Modeling of bidding prices using soft ComputingTechniques

Luis Gallego, Member, IEEE, Oscar Duarte, Member, IEEE,

Abstract- In this paper an strategy for discovering patternsover the continuous domain of bidding prices is proposed. Inparticular, the proposed method represents bidding functions aspoints in a multidimensional space where a clustering algorithm isapplied. On the other hand, as a result of this method, a dramaticreduction over the search space of bidding strategies is achieved.In addition, some relations of dominance over bidding strategiesare found, improving the pattern recognition process of agent'sbidding behavior. This method is applied on the bidding pricesdatabase for some GENCO's of the colombian power market.Furthermore, an application of some data mining algorithmsis presented with the purpose of quantifying some hypothesisfo~mulated on the effect of hydrology over spot and biddingprices.

Index Terms- Bidding prices, Electricity Market, Patternrecognition.

bidding prices that are assigned to the different portfolio ofgenerating plants. This search of patterns allows to assessbidding functions that are typically used by GENCOs duringtheir daily routine.

The proposed methodology is based on a popular clusteringalgorithm called k-means which has been extensively used inunsupervised learning tasks. However, the main contribution ofthis methodology is not the application of k-means algorithm,but the representation of bidding functions such as points ina multidimensional space where clustering algorithms mightbe applied. In addition, a new data mining algorithm isused to quantify the confidence in common knowledge aboutsome variables that are currently assumed to affect biddingstrategies.

where:mi is the assigned centroid of cluster Ci .

p is an object assigned to cluster Gi .

D represents the sum, over all clusters, of the within-clustersums of point-to-cluster-centroid distances.

A. The k-means algorithm.

Broadly speaking, the k-means algorithm consist in a clus­tering method that divide a space of n objects into k partitions,where each partition represents a cluster. These clusters areformed following the idea of optimizing a similarity function,so that objects within a cluster maximize this function andon the contrary, objects of different clusters minimize thi~function. In the case of k-means algorithm, this function iscommonly defined as a distance from any object that belongsto a particular cluster, to a point that is defined as the centroidof this particular cluster. Consequently, this algorithm returnstwo arguments as follows:

1) A vector of length k representing the centroids for thek clusters.

2) A vector of length n representing the assignment of eachobject to one of the k clusters.

The centroids are obtained by an iterative process attemptingto minimize the sum, over all clusters, of the within-clustersums of point-to-cluster-centroid distances. Therefore, the al­gorithm converge to a local optimum that consist in a partitionof points in which moving any single point to a' differentcluster increases the total sum of distances [1]. This sum isdefined as [2]:

I. THE PROBLEM

A MAJOR problem commonly faced in modeling the be­havior of bidding prices consists in determining the

most representative bidding strategies of market agents. Inparticular, the continuous nature of bidding prices domainmakes difficult to assess and predict competitors behavior.Consequently, in order to simulate the agents behavior moreaccurately, a methodology to characterize their behavior seemsto be mandatory. On the other hand, it is intuitive to think thatGeneration Companies (GENCOs) design their strategies overtheir own portfolio of generating plants, therefore, seems tobe logical the construction of bidding functions that representthe aggregated behavior of GENCOs.

Following this disaggregated approach, two possible ques­tions need to be properly addressed:

• What bidding prices are assigned by GENCOs to eachone of their generating plants?

• Does these bidding prices correspond to some particularbidding strategies?

• Is there a common knowledge about the main variablesthat drive these bidding strategies? If so, Can this knowl­edge be quantified by any means?

These are the questions that are solved with the proposedmethodology in this paper.

II. THE PROPOSED METHODOLOGY

With the aim of addressing the above mentioned issues,is necessary to find an strategy of pattern recognition among

L. Gallego and O. Duarte are professors of the National Univer­si~ of Colombia - and active researcher of group PAAS-UN (e­mall://lgallegov@unal.edu.co) http://www.paas.unal.edu.co

978-1-4244-2218-0/08/$25.00 ©2008 IEEE.

k

D == L L Ip - mi 1

2

i=l pECi

(1)

isolated process for each generating plant, but as a combinedbidding strategy of the agents over their diversified portfolios.

