Providing for transmission in times of scarcity: an ISO cannot do it all

Electrical Power and Energy Systems 21 (1999) 147–163

Providing for transmission in times of scarcity: an ISO cannot do it all Eric Allen, Marija IIic*, Ziad Younes Massachusetts Institute of Technology, Cambridge, MA 02139, USA

Abstract The main purpose of this paper is to point out issues critical for establishing a good transmission strategy in an energy market. First, it is suggested that a transmission strategy must be discussed in the context of a specific market structure. Responsibilities of a transmission system provider differ fundamentally, depending on the type of energy market it is supposed to serve. To show this, a summary of information assumed to be known to an Independent System Operator (ISO) in three energy market structures is given, i.e., (1) a mandatory ISO, (2) an entirely multilateral market and (3) a voluntary ISO. Next, it is suggested which of the transmission strategies proposed may be most suitable for each of these three markets. In particular, we argue that the proposal by Hogan (Hogan WW. Contract networks for electric power transmission, J Regulatory Economics, 1992; 4: 211– 242) naturally lends itself only to a market structure that is a mandatory pool in which all energy price bids are assumed to be known to the ISO. In contrast to this, the proposal by Wu and Varaiya (Wu FF and Varaiya P, Coordinated multilateral trades for electric power networks: theory and implementation, POWER Report PWR-031, University of California Energy Institute, June 1995) is well suited for a bilateral energy market, in which an ISO imposes no requirements on market participants regarding prices of specific energy transactions. Finally, the proposal by Ilic´ et al. (Ilic´ M, Graves F, Fink L and DiCaprio A. A framework for operations in a competitive open access environment, Electricity Journal, March 1996; Ilic´ M. A possible framework for implementing energy transactions into real-time system operation and pricing for system services. Proceedings of the EPRI Conference on Innovative Approaches to Electricity Pricing: Managing the Transition to Market-Based Pricing, La Jolla, CA, March 1996; Ilic´ M, Hyman L, Allen EH, Cordero R and Yu C-H, Interconnected system operations and expansion planning in a changing industry: coordination vs. competition. In: Topics in regulatory economics on policy series. Dordrecht: Kluwer Academic, 1997. pp. 307–332) lends itself to a voluntary ISO structure, in which some energy providers are scheduled on a price bid basis by an ISO and some are multilateral. The differences between these three proposals concerning an ISO’s responsibility for achieving systemwide efficiency and fair charges for transmission service, particularly at times of scarcity, are analyzed. It is shown that an ISO equipped with the present types of optimization tools for both reliability and efficiency is generally ‘blind’ to questions of fairness with respect to the individual market participants when providing transmission system support. In order to get around this problem, much more work will have to be done by the technical and regulatory communities. The only tools at an ISO’s disposal at present are used for systemwide objectives, such as systemwide reliability. While some of this work is under way, it will take some time to develop the actual ISO tools necessary for implementing the fairness criterion metrics (‘standards’), whichever ones the community arrives at. (Developing metrics of fair reliability contributions for the individual market participants is a nonunique process, and it may be very difficult to actually agree upon). Meanwhile, in order to have an ISO actively help energy markets in a fair and efficient way in realistic markets, which are likely to be voluntary ISOs, a system user must become an active part of decision making, indicating how much it wishes to use the system at times of scarcity and at which price. One possible way for doing this, based on the Ilic´ proposal, is described here. 䉷 1997 Elsevier Science Ltd. All rights reserved.

1. Introduction The main purpose of this paper is to point out issues critical for establishing a good transmission strategy in an energy market. A transmission strategy must be discussed in the context of a specific market structure, because the responsibilities of a transmission system provider differ fundamentally depending on the type of energy market it is supposed to serve. * Corresponding author..

Information assumed to be known to an Independent System Operator (ISO) is summarized for three market structures: (1) a mandatory ISO, (2) an entirely multilateral market 1 and (3) a voluntary ISO. A debate about transmission provision in times of scarcity has focused on ideas ranging from making the grid equally available and at the same price to all loads on a pro rata basis [1], through nodal price-based congestion pricing [2] and 1 The terms bilateral and multilateral are used interchangeably in this paper.

0142-0615/99/$ - see front matter 䉷 1997 Elsevier Science Ltd. All rights reserved. PII: S0142-061 5(97)00039-2

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market-based determination of transmission use subject to physical curtailment [3]. A relatively comprehensive approach to system transmission provision described in [4–6] has received little attention in the overall rush toward deregulation. Here we revisit these ideas by focusing on technical questions concerning the distinction between reliability and efficiency as a function of energy market structure and transmission strategy used. This is done primarily to point out that it is generally unrealistic to put the entire burden on an ISO to provide for efficiency and nondiscriminatory reliability-based curtailment. For the sake of completeness, we first define in Section 2 short-term efficiency as seen by individual market participants under competition. Next, in Section 3, we revisit the definition of efficient real-time dispatch in a regulated industry. This is followed in Section 4 by an analysis of efficient energy pricing in a competitive industry as a function of energy market structure adopted. It is concluded that systemwide efficiency is only computable by an ISO in a mandatory ISO market structure. In the other two structures, an ISO would lack the basic economic data necessary to compute systemwide efficiency at its level. Instead, systemwide efficiency is primarily a function of how well the energy market works. This section is intended only for clarification purposes and not to pursue arguments concerning which energy market structure should be adopted. It is essential to understand this material prior to bringing into the debate the transmission system-imposed constraints. We also point out that in none of the market structures does an ISO have a measure of fairness with respect to individual market participants; the general optimization tools currently used in tight power pools by a system operator 2 only allow decisions with regard to systemwide measures concerning both efficiency and reliability. It is unrealistic to expect an ISO to suddenly know what is ‘fair’. Next, in Section 5, we begin to analyze the imposition of system constraints on the energy market as a function of transmission strategy. Given that much information already exists in the literature with regard to the proposal by Hogan [2] 3, we do not analyze how transmission is provided and priced, except at a very conceptual level. Much more space is given to analyzing the impact of system provision in a market structure proposed by Wu and Varaiya [3]. The finding that the outcomes of individual market participants are highly dependent on the technical rules for (partial) power curtailment when not all can be served is illustrated in Section 6 on a typical three-bus power system and also on a more complicated four-bus power system. The four-bus power system is particularly interesting for its illustration of the interplay between energy market activities and the 2 These are assumed to be the kernel of the ISO’s tools, at least for the immediate future. 3 We recognise that the theoretical foundation for [2] already existed in the early work by Schweppe [7].

physical curtailment rules. In Section 7, a general result is developed concerning the minimum number of submarkets as a function of the number of active transmission constraints. Even in what may appear to be a not very restrictive proposal for transmission system support (i.e., an entirely bilateral market), at least three problems arise: (1) how to ration real-time system use without affecting the financial outcomes of market participants explicitly; (2) how to plan system use to avoid imposition of system constraints; and (3) how to provide for financial stability against system shortages. In this paper we show that the answer to the first question is a function of the market structure and the transmission strategy developed. The basic role of a system provider is fundamentally different in a completely coordinated energy market like a mandatory pool [8] and in an entirely competitive (bilateral) market. Energy markets evolving in the United States are likely to be a mix of both, and the role of an ISO must be carefully examined, keeping this in mind. To resolve the first question, we strongly recommend one possible iterative auctioning mechanism for system rationing in times of scarcity that does not require the ISO to explicitly prescribe a price-based or power-based procedure. The iterative auctioning mechanism for system provision described in Section 5.3 naturally lends itself to a competitive environment in which both bilateral and ISOschedules transaction (or network and point-to-point users) coexist. The second and third questions are not considered in this paper, interested readers could see [9]. 2. Energy market economics without congestion We begin by examining the economics in a power system that has no line-flow constraints. 4 We first describe energy market economics without congestion, as seen by individual competitive market participants (CMPs). Next, we analyze the relation between what is seen as optimal by individual CMPs and systemwide efficiency measured in terms of social welfare; particular emphasis is on conditions under which the two may become the same. At each generation bus Gi the cost to produce QGi units of power is assumed to be a quadratic function: Ci …QGi † ˆ aGi Q2Gi ⫹ bGi QGi ⫹ cGi

…1†

Similarly, at each load bus Li there is a quadratic utility function that measures the usefulness of using QLi units of power: Ui …QLi † ˆ ⫺aLi Q2Li ⫹ bLi QLi ⫹ cLi

…2†

The supply curve for the marketplace is an aggregation of the marginal cost curves for each of the individual 4

P represents price and Q represents a quantity of real power.

