Scheduling and Pricing of Coupled Energy and Primary, Secondary, and Tertiary Reserves

Transcription

1 Scheduling and Pricing of Coupled Energy and Primary, Secondary, and Tertiary Reserves FRANCISCO D. GALIANA, FELLOW, IEEE, FRANÇOIS BOUFFARD, STUDENT MEMBER, IEEE, JOSÉ M. ARROYO, MEMBER, IEEE, AND JOSÉ F. RESTREPO, STUDENT MEMBER, IEEE Invited Paper Current practice in some electricity markets is to schedule energy and various reserve types sequentially, first clearing the energy market, followed by the reserves needed. Since distinct reserve services can in fact be strongly coupled, and the heuristics required to bridge the various sequential markets can ultimately lead to loss of social welfare, simultaneous energy/reserves market-clearing procedures have been proposed and are in use. However, they generally schedule reserve services subject to exogenous rules and parameters that do not relate to actual operating conditions. The weaknesses of the current approaches warrant the investigation of alternatives. In that regard, we present a different methodology to the simultaneous market clearing of energy and reserve services. This approach avoids the pitfalls of the sequential procedures, while at the same time its basis for scheduling reserve services does no longer rely on some rules of thumb. The salient feature of the proposed approach is that, under marginal pricing, it yields a single price for all reserve types scheduled at a bus, unlike the current approaches. We show that this common price is given by the nodal marginal cost of security. We present two specific implementations of a simultaneous security-constrained market-clearing procedure, one deterministic and one probabilistic. An example of joint market clearing of energy with reserves required for primary and tertiary regulation illustrates how their strong coupling affects their schedule and prices. Keywords Electricity markets, energy and security pricing, marginal pricing, regulation, reserve, security-constrained market clearing, transmission constraints. Manuscript received October 1, 2004; revised June 1, This work was supported in part by the Natural Sciences and Engineering Research Council of Canada, and in part by the Fonds québécois de la recherche sur la nature et les technologies. The work of J. M. Arroyo was supported in part by the Ministry of Science and Technology of Spain under Grant CICYT DPI , and in part by the Junta de Comunidades de Castilla-La Mancha, Spain, under Grant GC F. D. Galiana, F. Bouffard, and J. F. Restrepo are with the Department of Electrical and Computer Engineering, McGill University, Montreal, QC H3A 2A7, Canada ( galiana@ece.mcgill.ca; francois.bouffard@mcgill.ca; jose.restrepo@mail.mcgill.ca). J. M. Arroyo is with the Departamento de Ingeniería Eléctrica, Electrónica y Automática, E.T.S.I. Industriales, Universidad de Castilla-La Mancha, Ciudad Real E-13071, Spain ( josemanuel. arroyo@uclm.es). Digital Object Identifier /JPROC I. INTRODUCTION This paper addresses the issue of how to schedule and price energy and various reserve services in security-constrained electricity markets. A common current practice is to schedule and price these services as if they were weakly coupled, using sequential market-clearing procedures that lead to correspondingly separate prices [1] [5]. Since distinct services can in fact be strongly interrelated, we argue in favor of simultaneous [3] [18] over sequential market-clearing procedures to avert the need for ad hoc operator intervention, uneconomical out-of-merit operation, the start-up of extra units, as well as unnecessary load shedding. Nevertheless, current simultaneous approaches are still not satisfactory, as they schedule reserve services according to rules whose robustness, in the face of a spectrum of disturbances, may be uncertain. The flexibility of the generators to alter their outputs under load following or to respond to major contingencies is what defines generation-side operating reserve services. Similarly, demand-side operating reserve is defined by the flexibility of the loads to alter consumption of their own accord. Involuntary decreases in consumption are termed load shedding. In this paper, we envision the future generations of shortterm security-constrained electricity markets for which in the event of any of a set of prespecified credible contingencies, enough generation and consumption flexibility relative to the predisturbance levels should be made available to balance power at every bus [19] [27]. Generally, the set of credible contingencies consists of outages of major lines and generators, as well as large demand variations. In addition, such security-constrained electricity markets must provide sufficient reserve for load following, that is, to regulate the area control error during normal load fluctuations with time [28]. Security can be defined in a deterministic [19], [29] or probabilistic sense [30] [33]. In the former, for each postdisturbance state, irrespective of its likelihood of occurrence, power must balance without load shedding. Alternatively, a /$ IEEE 1970 PROCEEDINGS OF THE IEEE, VOL. 93, NO. 11, NOVEMBER 2005

