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330 IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 17, NO. 2, MAY 2002

Comparison Between Closed-Loop and PartialOpen-Loop Feedback Control Policies in Long Term

Hydrothermal SchedulingL. Martinez and S. Soares

Abstract—Stochastic dynamic programming has been ex-tensively used in the optimization of long term hydrothermalscheduling problems due to its ability to cope with the nonlinearand stochastic characteristics of such problems, and the fact that itprovides a closed-loop feedback control policy. Its computationalrequirements, however, tend to be heavy even for systems with asmall number of hydroplants, requiring some sort of modelingmanipulation in order to be able to handle real systems. An alter-native to closed-loop optimization is an approach that combinesa deterministic optimization model with an inflow forecastingmodel in a partial open-loop feedback control framework. Ateach stage in this control policy, a forecast of the inflows duringthe period of planning is made, and an operational decision forthe following stage is obtained by a deterministic optimizationmodel. The present paper compares such closed-loop and partialopen-loop feedback control policies in long term hydrothermalscheduling, using a single hydroplant system as a case study tofocus the comparison on the feedback control performance. Thecomparison is made by simulation using data from historical andsynthetical inflow sequences in the consideration of three differentBrazilian hydroplants located in different river basins. Resultshave demonstrated that the performance of the partial open-loopfeedback control policy is similar to that of the closed-loop controlpolicy, and is even superior in dry streamflow periods.

Index Terms—Dynamic programming, hydroelectric powergeneration, nonlinear programming, power generation sched-uling, stochastic optimal control.

NOTATION

Index of month.Number of months in the planning period.Non-hydraulic cost function (in dollars).Hydroelectric generation function (in megawattmonths).Load demand (in megawatt months).Future operational cost (in dollars).Water storage in the reservoir (in cubic hectome-ters).Bounds on minimum and maximum reservoirstorage in the reservoir.Water discharge through the turbines (in cubic hec-tometers).

Manuscript received March 6, 2001; revised October 26, 2001. This workwas supported by the São Paulo State Foundation for the Support of Research(FAPESP) and the Brazilian National Research Council (CNPq).

The authors are with the Department of Engineering Systems, School of Elec-trical and Computer Engineering, State University of Campinas, Sao Paulo,Brazil (e-mail [email protected]; [email protected]).

Publisher Item Identifier S 0885-8950(02)03827-0.

Bounds on minimum and maximum water dischargethrough the turbines.Water spillage from the reservoir (in cubic hectome-ters).Constant factor (in MW/hm/month/m).Forebay elevation function (in meters).Tailrace elevation function (in meters).Average penstock head loss (in meters).Incremental water inflow (in cubic hectometers).Expected value.Minimum expected operational cost (in dollars).Probability density function.Large positive constant.Coefficients of the terms with the exponentof theforebay elevation function.Coefficients of the terms with the exponentof thetailrace elevation function.Month.Time [ , with being theyear and the month].Expected value.Standard deviation.Autocorrelation coefficient.Sequence of uncorrelated random variables.

I. INTRODUCTION

L ONG term hydrothermal scheduling is a complex problemdue to various aspects of the modeling involved, including

the randomness of inflows into the hydroplants, the interconnec-tion of hydroplants located in a cascade, and the nonlinearity ofhydro production and thermal cost functions.

Dynamic programming (DP) [1] has been extensively used inthe optimization of hydrothermal scheduling problems in par-ticular, and water resource systems in general. The popularityand success of this technique can be attributed to the fact thatthe nonlinear and stochastic features of such problems can beadequately handled by a DP formulation [2]. Moreover, the DPapproach is a closed-loop control policy designed for the obtain-ment of, not optimal numerical values for the decision variables,but rather an optimal rule for selecting, at each stage, the optimaldecision for each possible state of the system [3].

The usefulness of the DP technique, however, is limited by theso-called “course of dimensionality,” since the computationalburden increases exponentially with the number of state vari-ables. Various approaches have been suggested to overcome the

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MARTINEZ AND SOARES COMPARISON BETWEEN CLOSED-LOOP AND PARTIAL OPEN-LOOP FEEDBACK CONTROL POLICIES 331

problem of dimensionality, including the aggregation of the hy-droelectric system through a composite representation [4]–[9]and the use of stochastic dual dynamic programming, based onBender’s decomposition [10]–[12].