However, given the continuous domain of bidding prices andreservoir levels (hydrology), is necessary to find an strategy ofdiscretization over these domains. In this proposed methodol­ogy a fuzzy discretization is used by applying a fuzzy versionof k-means called fuzzy k-means. These type of clustersfollows the idea of fuzzy sets [3], where an element belongspartially or in a certain degree to an specific set. Therefore,the assignment of one object to a particular cluster is definedby a "membership" degree to that cluster, taking values in theinterval [0, I]. Figure 1 shows the type of fuzzy sets consideredover the variables domain in the mined association rules.

On the other hand, the search of association rules followingthis new fuzzy approach becomes a little bit different becausenow counting must consider the different membership degreesof objects. Fuzzy Association rules is a relatively new datamining algorithm. References [4] [5] introduce the conceptof Fuzzy Association Rules and present some applications onrelational databases. Reference [6] develops this algorithm in amultidimensional databases model and reference [7] apply thismethodology to search possible correlations between lightningparameters and some geographical and meteorological char­acteristics. In this context, with the purpose of evaluating the

C. Data mining Association Analysis on power market data­bases.

Broadly speaking, data mining consists in a large collectionof algorithms to extract or "mining" knowledge that mightbe interesting and innovative to the analyst. Among all thepossible analysis that can be achieved with these algorithms,there is one that may result interesting in power marketdatabases: association analysis. In general terms, associationanalysis is the discovery of association rules showing somedata attributes that occur frequently together in a given setof data [2]. In this paper this algorithm is used with thepurpose of quantifying several hypothesis about the effect ofsome variables over bidding and spot prices. In particular, thisalgorithm is used to quantify the confidence about the effect ofhydrological conditions on bidding and spot prices in a marketstrongly dependent on Hydro plants(the colombian case).

These association rules are of the form X =} Y in such away that hypothesis to be verified may be also represented byrules. In our case, rules may be in the form:

IF Hydrology is High =} Spot Price is LowIF Hydrology is High =} Bidding Price is Low

In short, the search for significative rules is based onthe estimation of two basic parameters called support andconfidence. The support (Sup) of a rule in the form X =} ymeasures the frequency of this rule on the data set, and theconfidence (Con f) of X =} Y measures the conditionalprobability of a consequent given a particular antecedent.Formalizing these relations:

(4)

(5)

Sup(X =} Y) = P(X U Y)

Conf(X =} Y) = P(Y!X)

(3)

Bidding functions represented in the form of equation 3,might be interpreted as points in a u-dimensional space, whereeach dimension represents a bidding price of a particular plantand where a clustering algorithm may be applied with theaim of finding certain regularities over this new space. It isimportant to highlight that this approach allows to summarizein a single point over a multidimensional space, the strategicbidding behavior of an agent. Even more, by following thisnew approach, bidding behavior is not considered as an

B. Clustering methods over the space of bidding prices.

The application of k-means algorithm over the continuousdomain of bidding prices starts from a suitable representationof bidding functions as points in a u-dimensional space, whereu corresponds to the number of generating plants that eachGENCO owns.

Formalizing, an agent a might own n generating plants ofq different technologies, and might assign a distinct biddingprice Pij to each one of the plants in his portfolio. Now,bidding functions (Ca ) of agent a must be a function ofbidding prices as follows:

Ca == f(Pij) where i E [1, n] y j E [1, q] (2)

It is possible to think that bidding functions also dependon quantities (MW) that are declared available to the SystemOperator (ISO). However, in the case studied here it is assumedthat agents have no incentives to declare themselves unavail­able. This assumption is based on the behavior exhibited byGENCOs in colombian market data, showing that there is nota significant variation in plants availability. On the contrary, asa result of the Price-Based Unit Commitment (PBUC) schemeof colombian power market, bidding prices exhibit a greatdynamics.