2 probabilistic security criterion may allow power to balance at every node with some amount of load shedding provided that the expected value of the load shed is either as small as possible or below some specified level. In addition, as all reserve services act with the common purpose of ensuring that the security criterion is met, we argue that all such services should be priced equally at each bus and furthermore that this nodal price should be the nodal marginal cost of security. This paper first surveys the current and recent industry practices regarding reserve services definitions, scheduling, and pricing. Based on their different time responses and control characteristics, we then adopt an alternate approach to the current reserve services definitions. This approach permits the time decomposition of the power balance requirements after any given contingency. We then describe specific details of the simultaneous security-constrained market-clearing problem which is based on this control time decomposition. Next, we establish the pricing result highlighted above through a general mathematical form, valid for both deterministic and probabilistic security criteria. Finally, a numerical example illustrates the strong coupling that may exist between the predisturbance schedule and the reserve services, as well as the influence that simultaneous market clearing has on security and energy prices. II. PAST AND CURRENT INDUSTRY PRACTICES The North American Electric Reliability Council (NERC) defines a number of operating reserve services which are further classified as spinning or supplemental [34]. Spinning reserves serve the purposes of: 1) automatically responding to important contingencies so as to keep the system-wide power balance that is, to provide primary regulation, and 2) following the normal load variations through automatic generation control (AGC) that is, to provide secondary regulation. The provision of supplemental reserves, on the other hand, is not limited by the synchronization state of the generators. That is, supplemental reserves may be provided by generators that are spinning and synchronized to the grid or by generators that are not synchronized yet. Supplemental reserves, sometimes called replacement reserves, serve the purpose of responding to disturbances but with a longer time lag of several minutes that is, to provide tertiary regulation. The deployment of supplemental reserves permits the repositioning of generation levels such that the faster spinning reserves (used mainly for primary and secondary regulation) become available to respond to further disturbances. Very often, quality attributes are attached to the different types of reserves. Spinning reserves are said to be of high quality because they are very valuable as they keep the power system in balance in the immediate instants following a disturbance. Classic steam and large hydro generators, constituting the majority of the base load capacity on-line most of the time, provide most of those fast response reserve services [35]. On the other hand, because their deployment occurs with some time lag from the initiating moment of a disturbance, supplemental reserves are said to be of lower quality. Gas turbines and pumped-storage hydro plants are well suited to provide supplemental reserve services because they can be started and ramped up very rapidly [35]. Past and current operating reserves scheduling strategies are many, of which we distinguish three main flavors which possess increasing levels of complexity. The simplest method uses a sequential reserve procurement approach scheduling the reserve services in a series of auctions run once energy schedules have been determined in a separate market. Here the sequence of auctions first schedules the high-quality services followed by the services of decreasing quality levels [2]. Each individual auction is either run on the basis of minimizing the total cost of procurement or the net social cost [3]. Generally, under these approaches, unused reserve capacity in a high-quality service auction is carried over to the next lower quality service auction in the sequence. This is based upon the principle that high-quality reserves are considered to be substitutes for lower quality reserves in the case when they are less expensive. The sequential procurement approach was taken in the first years of operation of the California Independent System Operator (ISO) and was proven to be a poor reserve market design prone to manipulations by the market players [3]. Moreover, the functional decoupling of the energy and reserves markets constitutes another weakness of this approach in cases when the original energy schedule does not lead to the commitment of sufficient capacity so as to meet the reserve requirements. In such instances, the system operator has to resort to out-of-market manipulations which, evidently, lead to losses of social welfare. The second reserve procurement method schedules simultaneously the various services subject to requirements for each, but is still separate from the energy market [3]. This approach embeds explicitly the reserve services substitution properties. The objective of the reserve procurement under this approach may be based on social cost minimization or on procurement cost minimization also known as the rational buyer approach [3], [11]. We note that this scheduling scheme also suffers from the decoupling between the energy and the reserve markets. The third way by which system operators schedule reserve services is through the simultaneous scheduling of energy and reserves. Here the scheduling of energy and reserves are optimized concurrently subject to: 1) a single or several nodal power balances and 2) requirements for each of the reserve services [4] [10], [14] [18], thus avoiding the energy/reserve markets separation pitfall. This approach is generally associated with the U.S. East Coast ISOs (PJM, ISO New England, and New York ISO). Very often this simultaneous approach is called security-constrained market clearing. In all three approaches, the scheduling procedures yield separate prices for the various reserve services [2] [18]. In procurement procedures where social cost minimization is used, the services are priced at their marginal costs. Under the rational buyer approach, the prices of the different services are those minimizing the procurement costs. Lastly, the pay-as-bid option remunerates the providers of reserve services in such a way that they recover their offered costs. In the GALIANA et al.: SCHEDULING AND PRICING OF COUPLED ENERGY AND PRIMARY, SECONDARY, AND TERTIARY RESERVES 1971

3 United Kingdom, under the New Electricity Trading Agreements (NETA) a pay-as-bid pricing scheme was adopted to compensate the providers of reserves in the energy balancing market [36]. In a quest for computational simplicity, the past and the current approaches have reserve requirements set by parameters or rules exogenous to the reserve scheduling procedure. This philosophy has the effect of putting the emphasis merely on the quantities of the different services rather than on their degree of adequacy to respond to a spectrum of system disturbances, which, in general, could lead to economically suboptimal and even infeasible operating states. III. TIME DECOMPOSITION OF RESERVE SERVICES ACTIONS In this section, we address reserve services from the point of view of their time of deployment following the occurrence of a contingency as well as the corresponding control strategies behind their deployment. This alternate approach is required by the fact that the notion of power balance1 is fundamental in the definition of security and the response time taken by generating units following a disturbance. Besides the predisturbance power balance, three other steady-state power balance relations can be distinguished: 1) that of the primary regulation interval occurring within seconds of a disturbance, where the goal is to keep the system frequency within a prescribed range through governor droop response [28], [35]; 2) that of the secondary regulation interval associated with normal load-following, within a time frame of several minutes, where the goal is to bring the area control error to zero through participation factors [28], [35], [37]; and finally 3) the power balance occurring about min after a contingency, that of the tertiary regulation interval, where the goal is to impose a new postcontingency set point with zero area control error that also meets all prescribed transmission flow constraints [2] [13], [27], [38]. These types of power balance equations happening in sequence following a disturbance form the defining elements of the system security criterion. A. Primary Regulation Interval During the primary regulation interval, spinning generators automatically adjust to disturbances as their speed governors respond to deviations from the nominal system frequency through their droop characteristics [28]. These automatic actions are fast, stabilizing the system frequency within 5 10 s. The level of the above-mentioned adjustments is bounded within a relatively narrow range around the predisturbance set point. This range is highly dependent on the type of generation unit [35] and, in general, the reliability standards require that every generating unit over 10 MW must participate in providing reserve during the primary regulation interval [39]. In addition, this type of regulating action has to respect a specified range of system frequency excursions with respect to the nominal frequency, failing 1 Although a nodal balance must exist for both real and reactive power, for the sake of clarity, in this paper we consider only active power balance. which, load shedding or generation tripping would follow, potentially leading to system instability. We note as well that as loads may respond to changes in system frequency, the consumers also adjust their load during the primary regulation interval, although they may usually do so unknowingly and unintentionally [28], [40]. It remains unclear whether demand-side primary regulation interval participation should be accounted for in clearing security-constrained electricity markets because of significant uncertainty in the frequency response characteristics of loads. B. Secondary Regulation Interval During the secondary or load-following regulation interval, spinning generators automatically vary their generation set points proportionally to some participation factors with the goal of nullifying the area control error in following normal load variations [28], [35], [39]. This control action typically has time constants of the order of a few minutes [28], [35]. The objective during the secondary regulation interval is to follow slow and small time-varying load fluctuations around the predisturbance demand, and not to correct the set points after major contingencies, a task assigned to the tertiary regulation For each generator, during the secondary regulation interval, both up and down adjustments are centered on the predisturbance set point and are within limits equal to or broader than those imposed by the primary regulation C. Tertiary Regulation Interval During the tertiary regulation interval, typically with time constants exceeding 10 min, a variation of the predisturbance set points is scheduled for each credible contingency within the prescribed generation, demand, and transmission flow limits, while ensuring that the area control error is zero and that power balances at every bus. Again, the bounds on tertiary regulation interval actions are centered on the predisturbance set points and are equal to or broader than the bounds imposed by the primary and secondary regulation intervals. During the tertiary regulation interval, it becomes necessary to impose the condition that, following any credible contingency, all line flows should lie within their steady-state operating limits. This line flow limit condition is not present in primary and secondary regulation intervals for two reasons: 1) lines may operate beyond their flow limits for short periods without damage and 2) during the primary and secondary regulation intervals, the controls generally do not have sufficient degrees of freedom to respect all flow limits under all disturbances. Tertiary regulation permits a wider participation of demand-side and nonspinning generation-side reserves due to the longer time lag allowed before their deployment than the few seconds and minutes available for the other two intervals. D. Offered Reserve Services The above-mentioned power balance relations cannot be considered to be generally independent, since they share a 1972 PROCEEDINGS OF THE IEEE, VOL. 93, NO. 11, NOVEMBER 2005

