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Energy and Reserve Pricing in Security and Network-Constrained Electricity Markets José M. Arroyo, Member, IEEE, and Francisco D. Galiana, Fellow, IEEE

Vector of upper elastic demand bounds, which are . less than or equal to Vector of lower elastic demand bounds, which are . greater than or equal to Vector of line power flow capacities in the precontingency state. Vector of line power flow capacities under contingency . Vector of generator capacities in the precontingency state. Vector of generator capacities under contingency . Vector of generator minimum power outputs in the precontingency state. Vector of generator minimum power outputs under contingency . Vector of rates offered by consumers to provide down-spinning reserve. Vector of rates offered by consumers to provide up-spinning reserve. Vector of rates offered by generators to provide down-spinning reserve. Vector of rates offered by generators to provide up-spinning reserve. Variation of load defining contingency .

Abstract—This paper analyzes some unresolved pricing issues in security-constrained electricity markets subject to transmission flow limits. Although the notion of separate reserve types as proposed by FERC can be precisely and unambiguously defined, when transmission constraints are active, the very existence of separate reserve prices and markets is open to question when the prices are based on marginal costs. Instead, we submit here that the only products whose marginal costs can be separately and uniquely defined and calculated are those of energy and security at each node. Thus, under marginal pricing, at any given network bus all scheduled reserve types should be priced not at separate rates but at a common rate equal to the marginal cost of security at that bus. Furthermore, we argue that nodal or area reserves cannot be prespecified but must be obtained as by-products of the market-clearing process. Simulations back up these conclusions. Index Terms—Contingency-constrained scheduling, demand-side reserve, electricity markets, security and energy pricing, transmission constraints, up and down-spinning reserve.

I. NOMENCLATURE Functions: Benefit function bid by load to consume in the precontingency state. Cost function offered by generator to produce in the precontingency state. Constants: Vector of linear offered cost coefficients. dc load flow matrix in the precontingency state. dc load flow matrix under contingency . Vector of fixed offered cost coefficients. Matrix relating power flows to nodal phase angles in the precontingency state. Matrix relating power flows to nodal phase angles under contingency . Vector of upper bounds on increased levels of power consumed for down-spinning reserve. Vector of lower bounds on decreased levels of power consumed for up-spinning reserve. Manuscript received February 26, 2004; revised May 27, 2004. 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, Québec. J. M. Arroyo was supported by McGill University through the Richard H. Tomlinson Postdoctoral Fellowship and by the Junta de Comunidades de Castilla–La Mancha, Spain. 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 (e-mail: [email protected]). F. D. Galiana is with the Department of Electrical and Computer Engineering, McGill University, Montreal, QC, H3A 2A7, Canada (e-mail: [email protected]). Digital Object Identifier 10.1109/TPWRS.2005.846221

Derating of generator defining contingency .

Variables: Vector of precontingency bus demands. Vector of bus demands under contingency . Vector of line flows in the precontingency state. Vector of line flows under contingency . Vector of generator power outputs in the precontingency state. Vector of generator power outputs under contingency . Revenue of load from the sale of reserves. Vector of down-spinning reserves provided by consumers. Vector of up-spinning reserves provided by consumers. Revenue of generator from the sale of reserves. Vector of down-spinning reserves provided by generators. Vector of up-spinning reserves provided by generators. Vector of 0/1 variables denoting the off/on status of the generators.

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ARROYO AND GALIANA: ENERGY AND RESERVE PRICING IN SECURITY AND NETWORK-CONSTRAINED ELECTRICITY MARKETS

Vector of nodal phase angles for the precontingency state. Vector of nodal phase angles under contingency . Vector of nodal marginal costs of energy. Vector of nodal marginal costs of security. Lagrange multipliers: Vector of Lagrange multipliers associated with the power balance constraints in the precontingency state. Vector of Lagrange multipliers associated with the power balance constraints under contingency . Sets: Set of indices of derated generators defining contingency . Set of indices of generators. Set of indices of loads. Set of indices of the credible contingencies. Set of indices of out-of-service generators defining contingency . Set of indices of loads subject to a sudden variation defining contingency . II. INTRODUCTION ECURITY, in a deterministic sense, is the capability of a power system to survive a specified set of credible contingencies such as generator and line outages or sudden variations of load, all without having to shed load beyond the limits of voluntary interruptibility.1 Maintaining a high level of security, which has historically been of major concern to the power industry, continues to be a high priority goal in the current context of competitive electricity markets. However, events such as the failure of the California market, the recurrence of price spikes due to transmission bottlenecks, and recent major blackouts worldwide reveal that a number of unresolved issues remain concerning the relation between system security and the rules governing electricity markets. In power systems, two kinds of security actions can be defined [1]. Preventive security actions ensure that in the event of a contingency enough resources are available for the quick execution of corrective security actions that guarantee the normal operation of the system once the contingency has taken place. Examples of preventive actions include the turning on of extra generating units or the redispatch of already committed units in the precontingency state. Corrective actions include the fast redispatching of generation or the curtailment of selected loads under a specific contingency. In addition, for certain types of slowly-developing contingencies, corrective actions may require turning on some standby generation. Reserves constitute the resources that enable the implementation of preventive and corrective security actions. FERC defines several types of reserves [2], [3] and proposes that they be treated as separate unbundled commodities with separate prices. This paper focuses on spinning or synchronized reserves, specifically up-spinning and down-spinning [4]; however, the main

S

1Probabilistic security allows for some load shedding provided that the expected load not served is below a certain threshold.

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conclusions reached also apply to other types such as standby reserves. One fundamental issue raised here is whether the resources offered by the independent agents to satisfy system security, namely, up and down-spinning reserves, can be separately priced. Before pursuing this question further, we first summarize the most relevant previous related work. The work of Schweppe and his colleagues [5] introduced the notion of spot pricing in electricity markets as defined by marginal cost theory. In [6], the marginal cost of up-spinning reserve was calculated through an iterative process within a decentralized market framework. In [7], the authors achieved a socially optimal level of system security in a decentralized marketplace through pricing incentives and information on system needs. Both of these contributions [6], [7] assumed a perfect market as well as advanced knowledge of probabilistic information about the contingencies. A geometric approach was used in [8] to analyze the behavior of marginal prices as the system operating point nears a feasibility boundary. Price signals were developed to encourage the elastic demand to adjust itself to avoid conditions such as voltage collapse. In [9], Kumar and Sheblé first proposed a joint energy and ancillary services market where each generator and consumer had the obligation to take part in the spinning reserve market by pre-specifying its fraction of participation. This is a step that, as stated by the authors, requires engineering judgment and off-line studies. Other authors [10]–[12] have also considered security-constrained models including the transmission network and flow constraints, with prespecified nodal or area reserves. The prespecification of local reserves or reserve participation levels in the above approaches permits the calculation of their marginal costs as the Lagrange multipliers of the nodal balance of the combined local and imported reserves. One shortcoming of these approaches is that since the local reserves or their fractional levels are prespecified, the market-clearing solution is generally suboptimal and may even be infeasible. We argue here that local reserves, being critical elements of the security-constrained market-clearing problem, must be treated as decision variables of the optimization process and not as prespecified parameters. Strbac and colleagues [13] proposed a security-constrained economic dispatch for a horizon of one year. The cost of corrective security was defined as the difference between the costs of the solutions with and without security constraints. This work allowed for demand-side corrective actions but supposed that preventive actions such as unit commitment were given. In [14], this work was extended from a dc load flow to an ac network model. Recently, a joint energy/reserve market model including demand-side reserve offers was developed for systems without network constraints [4]. Several reserve products were considered, including up and down-spinning reserves supplied by the generators and the loads. In this model with essentially a single bus, it was possible to associate each type of reserve directly and uniquely with one specific type of contingency. For example, up-reserve was defined by the loss-of-generation contingencies, while down-reserve was associated with a sudden decrease in demand. It was then possible to determine a unique marginal

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cost for each type of reserve through the corresponding Lagrange multipliers. When the market-clearing process models the transmission network and some flow limits become active, the association of each offered reserve type with a specific kind of contingency is no longer possible. As an example, when some lines are congested, the outage of a generator or a line may lead to corrective actions under which some of the remaining generating units increase their output, thus using up-reserve, while, simultaneously, other units decrease their outputs, thus using down-reserve. This category of problem requiring the simultaneous use of different reserve types is a critical issue that can occur with frequency when transmission congestion becomes active, yet it has not been adequately addressed and resolved in the literature. This paper therefore examines the short-term operation and pricing of the various products traded in a joint energy/reserve market while accounting for transmission network flow limits and security constraints. In this market, besides submitting offers to sell and bids to buy energy, the participants can also contribute toward system security by offering to sell both up and down-spinning reserves at different rates. The system operator clears such a market by scheduling all the energy and reserve offers and bids so as to maximize the system social welfare while satisfying all operational constraints including those imposed by the need to survive the set of credible contingencies. The model presented is a contingency and network-constrained market-clearing procedure where energy and two different types of spinning reserve are jointly dispatched. In essence, this model combines the contingency-constrained optimal power flow first proposed in [15] and a market-clearing procedure for both energy and reserve [9]–[12]. One difference with respect to previous contingency-constrained models [13]–[15] is that the preventive security measures include unit commitment decisions for the precontingency state. Corrective security actions are explicitly accounted for in the market-clearing process by ensuring that all operational constraints are satisfied under all credible contingencies. These corrective actions define the required levels of two distinct types of reserve, namely, up and down-spinning reserves, which constitute a salient feature with respect to previous joint energy/reserve market-clearing models [9]–[12]. The goal of this paper is to analyze such model in terms of pricing and feasibility of the allocation of reserves. The main contributions of this work are as follows: 1)