E. Allen et al. / Electrical Power and Energy Systems 21 (1999) 147–163

generators. For an individual supplier, the marginal cost MCi is the derivative of the cost with respect to quantity: dCi …QGi † ˆ 2aGi ⫹ bGi MCi …QGi † ˆ dQGi

pGi …QGi † ˆ PQGi ⫺ Ci …QGi †

…4†

assuming that all of the quantity produced is sold at the price P. The maximum of the profit function is found by differentiation: dpGi …QGi † ˆ P ⫺ MCi …QGi † ˆ 0 dQGi

…5†

Therefore, for an individual generator with a quadratic cost curve, the generator’s supply as a function of the price P is Si …P† ˆ QGi ˆ

P ⫺ bGi 2aGi

…6†

aS ˆ

NG X

bS ˆ

…14†

aD ˆ

NL X 1 2a Li iˆ1

…15†

bD ˆ

nL X bLi 2a Li iˆ1

…16†

The equilibrium point is the point at which the supply and demand curves meet. The price at equilibrium, denoted Pl , is

bD ⫹ bS aD ⫹ aS

Pl ˆ

…17†

It is frequently mentioned in the literature that social welfare is maximized at the equilibrium of the market. Welfare is defined as the total utility minus the total cost of all participants: Wˆ

NL X

Ui …QLi † ⫺

NG X

Ci …QGi †

…18†

iˆ1

Assuming that the marketplace is relatively competitive and stable, the price will converge to the equilibrium price.

…7† 3. Efficient use of generation in a regulated industry

1 2aGi

…8†

NG X bGi 2a Gi iˆ1

…9†

iˆ1

D…P† ˆ bD ⫺ aD P

iˆ1

The complete supply curve for the market is the sum of the supply curves for all generators: S…P† ˆ aS P ⫺ bs

and the total aggregate demand in the market is

…3†

If the prevailing price in the market is P, the optimal strategy for generator i is to continue producing until the marginal cost is equal to the price. This result is easily shown by noting that the profit p Gi is a function of the quantity:

149

Note that the supply curve is a linear function, since we have assumed quadratic cost functions. The demand curve is determined by the same procedure. An individual buyer continues to purchase until the marginal utility is equal to the price. A buyer’s net utility function p Li (in this marketplace, the loads are the buyers) is

pLi …QLi † ˆ Ui …QLi † ⫺ PQLi

…10†

which is maximized at the point where the derivative is zero: dpLi …QLi † ˆ MUi …QLi † ⫺ P ˆ 0 dQLi

…11†

The marginal utility MUi for the quadratic utility function as used here is dUX …QLi † ˆ bLi ⫺ 2aLi QLi MUi …QLi † ˆ dQLi

…12†

min

QG1 …;QGn

n X

Ci …Ggi †

…19†

iˆ1

given that total generation and demand balance 5 n X iˆ1

QGi ˆ

m X

QLi

…20†

jˆ1

and subject to generation capacity constraints

Therefore, the demand function for load i is b ⫺P Di …P† ˆ Li 2aLi

Here we first recall a well-known definition of systemwide short term efficiency used in real-time operations in a regulated industry, and analyze the basics of how this changes under open access. This should set the stage for analyzing the implications of various market structures on the efficient pricing of generation to reflect systemwide efficiency. In a regulated industry, the systemwide optimum is obtained by minimizing the total cost of generation needed to meet the given demand. This is how short-term scheduling is done, each 15 minutes or so, as demand changes. Assume a power system consisting of n generators and m loads. Real-time use of available generation is routinely carried out based on the results of (unconstrained) economic dispatch to calculate:

max Qmin G ⱕ QGi ⱕ QGi

…13†

5

For simplicity, we assume negligible transmission losses.

…21†

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It is a simple matter of static optimization to show that at the optimum all power plants operate at the same marginal cost, i.e., that dC1 dC2 dCn ˆ ˆ…ˆ ˆ lⴱ dQG1 dQG2 dQGn

…22†

This marginal cost of the plants at the optimum is often referred to as ‘system l ’. Note that it is typically assumed in present optimization tools that demand cost elasticity is negligible, unless explicitly stated otherwise. A straightforward generalization of these tools to account for demand price elasticity is possible [10].

4. Efficient pricing of generation under open access Under open access the basic definition of efficient pricing of generation becomes a more complicated matter, even in the simplest case when no transmission charges are accounted for. It is argued here that the impact of an ISO in charge of approving various transactions on systemwide efficiency becomes a function of market (industry) structure adopted. To clarify this, consider three typical market structures: (1) price bid-based scheduling by a mandatory ISO; (2) an entirely bilateral energy market, with an ISO providing transmission services; (3) price bid-based scheduling by a voluntary ISO.

It is interesting to observe that the profits obtained by individual market participants are expected to be the same under perfect market conditions as the profits that would be obtained if each participant were paid only its price bid. The analysis of these profits is relevant for long-run incremax mental financing so that generation capacity installed PGi can be planned for with the objective of minimizing the risk of investment, assuming that the profit resulting from the difference between the energy market price and the operating cost (presumably close to the bidding price) will go into the investment recovery. This problem is not a subject of this paper; see [9]. Moreover, an interesting transitional issue concerns different rules for compensating ‘old’ and ‘new’ plants. The old plants are still generally allowed to bid their cost, while the new generation could bid an arbitrary price, at least in principle. This has certain distorting effects not discussed here. The implications of requiring price bids to be made public are also not discussed here, but are a wellknown concern in their own right; see [8,11,9]. For the purpose of understanding systemwide efficiency as a function of energy market structure it suffices to recognize that an ISO can be held responsible for efficient realtime use of resources in the area under its control in a mandatory ISO structure, and that the measure of efficiency in this case is a well-defined notion.

4.2. Entirely bilateral market 4.1. Mandatory ISO Most bid-based ISOs evolving in the United States have proposed the so-called Dutch auction, similar to the mandatory poolco structure in place in the United Kingdom. Transactions are scheduled according to their price bids in merit order (the least expensive units are fully scheduled), and there is only one so-called marginal unit that is partly used. The price of this unit determines the energy clearing price which is paid to all, even to those with much lower bids than the clearing price. It is clear that under this structure, assuming that price bids are close to generation cost, an ISO has a clear measure (definition) of system efficiency under its jurisdiction. The short-term efficiency used by a mandatory ISO is identical to Eq. (19) used at present in a regulated industry, with the difference of cost being replaced by the price bid: min …

QG1 ; ;QGn

n X

PBi …QGi †

…23†

iˆ1

The justification for paying everyone the same (highest) price as the plant actually scheduled can be found in condition (22), which is met at the systemwide equilibrium. Condition (22) now becomes dPB1 dPB2 dPBn ˆ ˆ…ˆ ˆ lⴱ dQG1 dQG2 dQGn

…24†

An entirely bilateral market assumes an arbitrary structure in which market participants (or groups of market participants) directly negotiate energy prices only among the parties directly involved in energy sales and purchases. An ISO is not given prices of energy sales and purchases. Basic decision making in such markets is based on formulae described in Section 2. Under perfect market conditions all short-term marginal prices are the same; i.e., the economic equilibrium is defined by condition (24). For understanding the impact and responsibilities of an ISO in this case, it is important to recognize that bilateral markets are often far from the actual economic equilibrium defined in Eq. (24), in particular for an industry in transition [8]. Consequently, the short-run marginal price at which bilateral deals are established could vary drastically; i.e., no single energy market price is known. In a market structure like this, an ISO cannot be held responsible for efficient use and distribution of energy across the system, since it is not given basic economic data. Its main role is facilitating market needs, i.e., making the transactions implementable. In other words, as far as the ISO knows, it does not have a measure of systemwide efficiency and cannot be held responsible for achieving it. (The ideal achievable systemwide efficiency remains definition (22), independent of the market structure. The point here is that an ISO cannot

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influence systemwide efficiency explicitly in an entirely bilateral market.) Achieving systemwide efficiency in bilateral markets is influenced instead by the activities of individual market participants. At least in theory, in mature, large, sufficiently fluid markets the systemwide optimum is achievable without having to make prices public 6.

