604

IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 26, NO. 2, MAY 2011

Optimal Scheduling of a Price-Taker Cascaded Reservoir System in a Pool-Based Electricity Market F. Javier Díaz, Javier Contreras, Senior Member, IEEE, José Ignacio Muñoz, and David Pozo, Student Member, IEEE

Abstract—A mixed integer nonlinear programming (MINLP) model for scheduling of the short-term integrated operation of a series of price-taker hydroelectric plants (H-GENCO) along a cascaded reservoir system in a pool-based electricity market is presented. The objective of the H-GENCO can be either to maximize profit, taking into account technical efficiency, or to maximize technical efficiency, maintaining a profit level. In both cases, the efficiency can be accurately obtained using the “Hill diagram” supplied by turbine manufacturers. A multiple nonlinear regression analysis of the unit’s technical efficiency is estimated as a quadratic function of net head and water discharge. Several case studies of realistic dimensions are described, where results indicate that a profit-based MINLP produces better results compared to an MILP model, on the other hand, higher efficiencies and water savings are obtained in the efficiency-based model.

Slope of the standard discharge of reservoir . Electricity price in period [ /MWh]. Conversion factor from water discharge to water storage . Minimum profit in the efficiency-based model. Number of time periods under study. Maximum power or plant capacity [MW]. Maximum discharge of water at plant .

Index Terms—Hill diagram, hydroelectric generation, net head, technical efficiency, water discharge.

Minimum discharge of water at plant .

NOMENCLATURE

Start-up cost of power plant [ ].

The notation used in this paper is as follows:

Spillage of reservoir

A. Constants

.

Future water value in reservoir [ / Initial volume of reservoir

Efficiency coefficient as function of net head and water discharge. Maximum head of reservoir [m]. Minimum head of reservoir [m].

.

Maximum volume of reservoir

.

Minimum volume of reservoir

.

Forecast of the natural water inflow of the reservoir associated with plant in period .

Transition matrix with 0/1 values whose elements are equal to 1 if there is flow to the reservoir from reservoir , and 0 otherwise. Slope of the standard head for reservoir .

].

B. Variables

Slope as linear function of the volume of reservoir . Manuscript received August 04, 2009; revised November 26, 2009, February 02, 2010, and May 09, 2010; accepted July 25, 2010. Date of publication September 07, 2010; date of current version April 22, 2011. This work was supported in part by the Universidad Nacional de Colombia, sede Medellín, Facultad de Minas and the Dirección de Investigaciones de Medellín, DIME, under project 20201007010 and in part by the Ministry of Education and Science of Spain grant ENE2006-02664. Paper no. TPWRS-00608-2009. F. J. Díaz is with the Facultad de Minas, Universidad Nacional de Colombia, sede Medellín, Colombia (e-mail: [email protected]). J. Contreras, J. I. Muñoz and D. Pozo are with the Escuela Técnica Superior de Ingenieros Industriales, Universidad de Castilla La Mancha, 13071 Ciudad Real, Spain (e-mail: [email protected]; [email protected]; [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TPWRS.2010.2063042 0885-8950/$26.00 © 2010 IEEE

Start-up cost of plant [ ]. Efficiency of turbine in period

.

Net head of reservoir in period

.

Standard net head of reservoir in period . Operating income of plant [ ]. Revenue from future water value of reservoir [ ]. Power output of plant in period [MW]. Total profit of the reservoir system. Water discharge by plant .

in period

DÍAZ et al.: OPTIMAL SCHEDULING OF A PRICE-TAKER CASCADED RESERVOIR SYSTEM

Standard water discharge to plant in period . 0/1 variable which is equal to 1 if plant is online in period , and 0 otherwise. Spillage of the reservoir associated with . plant in period Water volume discharged by plant in the . time horizon Water volume in the reservoir associated . with plant in period 0/1 variable equal to 1 if plant is started-up at the beginning of period , 0 otherwise. 0/1 variable equal to 1 if plant is shut-down at the beginning of period ; and 0 otherwise. C. Sets Set of indices of the periods for the time horizon. Set of indices of the plants belonging to the H-GENCO. Set of efficiency coefficients as a function of net head and water discharge. Set of indices of reservoirs upstream reservoir . I. INTRODUCTION

I

N electric power generation systems, hydrothermal coordination is a complex problem requiring dynamic optimization. It is a stochastic, large-scale problem and related to decisions associated with electricity generation allocated to each plant, hydraulic and thermal, for each period of the planning horizon. Conditions such as temporal and spatial water balance, constraints of water storage in reservoirs and physical capabilities of both generation plants and transmission lines must satisfy the operating system conditions. Two basic approaches to the problem have been considered. In traditional schemes, based on monopolistic centralized systems, the minimum expected total cost for the planning horizon is considered in the objective function. A future cost function is used, based on hydrological scenarios, to determine the water amount required in the current period and to be saved for future use [1]. Deregulated systems, with competitive markets schemes, usually have as their main objective profit maximization for the generating company. These pool-based electricity markets have expanded around the world (South America [2], Spain [3], New England [4], Norway [5]). The short-term hydrothermal coordination problem is related to three problems, which in the context of an integrated system of cascaded reservoirs could be specified as: 1) the hydroelectric economic dispatch (HED), related to the question of how much to generate in each plant in an optimal way, considering the unit’s technical efficiency; 2) the hydro-unit commitment