An alternative approach to such closed-loop optimizationcombines a deterministic optimization of the hydrothermalscheduling problem with inflows furnished by a forecastingmodel. At each stage of the planning period this feedbackcontrol scheme determines an optimal decision based on thecurrent forecast of future values, and this decision is utilizeduntil a new forecast becomes available, this again based on thelatest available information in the system. When this new fore-cast becomes available, a new optimal decision is determinedwithin the framework of partial open-loop feedback control [3].

This partial open-loop approach is based on a deterministicoptimization model [13]–[15], thus permitting the represen-tation of the hydro system in detail, with the considerationof each hydro plant individually, including its operationalconstraints and nonlinear production characteristics. Moreover,the stochastic model considered for the representation of theinflows can be quite general, based on any methodology, andspecific for each hydro plant in the system. But the main featureof this approach is that it can be applied without simplificationsto multiple hydro plant systems.

Evaluations of the adequacy of the partial open-loop feed-back control policy for the hydroelectric systems of Turkey [16],New Zealand [17], and Brazil [18] have been carried out. Thefirst of these showed that the reservoir trajectory resulting fromsuccessive updates of forecasted inflows was similar to that ob-tained when assuming perfect foresight of the inflows duringthe planning period, thus indicating an efficient performanceof the partial open-loop approach. The second study, using theNew Zealand system, again obtained results not differing signif-icantly from those obtained by stochastic DP. A different con-clusion was reached for Brazil [18], suggesting that the perfor-mance of the partial open-loop approach might be dependenton the specific hydro system considered. This work, however,did not constitute a complete comparison, since the open-loopfeedback control scheme was implemented through the samestochastic dynamic programming model, where the inflow ineach stage is represented by its expected value instead of by itsconditional probability distribution.

The goal of the present paper is to compare these closed-loopand partial open-loop feedback control policies in long termhydrothermal scheduling, using the Brazilian system as a casestudy. The comparison involves systems composed of a singlehydro plant since the idea is to focus on the effectiveness ofthe different feedback control policies in coping with the ran-domness of inflow. Therefore, the modeling manipulations usu-ally necessary to implement a closed-loop approach for multiplehydro plant systems is not required.

The comparison was made for three hydro plants, each lo-cated in a separate Brazilian river basin. These plants were se-lected because they have large plurianual reservoirs and are lo-cated in the upstream part of the river basins, thus being espe-cially important hydro plants in the operation of the cascades towhich they belong.

A lag-one parametric autoregressive model, PAR(1), was ad-justed to the historical inflow records and used in the feedbackcontrol policies, both that of the closed-loop control policyfor providing the conditional probability density functions,and that of the partial open-loop one for providing the inflowforecasting. The approaches were then compared by simulationusing historical and synthetical inflow sequences generated bythe same PAR(1) model. Deterministic optimization assumingperfect foresight of inflow during the period of planning wasalso considered. The effectiveness of the approaches was mea-sured using the mean and standard deviation values for hydrogeneration and operational costs during the planning period.

The paper is structured as follows. Section II presentsthe formulation of the long term hydrothermal schedulingproblem. Sections III and IV present the closed-loop andpartial open-loop feedback control policies, respectively.Section V presents the comparison between the approaches andSection VI presents the conclusions of the study.

II. PROBLEM FORMULATION

The deterministic version of the long term hydrothermalscheduling of a single hydroelectric plant can be formulated asthe following nonlinear programming problem

(1)

subject to

(2)

(3)

(4)

(5)

(6)

given

(7)

The objective function (1) is composed of two terms whichrepresent the operational costs during the planning period andthe future costs associated with the final storage in the reservoir.The operational cost represents the minimum cost from com-plementary nonhydraulic sources such as thermoelectric gener-ation, imports from neighboring systems, and load shortage. Asa consequence of this minimization, is a convex decreasingfunction of the hydro generation and depends on the systemload demand ; the function is a terminal conditionwhich represents future operational cost as a function of the finalreservoir storage. This term is essential for the equilibrium be-tween the use of water during the planning period and its useafterwards.

Hydro generation in stage is a nonlinear function repre-sented by (2), where is the water storage in the reservoir,

is the water discharge through the turbines, andthe waterspillage from the reservoir. The constantis the product of


water density, gravity acceleration and average turbine/gener-ator efficiency, is the forebay elevation as a function ofwater storage, is the tailrace elevation as a function of totalwater release, and is the average penstock head loss.

The equality constraints in (3) represent the water balance inthe reservoir at each stage, where is the incremental waterinflow. Other terms such as evaporation and infiltration have notbeen considered for the sake of simplicity.