4 common predisturbance set point. Consequently, the various reserve services, given by the postcontingency or load-following power adjustments, are also coupled. The reserve services associated with the primary, secondary, and tertiary regulation intervals can be offered by either the generators or the loads. In addition, these reserve services can also be subdivided into up or down services. For example, following a major contingency, under transmission congestion the tertiary reserve deployment actions may require that some units provide up adjustments at the same time that other units supply down adjustments [27]. In this paper, we consider a market allowing distinct offers to be submitted for each of the following services: 1) primary up generation reserve; 2) primary down generation reserve; 3) secondary up generation reserve; 4) secondary down generation reserve; 5) tertiary up generation reserve; 6) tertiary down generation reserve; 7) tertiary up load reserve; 8) tertiary down load reserve. Note that this list does not distinguish between spinning and nonspinning reserve services. Intrinsically, already spinning generators provide reserves associated with the primary, secondary, and tertiary regulation intervals, while nonspinning generators may only provide reserves associated with the tertiary regulation Here reserves are characterized solely by their deployment time delay rather than by their predisturbance synchronization state. E. Toward More Rigorous Security-Driven Reserve Scheduling and Market Clearing Security imposes that power should balance at every node during all three regulation intervals. The severity of any disturbance and thus the reserve services that need to be deployed in response depend on how the system is scheduled. Strictly speaking, it is necessary to have reserve schedules that truly reflect the requirements imposed by the set of credible contingencies and load variations. This proposal is a departure from current practice which uses prespecified reserve requirements so as to keep the computational effort within reasonable limits. Yet in light of the current improvement trends in the state-of-the-art computing machinery and software, the explicit consideration of the postdisturbance operating conditions, which model reserve requirements and deployment, is within reasonable reach. IV. SPECIFICS OF SIMULTANEOUS MARKET-CLEARING MODEL Two specific implementations of the envisioned marketclearing formulation, one deterministic and the other probabilistic are now described. For illustrative purposes, a singleperiod formulation is considered, as this model is simpler to describe and analyze, yet bringing out the main features of this market-clearing procedure. The nomenclature corresponding to this Section is found in the Appendix. A. Deterministic Model The deterministic market-clearing formulation maximizes the predisturbance social welfare. This is equivalent to where and represent respectively the total generation cost and the total demand benefit functions, with and the respective predisturbance generation and demand set points. The term is the cost function of all the reserve services offered by the loads and generators, denoted by the vector. In this section, the vector denotes the predisturbance on/off generation commitment status, although more generally it can also model the postcontingency on/off status of standby units that may be turned on after contingencies,. The minimization is subject to a predisturbance nodal power balance relation to the nodal power balance under the primary regulation interval and to the nodal power balance during the tertiary regulation interval where and are respectively the pre- and postcontingency network susceptance matrices which are of the same dimensions as the number of nodes remains constant both in the pre- and postdisturbance states. In the case when contingency corresponds to a line failure, the matrices and differ, while they are identical for other disturbance types. In(3)and(4),(, )and(, )arerespectively the vectors of generation and demand increments relative to their respective set points during the primary and tertiary regulation intervals following contingency. As discussed in Section III, the demand-side primary regulation component is assumed to be nil, that is, for all. The load-following nodal power balance is calculated at the prescribed upper and lower limits of the normal load variations with respect to the predisturbance demand set point, (, ). Equivalently, we have Typically, the prescribed up/down system demand variations for secondary regulation purposes are of the order of (1) (2) (3) (4) (5) GALIANA et al.: SCHEDULING AND PRICING OF COUPLED ENERGY AND PRIMARY, SECONDARY, AND TERTIARY RESERVES 1973

