cf(Qf,s). A
general model, where the oligopolistic profit maximization(1.8)and cost minimization problems were solved simultaneously was proposed by[8]. Nevertheless, for clarity reasons, the sequel of this paper separates both problems.
Onlyhfremains unknown. The next section discusses several methodologies for estimating this parameter.
4. Conjectural variation estimation
Since residual demand slope (RDS) represents the firms’ mid-term strategies[9], its value has a high impact on the CVE results. Therefore, a methodology to estimate these parameters is essential to obtain credible results. Unfortunately, values for these parameters are not easily obtained from market public information. In this section two different estimation methodologies are illustrated. The main difference between both meth-odologies is the information that the estimated parameters provide. While the first procedure fits the market equilibrium conditions to real historical data, the second procedure finds the values of the parameter that exactly fit model’s outputs to historic results. Both of them are validated finding h values that fit models’ results to historical ones.
4.1. Explicit estimation
Attending to the definition of CV in this context as RDS for each agent, it could be computed from day-ahead market data. Nevertheless, CV stand for the firms’ mid-term strategies and their values cannot be com-puted from RDS short-term measures.
4.2. Implicit estimation
The implicit estimation (IE) approach (first exposed by[9]) finds the values for the conjectural variations that fit a certain CVE to historical scenarios. It is based on Eq.(1.9), which relates firms’ output and market clearing price in a CVE context. The IE estimator ofhf, for past scenarios, can be obtained by rewriting Eq.
(1.9)usinghf:
^ hf ¼
pMRf
qf hf
¼pMCf
qf hf
. ð1:10Þ
When fitting past scenarios, market prices and firms’ outputs can be assumed to be optimal. Therefore, mar-ginal revenue and marmar-ginal cost have been considered equal and MRfhas been substituted by MCfat
expres-sion(1.10).
From(1.10)an estimator for the multiperiod case can easily be extrapolated
^ hf;s¼
psMCf;s
qf;shf;s
. ð1:11Þ
While values for the market clearing priceps, and each firm’s production in past scenarios can be easily
ob-tained from public market data, marginal costs values MCf,scannot be obtained this way. In power markets,
marginal costs consist mainly of fossil–fuel prices and thermal efficiencies. However, the influence of time fac-tors such as hydrothermal coordination, the cost of power plants’ shutting down, over-sized take-or-pay contracts or capacity payments must also be considered. Estimation at[11,12] is based on computing fuel costs’-based marginal costs and hypothesizing about how these should be adjusted in order to reflect time fac-tors’ influence. A more detailed methodology has been used in our approach. In that case a model such(1.8)is being fitted and that the production schedule of each firm^qf;swas available for past scenarios, the marginal costs MCf,s could be obtained from a cost minimization problem such as(1.12)
min
QUf;s
CfðQUu;sÞ;
s.t.
P u2f
QUu;s¼^qf;s:MCf;s 8s;
Additional constraints:
8 < :
ð1:12Þ
The problem’s objective is minimizing firmf’s total costs, which depend on the generation schedule QUu,sof
generation schedule^qf;sof firmf. Its dual variable represents the marginal costs MCf,s. Additional constraints
can be used in order to model hydro-thermal coordination, fuel take-or-pay contracts, capacity payments or any other element involved in the firm’s costs minimization problem (seeAppendix A). Notice that, since the objective function in(1.12)is expressed in terms of costs, incomes must be substracted from it. Note also that despite the fact that a problem such as (1.12) should be solved for each firm f, all problems can be solved simultaneously.
Once marginal costs and CV-values have been computed for past scenarios, mid-term estimates must be computed. Several data-analysis methodologies could be addressed at this point. They can be classified as fol-lows[9]:
• Relational models: The variable forecasted, hf,s, is related to various other explanatory variables whose
long-term values are known.
• Classification models: The variables are categorized by different levels of several discrete factors. Variance analysis and clustering are some statistical methodologies based on classification processes.
• Time series models: The variable’s temporal evolution is analyzed as a time series in order to find its mid-term trend and seasonal behavior. Explanatory variables can be added to this kind of models.