Lower and upper bounds on variables, expressed by con-straints (4)–(6), are imposed by the physical operationalconstraints of the hydro plant, as well as other constraintsassociated with the multiple uses of water, such as irrigation,navigation, and flood control.

In the following sections, the closed-loop and partialopen-loop feedback control policies in the long term hy-drothermal scheduling problem (1)–(7) are presented.

III. CLOSED-LOOPFEEDBACK CONTROL

Closed-loop optimization is the central characteristic of thestochastic dynamic programming technique. The goal of theclosed-loop feedback control (CLFC) policy is to determine arule for decision-making at each stage of the planning periodwhich provides the optimal decision for each possible state ofthe system. Mathematically, the CLFC finds a sequence of de-cision functions mapping the states into decisions so as to min-imize the expected costs.

In some applications, the system state is constituted only bythe storage variable, which is the case when the stochastic vari-able is considered independent in time. In other situations, how-ever, when the stochastic variable is modeled by autoregressivemodels, the system state must be increased to include the waterinflows from previous stages in order to represent the time de-pendence of the inflows, a procedure which makes the “courseof dimensionality” even more crucial to this approach.

It is assumed, in this paper, that the stochastic variable rep-resenting the inflow in stagedepends only on the inflow fromthe previous stage . This means that the inflows are repre-sented by a PAR(1) model, describing the stochastic process ofthe hydrologic variable as a Markov chain [3]. For reservoiroperation, the state variables are the water stored in the reservoirat the beginning of each stage and the water inflow duringthe previous stage which represents the hydrological trend.The control variables are the amount of water dischargedandspilled from the reservoir during the time stage. The longterm hydrothermal scheduling problem, in its stochastic version,can be formulated as

(8)

subject to the constraints in (2)–(6), where is theexpected value with respect to the inflow during stagecondi-tioned by the inflow during stage .

At each stage, decisions are ranked based on the minimiza-tion of the sum of the present cost plus that of the expected fu-ture cost, assuming optimal decision-making for all subsequentstages. This cost function is additive in the sense that the cost in-

curred at time accumulates over time. According to Bellman’sOptimality Principle [1], the optimal decision is obtained bysolving the following recursive equation

(9)

with

where/subject to (2)–(6);

represents the minimum expected operationalcost from stage till the end of the planningperiod , assuming that the system is at thestate ( );probability density function of the inflow instage conditioned by the inflow in the pre-vious stage .

The resolution of (9) requires the discretization of the stateand control variables and the conditioned probability densityfunction of the inflows, which leads to the “course of dimen-sionality” in DP, as already commented.

IV. PARTIAL OPEN-LOOPFEEDBACK CONTROL

In the partial open-loop feedback control (OLFC) approach,the randomness of inflows is implicit when stochastic variablesare assigned to their expected values, which are provided byinflow forecasting models. The deterministic version of theproblem is then solved by a deterministic optimization model,and the optimal decision variable associated with the first stageis then implemented. In order to avoid error propagation, thescheme must be repeated at each stage throughout the planningperiod.

One major issue in the design of the OLFC approach is the ter-minal condition of the deterministic optimization model.The terminal condition is known to establish a trade-off betweenthe benefits associated with the use of water for hydro genera-tion during the planning period and the expectation of futurebenefits deriving from storage at the end of the planning period,both measured in terms of nonhydraulic generation economy.Therefore, the terminal condition is a critical aspect of the OLFCapproach and has a great influence on the overall performance;moreover, an adequate terminal condition is a necessity for im-plementing efficient OLFC approaches.

One way of overcoming the problem of obtaining a properterminal condition is to extend the end of the planning period sothat the influence of on the decision during the first stagebecomes negligible. This turns out to be rather inconvenient,however, since the extension of the planning period increasesthe errors in inflow forecasting, thus reducing the performanceof the OLFC approach.

On the other hand, establishing a shorter planning period sothat the forecasting model will be able to improve its perfor-mance would require a precise estimation of the expected future


Fig. 1. Water level curve for reservoir storage in end of April.

operational cost, since in this case the influence of the terminalcondition on the decision of the first stage is crucial.