4.3. Voluntary ISO A typical bid-based pool structure under serious consideration in many parts of the United States can be characterized as a mixture of at least three types of economic transactions: (1) price-based bids directly to an ISO and scheduled according to merit; (2) strictly bilateral transactions which, given the permission of the ISO, physically inject power into one location and take out the same amount; the price of these transactions is not made known to the ISO 7; and (3) transactions between a power producer and an ISO, in which the ISO schedules and determines how much the transaction is worth but the market participant may have another, strictly financial, agreement with someone else to hedge against variations in the price determined by the ISO 8. All utility-owned units belong to category (1), and since they are still regulated according to Federal Energy Regulatory Commission (FERC) rules, they are currently paid only their operating (fuel) costs 9. The other two categories are typically some sort of independent power producers (IPPs) and are paid the full clearing price in Eq. (24), as long as they are scheduled. This market structure can be viewed as being ‘nested’ relative to the traditional horizontal structures in a regulated utility organization [12]. An ISO that is in charge of providing system services to all only knows price bids of parties belonging to categories (1) and (3). Again, as in the case of an entirely bilateral structure, the ISO is not given energy prices at which bilateral trades take place. It should be clear, based on the prior discussion, that an ISO can clearly understand only the impact on systemwide efficiency of participants from categories (1) and (3). Thus, it cannot be held responsible for systemwide efficient operation. There is no well-established definition of efficiency in this case, except in the extreme case of perfect markets. In this case an ISO is just a particular submarket in a larger market structure. The dynamics of a voluntary ISO have not been studied at all with respect to their impact on achieving 6 In this paper only short-term efficiency is discussed; for long-term efficiency see [9]. 7 In NEPOOL these are referred to as self-scheduled transactions. 8 Regional jargon for such markets may vary; e.g., in the New England Power Pool (NEPOOL) they are referred to as bilateral. 9 Exceptions to this take place only if utility-owned plants demonstrate that the transmission system with which they are vertically integrated does not treat them differently than the non-utility-owned units; one successful case is the Southern Companies.

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systemwide efficiency and incentives to market participants to do so. 4.4. Summary: ISO’s potential impact on systemwide efficiency as a function of market structure For what follows, we summarize that only in a mandatory ISO market structure can an ISO be fully in charge of system efficiency. In other market structures, even the basic definition is conditional on how perfect the bilateral market is. It is essential to understand this prior to attempting to understand the role of an ISO in providing efficient and non-discriminatory transmission service to all. Inefficiencies could be caused by inadequate transmission strategies but are equally likely to be heavily influenced by the ISO’s inability to actually determine systemwide efficiency. 4.5. Role of the ISO in determining ‘fair’ profit allocation In a mandatory ISO market structure, an ISO schedules use of price-bid-based generation to meet anticipated demand by using the cheapest units first. The energy clearing price is determined by the highest bid power plant that actually gets scheduled. As long as a plant is used, it is generally paid more than what its price bid is. Since an ISO schedules power plants so that the total (systemwide) price of meeting the demand is minimized, this is the best that can be done. According to the basic laws of competitive economics, the system equilibrium condition (19) obtained by the ISO’s coordinated price minimization and the equilibrium obtained in an entirely bilateral market Eq. (5) result in the same market solution. The highly theoretical result has been the subject of long-term investigations and is not a subject of this paper. Observe also that, provided there exists a unique market equilibrium, profit allocation to individual market participants, at which individual (dp Gi)/(dQGi) ˆ 0, is the same as profit allocation determined by the ISO’s coordinated computation, at which all short-run marginal prices are the same according to Eq. (24). Consequently, the ISO-based profit allocation is equivalent to the profits obtainable on bilateral markets. The question of fairness is moot in this case. This is, of course, true prior to accounting for transmission system charges. To summarize, under ideal assumptions both ISO-coordinated scheduling and bilateral market-determined energy deals lead to the same systemwide efficiency and profit allocation. The role of an ISO becomes much more delicate where there is insufficient system support to facilitate the most desirable transactions arrived at through energy market mechanisms. In reality, there is a problem with assuming an ideal market. In point of fact, there will be decided information, knowledge and other advantages for some participants in a bilateral-only market—especially one with system

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constraints—so that making an assumption of ideality is less than useful. However, an ISO-coordinated scheduling has its own share of possible problems. Potentially, the most critical one deals with collusion problems leading to cartel-like arrangements. Learning effective patterns for collusion of this type is much easier in a ISO-based energy market which provides all CMPs in a repetitive manner with the economic information needed to do so [9,11,13]. The problems related to non-ideal energy market conditions are not the subject of this paper; they are only discussed to highlight the fact that it is not realistic to rely on the ISO to deal with nonperfect market issues. Most of what follows is concerned with the imposition of transmission constraints on market outcomes, and with good transmission strategy. 5. Imposition of congestion constraints on the marketplace: how much freedom does a system user effectively see? In this section, we briefly look at methods of transmission/system support with regard to the type of fundamental restrictions they impose on system users. In many networks, the equilibrium operating point that results from market-driven forces produces line flows that exceed allowable operating limits. In these cases, it is necessary for an ISO to curtail some or all of the power transactions in order to permit safe, reliable system operation. Accounting for these system constraints is very much a function of the underlying energy market structure. 5.1. Transmission constraints in a mandatory ISO A transmission strategy promoted by Hogan [2] is a generalization of the unconstrained price bid-based generation scheduling defined in Eqs. (23) and (24), subject to transmission system constraints relevant for technical feasibility of the proposed sales. In the simplest case, when transactions do not cause any dynamic or voltage problems, these constraints are expressed by requiring that real power quantities balance at each bus in the system; i.e., the power quantity injected into the bus equals the power flow taken by transmission lines directly connected to this bus: X Ti⫺j …25† QGi ˆ j僆Ki

where Ki is the set of buses directly connected to bus i, and Ti⫺j is the line flow in a transmission line from bus i to bus j. The transmission line flows are, furthermore, subject to so-called congestion constraints (typically, implied thermal limits): max Ti⫺j ⱕ Ti⫺j

…26†

Note that this formulation is for purposes of simplicity; a more exact formulation of operating constraints would include reactive power/voltage and dynamic security constraints [14].

The basic signals for transmission pricing as proposed by Hogan are based on nonidentical short-run marginal prices (dPGi/dQGi) at various buses when a constrained optimization problem is defined by optimizing Eq. (23) subject to both Eqs. (25) and (26). The proposal suggests that a generator at bus i should be paid (dPGi/dQGi) and a load at bus j should pay (dPLj/dQLj). In this paper we do not discuss the marginal surplus (MS) created by such pricing 10 or other much-debated aspects of the strategy. Instead, it is important for further discussion to observe that: (1) A bundled price is being paid to a generation provider, and paid by an energy user; accounted for by traditional coordinated optimal power flow (OPF) scheduling methods, in which cost is replaced by price. (2) An ISO is no longer capable of allocating charges related to transmission system constraints in a ‘fair’ way. Fair is now an undefined term. (3) Finally, an ISO can carry out an optimization of this sort provided all market participants make their price bids known 11. (4) An ISO is not capable of differentiating actions for reliability (meeting constraints such as Eqs. (25) and (26)) from actions for efficiency Eq. (19), since only a constrained optimization is performed 12. (5) The solution obtained by solving this problem is no longer the same solutionas would result froma bilateral market, primarily in terms of profit allocation to individual market participants. This is one of the main themes of this paper. 5.2. Transmission constraints in a entirely bilateral market It is presumed by Wu and Varaiya [3] that only after all sales have been made does an ISO check whether the total set of transactions is feasible on the network, i.e., meets all line-flow constraints. If the transactions are feasible, then nothing further need be done. However, if at least one lineflow exceeds the maximum allowed limit, then the allowed transactions are curtailed to a point q0 ˆ 0 0 0 0 T QG1 …QGNG QL1 …QLNL Š which is feasible. The method proposed in [3] has three basic steps: (1) All buyers and sellers negotiate in an open energy market in order to agree to trades that benefit all participants. The resulting price in the market is the uncongested price of electricity (energy clearing price). 10 MS is the difference between the net price paid to suppliers and the net price paid by demand; in an unconstrained system at equilibrium, MS is identically equal to zero. 11 As pointed out above, this is not the case in any structure but a mandatory ISO. 12 This becomes particularly important to understand when attempting to implement the proposed transmission strategy in market structures other than a mandatory ISO; one often hears questions as to whether a particular action would be taken for reliability or for efficiency. Any solution that meets the system constraints is a reliable solution, and there are conceptually many combinations of system inputs which meet them. Efficiency is only determined by optimizing systemwide efficiency without any consideration of the implications for the profits of individual CMPs.

