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2842

IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 28, NO. 3, AUGUST 2013

A Chance-Constrained Unit Commitment With an Security Criterion and Signi?cant Wind Generation

David Pozo, Student Member, IEEE, and Javier Contreras, Senior Member, IEEE

Abstract—This paper presents a new approach for the joint energy and reserves scheduling and unit commitment with reliability constraints for the day-ahead market. The proposed criterion where demand must method includes a novel be met with a speci?ed probability under any simultaneous loss of generating units. A chance-constrained method is proposed with an -quantile measure to determine the con?dence level to meet the demand under simultaneous contingencies. The chance-constrained optimization problem is recast as a mixed integer linear programming optimization problem. Wind and demand uncertainty are included into the model. The methodology proposed is illustrated with several case studies where the effect of increasing wind power penetration is analyzed showing the performance of our model. Index Terms—Chance-constrained programming, conditional value at risk, stochastic security constrained unit commitment.

Net load in period depending on the random set . Parameters Number of unavailable generator units. Total number of generator units. Con?dence level of the ELNS/EWS probabilistic constraint in period . Typically takes values between 0.95 and 1. ELNS/EWS limits for period . Parameters related with the whole time horizon. Ramp-down/ramp-up limit for unit . for

NOMENCLATURE The mathematical symbols used throughout this paper are described below. The operators and are used as expectation and probability of a random function, respectively. The bold-typed symbol is de?ned as a random function, and represents a particular realization or scenario. Index Generating unit. Time period scheduling interval. Index for the scenario realization . Random Functions Demand in period depending on the random set . Wind power generation in period depending on the random set .

Manuscript received May 31, 2012; revised September 27, 2012; accepted November 03, 2012. Date of publication December 20, 2012; date of current version July 18, 2013. This work was supported in part by the Junta de Comunidades de Castilla-La Mancha Formación del Personal Investigador (FPI) under Grant 402/09 and by the Spanish Ministry of Science and Innovation under Grant ENE2009-09541. Paper no. TPWRS-00585-2012. The authors are with the Escuela Técnica Superior de Ingenieros Industriales,Universidad de Castilla-La Mancha, 13071 Ciudad Real, Spain (e-mail: David.Pozo@uclm.es; Javier.Contreras@uclm.es). Color versions of one or more of the ?gures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identi?er 10.1109/TPWRS.2012.2227841

Shutdown/start-up ramp limit for unit . Minimum/maximum power output for unit . Scheduled down/up reserve upper bound for unit . Demand/wind production/net load expected value in period . Demand/wind production/net load standard deviation in period . Fixed/linear cost for unit . Linear down/up reserve cost for unit . Unit cost for the load energy not served in $/MWh. Unit cost for the wind energy spilled in $/MWh. Outage probability for unit . Probability of units outages. Binary Variables On/off status of unit in period . Equal to 0 if unit has a contingency in period , 1 otherwise. Positive Variables Scheduled energy for unit in period . simultaneous generating

0885-8950/$31.00 ? 2012 IEEE

POZO AND CONTRERAS: CCUC WITH AN

SECURITY CRITERION AND SIGNIFICANT WIND GENERATION

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TABLE I ENERGY AND RESERVE DISPATCHING WITH SECURITY CRITERIA MODELS

Independent and joint

security criteria with an

con?dence level.

Scheduled down reserve for unit in period . Scheduled up reserve for unit in period . Auxiliary variable. Auxiliary variable. Lagrange multipliers associated with the worst-contingency problem. ELNS EWS Expected load not served. Expected wind spillage.

basically replaces the true distribution by an empirical distribution corresponding to a Monte Carlo sample, , as

I. INTRODUCTION TOCHASTIC optimization has mainly focused on maximizing or minimizing expectations [1]. Within this framework some decisions are made before considering uncertainty, and other decisions are made when the uncertainty is realized. A particular case of stochastic optimization is the so-called twostage stochastic optimization. Robust optimization [2], [3] is another line of research dealing with uncertainty. It is based on the worst case outcomes and is criticized as being a too conservative formulation for some kinds of problems. Probabilistic optimization or chance-constrained optimization is another methodology to deal with random processes where one or several constraints or an objective function must be satis?ed with high probability, de?ned as (1) Chance-constrained problems are still largely intractable except for some special cases. The main reasons are: 1) in general, , computing for any feafor any given sible can be hard and 2) the feasible region de?ned by a chance constraint is generally not convex even if is convex in for every possible realization of . To overcome these problems, a sample average approximation (SAA) can be used to obtain good candidate solutions. SAA