To resolve the conflict between the terminal condition and theoptimization horizon in the OLFC design, it is suggested herethat the reservoir should be as full as possible at the beginningof the dry season. This decision is supported by the fact that theoptimal deterministic solution for the historical inflow recordsindicates that the storage of the hydro plant should almost al-ways be at maximum level at the beginning of the dry season.Indeed, the deterministic optimization based on historical in-flow records (1931 to 2000) for the three Brazilian hydro plantsconsidered in this paper, shows that the reservoirs are generallyfull at the end of April, the beginning of the dry season, as canbe seen in the duration curves of reservoir storage in Fig. 1.

Based on these results, the OLFC approach proposed herewill try to maintain the storage reservoir of the hydro plant full atthe beginning of May, which is the beginning of the dry season.For this reason, the optimization horizon should be variablethroughout the planning period so that the final stage will al-ways be April. For example, the OLFC decision for the month ofSeptember is obtained by optimizing the operation for a periodof eight months, whereas in the month of January, the OLFCdecision is obtained by optimizing the operation for a period offour months. The average optimization horizon in this schemewill be six months, which is a reasonable period for inflow fore-casting.

Assuming that represents the next month of April, the so-lution of the problem (1)–(7) is obtained by considering the ter-minal condition , where is a positiveconstant large enough to ensure that the terminal condition pre-vails over the remaining objective function.

In this paper, the deterministic solution of the optimizationproblem is obtained by a nonlinear network flow algorithm spe-cially developed for hydrothermal scheduling [15] and the in-flow forecasting model is the PAR(1) model.

V. TEST RESULTS

This section provides a comparative analysis of CLFC andOLFC policies in long term hydrothermal scheduling throughsimulation. The two approaches have been applied to the spe-cific case of systems comprising a single hydro plant.

The three different hydro plants located in different Brazilianriver basins selected for the study were Furnas, located on the

TABLE IHYDRO PLANT CHARACTERISTICS

TABLE IIHYDRO GENERATION CHARACTERISTICS

Grande river, Emborcação on the Paranaiba river, and Sobrad-inho on the São Francisco river. The main operational charac-teristics of these hydro plants are given in Table I.

As is standard for planning studies in the Brazilian powersystem, the forebay elevation and the tailrace elevation

are fitted by fourth degree polynomial functions of thewater storage and discharge in the reservoir, whereand

are the coefficients of the terms with the exponentofeach polynomial function, respectively. Table II gives thepolynomial coefficients, the value of the constant, given in(MW/(hm /month)m), and the average penstock head loss, inmeters, respectively, for each hydro plant considered.

The operational cost is, in general, obtained by the op-timal dispatch of the nonhydraulic sources available. Optimiza-tion ranks these sources according to their marginal costs, whichresults in a convex increasing operational cost function. For thenonhydraulic aspects of the Brazilian system, an estimate of theoperational cost is given by the following quadratic function

(10)

The load demand in (10) was considered both constantduring the planning period and equal to the installed capacity ofthe hydro plant. This assures a balanced hydrothermal systemsince, for these three hydro plants, the firm energy is approxi-mately 50% of the installed capacity.

These assumptions (load demand and operational cost), al-though arbitrary, do not change the essential nature of the con-clusions, since optimal scheduling tries to distribute the hydrogeneration throughout the planning period in order to equalizethe marginal operational costs, whatever they may be. As it isshown in [19], different forms of composite thermal marginalcosts will lead to the same hydro production scheduling, beingsufficient to consider a linear thermal marginal cost as suggested


in (10). A linear hydro marginal profit would also be sufficientfor obtaining the optimal hydro production scheduling, whichmeans that the control policies here compared would be ap-propriated for owners of independent hydro plants under bothderegulation and competition environment.

For both control policies analyzed, the coordination betweenlong and short term hydrothermal scheduling is established bygeneration targets imposed to each hydro plant in the short termhorizon. These targets are the hydro generation decisions of thelong term model for the first time interval.

The comparison between CLFC and OLFC policies has beenmade using a simulation model which reproduces the behaviorof the hydrothermal system. This simulation model provides theresponse of the system for specific inflow sequences, accordingto the control policy adopted, thus allowing the comparison ofthe two techniques for the same computational environment.

Two different scenarios, in terms of inflow sequences, weresimulated. The first corresponded to a synthetic inflow sequenceof 1000 years, generated by a PAR(1) model, and the secondcorresponded to the historical inflow records for the 70 yearsbetween 1931 and 2000.

The PAR(1) model was applied to the actual historical inflowrecords after normalization of the series by subtracting the ex-pected value and dividing by the standard deviation. As a result,the PAR(1) model is represented by

(11)

whereinflow at time , with beingthe year and the month;expected value of the inflow for month;standard deviation of the inflow for month;autocorrelation coefficient of the normalized series;sequence of uncorrelated random variables with dis-tribution , .