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sensitive [14]. It is critical to provide the ISO as soon as possible with some guidelines concerning equal (physical) access in the changing industry. Of course, the issues become even more difficult once the temporal aspects of technical feasibility enter the problem. The same concern is related to the required computation of socalled available transmission capacity (ATC); e.g., ATC for a bilateral transaction is different than ATC for the same amount of power for the transactions scheduled by an ISO [14]. 5.3. Transmission constraints in a voluntary ISO

Fig. 1. Iterative process of accommodating energy market from the ISO level.

(2) An ISO considers the complete set of proposed transactions and determines whether the set is feasible, i.e., does not produce any transmission line-flow exceeding maximum limits. If the set is feasible, then no further action is needed, and all proposed transactions are made. However, if the line-flow limit is violated, then the ISO uses a protocol to curtail some or all of the proposed transactions so that the line-flow limit is not exceeded. The transactions that are permitted are made at the price originally agreed upon, the uncongested price. The ISO will also supply information to all market participants on how further trades may be made without violating line-flow limits. (3) The information from the ISO would, for example, allow a load to purchase more power from a generator, but only if that load simultaneously sells a given fraction of that power to a different generator. In this manner, submarkets for electricity in a postcongested network are formed. Trades take place in the new submarkets, and the resulting transactions bring the system to a point of optimum social welfare. The final outcomes in the two-tiered market are not affected by the ISO’s choice of curtailment levels; the curtailment affects only how much power is priced at the uncongested price and how much is priced according to the postcongested market. Observe that this method amounts to (power) quantity control of the market by the ISO, since the actual profit allocation to individual market participants is a function of the physical curtailment procedure selected. The customer, although seemingly free to trade on the market, remains dependent on actions of the ISO. When a set of proposed transactions becomes constrained, it is up to an ISO to decide how to ration system service. This is a function of the type of information available to the ISO; if no economic information is given, the only basis for reduction is strictly technical. It cannot be overemphasized that the technical (reliability) criterion is not uniquely defined at present and that much room is left for different solutions, ranging from simply rejecting everyone the same amount to rejecting the transactions to which the physical constraints are most

In an industry structure characterized by a mix of technical (power quantities) and economic (price bids) information made available to the ISO, the question of equal access become even more complex. Should the ISO declare certain power quantities to be feasible based strictly on technical considerations, or should the ISO compromise between economics and reliability (technical feasibility)? Further, the industry already differentiates in its proposals for restructuring, between load curtailment procedures as financial rights used for accounting purposes only (which are typically pro rata load share-based) and the curtailment to be done in real-time operations by an ISO. It is not clear where these two meet. We conclude here that even if all are charged equally per MW share of transmission system use, one must define the reliability (technical feasibility) tests uniquely as a guide to the ISO in order for this procedure to be consistent. Because of the ISO’s fundamental inability to compute efficiency in a voluntary ISO or to curtail in a ‘fair’ way, we next consider a transmission market mechanism involving the ISO and the CMPs in which only information exchange essential for dealing with system constraints is required. 5.3.1. A two-level iterative market for transmission provision A two-level market structure described in [4–6,12] is shown in Fig. 1. Here are the basic steps: (1) The energy market evolves at the first level and is only concerned with the financial arrangements for power purchasing in the most beneficial way, without any consideration of charges for system support 13. The prices of various transactions are not public 14. 13 This framework considers transmission congestion as just one component of a more general system support necessary to make the requested transaction feasible. 14 This transmission provision strategy allows for arbitrary forms of energy markets, ranging from coordinated power exchanges through a mix of bilateral and power exchange-based and entirely bilateral markets. It is important, however, to impose functional separation between this level and the level of transmission provision; this is why we take a two-level approach. If the separation is not imposed, the system provider may be viewed as being biased in favor of some transactions. This issue is particularly critical given the earlier observations in this paper concerning the ISO’s domain within which it can make competent decisions.

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(2) In an asynchronous way, without waiting for the energy market to reach an equilibrium, various transactions approach the ISO at the secondary level and request the implementation of their specific transactions. (3) The ISO, based on the information about (a) transaction location, (b) duration and starting time and (c) firmness, estimates the total expected charge for affecting system reliability. This information is sent back to the potential system user, and the user reevaluates his intent to make the specific transaction, using the information about the system charge as one component in its cost estimate. The user is likely to iterate this information with the ISO several times before actually settling on the amount and type of transaction to be actually served by the system. It was shown in [6] that this process rather easily converges to the same systemwide optimal solution as the one determined in [2] without having to mandate the congestion price nor the information about the sale 15. While some of the differences may be too subtle to appreciate, we point out that this proposal takes into consideration the expected cost of the transaction ex ante [12] and is capable of capturing intertemporal effects when serving transactions of various types. Moreover, it is intended to allow full freedom to the system user to make decisions regarding both quantity and price of the overall transaction, including the charge for system support, without mandating a prespecified pattern of power or price control 16. 5.3.2. Relative charge for affecting reliability The premise of this proposal is that a transmission provider (ISO) will have to plan for expected changes in system inputs so that the level of reliability is maintained unchanged as various system users are served. For the proposed two-level iterative system provision to function in meaningful way, at least two measures must be clearly defined: 5.3.2.1. Optimal level of reliability for a system under the ISO’s jurisdiction This is necessary in order to anchor the reliability measure relative to a pre-agreed-upon reference for reliability deviations caused by transactions requesting system use. Determining this may be system-dependent and will require some work; however, at least in principle, this level of reliability should reflect system conditions in response to what is considered normal operation in the given system. This general information is obtainable by running a constrained economic dispatch assuming costbased optimization of all system users Eq. (22). One possible piece of information to be used could be the optimal reliability status of each equipment component 15 It is important to understand that an extreme case of a multilateral transaction is actually a bid-based pool a` la Hogan, with functional separation between energy market sales (PX) and transmission provision (ISO). 16 It is important to understand the temporal effect of this process, because, depending on the type of transaction, the ISO will see requests for different times of use, different degrees of firmness, etc.

when the system is subject to a typical historic supply/ demand pattern. For a transmission line connecting buses i and j, this could be the line-flow Tijopt corresponding to the optimal level of systemwide reliability agreed upon. This should be known for all lines 17. 5.3.2.2. Relative impact of particular equipment status on systemwide reliability It is necessary for an ISO to have an estimate of the relative importance of constraints reached on specific equipment (e.g., transmission line ‘congestion’) on systemwide reliability. It is well known that when certain transmission lines in a given system are congested, their impact on the severity of the overall system situation may be much greater than the impact of some other lines. These relating weighting factors aij for each transmission line i ⫺ j can be determined ahead of time and even recomputed periodically as system conditions change [15]. With the information about the optimal status of each equipment component Tijopt and the relative importance of each component aij, a system provider is in a position to compute the relative charge for each transaction K requesting system use by simply establishing the charge for its relative impact on systemwide reliability: SCK ˆ

…nX ⫹ m† …nX ⫹ m† iˆ1

aij …Tij ⫺ Tijopt †2 I…Tij ⭓ Tijopt †

…27†

jˆ1

Here Tij are actual line flows caused by the transaction K superposed on the optimal operating conditions A. One should contrast this way of calculating relative impact of a transaction K on systemwide reliability to the approach originally described in [6]. The latter penalizes for the relative impact of a transaction K on systemwide reliability by computing SCK ˆ

…nX ⫹ m† …nX ⫹ m† iˆ1

aij …Tij ⫺ T ij †2 I…Tij ⭓ Tij †

…28†

jˆ1

where T ij represents the transmission line limit, instead of the optimal value Tijopt. This modification is introduced because it is potentially more justifiable to charge each transaction for its relative impact on deviations from optimal, as though no other transactions were present, rather than charge for incremental impact, which is dependent on what everyone else is doing on the system. In the modification proposed here, the order in which transactions are served is not critical when sending the signal about the system charge. The general idea is quite simple. For the expected market needs, a system provider ought to plan and put in new enhancements so that reliability is not affected in any major way. As various users get in the system, they pay their share of the impact on reliability. If the estimated system charge is too high, the CMP will modify its request by means of a two-level auctioning scheme. 17

Similar measures must be established for voltages at buses, etc.