5 10%. The vector of participation factors is an output of the market-clearing process given by the sensitivity of the secondary regulation with respect to the demands (6) 5) Generation limits during the secondary regulation interval: (13) where. Note that all power balance equations, (2) (5), contain the predisturbance generation and demand set points and as common variables, thus coupling all regulation intervals. The power balance relations, (2) (5), are based on the dc load flow but more generally a nonlinear network formulation can be used. We also note that it is not necessary to use a full network model for the primary and secondary regulation intervals since, as discussed in Section III, line flow limits are not enforced during these intervals. Here, we use network models for all regulation intervals for the sake of generality and clarity of presentation. The numerical implementation of the market-clearing solution does, however, use single node power balance models for both the primary and secondary regulation intervals. In what follows, we describe the constraints applying to the variables specific to each of the regulation intervals. 1) Line flow limits during the predisturbance state and the tertiary regulation interval: (7) (8) As explained before, if under contingency, a particular line is out of service, then the corresponding flow limit is set to zero and the corresponding network matrices and are adjusted accordingly. 2) Limits on system frequency excursions during the primary regulation interval: (9) 3) Limits on predisturbance generation and demand set points: (10) (11) 4) Generation limits during the primary regulation interval: (12) 6) Generation and demand limits during the tertiary regulation interval: (14) (15) The above sets of constraints (items 3 6 above) reflect the physical output limitations given the specific capacities of the generating units and the demands. Next, we consider the noncapacity limits associated with the specific deployment delays of the different regulation intervals. 7) Generation adjustment limits during the primary regulation interval: (16) where for online units, if the primary generation adjustment is within the above limits, it is governed by [28] (17) 8) Generation adjustment limits during the secondary regulation interval: (18) 9) Generation adjustment limits during the tertiary regulation interval: (19) Note that ramping limits are implicitly incorporated in the above regulation bounds. Finally, the demand-side adjustments during the tertiary regulation interval are bounded by (20) 1974 PROCEEDINGS OF THE IEEE, VOL. 93, NO. 11, NOVEMBER 2005

6 The items that follow define the various reserve services being traded, noting that these are nonnegative quantities defined by the worst case disturbances. Each reserve service is offered at a possibly different cost, the sum of which makes up the objective function component. 10) Up and down reserve services associated with the primary regulation interval: regulation during normal load following, although small, could also be incorporated in the predisturbance state, but in this section, for simplicity, it has been neglected. Finally, since the probabilistic model permits load shedding to be applied during the tertiary regulation interval, the objective function also includes a corresponding expected cost of load shed (21) 11) Up and down reserve services associated with the secondary regulation interval: (24) (22) 12) Up and down reserve services associated with the tertiary regulation interval: where the parameter vector is the estimated rate at which the load shed,, is valued. The minimization is subject to the same set of equalities and inequalities as the deterministic problem with the exception of the tertiary power balance that now includes load shedding (25) (23) Since the market-clearing process maximizes social welfare including the cost of providing reserve services, the optimization automatically bounds the reserve levels from above. Lastly, we recall that multiperiod unit commitment requires also the satisfaction of other constraints such as minimum up- and downtime as well as interhour ramp limits [28]. B. Probabilistic Model In the specific market-clearing problem being considered here, the probabilistic model differs from the deterministic in three fundamental ways: 1) the objective function; 2) the presence of load shedding variables; and 3) the assumption that each contingency has a known probability of occurrence, where is the probability that no contingency occurs. Only the set of credible contingencies is considered, all other contingencies being assumed to have zero probability. The objective function now seeks to maximize the expected social welfare taking into account the predisturbance welfare as well as the welfare during the tertiary postcontingency states. We exclude the postdisturbance cost of reserves, since the reassessment of reserves and set points following a permanent outage would be carried out by rebalancing the market. Also, in this section, we have neglected the costs of primary regulation in the postcontingency states, as these costs are very small compared to that incurred during the tertiary regulation The cost of secondary where. C. Computational Issues This envisioned market-clearing model in both the deterministic and probabilistic flavors is clearly more complex than current market-clearing methods being used in the world today. First, its scope, as it covers simultaneously the preand postdisturbance states, is much broader thus requiring the definition of many more optimization variables and constraints. This issue is exacerbated by the intrinsic size of real power systems. However, the steady progress in computing technologies could well curb these difficulties within the coming years. It is always interesting to remind ourselves that the computing power required to run the current large control areas covered by some of the ISOs in the United States would have been considered to be out of practical reach 20 years ago. Second, questions regarding the existence of feasible schedules arise in any market-clearing formulation, this one not being an exception. There are a number of methodologies used in practice to circumvent this shortcoming; these include constraint relaxations and constraint violation penalizations added to the objective function. It is interesting to note that the probabilistic formulation described above is less prone to infeasibility, as it optimizes explicitly over possible load shedding actions for cases when the system has to be scheduled at or very close to its feasibility boundaries. Third, an issue also present in current markets is the heavy reliance on external data sources, which questions the robustness of any market-clearing formulation. GALIANA et al.: SCHEDULING AND PRICING OF COUPLED ENERGY AND PRIMARY, SECONDARY, AND TERTIARY RESERVES 1975