If local reserves are prespecified, the solution is suboptimal and can even be infeasible. Instead, local reserves are by-products of the market-clearing optimization process. 2) The notion of marginal costs for the individual reserve types such as up and down-spinning reserves is shown not to exist. Instead, the only products whose marginal costs can be uniquely defined and calculated at each node are energy and security, the latter defining, under marginal pricing, the nodal prices for any type of reserve product being traded at that bus. A possible extension of this work is the consideration of probabilistic measures of reliability. To derive probabilistic indices

IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 20, NO. 2, MAY 2005

such as loss-of-load probability (LOLP) and expected unserved energy (EUE), a number of approaches have been proposed in the literature: the widely used decision tree methods [16], [17] and a probabilistic unit commitment method [18]. The remainder of this paper is organized as follows. In Section III, the mathematical formulation of the proposed model is presented. The nodal marginal costs of energy and security are defined in Section IV. Pricing issues are discussed in Section V. In Section VI, results from two case studies are provided and analyzed. In Section VII, some relevant conclusions are drawn. Finally, an Appendix is included to show the general derivation of the nodal marginal costs of energy and security. III. FORMULATION The market-clearing procedure is given by Min (1) subject to (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12) (13) where the superscript denotes the transpose operator. The objective function to be minimized (1) consists of the sum of the offered cost functions for generating power minus the sum of the bid benefit functions for consuming power plus the cost of all up and down-spinning reserves offered by the generators and the loads. Note that, in the proposed model, no term is incorporated in the objective function to account for the extra cost of the corrective actions in the event of a contingency, that is, for the actual use of the reserve. Thus, if called to provide reserve, the market participants would have to modify their respective production and consumption without any further remuneration. We suggest that this is reasonable because the probability of occurrence of a contingency is very low and the market participants already profit from their willingness to provide these corrective actions. Using a dc load flow model, constraints (2) and (3) represent the nodal power balance equations under the precontingency deand contingency states, respectively. The variables and note the vectors of the corresponding Lagrange multipliers.

ARROYO AND GALIANA: ENERGY AND RESERVE PRICING IN SECURITY AND NETWORK-CONSTRAINED ELECTRICITY MARKETS

Constraints (4) and (5) express the line flows in terms of the nodal phase angles under the normal and contingency states, respectively, while constraints (6) and (7) enforce the corresponding line flow capacity limits. Constraints (8) and (9) set the generation limits for the precontingency and contingency states, respectively. Note that the vector of on/off variables is the same under both states, indicating that only those generators that are scheduled on in the precontingency state can participate in corrective actions during contingency states. In the case of standby reserve, corrective actions may also allow some units to be turned on after a slowlydeveloping contingency however this case has not been dealt with here. Constraints (10) set the elasticity limits of the demand, and . In addition, it is assumed that customers may be willing to alter their consumption during contingencies thereby also contributing to system security. Thus, by announcing their willingness to curtail, the demands would contribute to up-spinning reserve, while by agreeing to increase consumption they would contribute to down-spinning reserve. Constraints (11) set and , inside of which the consumers are ready the limits to modify their demand relative to their normal level, . Note that these limits are less restrictive than the elasticity limits imposed in (10). Finally, constraints (12) and (13) relate the up and down-spinning reserve contributions to the power levels produced and consumed under the precontingency and contingency states. The framework presented here is general enough to take into account any contingency of the following types: ; a) loss of a sub-set of generators b) derating of a sub-set of generators ; sudden variation of a sub-set of loads ; d) , and ). loss of a sub-set of lines (modeled by The notion of system security is defined with respect to a set of pre-defined credible contingencies of the above types, keeping in mind that some of the constraints in (2)–(13) must be modified to meet the definition of the contingency as given by items a)–d) above. Preventive security is realized by redispatching the generation levels at the precontingency state and by requiring that the commitment status of the generators be the same under both the precontingency and contingency states. Corrective security actions are implemented after each contingency by redispatching and . the generation and demand to new levels defined by c)

IV. NODAL MARGINAL COSTS OF ENERGY AND SECURITY The model presented in the previous section identifies three separate products at each node, namely energy, up-spinning reserve and down-spinning reserve, each of which is offered for sale by each agent at a possibly different rate. In addition, although only generators sell energy, both generators and loads can offer to sell reserves. The fact that three distinct commodities can be traded suggests the possibility that three distinct nodal marginal costs exist. However, we argue here that in general this is not feasible and that only two marginal costs can in

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fact be defined per node: 1) the marginal cost of energy and 2) the marginal cost of security. As discussed in the Introduction, when network constraints are ignored, it is possible to associate one type of contingency with one kind of reserve (e.g. up-spinning reserve with generator outages). The marginal cost of generator outages could then be defined as the marginal cost of the level of up-spinning reserve. When transmission congestion is considered, however, the corrective actions following a contingency may require the simultaneous use of a combination of up and down-spinning reserves at different buses. It is then generally impossible to identify a set of contingencies that would require the use of one type of reserve at all buses at the exclusion of other reserve types. Therefore, it is not possible to derive a separate and unique marginal cost for each type of reserve. What can however be defined uniquely and under the most general circumstances, are the nodal marginal costs of meeting the power balance for all credible contingencies (3). We submit here that these marginal costs, denoted here by the vector , define the nodal marginal costs of security. As shown in the Appendix, from the Karush–Kuhn–Tucker optimality conditions, it readily follows that (14) where the are the Lagrange multipliers associated with (3). Moreover, since the purpose of reserves is to maintain system security, under marginal pricing the vector denotes the nodal prices applied to all forms of reserves, whether up or down, whether supplied by generators or demands, and whether sold or bought. In a similar way, we can define the nodal marginal costs of meeting the power balance for all credible contingencies (3) plus the precontingency state (2). These marginal costs, denoted here , also readily follow from the Karush–Kuhn–Tucker opby timality conditions as shown in the Appendix (15) where the are the Lagrange multipliers associated with (2). Similarly, we argue that defines the nodal marginal costs of energy, which under marginal pricing, defines the nodal prices of electricity in the precontingency state. As seen in the Appendix, the marginal cost of security can be interpreted in terms of the change in the optimum value of the objective function due to an infinitesimal change in the power balance equations under all contingencies. In addition, the marginal cost of energy can be interpreted in terms of the change in the optimum value of the objective function due to an infinitesimal change in all the power balance equations, including that of the precontingency state. It is interesting to observe that as the list of credible contingencies grows, so does the possibility of increasing the marginal costs of security and energy. However, these prices will increase only if the contingency added is “more severe” than the existing ones. If the new contingency is “less severe” than the ones to

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which the system can already survive, then the corresponding will be zero. Lagrange multiplier V. REVENUES AND PAYMENTS UNDER MARGINAL PRICING The market-clearing process defined by the optimization problem (1)–(13), yields the scheduled values of power proand , duced and consumed by each agent, respectively, as well as the reserves provided by each participant, namely, , , , and . From the nodal prices of energy and security derived in (14) and (15), we can define the corresponding revenues and payments for all market participants. For example, the revenue of generator from the sale of reserves becomes

Fig. 1. Three-bus example. TABLE I GENERATOR DATA—THREE-BUS SYSTEM

(16) while the revenue of load from the sale of reserves becomes (17) This paper does not deal with the settlement of these reserve payments, however this could be done by charging each generator and load in proportion to the amounts of power generated or consumed.

TABLE II OPTIMAL PARTICIPANT SCHEDULES (IN MEGAWATTS)—THREE-BUS SYSTEM

VI. NUMERICAL RESULTS Results from two test cases are presented in this section. The first case consists of three buses, while the second is based on the IEEE Reliability Test System [19]. In both cases, a single time period is studied. The model has been implemented on a Pentium IV, 2.66-GHz processor with 512 MB of RAM using CPLEX 8.1 under GAMS [20].