E. Allen et al. / Electrical Power and Energy Systems 21 (1999) 147–163

Note that transactions requesting use of the system over longer time horizons and on a firm basis would put different requirements on system support than shorter, nonfirm transactions. Moreover, given the iterative nature of deciding on the charge for the desired transaction, an ISO and potential system user could consider a variety of options that take into account the intertemporal aspects of system reliability. For example, a transaction more receptive to the ISO’s suggestions about scheduled maintenance may end up with a lower charge for system use than an otherwise identical transaction with a predetermined, non-negotiable maintenance schedule. The general setup of the proposed two-level iterative scheme allows for developing adaptive schemes of this type when deciding on the actual amount of power to be served and the system charge for this service. We recognize that a system provider will be planning for system enhancements and charging ex ante for relative impact on system reliability in a dynamic manner, and not in response to the first transaction that perturbs the level of reliability. Of course, every single transaction is likely to have a somewhat negative impact on the optimal reliability 18. An ISO will have to develop tools which, based on the relative impact of transactions on reliability and the frequency of critical requests, guide investment into those system enhancements most likely to maintain reliability as desired, i.e., close to the optimal level. Ex ante pricing mechanisms for transmission charges most likely to recover this investment must be adopted, so that after transactions of various duration and firmness have been served, the charges come close to recovering the investment in system enhancements. Given the discrete nature and lumpiness of the investment, this must be done carefully to reflect long-run effects of transactions on system conditions, not short-run marginal cost signals. We illustrate the method here only in the context of thermal limits. Generalizations are possible taking into consideration other static and dynamic reliability concerns.

6. Numerical illustrations to highlight key features of the three proposals This section uses the framework proposed by Wu and Varaiya to illustrate that under perfect energy market conditions: (1) systemwide efficiency does not depend on the type of physical curtailment done by an ISO in order to meet operating constraints; and (2) individual profits of CMPs strongly depend on how much is curtailed. In Section 7 generalized proofs of these results are provided. These results are critical for dealing with the type of questions to which ISOs will have to respond in the future 18 This assumes careful computation of the optimal reference level of reliability.

155

Fig. 2. Three-bus example.

when proposing specific strategies for transmission system provision and pricing. 6.1. Three-bus example In order to illustrate the proposed marketplace for a congested network, we begin with a simple three-bus system with two generators and a single load shown in Fig. 2. The cost and utility functions are C1 …QG1 † ˆ Q2G1 ⫹ QG1 ⫹ 0:5

…29†

C2 …QG2 † ˆ 2Q2G2 ⫹ 0:5QG2 ⫹ 1

…30†

U1 …QL1 † ˆ 214:1667QL1 ⫺ 10Q2L1

…31†

Furthermore, the transmission lines are assumed to be lossless and have the same impedance. In this example, a D ˆ 0.05, b D ˆ 10.7083, a S ˆ 0.75, and b S ˆ 0.624, resulting in an equilibrium price of Pl ˆ 14.1667, generation levels of QG1 ˆ 6.5833, QG2 ˆ 3.4167, and a load of QL1 ˆ 10. We assume that the generators and load at each bus are an aggregate of a large number of smaller generators and loads, so that the marketplace may be regarded as reasonably competitive and not characterized by oligopoly or monopsony. The line flow from G1 to L1 may be approximated as TG1⫺L1 ⬇

2 1 Q ⫹ QG2 3 G1 3

…32†

This approximation is derived by observing that two-thirds of the current flow from G1 to L1 travels through this line, since the direct path has half the impedance of the path from G1 to L1 via G2. Similarly, one-third of the current flow from G2 to L1 travels through the G1–L1 link. A more

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precise approximation can be developed using partial derivatives calculated at the operating point. The line from G1 to L1 has an upper flow limit of 5j. however, the proposed marketplace transactions yield a flow TG1–L1 ⬇ 5.5277. Since both transactions contribute toward the congestion, the curtailment strategy we initially consider is to reduce both QG1 and QG2 by the amount of the excess line flow (0.5277). The curtailment in QG1 reduces two-thirds of the excess line-flow, while the reduction in QG2 cuts the other third. The allowed operating point is now QG1 0 ˆ 6.0556, QG2 0 ˆ 2.8890, and QL1 0 ˆ 8.9446. We consider the effects of different curtailment protocols later. The ISO now provides the necessary information to establish a trilateral marketplace for additional trades. In this case, if QG2 is increased at twice the rate of decrease in QG1, then the flow in the congested line is unchanged. Therefore, G1 can sell additional power to the load if G1 simultaneously buys twice as much power from G2. The choice of a middleman does not matter; the resulting transactions at equilibrium will be the same [3]. The trilateral marketplace is thus in fact two coordinated bilateral markets: one for exchanges between G1 and L1, and one between G1 and G2. The two markets are coordinated because the price in one market affects the demand or supply in the other. The load’s demand for power in the G1–L1 market is determined by the same strategy as before: continue to buy power until the price is equal to the marginal utility. Similarly, G2 continues to sell power to G1 as long as the price is greater than the marginal cost. The strategy for G1 is a little more complicated. The profit for G1 in the trilateral market is a function of the prices PG1–L1 and PG1–G2 in the two post-congestion markets and the quantity Q~ G1 that is sold to the load after curtailment:

p~ G1 ˆ PG1–L1 Q~ G1 ⫺ 2PG1–G2 Q~ G1 ⫹ aG1 QG1 ⫹ bG1 QG1 02

0

The supply in the G1–L1 market and the demand in the G1– G2 market are thus both defined by P ⫺ 2PG1–G2 ⫹ bG1 Q~ G1 ˆ G1–L1 ⫹ QG1 0 2aG1

The demand of L1 and the supply from G2 are the same as in the marketplace before congestion: DG1–L1 …PG1–L1 † ˆ

bL1 ⫺ PG1–L1 ⫺ QL1 0 2aL1

…36†

SG1–G2 …PG1–G2 † ˆ

PG1–G2 ⫺ bG2 ⫺ QG2 0 2aG2

…37†

If the starting price in both markets is Pl , then L1 is willing to buy additional power and G2 is interested in selling power, but G1 is not interested in making any trades, since they would be unprofitable. Remember that the congestion constraint forbids G2 from selling directly to L1. However, G1 would ask for a higher price when selling to L1 and a lower price when buying from G2, and since both G2 and L1 can still make profitable deals, it would be expected that the two prices would converge to an equilibrium. The equilibrium of the trilateral marketplace can be found by first rearranging Eqs. (36) and (37): PG1–L1 ˆ bL1 ⫺ 2aL1 QL1 0 ⫺ 2aL1 Q~ G1

…38†

PG1–G2 ˆ bG2 ⫹ 2aG2 QG2 0 ⫹ 4aG2 Q~ G1

…39†

where we have substituted Q~ G1 ˆ DG1–L1 …PG1–L1 † and 2Q~ G1 ˆ SG1–G2 …PG2–L2 †. Substituting into Eq. (35), we have Q~ G1 ˆ aG1 QG1 0 ⫺ 2aG2 QG2 0 ⫺ aL1 QL1 0 ⫹

2

…40†

0

⫹ cG1 † …33† The profit to G1 includes cost savings from reducing generation; of the power bought from G2, half is resold to G1 at the price PG1–L1, and the other half is used to supply a portion of the original QG1 0 units of power that was sold in the pre-congestion market. Note that G1 must buy 2Q~ G1 from G2; G1 could buy more than this amount from G2, but doing so would not be sensible, since half of the extra power purchased can be sold to the load at a price that exceeds the marginal cost savings from reducing generation at G1. At the given prices, the maximum profit from G1 is achieved by selling QG1 to L1 such that dp~ G1 ˆ PG1–L1 ⫺ 2PG1–G2 ⫹ 2aG1 …QG1 0 ⫺ Q~ G1 † ⫹ bG1 ˆ 0 dQ~ G1 …34†

1 1 bG1 ⫺ bG2 ⫹ bL1 2 2

aG1 ⫹ 4aG2 ⫹ aL1

⫹ cG1 ⫺ …aG1 …QG1 ⫺ Q~ G1 † ⫹ bG1 …QG1 ⫺ Q~ G1 † 0

…35†

Recall that the actual generation and load quantities are a sum of the amount traded in the trilateral agreements and the curtailed amounts set by the ISO: QG1 ˆ QG1 0 ⫺ Q~ G1