S

(2) Chance-constrained optimization has been applied to solve the unit commitment (UC) problem with wind power [4] or demand uncertainties [5]. In [5], they solve a chance-constrained UC using an iterative sequence of deterministic versions of the UC. However, [4] applies the SAA methodology based on the approach shown in [6] to deal with probabilistic constraints. That paper reformulates the probabilistic constraints as mixed integer constraints including a binary variable for each scenario coming from the SAA. The complexity of the problem grows very fast with the number of scenarios. In our approach, a second-order -quantile measure is used based on the expected value of a random distribution tail. A linear approach based on the Rockafellar and Uryasev’s Conditional Value at Risk (CVaR) de?nition [7] is applied to convert the chance constraint into an equivalent deterministic set of equations without binary variables. In power systems operations, security criteria are needed to attain reliability and stability. An security constraint is the common criterion [8]–[13] to keep the system stable in case of one outage of a generating unit or line, where reserves planning is justi?ed to compensate for possible outages. Therefore, both energy dispatch and reserve scheduling can be jointly optimized. Several references have proposed joint energy and reserve scheduling models focusing on the day-ahead market. Table I summarizes all of these references whose formulation has been stated as MILP. The criteria for classi?cation of these models is as follows. ? A ?rst classi?cation criterion is based on considering the contingencies as scenarios of a two-stage optimization problem, where it is necessary to know the probability of occurrence of each contingency. The size of this problem grows very fast, and it becomes intractable because of the combinatorial nature of the problem. Some authors limit the scenarios of contingencies to an umbrella of credible contingencies [8]–[11], [13], [14]. The work in [15] and [16] propose a robust optimization model where

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the system remains stable in the -worst contingencies. A robust or worst-case formulation is more proper for reliability analysis due to the conservative nature of this formulation. ? A second criterion is based on the number of simultaneous contingencies included in the optimization problem. The criterion is the most common method, and the criterion is applied in some works, but more strict criteria increase the complexity of the proposed models. The robust model proposed by [15] and [16] allows us to include a security criterion in a moderately sized problem. ? A third criterion is based on modeling the uncertainty of contingencies using stochastic variables. In [12] and [14], they include demand and wind power production as stochastic variables within a small number of scenarios. For example, in [14], the authors apply their model to a case study limited to three wind scenarios and three demand scenarios. The work in [12] uses 13 joint demand/wind scenarios where contingencies are not modeled as stochastic variables, reformulating the problem with binary variables within a few scenarios. ? A fourth criterion is based on modeling the transmission network. Although the network is needed for energy delivery and line outages may occur, some works have not considered its representation due to the complexity involved. However, even if the network is not represented, the joint energy and reserve dispatch with contingencies is a valuable tool for the independent system operator (ISO). A. Aims and Contributions The chance-constrained unit commitment (CCUC) problem is formulated to meet real-time demand. To do this, joint generation and reserves scheduling for the day-ahead market is required. This problem is solved by imposing that demand must be met in any plausible scenario. However, extreme scenarios, e.g., the highest demand level with the lowest wind production, require high amounts of reserves. Note that extreme scenarios have low probabilities, and they are the tails of some random distributions. In this paper, a chance-constrained formulation is proposed for the stochastic CCUC, where demand is not met in those extreme scenarios in order to reduce the cost of the energy dispatched. The chance constraints proposed are the risk measure of not meeting the demand with a certain con?dence level, . The formulation is based on limiting the CVaR [7], [18] by an -quantile. The main contributions of our paper are given here. 1) A chance-constrained optimization problem is formulated subject to stochastic demand for the scheduling of energy and spinning reserves with an -quantile and two criteria: independent and joint. 2) Different levels of penetration of intermittent energy sources within a multiperiod framework are included in the model. 3) Conditions for the global optimum are derived using standard duality theory rewriting the chance-constrained problem as an equivalent deterministic mixed integer linear programming (MILP) with a reduced number of binary variables.