A number of estimates for the periodic function have beensuggested by statisticians [20]. In this work, the Maximum Like-lihood Estimate method was used to determine the autocorrela-tion function in the PAR(1) model [21]. Table III shows the au-tocorrelation coefficients for the hydro plants underconsideration.

The two control policies were simulated, the first (CLFC)adopting the PAR(1) model for providing the conditional proba-bility density functions of the inflows, while the second (OLFC)adopting the same PAR(1) model for forecasting of the inflows.In this way, the comparison of the two approaches in the syn-thetical scenario, also generated by the same model, focus onthe effectiveness of the feedback control schemes under idealconditions where both approaches have the same, and exact, in-terpretation of the stochastic behavior of the inflow.

In the CLFC policy, the probability matrix of the inflow is es-timated from the synthetical inflow sequence generated by thePAR(1) model. The state variable is discretized into ten pos-sible intervals and the transition probability of each state is cal-culated by computing the frequency of occurrence of the inflowof each month, as a function of the inflow of the previous month.The probability transition matrix for Furnas hydro plant, for the

TABLE IIICOEFFICIENTS� =�

Fig. 2. Decision table of Furnas hydro plant for January.

month of January, is presented in Table IV. The state variableis discretized into 100 possible values. The optimal decision ruleis the result of decision tables which provide the optimal hydrogeneration decision and the future expected operational cost foreach possible state of the system. Fig. 2 shows the decision tablefor the month of January obtained by the CLFC policy for thisplant, obtained through the resolution of the recursive equation(9), with 100 states for the reservoir storage and ten states forthe last month inflow.

Assuming that the inflows are known exactly for the plan-ning period, the deterministic optimization of the scheduling ofthe system is also conducted [optimal solution (OS)]. The sta-tistics of interest for the simulations and OS are the values ofthe mean and standard deviation of hydroelectric generation andoperational cost. The results obtained in the simulations usingsynthetical and historical inflow scenarios for the three hydroplants of Furnas, Emborcação, and Sobradinho are presented inTables V–VII, respectively.

The results revealed higher average hydroelectric generationwith the use of the OLFC for all simulations and all hydroelec-tric plants considered. The standard deviation, however, is alsohigher, which indicates greater fluctuation in hydro generation.Since operational costs are convex and increasing, such fluctu-ations provoke cost increases which, not being compensated forby a slightly higher average generation, lead to higher final op-erating costs.

Overall, the CLFC policy was more efficient for the case ofthe Furnas plant, but this efficiency diminished somewhat for the


TABLE IVINFLOW TRANSITION PROBABILITY MATRIX FOR FURNAS, DECEMBERTHROUGH JANUARY

TABLE VHYDROELECTRIC GENERATION AND OPERATIONAL COST OFFURNAS

HYDRO PLANT

TABLE VIHYDROELECTRICGENERATION AND OPERATIONAL COST OFEMBORCAÇÃO

HYDRO PLANT

TABLE VIISTATISTICS OFSIMULATIONS FOR SOBRADINHO HYDRO PLANT

Emborcação plant, and even more for that of Sobradinho. Theadvantages observed with the synthetical simulations, however,reduced with the historical simulations, when the two controlpolicies led to almost equivalent performances. Indeed, for his-torical simulations, the two control policies were equivalent forthe Emborcação and Sobradinho plants (differences less than1%), while for the Furnas plant the CLFC was more efficient(around 4%).

The superior performance of the CLFC policy in the case ofsynthetical simulations confirms the optimality of such controlpolicy when the stochastic process of inflows is exactly repre-sented by a PAR(1) model. In the real case, however, when in-flows behave as in the historical records, the two control policies

Fig. 3. Storage level trajectories of Furnas plant from 1950 to 1960.

produce similar results, indicating that the stochastic process ofinflows is not adequately represented by the PAR(1) model.

It can thus be concluded that the two control policies pro-duce similar results in the real case when using the same PAR(1)model. If, however, more efficient techniques are used for fore-casting the inflows, such as models based on neurofuzzy net-works [22], the performance of the OLFC policy may surpassthat of the CLFC, since the latter is limited to autoregressivemodels and the “course of dimensionality” does not allow theuse of models of a superior order.