TABLE III CORRECTIVE ACTIONS (IN MEGAWATTS)—THREE-BUS SYSTEM

A. Three-Bus Example The topology of this example is shown in Fig. 1. Line reactances are all 0.63 p.u on a base of 100 MVA and 138 kV. The capacities of lines 1–2 and 1–3 are 100 MVA, whereas the flow of line 2–3 is limited to 30 MVA. Data for the generators are given in Table I. For the sake of simplicity, the generators offer linear cost functions of the form . An elastic load with lower and upper elastic bounds of 50 and 80 MW, respectively, is located at bus 3 (see Fig. 1). This load bids to buy energy at $4/MWh and offers to sell up and down-spinning . The load can be parreserves at tially curtailed below its lower elasticity limit of 50 MW down to 20 MW in order to provide additional up-spinning reserve. If needed, this load can increase its consumption beyond its upper elasticity limit of 80 MW up to 100 MW in order to contribute to down-spinning reserve. Four credible contingencies are considered here, defined by the outage of any single generator and of line 1–3. Table II lists the optimal power produced and consumed for the case without security, i.e., when only energy is cleared, with all contingency and reserve constraints removed from the model (1)–(13). Here, only generators 2 and 3 are committed and the load is at its lower elastic bound of 50 MW. In order to meet the power flow capacity of line 2–3, the cheapest generator 2

limits its production to 45 MW, so that the remaining 5 MW are supplied by generator 3. The value of the objective function is $40. Table II also shows the optimal participant schedules for the security-constrained case. The load still consumes 50 MW, however the schedule of the generators has changed (preventive security action), requiring the commitment of the expensive generator 1. This unit operates at its minimum power output of 2.4 MW, also providing 7.6 MW of up-spinning reserve. In addition, preventive security requires that the power outputs of the cheapest generators (2 and 3) be reduced with respect to the unconstrained case. Generator 2 also provides down-reserve while generator 3 produces up-reserve. The value of the objective function for this security-constrained case is $175.78, which is considerably higher than the $40 that it costs to operate the system without security. Finally, because of its high offer, no reserves are allocated to the load. Table III shows the power supplied and consumed by each participant under all four contingencies considered (corrective security actions). The outage of generator 1 requires that generator 3 use up-spinning reserve by increasing its output to 6.2

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TABLE IV NODAL RESERVES (IN MEGAWATTS)—THREE-BUS SYSTEM

TABLE VI NODAL PRICES (IN DOLLARS PER MEGAWATTHOUR)—THREE-BUS SYSTEM

TABLE V LAGRANGE MULTIPLIERS (IN DOLLARS PER MEGAWATTHOUR)—THREE-BUS SYSTEM

TABLE VII REVENUES FROM THE SALE OF RESERVES (IN DOLLARS)—THREE-BUS SYSTEM

MW from its precontingency level of 3.8 MW (see Table II). Similarly, the outage of generator 2 forces generators 1 and 3 to increase their production with respect to the precontingency state. The outage of generator 3 is significant because, in order to survive this contingency, unit 1 uses up-spinning reserve by increasing its output to 10 MW from its precontingency level of 2.4 MW. Simultaneously, unit 2 decreases its power output to 40 MW from its precontingency level of 43.8 MW, thus using down-spinning reserve. This is an example where, due to the congestion of line 2–3, a loss-of-generator contingency requires the simultaneous use of different types of reserves. Finally, in order to survive the loss of line 1–3, generators 2 and 3 make use of down and up-spinning reserve, respectively. Methods that ignore transmission constraints cannot identify or model this type of mixed reserve requirements [3], [4]. Table IV summarizes the nodal spinning reserves necessary to guarantee system security. These values, found by the optimization process presented here, would be difficult to pre-specify as required by other approaches [10]–[12]. The Lagrange multipliers associated with the power balance equations of the precontingency and contingency states are listed in Table V. For the loss of generator 1, the corresponding multipliers are all zero because the reserves required for the other contingencies cover this case (see Table III). Moreover, since the system can survive the outages of generator 1 or 2 without congesting line 2–3, their multipliers are identical. However, since the loss of generator 3 congests line 2–3 the nodal multipliers are all different. Finally, since line 2–3 is congested during the outage of line 1–3, the associated Lagrange multipliers are not only dissimilar but two of them are negative. This result is in agreement with [8]. The sum per bus of the Lagrange multipliers associated with the power balance of the precontingency and contingency states represents the nodal energy price under security constraints. Analogously, the sum per bus of the Lagrange multipliers associated with the power balance under contingencies represents the nodal security price. Both energy and security nodal prices are presented in Table VI, which also shows the energy prices obtained when security is not considered. From Table VI, it can be inferred that security slightly increases the energy prices with respect to the unconstrained case. As can be seen, the location

of the single load, i.e., bus 3, determines the highest nodal security price. Note that although the energy price of bus 1 is lower than the rate offered by generator 1 located at the same bus, this result is not unusual under marginal pricing [5]. The lack of revenue reconciliation in this example is due to the relatively low reserve rate offered by the generator which is then scheduled at its minimum power output so as to provide cheap up-spinning reserve (see Table II). Finally, Table VII shows the distribution of revenues among the participants from the sale of reserves. Each participant, whether supplying up or down-spinning reserve, gets remunerated at the same rate, namely the nodal price of security. The more significant contribution to security provided by generator 3 allows it to earn the highest revenue from the sale of reserves. B. IEEE RTS Based Case The case based on the 24-bus IEEE Reliability Test System [19] comprises 26 generators and 17 inelastic loads. The data for the generators can be found in [4]. Generator quadratic energy offers have been approximated by piecewise linear curves with 500 blocks each. The load profile corresponds to a weekday of a winter week at 18:00 [19]. All loads offer identical rates of $20/MW/h and $1000/MW/h for up and down-spinning reserves, respectively. Loads can be curtailed from their normal levels all the way to zero to provide up-spinning reserve. If needed, the demands can also increase by up to 10% to contribute to down-spinning reserve. The only modification with respect to the network data listed in [19] consists in the reduction of the capacities of lines 11–13, 15–16 and 15–24 from 500 MVA to 175 MVA, 60 MVA, and 175 MVA, respectively. The credible contingencies considered are the outage of any single generator and the outage of line 14–16. The optimal solution was achieved in 5.5 s of CPU time including unit commitment, and the optimum value of the objective function was $67 176.71. Tables VIII and IX provide the schedule of energy and reserves for generators and loads, respectively. As can be noted, generators 1–7 provide only up-spinning reserve, generators 17, 22 and 25 provide only down-spinning reserve, and generators 21 and 23 contribute to security with both types of reserves. In addition, loads 3, 12

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TABLE VIII OPTIMAL GENERATOR SCHEDULE (IN MEGAWATTS)—24-BUS SYSTEM

TABLE X NODAL RESERVES (IN MEGAWATTS)—24-BUS SYSTEM

TABLE XI LAGRANGE MULTIPLIERS (IN DOLLARS PER MEGAWATTHOUR)—24-BUS SYSTEM

TABLE IX OPTIMAL SCHEDULE OF THE LOADS (IN MEGAWATTS)—24-BUS SYSTEM

and 13 contribute to security with up-reserves equal to 115.6, 138.3, and 77.48 MW, respectively. As in the previous example, there are contingencies requiring the simultaneous use of up and down-spinning reserves. For instance, in order to survive the outage of generator 24, generator 17 decreases its power output to 150.5 MW, making use of 4.5 MW of down-spinning reserve, whereas generators 21 and 23 increase their production up to 156.46 MW and 197 MW, by using 28.43 MW and 70.16 MW of up-spinning reserve, respectively. Table VIII also shows the optimal solution for the non security-constrained case that differs from the security-constrained case not only in the dispatch but also in the scheduling of the

generators (see generators 1–7 and 21–23). The objective function value of the optimal solution without security is $59 797.12 and was obtained in 1.7 s of CPU time. Table X summarizes the nodal spinning reserves that guarantee the security of the system as determined by the optimal market-clearing process. As several generating units are located at the same bus [19], the values shown in the “Generation” columns denote the total reserve provided by all units connected to the bus. Only those buses with nonzero reserve contributions are shown in Table X. We emphasize once more that the a priori estimation of the optimal nodal reserves is a difficult task. Market-clearing algorithms that rely on such a prespecification are therefore prone to suboptimality and could lead to infeasible market equilibrium [10]–[12]. The Lagrange multipliers associated with the power balance equations of the precontingency and contingency states are listed in Table XI. Only the outages of generators 24 and 26 as well as the loss of line 14–16 affect the optimal solution, i.e., the Lagrange multipliers for the remaining contingencies are all zero and the corresponding constraints could have been removed from the model without altering the optimal solution.