On the other hand, a meaningful difference is commonlyobserved among bidding prices of different generation tech­nologies, probably explained by the distinctive operationalcosts associated to thermal and Hydro technologies. Thiscondition might cause that the application of clustering al­gorithms such as kmeans results in fake patterns that falls ina space between bidding prices of different technologies. Toavoid these unexpected results, a single bidding function isconstructed for each generation technology in the portfolioof agents. Following this approach, a bidding function foreach technology (Caj) is represented by a vector of lengthu, formed by bidding prices Pij of the u generating plantsthat belongs to a particular technology j. Thus,

Algorithm 1 Clustering algorithm k-means

Finally, this simple clustering algorithm follows 5 steps thatare shown in algorithm 1.

1: Initialize k II Number of partitions2: Select randomly k objects as initial centroids3: (Re)assign each object to a cluster whose centroid is nearest.4: (Re)calculate centroids in each cluster no change in centroids is

observed

""Plant 1

~Plant 2

800 1000Declared Availability (MW)

\Plant 3

Fig. 1: Type of fuzzy sets considered in the mined association rulesFig. 2: An example of a Hydro bidding function of a colombain GENCO

mined fuzzy association rules a new parameter called certaintyfactor is defined. The certainty factor takes values in [1,-1]and tries to measure not only the presence of the elements ina rule, but also the absence. To illustrate this situation it ispossible to evaluate a rule in the form:

I F Hydrology is Low THEN Bidding Price is High

but also we may ask about the rule that imply the absence ofthe consequent and the antecedent as:

space of this figure, representing that some bidding strategiesare used more often than others in a distinctive manner.Precisely, proper identification of these clusters is the maingoal of k-means algorithm and even more, the centroids ofthese clusters may be considered as the bidding patterns of theagent in consideration. Therefore, as a result of the applicationof k-means algorithm, k bidding patterns are obtained foreach type of plants portfolio (Thermal, Hydro, Nuclear) ofthe agent.

Fig. 3: Representation of bidding functions in a n-dimensional space for aparticular agent

To illustrate the application of k-means algorithm, theobtained clusters for the agent in figure 3 are shown in differentcolors in figure 4 for a value of k == 6.

Bidding Price Plant 2 ($/KWh)

..as

30 40

..... .~~

Soma typicalclusterlngsara

eKhltlted •

Each point In this spacarapresantsablddlngfunctlOhovarthahydroplantsportfollo-............. x- .. "

(Bidding function In FIgure 1) -.............. ~::;;:

Bidding Price Plant 1 ($/KWh)

3070

IF Hydrology is not Low THEN Bidding Price is not High?

In this sense, the rule must be considered very strong if bothrules are also strong. On the other hand, the sign of certaintyfactor explain the dependence relation between antecedent andconsequent. So, it is positive when the dependence betweenand is positive, 0 when there is independence, and a negativevalue when the dependence is negative [4]. In other words, apositive certainty factor means that the presence of antecedentimplies the presence of consequent, and on the contrary, anegative certainty factor means that the presence of antecedentimplies the absence of consequent.

Finally, this data ming algorithm was applied to colombianpower market database in order to search for associationrules between Hydrology and Bidding prices. The values ofconfidence, support and certainty factors are used to quantifythe confidence and truthfulness of some hypothesis about theeffect of hydrology on bidding prices.

As it was mentioned before, the methodology was applied to

Fig. 4: Application of k-means algorithm over a three-dimensional space ofbidding functions

lidding Price. P1..,t 1 ($/KWh)

20 08Iddlng PrIces Plant 2(SIKWh)

..~-\'.' ._.• ~..•.~-: ::~_.!e•.e. ~_ •. ••

t;rM'\,8 a a ..

':; -=. t ••100 • a ••

80eo

III. AN EXAMPLE OF APPLICATION

The above described methodology was applied to ten of themost representative GENeOs of the colombian power market.With the purpose of illustrating the proposed methodology, aparticular bidding of one of the analyzed agents is shown infigure 2. This bidding function is constructed based on biddingprices of three hydro plants portfolio of this particular agentwhich can be represented as a point in a three-dimensionalspace. This particular point representing the bidding functionin figure 2 is shown in figure 3.