7 V. PRICING OF ENERGY AND RESERVES UNDER SIMULTANEOUS MARKET CLEARING The market-clearing formulation exposed in the previous section can be reexpressed more compactly in the following general mathematical form: subject to (26) (27) (28) (29) (30) (31) Above, the vector represents all continuous decision variables during the predisturbance state as well as during the primary, secondary, and tertiary regulation intervals. It includes bus generation and elastic demand levels, bus voltages, and load shedding if needed. The vector represents all binary decision variables such as unit commitment variables. The objective function denoted by represents a general measure of social welfare over the pre- and postdisturbance states. The equality constraints represent the nodal power balance requirements under the predisturbance state (27), the two postcontingency regulation intervals [primary (28) and tertiary (29)], and the up and down load-following secondary regulation interval (30). Note that all the power balance equations [(27) (30)] are defined by the corresponding load flow equations [(2) (5)] and therefore are of the same dimension. The same set of credible contingencies defines the primary and tertiary regulation interval power balance equations, since after each major outage both primary and tertiary control actions must be applied. As secondary regulation is scheduled with the objective of following normal load variations, major outages are not considered in its definition (30). Relation (31) represents all inequality constraints under the predisturbance state, the postcontingency primary and tertiary regulation intervals, and the load-following secondary regulation The vectors, and for, and (, ) are the Lagrange multipliers corresponding to the power balance relations (27) (30), while is the vector of multipliers associated with the inequality constraints (31). Next, we derive two basic quantities, the nodal marginal costs of energy and the nodal marginal costs of security, which under marginal pricing [41], [42] define the nodal prices of energy and security for the envisioned market-clearing formulation. The marginal costs2 of energy are the sensitivities of the optimum objective function (26) to a perturbation of the right-hand sides of the power balance equations (27) (30) by a common infinitesimal parameter vector. In this definition, we argue that if we perturb the power balance relation in the predisturbance state by, then the power balance relations after the contingencies and during load-following must also be identically perturbed. We also argue that since security is defined by the ability to satisfy all postdisturbance power balance relations, the marginal costs of security should be defined by the sensitivities of the optimum objective function (26) to a perturbation of the right-hand sides of the power balance equations, (28) (30), by a common infinitesimal increment vector. The same perturbation must apply to the primary, secondary, and tertiary regulation intervals so that the significance of power balance remain consistent across all regulation intervals. The fact that and are not necessarily the same means that we can incrementally adjust the strictness of the postdisturbance power balances (or security criterion) without necessarily modifying the strictness of the predisturbance power balance. Following the mathematical argument based on the application of those infinitesimal perturbations to the Karush Kuhn Tucker optimality conditions put forward in [27], we obtain the following infinitesimal increment in the scheduling cost (32) which, consequently, determines the nodal marginal costs of energy and the nodal marginal costs of security (33) (34) Some comments can now be made regarding this result. 1) According to marginal pricing theory [41], [42] the vectors of nodal energy and security prices are, respectively, the vectors of nodal marginal costs of energy and security, and. This marginal pricing rule is motivated economically from the fact that under such regime the prices of energy and security equal those of the next unit of energy and 2 These sensitivities should really be called marginal social welfare, but we have opted for the more commonly used marginal cost PROCEEDINGS OF THE IEEE, VOL. 93, NO. 11, NOVEMBER 2005

8 security respectively. This pricing scheme contrasts with a pay-as-bid pricing regime under which, from the consumers point of view, the prices represent the average costs of energy and security. 2) As stated in Section IV-A, each type of reserve is offered for sale by each agent at a possibly different rate. However, since all reserve types are securityenhancing services, they are all remunerated at the corresponding nodal marginal price of security. This result excludes cases where, for example, primary and tertiary reserve services have different prices, which may be the case of markets where reserve services are either sequentially procured or simultaneously dispatched to meet prespecified requirements. This different remuneration scheme will likely require that market participants develop new bidding strategies. 3) Since during the predisturbance state the energy produced by the generators and consumed by the loads satisfies system security, the price paid to the generators and charged to the loads during the predisturbance state is the marginal cost of satisfying all pre- and postdisturbance power balance relations. 4) The marginal cost of energy is equal to the marginal cost of security plus the Lagrange multipliers associated with the predisturbance power balance equations. This implies that when both energy and security prices are positive (the usual case), the marginal cost of energy is always greater than the marginal cost of security. Then, the price of reserve cannot be higher than the price of energy at any given bus, thus excluding those cases where the opposite behavior was reported [3]. This result is in agreement with the fact that the main commodity being traded in a power market is energy, while reserve services are complementary commodities that guarantee a secure delivery of energy. In unusual transmission-congested cases where some predisturbance Lagrange multipliers are negative, it is possible for some nodal energy prices to be lower than the corresponding security price. 5) Since the expressions for the marginal costs of energy and security in (33) and (34) are summations, they may be unbundled into separate terms representing the sensitivity of the optimum objective function to the various pre- and postdisturbance states. These unbundled sensitivities are significant, as they characterize the relative importance of the primary, secondary, and tertiary services on the total security price. However, we cannot consider these unbundled quantities as the marginal costs of the separate reserve services. This interpretation would be illogical, as it would violate the argument made earlier that the same power balance criterion must be applied across all regulation intervals. 6) In a deterministic security-constrained market, many postdisturbance Lagrange multipliers are zero, indicating that the corresponding disturbance is already Fig. 1. Table 1 Generator Data Three-bus system. compensated for by more restrictive umbrella disturbances [27], [33]. Incrementing the list of credible contingencies may add to the set of umbrella disturbances and increase the prices of security and energy [27]. In contrast, in a probabilistic security-constrained market, postdisturbance Lagrange multipliers are nonzero, as every disturbance has a finite probability of becoming active and reserve deployment costs are explicitly accounted for in the objective function for each of the credible disturbances [33]. VI. EXAMPLES We now examine the simultaneous scheduling of energy and reserve services associated with different regulation intervals. Consider the three-bus system of Fig. 1 with all line reactances equal to 0.63 per unit on a base of 100 MVA and 138 kv. The flows in lines 1 2 and 1 3 are limited to 100 MVA, while that in line 2 3 is limited to 25 MVA. Technical and economic generator data are given in Table 1. The generators offer energy at linear costs of the form. All generators offer primary and tertiary reserve services limited by their generation limits at the rates shown in Table 1: for primary up reserve, for primary down reserve, for tertiary up reserve, and for tertiary down reserve. The speed-droop constant is given in mhz/mw. The inelastic load of 50 MW located at bus 3 offers to sell tertiary regulation reserve services at the rates of MW/h by agreeing to decrease its consumption down to 20 MW or increase it up to 100 MW after a contingency. Four credible contingencies are considered, defined by the outage of any single generator. The system frequency excursion is limited during the primary regulation interval to 480 mhz. GALIANA et al.: SCHEDULING AND PRICING OF COUPLED ENERGY AND PRIMARY, SECONDARY, AND TERTIARY RESERVES 1977