…41†

QG2 ˆ QG2 0 ⫹ 2Q~ G1

…42†

QL1 ˆ QL1 0 ⫹ Q~ G1

…43†

For the numerical example given earlier, Q~ G1 ˆ 0:6389, PG1–L1 ˆ 22.5, and PG1–G2 ˆ 17.167. At these price levels G1 will produce 5.4166 units of power and G2 will produce 4.1667 units. The load purchases 9.5833 units. The trilateral marketplace equilibrium is also the point at which total social welfare is maximized. To see this, note that when the power flow TG1–L1 is constrained at its maximum limit, total social welfare is a function of only one

E. Allen et al. / Electrical Power and Energy Systems 21 (1999) 147–163

157

variable, Q~ G1 , since the other generator and load quantities are constrained by the power balance requirement and the line-flow constraint. These two constraints leave only one degree of freedom. Differentiation of Eq. (18) with respect to Q~ G1 gives dW ˆ ⫺2aG1 …QG1 0 ⫺ Q~ G1 † ⫺ bG1 dQ~ G1 ⫹ 4aG2 …QG2 0 ⫹ 2Q~ G1 † ⫹ 2bG2 ⫹ 2aL1 …QL1 0 ⫹ Q~ G1 † ⫺ bL1 ˆ 0

…44†

It is straightforward to show that optimal social welfare requires that the line flow reach its limit. Since the unconstrained optimum violates the congestion constraint, both generators are generating at a point where the marginal cost is below Pl while the load’s marginal utility is above Pl . If the line-flow is below the maximum limit, then either generator can sell more power to the load at a cost less than utility to the load, increasing total welfare; therefore, the line-flow must reach the limit in order to maximize social welfare under the congestion constraint. Since the trilateral market equilibrium is the point that maximizes welfare, it follows that the generation and load levels will be the same regardless of which curtailment protocol is used. To see this, substitute Eq. (40) into Eq. (41) to find the total generation for G1 at equilibrium: QG1 ˆ

2aG2 …2QG1 0 ⫹ QG2 0 † ⫹ aL1 …QG1 0 ⫹ QL1 0 † aG1 ⫹ 4aG2 ⫹ aL1 1 1 bG1 ⫺ bG2 ⫹ bL1 2 ⫺ 2 aG1 ⫹ 4aG2 ⫹ aL1

…45†

Using the line-flow approximation of Eq. (32) and the following relation, derived from the power balance requirement QG1 0 ⫹ QL1 0 ˆ 2QG1 0 ⫹ QG2 0

…46†

we can rewrite Eq. (45) as

QG1 ˆ

1 1 bG1 ⫹ bG2 ⫺ bL1 2 2 aG1 ⫹ 4aG2 ⫹ aL1

3…2aG2 ⫹ aL1 †TG1–L1 ⫺

…47†

Since TG1–L1 is fixed at its maximum limit (5, in our example), QG1 is independent of the curtailment level. Since QG1 uniquely determines QG2 and QL1 through the power balance requirement and the line-flow constraint, these quantities are also independent of the curtailment level. Furthermore, by substituting Eqs. (42) and (43) into Eqs. (38) and (39), we see that the prices in the trilateral market at equilibrium are also independent of the curtailment. The choice of curtailment will affect only how much power is priced at Pl and

Fig. 3. Four-bus example.

how much is priced by the trilateral prices PG1–L1 and PG1–G2, not what the numerical values of these prices are. 6.2. Four-bus example We now illustrate how a second-tier market can operate in a four-bus system with two generators and two loads, as shown in Fig. 3. The cost and utility functions are C1 …QG1 † ˆ Q2G1 ⫹ QG1 ⫹ 0:5

…48†

C2 …QG2 † ˆ 2Q2G2 ⫹ 0:5QG2 ⫹ 1

…49†

U1 …QL1 † ˆ 94:1667QL1 ⫺ 10Q2L1

…50†

U2 …QL2 † ˆ 158:1667QL2 ⫺ 12Q2L2

…51†

The market equilibrium is similar to the last example; G1 sells 6.5833, G2 sells 3.4167, L1 consumes 4 and load 2 consumes 6, all at a price of Pl ˆ 14.1667. The transmission line connecting G1 to L1 has a maximum flow limit of 3.7. The flow through this line may be expressed as: TG1⫺L1 ⬇

1 1 1 Q ⫹ QG2 ⫹ QL1 2 G1 8 8

…52†

At the market equilibrium, TG1–L1 ˆ 4.2187; therefore the ISO cannot allow all of the transactions proposed in the open market. If the ISO chooses to curtail all market participants by the same amount, that amount is equal to 3/2 of the excess line flow of 0.5187, resulting in curtailed operating levels of QG1 0 ˆ 5.8917, QG2 0 ˆ 2.7251, QL1 0 ˆ 3.3084, and QL2 0 ˆ 5.3084. To form a multilateral (in this case, quadrilateral) second-level market, the ISO can suggest that L1 may buy 1 unit from G2 by simultaneously selling

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Table 1 Profits for curtailment method A Participant

Amt at Pl ˆ 14.1667

Amt in second market

G1 G2 L1 L2

5.8917 2.7251 3.3084 5.3084

⫺ 0.4364 1.3483 0.3975 0.5143

11.9106 16.7936 ⫺ 20.0487 ⫺ 18.4219

2/5 unit to G1. Similarly, L2 may buy 1 unit from G2 as long as 1/4 unit of power is sold to G1. This will again create two bilateral markets: one between the load and G1, and one between the loads and G2. Assuming that the two markets are competitive, G2 will offer to sell SG2(PG2) units if the price is PG2: SG2 …PG2 † ˆ

PG2 ⫺ bG2 ⫺ QG2 0 2aG2

Second market price

…53†

Utility/cost

Net profit

⫺ 35.72 ⫺ 36.22 211.64 514.11

42.55 25.03 156.80 429.34

This profit is maximized when L1 purchases Q~ L1

2 5 PG1 ⫺ PG2 ⫹ bL1 3 3 ˆ ⫺ QL1 0 2aL1

…58†

L1 pays, in effect, a price of (5/3)PG2 ⫺ (2/3)PG1 for each unit of power in the postcongestion market. L2 will maximize profit by purchasing 1 4 PG1 ⫺ PG2 ⫹ bL2 3 ˆ 3 ⫺ QL2 0 2aL2

G1 will realize a profit of p~ G1 on the purchase of Q~ G1 units at a price PG1:

Q~ L2

p~ G1 ˆ ⫺PG1 Q~ G1 ⫹ aG1 QG1 02 ⫹ bG1 QG1 0 ⫹ cG1

since L2 must sell 1/3 unit to G1 and buy 4/3 unit from G2 to increase load usage by 1 unit. The effective price for L2 is (4/ 3)PG2 ⫺ (1/3)PG1. The equilibrium point for the two markets is reached at

⫺…aG1 …QG1 0 ⫺ Q~ G1 †2 ⫹ bG1 …QG1 0 ⫺ Q~ G1 † ⫹ cG1 † …54† Given the price, the optimal quantity for G1 to purchase is determined by dp~ G1 ˆ ⫺PG1 ⫹ 2aG1 …QG1 0 ⫺ Q~ G1 † 0 ⫹ bG1 ˆ 0 dQ~ G1

…55†

so the demand in the G1 electricity market is given by DG1 …PG1 † ˆ

bG1 ⫺ PG1 ⫹ QG1 0 2aG1

…56†

The optimal strategy for the loads depends on both PG1 and PG2. If L1 wishes to increase its consumption by 1 unit, it must buy 5/3 from G2 and sell 2/3 to G1 to stay within the line-flow constraint. The profit for L1 is

p~ L1 ˆ

2 5 P Q~ ⫺ PG2 Q~ L1 ⫺ …bL1 QL1 0 ⫺ aL1 QL1 02 † 3 G1 L1 3 ⫹ bL1 …QL1 0 ⫹ Q~ L1 † ⫺ aL1 …QL1 0 ⫹ Q~ L1 †2

…57†

…59†

2 1 Q~ G1 ˆ Q~ L1 ⫹ Q~ L2 3 3

…60†

5 4 Q~ G2 ˆ Q~ L1 ⫹ Q~ L2 3 3

…61†

For our example, this equilibrium is reached at Q~ G1 ˆ 0:4364 and Q~ G2 ˆ 1:3483, implying Q~ L1 ˆ 0:3975 and Q~ L2 ˆ 0:5143. The final generation levels are 5.4553 for G1 and 4.0734 for G2. L1 uses 3.7059, and L2 uses 5.8227. It can be shown that this operating point maximizes total social welfare. The equilibrium prices in the postcongestion market are PG1 ˆ 11.9106 and PG2 ˆ 16.7936. In this scenario, G1 will sell QG1 0 ˆ 5.8917 units at a price of 14.1667 and then buy back Q~ G1 ˆ 0:4364 units at a price of 11.9106. Since G1 generates 5.4553 units at a cost of 35.72, it obtains a net profit of 42.55. The costs and profits of all market participants are shown in Table 1. To illustrate the effects of a different curtailment protocol, let us now assume that sales by G2 are not affected by curtailment, under the theory that G2 has little effect on