On the other hand, the rest ofpoints are also a representationof different bidding functions, capturing the behavior of thisparticular agent in the period from January to Febuary of 2004.In addition, some typical clusters may be identified in the

ten of the most representative agents of the colombian powermarket in a sequence of steps that are shown in figure 5 andmay be summarized in four steps as follows:

1) Achieve SQL queries about bidding prices of all agentsportfolios on a new market database. A new colombianpower market database was constructed with the purposeof organizing and combining several sources of informa­tion about this power market. This new database allowsto correlate essential information about agents that iscommonly dispersed.

2) Classify bidding prices by different technologies in orderto obtain the desired bidding functions.

3) Apply k-means algorithm in an iterative manner. In thiscase, an iteration ends when a partioning of the spacein k clusters is obtained. However, this application isiterative in the sense of progressively increasing thenumber k of clusters until any stop criterion is reached.In the proposed methodology, this stop criterion consistin a inter-centroids distance, which is calculated on eachiteration until 10% of these inter-centroids distances areless than 10% of the maximum inter-centroids distancecalculated in the current iteration. The underlying ideaof this criterion is to search for a number of k partitionsover the space, so that centroids are not close enough tobe considered as a single cluster.

4) If the stop criterion is fulfilled, current centroids aretaken as the k bidding function patterns.

For each agentk patterns of bidding

functions are obtained

Fig. 5: Flow chart of the application of k-means to obtain bidding patternfunctions

IV. RESULTS

A. Clustering over bidding prices space

As it was mentioned before, the proposed clustering algo­rithm was applied to ten of the most representative GENCOs

4

Strategy Plant Plant Plant Plant Plant Plant1 2 3 4 5 6

60.8 70.9 57.3 60.2 71.5 471.3

2 47.1 281.2 118.8 43.9 258.0 482.5

3 42.8 37.6 37.4 38.1 39.0 52.4

4 55.2 57.3 46.6 54.6 56.9 80.4

TABLE I: Details of bidding function patterns for an hydro plant portfolio.*All prices are in ($/KWh) at constant prices of December-2004.

in the colombian market. As a result, each GENCO wascharacterized by k bidding pattern functions that must beinterpreted as the most representative and distinctive biddingstrategies of the agent considered. An example of these biddingpatterns is shown in table I for an Hydro plant portfolio of oneof the most complex GENCOs in the colombian power market.

On the other hand, it is important to notice that agents witha diversified portfolio, may offer a combination of biddingfunctions by technology. This type of strategies are consideredmixed strategies depending on the number of bidding patternsfound by technology. However, one of the most interestingresults shows that not necessarily all the possible combinationsof mixed strategies are used by agents and also, not all thestrategies are used with the same frequency. This result showsthat it is possible to establish an ordering or dominance overthe strategies space in terms of their frequency ofuse. To illus­trate this dominance relation it is shown in table II for one ofthe most complex GENCO of colombian market a descriptionof used strategies and their frequency of use. First of all, it isimportant to say that in spite of the diversity of its portfolio,a big reduction is achieved over the continuous domain ofbidding prices (16 pattern strategies for this particular agent).Second, not all the possible combinations are used Gust 8 of 16possible mixed strategies are used). Third, a set of 4 strategiesare employed most of the time (90% of time). Therefore,from a infinite set of possibilities in the continous domainof bidding prices, it is possible to model the bidding behaviorof this agent by 4 strategies, which is an exceptional reductionover the bidding prices space. Finally, table III summarize the

Mixed Hydro Thermal FrequencyStrategy Portfolio Portfolio

1

2 4 3 5

3 2 4 26

4 4 2 32

5 3 55

6 3 2 207

7 4 289

8 4 370

TABLE II: Mixed strategies for agents with diversified portfolios

mentioned reduction describing the number of bidding patternstrategies over the possible, the used and the dominant spacewhere in some cases this reduction may take values around50 %.'