9 Table 2 Optimal Generator Schedules (MW) Without Security Table 4 Optimal Schedules (MW) Table 3 Nodal Energy Prices ($/MWh) Without Security Table 5 Frequency Excursions (mhz) This system is analyzed under the deterministic marketclearing model. For the sake of conciseness, it is assumed that the load does not fluctuate, rendering unnecessary the deployment of secondary regulation reserve services. First, we discuss the implications of sequentially scheduling energy, followed by the primary and tertiary reserve services. This is followed by a detailed analysis of simultaneously clearing energy with primary and tertiary reserves. A. Discussion of Sequential Market Clearing Table 2 lists the generation schedule after the first step of a sequential market-clearing procedure in which energy is scheduled without considering primary and tertiary regulation interval security constraints. At this step, only generators 2 and 3 are committed. To respect the power flow capacity of line 2 3, the cheapest generator 2 limits its production to 37.5 MW with the remaining 12.5 MW being supplied by generator 3. Table 3 lists the nodal energy prices of this example, which are all different due to the congestion of line 2 3. The schedule shown in Table 2 meeting only the predisturbance generation and network constraints forms the starting point for sequential market-clearing schemes. A simple analysis shows that the loss of generator 2 would force the remaining generator 3 to provide all the reserve services under the primary regulation interval, leading to an infeasible steady state with a system frequency excursion of mhz. In addition, with this schedule, the power balance under the tertiary regulation interval can only be satisfied after the loss of generator 3 by scheduling 12.5 MW of up-demand reserve at bus 3 (i.e., by decreasing the demand by 12.5 MW) at the very high cost of $1250. Sequential scheduling of primary and tertiary regulation reserve services would therefore require committing some of the generating units that are scheduled off after the first step. In this example, a feasible solution meeting the primary and tertiary regulation requirements can be readily found by turning on either generator 1 or 4 and redispatching units 2 and 3. For larger systems, however, finding a new unit commitment is in general nontrivial. The final prices of energy and reserves moreover will depend on the specific sequential approach taken. For example, the redispatching method could be based on minimizing the deviations from the set point found in the first sequential steps or on some other objective [5]. In general, the sequential approach has several drawbacks: 1) if a complete recommitment of all units is permitted in the sequential steps, then these intermediate problems are as difficult to solve as a single simultaneous market clearing; 2) if units committed in a previous sequential step must remain committed, then the problem may be infeasible; and 3) the sequential solution always yields a worse level of social welfare. Moreover, under the sequential approach, the prices of primary and tertiary reserve services are inherently different, a result which contradicts the argument made here that, since all reserve products contribute to meeting security, they should be valued equally. Finally, under sequential market clearing, the energy prices shown in Table 3 do not reflect the effect of the primary and tertiary regulation intervals as is the case in the envisioned simultaneous approach. These prices therefore do not reflect the true cost of energy under all security constraints. B. Simultaneous Market-Clearing Procedure With Primary and Tertiary Security Constraints Table 4 shows the optimal participant schedules for the proposed simultaneous market clearing of energy with primary and tertiary reserve services. Since units 1 and 4 offer energy at the relatively high rates of 52 and $50/MWh, respectively, to meet the primary regulation interval requirements generator 4 is turned on because of its cheaper primary reserve offers while generator 1 remains off-line. Furthermore, since unit 4 has a technical minimum of 2.4 MW, the power schedule of generators 2 and 3 is modified relative to the regulation-unconstrained schedule of Table 2. In addition, the primary regulation interval also affects the predisturbance power outputs of generators 2 and 3 because, as shown in Table 5, the constraint limiting the frequency excursion is binding for the outage of generator 2. Consequently, the allocation of tertiary reserves to generators 2 and 3 is also affected by the primary reserve requirements. The load does not contribute to tertiary regulation due to its high offer of $100/MW/h. Table 5 lists the frequency excursions due to primary regulation following each contingency. Since generator 1 is not 1978 PROCEEDINGS OF THE IEEE, VOL. 93, NO. 11, NOVEMBER 2005

