Cournot Equilibrium in Two-settlement Electricity Markets: Formulation and Computation. Jian Yao

Cournot Equilibrium in Two-settlement Electricity Markets: Formulation and Computation by Jian Yao A dissertation submitted in partial satisfaction o...
0 downloads 0 Views 963KB Size
Cournot Equilibrium in Two-settlement Electricity Markets: Formulation and Computation by Jian Yao

A dissertation submitted in partial satisfaction of the requirements for the degree of Doctor of Philosophy in Engineering - Industrial Engineering and Operations Research in the GRADUATE DIVISION of the UNIVERSITY OF CALIFORNIA, BERKELEY

Committee in charge: Professor Shmuel S. Oren, Chair Professor Ilan Adler Professor Richard J. Gilbert Spring 2006

The dissertation of Jian Yao is approved:

Chair

Date

Date

Date

University of California, Berkeley

Spring 2006

Cournot Equilibrium in Two-settlement Electricity Markets: Formulation and Computation

Copyright 2006 by Jian Yao

1

Abstract

Cournot Equilibrium in Two-settlement Electricity Markets: Formulation and Computation by Jian Yao Doctor of Philosophy in Engineering - Industrial Engineering and Operations Research University of California, Berkeley Professor Shmuel S. Oren, Chair

Sufficient and efficient electricity supply is critical to the steady growth of economies. In the last two decades, many electricity sectors around the world have been undergoing a reform from a command-and-control industry to competitive markets. A major obstacle to a successful reform is locational market power. To address this issue, many markets have been following a multiple-settlement approach where forward transactions, day-ahead transactions, and real-time transactions are settled sequentially at different prices. In this thesis, we focus on centrally-dispatched markets, and explore the effect of forward contracting on energy prices and consumer benefits. In the first part, we develop two Nash-Cournot models of spot electricity markets. In these two models, the strategic variables for the independent system operator (ISO) are assumed to be the nodal imports/exports and the locational price premiums, respectively. Both models can be represented as linear complementarity problems (LCPs) with a positive semi-definite matrix. The second part extends the two spot market models to two-settlement (forward

2 and spot) markets, which account for flow congestion, demand uncertainty and system contingencies. In both two-settlement models, we view the settlements as a two-stage Nash-Cournot game, and formulate its subgame perfect Nash equilibrium as a stochastic equilibrium problem with equilibrium constraints (EPEC), in which each firm solves a stochastic mathematical program with equilibrium constraints (MPEC). In each MPEC, the upper level is the firm’s utility-maximization problem in the forward market, and the lower level, which is shared by all MPECs, is a parametric LCP characterizing the equilibrium outcomes in the spot market. We apply the models to a stylized version of the Belgian electricity network, and observe that, with two settlements, suppliers are willing to engage in forward transactions despite the mitigating effect of such transactions on their market power; forward contracting reduces spot prices and increases social welfare; and producers commit more forward contracts when the markets become less concentrated. Next, we introduce two alternatives for capping prices. A spot price cap is a regulatory approach to truncate price spikes in the real-time market. A forward price cap is a proxy for competitive entry: when the forward prices exceed the long-run amortized cost, new capacity will be invested through entry which will effectively cap forward prices. Compared to the system without caps, we find that a spot price cap decreases firms’ incentives for forward contracting, whereas a forward price cap raises such incentives. Both caps, however, reduce spot prices and increase consumer welfare. The dimension and complexity of the EPEC model present a computational challenge. To solve the EPEC problems, the last part of this thesis introduces new MPEC/EPEC algorithms. The MPEC algorithm represents the state variables as piece-wise linear functions of the design variables, and searches for a B-stationary point via parametric LCP pivoting and finitely many quadratic programs. The MPEC algorithm maintains the satisfaction of the linear complementarity constraints and guarantees monotone objective values. We further develop an EPEC algorithm by solving the MPEC problems iteratively. Nu-

3 merical examples demonstrate that such algorithms are effective for solving both randomly generated MPEC test problems and real-world EPEC problems derived from our specific application.

Professor Shmuel S. Oren Dissertation Committee Chair

i

Contents List of Figures

iv

List of Tables

v

1 Introduction 1.1 Electricity Restructuring . . . . . . . . . 1.2 Market Power and Multiple Settlements 1.2.1 Market Power . . . . . . . . . . . 1.2.2 Multiple Settlements . . . . . . . 1.3 About this Thesis . . . . . . . . . . . . . 1.3.1 Scope of this Thesis . . . . . . . 1.3.2 Outline of this Thesis . . . . . . 2 The 2.1 2.2 2.3 2.4

2.5

2.6 2.7 2.8 3 The 3.1 3.2 3.3

. . . . . . .