605

(HUC), associated with the question of when to generate, i.e., deciding when to start up and shut down; and 3) the design or preparation of energy offers to the day ahead market. This paper addresses the first two problems integrally from the perspective of the hydroelectric plant owner, process known as the scheduling problem, which has been solved using different methodologies, and whose results can be useful to solve 3). Most models ignore the variable head effect in the performance curves of the hydroelectric generating units that represent the power generated as a water discharge function to avoid the nonlinearity and nonconcavities that could lead to a local optimum. These simplifications can, however, lead to inaccuracies. Simple performance curves have been used, through a concave piecewise linear approximation [6], [7] or the modelling of the so-called local best efficiency points [8], [9]. These problems have been addressed, among others, with linear programming techniques. By exploiting some special features of certain constraints, the problem has been formulated as a network flow program, whose solution is reached faster than using standard linear programming algorithms [10], [11]. Mixed integer linear programming (MILP), which allows us to combine continuous, integer and binary variables in a unique model, is a good technique to formulate and solve these problems. Binary variables can be used in piecewise linear approximations to represent linear segments [12], [13]. Besides other traditional optimization methods, heuristic techniques and some combinations of both have been used [14], [15]. Benders’ cuts decomposition strategy [16] is appropriate when there are complicating variables, i.e., those whose presence adds significant complexity to the problem. Stochastic dual dynamic programming (SDDP) [1] is a recursive function, updated from linear programming dual solutions. In this way approximations to the recursive function are obtained avoiding the well-known dynamic programming “curse of dimensionality”. Lagrange relaxation (LR) is appropriate where there are complicating constraints, i.e., their presence makes the problem significantly more difficult to solve [17]–[23]. In the “price decomposition” strategy the multipliers associated with demand constraints are used to estimate electricity prices. One LR disadvantage is that slight variations in the multipliers can generate oscillations in the solution of the sub-problems [18], [23]. The HUC problem formulated in [19] as an MINLP model, solved with LR, takes the water discharge at each stage for each plant as an input, unlike this paper, where turbine water discharge is a major decision variable. Hydroelectric power is a nonlinear and nonconvex function of the water discharge [20], [24]–[26]. A piecewise linear approximation overestimates the generation among the best efficiency points. In [24] a model of the British Columbia Hydro Power Authority (BCH) system in Canada is developed and applied to a 2700 MW plant with ten units of four types. Through an example, this model shows that allocating all available units in a plant could produce a 15% efficiency loss or an energy loss equivalent to 80 MWh. In [25] a function for each hydraulic head is used and quadratic approximations for the relationship between power and discharge. In [26] the reservoir gross head is considered as a cubic polynomial of the water volume and a

606

quadratic one of the discharge; the unit efficiency is assumed as a 2nd degree concave polynomial of the water discharge, and the power as a 4th degree polynomial. Variable head effects have been studied in [14] and [27]–[31]. In [29], they are considered in a profit-based model with two risk-aversion criteria, minimum profit and minimum conditional value-at risk (CVaR). The problem is solved through an iterative procedure with piecewise linear approximations. In [31] it is shown that the short-term behavior of a reservoir depends on both its relative position in the system and on the hydro-chain physical parameters. The plant efficiency is assumed as a linear function of the head, and the head as a linear function of the stored volume. In this way, the power generation is a nonlinear function of the water discharge and the volumes stored in the same reservoir and in the next one; the solution relies on indefinite quadratic programming. Through a three-cascaded reservoir case study, it is shown that the optimization process defers the power generation in the initial periods to achieve high storage levels in plants with increased efficiency. Results, compared with a linear programming method, which ignores the head dependence, show that the benefits are increased by between 3.21% and 7.86%, with a similar computational effort. Our scheduling problem is similar to one of the sub-problems generated by LR in solving the short-term hydrothermal coordination problem. The proposed model allows the hydroelectric generation company, H-GENCO, to determine its scheduling in the short-term market (one week) and can be efficiently solved by commercial packages [32]. This paper’s main motivation is to provide an H-GENCO with a short-term scheduling tool to achieve either: 1) maximum profit from selling power in the day-ahead market while considering all its operating constraints, including technical efficiency [33] and the generation units’ start-up costs [8], [34], or 2) maximum efficiency for a desired profit level. It is an approach suitable for a hydro generating company (H-GENCO) which seeks to maximize its profits, whose problem is to determine the optimal scheduling in the short term. A mixed-integer nonlinear programming model (MINLP) is developed for the integrated operation of a series of hydroelectric plants along a cascaded reservoir system. A case study is analyzed in detail for a price-taker H-GENCO of realistic size. Since its production schedule does not alter the market clearing prices, prices are assumed known or forecasted [35]–[37]. An H-GENCO that develops bid strategies in perfect competitive markets has incentives to bid at marginal costs, because many small producers are involved, and it is not possible for just one of them to influence the market price. Thus, each producer must accept this price, which is assumed to fully describe the market conditions [38]–[41]. On the other hand, in oligopolistic markets, the optimal offering strategy design is a very complex problem for a producer who must make decisions with imperfect information on markets formed by interactions among different agents, producers and consumers, whose behavior is unknown. A price-maker is a participant whose bidding decisions have an impact on the market; generally, it is a big company in a market with only a small number of competitors. The case of an H-GENCO with market power would be more realistic but it is beyond the scope

IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 26, NO. 2, MAY 2011

of this study; it would require to model the market clearing price changes with the H-GENCO total production. This would be possible estimating and incorporating to the models, a residual demand curve only considering the consequences of the main player’s decisions on the market price [42], [43]. In this case, the producer acts as a “leader” in a Stackelberg game, facing a residual demand function, obtained by the aggregation of the competitors’ demand and supply offers. Finally, in more complex cases, a game theory approach can take into account the strategic behavior of the other market participants, who also act so as to maximize their own profits. Besides, simulation models are also based on the assumptions about the behavior of the agents. Other price-maker models are presented in [44]–[53]. The main contributions of this paper are the following: 1) The technical efficiency estimation of a Francis turbine as a net head and water discharge quadratic function from the so-called “Hill diagram” supplied by the manufacturer. This procedure is proposed, but not developed, in [19]. References [19], [20] and [54] present numerical values for these coefficients that must be estimated beforehand. To the authors’ knowledge, these “Hill diagrams” have not been used as part of an optimization process. In such case it is possible to obtain better results, since the points are taken from specific machine operation modes. To illustrate this, we compare our model with the MILP model in [12], where our model produces better results. 2) The representation of hydro unit performance curves as continuous, nonlinear and nonconcave functions, incorporating the technical efficiency to overcome the inaccuracies of the discrete approximations. 3) The modelling of the variable head effect in the power generation function. 4) The comparison between profit-based versus efficiency based approaches for an H-GENCO. The remainder of this paper is organized as follows. In Section II, the MINLP is formulated in detail for the short-term integrated operation of the hydroelectric system. In Section III, results of a case study of realistic dimensions are provided and discussed. Finally, conclusions are presented in Section IV. II. MODEL FORMULATION In the following an H-GENCO scheduling problem is formulated as an MINLP model. The first subsection presents the technical efficiency and the power generation function, the next one shows the profit-based model formulation, and the last one shows an alternative objective function based on efficiency. A. Technical Efficiency and Power Generation Function Hydroelectric generating units have a complex operational performance. Their power generation depends on three variables: the net head, the water discharge to the turbines and the technical efficiency of the turbine-generator set. The net head is a nonlinear function of variables such as the gross head (associated with the reservoir water volume), the water discharge and the water level in the reservoir tail. Moreover, the turbine efficiency is a nonlinear function of the net head and the water discharge.