10 Table 6 Generation After Primary Regulation Interval (MW) Table 9 Lagrange Multipliers ($/MWh) for Tertiary Regulation Interval Table 7 Generation and Demand After Tertiary Regulation Interval (MW) Table 10 Nodal Prices of Energy and Security and Their Components ($/MWh) Table 8 Lagrange Multipliers ($/MWh) for Predisturbance State and Primary Regulation Interval committed in the predisturbance state, its outage does not require primary reserve. For the loss of units 3 and 4, the frequency constraint of 480 mhz is not binding; however, the loss of unit 2 leads to the maximum possible frequency drop. Table 6 shows the power supplied by each generator during the primary regulation interval due to the speed-droop characteristics. It should be noted that under the loss of generator 3, line 2 3 is overloaded during the primary regulation This overload is eventually eliminated by the tertiary regulation actions which enforce all line power flow limits. Table 7 shows, for all contingencies, the power supplied and consumed by each participant after the tertiary regulation The Lagrange multipliers associated with the power balance equations under the predisturbance, primary, and tertiary states are listed in Tables 8 and 9. The multipliers associated with the predisturbance state are all equal to $4.85/MWh because there is no line congestion in this state. The umbrella contingencies under both the primary and tertiary states are the outages of generators 2 and 3. Referring to Table 8, the outage of unit 3 leads to nonzero Lagrange multipliers ($0.15/MWh) during the primary regulation interval even though the system frequency deviation here ( 171 mhz) is not at a limit. This may seem counterintuitive but it is in accordance with the fact that the loss of generator 3 is the contingency that sets the level of primary regulation provided by generator 2 (8.13 MW in Table 4). In comparison, the outage of unit 2 results in much higher multipliers of $9.05/MWh. This is because under this outage the system frequency deviation is at its lower bound of 480 mhz, which means that the predisturbance set point is affected at a correspondingly higher cost. Note in Table 8 that since transmission constraints, unlike during the tertiary regulation interval, are not enforced during primary regulation, it is normal for all Lagrange multipliers to be equal under that regulation Referring to Table 9, the congestion of line 2 3 during the loss of generator 3 causes the observed differences in Lagrange multipliers associated with this contingency during the tertiary regulation The negative Lagrange multiplier at bus 2 is an economic signal that there is an excess of generation at bus 2 which is creating the observed congestion. Referring to Table 10, we see that: 1) the nodal prices of energy and security are all different due to line congestion; 2) the nodal prices of energy are all greater than the corresponding nodal security prices; and 3) the nodal prices of energy and security are highest at bus 3 since the entire load is located there and line 2 3 feeding this bus is congested in one of the postcontingency states (loss of generator 3). The economic signal sent by the high energy price at bus 3 is that there is a lack of generation or an excess of demand at that bus. The economic signal sent by the higher security price at bus 3 is that there is a lack of security-enhancing services at that bus compared to other buses. From the component,, at bus 3 ($34.00/MWh), we see that there is a greater need for low-cost tertiary reserves than for low-cost primary reserves whose corresponding price component is $9.20/MWh. This is consistent with the fact that primary reserves can come from any source in the network, whereas in this case the transmission constraint limits the transfer of tertiary reserves. The opposite behavior is observed at bus 2 where the security price tertiary reserve component,, is only $4.00/ MWh, indicating that there is a relatively good supply of tertiary reserve at that bus, either local or available from other buses. VII. CONCLUSION We have considered electricity markets which schedule and price energy as well as reserve services required to balance power during primary, secondary, and tertiary regulation intervals. We have argued that simultaneous market- GALIANA et al.: SCHEDULING AND PRICING OF COUPLED ENERGY AND PRIMARY, SECONDARY, AND TERTIARY RESERVES 1979

11 clearing procedures for these services offer several advantages over sequential markets: 1) simultaneous schemes explicitly model the coupling that exists between the various reserve services and energy; sequential methods assume that this coupling is weak by first scheduling energy, usually followed by the reserve services needed for the primary, then secondary, and finally tertiary regulation intervals. To bridge these sequential steps a number of heuristics are used, such as committing new units on top of those already committed in the previous steps, or redispatching subject to a minimum deviation scheme relative to the previous steps; 2) simultaneous schemes result in a higher level of social welfare; sequential schemes may be infeasible in cases where the simultaneous schedule is feasible; and 3) current sequential market-clearing schemes are unsatisfactory from the viewpoint that they schedule reserves based upon rules and parameters not necessarily appropriate to meet rigorous security criteria. We have presented what we envision as a possible future general simultaneous market-clearing problem subject to rigorous security criteria based on the explicit modeling of the power system in the aftermath of credible disturbances. The security criteria are valid for both deterministic and probabilistic instances of the market-clearing procedure. The probabilistic model is shown to differ only slightly with respect to the deterministic one, affecting only the tertiary regulation interval power balance equation through the consideration of load shedding. The objective function is also modified by incorporating the likelihood of the contingencies and by considering both pre- and postcontingency social welfare. We have then shown that, under marginal pricing, energy is priced at the nodal marginal cost of energy, and that all reserve services provided at each bus are priced at the same nodal marginal cost of security. The nodal prices of energy are shown to be equal to the sum of the Lagrange multiplier vectors associated with the nodal power balance equations in the predisturbance state as well as in the primary, secondary, and tertiary regulation states. On the other hand, the nodal prices of security are equal to the sum of the Lagrange multiplier vectors associated with the nodal power balance equations in the security-defining primary, secondary, and tertiary regulation intervals. One interesting conclusion is that, in the most common cases, the nodal price of energy is always greater than or equal to the corresponding nodal price of security. Another noteworthy conclusion is that all reserve services, whether primary, secondary, or tertiary, are remunerated at the same marginal rate equal to the nodal price of security. Since the expressions for marginal cost of energy and security are summations of the Lagrange multipliers associated with the various power balance relations, one may unbundle them into separate terms representing the sensitivity of the optimum objective function to the various preand postdisturbance services. These unbundled sensitivities are significant as they characterize the relative importance of the primary, secondary, and tertiary services on the total security price. However, we cannot consider these unbundled quantities as the marginal costs of the separate reserve services. This interpretation would be illogical, as it would violate the argument made here that the same power balance criterion must be applied across all regulation intervals. A numerical example has illustrated the strong coupling that may exist among the various reserve services and the complexities that would have to be overcome by sequential methods to find feasible schedules with the same high levels of social welfare. The example illustrates how the scheduled power and reserve services are used under predisturbance and postcontingency conditions. The example also shows how the nodal prices of energy and security are built from the Lagrange multipliers of the associated power balance equations, identifying the so-called umbrella contingencies and revealing a degree of coupling between primary and tertiary regulation intervals that is not readily apparent. We recognize that there are advantages to sequential markets, primarily the lower complexity of the approach that renders the process more transparent. Also, the simultaneous market-clearing method may be seen as a black box where the schedules and prices obtained are harder to explain and justify. We nevertheless stress that the envisioned simultaneous market clearing of reserve services and energy leads to schedules with higher social welfare, to more rigorous security-driven reserve scheduling, and to a pricing scheme that is based on marginal pricing theory. APPENDIX NOMENCLATURE A. Indexes Index of generators running from 1 to. Index of credible contingencies running from 1 to. B. Parameters 1) Generation-Side Parameters: Vector of maximum power outputs. Vector of minimum power outputs. Vector of maximum power outputs after contingency. Vector of minimum power outputs after contingency. Vector of maximum adjustments after contingency during the primary regulation Vector of minimum adjustments after contingency during the primary regulation Vector of maximum adjustments during the secondary regulation Vector of minimum adjustments during the secondary regulation 1980 PROCEEDINGS OF THE IEEE, VOL. 93, NO. 11, NOVEMBER 2005