1 1 3 3 5 7 7 9

. . . . . . . . . . . . . .

11 11 12 14 17 17 20 25 26 26 27 30 31 37 39

Model T1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

41 41 43 46

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

Spot Market Models Introduction . . . . . . . . . . . . . . . . . . . . . . . Literature Review . . . . . . . . . . . . . . . . . . . The ISO’s Problem . . . . . . . . . . . . . . . . . . . The First Spot Market Model: Model S1 . . . . . . . 2.4.1 The Firms’ Problems . . . . . . . . . . . . . . 2.4.2 Market Equilibrium Conditions . . . . . . . . 2.4.3 Difficulties of Model S1 . . . . . . . . . . . . The Second Spot Market Model: Model S2 . . . . . 2.5.1 The Firms’ Problems . . . . . . . . . . . . . . 2.5.2 Market Equilibrium Conditions . . . . . . . . 2.5.3 Difficulties of Model S2 . . . . . . . . . . . . A Numerical Example: the Stylized Belgian Network Numerical Results . . . . . . . . . . . . . . . . . . . Summary . . . . . . . . . . . . . . . . . . . . . . . . First Two-settlement Introduction . . . . . . . Literature Review . . . Pricing Schemes . . . .

Model: . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . .

ii 3.4

. . . . . . . . . .

. . . . . . . . . .

. . . . . . . . . .

. . . . . . . . . .

. . . . . . . . . .

. . . . . . . . . .

. . . . . . . . . .

. . . . . . . . . .

. . . . . . . . . .

. . . . . . . . . .

. . . . . . . . . .

. . . . . . . . . .

. . . . . . . . . .

. . . . . . . . . .

. . . . . . . . . .

. . . . . . . . . .

. . . . . . . . . .

. . . . . . . . . .

47 48 48 49 51 52 52 54 57 58

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

61 61 62 63 64 71 72 78

T2 . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . .

. . . . . . . .

. . . . . . . .

. . . . . . . .

. . . . . . . .

. . . . . . . .

. . . . . . . .

. . . . . . . .

. . . . . . . .

. . . . . . . .

. . . . . . . .

. . . . . . . .

. . . . . . . .

. . . . . . . .

. . . . . . . .

79 79 79 80 81 82 83 85 86

6 The Algorithms 6.1 Complementarity Problem . . . . . . . . . . . . . . . . . . . . . . . 6.1.1 Linear Complementarity Problem . . . . . . . . . . . . . . . 6.1.2 Mathematical Program with Equilibrium Constraints . . . 6.1.3 Equilibrium Problem with Equilibrium Constraints . . . . . 6.2 A Compact Presentation of Model T2 . . . . . . . . . . . . . . . . 6.2.1 Compact Presentation of the Inner Problem in Model T2 . 6.2.2 Compact Presentation of the MPEC Problems in Model T2 6.3 Solution Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.1 Properties of the MPEC Problems . . . . . . . . . . . . . . 6.3.2 The MPEC Algorithm . . . . . . . . . . . . . . . . . . . . . 6.3.3 The EPEC Scheme . . . . . . . . . . . . . . . . . . . . . . . 6.4 Computational Results . . . . . . . . . . . . . . . . . . . . . . . . . 6.4.1 Tests of the MPEC Algorithm . . . . . . . . . . . . . . . . 6.4.2 Test the EPEC algorithm . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . .

. . . . . . . . . . . . . .

. . . . . . . . . . . . . .

. . . . . . . . . . . . . .

. . . . . . . . . . . . . .

87 88 88 89 91 92 94 96 97 97 98 106 108 108 110

3.5 3.6

3.7

The Spot Market . . . . . . . . . . . . . . . 3.4.1 The ISO’s Problem . . . . . . . . . . 3.4.2 The Firms’ Problems . . . . . . . . . 3.4.3 Spot Market Equilibrium Conditions The Forward Market . . . . . . . . . . . . . Numerical Results . . . . . . . . . . . . . . 3.6.1 Forward Commitment Values . . . . 3.6.2 Spot Prices and Social Welfare . . . 3.6.3 Flow Congestion . . . . . . . . . . . Summary . . . . . . . . . . . . . . . . . . .