4.3. Advanced implicit estimation
IE techniques fit the market equilibrium model to historical scenarios adjusting values for the conjectural variations. Nevertheless, real data fitting is a very ambitious task because every factor at firms’ decision-taking must be considered. Some of them, such as changes on firms’ strategies or energy policy actions, cannot be included in market equilibrium models. However, they have an impact on market price.
A new approach to fithftaking into account the model’s inadequacy to represent all factors involved in the
market has been developed. It is called advanced implicit estimation (AIE) and is based on a procedure where
hf,sis updated iteratively. Then, market price results meet real past scenarios even if the model does not
con-sider all factors that influence firms’ decision-making.
Let hkf;s behf,svalue at iterationkin problem(1.8). Let this problem’s results at iterationkbeqkf;s, MC k f;s andpk
s. Since these values determine the market equilibrium,(1.6)can be used. The following expression has been obtained writing(1.9)usingpk
s:
pks¼MCk f;sþh
k f;s ðq
k
f;shf;sÞ. ð1:13Þ
Let ek
s be the price error defined as the difference between historical value^ps and price estimatepks
ek
s ¼^pspks. ð1:14Þ
Regarding(1.13), it can be inferred that, assuming thatqk
f;sand MC k
f;sare constants, a deviation ofh k
f;swould result in a proportional variation ofpk
s in the next iteration. Thus, the incrementDh k
f;sindicates the direction wherehf,sshould be updated in the next iteration if there is a small difference between price estimation at the
next iterationpkþ1
s and its historical value
Dhk f;s¼
^
pspks
qk f;shf;s
. ð1:15Þ
Table 1 AIE procedure
1 Initialization:k¼0;hf;s¼h0f;s
2 Solve(1.8)and keep the solutionsqk
f;s, MCkf;sandpks
3 If maxsjeksj6epstop the iterations and go to step 5, else go to step 4
4 Update:hkþ1
f;s ¼hkf;sþaDhkf;s;k¼kþ1 Next to 2
Whileqk
f;sand MC k
f;schange in consecutive iterations, it can be expected that their rate of variation is lower than that ofhkf;sover consecutive iterations. An updating computational coefficientathat reduces the step size is used in order to prevent oscillations.Table 1shows all the steps of this procedure.
Note that the convergence criterion has been expressed in terms of price. The algorithm stops when the maximum price error maxsðjek
sjÞis lower than a certain thresholdep. Initial valuesh
0
f;saffect procedure conver-gence. Authors consider those obtained by IE the best initial values.
Since this procedure estimateshf,s values that fit a certain model’s results to prices in real past scenarios,
these parameters provide information about factors that are not being included into the model but do have an influence in the market. Temporal evolution of these elements and their relationship with some other vari-ables can be statistically analyzed in order to forecast mid-term values. Time series models, based on finding the correlation between elapsed values of a sequential variable, can be used to perform this analysis. Addition-ally, explanatory variables such as power demand, generation contracts or fuel price evolution, or any other variable whose values can be easily foretold during the desired forecast scope, can be applied to forecasthf,s
values in future scenarios.
Fig. 2shows a graphical summary of the proposed methodology.
5. Implementation
In this section an illustrative example of conjectural variation inference is shown. The objective is to fit firms’ behavior in a historical scenario consisting on four periods, P1–P4, each of durationts= 1000 hours.
Only two competitive firms have been considered: AG1 and AG2. Generation contracts’ influence has been ignored in this example. Let the values for demand (d), market clearing price (p) and each firm’s generation (qf) atTable 2be the values to be fitted.
Each firm owns two generation units: CC and HI belong to the first one and CO and NU to the second one. CC and CO resemble two thermal generation units based on different technologies, and therefore, different
Fig. 2. Overview of the proposed methodology.