ARROYO AND GALIANA: ENERGY AND RESERVE PRICING IN SECURITY AND NETWORK-CONSTRAINED ELECTRICITY MARKETS

TABLE XII NODAL PRICES (IN DOLLARS PER MEGAWATTHOUR)—24-BUS SYSTEM

TABLE XIII REVENUES FROM THE SALE OF RESERVES (IN DOLLARS)—24-BUS SYSTEM

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VII. CONCLUSIONS Recent blackouts throughout North America and Europe call into question the compatibility of the rules governing electricity markets and the secure operation of power systems. In particular, when transmission flow limits are active, the scheduling of nodal reserves to provide sufficient flexibility to survive a set of credible contingencies is a difficult problem to which we have made several contributions in this paper. 1) We have formulated a general market-clearing process in the form of an optimization problem that accounts for transmission flow limits, preventive and corrective security, unit commitment, and two types of reserves offered by both generators and loads. 2) The notion of marginal costs of the individual reserve types such as up and down-spinning reserves is shown not to exist through numerical counterexamples. Instead, the only products whose marginal costs can be separately and uniquely defined and calculated at each node are energy and security, the latter defining, under marginal pricing, the nodal prices for any type of reserve product being traded at that bus. 3) Finally, we show through some numerical examples that local (or area) reserve requirements cannot be specified a priori without sacrificing optimality and possibly rendering the market equilibrium infeasible. Instead, local reserves are determined as by-products of the proposed market-clearing process. Simulation results from two case studies, including the IEEE 24-bus RTS, have been presented to illustrate the main issues discussed here. APPENDIX DERIVATION OF NODAL MARGINAL COSTS

As can be seen, the outage of generator 26 has the greatest impact on system security. Table XII presents the nodal prices of energy and security. The highest nodal prices for security occur at buses 11 and 24. This can be explained by the fact that these two buses split the system into two areas: the upper area with excess of generation (buses 13–24) and the lower area with deficit of generation (buses 1–10). Other buses with relatively high security prices are 3, 14, and 15, which together with buses 11 and 24 are geographically near the congested lines and the line 14–16 defining one of the credible contingencies. Table XII also presents a comparison with the energy prices obtained when security is neglected. Note that security causes an increase in the nodal energy price of over 10% in most buses. Buses 7 and 13 constitute an exception experiencing a reduction in the energy price when security is accounted for. Finally, Table XIII shows the distribution of nodal revenues between generation and demand from the sale of up and down-spinning reserves. Only those buses with revenues different from zero are listed. Those loads connected to buses 3, 14 and 15 with relatively high security prices (see Table XII), earn the highest reserve revenues, followed by generator 25 which provides 95.01 MW of down-spinning reserve at bus 18.

The market-clearing problem formulated in this paper has the following general form:

subject to

(A1) where represents the vector of binary decision variables, is is the objecthe vector of continuous decision variables, represents the nodal power balance equative function, tions in the precontingency state, denotes the nodal power balance equations under the credible contingency , and represents all remaining inequality constraints. The vecand are the Lagrange multipliers corresponding to tors , the previous relations. Consider the Lagrangian function evaluated at , the optimum vector of binary variables (A2)

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The optimal solution of (A1) must satisfy Karush–Kuhn–Tucker necessary optimality conditions

the

while the nodal marginal costs of security are:

(A10)

REFERENCES

(A3) In order to calculate the nodal marginal costs of energy, let the power balance equations in the precontingency and contingency . states be incremented by an infinitesimal parameter vector, Similarly, in order to derive the nodal marginal costs of security, let the power balance equations under all contingencies be incremented by another infinitesimal parameter vector, . These incremental changes to the problem equalities result in an infinitesimal change to the continuous variables of the optimum to (A1), denoted by . The equalities in (A1) now must satisfy

(A4) Similarly, the optimum value of the objective function changes by (A5) Using (A3),

can be expressed as

(A6)

Furthermore (A7) Applying (A4) and (A7) to (A6) yields (A8)

Consequently, the nodal marginal costs of energy are (A9)

[1] T. E. Dy Liacco, “The adaptive reliability control system,” IEEE Trans. Power Apparatus and Systems, vol. PAS-86, no. 5, pp. 517–531, May 1967. [2] Order 888: Promoting Wholesale Competition Through Open Access Nondiscriminatory Transmission Services by Public Utilities, Federal Energy Regulatory Commission. (1996, Apr.). http://www.ferc.gov [Online] [3] N. S. Rau, “Optimal dispatch of a system based on offers and bids—a mixed integer LP formulation,” IEEE Trans. Power Syst., vol. 14, no. 1, pp. 274–279, Feb. 1999. [4] J. Wang, N. Encinas Redondo, and F. D. Galiana, “Demand-side reserve offers in joint energy/reserve electricity markets,” IEEE Trans. Power Syst., vol. 18, no. 4, pp. 1300–1306, Nov. 2003. [5] F. C. Schweppe, M. C. Caramanis, R. D. Tabors, and R. E. Bohn, Spot Pricing of Electricity. Norwell, MA: Kluwer, 1988. [6] M. C. Caramanis, R. E. Bohn, and F. C. Schweppe, “System security control and optimal pricing of electricity,” Elect. Power Energy Syst., vol. 9, pp. 217–224, Oct. 1987. [7] R. J. Kaye, F. F. Wu, and P. Varaiya, “Pricing for system security,” IEEE Trans. Power Syst., vol. 10, no. 2, pp. 575–583, May 1995. [8] R. Rajaraman, J. V. Sarlashkar, and F. L. Alvarado, “The effect of demand elasticity on security prices for the Poolco and multi-lateral contract model,” IEEE Trans. Power Syst., vol. 12, no. 3, pp. 1177–1184, Aug. 1997. [9] J. Kumar and G. Sheblé, “Framework for energy brokerage system with reserve margin and transmission losses,” IEEE Trans. Power Syst., vol. 11, no. 4, pp. 1763–1769, Nov. 1996. [10] M. Aganagic, K. H. Abdul-Rahman, and J. G. Waight, “Spot pricing for generation and transmission of reserve in an extended Poolco model,” IEEE Trans. Power Syst., vol. 13, no. 3, pp. 1128–1135, Aug. 1998. [11] T. Alvey, D. Goodwin, X. Ma, D. Streiffert, and D. Sun, “A security-constrained bid-clearing system for the New Zealand wholesale electricity market,” IEEE Trans. Power Syst., vol. 13, no. 2, pp. 340–346, May 1998. [12] K. W. Cheung, P. Shamsollahi, D. Sun, J. Milligan, and M. Potishnak, “Energy and ancillary service dispatch for the interim ISO New England electricity market,” IEEE Trans. Power Syst., vol. 15, no. 3, pp. 968–974, Aug. 2000. [13] G. Strbac, S. Ahmed, D. Kirschen, and R. Allan, “A method for computing the value of corrective security,” IEEE Trans. Power Syst., vol. 13, no. 3, pp. 1096–1102, Aug. 1998. [14] A. Jayantilal and G. Strbac, “Load control services in the management of power system security costs,” IEE Proc.–Gener., vol. 146, no. 3, pp. 269–275, May 1999. [15] B. Stott, O. Alsac, and J. L. Marinho, “The optimal power flow problem,” in Proc. SIAM Int. Conf. Elect. Power Problems: Mathemat. Challenge, Seattle, WA, Mar. 1980, pp. 327–351. [16] J. Kumar and G. Sheblé, “A decision analysis approach to the transaction selection problem in a competitive electric market,” Elect. Power Syst. Res., vol. 38, pp. 209–216, Sep. 1996. [17] EPRI, “MIDAS, multi objective integrated decision analysis system,” EPRI J., vol. 12, pp. 57–58, 1987. [18] F. Bouffard and F. D. Galiana, “An electricity market with a probabilistic spinning reserve criterion,” IEEE Trans. Power Syst., vol. 19, no. 1, pp. 300–307, Feb. 2004. [19] Reliability Test System Task Force, “The IEEE reliability test system—1996,” IEEE Trans. Power Syst., vol. 14, no. 3, pp. 1010–1020, Aug. 1999. [20] GAMS: A User’s Guide, A. Brooke, D. Kendrick, A. Meeraus, and R. Raman. (2003). http://www.gams.com/ [Online]

ARROYO AND GALIANA: ENERGY AND RESERVE PRICING IN SECURITY AND NETWORK-CONSTRAINED ELECTRICITY MARKETS

José M. Arroyo (S’96–M’01) received the Ingeniero Industrial degree from the Universidad de Málaga, Málaga, Spain, in 1995 and the Ph.D. degree in power systems operations planning from the Universidad de Castilla–La Mancha, Ciudad Real, Spain, in 2000. From June 2003 through July 2004, he held a Richard H. Tomlinson Postdoctoral Fellowship at the Department of Electrical and Computer Engineering of 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.

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Francisco D. Galiana (F’91) received the B.Eng. (Hon.) degree from McGill University, Montreal, QC, Canada, in 1966 and the S.M. and Ph.D. degrees at the Massachusetts Institute of Technology, Cambridge, MA, 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 Harbor. He joined the Department of Electrical and Computer Engineering at McGill University in 1977, where he is currently a Full Professor.