B. Paper Organization The remainder of this paper is organized as follows. In Section II, wind and demand scenario generation are described, as well as the decision framework and the chance of not meeting the demand for all scenarios. In Section III, the CCUC problem is formulated. Section IV presents several case studies using the proposed methodology. The main conclusions are summarized in Section V. II. METHODOLOGY A. Wind and Demand Scenario Generation Demand and wind generation are considered here as random functions depending on the random sets and , respectively. In our paper, these random functions are dependent on the time period , which is assumed to be 1 h, but a smaller time period can be considered to simulate real-time stochastic demand and wind generation [18]. Several models have been proposed for forecasting wind [19] where all of them have an associated forecasting error. In general, the errors increase as the forecasting horizon increases. Network constraints are not included in the model proposed in this paper. Therefore, the demand and wind production random functions can be described by normal distributions with zero-mean normally distributed errors for each time period due to the central limit theorem. Other statistical distributions can be considered in the model proposed. Then, the demand distribution function is de?ned as , where is the predicted load demand and the term is the zero-mean normally distributed forecast error of the random demand. The random wind power generation function is de?ned as a Normal distribution: , where the term is the zero-mean normally distributed forecast error of the random wind power generation. Since wind generation is considered a nondispatchable energy source, the net load is de?ned as the difference between the demand and the wind generation: . The net load is a normal random function with zero mean error. Assuming both demand and wind power forecast errors are uncorrelated, the net load is de?ned as , where and the deviation is given by . A Monte Carlo simulation is used to generate a suf?cient number of net load scenarios based on a Normal distribution for each hour. A Latin hypercube sampling technique has been used [20], yielding a good random sampling ?tted to the Normal distribution. B. Decision Framework A day-ahead market operation is considered where the ISO schedules the generators and reserves with economic and reliability criteria. Under economic criteria, the ISO maximizes the social welfare in a pool-based market in the so-called market clearing process. Wind generators submit their estimated productions , along with conventional generators, to the ISO. The ISO clears the market at demand level every hour for a 24-h time horizon; see Fig. 1. Because the estimated wind and demand have a forecast error, the ISO needs to schedule reserves to

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In the right-hand side tail (RHST), with high values of demand realization and low values of wind production, the scheduled up spinning reserve has to be deployed to meet the demand. is de?ned as a reliability distribution function

Fig. 1. Decision framework.

(4) for any scenario. In A nonrisky stochastic UC makes case of load shedding, is allowed for those scenarios. The value of the load shedding is known as loss of load (LOL), and its probability of occurrence is de?ned as LOL probability (LOLP) as (5) The expected value of LOL is called expected load not served (ELNS), which is de?ned as (6)

Fig. 2. Wind energy spillage and load shedding.

ensure that there is enough power available to meet the demand using all wind generators. The energy reserves can be planned considering the generating units or the demands. In this paper, the reserves are scheduled for conventional generation units only. In addition, only spinning reserves are considered, disregarding nonspinning reserves. The model is formulated as a two-stage optimization problem. In the ?rst stage, a UC is solved with up and down reserve scheduling. In the second stage, the real demand has to be met with high probability. Reserves are deployed to meet real demand and load shedding or wind spillage are required in case the real demand cannot be met with the scheduled variables. simultaneous contingencies can take place during each hour of the scheduled day, where each contingency is cleared in the next hour. C. Chance of Not Meeting the Demand Fig. 2 represents the probability distribution function (PDF) of the net load for a particular time . The scheduled reserves must be deployed to satisfy the energy balance in case of any deviation from the scheduled net load to a random net load . A robust formulation schedule entails suf?cient up and down reserves to satisfy any net load, given as (3) The tails of the random functions have low probabilities, therefore there is no need to schedule reserves to satisfy any net load. However, there is a chance of not meeting the demand with the scheduled reserves and productions. In this case, there are demand or wind production curtailments: right- and left-hand side tails, respectively.