Table 2
Results of a past scenario
S d(GW h) p(€/MW h) qAG1(GW h) qAG2(GW h)
P1 50.00 45.00 25.00 25.00
P2 45.00 39.45 24.00 21.00
P3 30.00 20.00 10.00 20.00
variable costs. HI resembles an hydraulic generation unit where a maximum amount of energy can be gener-ated all over the periods of the scenario. Technical characteristics of NU are similar to those of a nuclear gen-eration unit. It has low variable costs and, therefore, usually operates close to its maximum capacity.Table 3
shows generation cost (gc) and maximum generation capacity (pmax) of each unit. Available hydraulic energy for the scenario is fixed to caphi= 15 MW h.
The following mathematical model is used to compute the perfect competition equilibrium that minimizes systems’ total costs for this scenario. Results for the scenario presented in Table 2are presented inTable 4.
min
QUu;s
X
u;s
gcuQUu;s;
s.t.
P u
QUu;s¼ds:ps 8s; P
s
QUhi;s6caphi;
QUu;s6tspmax
u 8u;s: 8 > > > < > > > :
ð1:16Þ
Table 4shows the results for units’ production (QUu,s), firms’ production (Qf,s) and market price (p) for this
scenario. Notice that even though firms’ outputs are close to those provided byTable 2, market clearing prices for each period are slightly different. An explanation for this is the existence of oligopolistic competition in this market. An oligopolistic equilibrium can be computed using the next formulation, a particularization of(1.8)
for this example
min
QUu;s
X
u;s
gcuQUu;sþX a;s
hf;s 2 Q
2
f;s;
s.t.
P u2f
QUu;s¼Qf;s 8f;s; P
f
Qf;s¼ds 8s:ps; P
s
QUhi;s6caphi;
QUu;s6tspmaxu 8u;s: 8 > > > > > > > < > > > > > > > :
ð1:17Þ
Table 3
Technical information
Firm Unit pmax (MW) gc (€/MW h)
AG1 hi 8.00 0.00
cc 17.00 20.00
AG2 nu 20.00 10.00
co 22.00 30.00
Table 4
Market results in a perfect equilibrium context
Firm Unit P1 P2 P3 P4
QUu,s(GW h)
AG1 hi 7.00 8.00 0.00 0.00
cc 17.00 17.00 10.00 0.00
QAG1,s(GW h) 24.00 25.00 10.00 0.00
AG2 nu 20.00 20.00 20.00 20.00
co 6.00 0.00 0.00 0.00
QAG2,s(GW h) 26.00 20.00 20.00 20.00
hf,svalues that fit problem(1.17)’s results to those inTable 2should be inferred. Eq.(1.11)provides an
esti-mator of these parameters where all necessary information excluding MCf,sis available. Nevertheless, MCf,s
can be obtained by particularizing(1.12)to this example and solving it for each firmf.Table 5shows values forhf,s,Qf,s,and MCf,s obtained using this methodology.
Based onTable 7, CVE results are similar to those given inTable 2not only in terms of firms’ production, but also in terms of market price. The same feature can be observed following the AIE procedure (Table 8), where initial values for hf,s, the maximum admissible price error and the updating coefficient were fixed to
h0f;s¼0€=MW h2,ep= 0.001€/MW h anda= 0.5, respectively.