12 Vector of maximum adjustments after contingency during the tertiary regulation Vector of minimum adjustments after contingency during the tertiary regulation Frequency droop characteristic of generator (for primary regulation). 2) Demand-Side Parameters: Vector of the upper limits on power demand elasticity. Vector of the lower limits on power demand elasticity. Vector of maximum power consumptions after contingency. Vector of minimum power consumptions after contingency. Vector of the upper limits of the normal load variation. Vector of the lower limits of the normal load variation. Vector of maximum adjustments after contingency during the tertiary regulation Vector of minimum adjustments after contingency during the tertiary regulation Vector of the values of lost load. 3) Network Parameters: Network susceptance matrix in the predisturbance state. Network susceptance matrix after contingency. Network flow sensitivity matrix in the predisturbance state. Network flow sensitivity matrix after contingency. Vector of line flow limits in the predisturbance state. Vector of line flow limits after contingency. Maximal postcontingency frequency deviation. 4) Probabilities: Probability of occurrence of contingency. Probability that none of the considered contingencies occur,. C. Variables 1) Generation-Side Variables: Vector of binary variables whose element equals one if generator is committed; equals zero otherwise. Vector of power outputs. Vector of adjustments after contingency associated with the primary regulation Vector of up adjustments associated with the secondary regulation Vector of down adjustments associated with the secondary regulation Vector of adjustments after contingency associated with the tertiary regulation Vector of primary up reserve. Vector of primary down reserve. Vector of secondary up reserve. Vector of secondary down reserve. Vector of tertiary up reserve. Vector of tertiary down reserve. 2) Demand-Side Variables: Vector of power consumptions. Vector of adjustments after contingency associated with the primary regulation Vector of adjustments after contingency associated with the tertiary regulation Vector of tertiary up reserve. Vector of tertiary down reserve. Vector of involuntary load shedding applied during the tertiary regulation interval after contingency. 3) Network Variables: Steady-state frequency error after contingency associated with the primary regulation rval. Vector of bus voltage angles in the predisturbance state. Vector of steady-state bus voltage angles after contingency associated with the primary regulation Vector of steady-state bus voltage angles associated with the secondary up regulation GALIANA et al.: SCHEDULING AND PRICING OF COUPLED ENERGY AND PRIMARY, SECONDARY, AND TERTIARY RESERVES 1981

13 Vector of steady-state bus voltage angles associated with the secondary down regulation Vector of steady-state bus voltage angles after contingency associated with the tertiary regulation REFERENCES [1] B. R. Barkovich and D. V. Hawk, Charting a new course in California, IEEE Spectr., vol. 33, no. 7, pp , Jul [2] H. Singh and A. Papalexopoulos, Competitive procurement of ancillary services by an independent system operator, IEEE Trans. Power Syst., vol. 14, no. 2, pp , May [3] S. S. Oren, Design of ancillary services markets, presented at the 34th Hawaii Int. Conf. System Sciences, Maui, HI, [4] A. Papalexopoulos and H. Singh, On the various design options for ancillary services markets, presented at the 34th Hawaii Int. Conf. System Sciences, Maui, HI, [5] M. Shahidehpour, H. Yamin, and Z. Li, Market Operations in Electric Power Systems: Forecasting, Scheduling, and Risk Management. New York: IEEE-Wiley, [6] J. Kumar and G. 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Green, Microeconomic Theory. New York: Oxford Univ. Press, [42] F. C. Schweppe, M. C. Caramanis, R. D. Tabors, and R. E. Bohn, Spot Pricing of Electricity. Boston, MA: Kluwer, Francisco D. Galiana (Fellow, IEEE) received the B.Eng. (Hon.) degree from McGill University, Montreal, QC, Canada, in 1966 and the S.M. and Ph.D. degrees from the Massachusetts Institute of Technology, Cambridge, in 1968 and 1971, respectively. He spent some years at the Brown Boveri Research Center, Baden, Switzerland, and held a faculty position at the University of Michigan, Ann Arbor. He joined the Department of Electrical and Computer Engineering at McGill University in 1977, where he is currently a Full Professor PROCEEDINGS OF THE IEEE, VOL. 93, NO. 11, NOVEMBER 2005

14 François Bouffard (Student Member, IEEE) received the B.Eng. (Hon.) degree in electrical engineering from McGill University, Montreal, QC, Canada, in He is currently working toward the Ph.D. degree at McGill University. He was a Visiting Scholar at the Universidad de Castilla-La Mancha, Ciudad Real, Spain, in 2003 and He was recently a Faculty Lecturer in the Department of Electrical and Computer Engineering at McGill University. His research interests are in the fields of power systems economics, reliability, control, and optimization. José M. Arroyo (Member, IEEE) received the Ingeniero Industrial degree from the Universidad de Málaga, Málaga, Spain, in 1995, and the Ph.D. degree in power system operations planning from the Universidad de Castilla-La Mancha, Ciudad Real, Spain, in From June 2003 to July 2004, he held a Richard H. Tomlinson Postdoctoral Fellowship in the Department of Electrical and Computer Engineering, McGill University, Montreal, QC, Canada. He is currently an Associate Professor of Electrical Engineering at the Universidad de Castilla-La Mancha. His research interests include operations, planning and economics of power systems, as well as optimization and parallel computation. José F. Restrepo (Student Member, IEEE) received the Ingeniero Electricista degree from Universidad Pontificia Bolivariana, Medellín, Colombia, in 2003 and the M.Eng. degree from McGill University, Montreal QC, Canada, in He is currently working toward the Ph.D. degree at McGill University. His research interests include security, control, and optimal operation of power systems. GALIANA et al.: SCHEDULING AND PRICING OF COUPLED ENERGY AND PRIMARY, SECONDARY, AND TERTIARY RESERVES 1983