4 Price Caps 4.1 Introduction . . . . . . . . . . . . . . . . 4.2 The Model . . . . . . . . . . . . . . . . 4.2.1 Pricing Schemes with Price Caps 4.2.2 The Spot Market . . . . . . . . . 4.2.3 The Forward Market . . . . . . . 4.3 Numerical Results . . . . . . . . . . . . 4.4 Summary . . . . . . . . . . . . . . . . .

. . . . . . .

. . . . . . .

5 The Second Two-settlement Model: Model 5.1 Introduction . . . . . . . . . . . . . . . . . . 5.2 The Spot Market . . . . . . . . . . . . . . . 5.2.1 The ISO’s Problem . . . . . . . . . . 5.2.2 The Firms’ Problems . . . . . . . . . 5.2.3 Spot Market Equilibrium Conditions 5.3 The Forward Market . . . . . . . . . . . . . 5.4 Numerical Results . . . . . . . . . . . . . . 5.5 Summary . . . . . . . . . . . . . . . . . . .

iii 6.5

Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

111

7 Concluding Remarks

113

Bibliography

115

A Model Notations 120 A.1 Notations of Single-settlement Models . . . . . . . . . . . . . . . . . . . . . 120 A.2 Notations of Two-settlement Models . . . . . . . . . . . . . . . . . . . . . . 121

iv

List of Figures 2.1 2.2 2.3 2.4 2.5

Belgian high voltage network . . . . . . . . . . . . . . . . . . . . . . . . . . 31 Duopoly nodal prices with on-peak demands ($/MWh) . . . . . . . . . . . . 37 Duopoly nodal prices with shoulder demands ($/MWh) . . . . . . . . . . . 38 Duopoly nodal prices with low demands($/MWh) . . . . . . . . . . . . . . 39 Nodal prices under different resource structures with shoulder demands ($/MWh) 40

3.1 3.2 3.3 3.4 3.5 3.6

The model dynamics . . . . . . . . . . . . . . . . . . . . . Total forward commitment with different numbers of firms Expected spot generation (MW) . . . . . . . . . . . . . . Expected spot nodal prices ($/MWh) . . . . . . . . . . . Spot nodal prices with two settlements ($/MWh) . . . . . Forward commitments and spot zonal outputs (MW) . . .

. . . . . . . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

42 53 55 56 56 58

4.1 4.2 4.3 4.4 4.5 4.6

Spot price cap . . . . . . . . . . . . . . . . . . . . The ISO’s objective when qic + ric ≤ v¯ic . . . . . . The ISO’s objective when qic + ric > v¯ic . . . . . . Comparison of total forward contracting (MW) . Total forward contract with spot caps (MW) . . Total forward contract with forward caps (MW)

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

67 68 69 76 77 77

5.1

Expected spot nodal prices ($/MWh) . . . . . . . . . . . . . . . . . . . . .

86

6.1

A typical partition of Xg . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

101

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

v

List of Tables 2.1 2.2 2.3 2.4 2.5

States of the Belgian spot market . . . . . . . . . . . Node information in the Belgian network . . . . . . Node information in the Belgian network (continued) Transmission lines in the Belgian network . . . . . . Resource ownership structures . . . . . . . . . . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

32 33 34 35 36

3.1 3.2 3.3 3.4 3.5

Forward commitments with two firms (MW) Forward commitments with four firms (MW) Spot zonal prices ($/MWh) . . . . . . . . . . Surpluses ($/h) . . . . . . . . . . . . . . . . . Flow congestion . . . . . . . . . . . . . . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

53 53 57 57 59

4.1 4.2 4.3

Spot zonal prices of zone 1 ($/MWh) . . . . . . . . . . . . . . . . . . . . . . Spot zonal prices of zone 2 ($/MWh) . . . . . . . . . . . . . . . . . . . . . . Total spot generation (MW) . . . . . . . . . . . . . . . . . . . . . . . . . . .

73 74 74

5.1 5.2

Forward commitments with two firms (MW) . . . . . . . . . . . . . . . . . Forward commitments with three firms (MW) . . . . . . . . . . . . . . . . .

85 85

6.1 6.2 6.3 6.4

Test results of the MPEC algorithm . . . . . . . . . . . . . Test results of the EPEC algorithm . . . . . . . . . . . . . . Iterations of firms’ total forward commitments (two firms) . Iterations of firms’ total forward commitments (three firms)

. . . . .

. . . . .

. . . . .

. . . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

109 111 111 112

vi

Acknowledgments I would like to express my deepest thanks to Dr. Shmuel S. Oren and Dr. Ilan Adler for their knowledge, supervision and encouragement throughout my Ph.D. studies, and for their active involvement in the preparation and writing of this thesis. Their rich experience and contributions in the spheres of operation research and economics led me to aspire to incorporate a similar portfolio in my own career. Since the genesis of this work, Dr. Oren, chair of my thesis committee and an exceptional mentor, has provided an optimal mix of timely and generous feedback, keen insight, and unflagging spiritual and financial support towards my goal. I am indebted to Dr. Adler, co-chair of my thesis committee and the department chair, for steering me out of blind alleys – also for sharing with me his 8th-grade son’s problems in the 12thgrade mathematics contests. His brilliant insights, moreover, have helped me develop the algorithms in this thesis. It is not possible to do justice here to my enormous intellectual and personal debt to Shmuel and Ilan, but this thesis would not have been written without them. I am also grateful to the outside member of my thesis committee, Dr. Richard Gilbert, the former chair of the Economics Department. He monitored my work and made every effort to provide me with valuable comments on my workshop and to read earlier versions of this thesis. I am thankful to my colleagues, past and present, for facilitating and encouraging my thesis research and, more broadly, for creating and sustaining a forum of research on electricity markets, for their lively camaraderie, for the potluck parties and for their continuing friendship. I would also thank Dr. Bert Willems of Katholieke Universiteit Leuven for providing the network data of the Belgian electricity system. A special thanks goes to Mei Yang, my wife. I would not have accomplished this work without her constant sacrifice, support and encouragement. Throughout, Mei labored

vii as hard as, if not harder than, I did for the Ph.D., commuting two hours to her daily work in order to save my transportation time. This research is supported by the National Science Foundation Grant ECS-0224779, by the Power System Engineering Research Center (PSERC), by the University of California Energy Institute (UCEI) and by the Department of Energy through a grant administered by the Consortium for Electric Reliability Technology Solutions (CERTS). While I gratefully acknowledge the support of my advisors and my colleagues for this work, the results and views contained herein are solely my own and do not necessarily represent any of them.

1

Chapter 1

Introduction 1.1

Electricity Restructuring In the last two decades, many electricity sectors around the world have been un-

dergoing a major restructuring, or reform, from a command-and-control industry to competitive markets. Such restructuring stems from a new viewpoint towards electricity supply chains. Before the restructuring, a prevailing belief was that electricity is a national asset, and that maintaining reliable power supply requires public operation. Hence, electricity sectors in most countries were composed of vertically integrated enterprises, and were subject to full regulation. The generation, transmission, distribution and retail segments were integrated within natural monopolies. State governments played a dual role as electricity service providers and as regulators. Electricity consumers were confined to utilities which were endowed with exclusive privileges to supply electricity within their dominant regions. As economic growth becomes more and more dependent on adequate electricity supplies, many governments have started to realize that this growth may be impeded by fully regulated and vertically integrated electricity sectors due to their slow response to technological progress in electricity operations and to changes in economics, social and environmental policies. Moreover, the successful deregulation in other previously regulated

2 industries, such as the oil and gas, brings the belief that the provision of electricity is a service, and that it may best be accomplished by competitive markets. Electricity reform has been aiming at enhancing long-term consumer welfare. This enhancement relies on competitive wholesale power markets to draw efficient investments in new generation capacity and to manage better the operation of existing capacity. The introduction of competition has provided a strong platform for drawing private investors into power development. In addition, retail competition will grant consumers the flexibility of choosing power suppliers based on their price/service preferences. Competitive retailers are then expected to meet individual consumer needs by providing enhanced portfolios of retail service products, risk management, demand management and service/quality combinations. Electricity restructuring in different markets has been following different blueprints due to their heterogeneous circumstances, such as income levels, generation/consumption patterns, government roles, legislation frameworks and resource ownerships. The typical architecture of electricity restructuring, however, comprises several key components (see [57]): “1) a withdrawal from the electricity sector of the direct influence of government (beyond establishing the legal and regulatory framework within which the market operates), especially changes in ownership of electricity companies into the private sector; 2) a legal unbundling of the industry into its component sectors - generation, transmission, distribution and retail; and 3) the presence of mechanisms, such as a power pool and open access to transmission and distribution facilities, to enable competition”. In the US, electricity reform takes place at both the federal and state levels, including the creation of wholesale power pools, the divestment of generation assets and the introduction of competition in generation and retailing segments. On the wholesale level, the Energy Policy Act of 1992 (EPAct) catalyzed the development of “open access” to transmission grids. Orders 888 and 889 [15, 16] issued by the Federal Energy Regulatory Commission (FERC) has encouraged further wholesale competition. The FERC requires all public utilities that own, control or operate interstate transmission grids to: 1) file open