Unfortunately, most of market results are not so easily fitted as those fromTable 2. Important factors in market behavior cannot be included in the CVE. AIE procedure should be used in these scenarios in order to obtain values ofhf,sthat exactly fit the model’s market price results to real data. Assume that a different sce-Table 5
Market results in an imperfect competition context
s P1 P2 P3 P4
hAG1,s(€/MW h2) 0.70 0.50 0.00 0.00
hAG2,s(€/MW h2) 0.60 0.45 0.20 0.20
QUhi,s(GW h) 8.00 7.00 0.00 0.00
QUcc,s(GW h) 17.00 17.00 10.00 0.00
QAG1,s(GW h) 25.00 24.00 10.00 0.00
QUnu,s(GW h) 20.00 20.00 20.00 20.00
QUco,s(GW h) 5.00 1.00 0.00 0.00
QAG2,s(GW h) 25.00 21.00 20.00 20.00
ds(GW h) 50.00 45.00 30.00 20.00
MCAG1,s(€/MW h) 27.50 27.45 20.00 16.06
MCAG2,s(€/MW h) 30.00 30.00 16.00 12.00
ps(€/MW h) 45.00 39.45 20.00 16.06
Table 6
New scenario to be fitted
S d(GW h) p(€/MW h) qAG1(GW h) qAG2(GW h)
P1 50.00 45.00 17.00 33.00
P2 45.00 39.45 13.00 32.00
P3 30.00 20.00 12.00 18.00
P4 20.00 16.06 12.00 8.00
Table 7 Results of IE
s P1 P2 P3 P4
hAG1,s(€/MW h2) 0.76 0.61 0.00 0.00
hAG2,s(€/MW h2) 0.88 0.73 0.83 0.51
QUhi,s(GW h) 8.00 7.00 0.00 0.00
QUcc,s(GW h) 17.00 17.00 17.00 0.20
QAG1,s(GW h) 25.00 24.00 17.00 0.20
QUnu,s(GW h) 20.00 20.00 13.00 19.80
QUco,s(GW h) 5.00 1.00 0.00 0.00
QAG2,s(GW h) 25.00 21.00 13.00 19.80
ds(GW h) 50.00 45.00 30.00 20.00
MCAG1,s(€/MW h) 33.12 30.68 20.83 20.00
MCAG2,s(€/MW h) 30.00 30.00 10.00 10.00
nario that was to be fitted. Let values of demand, market clearing price and each firm’s generation be those provided atTable 6. In this case, the IE and AIE procedures would respectively provide the results provided byTables 7 and 8. Notice that while AIE is capable of fitting market price results, IE is not. It is also remark-able that neither has fitted productions to the expected values. It can be concluded from these results that AIE should be applied to infer the historical values of hf,s.
6. Case study
In order to validate the fitting procedures described in previous sections, a CVE oligopolistic model such as
(1.8)has been applied to the Spanish Wholesale Electricity Market.
6.1. System description
The Spanish Electricity Market was established in 1998 and six main competitors are nowadays within: Endesa (GE), Iberdrola (IB), Unio´n Fenosa (UF), Hidrocanta´brico (HC), Viesgo (VIE) and Gas Natural (GN). GE and IB are generally considered the two big oligopolistic competitors because they have important market shares. All other firms can be considered to be price-takers, and therefore, their RDS have been con-sidered null. The system meets a maximum peak close to 37,212 MW.3The system’s yearly energy demand added up to 225,688 GW h1and the average hydro energy available amount to 30,176 GW h.Table 9shows the generation capacity of each firm and technology, where FF stands for fossil fuels (coal, gas, petrol), NU for nuclear generation, HI for hydraulic generation and BT for pump-turbine generation. Note that GE’s share in the market stems mostly from its coal units. However, IB holds an important hydro share into its portfolio. Table 8
Results of AIE
s P1 P2 P3 P4
hAG1,s(€/MW h2) 0.48 0.37 0.00 0.00
hAG2,s(€/MW h2) 0.60 0.45 0.77 0.19
QUhi,s(GW h) 8.00 7.00 0.00 0.00
QUcc,s(GW h) 17.00 17.00 16.99 0.00
QAG1,s(GW h) 25.00 24.00 16.99 0.00
QUnu,s(GW h) 20.00 20.00 13.00 20.00
QUco,s(GW h) 5.00 1.00 0.00 0.00
QAG2,s(GW h) 25.00 21.00 13.00 20.00
ds(GW h) 50.00 45.00 30.00 20.00
MCAG1,s(€/MW h) 33.12 30.68 20.00 16.06
MCAG2,s(€/MW h) 30.00 30.00 10.00 12.37
ps(€/MW h) 45.00 39.45 20.00 16.06
3
Data for year 2003. Table 9
Installed generation capacity of each firm and technology (GW)
Firm FF NU HI BT Total
GE 9.431 3.491 3.871 1.409 18.202
IB 8.136 3.191 5.652 2.101 19.08
UF 4.665 0.7 1.257 0.208 6.83
HC 2.24 0.155 0.281 0.133 2.809
VIE 1.516 0.306 0.36 2.182
GN 1.515 1.515
Every main thermal generation unit that sells energy to the market has been considered in the CVE model. Hydraulic and pump-turbine generation units have been grouped into ten, one for each firm and technology. The model has been fitted to a test data interval including four years (2000–2003) which have been divided into periods, one for each four weeks. Each period has also been divided into eight different load levels.