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Energy and Reserve Pricing in Security and Network-Constrained Electricity Markets José M. Arroyo, Member, IEEE, and Francisco D. Galiana, Fellow, IEEE

Vector of upper elastic demand bounds, which are . less than or equal to Vector of lower elastic demand bounds, which are . greater than or equal to Vector of line power flow capacities in the precontingency state. Vector of line power flow capacities under contingency . Vector of generator capacities in the precontingency state. Vector of generator capacities under contingency . Vector of generator minimum power outputs in the precontingency state. Vector of generator minimum power outputs under contingency . Vector of rates offered by consumers to provide down-spinning reserve. Vector of rates offered by consumers to provide up-spinning reserve. Vector of rates offered by generators to provide down-spinning reserve. Vector of rates offered by generators to provide up-spinning reserve. Variation of load defining contingency .

Abstract—This paper analyzes some unresolved pricing issues in security-constrained electricity markets subject to transmission flow limits. Although the notion of separate reserve types as proposed by FERC can be precisely and unambiguously defined, when transmission constraints are active, the very existence of separate reserve prices and markets is open to question when the prices are based on marginal costs. Instead, we submit here that the only products whose marginal costs can be separately and uniquely defined and calculated are those of energy and security at each node. Thus, under marginal pricing, at any given network bus all scheduled reserve types should be priced not at separate rates but at a common rate equal to the marginal cost of security at that bus. Furthermore, we argue that nodal or area reserves cannot be prespecified but must be obtained as by-products of the market-clearing process. Simulations back up these conclusions. Index Terms—Contingency-constrained scheduling, demand-side reserve, electricity markets, security and energy pricing, transmission constraints, up and down-spinning reserve.

I. NOMENCLATURE Functions: Benefit function bid by load to consume in the precontingency state. Cost function offered by generator to produce in the precontingency state. Constants: Vector of linear offered cost coefficients. dc load flow matrix in the precontingency state. dc load flow matrix under contingency . Vector of fixed offered cost coefficients. Matrix relating power flows to nodal phase angles in the precontingency state. Matrix relating power flows to nodal phase angles under contingency . Vector of upper bounds on increased levels of power consumed for down-spinning reserve. Vector of lower bounds on decreased levels of power consumed for up-spinning reserve. Manuscript received February 26, 2004; revised May 27, 2004. 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, Québec. J. M. Arroyo was supported by McGill University through the Richard H. Tomlinson Postdoctoral Fellowship and by the Junta de Comunidades de Castilla–La Mancha, Spain. 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 (e-mail: [email protected]). F. D. Galiana is with the Department of Electrical and Computer Engineering, McGill University, Montreal, QC, H3A 2A7, Canada (e-mail: [email protected]). Digital Object Identifier 10.1109/TPWRS.2005.846221

Derating of generator defining contingency .

Variables: Vector of precontingency bus demands. Vector of bus demands under contingency . Vector of line flows in the precontingency state. Vector of line flows under contingency . Vector of generator power outputs in the precontingency state. Vector of generator power outputs under contingency . Revenue of load from the sale of reserves. Vector of down-spinning reserves provided by consumers. Vector of up-spinning reserves provided by consumers. Revenue of generator from the sale of reserves. Vector of down-spinning reserves provided by generators. Vector of up-spinning reserves provided by generators. Vector of 0/1 variables denoting the off/on status of the generators.

0885-8950/$20.00 © 2005 IEEE

ARROYO AND GALIANA: ENERGY AND RESERVE PRICING IN SECURITY AND NETWORK-CONSTRAINED ELECTRICITY MARKETS

Vector of nodal phase angles for the precontingency state. Vector of nodal phase angles under contingency . Vector of nodal marginal costs of energy. Vector of nodal marginal costs of security. Lagrange multipliers: Vector of Lagrange multipliers associated with the power balance constraints in the precontingency state. Vector of Lagrange multipliers associated with the power balance constraints under contingency . Sets: Set of indices of derated generators defining contingency . Set of indices of generators. Set of indices of loads. Set of indices of the credible contingencies. Set of indices of out-of-service generators defining contingency . Set of indices of loads subject to a sudden variation defining contingency . II. INTRODUCTION ECURITY, in a deterministic sense, is the capability of a power system to survive a specified set of credible contingencies such as generator and line outages or sudden variations of load, all without having to shed load beyond the limits of voluntary interruptibility.1 Maintaining a high level of security, which has historically been of major concern to the power industry, continues to be a high priority goal in the current context of competitive electricity markets. However, events such as the failure of the California market, the recurrence of price spikes due to transmission bottlenecks, and recent major blackouts worldwide reveal that a number of unresolved issues remain concerning the relation between system security and the rules governing electricity markets. In power systems, two kinds of security actions can be defined [1]. Preventive security actions ensure that in the event of a contingency enough resources are available for the quick execution of corrective security actions that guarantee the normal operation of the system once the contingency has taken place. Examples of preventive actions include the turning on of extra generating units or the redispatch of already committed units in the precontingency state. Corrective actions include the fast redispatching of generation or the curtailment of selected loads under a specific contingency. In addition, for certain types of slowly-developing contingencies, corrective actions may require turning on some standby generation. Reserves constitute the resources that enable the implementation of preventive and corrective security actions. FERC defines several types of reserves [2], [3] and proposes that they be treated as separate unbundled commodities with separate prices. This paper focuses on spinning or synchronized reserves, specifically up-spinning and down-spinning [4]; however, the main

S

1Probabilistic security allows for some load shedding provided that the expected load not served is below a certain threshold.

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conclusions reached also apply to other types such as standby reserves. One fundamental issue raised here is whether the resources offered by the independent agents to satisfy system security, namely, up and down-spinning reserves, can be separately priced. Before pursuing this question further, we first summarize the most relevant previous related work. The work of Schweppe and his colleagues [5] introduced the notion of spot pricing in electricity markets as defined by marginal cost theory. In [6], the marginal cost of up-spinning reserve was calculated through an iterative process within a decentralized market framework. In [7], the authors achieved a socially optimal level of system security in a decentralized marketplace through pricing incentives and information on system needs. Both of these contributions [6], [7] assumed a perfect market as well as advanced knowledge of probabilistic information about the contingencies. A geometric approach was used in [8] to analyze the behavior of marginal prices as the system operating point nears a feasibility boundary. Price signals were developed to encourage the elastic demand to adjust itself to avoid conditions such as voltage collapse. In [9], Kumar and Sheblé first proposed a joint energy and ancillary services market where each generator and consumer had the obligation to take part in the spinning reserve market by pre-specifying its fraction of participation. This is a step that, as stated by the authors, requires engineering judgment and off-line studies. Other authors [10]–[12] have also considered security-constrained models including the transmission network and flow constraints, with prespecified nodal or area reserves. The prespecification of local reserves or reserve participation levels in the above approaches permits the calculation of their marginal costs as the Lagrange multipliers of the nodal balance of the combined local and imported reserves. One shortcoming of these approaches is that since the local reserves or their fractional levels are prespecified, the market-clearing solution is generally suboptimal and may even be infeasible. We argue here that local reserves, being critical elements of the security-constrained market-clearing problem, must be treated as decision variables of the optimization process and not as prespecified parameters. Strbac and colleagues [13] proposed a security-constrained economic dispatch for a horizon of one year. The cost of corrective security was defined as the difference between the costs of the solutions with and without security constraints. This work allowed for demand-side corrective actions but supposed that preventive actions such as unit commitment were given. In [14], this work was extended from a dc load flow to an ac network model. Recently, a joint energy/reserve market model including demand-side reserve offers was developed for systems without network constraints [4]. Several reserve products were considered, including up and down-spinning reserves supplied by the generators and the loads. In this model with essentially a single bus, it was possible to associate each type of reserve directly and uniquely with one specific type of contingency. For example, up-reserve was defined by the loss-of-generation contingencies, while down-reserve was associated with a sudden decrease in demand. It was then possible to determine a unique marginal

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cost for each type of reserve through the corresponding Lagrange multipliers. When the market-clearing process models the transmission network and some flow limits become active, the association of each offered reserve type with a specific kind of contingency is no longer possible. As an example, when some lines are congested, the outage of a generator or a line may lead to corrective actions under which some of the remaining generating units increase their output, thus using up-reserve, while, simultaneously, other units decrease their outputs, thus using down-reserve. This category of problem requiring the simultaneous use of different reserve types is a critical issue that can occur with frequency when transmission congestion becomes active, yet it has not been adequately addressed and resolved in the literature. This paper therefore examines the short-term operation and pricing of the various products traded in a joint energy/reserve market while accounting for transmission network flow limits and security constraints. In this market, besides submitting offers to sell and bids to buy energy, the participants can also contribute toward system security by offering to sell both up and down-spinning reserves at different rates. The system operator clears such a market by scheduling all the energy and reserve offers and bids so as to maximize the system social welfare while satisfying all operational constraints including those imposed by the need to survive the set of credible contingencies. The model presented is a contingency and network-constrained market-clearing procedure where energy and two different types of spinning reserve are jointly dispatched. In essence, this model combines the contingency-constrained optimal power flow first proposed in [15] and a market-clearing procedure for both energy and reserve [9]–[12]. One difference with respect to previous contingency-constrained models [13]–[15] is that the preventive security measures include unit commitment decisions for the precontingency state. Corrective security actions are explicitly accounted for in the market-clearing process by ensuring that all operational constraints are satisfied under all credible contingencies. These corrective actions define the required levels of two distinct types of reserve, namely, up and down-spinning reserves, which constitute a salient feature with respect to previous joint energy/reserve market-clearing models [9]–[12]. The goal of this paper is to analyze such model in terms of pricing and feasibility of the allocation of reserves. The main contributions of this work are as follows: 1)