DÍAZ et al.: OPTIMAL SCHEDULING OF A PRICE-TAKER CASCADED RESERVOIR SYSTEM

The performance curves of the hydroelectric generating units are nonlinear and nonconcave [8], [34]. More generally, however, technical efficiency and power output could be formulated as higher level polynomial functions of these two variables, net head and water discharge [20]. In [12], an MILP model that assigns to the units’ performance curve three different levels of water stored in the reservoirs (low, medium and high) is formulated and solved; binary variables are used to select the performance curve corresponding to each of these three levels and to model each of these nonconcave curves. In [13], for a single reservoir with a multiunit pump-storage hydro plant, an improvement in two steps is proposed: 1) an extension of [12] to slightly generalize its approach to a parametric number of water volumes, and 2) an enhanced linearization with a more precise estimation of the upper bound on the power production through a convex combination method considering both volumes and discharges. In this paper these curves are modelled by a continuous function that allows us to represent the nonconcavities where, unlike [8] and [34], the plant operation is not restricted to the local best efficiency points, i.e., the curves are appropriate to represent the global nonconcavity. The power output of a hydroelectric generating unit in time period , expressed in MW, can be written as in (1), where represents the unit efficiency in percent, the , and the net head water discharge to the turbine in in m:

607

Fig. 1. Francis turbine efficiency as a net head and water discharge function.

To derive the actual analytic expression of the efficiency we take a sample of 378 points from the curves in Fig. 1. With the statistical package R, version 2.4.1, we develop an algorithm to transform these points to the original scale, and to estimate , in (2). The Appendix shows the the parameters , numerical details to reach (3):

(1) Many related works consider the net head and the water discharge as parameters with average values for the time horizon [30], [31], [55] and [56]. However, using the variable net head to calculate the power generated in a hydroelectric plant operation can have a significant effect. In [12], Hill diagrams are represented as a family of nonlinear and non concave curves, known as performance curves, each for a head-specific value. In this paper, a new nonlinear model to represent a turbine operation is formulated and the power output is taken as a nonlinear function. The turbine is projected to operate with design parameters so that its performance is optimal. Operating far from these parameters will make inefficient the electrical power production process. We begin with the information available from the manufacturers, usually denoted as efficiency graphs at different levels, depending on the net head and water discharge, known as Hill diagrams, such as that illustrated in Fig. 1 adapted from [19] and [57] in [58]. In this figure, for any pair (net head, water discharge) associated with any point of the feasible operation region, it is possible to read the turbine efficiency (and the power output). Besides, there is a “maximum efficiency” point, about 0.94 in this case, with the optimal performance values for some design parameters, called design head, , and design discharge, , with values of 41.67 , respectively. We analyze a general reaction m and turbine; in this case, a Francis turbine efficiency is considered as a quadratic function of the net head and water discharge, as presented in (2), irrespective of the subscripts: (2)

(3) For any pair (net head, water discharge) associated with any point of the feasible operation region, (3) can be used to estimate the turbine efficiency, from which it is possible to calculate the output power, in MW, using (1). Water discharge to the unit in period is regarded as the decision or control variable, and the net head as an endogenous variable, dependent on the reservoir water volume. Note that the head-volume relationship is usually represented by level curves for each reservoir, according to its topology. In this paper, the head is taken as a linear function of the volume, according to the results of a preliminary study where a linear regression presents an approximate 96% fit. B. Profit-Based Model Formulation The purpose of any H-GENCO in an electricity market is to maximize its own profit, computed as the difference between its revenues and its total operation costs. For an H-GENCO, the production costs are negligible; the most significant costs which represent a real impact on its short-term scheduling are the start-up costs of the units, caused mainly by the additional maintenance required for the mechanical equipment and the control equipment malfunction with frequent start-ups. So, the objective function to maximize can be expressed as (4):

(4)

608

IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 26, NO. 2, MAY 2011

where

with its maximum capacity. In (13), the total volume discharged in each reservoir is computed over the programming horizon: (5) (6) (7)

In (4), we compute the system total profit as the sum of the profit from each of its plants, where: 1) the first term represents operational income, 2) the second one is the income from the future value of stored water in the reservoir at the end of the programming period, as defined by a medium- or long-term model, and 3) the last term represents the start-up cost, dependent on the type of plant [8], [34]. In (5), the operational income of each plant belonging to the H-GENCO is computed as the sum of the electricity price times the power output in all time periods. In (6), income from the future water value is obtained as the volume stored in the last time period times the constant future water value, VFA(i). In (7) the start-up cost in the programming horizon is the sum for all time periods of the start-up cost times whenever the hydro units are started. The constraints to the problem are presented as follows. The reservoir continuity equation is formulated in (8). The water volume in the current period is computed as the volume in the previous one plus the total inflows to the reservoir, both the natural inflows and the contributions from upstream ones, less the water discharge to the turbines less spillage by each reservoir. is equal to 0.0036. The water travel The conversion factor time from one reservoir to another, downstream, is assumed as takes 0/1 binary one hour. The transition matrix values, which are equal to 1 if the reservoir get flows from the reservoir , upstream, and 0 otherwise:

(14)

(15) In (14), the reservoir water height is estimated as a linear function of its volume, where the head-volume relationship slope is calculated as in (15) for each reservoir. We assume a minimum height associated with the minimum volume:

(16) (17) We also assume that each plant operates with a Francis turbine, or its equivalent, with a Hill diagram similar to the one shown in Fig. 1, which henceforth will be called “standard turbine”, where each turbine has its own parameters. For this reason, in (16) a change of scale in the head axis is made by , to its a linear transformation from its original values, . We suppose there equivalents in the standard turbine, is a minimum value corresponding to the turbine standard min, and the slope, , for each reservoir imum, is calculated as in (17):

(18)

(8) Equations (9)–(24) represent the reservoir constraints. In the following, they are presented in blocks:

(9) (10) (11) (12) (13) In (9), the volume of reservoir during period must remain between its lower and upper limits. In (10), the net head is bounded between its lower head and its higher one. In (11), as set by the , the turbine water discharge associated binary variable, with plant in period must remain between its lower and upper ; but, if the turbine is limits, if the turbine is online, , the discharge will be 0. In (12), the spillage offline, of the reservoir associated with plant is limited in accordance

(19)