All the information that has been used in this case study is in the public domain. Units’ productions and market prices have been obtained through the Spanish Market Operator. Fuel costs and units’ thermal rates are respectively based on international fuel prices and standard technology rates.
6.2. AIE for past scenarios
Fig. 3shows the estimate of average market price that has been attained through AIE methodology during period 2000–2003. The bold and dotted lines represent, respectively, the historical and estimated real market prices in (€/MW h). Note that price estimation error is rarely over 5%. Only summer months of year 2002 have surpassed this threshold. This can be explained by the fact that 2002 was a year of mayor regulatory uncer-tainty and this factor cannot be included into the model.
6.3. Time series model
Provided hf,s values for past scenarios, a statistical model was used for forecasting future values. An
ARIMA time-series model with explanatory variables was used for this task.hfvalues were fitted over four
consecutive years (2000–2003) that were divided into 42 periods, 13 for each year. Values were grouped attend-ing to the load level and firm. This time-series model was then used to forecast values for year 2004.
Unfortunately, implicitly estimatedhfwas not found a smooth variable. Not only were its values spread
over a wide interval (always positive), but they also had a seemingly random rate of variation. Fortunately, it can be verified that the sensitiveness of the equilibrium to increments ofhfis not the same all over the
inter-val of possible inter-values, i.e., the price’s variation would not be as high if it changed from 0 to 50€/MW h2than if it changed from 300 to 350€/MW h2. In order to benefit from this feature the napierian logarithm Ln(RD) of
hfwas fitted instead ofhfitself. This way, not only was the random variations’ (‘‘noise’’) around the highest
values smoothed but also the outlayer values’ bad influence on the fitting was cancelled.
0 10 20 30 40 50 60 70 80
1 2000
4 7 10 1 2001
4 7 10 1 2002
4 7 10 1 2003
4 7 10
Year/Month
Market price (Euro/MWh)
PREAL PCALC
Figs. 4 and 5show the results obtained by fitting an ARIMA model to Ln(RD) for firms GE and IB in two different load levels, respectively. Bold and solid lines stand for the real and fitted values of Ln(RD), respec-tively. Dotted lines represent the estimation extremes of the estimate’s 95% confidence interval. The figures also show the model’s forecast for year 2004. Note that the confidence interval grows over the forecasting interval. A model with order 1 autoregressive component and without seasonal or differentiation components was found as the most suited one. Three explanatory variables were found by the model as the most correlated ones: coal price, energy supply contracts and a measure of market’s available generation. Many other vari-ables were evaluated and rejected due to collinearity or lack of explanatory value.
6.4. Market price forecast
The CVE model has been run with the CV estimates for future scenarios in order to forecast market price during the year 2004.Fig. 6shows the difference between estimated and real average price during this period.
0 1 2 3 4 5 6 7 8 9
1 2000
4 7 10 1
2001
4 7 10 1
2002
4 7 10 1
2003
4 7 10 1
2004
4 7 10
Year/Month
Ln(RD) - Ln( \MWh
2)
Ln(RD)
Ln(RD)-fit
LB95
UB95
Fig. 4. Napierian logarithm of RDS for GE.
0 1 2 3 4 5 6 7 8
1 2000
4 7 10 1
2001
4 7 10 1
2002
4 7 10 1
2003
4 7 10 1
2004
4 7 10
Year/Month
Ln(RD) - Ln( \MWh
2)
Ln(RD)
UB95
LB95 Ln(RD)-fit
Note that price estimation error is rarely over 15%. Attending to this result it can be concluded that the pro-posed methodology provides an accurate forecast of firms’ behavior and market clearing price. Note also that estimation error grows during the last four months of the year. This is due to the fact that CV estimates’ con-fidence interval increases over time.