If local reserves are prespecified, the solution is suboptimal and can even be infeasible. Instead, local reserves are by-products of the market-clearing optimization process. 2) The notion of marginal costs for the individual reserve types such as up and down-spinning reserves is shown not to exist. Instead, the only products whose marginal costs can be uniquely defined and calculated at each node are energy and security, the latter defining, under marginal pricing, the nodal prices for any type of reserve product being traded at that bus. A possible extension of this work is the consideration of probabilistic measures of reliability. To derive probabilistic indices

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such as loss-of-load probability (LOLP) and expected unserved energy (EUE), a number of approaches have been proposed in the literature: the widely used decision tree methods [16], [17] and a probabilistic unit commitment method [18]. The remainder of this paper is organized as follows. In Section III, the mathematical formulation of the proposed model is presented. The nodal marginal costs of energy and security are defined in Section IV. Pricing issues are discussed in Section V. In Section VI, results from two case studies are provided and analyzed. In Section VII, some relevant conclusions are drawn. Finally, an Appendix is included to show the general derivation of the nodal marginal costs of energy and security. III. FORMULATION The market-clearing procedure is given by Min (1) subject to (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12) (13) where the superscript denotes the transpose operator. The objective function to be minimized (1) consists of the sum of the offered cost functions for generating power minus the sum of the bid benefit functions for consuming power plus the cost of all up and down-spinning reserves offered by the generators and the loads. Note that, in the proposed model, no term is incorporated in the objective function to account for the extra cost of the corrective actions in the event of a contingency, that is, for the actual use of the reserve. Thus, if called to provide reserve, the market participants would have to modify their respective production and consumption without any further remuneration. We suggest that this is reasonable because the probability of occurrence of a contingency is very low and the market participants already profit from their willingness to provide these corrective actions. Using a dc load flow model, constraints (2) and (3) represent the nodal power balance equations under the precontingency deand contingency states, respectively. The variables and note the vectors of the corresponding Lagrange multipliers.

ARROYO AND GALIANA: ENERGY AND RESERVE PRICING IN SECURITY AND NETWORK-CONSTRAINED ELECTRICITY MARKETS

Constraints (4) and (5) express the line flows in terms of the nodal phase angles under the normal and contingency states, respectively, while constraints (6) and (7) enforce the corresponding line flow capacity limits. Constraints (8) and (9) set the generation limits for the precontingency and contingency states, respectively. Note that the vector of on/off variables is the same under both states, indicating that only those generators that are scheduled on in the precontingency state can participate in corrective actions during contingency states. In the case of standby reserve, corrective actions may also allow some units to be turned on after a slowlydeveloping contingency however this case has not been dealt with here. Constraints (10) set the elasticity limits of the demand, and . In addition, it is assumed that customers may be willing to alter their consumption during contingencies thereby also contributing to system security. Thus, by announcing their willingness to curtail, the demands would contribute to up-spinning reserve, while by agreeing to increase consumption they would contribute to down-spinning reserve. Constraints (11) set and , inside of which the consumers are ready the limits to modify their demand relative to their normal level, . Note that these limits are less restrictive than the elasticity limits imposed in (10). Finally, constraints (12) and (13) relate the up and down-spinning reserve contributions to the power levels produced and consumed under the precontingency and contingency states. The framework presented here is general enough to take into account any contingency of the following types: ; a) loss of a sub-set of generators b) derating of a sub-set of generators ; sudden variation of a sub-set of loads ; d) , and ). loss of a sub-set of lines (modeled by The notion of system security is defined with respect to a set of pre-defined credible contingencies of the above types, keeping in mind that some of the constraints in (2)–(13) must be modified to meet the definition of the contingency as given by items a)–d) above. Preventive security is realized by redispatching the generation levels at the precontingency state and by requiring that the commitment status of the generators be the same under both the precontingency and contingency states. Corrective security actions are implemented after each contingency by redispatching and . the generation and demand to new levels defined by c)

IV. NODAL MARGINAL COSTS OF ENERGY AND SECURITY The model presented in the previous section identifies three separate products at each node, namely energy, up-spinning reserve and down-spinning reserve, each of which is offered for sale by each agent at a possibly different rate. In addition, although only generators sell energy, both generators and loads can offer to sell reserves. The fact that three distinct commodities can be traded suggests the possibility that three distinct nodal marginal costs exist. However, we argue here that in general this is not feasible and that only two marginal costs can in

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fact be defined per node: 1) the marginal cost of energy and 2) the marginal cost of security. As discussed in the Introduction, when network constraints are ignored, it is possible to associate one type of contingency with one kind of reserve (e.g. up-spinning reserve with generator outages). The marginal cost of generator outages could then be defined as the marginal cost of the level of up-spinning reserve. When transmission congestion is considered, however, the corrective actions following a contingency may require the simultaneous use of a combination of up and down-spinning reserves at different buses. It is then generally impossible to identify a set of contingencies that would require the use of one type of reserve at all buses at the exclusion of other reserve types. Therefore, it is not possible to derive a separate and unique marginal cost for each type of reserve. What can however be defined uniquely and under the most general circumstances, are the nodal marginal costs of meeting the power balance for all credible contingencies (3). We submit here that these marginal costs, denoted here by the vector , define the nodal marginal costs of security. As shown in the Appendix, from the Karush–Kuhn–Tucker optimality conditions, it readily follows that (14) where the are the Lagrange multipliers associated with (3). Moreover, since the purpose of reserves is to maintain system security, under marginal pricing the vector denotes the nodal prices applied to all forms of reserves, whether up or down, whether supplied by generators or demands, and whether sold or bought. In a similar way, we can define the nodal marginal costs of meeting the power balance for all credible contingencies (3) plus the precontingency state (2). These marginal costs, denoted here , also readily follow from the Karush–Kuhn–Tucker opby timality conditions as shown in the Appendix (15) where the are the Lagrange multipliers associated with (2). Similarly, we argue that defines the nodal marginal costs of energy, which under marginal pricing, defines the nodal prices of electricity in the precontingency state. As seen in the Appendix, the marginal cost of security can be interpreted in terms of the change in the optimum value of the objective function due to an infinitesimal change in the power balance equations under all contingencies. In addition, the marginal cost of energy can be interpreted in terms of the change in the optimum value of the objective function due to an infinitesimal change in all the power balance equations, including that of the precontingency state. It is interesting to observe that as the list of credible contingencies grows, so does the possibility of increasing the marginal costs of security and energy. However, these prices will increase only if the contingency added is “more severe” than the existing ones. If the new contingency is “less severe” than the ones to

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which the system can already survive, then the corresponding will be zero. Lagrange multiplier V. REVENUES AND PAYMENTS UNDER MARGINAL PRICING The market-clearing process defined by the optimization problem (1)–(13), yields the scheduled values of power proand , duced and consumed by each agent, respectively, as well as the reserves provided by each participant, namely, , , , and . From the nodal prices of energy and security derived in (14) and (15), we can define the corresponding revenues and payments for all market participants. For example, the revenue of generator from the sale of reserves becomes

Fig. 1. Three-bus example. TABLE I GENERATOR DATA—THREE-BUS SYSTEM

(16) while the revenue of load from the sale of reserves becomes (17) This paper does not deal with the settlement of these reserve payments, however this could be done by charging each generator and load in proportion to the amounts of power generated or consumed.

TABLE II OPTIMAL PARTICIPANT SCHEDULES (IN MEGAWATTS)—THREE-BUS SYSTEM

VI. NUMERICAL RESULTS Results from two test cases are presented in this section. The first case consists of three buses, while the second is based on the IEEE Reliability Test System [19]. In both cases, a single time period is studied. The model has been implemented on a Pentium IV, 2.66-GHz processor with 512 MB of RAM using CPLEX 8.1 under GAMS [20].