For the sake of simplicity the dependence of with respect to the ?rst-stage problem is omitted: on/off variables , and energy and scheduled reserve variables represented by vector . Thus, and, therefore, LOLP and ELNS are also functions of the ?rst-stage decisions, which are omitted for the same reasons. In the LHST of the net load distribution function there are low-demand scenarios with high-wind production (see Fig. 2). Similar to , the reliability distribution function is de?ned as (7) for all scenarios. A nonrisky stochastic UC makes In the case of wind spillage, for those scenarios. The wind spillage probability (WSP), de?ned as follows: (8) is de?ned as the probability of having wind spillage. The expected value of WSP is de?ned as expected wind spillage (EWS) as follows: (9) Note that a minus sign is added in (9) to have positive values of EWS. and are related to the economic interpretation of CVaR referring to losses and gains, respectively. The advantage of using CVaR is that it represents a coherent risk measure with good properties. The balance equation (3) de?ned for all scenarios may not be satis?ed in two cases, given here. 1) Uncoupled probabilistic period-to-period constraints: (10) (11)

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2) A single probabilistic constraint for all periods:

(12) (13) Equations (10) and (11) de?ne the individual probabilistic constraints for each time period in the load shedding and wind spillage cases, respectively. In the load shedding case (10), the probability of is bounded by and must be greater than . In other words, ELNS is bounded by with a high probability, . A similar interpretation can be used in (11) for the wind spillage limits. On the other hand, (12) and (13) set a single probabilistic constraint for the entire time horizon. III. PROBLEM FORMULATION The UC problem has a deterministic level of demand , with the scheduling of spinning reserves to meet any demand , and with a wind distribution function or, equivalently, a net load . The chance-constrained two-stage UC formulation without contingencies is de?ned at the ?rst-stage problem by the minimization of the cost of the energy dispatched, the up and down spinning reserves scheduling, and the expected cost of the second-stage. It is given by

cost of the energy dispatched composed of the energy commitment , the up and down spinning reserves , and the expected second-stage resource cost . For the sake of simplicity the costs are de?ned by linear functions: ; . Equation (15) represents the energy balance for the day-ahead market where the load and wind power have sent their expected demands and wind productions to the ISO, respectively. is the expected net load obtained from the difference between the expected demand and the expected wind generation. Equation (16) shows that the energy committed must be bounded by the generating unit limits and . Equations (17) and (18) are the ramp-down/ show the ramping limits, where ramp-up limits and are the shutdown/startup ramp limits for unit . Constraints (19) and (20) set the technical limits for the units including the up and down spinning reserves in the same way as (16). Equations (21) and (22) set the maximum up and down spinning reserves. Equations (23) and (24) show the ramping limits including the reserve scheduling in the same way that (17) and (18) do. The second-stage problem

(26) (27) is de?ned after uncertainty is realized given the ELNS or EWS limits, where reserve deployment acts as a corrective action. A linear objective function (26) penalizes ELNS and EWS with a cost of and , respectively. In general, load shedding has worse consequences than wind spillage and most literature has paid attention to load shedding modeling disregarding wind spillage. Therefore, . The value of should be close to the market clearing price and it represents the cost of opportunity to produce with the spilled wind energy. The chance-constrained two-stage UC model is modeled as an MILP except for the two probabilistic constraints at the second-stage. In the next sub-section the chance constraint is reformulated as a linear set of constraints. A. Linear Chance Constraint Formulation

(14) subject to the deterministic scheduling constraints (15) (16) (17) (18) (19) (20) (21) (22) (23) (24) (25) where variable de?nes the energy committed for generator at time , is the on/off binary variable for the commitment of the units, and are the scheduled up/down spinning reserve variables. The objective function (14) minimizes the total

For any probability level

,

can be formulated as

(28) where and de?nes the probability of the distribution. SAA is used for the distribution with samples, . Then, given , the chance constraint (10) is de?ned as [7]

(29) (30) where the optimal value tween 0 and . is the highest possible value be-

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Likewise, if a single chance constraint is used for all periods (12), then (29) is substituted by (31) because of the linear de?nition of the chance constraint, i.e. . In a similar way SAA is de?ned for EWS with (32) (33) where the optimal value is the highest possible value between 0 and . Then, the chance-constrained UC problem is de?ned as an MILP problem with the objective function (14) subject to the ?rst-stage constrains (15)–(25) and the second-stage constraints given by (30)–(33). Note that this linear approach deals with probabilistic constraints without including any binary variable. B. Chance-Constrained Unit Commitment With Contingencies Given the scheduled energy and spinning up and down reserves from the ?rst stage, system stability needs to be ensured for any contingency of the conventional generators. First, the RHST of the case of loss of load is studied, and, later, the LHST of the case of wind spillage is studied. Since the network is not modeled, both events (LOL and WS) cannot happen simultaneously. The worst possible contingencies are those where the demand is not met even if the up spinning reserves are deployed, . Assuming that takes the value zero when unit has a contingency at time , the worst case contingency problem can be de?ned for the RHST of the random function as shown in (34)

variables since the solutions are always at one vertex of the polyhedron formed by the feasible region [21]: (37)