3 access nondiscriminatory transmission tariffs containing minimum terms and conditions, 2) take transmission service (including ancillary services) for their own new wholesale purchases and sales of electricity under open access tariffs, 3) develop and maintain a same-time information system that will give existing and potential users the same access to transmission information that the public utility enjoys, and 4) separate transmission from generating and marketing functions and communications. Later, the FERC’s Order 2000 [17] urged transmission owners to hand over the control of transmission facilities to regional transmission organizations. On the retail level, California has pioneered the state restructuring initiatives with the “Yellow Book” [8] in 1993 and the “Blue Book” (the Order Instituting Rule making and Order Instituting Investigation R.94-04-031/I.04-04-032) in 1994. These two books propose gradually granting voluntary and direct access to the market for electric generation services to all California consumers. Other restructured markets in the US are the northeastern power pools including Pennsylvania-New Jersey-Maryland (PJM) Interchange, New York and New England, as well as the ERCOT market in Texas. As of 2004, twenty-four states and the District of Columbia had enacted legislation or issued regulatory orders to permit retail access to competitive electricity suppliers – seven of these states delayed or suspended their plans for retail access, mainly due to the California energy crisis in 2001.

1.2 1.2.1

Market Power and Multiple Settlements Market Power The goal of long-run consumer welfare enhancement by way of electricity reform

cannot be achieved without removing obstacles, such as market power, from its path. Market power is defined as the ability of a supplier to raise prices above competitive levels and maintain those prices for a significant period of time. Concerns regarding market power abuse have been widely examined in economics literature and in antitrust practice across

4 a broad range of industries. The exploitation of market power can significantly erode consumer benefits that would be brought by the electricity reform. Two types of market power have been identified by economists: vertical and horizontal. Vertical market power is exercised when a firm with assets in two or more segments, such as electricity generation and transmission, uses its dominance in one segment to raise prices and to increase profits for the overall enterprise. Horizontal market power is exercised when a firm raises prices through the control of its own activity, such as reducing electricity generation, where it possesses a large enough share of the entire market. In most restructured electricity markets, vertical market power has been substantially mitigated through the unbundling of generation, transmission and distribution segments, and through “open access” to transmission grids. Horizontal or locational market power, however, remains an important issue to policy makers due to the non-storability of electricity, the lack of demand elasticity, the high market concentration and the limited transmission network capacities. Economists have developed two market power measures: the Herfindahl-Hirschmann Index (HHI) and the Lerner Index (LI). The former measures market concentration, whereas the latter measures how market prices have been marked up above producers’ marginal cost. The higher the HHI and LI are, the more market power suppliers possess. Using the 1994 data and a narrower definition of the geographic scope of electricity markets, Cardell et al. [9] calculate the HHI values for 112 regions based on state boundaries and North American Electric Reliability Council (NERC) subregions, and find that approximately 90 percent of the markets had HHI values above 2500. This suggests that electricity markets in the US are still highly concentrated. Other studies of the California and UK markets in [58, 59, 60] show that electricity markets are vulnerable to the exercise of market power. A number of factors, such as transmission capacities, market concentration and market designs, affect producers’ ability to exercise market power. One way to reduce market power is to increase the competition among geographically separate markets by

5 augmenting interconnection capacities. Borenstein, Bushnell and Stoft [5] show that, with sufficient transmission capacity, an attempt to restrict output attracts supply from neighboring regions and thus fails to raise locational prices. In fact, the threat of competitive imports can be sufficient to forestall market power exercise without the actual use of transmission capacities. In a study of British electricity markets, Green [27] argues that an electricity pool could be modelled as if generators compete through supply functions, and shows that a concentrated market yields equilibrium prices well above marginal costs. von der Fehr and Harbord [18] reach similar conclusions via an auction approach. They find the connection between spare capacity and prices: prices will be much higher if no firm is capable of meeting the entire demand. Wolfram [59, 60] finds that the mark-ups are higher when demand is above the average level, and that generators with greater infra-marginal capacities tend to submit higher bids. There are also studies of the impact of different market rules. Bower and Bunn [6] use a simulation approach and show that moving from the Pool system to New Electricity Trading Arrangements (NETA), which are based on a discriminatory “pay-as-bid” auction, could lead to higher prices. In contrast, Fabra, von de Fehr and Harbord [13] show in a theoretical model that a discriminatory auction is less vulnerable to market power exercise than a uniform-price auction, because the latter allows generators to receive a high price while still submitting low bids. They show that uniform price auctions are (weakly) more efficient, but discriminatory auctions are (weakly) better for consumer surplus; in other words, the switch to NETA might reduce prices.