TABLE III CORRECTIVE ACTIONS (IN MEGAWATTS)—THREE-BUS SYSTEM

A. Three-Bus Example The topology of this example is shown in Fig. 1. Line reactances are all 0.63 p.u on a base of 100 MVA and 138 kV. The capacities of lines 1–2 and 1–3 are 100 MVA, whereas the flow of line 2–3 is limited to 30 MVA. Data for the generators are given in Table I. For the sake of simplicity, the generators offer linear cost functions of the form . An elastic load with lower and upper elastic bounds of 50 and 80 MW, respectively, is located at bus 3 (see Fig. 1). This load bids to buy energy at $4/MWh and offers to sell up and down-spinning . The load can be parreserves at tially curtailed below its lower elasticity limit of 50 MW down to 20 MW in order to provide additional up-spinning reserve. If needed, this load can increase its consumption beyond its upper elasticity limit of 80 MW up to 100 MW in order to contribute to down-spinning reserve. Four credible contingencies are considered here, defined by the outage of any single generator and of line 1–3. Table II lists the optimal power produced and consumed for the case without security, i.e., when only energy is cleared, with all contingency and reserve constraints removed from the model (1)–(13). Here, only generators 2 and 3 are committed and the load is at its lower elastic bound of 50 MW. In order to meet the power flow capacity of line 2–3, the cheapest generator 2

limits its production to 45 MW, so that the remaining 5 MW are supplied by generator 3. The value of the objective function is $40. Table II also shows the optimal participant schedules for the security-constrained case. The load still consumes 50 MW, however the schedule of the generators has changed (preventive security action), requiring the commitment of the expensive generator 1. This unit operates at its minimum power output of 2.4 MW, also providing 7.6 MW of up-spinning reserve. In addition, preventive security requires that the power outputs of the cheapest generators (2 and 3) be reduced with respect to the unconstrained case. Generator 2 also provides down-reserve while generator 3 produces up-reserve. The value of the objective function for this security-constrained case is $175.78, which is considerably higher than the $40 that it costs to operate the system without security. Finally, because of its high offer, no reserves are allocated to the load. Table III shows the power supplied and consumed by each participant under all four contingencies considered (corrective security actions). The outage of generator 1 requires that generator 3 use up-spinning reserve by increasing its output to 6.2

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TABLE IV NODAL RESERVES (IN MEGAWATTS)—THREE-BUS SYSTEM

TABLE VI NODAL PRICES (IN DOLLARS PER MEGAWATTHOUR)—THREE-BUS SYSTEM

TABLE V LAGRANGE MULTIPLIERS (IN DOLLARS PER MEGAWATTHOUR)—THREE-BUS SYSTEM

TABLE VII REVENUES FROM THE SALE OF RESERVES (IN DOLLARS)—THREE-BUS SYSTEM

MW from its precontingency level of 3.8 MW (see Table II). Similarly, the outage of generator 2 forces generators 1 and 3 to increase their production with respect to the precontingency state. The outage of generator 3 is significant because, in order to survive this contingency, unit 1 uses up-spinning reserve by increasing its output to 10 MW from its precontingency level of 2.4 MW. Simultaneously, unit 2 decreases its power output to 40 MW from its precontingency level of 43.8 MW, thus using down-spinning reserve. This is an example where, due to the congestion of line 2–3, a loss-of-generator contingency requires the simultaneous use of different types of reserves. Finally, in order to survive the loss of line 1–3, generators 2 and 3 make use of down and up-spinning reserve, respectively. Methods that ignore transmission constraints cannot identify or model this type of mixed reserve requirements [3], [4]. Table IV summarizes the nodal spinning reserves necessary to guarantee system security. These values, found by the optimization process presented here, would be difficult to pre-specify as required by other approaches [10]–[12]. The Lagrange multipliers associated with the power balance equations of the precontingency and contingency states are listed in Table V. For the loss of generator 1, the corresponding multipliers are all zero because the reserves required for the other contingencies cover this case (see Table III). Moreover, since the system can survive the outages of generator 1 or 2 without congesting line 2–3, their multipliers are identical. However, since the loss of generator 3 congests line 2–3 the nodal multipliers are all different. Finally, since line 2–3 is congested during the outage of line 1–3, the associated Lagrange multipliers are not only dissimilar but two of them are negative. This result is in agreement with [8]. The sum per bus of the Lagrange multipliers associated with the power balance of the precontingency and contingency states represents the nodal energy price under security constraints. Analogously, the sum per bus of the Lagrange multipliers associated with the power balance under contingencies represents the nodal security price. Both energy and security nodal prices are presented in Table VI, which also shows the energy prices obtained when security is not considered. From Table VI, it can be inferred that security slightly increases the energy prices with respect to the unconstrained case. As can be seen, the location

of the single load, i.e., bus 3, determines the highest nodal security price. Note that although the energy price of bus 1 is lower than the rate offered by generator 1 located at the same bus, this result is not unusual under marginal pricing [5]. The lack of revenue reconciliation in this example is due to the relatively low reserve rate offered by the generator which is then scheduled at its minimum power output so as to provide cheap up-spinning reserve (see Table II). Finally, Table VII shows the distribution of revenues among the participants from the sale of reserves. Each participant, whether supplying up or down-spinning reserve, gets remunerated at the same rate, namely the nodal price of security. The more significant contribution to security provided by generator 3 allows it to earn the highest revenue from the sale of reserves. B. IEEE RTS Based Case The case based on the 24-bus IEEE Reliability Test System [19] comprises 26 generators and 17 inelastic loads. The data for the generators can be found in [4]. Generator quadratic energy offers have been approximated by piecewise linear curves with 500 blocks each. The load profile corresponds to a weekday of a winter week at 18:00 [19]. All loads offer identical rates of $20/MW/h and $1000/MW/h for up and down-spinning reserves, respectively. Loads can be curtailed from their normal levels all the way to zero to provide up-spinning reserve. If needed, the demands can also increase by up to 10% to contribute to down-spinning reserve. The only modification with respect to the network data listed in [19] consists in the reduction of the capacities of lines 11–13, 15–16 and 15–24 from 500 MVA to 175 MVA, 60 MVA, and 175 MVA, respectively. The credible contingencies considered are the outage of any single generator and the outage of line 14–16. The optimal solution was achieved in 5.5 s of CPU time including unit commitment, and the optimum value of the objective function was $67 176.71. Tables VIII and IX provide the schedule of energy and reserves for generators and loads, respectively. As can be noted, generators 1–7 provide only up-spinning reserve, generators 17, 22 and 25 provide only down-spinning reserve, and generators 21 and 23 contribute to security with both types of reserves. In addition, loads 3, 12

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TABLE VIII OPTIMAL GENERATOR SCHEDULE (IN MEGAWATTS)—24-BUS SYSTEM

TABLE X NODAL RESERVES (IN MEGAWATTS)—24-BUS SYSTEM

TABLE XI LAGRANGE MULTIPLIERS (IN DOLLARS PER MEGAWATTHOUR)—24-BUS SYSTEM

TABLE IX OPTIMAL SCHEDULE OF THE LOADS (IN MEGAWATTS)—24-BUS SYSTEM

and 13 contribute to security with up-reserves equal to 115.6, 138.3, and 77.48 MW, respectively. As in the previous example, there are contingencies requiring the simultaneous use of up and down-spinning reserves. For instance, in order to survive the outage of generator 24, generator 17 decreases its power output to 150.5 MW, making use of 4.5 MW of down-spinning reserve, whereas generators 21 and 23 increase their production up to 156.46 MW and 197 MW, by using 28.43 MW and 70.16 MW of up-spinning reserve, respectively. Table VIII also shows the optimal solution for the non security-constrained case that differs from the security-constrained case not only in the dispatch but also in the scheduling of the

generators (see generators 1–7 and 21–23). The objective function value of the optimal solution without security is $59 797.12 and was obtained in 1.7 s of CPU time. Table X summarizes the nodal spinning reserves that guarantee the security of the system as determined by the optimal market-clearing process. As several generating units are located at the same bus [19], the values shown in the “Generation” columns denote the total reserve provided by all units connected to the bus. Only those buses with nonzero reserve contributions are shown in Table X. We emphasize once more that the a priori estimation of the optimal nodal reserves is a difficult task. Market-clearing algorithms that rely on such a prespecification are therefore prone to suboptimality and could lead to infeasible market equilibrium [10]–[12]. The Lagrange multipliers associated with the power balance equations of the precontingency and contingency states are listed in Table XI. Only the outages of generators 24 and 26 as well as the loss of line 14–16 affect the optimal solution, i.e., the Lagrange multipliers for the remaining contingencies are all zero and the corresponding constraints could have been removed from the model without altering the optimal solution.