Table 2 Profits for curtailment method B Participant

Amt at Pl ˆ 14.1667

Amt in second market

G1 G2 L1 L2

5.6401 3.4167 3.6227 5.4341

⫺ 0.1848 0.6567 0.0832 0.3886

Second market price 11.9106 16.7936 ⫺ 20.0487 ⫺ 18.4219

Utility/cost

Net profit

⫺ 35.72 ⫺ 36.22 211.64 514.11

41.98 23.21 158.65 429.97

E. Allen et al. / Electrical Power and Energy Systems 21 (1999) 147–163

the flow through the congested line. Such a protocol would result in a curtailment shown in the second column of Table 2. The profits of the market participants are also shown in Table 2. Note that the final generation and load levels and secondary market prices are the same for both curtailments; however, the individual profits are affected by the curtailment method that is chosen. 7. Zones as a minimum number of submarks for given active constraints

linear approximations NG X

akj QGj ⫹

NX L ⫺1

jˆ1

bkj QLj ˆ Tkmax

1 ⱕ k ⱕ NB

…62†

jˆ1

where Tkmax is the maximum line-flow for the kth constrained line. We choose bkNL to be zero because QLNL is uniquely determined by the power balance relation: NG X

QGj ⫺

jˆ1

It is interesting to address in the context of energy markets a very much debated question concerning zonal pricing for system transmission [16]. Given the introductory analysis in this paper, it should be clear that under perfect market conditions the entire system should see the same energy market price prior to considering charge for system support. The possibility of aggregating subareas (zones) within such an energy market so that they pay similar charges for system support has been discussed a great deal because of its appealing simplicity compared to considering a large number of system users separately. In the following, some interesting results are established concerning the minimal number of submarkets (zones) as a function of number of active transmission constraints. This result points out that, since the type and number of active system constraints vary considerably, these zones must be viewed as dynamic and must be recomputed periodically.

159

NL X

QLj ˆ 0

…63†

jˆ1

Then if generator i, where NB ⫹ 2 ⱕ i ⱕ NG, wishes to conduct further transactions following curtailment, it can sell Q~ Gij additional units of power to each generator from j ˆ 1 to j ˆ NB ⫹ 1 if it satisfies the following relations, which require that the flow in line k remains unchanged: NX B ⫹1

…aki ⫺ akj †Q~ Gij ˆ 0

1 ⱕ k ⱕ NB

…64†

jˆ1

Q~ Gij may be negative, which corresponds to a purchase from generator j. Solving the system of equations defined by Eq. (64) is in general a problem of finding a one-dimensional basis for the nullspace of a matrix. Since the basis vector is not uniquely defined, we add a constraint to choose the basis vector whose elements sum to 1: NX B ⫹1

Q^ Gij ˆ 1

…65†

jˆ1

7.1. The general case For an arbitrary network with NG generators and NL loads, the secondary market is established by creating NB ⫹ 1 bilateral markets for power, where NB is the number of transmission constraints that are binding. Each market consists of power sold to or from one of NB ⫹ 1 nodes. The choice of nodes does not affect the final prices reached in the secondary market [3]; we will assume that the nodes for the postcongestion market are generators 1 through NB ⫹ 1. Each of these generators buys or sells power with the remaining NG ⫺ NB ⫺ 1 generators and all loads but not with any of the other generators numbered between 1 and NB ⫹ 1. The remaining generators and all loads may buy or sell additional power if they trade in a specific ratio in all NB ⫹ 1 markets, where the ratio is determined by the binding transmission constraints. There are NB ⫹ 1 submarkets because any combination of NB ⫹ 2 participants can agree on a transaction which does not change any of the flows on the critical lines. As shown earlier in the three- and four-bus examples, a different price is needed for sales to and from different generators because the congestion constraints cause the different generators to each sell in effect a different commodity; a purchase from G1 affects congested lines differently than a purchase from G2. Mathematically, the NB constraints are represented by the

NX B ⫹1

…aki ⫺ akj †Q^ Gij ˆ 0

1 ⱕ k ⱕ NB

…66†

jˆ1

The solution to the system of equations defined by Eq. (65) and Eq. (66) is unique (assuming it exists); Q^ Gij indicates how much power generator i must sell to generator j if generator i wishes to sell a total of 1 unit. If the price of power sold to generator j is Pj, then the marginal revenue received by generator i for selling 1 unit of power, denoted as PMGi, is PMGi ˆ

NX B ⫹1

Q^ Gij Pj

…67†

jˆ1

PMGi is the price seen by generator i (NB ⫹ 2 ⱕ i ⱕ NG); it is a function of the NB ⫹ 1 bilateral market prices that exist in the multilateral marketplace. The marginal price for 1 unit of power purchased by load i is derived in the same manner. In order to purchase 1 unit of power, load i must purchase Q^ Lij units from generator j, where 1 ⱕ j ⱕ NB ⫹ 1 and Q^ Lij satisfies NX B ⫹1 jˆ1

Q^ Lij ˆ 1

…68†

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NX B ⫹1

…bki ⫹ akj †Q^ Lij ˆ 0

1 ⱕ k ⱕ NB

…69†

jˆ1

solution of the system of equations given by MCi …QGi † ˆ l ⫹

PMLi ˆ

Q^ Lij Pj

…70†

MUi …QLi † ˆ l ⫺

7.2. Proof of optimal total social welfare We now prove that the equilibrium of the two-tiered market scheme from [3] is the point at which total social welfare is maximized. First, we determine, using the linear approximation, what the generation and load levels are at the point of optimal social welfare. Next, we determine what the equilibrium prices are, and then we conclude by showing that the marginal price for all generators and loads in the secondary market are identical to the marginal costs and utilities at the point of optimal social welfare.

7.2.1. Point of maximum social welfare The total welfare is the total utility of all loads minus the total costs of all generation: Wˆ

Ui …QLi † ⫺

iˆ1

NG X

Ci …QGi †

LˆW ⫹l

NG X

QGi ⫺

iˆ1

⫹

kˆ1

…75†

…71†

NB X

lj bji

1 ⱕ i ⱕ NL

…76†

7.2.2. Determination of equilibrium prices in second-tier market As we showed earlier, each generator in an open market will sell power until the marginal cost of power reaches the price. For generators 1 to NB ⫹ 1, which each sell power at prices P1 to PNB ⫹1 , the equilibrium prices must then satisfy Pi ˆ MCi …QGi 0 ⫹ Q~ Gi †

1 ⱕ i ⱕ NB ⫹ 1

…77†

Meanwhile, the remaining generators will choose a production level such that the marginal cost is equal to the marginal price PMGi, so that at equilibrium PMGi ˆ MCi …QGi 0 ⫹ Q~ Gi †

NB ⫹ 2 ⱕ i ⱕ NG

…78†

Similarly, all loads will purchase power until the marginal price PMLi equals the marginal utility: PMLi ˆ MUi …QLi 0 ⫹ Q~ Li †

1 ⱕ i ⱕ NL

…79†

At equilibrium, the prices P1 through PNB ⫹1 satisfy Eqs. (77)–(79).

iˆ1

We wish to find the maximum of W under the constraints of Eqs. (62) and (63). To find this maximum, we write the Lagrangian L:

NB X

1 ⱕ i ⱕ NG

jˆ1

jˆ1

NL X

lj aji

jˆ1

PMLi, the price for a unit of power for load i, is then NX B ⫹1

NB X

0

lk @

NG X jˆ1

NL X

! QLi

iˆ1

akj QGj ⫹

NX L ⫺1

1 bkj QLj ⫺ Tkmax A

…72†

jˆ1

7.2.3. Equality of second-tier equilibrium and optimal social welfare Notice that the optimal social welfare solution has NB ⫹ 1 shadow prices (l and l 1 through lNB ), while the two-tier marketplace results in NB ⫹ 1 prices. We now show that the two sets of prices are linear transformations of each other. First, note that if the two solutions are equal, then Eqs. (75) and (77) are equal for 1 ⱕ i ⱕ NB ⫹ 1: Pi ˆ l ⫹