(20) (21) (22) In (18), a change of scale in the discharge axis, is applied to transform the discharge, , to its equivalent standard dis, where the slope, , is calculated as charge, in (19). These scale changes on head and discharge axes allow computing for each generation unit the efficiency through (2), as shown in (20), and the power output using (1), as shown in (21). In (22), the maximum power generation is limited to the plant capacity:

(23) (24)

DÍAZ et al.: OPTIMAL SCHEDULING OF A PRICE-TAKER CASCADED RESERVOIR SYSTEM

609

TABLE I HYDRO SYSTEM DATA

(25) (26) (27) The nonnegative variables used in the model are listed in (23)–(24). Constraints (25)–(27) use binary variables to model the start-up and shut-down of the plants along the time horizon may seem superfluous, since they only [11]. The variables appear in these two restrictions. However, numerical simulations have proved their ability in reducing computing time sig, standard nificantly [12]. The variables named discharge, , efficiency, , and power, , are discharge, equal to 0 if plant is offline in period , . To conclude, we propose an MINLP model where (18), (20) and (21) are nonlinear functions representing the standard discharge, the efficiency and the output power, respectively. There are binary variables in (11) and (22) to put limits to the discharge and the power output, in (18) and (20) to calculate the standard discharge and the efficiency, and in (25) and (26) to control the start-up and the shut-down of the plants. In this case, appears a binary variable multiplying a linear function in (18) and a nonlinear one in (20). C. Alternative Objective Function Formulation Based on Efficiency Operating a turbine close to its design point produces high technical efficiency, which has many advantages such as, to save on both, used water in power generation and maintenance costs, to enlarge the mechanical equipment lifetime, to avoid malfunctions in the control equipment and wear and tear of the windings. On the contrary, operating the turbine far from its design point, can cause cavitation and mechanical vibration with severe consequences to the turbine, low performance, mechanical wear, components damage, too much noise and power output oscillations. Hydraulic turbine cavitation can result critical for the electrical energy offer. Through experimental data, [59] reports the costs of its detection in the hydroelectric plant “Raúl León” known as “El Guri Reservoir” in Venezuela, the second biggest in the world (after Itaipú in Brazil) with 20 generation units each of 500 MW and 10 000 MW total capacity. In [59], the failure of a unit is computed as an average value of 20 000 /h: 500 000 kW * 0.04 /kWh. The average electricity price in the Spanish-Portuguese market was 0.04 /kWh in both years 2004 and 2009 [60]. In this second model we propose as an alternative objective function, , the weighted technical efficiency maximization, as in (28), where we take the power generated by each hydro plant per period as the weighting factor. To the authors’ knowledge, this approach has not been proposed before nor applied on any real system and it is a contribution of this paper. The alternative scheduling problem maximizes (28) subject to the same constrains (5)–(27) but, in this case, it is convenient to add an additional condition, as (29), expressing the minimum

profit that the producers want to obtain, which should be at least equal to the one obtained in (4):

(28) (29) The main purposes of this alternative model, conformed only by physical variables without considering prices, are twofold: 1) to help finding an alternative optimal solution that maximizes the technical efficiency of the turbines without decreasing the producer profits, as shown ahead in Fig. 9, where a new hydro generation scheduling program that maintains the same profit level allows to increase the system overall efficiency (in order to prevent the machinery deterioration. This solution would be an alternative optimum to the maximum profit scheduling); and ii) to allow scenario analysis with two objectives: maximum profit and maximum efficiency. Figs. 9 and 10 show how small decreases of the first objective produce significant increases of the latter one, which could be generalized as a multi-objective analysis case (not developed in this paper). An H-GENCO would pursue the efficiency maximization objective if sacrificing a small part of its profits would greatly improve its overall efficiency in order to save future maintenance costs and cavitation issues. III. CASE STUDIES The aim of this paper, as stated in the introduction, is to obtain the optimal hydroelectric generation as part of the design of offers to the day-ahead market. In this section, the results of two realistic case studies are compared and discussed. The system comprises eight cascaded plants along the Duero river basin in Spain, whose data are shown in Table I, as taken from [12]. The models are implemented on a Sun FIRE v20z, biprocessor AMD Opteron Dual Core 270, 2.0 Ghz, 8 GB RAM memory using SBB under GAMS [32]. In the first case study spanning 24 h, the system contains 4235 equations and 2717 variables, where 2069 are continuous, and 648 are discrete ones. The optimal solution is obtained in 1.19 min of CPU time. In the second case, weekly, the system contains 30 252 equations and 16 257 variables, where 12 201 are continuous, and 4056 are discrete variables. The optimal solution is obtained in 40.7 min of CPU time. A. First Case Study We run our MINLP model for one day split into 24 h according to the original data of the MILP model in [12]. The re-

610

IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 26, NO. 2, MAY 2011

TABLE II OPTIMAL PRODUCTION SCHEDULE: MILP [12] VERSUS MINLP METHOD. ONE DAY

Fig. 2. Hydraulic topology of the river basin.

sults obtained in [12] are shown and compared to our MINLP model in Table II. Both models consider the final water volume equal to the initial one for all reservoirs, and, therefore the dis, are equal, too. In the MINLP model, charge volumes, the optimal value of the objective function is $597 751.2, i.e., a 7.01% higher than $558 587.1 achieved in [12], showing the advantage of more detailed efficiency modelling. The operating income also increases from $565 647.1 in the MILP model to $601 950.3 in the MINLP one, a 6.41% increase. B. Second Case Study The case study is also taken from [12], but for a period of one week and the year 2008. The information and relevant assumptions are completed and updated, as presented below, to formulate and solve the model as an MINLP. The natural water inflow to each reservoir is assumed constant during the time horizon, in our case, one week divided into 168 hourly periods. The initial and final reservoir water volumes are known at the beginning of the study period. The latter ones, as well as the future water value, VFA, are usually obtained by medium-or long-term planning models. For the sake of comparison of operation policies, we develop two scenarios for the final volume: fixed and free volume. In the first one, without loss of generality, for each reservoir, we assume a basic scenario with a final water volume equal to the initial one; therefore, the production schedule is indifferent to the , at the end of the period prestored water future value, viously defined. A proportional change in all does not , affect the production schedule (volume discharged, ), although the revenue from and operative income, future water value, , and the profit, , change. Fig. 2 shows the spatial coupling between reservoirs. We show in Table III the following cases summarized in terms of the main variables involved in each of the tables that follow. In the free volume case, the maximum efficiency model (comand pared with the maximum profit one) has lower higher efficiency. In the fixed volume case, the maximum efficiency model (compared with the maximum profit one) has the but higher efficiency. same Table IV presents the optimal production schedule with fixed volume when profit is maximized as in (4). In general, the power output of the eight plants follows the hourly prices profile for the Spanish electricity market obtained from [59] for the week