(38) (39) The dual variables are shown on the right-hand side of the constraints. The ELNS constraint (6) is now calculated with the worst contingency random function de?nition (37) and constraints (38) and (39). This formulation contains a min operator which makes ?nding the resolution more dif?cult since, if the KKT conditions are rewritten, the linear properties disappear. Duality theory has been used to reformulate the problem as a set of linear constraints, described in the Appendix. For the LHST of the random distribution, outages of one or more units may decrease wind spillage since the scheduled production or the down reserves may be replaced by the wind spilled or by the scheduled up reserves. Hence, the worst case for the LHST under outages is given by the linear constraints (32) and (33). Therefore, the independent CCUC problem is de?ned by the objective function (14) subject to the ?rst-stage constraints (15)–(25) and second-stage constraints with contingencies given by the set (32) and (33) and (51)–(55). The problem is stated as a MILP where multiple criteria at different con?dence levels can be added. C. Probability of Simultaneous Contingencies In the previous subsection, a MILP formulation is proposed to security criteinclude various levels of reliability with an rion. Assuming that all contingencies have the same probability of occurrence, the probability of a single contingency is greater than having two simultaneous contingencies and also greater than three simultaneous contingencies. Therefore, a single equation for multiple security criteria for ELNS can be formulated as

(35) (36) where in (34) represents the LOL for the worst case contingencies in period , and it is a random function of . The down reserves have not been included in the objective function because they are not deployed in the RHST. Equation (35) limits the number of contingencies to simultaneous contingencies in each period. It is assumed that the contingencies are cleared in the same period, . For the next to periods a new set of worst contingencies can happen. This assumption is more conservative than assuming that the contingency is not cleared during the scheduling period. The optimization variable is the worst contingency, , de?ned as a binary variable (36). Since the coef?cients of all constraints have unitary values for the decision variables, the entire binary linear problem can be transformed into a linear problem with continuous decision (40) is de?ned as the set of constraints [cf. where (52)–(55)] parameterized by contingencies with a con?dence level. is the probability of occurrence of simultaneous contingencies and represents the probability of no contingency, which should be close to 1 for reliable systems. The values to are obtained from the probability of failure of each of the generating units. If a generating unit has a prede?ned MTBF (mean time between failures) obtained from historical or simulated data, then, the probability of failure of a generating unit is . Then, the probability of a single contingency in the system is (41)

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TABLE II ORDER OF COMPLEXITY