Fig. 3 shows the trajectories of water storage in the reservoirsobtained by the two feedback control policies in the simula-tion of the Furnas hydroelectric plant during the period of 1950to 1960. It can be noted that in years with average streamflowsuch as 1950/1951, 1951/1952, and 1959/1960, the differencesbetween CLFC and OLFC policies are reduced, in contrast tothe large difference verified for the critical period of 1952 to1956, when water was scarcer and the OLFC approach was su-perior. This can be explained by the fact that although the inflowis critical and that the PAR(1) model cannot correctly estimateit in both control policies, the terminal condition considered inthe OLFC policy is responsible for maintaining a higher levelof storage in the reservoir, thus leading to higher values for thewater head, which increases the productivity of the plant andtherefore improves its efficiency. This is an interesting featureof the OLFC policy since dry streamflow periods are those ofhigher operational costs and shortage risks and so more impor-tant for operation planning purposes.

This paper has shown that it is possible to implement efficientOLFC schemes which similar performance compared to tradi-


tional stochastic dynamic programming, even though restrictedto the use of the same stochastic inflow model.

VI. CONCLUSION

This paper has compared the partial open-loop andclosed-loop feedback control policies in long term hy-drothermal scheduling for hydrothermal systems composed ofa single hydro plant. The idea was to focus the comparisonon the effectiveness of the two feedback control policies forcoping with the randomness of inflow.

The comparison was made through simulation using histor-ical inflow records, as well as synthetical inflow sequences gen-erated by a lag-one parametric autoregressive model. The samemodel was used in the simulations with both feedback controlpolicies, in the closed-loop one for providing the conditionalprobability density functions of the inflows, and in the partialopen-loop one for providing the forecasting of the inflows. Thesimulations were made considering three hydro plants locatedin different Brazilian river basins.

The results revealed that the partial open-loop feedbackcontrol policy provided somewhat higher average and standarddeviation for hydroelectric generation in all simulations per-formed. The higher standard deviation provided, however, notbeing compensated for by a slightly higher average generation,lead to higher final operating costs.

The closed-loop feedback control policy was more efficientin the synthetical simulations. This advantage, however, reducedwith the historical simulations, when the different control poli-cies led to almost equivalent performances.

In dry streamflow periods, which are important for opera-tion planning purposes, the partial open-loop feedback controlpolicy revealed to be more efficient than the closed-loop feed-back control policy. In this critical situation, where water supplyis very limited and the stochastic model cannot correctly esti-mate it, the partial open-loop approach was more efficient onaccount of the terminal condition considered in the determin-istic optimization model used to provide the control policy.

Other possible benefits associated with the partial open-loopfeedback control should be resulting from a detailed system rep-resentation in the case of multiple hydro plant systems and theuse of models more efficient than the lag-one parametric au-toregressive model to represent the actual stochastic process ofthe inflows. These benefits should make the performance of thepartial open-loop control policy surpass that of the closed-loopcontrol policy, since the latter cannot cope with these kind ofrepresentations on account of the “course of dimensionality” as-sociated with its feedback control policy.

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[22] R. Ballini, M. Figueiredo, S. Soares, M. Andrade, and F. Gomide, “Aseasonal streamflow forecasting model using neurofuzzy network,”in Information, Uncertainty, and Fusion, B. Bouchon-Meunier, R. R.Yager, and L. Zadeh, Eds. Boston, MA: Kluwer, 2000, pp. 257–276.

L. Martinez was born in Brazil in 1971. She received the B.Sc. degree in mathe-matics and the M.Sc. degree in computational mathematics both from the StateUniversity of São Paulo (USP), São Paulo, Brazil, in 1994 and 1996, respec-tively. She is currently pursuing the Ph.D. degree in electrical engineering fromthe State University of Campinas, Sao Paulo, Brazil.

Her area of research interest is power systems planning and operation re-search.

S. Soareswas born in Brazil in 1949. He received the B.Sc. degree in mechanicalengineering from the Aeronautics Institute of Technology (ITA), Brazil, and theM.Sc. and Ph.D. degree in electrical engineering from the State University ofCampinas, Sao Paulo, Brazil, in 1972, 1974, and 1978, respectively.

He joined the staff at UNICAMP in 1976. From 1989 to 1990, he was withthe Department of Electrical Engineering at McGill University in Montreal, QC,Canada, in a post-doctoral program. Currently, he is a Professor with the Facultyof Electrical Engineering at UNICAMP. His area of research interest is planningand operation of electric energy systems.

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