TABLE III SUMMARY TABLE OF THE SECOND CASE STUDY. ONE WEEK (168 h)

of October 9–15, 2008. Fig. 3 shows this trend for plant 1. For a detailed analysis we select plant 1, for which the water discharge and price evolution are presented in Fig. 4, similar to Fig. 3, as expected. The reservoir volume evolution associated with plant 1, and its discharge, are shown in Fig. 5. Plants 1, 2, 4 and 7 are on at the beginning of the period under study. Its discharge (and therefore its generation) can go down to its minimum limit, given in Table II, without shutting down the turbine. Table IV and Figs. 3–5 show the maximum profit scheduling by keeping unit 1 (same for unit 4) operational all week long, so as to avoid its start-up cost. Unit 1 is operating close to 0.5 MW power output during many time periods, i.e., away from its best efficiency point, with possible cavitation issues. In this case, it would be convenient to modify the operation policy, es, to pecially the minimum water discharge allowable, prevent undesirable results. Increasing the water discharge in units 1 and 2 would lead to higher volumes in unit 3, whose initial volume is approximately 66% of its storage capacity, and whose inflows are nearly ten times the ones in plants 1 and 2, as shown in Table I, causing a high risk of spillage. Note that plant 3 is the only one with an initial volume 50% higher than its own storage capacity. The maximum efficiency scheduling gives an alternative optimal solution, with the same profit, but with better overall efficiency, as shown in Fig. 8. Fig. 6 shows the negative correlation (with the exception of the minimum points in this case) between water discharge and efficiency. This can be explained as follows: In Fig. 1, it can be seen that for a given net head, from the minimum discharge, the efficiency increases with the discharge (positive correlation) until a “relative efficiency maximum discharge” point, and from this point the efficiency decreases with the discharge (negative correlation). For each net head there is a point associated with the “relative efficiency maximum discharge”. Uniting all these

DÍAZ et al.: OPTIMAL SCHEDULING OF A PRICE-TAKER CASCADED RESERVOIR SYSTEM

611

Fig. 6. Plant 1: Negative correlation between efficiency and discharge. Fig. 3. Plant 1 power output versus electricity prices for October 9–15, 2008.

Fig. 4. Plant 1 water discharge versus electricity prices for October 9–15, 2008.

Fig. 5. Plant 1 volume and water discharge for October 9–15, 2008.

TABLE IV MAXIMIZING PROFIT WITH FIXED VOLUME. OPTIMAL PRODUCTION SCHEDULE. ONE WEEK

points, we can obtain a “relative efficiency maximum trajectory” that divides the feasible operation region into two sub-regions, one above this “trajectory”, with negative correlation, and another below, with positive correlation. Fig. 7 shows the positive correlation between technical efficiency and productivity, the , latter defined as the power generated per discharge unit, which with a simple transformation could become energy per . unit of water discharged, so:

Fig. 7. Plant 1: Positive correlation between efficiency and productivity.

We now compare the results maximizing profit, in Table IV, with the ones obtained maximizing efficiency, shown in Table V, assuming we want to keep the same level of profits, at least equal to (4), without modifying the other constraints (5)–(27). There are no differences in the overall results in terms of income, future water value, water discharge, and start-up costs. If the volume is constrained in the maximum efficiency model, the water discharge is the same as in the maximum profit one, but the discharge is developed in different time periods when the efficiency is higher. These results show that the maximum efficiency solution is an alternative optimum to the maximum , and profit one. However, the operational incomes, , decrease for plants 4, 5 and 7, increasing for profits, the rest. The decrease for these plants connected in series is partially compensated by the increase for plant 6 (in parallel with plant 5; see Fig. 2), whose natural water inflow of the reservoir is nearly four times the one of plants 4 and 5 (see Table II); besides, its efficiency is very low and it increases meaningfully (Fig. 8). Fig. 8 shows a comparison between unit technical efficiencies in both profit maximization and efficiency maximization models, as in Table III. Several cases ranging from efficiency maximization without a desired profit level (unconstrained profit) to efficiency maximization with a 100% profit level are studied. It is remarkable the dramatic efficiency rise of turbines 5 to 8 in all cases. Efficiency increases significantly for 90%–97% profit levels, which are close enough to the best solution in terms of profit. For the sake of completion, the same exercise is repeated assuming that the final water volume of each reservoir is not fixed to its initial value, but it varies freely. Tables VI and VII show the results, where the solution in Table VII is an alternative optimum to the one in Table VI. Comparing results of the maximum profit models, with free and constrained volume,

612

IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 26, NO. 2, MAY 2011

TABLE VII MAXIMIZING EFFICIENCY WITH FREE VOLUME. OPTIMAL PRODUCTION SCHEDULE. ONE WEEK

Fig. 8. Efficiency results: maximizing profit versus maximizing efficiency. Fixed volume.

TABLE V MAXIMIZING EFFICIENCY WITH FIXED VOLUME. OPTIMAL PRODUCTION SCHEDULE. ONE WEEK

Fig. 9. Efficiency results: maximizing profit versus maximizing efficiency. Free volume.

TABLE VI MAXIMIZING PROFIT WITH FREE VOLUME. OPTIMAL PRODUCTION SCHEDULE. ONE WEEK

we can observe that Table VI with free volume, gives higher values versus the water-constrained case (Table IV), in all the , and for the plants, both for the volume discharge, , as expected. Besides, revenue operating income, from future water value, , is higher too with free volume (approximately 1.7 versus 1.5 M ). How is it possible that higher water discharges can result in better revenues for water storage? The explanation follows. The constrained volume model takes the final storage given to all the plants in the overall period and, therefore, a given water amount is optimally discharged in each hour. However, in the free volume model, the scheduling program can discharge water volume freely in an optimal way and, therefore, can storage it