TABLE III CONVENTIONAL GENERATOR DATA FOR A TEN-UNIT SYSTEM

and the probability of two simultaneous contingencies ignoring the dependence between them, e.g. cascade failures, is given by (42) where the right-hand side is justi?ed in case of a large number of units, . The probability of not having a contingency is calculated in (43) given a joint constraint of simultaneous contingencies. Note that a joint EWS has not been de?ned because it does not depends on the number of the contingencies [see (32) and (33)]. The joint CCUC problem is de?ned by (14)–(25), sets of constraints [cf. (52)–(55)], the set (32) and (33), and the joint chance constraint (40). The resulting problem is stated as an MILP. The peculiarity of the independent CCUC model is that it is solved assuming that simultaneous contingencies will happen but the system will remain stable. However, the joint CCUC model assumes that the probability of simultaneous contingencies ranges between 0 and . Therefore, the joint CCUC is penalized in the objective function and in the chance constraint (40) with the probability vector , resulting in a lower value of scheduled reserves. D. Computational Issues Joint energy and reserve unit commitment with security criteria models are generally dif?cult to solve because the combinatorial number of contingency scenarios makes the problem intractable. The computational order of the independent CCUC model proposed here does not depend on the number of contingencies, , and the joint CCUC model depends on a polynomial complexity, as shown in Table II. Table II illustrates the comparison between the order of complexity of the two proposed models with two other models. The ?rst one is an equivalent robust optimization model using stochastic wind and demand based on [16]. The second one is a contingency scenario-dependent model where the contingencies are treated as scenarios based on [10]. Symbol is used in Table II to de?ne the total number of periods, is used for the total number of generators, for the total number of wind and demand scenarios, for the total number of simultaneous contingencies, and for the total number of combinations of contingencies, i.e., . The second column shows the order of the binary variables, the third and fourth columns show the order of the continuous variables and constraints, respectively. The order of complexity for the models proposed in this work has advantages over the scenario-dependent contingency models, where the combinatorial number of contingency scenarios, , imposes a substantial computational load, making the problem intractable for large systems. The robust optimization model in [16] has the same order of complexity as the independent CCUC model proposed here. Additionally, the joint CCUC model has polynomial complexity depending on the number of simultaneous contingencies, . Because is usually small, the expected computational load for solving both models should be similar. IV. CASE STUDIES For the purpose of illustrating our methodology the system presented in [15] is considered. The original problem consists of a ten-unit system that has been scaled four times replicating the ten-unit system and scaling the demand accordingly. Therefore the demand is stochastic and ?tted to a normal distribution with a mean of 2800 MW and 10% of standard deviation. 100 scenarios have been generated with a Latin hypercube sampling technique. The original data [15] is presented in Table III. The second and third columns show the minimum and maximum capacities of each generating unit. The fourth column shows the upper and lower limits of the spinning reserves. The ?fth to seventh columns show the ?xed cost of switching on the generating unit, the marginal cost of energy production and the cost of the up and down reserves, respectively. The probability of unit failure is the same for all units, 1/1000 (hour/hour). We have assumed the contingency probability for each unit to be independent from the other units’ and independent from the other contingency states. The values of the unit cost of the energy not served, , and the unit cost of the wind energy spilled, , are $100/MWh and $10/MWh, respectively. All case studies have been solved using CPLEX 11 under GAMS [22]. An Intel Core Duo E7500 computer at 2.93 GHz and 4 GB of RAM has been used. A. Single Period Case Wind power generation is included in a 40-unit system and 10% wind power penetration factor. Then, 10% of wind power

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Fig. 3. Total cost versus wind energy penetration. Top: independent criterion. Bottom: joint criterion.

Fig. 4. Scheduled up reserves versus wind energy penetration. Top: indepencriterion. Bottom: joint criterion. dent

penetration means that 10% of the demand is dispatched with wind energy and 90% with conventional energy. The demand is scaled to the same proportion of the wind penetration to simulate that the demand growth is covered with the new wind generators. Uncertainty in wind power generation is modeled using a normal distribution with a mean value obtained from the wind power penetration and a standard deviation equal to 50% of the mean value, assuming that the wind power costs are null. Figs. 3–5 show the total costs and the up and down reserves scheduled for different wind penetrations ranging from 0% to 90% levels and an security criterion. The con?dence level has been chosen as 0.97 for all cases, and the ELNS and EWS are limited to 1% of the conventional productions. The total cost increases with the wind penetration because the up and down reserves scheduled increase with the wind energy penetration (Figs. 4 and 5). The higher the wind energy penetration is, the higher the up and down reserves are due to the net load deviation increase. Note that all 40 units are connected in the case of high wind penetration (higher deviations). Therefore, they are responsible for satisfying the demand with random wind scheduling, and more up and down reserves are necessary than in the case of low wind penetration. The total cost increases in the same way that the scheduled reserves do. For the independent security criterion case, the up reserves increase about 455 MW for each plausible outage, which is equivalent to lose the biggest scheduled unit. This is because the 455-MW units have the cheapest energy and reserves costs. Hence these units are scheduled in the optimal solution. However, an outage of one of these units has a major impact on the system since the energy previously committed by these units must be supplied by other generators’ scheduled reserves. As seen on top of Fig. 4, the up reserves values are almost parallel

Fig. 5. Scheduled down reserves versus wind energy penetration. Independent criteria. and joint

for each reliability criteria, , due to the successive loss of a 455-MW generating unit each time increases. For the joint security criterion case, the up reserves increase proportionally to the probability of having simultaneous contingencies. As in the independent criterion, the worst contingency occurs when a scheduled 455-MW generating unit has an outage. However, in the joint criterion, a probability vector measures the effect of multiple outages following a geometric series. For this case of study, the probability of having a single contingency in an hour is , and the probability of having two simultaneous contingencies in an hour is , for three simultaneous contingencies the probability is , for four simultaneous contingencies is and for ?ve simultaneous contingencies is . Therefore, from onwards, the up reserves remain almost the same. The total costs (Fig. 3) for the independent and joint criteria maintain a pro?le similar to the up reserves.