GENCO No. Possible No. Used No. DominantStrategies Strategies Strategies (90%

)

EPM 16 8 4

ISAGEN 24 24 14

EMGESA 30 25 15

EPSA 24 22 12

BETANIA 4 4

CHIVOR

EBSA 4

T/JERO 4 4 4

URRA

CORELCA 6 4

TABLE III: Comparison between possible and dominant strategies for theagents considered in the colombian power market

B. Fuzzy Association Rules

A data mining algorithm based on fuzzy association ruleswas applied to colombian power market database in order toquantify the cOl1:fidence and truthfulness of some hypothesisabout the effect of hydrology on bidding prices. In particu­lar, this hypothesis has been traditionally formulated in thecolombian power market due to the great dependence of thesystem on hydro plants. Historically, it has been consideredthat exists an inverse relation between bidding prices andhydrology, probably supported in the distinctive operationalcosts associated to thermal and hydro technologies.

In this section, the parameters of confidence, support andcertainty factors are used to quantify the veracity of thishypothesis by two different approaches. The first approachconsists in analyze this hypothesis in an aggregated form,'mining" the relation between daily spot price and nationalbidding reservoir. The second approach analyze this hypothesisin a disaggregated form, "mining" the relation between biddingprices for a particular plant and its bidding reservoir. As usual,this hypothesis is formulated in the form of a rule X =} Y.

1) Spot prices and national bidding reservoir: The resultsof this relation were obtained with the tool Fuzzy Text DataMiner developed in reference [8]. The results are shown intable IV. Given that in association analysis the first choice is toselect rules that exhibit high values of support, these rules areahighlighted in blue color. As it was mentioned before these arethe rules that are candidates to be strong rules. Therefore, inthis subset some important rules has been disregarded. Forinstance, the rule bidreservoirlow =} pricehigh has lowvalues of support. However, it results interesting to observethe confidence and certainty factor values for this rule, whichare also low (24.23% and 0.15, respectively). This valuesmay be interpreted as a low confidence in the belief thatnecessarily low national bidding reservoirs implies high spotprices. Hence, this information quantified in that form maybecome an important tool in the decision making process ofGENCOs.

On the contrary, focusing on rules with high support, it canbe seen that some rules also exhibit a high value of confidence,which are highlighted in red color. Regarding this subset, tworules seems to be interesting.

The first rule is bidreservoirmedium =} pricemediumwith confidence and certainty factor values of 53.48% and0.17, respectively. At first sight, seems that exists a probabilityaround 50% that being the national bidding reservoir in amedium level, the spot prices are also medium. Furthermore,checking the certainty factor values, seems that there is a weakrlation that implies the presence of both, medium prices andmedium reservoirs at the same time.

The second rule is bidreservoirhigh =} pricelow withconfidence and certainty factor values of 70.05% and 0.45,respectively. Apparently, this rule is the strongest rule mined,implying that exists a probability of 70% that being thenational bidding reservoir in a high level, the spot prices arelow. However, certainty factors are not as high as it can beexpected (0.45), what may suggest that not necessarily thepresence of high national bidding reservoir levels implies thepresence of low spot prices. This values may be supported inthe data behavior shown in figure 6 where most of spot pricesmay also be medium when national reservoirs are low. To sumup, despite the high levels of confidence, this rule is not asstrong as has been historically stated. Again, the main idea isthat by using an association analysis, the truthfulness of thishypothesis may be quantified, which is a huge contribution tomarket data analysis.

To conclude, it seems to exist an asymmetry in the hy­pothesis historically formulated about the effect of hydrologyin spot prices. Apparently, there is more confidence in thebelief that high bidding reservoir levels implies low spot pricesthat in the belief that spot prices are high when biddingreservoirs are low. These asymmetries found by applyingthe proposed methodology give more information about thebidding behavior of agents in power markets.