in reservoirs with better future water values. This is the case of the first three transfers between the first four plants, where ranges from 84.0 to 221.2, 630.0, and 900.2 (Table II). Then, decreases between reservoirs 4 and 5, increasing between reservoirs 5 and 7. Therefore, water flows from reser, voir 4 (going down quickly to its minimum volume, , and low operating income, ) with low efficiency, to reservoir 5, in order to go later from 5 to 7, whose is , to higher. Plant 6 generates at its maximum capacity, also increases. Reservoirs 7 and 8 transfer water, since reach quickly their maximum volumes. Comparing Tables VI and VII, with free volume, it can be , decreases a little seen that the volume discharged, in the maximum efficiency model, saving water, but not much, since the level of profits is maintained in both cases. Similarly to Figs. 8 and 9 depicts the efficiency results for a range of profit levels. As in the previous case, efficiency improves significantly for turbines 5 to 8. The overall efficiency ranges from 0.8132 (100% profit maximization) to 0.886 (unconstrained profit), as shown in Table III. It is also noted the decrease in efficiency with respect to Fig. 9 for the 100% profit case, particularly for turbines 1 and 2. In general, these results show that optimal scheduling with free volume gives higher profits, but lower overall technical efficiency. Finally, note that in the model that maximizes efficiency with free volume, decreasing the profit 10% (from 100% to 90%) produces 13.6% savings in the discharged water (from

DÍAZ et al.: OPTIMAL SCHEDULING OF A PRICE-TAKER CASCADED RESERVOIR SYSTEM

613

TABLE IX PARAMETER ESTIMATION AND SIGNIFICANCE TEST

TABLE X ANOVA MODEL

Fig. 10. Sample points of the efficiency curves (first section) and fitting model (second and third sections).

TABLE VIII DISTRIBUTION AND SELECTION OF POINTS

for the H-GENCO operation, either profit maximization or technical efficiency maximization, subject to a fixed profit level. The technical efficiency estimation is a quadratic function of the net head and water discharge resulting from a Hill diagram supplied by the turbine manufacturer. A general formulation of the unit performance curve as a continuous nonlinear and nonconcave function, incorporating the technical efficiency, is provided to overcome the inaccuracies of the discrete approaches. The hydroelectric generating unit characteristics are modeled in detail in order to obtain a good approximation to the relation between the net head, the water discharge and the technical efficiency. The mathematical functions formulated allow the adequate treatment of nonlinear and nonconcave unit performance curves. Two case studies of realistic dimensions are analyzed in detail, also comparing the profit maximization approach versus the efficiency maximization one. In particular, we show that a more accurate MINLP model outperforms an MILP model in the first case study and we compare a profit—versus an efficiency-based model in the second case study, where higher values of efficiency and water savings are obtained in the latter model. APPENDIX STATISTICAL SAMPLING METHOD OF THE EFFICIENCY CURVES

to ), which could be desirable in critical conditions such as long-term drought to avoid rationing or water needs such as farming, irrigation or human consumption. IV. CONCLUSIONS The main motivation of this paper is to provide a price-taker H-GENCO with a short-term scheduling tool in a pool-based electricity market. We define and compare two possible goals

Table VIII shows the distribution and selection of the points, Table IX presents the estimated coefficients, standard errors and p values in the t-student distribution, and Table X shows the ANOVA model. The p values in the t and F tests (Tables IX and X), indicate that all the parameters are significantly different from zero, with an adjusted equal to 0.82, meaning that the model explains approximately the data variability 82%. From this statistical procedure, the adjusted model in (3) estimates the technical efficiency expected value for net head and water discharge given values. Fig. 10 shows the sample points (first section) and the fitting model (second and third sections). REFERENCES [1] M. Pereira and L. Pinto, “Multistage stochastic optimization applied to energy planning,” Math. Program., vol. 52, pp. 359–375, 1985.

614

IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 26, NO. 2, MAY 2011

[2] T. J. Hammons, H. Rudnick, and L. A. Barroso, “Latin America: Deregulation in a hydro-dominated market,” HRW, vol. 10, no. 4, pp. 20–27, Sep. 2002. [3] J. J. González and P. Basagoiti, “Spanish power exchange market and information system. Design, concepts, and operating experience,” in Proc. IEEE 21st Int. Conf. Power Industry Comput. Applicat., Santa Clara, CA, May 1999, pp. 245–252. [4] K. W. Cheung, P. Shamsollahi, D. Sun, J. Milligan, and M. Potishnak, “Energy and ancillary service dispatch for the interim ISO New England electricity market,” IEEE Trans. Power Syst., vol. 15, no. 3, pp. 968–974, Aug. 2000. [5] O. B. Fosso, A. Gjelsvik, A. Haugstad, B. Mo, and I. Wagensteen, “Generation scheduling in a deregulated system. The Norwegian case,” IEEE Trans. Power Syst., vol. 14, no. 1, pp. 75–81, Feb. 1999. [6] H. Habibollahzadeh and J. A. Bubenko, “Applications of decomposition techniques to short-term operation planning of hydrothermal power systems,” IEEE Trans Power Syst., vol. 1, no. 1, pp. 41–47, Feb. 1986. [7] G. W. Chang, M. Aganagic, J. G. Waight, J. Medina, T. Burton, S. Reeves, and M. Christoforidis, “Experiences with mixed integer linear programming based approaches on short-term hydro scheduling,” IEEE Trans. Power Syst., vol. 16, no. 4, pp. 743–749, Nov. 2001. [8] O. Nilsson and D. Sjelvgren, “Variable splitting applied to modeling of start-up costs in short term hydro generation scheduling,” IEEE Trans. Power Syst., vol. 12, no. 2, pp. 770–775, May 1997. [9] O. Nilsson, L. Söder, and D. Sjelvgren, “Integer modeling of spinning reserve requirements in short term scheduling of hydro systems,” IEEE Trans. Power Syst., vol. 13, no. 3, pp. 959–964, Aug. 1998. [10] X. Guan, A. Svoboda, and C. A. Li, “Scheduling hydro power systems with restricted operating zones and discharge ramping constraints,” IEEE Trans. Power Syst., vol. 14, no. 1, pp. 126–131, Feb. 1999. [11] H. Brännlund, J. A. Bubenko, D. Sjelvgren, and N. Andersson, “Optimal short term operation planning of a large hydrothermal power system based on a nonlinear network flow concept,” IEEE Trans. Power Syst., vol. 1, no. 1, pp. 75–82, Feb. 1986. [12] A. J. Conejo, J. M. Arroyo, J. Contreras, and F. A. Villamor, “Scheduling of a hydro producer in a pool based electricity market,” IEEE Trans. Power Syst., vol. 17, no. 4, pp. 1265–1272, Nov. 2002. [13] A. Borghetti, C. D’Ambrosio, A. Lodi, and S. Martello, “An MILP approach for short-term hydro scheduling and unit commitment with head-dependent reservoir,” IEEE Trans. Power Syst., vol. 23, no. 3, pp. 115–1124, Aug. 2008. [14] C. Li, R. B. Johnson, and A. J. Svoboda, “A new unit commitment method,” IEEE Trans. Power Syst., vol. 12, no. 1, pp. 113–119, Feb. 1997. [15] C. Li, A. J. Svoboda, T. C. Li, and R. B. Johnson, “Hydro unit commitment in hydro thermal optimization,” IEEE Trans. Power Syst., vol. 12, no. 2, pp. 764–769, May 1997. [16] J. F. Benders, “Partitioning procedures for solving mixed-variables programming problems,” Numer. Math., vol. 4, pp. 238–252, 1962. [17] A. J. Conejo and N. J. Redondo, “Short-term hydro-thermal coordination by relaxation Lagrangian: Solution of the dual problem,” IEEE Trans. Power Syst., vol. 14, no. 1, pp. 89–95, Feb. 1999. [18] X. Guan, P. B. Luh, and L. Zhang, “Non-linear approximation method in Lagrangian relaxation-based algorithms for hydrothermal scheduling,” IEEE Trans. Power Syst., vol. 10, no. 2, pp. 772–778, May 1995. [19] E. C. Finardi and D. A. da Silva, “Unit commitment of single hydroelectric plant,” Elect. Power Syst. Res., vol. 75, pp. 116–123, Aug. 2005. [20] E. C. Finardi and D. A. da Silva, “Solving the hydro unit commitment problem via dual decomposition and sequential quadratic programming,” IEEE Trans. Power Syst., vol. 21, no. 2, pp. 835–844, May 2006. [21] A. Merlin and P. Sandrin, “A new method for unit commitment at Électricité de France,” IEEE Trans Power App. Syst., vol. PAS-102, no. 1, pp. 1218–1225, Jan. 1983. [22] X. Guan, A. Svoboda, and C. Li, “Scheduling hydro power systems with restricted operating zones and discharge ramping constraints,” IEEE Trans. Power Syst., vol. 14, no. 1, pp. 126–131, Feb. 1999. [23] E. Ni, X. Guan, and R. Li, “Scheduling hydrothermal power systems with cascaded and head-dependent reservoirs,” IEEE Trans. Power Syst., vol. 14, no. 3, pp. 1127–1131, Aug. 1999. [24] T. Siu, G. Nash, and Z. Shawwash, “A practical hydrodynamic unit commitment and loading model,” IEEE Trans. Power Syst., vol. 16, no. 2, pp. 301–306, May 2001.