Confidence interval estimation for price forecast is not a simple matter. Two approaches could be pro-posed. First, fitting period’s maximum price estimation error could be extended over the forecasting period. While this procedure is simple, it is not realistic because the confidence interval for CV estimate grows along the forecasting period. A second procedure would run the model with CV’s confidence interval extreme values. The results of each run would provide the extreme values on price estimate’s confidence interval.
Regarding the results obtained from the case study, the proposed methodology has been validated as a flex-ible and accurate tool to infer and forecast firms’ behavior in a mid-term scope.
7. Conclusions
The main contribution of this paper is a parameter inference methodology for conjectural variations based-models called ‘‘advanced inference estimation’’. It consists of an implicit and iterative estimation process based on public information. Its main advantage compared to other methodologies is that it provides para-meter values that fit mathematical models’ market prices to historical ones. These parapara-meters provide infor-mation about all elements that are not being considered by the model but do have an influence in the market. The temporal evolution of these elements and the relationship with some other variables can be statistically analyzed in order to forecast future mid-term values of this parameter. Therefore, these parameters can be used to cover up the model’s inadequacy to represent every aspect of the electricity market.
An implementation for a small problem has been developed as an illustrative example of this inference cedure. Also, a case study applied to the Spanish Electricity Market has been presented to show how the pro-posed methodology fits a real-sized model to historical data and forecasts conjectural variations in a mid-term scope.
Acknowledgements
The authors gratefully acknowledge the contributions of M. Ventosa and I. Hierro for their work on the original version of this document. They also wish to thank the referees for their contributions to the final ver-sion of this article.
10 15 20 25 30 35 40
jan feb mar apr may jun jul aug sep oct nov dec
Month
Price ( /MWh)
REAL ESTIM
Appendix A
Previous sections assume that some market factors can be included in the market equilibrium model through ‘additional constraints’. In this section illustrative examples of how some of these factors can be mod-eled are showed. Optimization problem(1.17) finds the CVE taking into account, not only units’ maximum generation capacity but also a simple version of hydrothermal programming. Additional constraints and terms can be included into this model in order to reflect capacity payments and take-or-pay contracts’ influence on the market.
Capacity payments are additional remuneration a regulator provides in order to promote generation invest-ment in guarantee of supply (new generating units). Spanish regulation established that a minimal amount of energy had to be generated by a unit over a year if this unit’s owner was to achieve this year’s additional remu-neration. This factor can be included in the model as follows:
min
QUu;s
X
u;s
gcuQUu;sþ X
a;s
hf;s 2 Q
2
f;sþ X
u
cpuCPSu;
s.t.
Constraints in Eq:(1.17), P
s
QUu;sPccpuCPSu 8u;
CPSu2 f0;1g 8u;
8 > > < > > :
ð1:18Þ
where CPSuis a binary variable that stands for the capacity payment condition’s satisfaction and ccpuis the
amount of energy generation that unitu has to supply in order to achieve the capacity payment cpu.
Fuel take-or-pay contracts can be defined as agreements between a buyer and seller in which the buyer will still pay some amount even if the product or service is not provided. When generation units’ fuel is provided under this type of contract it is assumed as an amount of fuel a firm has already paid for even if it is not deliv-ered. So, once the contract has been signed only one decision remains: when this fuel should be burnt. The influence of a take-or-pay contract for an equivalent amount of energy tpecc for unit cc could be modeled
as follows:
min
QUu;s
X
u;s
gcuQUu;sþ X
a;s
hf;s 2 Q
2
f;sgcccQTPcc;
s.t.
Constraints in Eq:(1.17), QTPcc6P
s
QUcc;s;
QTPcc6tpecc; 8
> < > :
ð1:19Þ
where QTPccis the equivalent amount of energy actually delivered. Note that from this ‘a posteriori’ point of
view, fuel take-or-pay contracts can be envisioned as an amount of fuel that a unit can burn for free. Once this limit has been surpassed additional consumption is reflected in the firms’ costs.
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