Rule Sup (%) Conf (0,10) CF

bidreservoirlow => pricelow 1.82 7.15 -0.84

pricelow => bidreservoirlow 1.82 2.92 -0.81

bidT'e-.'H;" no;,. me-dill rn => price!olo 14.37 29.09 -0.36

p"';celollJ =} bid,.e8t>'T'voil"mediurn 14.37 25.89 -0.30

bid,.e8t ',·poi,.high =} p,.iedoll' 34.84 70.05 0.45

p,.ice[o'w =} bidn;8eT'I'oirhigh 34.84 74.47 0.51

bidn~8el'l'oi,.zo/l.' =} p,.icemediu..", 11.90 65.54 0.38

pf'iceTHnlilllTl =} bidre.w'l' l.'oi T'l01l' 11.90 21.86 0.08

bid f'e8cf'poif'rnedi II rn =} price.", edi mn 24.11 53.48 0.17

p',.icf'rnedilllTl =} bidf'e8f'fPO;,.rnerlill1n 24.11 44.40 0.12

bid'l't'8ef'l'oiT'high =} p,.icernuLi 11m 17.10 30.76 -0.31

p,.iCf;merlilUlI =} bid,.e~H;r·I'oi,.high 17.10 31.72 -0.33

bidreservoirlow => pricehigh 5.36 24.23 0.15

pricehigh => bidreservoirlow 5.36 33.09 0.21

bidreservoirmedium => pricehigh 7.30 18.84 0.09

pricehigh => bidreservoirmedium 7.30 65.55 0.45

bidreservoirhigh => pricehigh 0.87 1.35 -0.87

pricehigh => bidreservoirhigh 0.87 3.72 -0.92

TABLE IV: Resulting Fuzzy Association Rules for daily spot prices andnational bidding reservoir

Highdlspe,sion

./lntll'V81 140-80SlKWhl

0.5 0.55NATIONAL BIDDING RESERVOIR (%)

Fig. 6: Relation between National Bidding reservoir and daily spot prices

2) Bidding prices and Bidding reservoirs: In like manner,fuzzy association rules were used to "mining" the disaggre­gated relation between Bidding prices and Bidding reservoirs.The results are shown in table V. It is important to distinguishthat, in this case, the bidding reservoir corresponds to aparticular plant as well as bidding prices. The results shown intable V belongs to the bidding behavior of one of the biggesthydro plants of Colombia.

0.5 0.6BIDDING RESERVOIR ("to)

Fig. 7: Relation between Bidding reservoir and bidding prices for a majorhydro plant in the colombian market

In general terms, results of table V show that there are noclearly distinguishable rules, evidenced in the lower values ofconfidence and support. However, within the rules that exhibitsthe higher values of support and confidence (but not highenough), two rules seems to be interesting.

The first rule is bidreservoirhigh => pricelow withconfidence and certainty factor values of 41.5% and 0.37, re­spectively. The second rule is bidreservoirhigh =} pricelowwith confidence and certainty factor values of 34.3% and 0.27,respectively. In conclusion, seems to be that the hypothesisabout the effect of bidding reservoirs on bidding prices isreally weak, finding, at most, values of confidence around42%. This quantification may suggest that at a disaggregatedlevel by plants, the agents has other incentives different thanhydrological conditions as it can be supported by the greatdispersion of data shown in figure 7.

Y. CONCLUSIONS

The main contribution of this paper is a methodology toreduce the search space over the continuous domain of bidding

Rule Sup(%) Conf(%) CF

bidreservoirlow =} pricelow 1.41 5.40 -0.72

pricelow => bidreservoirlow 1.41 3.12 -0.73

bid I't'S£: 'T'/loi T' TTl ed ill TTl => pl'i('doll' 12.18 25.40 -0.24

prin loU' => bidT't'.'U-'I'/'oinnf'di lilT/ 12.18 26.40 -0.28

hid n' 8 fT /.' 0 i I'll i g 11 => pricf:'!o/ll 26.70 41.50 037

p'l'i('f:'lo/ll => bidn Sf f'/:oil'high 26.70 4320 0.35

bidT't'st'T'/'oirlu/L' => prinTnf'diuTTI 13.10 32.30 0.35

pT i(,f'moli tl TTl => bidT'f:St'T'/'oirlolL' 13.10 23.50 0.09

bidre,'·wrl'oir",ediuIT/ => priCf'",ediuTn 26.40 3430 027

pl'i('£: ",p(liuTn => bidns£:'T'PoiI'TTlediuTTI 26.40 41.40 o IS

bidn'8f.l'uoil'high => pri('fTTlt-:di urn 15.60 33.10 -0.33

pricunediurn => bidl'eStTUoirlligh 15.60 36.20 -0.36

bidreservoirlow => pricehigh 6.20 27.10 0.21

pricehigh => bidreservoirlow 6.20 35.60 0.24

bidreservoirmedium =} pricehigh 8.10 17.40 0.03

pricehigh =} bidreservoirmedium 8.10 53.20 0.52

bidreservoirhigh => pricehigh 1.40 1.35 -0.89

pricehigh => bidreservoirhigh 1.40 3.72 -0.76

TABLE V: Resulting Fuzzy Association Rules for bidding prices and biddingreservoir