[25] J. Wang, X. Yuan, and Y. Zhang, “Short-term scheduling of largescale hydroelectric systems for energy maximization,” J. Water Resour. Plan. Manage., vol. 130, pp. 198–205, 2004. [26] A. B. Ferrer, “Applicability of deterministic global optimization to the short-term hydrothermal coordination problem,” Ph.D. dissertation, Univ. Politécnica de Cataluña, Barcelona, Spain, 2004. [27] J. García-González, E. Parrilla, J. Barquín, J. Alonso, A. SaizChicharro, and A. González, “Under-relaxed iterative procedure for feasible short-term scheduling of a hydro chain,” in Proc. IEEE PowerTech, Bologna, Italy, Jun. 23–26, 2003. [28] E. Parrilla and J. García-González, “Improving the B&B search for large-scale hydrothermal weekly scheduling problems,” Int. J. Elect. Power Elect. Syst., vol. 28, pp. 339–348, Jun. 2006. [29] J. García-González, E. Parrilla, and A. Mateo, “Risk-averse profit-based optimal scheduling of a hydro-chain in the day-ahead electricity market,” Eur. J. Oper. Res., vol. 181, pp. 1354–1369, Sep. 2007. [30] J. P. S. Catalão, S. J. P. S. Mariano, V. M. F. Mendes, and L. A. F. M. Ferreira, “Scheduling of head-sensitive cascaded hydro systems: a nonlinear approach,” IEEE Trans. Power Syst., vol. 24, no. 1, pp. 337–346, Feb. 2009. [31] J. P. S. Catalão, S. J. P. S. Mariano, V. M. F. Mendes, and L. A. F. M. Ferreira, “Parameterisation effect on the behavior of a head dependent hydro chain using a Non-linear model,” Elect. Power Syst. Res., vol. 76, pp. 404–412, Apr. 2006. [32] A. Brooke, D. Kendrick, A. Meeraus, and R. Raman, GAMS/Cplex 7.0 User Notes. Washington, DC: GAMS Development, 2000. [33] R. A. Ponrajah, J. Witherspoon, and F. D. Galiana, “Systems to optimize conversion efficiencies at Ontario hydro’s hydro-electric plants,” IEEE. Trans. Power Syst., vol. 13, no. 3, pp. 1044–1050, Aug. 1998. [34] O. Nilsson and D. Sjelvgren, “Hydro unit start-up costs and their impact on the short term scheduling strategies of Swedish power producers,” IEEE Trans. Power Syst., vol. 12, no. 1, pp. 38–44, Feb. 1997. [35] F. J. Nogales, J. Contreras, A. J. Conejo, and R. Espínola, “Forecasting next-day electricity prices by time series models,” IEEE Trans. Power Syst., vol. 17, no. 2, pp. 342–348, May 2002. [36] A. Angelus, “Electricity price forecasting in deregulated markets,” Electricity J., vol. 14, pp. 32–41, Apr. 2001. [37] B. R. Szkuta, L. A. Sanabria, and T. S. Dillon, “Electricity price shortterm forecasting using artificial neural networks,” IEEE Trans. Power Syst., vol. 14, no. 3, pp. 851–857, Aug. 1999. [38] D. Ladurantaye, M. Gendreau, and J. Potvin, “Strategic bidding for price-taker hydroelectricity producers,” IEEE Trans. Power Syst., vol. 22, no. 4, pp. 2187–2203, Nov. 2007. [39] S. Fleten and E. Pettersen, “Constructing bidding curves for a pricetaking retailer in the Norwegian electricity market,” IEEE Trans. Power Syst., vol. 20, no. 2, pp. 701–78, May 2005. [40] A. J. Conejo, F. J. Nogales, and J. M. Arroyo, “Price-taker bidding strategy under price uncertainty,” IEEE Trans. Power Syst., vol. 17, no. 4, pp. 1081–1088, Nov. 2002. [41] G. Gross and D. Finlay, “Generation supply bidding in perfectly competitive electricity markets,” Comput. Math. Org. Theory, vol. 6, pp. 83–98, 2000. [42] A. Baíllo, M. Ventosa, M. Rivier, and A. Ramos, “Optimal offering strategies for generation companies operating in electricity spot markets,” IEEE Trans. Power Syst., vol. 19, no. 2, pp. 745–753, May 2004. [43] R. Rajaraman and F. Alvarado, Optimal Bidding Strategies in Electricity Markets Under Uncertain Energy and Reserve Prices. Tempe, AZ: Power Systems Engineering Research Center (PSERC), 2003. [44] T. Li, M. Shahidehpour, and Z. Li, “Risk-constrained bidding strategy with stochastic unit commitment,” IEEE Trans. Power Syst., vol. 22, no. 1, pp. 449–458, Feb. 2007. [45] H. Niu, R. Baldick, and G. Zhu, “Supply function equilibrium bidding strategies with fixed forward contracts,” IEEE Trans. Power Syst., vol. 20, no. 4, pp. 1859–1867, Nov. 2005. [46] A. R. Kian, J. B. Cruz, and R. J. Thomas, “Bidding strategies in oligopolistic dynamic electricity double-sided auctions,” IEEE Trans. Power Syst., vol. 20, no. 1, pp. 50–58, Feb. 2005. [47] L. A. Barroso, R. D. Carneiro, S. Granville, M. V. Pereira, and M. H. Fampa, “Nash equilibrium in strategic bidding: A binary expansion approach,” IEEE Trans. Power Syst., vol. 21, no. 2, pp. 629–638, May 2006. [48] M. A. Plazas, A. J. Conejo, and F. J. Prieto, “Multimarket optimal bidding for a power producer,” IEEE Trans. Power Syst., vol. 20, no. 4, pp. 2041–2050, Nov. 2005.