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TABLE IV CPU TIME (s)

Independently constrained.,

Jointly constrained

Fig. 6. Multiperiod scheduled generation and reserves for the 40-unit system.

security criteria: independent and joint and signi?cant installed wind capacity. The chance constraints are de?ned as reliability constraints of the system limiting the ELNS and EWS. The model is reformulated as MILP solving the joint energy and reserve scheduling and UC for the day-ahead market, considering: 1) a linear approach for the chance constraint based on CVaR de?nition and 2) a dual formulation of the second-stage problem allowing to recast the problem as a linear set of constraints for the worst contingencies. The weak duality condition has replaced the maximum operator in the second-stage for a new upper boundary function. The proposed model can handle problems that are essentially intractable for multiperiod contingencies. Additionally, the model ensures high utilization of intermittent renewable energy sources with high reliability and robustness. Several illustrative examples show the performance of our model which is solved in a moderate computation time. The proposed model can be extended including the network representation into the constraints and lines outages. A new energy and reserve pricing scheme under security criteria can be analyzed. In addition, speci?c scenario reduction techniques can be applied to focus on the tail of the random distributions. These topics are left for future work. APPENDIX SECOND STAGE WITH CONTINGENCIES: RHST LINEAR REFORMULATION Given the second-stage problem as the worst case contingency of the LOL problem (37)–(39), the dual problem is stated as

The scheduled down reserves (Fig. 5) are based on the no-contingency case for the independent and joint criteria because contingencies help to reduce EWS. Note that for higher wind energy penetration the down reserves increase because the forecast errors are higher. Table IV shows the running times required for solving the problems for , 1, 3, and 5 for a case of 10% of wind penetration with the independent and joint criteria, JC and IC, respectively. B. Multiperiod Case This section presents the results for a 24-h case study based on the same 40-unit system. With 10% wind penetration, the net load demand is generated and computed in a multiperiod framework (see the gray lines in Fig. 6). The forecast errors increase along with time, therefore demand errors ranging from 2 to 10% and wind errors ranging from 5% to 50% have been chosen. The con?dence level is ?xed at 95% and ELNS and EWS are limited to 1% of the total net load for the 24 h. Fig. 6 shows the resulting demand and the scheduled up and down reserves for the independent CCUC model with security criterion where ranges from 0 to 3. Note that, for high demand levels, the up reserves increase signi?cantly because the 455-MW generating units are fully scheduled. The up reserves from other generating units need to be scheduled due to the outages of the 455-MW units. However, for low demand levels, the up reserves are quite similar because the generating units are not fully dispatched and an outage of a unit needs less up reserves. The CPU time used has been {2889, 2945, 3242, 3134} s for solving the cases 0, 1, 2, 3, respectively. V. CONCLUSION This paper presents a two-stage chance-constrained stochastic model for the UC problem with -quantile two

(44)

(45) (46) According to the weak duality condition, a vector of feasible dual variables always meets the following constraint: (47) where is the optimal solution of the primal (original) problem. Then, the upper bound of the optimal global solution of the problem (37)–(39) is de?ned as follows:

(48) as ELNS for period with simultaneous contingencies is de?ned based on the probability distribu. Similarly, is detion function, ?ned. Because constraint (49) holds by the duality theory and the ELNS is a coherent risk measure (i.e., the monotonicity

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axiom holds [23, Axiom M]), then the following inequality is satis?ed: (49)

(50) Note that inequality conservative than is more . Then (51) (52)

(53) (54) (55) Then, the probabilistically constrained problem for an security criterion is equivalent to the set of constraints (51)–(55) parameterized in and with con?dence level . REFERENCES