prices. The proposed representation of bidding functions aspoints in a multidimensional space, allows to find biddingpatterns by applying a clustering algorithm. As a result, adramatic reduction over the search space of bidding strategiesis achieved. This approach allows to summarize in a singlepoint over a multidimensional space, the strategic biddingbehavior of an agent. In addition, this bidding behavior is nolonger considered as an isolated process for each generatingplant, but as a combined bidding strategy over its diversifiedportfolio.

On the other hand, some relations of dominance overbidding strategies are found, by establishing an ordering interms of the frequency of use of these patterns. This orderingover bidding strategies may be the first step in the constructionof some analytical models supported in empiric evidence (p.e.non-cooperative games with mixed strategies). Even more, thispatterns may be used in a multi-agent model of power marketsthat involves learning capabilities of the agents, where theproblem of dimensionality is always present [9].

Furthermore, an application of a data mining algorithmbased on fuzzy association rules is presented. In this regard,the main contribution consists in a methodology to quantifythe confidence and truthfulness of some hypothesis aboutthe effect of hydrology on bidding prices. In particular, thismethodology allows to analyze rules over a partitioned domainof the variables involved. Consequently, some asymmetriesin these hypothesis may arise from the association analysis,which gives more significance to the relations studied. Inparticular, the relation between spot prices and national hy­drology exhibits an apparent asymmetry implying that thereis more confidence in the belief that high bidding reservoirlevels implies low spot prices that in the belief that spotprices are high when bidding reservoirs are low. In addition,

the association analysis in a disaggregated level, shows thatthe hypothesis historically stated about the inverse relation ofbidding prices and bidding reservoir levels is too weak in thecolombian case.

ACKNOWLEDGMENT

The authors would like to thank to the research group PAASUN of National University of Colombia and National ScienceAgency of Colombia COLCIENCIAS for the supporting ofthis work, as well as XM SA, the Colombian IndependentOperator, and UPME for providing the historical market dataand the colombian power system data, respectively.

REFERENCES

[1] G. Seber, Multivariate Observations. Wiley, 1984.[2] J. Han, Data Mining: Concepts and Techniques. Morgan Kaufmann

Publishers, 2001.[3] L. Zadeh, "Fuzzy sets," Information and control, vol. 8, pp. 338-353,

1965.[4] D. Sanchez and M. Delgado, "Fuzzy association rules: General model

and applications," IEEE Transactions on Fuzzy Systems, vol. 11, no. 2,pp. 214-225, April 2003.

[5] D. Sanchez, "Adquisici6n de relaciones entre atributos en bases de datosrelacionales," Ph.D. dissertation, Department of Computer Science andArtificial Inteligence University of Granada Spain, 1999.

[6] C. Molina, "Imprecision e incertidumbre en el modelo multidimensional:Aplicaci6n a la mineria de datos," Ph.D. dissertation, Universidad deGranada, Julio 2005.

[7] C. Younes and O. Duarte, "Metodologias para la correlacion depanimetros del rayo con caracteristicas geognificas y meteorologicas.caso colombiano," Ph.D. dissertation, Universidad Nacional de Colombia,Junio 2006.

[8] 1. Serrano and M. Vila, "Mining web documents to find additional queryterms using fuzzy association rules," Fuzzy Sets and Systems, vol. 148,no. 1, pp. 85-104, November 2004.

[9] L. Gallego and A. Delgadillo, "Agent learning methodology for generatorsin an electricity market," IEEE Power Engineering Society. IEEE PESGeneral Meeting, July 2008.