DÍAZ et al.: OPTIMAL SCHEDULING OF A PRICE-TAKER CASCADED RESERVOIR SYSTEM

[49] M. V. Pereira, S. Granville, M. C. Fampa, R. Dix, and L. A. Barroso, “Strategic bidding under uncertainty: A binary expansion approach,” IEEE Trans. Power Syst., vol. 20, no. 1, pp. 180–188, Feb. 2005. [50] D. Das and B. F. Wollenberg, “Risk assessment of generators bidding in day-ahead market,” IEEE Trans. Power Syst., vol. 20, no. 1, pp. 416–424, Feb. 2005. [51] E. Ni, P. B. Luh, and S. Rourke, “Optimal integrated generation bidding and scheduling with risk management under a deregulated power market,” IEEE Trans. Power Syst., vol. 19, no. 1, pp. 600–609, Feb. 2004. [52] A. J. Conejo, J. Contreras, J. M. Arroyo, and S. de la Torre, “Optimal response of an oligopolistic generating company to a competitive pool based electric power market,” IEEE Trans. Power Syst., vol. 17, no. 2, pp. 424–430, May 2002. [53] J. Contreras, O. Candiles, J. I. de la Fuente, and T. Gómez, “A cobweb bidding model for competitive electricity markets,” IEEE Trans. Power Syst., vol. 17, no. 1, pp. 148–153, Feb. 2002. [54] F. J. Heredia and N. Nabona, “Optimum short-term hydrothermal scheduling with spinning reserve through network flows,” IEEE Trans. Power Syst., vol. 10, no. 3, pp. 1642–1651, Aug. 1995. [55] J. M. Velázquez, P. J. Restrepo, and R. Campo, “Dual dynamic programming. A note on implementation,” Water Resour., vol. 35, pp. 2269–2272, 1999. [56] R. Campo and P. Restrepo, “Estudio de optimalidad del programa MPODE,” Mundo Eléctrico, vol. 18, no. 54, pp. 32–35, 2004. [57] E. C. Finardi, “Alocação de Unidades Hidrelétricas em Sistemas Hidrotérmicos Utilizando Relaxação Lagrangeana e Programação Quadrática Sequencial,” Ph.D. dissertation, Univ. Federal de Santa Catarina, Florianópolis, Brazil, 2003. [58] F. J. Díaz, “La eficiencia técnica COMO un nuevo criterio de optimización para la generación hidroeléctrica a corto plazo,” Dyna, vol. 76, no. 157, pp. 91–100, Mar. 2009. [59] A. J. Pedro, “Detección de Cavitación en Turbinas Francis,” M.S. thesis, Univ. Politécnica de Cataluña, Barcelona, Spain, 2004. [60] Market Operator of the Electricity Market of Mainland Spain. [Online]. Available: http://www.omel.es. F. Javier Díaz was born in San Vicente, Antioquia, Colombia, in 1956. He received the B.Sc. degree in industrial engineering and the M.Sc. and Ph.D.(C) degrees in systems engineering from the Facultad de Minas, Universidad Nacional de Colombia, Medellín, in 1982, 1992, and 2008, respectively. His research interests include power systems planning, operations and economics, and optimization and simulation. He is currently an Associate Professor at the Universidad Nacional de Colombia, Medellín.

615

Javier Contreras (SM’05) was born in Zaragoza, Spain, in 1965. He received the B.Sc. degree in electrical engineering from the University of Zaragoza, the M.Sc. degree from the University of Southern California, Los Angeles, and the Ph.D. degree from the University of California, Berkeley, in 1989, 1992, and 1997, respectively. His research interests include power systems planning, operations and economics, and electricity markets. He is currently a Full Professor at the Universidad de Castilla—La Mancha, Ciudad Real, Spain.

José Ignacio Muñoz received the B.S. degree in industrial engineering from the University of Navarra, Pamplona, Spain, in 1998 and the M.Sc. and Ph.D. degrees from the University of País Vasco, Bizkaia, Spain, in 2003 and 2009, respectively. His research interests include power systems forecasting, operations and economics, and project management. He has been working as a Project Manager in several engineering firms and is, currently, an Assistant Professor at the University of Castilla—La Mancha, Ciudad Real, Spain.

David Pozo (S’09) received the B.S. degree in electrical engineering from the University of Castilla—La Mancha, Ciudad Real, Spain, in 2006, where he is currently pursuing the Ph.D. degree. His research interests include power systems economics and electricity markets.