[1] J. Birge and F. Louveaux, Introduction to Stochastic Programming. Berlin, Germany: Springer-Verlag, 1997. [2] A. Ben-Tal, L. El Ghaoui, and A. Nemirovski, Robust Optimization. Princeton, NJ: Princeton Univ., 2009. [3] D. Bertsimas and M. Sim, “The price of robustness,” Oper. Res., pp. 35–53, 2004. [4] Q. Wang, Y. Guan, and J. Wang, “A chance-constrained two-stage stochastic program for unit commitment with uncertain wind power output,” IEEE Trans. Power Syst., vol. 27, no. 1, pp. 206–215, Feb. 2012. [5] U. Ozturk, M. Mazumdar, and B. Norman, “A solution to the stochastic unit commitment problem using chance constrained programming,” IEEE Trans. Power Syst., vol. 19, no. 3, pp. 1589–1598, Aug. 2004. [6] S. Ahmed and A. Shapiro, “Solving chance-constrained stochastic programs via sampling and integer programming,” in Proc. Tutorials in Operations Research: State-of-the-Art Decision-Making Tools in the Information-Intensive Age, Washingont, DC, Oct. 15, 2008, pp. 261–268. [7] R. Rockafellar and S. Uryasev, “Optimization of conditional value-atrisk,” J. Risk, vol. 2, pp. 21–42, 2000. [8] J. Arroyo and F. Galiana, “Energy and reserve pricing in security and network-constrained electricity markets,” IEEE Trans. Power Syst., vol. 20, no. 2, pp. 634–643, May 2005. [9] F. Bouffard, F. Galiana, and J. Arroyo, “Umbrella contingencies in security-constrained optimal power ?ow,” in Proc. 15th Power Systems Computation Conf., Liege, Belgium, Aug. 22–26, 2005, pp. 1–7, Session 1, Paper 4.

[10] F. Bouffard, F. Galiana, and A. Conejo, “Market-clearing with stochastic security-part I: Formulation,” IEEE Trans. Power Syst., vol. 20, no. 4, pp. 1818–1826, Nov. 2005. [11] F. Bouffard, F. Galiana, and A. Conejo, “Market-clearing with stochastic security-part II: Case studies,” IEEE Trans. Power Syst., vol. 20, no. 4, pp. 1827–1835, Nov. 2005. [12] J. F. Restrepo and F. D. Galiana, “Assessing the yearly impact of wind power through a new hybrid deterministic/stochastic unit commitment,” IEEE Trans. Power Syst., vol. 26, no. 1, pp. 401–410, Feb. 2011. [13] E. Karangelos and F. Bouffard, “Towards full integration of demand-side resources in joint forward energy/reserve electricity markets,” IEEE Trans. Power Syst., vol. 27, no. 1, pp. 280–289, Feb. 2012. [14] F. Bouffard and F. Galiana, “Stochastic security for operations planning with signi?cant wind power generation,” IEEE Trans. Power Syst., vol. 23, no. 2, pp. 306–316, May 2008. [15] A. Street, J. Arroyo, and F. Oliveira, “Energy and reserve scheduling security criterion via robust optimization,” in Proc. under an 17th Power Syst. Computation Conf., Stockholm, Sweden, Aug. 22–26, 2011, vol. 1, pp. 624–630. [16] A. Street, F. Oliveira, and J. Arroyo, “Contingency-constrained unit security criterion: A robust optimization apcommitment with proach,” IEEE Trans. Power Syst., vol. 26, no. 3, pp. 1581–1590, Aug. 2011. [17] F. Bouffard and F. Galiana, “An electricity market with a probabilistic spinning reserve criterion,” IEEE Trans. Power Syst., vol. 19, no. 1, pp. 300–307, Feb. 2004. [18] P. Varaiya, F. Wu, and J. Bialek, “Smart operation of smart grid: Risklimiting dispatch,” Proc. IEEE, vol. 99, no. 1, pp. 40–57, Jan. 2011. [19] G. Giebel, R. Brownsword, G. Kariniotakis, M. Denhard, and C. Draxl, The state-of-the-art in short-term prediction of wind power: A literature overview,” 2011, Project ANEMOS. [20] A. Olsson, G. Sandberg, and O. Dahlblom, “On Latin hypercube sampling for structural reliability analysis,” Structural Safety, vol. 25, no. 1, pp. 47–68, 2003. [21] G. Nemhauser and L. Wolsey, Integer and Combinatorial Optimization. New York: Wiley, 1988, vol. 18. [22] A. Brooke, D. Kendrick, A. Meeraus, and R. Raman, “GAMS/CPLEX 9.0,” User Notes 2003. [23] P. Artzner, F. Delbaen, J. M. Eber, and D. Heath, “Coherent measures of risk,” Math. Finance, vol. 9, no. 3, pp. 203–228, 1999. 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 working toward the Ph.D. degree. His research interests include power systems economics and electricity markets.

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

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