ARROYO AND GALIANA: ENERGY AND RESERVE PRICING IN SECURITY AND NETWORK-CONSTRAINED ELECTRICITY MARKETS

TABLE XII NODAL PRICES (IN DOLLARS PER MEGAWATTHOUR)—24-BUS SYSTEM

TABLE XIII REVENUES FROM THE SALE OF RESERVES (IN DOLLARS)—24-BUS SYSTEM

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VII. CONCLUSIONS Recent blackouts throughout North America and Europe call into question the compatibility of the rules governing electricity markets and the secure operation of power systems. In particular, when transmission flow limits are active, the scheduling of nodal reserves to provide sufficient flexibility to survive a set of credible contingencies is a difficult problem to which we have made several contributions in this paper. 1) We have formulated a general market-clearing process in the form of an optimization problem that accounts for transmission flow limits, preventive and corrective security, unit commitment, and two types of reserves offered by both generators and loads. 2) The notion of marginal costs of the individual reserve types such as up and down-spinning reserves is shown not to exist through numerical counterexamples. Instead, the only products whose marginal costs can be separately and uniquely defined and calculated at each node are energy and security, the latter defining, under marginal pricing, the nodal prices for any type of reserve product being traded at that bus. 3) Finally, we show through some numerical examples that local (or area) reserve requirements cannot be specified a priori without sacrificing optimality and possibly rendering the market equilibrium infeasible. Instead, local reserves are determined as by-products of the proposed market-clearing process. Simulation results from two case studies, including the IEEE 24-bus RTS, have been presented to illustrate the main issues discussed here. APPENDIX DERIVATION OF NODAL MARGINAL COSTS

As can be seen, the outage of generator 26 has the greatest impact on system security. Table XII presents the nodal prices of energy and security. The highest nodal prices for security occur at buses 11 and 24. This can be explained by the fact that these two buses split the system into two areas: the upper area with excess of generation (buses 13–24) and the lower area with deficit of generation (buses 1–10). Other buses with relatively high security prices are 3, 14, and 15, which together with buses 11 and 24 are geographically near the congested lines and the line 14–16 defining one of the credible contingencies. Table XII also presents a comparison with the energy prices obtained when security is neglected. Note that security causes an increase in the nodal energy price of over 10% in most buses. Buses 7 and 13 constitute an exception experiencing a reduction in the energy price when security is accounted for. Finally, Table XIII shows the distribution of nodal revenues between generation and demand from the sale of up and down-spinning reserves. Only those buses with revenues different from zero are listed. Those loads connected to buses 3, 14 and 15 with relatively high security prices (see Table XII), earn the highest reserve revenues, followed by generator 25 which provides 95.01 MW of down-spinning reserve at bus 18.

The market-clearing problem formulated in this paper has the following general form:

subject to

(A1) where represents the vector of binary decision variables, is is the objecthe vector of continuous decision variables, represents the nodal power balance equative function, tions in the precontingency state, denotes the nodal power balance equations under the credible contingency , and represents all remaining inequality constraints. The vecand are the Lagrange multipliers corresponding to tors , the previous relations. Consider the Lagrangian function evaluated at , the optimum vector of binary variables (A2)

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The optimal solution of (A1) must satisfy Karush–Kuhn–Tucker necessary optimality conditions

the

while the nodal marginal costs of security are:

(A10)

REFERENCES

(A3) In order to calculate the nodal marginal costs of energy, let the power balance equations in the precontingency and contingency . states be incremented by an infinitesimal parameter vector, Similarly, in order to derive the nodal marginal costs of security, let the power balance equations under all contingencies be incremented by another infinitesimal parameter vector, . These incremental changes to the problem equalities result in an infinitesimal change to the continuous variables of the optimum to (A1), denoted by . The equalities in (A1) now must satisfy

(A4) Similarly, the optimum value of the objective function changes by (A5) Using (A3),

can be expressed as

(A6)

Furthermore (A7) Applying (A4) and (A7) to (A6) yields (A8)

Consequently, the nodal marginal costs of energy are (A9)

[1] T. E. Dy Liacco, “The adaptive reliability control system,” IEEE Trans. Power Apparatus and Systems, vol. PAS-86, no. 5, pp. 517–531, May 1967. [2] Order 888: Promoting Wholesale Competition Through Open Access Nondiscriminatory Transmission Services by Public Utilities, Federal Energy Regulatory Commission. (1996, Apr.). http://www.ferc.gov [Online] [3] N. S. Rau, “Optimal dispatch of a system based on offers and bids—a mixed integer LP formulation,” IEEE Trans. Power Syst., vol. 14, no. 1, pp. 274–279, Feb. 1999. [4] J. Wang, N. Encinas Redondo, and F. D. Galiana, “Demand-side reserve offers in joint energy/reserve electricity markets,” IEEE Trans. Power Syst., vol. 18, no. 4, pp. 1300–1306, Nov. 2003. [5] F. C. Schweppe, M. C. Caramanis, R. D. Tabors, and R. E. Bohn, Spot Pricing of Electricity. Norwell, MA: Kluwer, 1988. [6] M. C. Caramanis, R. E. Bohn, and F. C. Schweppe, “System security control and optimal pricing of electricity,” Elect. Power Energy Syst., vol. 9, pp. 217–224, Oct. 1987. [7] R. J. Kaye, F. F. Wu, and P. Varaiya, “Pricing for system security,” IEEE Trans. Power Syst., vol. 10, no. 2, pp. 575–583, May 1995. [8] R. Rajaraman, J. V. Sarlashkar, and F. L. Alvarado, “The effect of demand elasticity on security prices for the Poolco and multi-lateral contract model,” IEEE Trans. Power Syst., vol. 12, no. 3, pp. 1177–1184, Aug. 1997. [9] J. Kumar and G. Sheblé, “Framework for energy brokerage system with reserve margin and transmission losses,” IEEE Trans. Power Syst., vol. 11, no. 4, pp. 1763–1769, Nov. 1996. [10] M. Aganagic, K. H. Abdul-Rahman, and J. G. Waight, “Spot pricing for generation and transmission of reserve in an extended Poolco model,” IEEE Trans. Power Syst., vol. 13, no. 3, pp. 1128–1135, Aug. 1998. [11] T. Alvey, D. Goodwin, X. Ma, D. Streiffert, and D. Sun, “A security-constrained bid-clearing system for the New Zealand wholesale electricity market,” IEEE Trans. Power Syst., vol. 13, no. 2, pp. 340–346, May 1998. [12] K. W. Cheung, P. Shamsollahi, D. Sun, J. Milligan, and M. Potishnak, “Energy and ancillary service dispatch for the interim ISO New England electricity market,” IEEE Trans. Power Syst., vol. 15, no. 3, pp. 968–974, Aug. 2000. [13] G. Strbac, S. Ahmed, D. Kirschen, and R. Allan, “A method for computing the value of corrective security,” IEEE Trans. Power Syst., vol. 13, no. 3, pp. 1096–1102, Aug. 1998. [14] A. Jayantilal and G. Strbac, “Load control services in the management of power system security costs,” IEE Proc.–Gener., vol. 146, no. 3, pp. 269–275, May 1999. [15] B. Stott, O. Alsac, and J. L. Marinho, “The optimal power flow problem,” in Proc. SIAM Int. Conf. Elect. Power Problems: Mathemat. Challenge, Seattle, WA, Mar. 1980, pp. 327–351. [16] J. Kumar and G. Sheblé, “A decision analysis approach to the transaction selection problem in a competitive electric market,” Elect. Power Syst. Res., vol. 38, pp. 209–216, Sep. 1996. [17] EPRI, “MIDAS, multi objective integrated decision analysis system,” EPRI J., vol. 12, pp. 57–58, 1987. [18] F. Bouffard and F. D. Galiana, “An electricity market with a probabilistic spinning reserve criterion,” IEEE Trans. Power Syst., vol. 19, no. 1, pp. 300–307, Feb. 2004. [19] Reliability Test System Task Force, “The IEEE reliability test system—1996,” IEEE Trans. Power Syst., vol. 14, no. 3, pp. 1010–1020, Aug. 1999. [20] GAMS: A User’s Guide, A. Brooke, D. Kendrick, A. Meeraus, and R. Raman. (2003). http://www.gams.com/ [Online]

ARROYO AND GALIANA: ENERGY AND RESERVE PRICING IN SECURITY AND NETWORK-CONSTRAINED ELECTRICITY MARKETS

José M. Arroyo (S’96–M’01) received the Ingeniero Industrial degree from the Universidad de Málaga, Málaga, Spain, in 1995 and the Ph.D. degree in power systems operations planning from the Universidad de Castilla–La Mancha, Ciudad Real, Spain, in 2000. From June 2003 through July 2004, he held a Richard H. Tomlinson Postdoctoral Fellowship at the Department of Electrical and Computer Engineering of 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.

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Francisco D. Galiana (F’91) received the B.Eng. (Hon.) degree from McGill University, Montreal, QC, Canada, in 1966 and the S.M. and Ph.D. degrees at the Massachusetts Institute of Technology, Cambridge, MA, 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 Harbor. He joined the Department of Electrical and Computer Engineering at McGill University in 1977, where he is currently a Full Professor.