NB X

lk aki

1 ⱕ i ⱕ NB ⫹ 1

…80†

kˆ1

The maximum occurs at the point where the partial derivatives of L are all zero. Setting (2L/2l ) and (2L/2l k) equal to zero results in Eqs. (62) and (63), respectively. The partial derivatives with respect to quantity are NB X 2L ˆ ⫺MCi …QGi † ⫹ l ⫹ lj aji ˆ 0 2QGi jˆ1

…73†

We now show that if l and l i are derived from the linear transformation in Eq. (80), Eqs. (75) and (76) are satisfied for all i, indicating that the equilibrium prices Pi occur at the point of maximum total welfare. First, we consider generators NB ⫹ 2 through NG, which at equilibrium choose a total generation level QGi ˆ QGi 0 ⫹ Q~ Gi according to MCi …QGi † ˆ

NX B ⫹1

Q^ Gij Pj

…81†

jˆ1 NB X 2L ˆ MUi …QLi † ⫺ l ⫹ lj bji ˆ 0 2QLi jˆ1

…74†

The point of optimal social welfare is then equal to the

Substituting Eq. (80) for Pj " # NX NB B ⫹1 X ^ QGij l ⫹ MCi …QGi † ˆ lk akj jˆ1

kˆ1

…82†

E. Allen et al. / Electrical Power and Energy Systems 21 (1999) 147–163

7.3. Discussion of the optimal operating point

This can be rearranged to MCi …QGi † ˆ l

NX B ⫹1

NB X

Q^ Gij ⫹

jˆ1

lk

kˆ1

NX B ⫹1

akj Q^ Gij

…83†

jˆ1

From Eq. (65), the first term is simply l because the summation is equal to 1. Using Eq. (66), the inner (j-indexed) summation of the second term is equal to NX B ⫹1

akj Q^ Gij ˆ

jˆ1

NX B ⫹1

aki Q^ Gij

…84†

jˆ1

which, from Eq. (65), is NX B ⫹1

akj Q^ Gij ˆ aki

…85†

jˆ1

Therefore, Eq. (83) reduces to MCi …QGi † ˆ l ⫹

NB X

lk aki

…86†

kˆ1

which is identical to Eq. (75). The proof for loads is very similar. Each load operates at a utility level given by MUi …QLi † ˆ

NX B ⫹1

Q^ Lij Pj

…87†

jˆ1

Again, we substitute for Pj using Eq. (80) " # NX NB B ⫹1 X Q^ Lij l ⫹ lk akj MUi …QLi † ˆ jˆ1

…88†

NX B ⫹1

Q^ Lij ⫹

jˆ1

NB X

lk

kˆ1

NX B ⫹1

8. Conclusions akj Q^ Lij

…89†

jˆ1

From Eq. (69), we know that NX B ⫹1

akj Q^ Lij ˆ ⫺

jˆ1

NX B ⫹1

bki Q^ Lij

…90†

jˆ1

which reduces to NX B ⫹1

akj Q^ Lij ˆ ⫺bki

…91†

jˆ1

Eq. (89) is therefore the same as MUi …QLi † ˆ l ⫺

We can see from the two-tier market that the point of optimal social welfare is composed of two parts: a feasible solution given by [QGi 0 QLi 0 ] T and a vector to reach the optimal point given by ‰Q~ Gi Q~ Li ŠT which does not affect the flows in critical lines. Once the line limits are reached, the market can, given information about the physical system, move to a point of optimal efficiency; however, the starting point on the boundary of secure operation will affect the profits received by individual market participants. If the starting point is in the security boundary and then moves with time to the boundary, then earlier transactions will receive priority and higher profits over later transactions that bring the system to the limit of security. It is important to understand that the proposal by Ilic´ et al. provides flexibility to market participants to decide on their own rationing in response to the (strictly) technical information given by the ISO regarding their relative impact on the proximity to the system constraints. As with the approach by Wu and Varaiya, it is not necessary for market participants to provide any information about the price at which energy trades are done. The only requirement is providing information about the location and the amount of power required to be accommodated by the transmission system. The premise is that market participants will incorporate the information of Eqs. (27) and (28) as an additional cost into their profit maximization objective in Eq. (4) and engage in direct iterations (negotiations) with the ISO until they agree how much power will be served, and at which charge.

kˆ1

This is equivalent to MUi …QLi † ˆ l

161

NB X

lk bki

…92†

kˆ1

which is identical to Eq. (76). Therefore, the equilibrium of the two-tier market is the point at which total social welfare is maximized.

Tools for system provision and cost minimization have historically been designed with the objective of systemwide performance improvement rather than facilitating specific patterns of power transactions. This approach has been adopted for real-time operations under open access. An ISO is expected to provide system support that at least ensures systemwide integrity (reliability) as various competitive transactions are implemented. It is less clear if an ISO is also responsible for most efficient operations. This becomes particularly confusing in an environment in which some transactions make their energy prices known to the operator (bid-based scheduling) and the others choose to keep the transaction confidential between directly involved parties and only request approval for using the system. At present, no unique technical definition of equal access when some transactions are not implementable is provided. An ISO could either curtail all an equal amount or only curtail the transactions that have the most impact on deterioration of system conditions. Moreover, a sequence of

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E. Allen et al. / Electrical Power and Energy Systems 21 (1999) 147–163

bilateral transactions requesting management of X MW does not have the same impact on system conditions as a set of transactions totalling the same X MW, but managed simultaneously by an ISO. It is absolutely necessary to provide unique feasibility tests applicable to all transactions in order not to discriminate against individual market participants. In this paper, we propose an iterative auction mechanism for transmission system provision that is based on first come/first served and an ex ante estimated charge for each specific transaction. Each transaction will be charged for its relative impact on deterioration in system conditions, reflecting the cost of system enhancements to bring the system back to the optimal reliability level. If a subset of transactions simultaneously request system provision, the total charge will be estimated using the same criterion. The only assumption here is that for an open access network, the optimal reliability level is defined, e.g., all voltages are close to 1 p.u. and phase angles across any transmission line are no larger than Y degrees 19. All deviations are computed relative to this level as the anchoring point for reliability pricing. No requirements for price knowledge are imposed. A CMP decides on its own, with this estimate, if it wishes to modify the amount of power for which it is requesting system services. By this method, the system user decides how much and at which price it would like to use the system. It is essential to give this choice to the user, since it is impossible to curtail transactions without affecting relative profits. The social welfare is achievable without having to release knowledge of individual energy prices. Insurance against unexpected changes in operating conditions and system support should be a matter between CMPs and various insurance companies. For these to evolve and be sustainable, both sides would have to engage in some fairly sophisticated tools typical of futures markets. As far as the fundamental use of the system goes, it would be discriminatory for specific system users to receive the initial rights at no cost, as suggested by some industry groups. Moreover, the involvement of an ISO in issuing these rights would be detrimental and indefensible; our basic observation has been that an ISO is capable only of optimizing social welfare and not of allocating profits in a fair economic way. A ‘fair’ system economics has not been defined, and because of this, an ISO should not interfere with users’ profits. The ISO’s main objective of optimizing social welfare can be achieved without any knowledge of energy prices of individual market participants (as shown in both [3] and [6]). Moreover, an ISO should not even actively curtail transactions, since this could affect individual profits. 19 It has some merit to define this in a non-uniform, system-specific manner; control areas more vulnerable to dynamic problems would have different requirement on phase angle differences than control areas with very little dynamic problems.

System users must become active CMPs; in order to be most profitable, they must evaluate for themselves the trade-off between potential profit and the cost of system service. The same way as CMPs go through auctioning for energy, they will do best to iterate their information with the system provider for system support and charges. Moreover, if they wish to insure themselves against changes in operating conditions, and they can find someone to sell this insurance at an acceptable price, they can buy it. Such insurance should not be required.

Acknowledgements The first author greatly appreciates a discussion with M.I.T. Professor Paul Joskow that stressed the need to clearly define the reference reliability level (anchor point) when charging for reliability in the proposal by Ilic´ et al. The first author also acknowledges several discussions with Leonard Hyman of SmithBarney, Inc. Partial financial support for this work was provided by the Transmission Provision and Pricing Consortium in the M.I.T. Energy Laboratory. This support is very much appreciated.

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