1.2.2

Multiple Settlements Among the many economic tools for mitigating market power is a multi-settlement

approach where forward transactions, day-ahead transactions, and real-time balancing transactions are settled sequentially at different prices. The California energy crisis in 2001 and

6 the collapse of ENRON have drawn more attention to the role of forward contracting in market power mitigation and risk management in electricity supply chains. A multi-settlement approach consists of at least a forward market and a spot (realtime) market, each producing its own financial settlements. Multiple settlements mean that charges for demand and payments to supply can occur in both the forward and the spot market. The forward market schedules production and consumption before the operating day. The real-time market reconciles any differences between the schedule in the forward market and the real-time load. Generation and demand bids that clear the forward market are settled at forward prices. These bids are applied to each operating period of the day and to each location in the system. (Also, increment offers and decrement bids can be submitted, which indicate the prices at which the supply and demand sides are willing to increase or decrease their injections or withdrawals in the system. These “incs” and “decs” are the tools for market participants to adjust their positions in the forward market.) The settlements in the forward market can be either financially or physically binding. At the spot, scheduled suppliers must produce the committed quantity or, otherwise, buy power to balance their positions. Likewise, wholesale buyers of electricity pay and lock in their rights to consume the quantities cleared at the forward prices. While the forward market settles the schedule and financial terms of energy production and consumption on the operating day, the quantities that clear the forward market do not necessarily clear the spot market. For generators, the spot market provides additional opportunities of offering energy into the market. Thus, commitments in the forward market are compensated or paid at realtime prices for the megawatts over- or under-produced. Multiple settlements also allow market participants to secure day-ahead prices and to reduce their vulnerability to price fluctuations in the spot market. Spot price volatility stems from unforeseen events, such as demand uncertainty and transmission and generation outages. Because the forward market is not significantly affected by these events, it

7 produces more stable prices than the spot market. Therefore, forward contracts can help market participants hedge spot price risks. Moreover, because forward prices influence realtime behaviors, they generate more stable profit for generators and provide incentives to wholesale consumers to better manage their consumption patterns. Finally, with the horizon for energy transactions expanded to multiple settlements, the market becomes more liquid, more competitive and more economically efficient. Thus, opportunities and incentives for participants to manipulate the markets are limited.

1.3

About this Thesis Theoretical studies in [2, 3, 37] suggest that forward contracting reduces the in-

centives of sellers to manipulate the markets. Although these results are appealing, the analysis ignores certain complexities of electricity systems, such as flow congestion, demand volatility and system contingencies. The main objective of this research is to study, when more realistic factors are considered, whether the generation firms are still willing to engage in forward contracts and, if so, whether their incentives for committing forward contracts will increase social surplus and reduce spot prices.

1.3.1

Scope of this Thesis Centrally-dispatched energy market. In this thesis, we will focus on the so-

called centrally-dispatched markets, a prevailing form of US electricity markets. This type of market usually consists of a pool run by an independent system operator (ISO) that serves as a trading broker or auction center for spot market transactions in the wholesale marketplace. Cournot competition. Cournot and supply function competition are two types of conjectural variations models used to represent energy markets. Unlike [4, 28, 42] where firms bid supply functions into the spot market, this research assumes Cournot competition

8 in both the spot and the forward markets. Single product. Multiple products and markets are another complexity of restructured electricity markets. For example, in the early restructured California market, there was a total of 11 markets for energy and ancillary services. The present work assumes a simple market architecture, with a forward energy market for each zone and a single spot energy market. Ignoring inter-temporal constraints. Electricity generation is subject to various inter-temporal constraints such as startup cost, minimum startup/shutdown time, minimum run times and maximum ramp rates. Incorporating these constraints would significantly complicate the analysis because they introduce non-convexities in firms’ decision making. In this research, we simplify our analysis by ignoring inter-temporal constraints. This will separate market participants’ decision making in each operating period. Including transmission constraints and system contingencies. Transmission constraints and loop flow congestion limit the amount of electricity that is allowed to be transported within a network. Such limitation grants producers more locational market power to extract surpluses from consumers. Unlike [2, 3, 4] ignoring network constraints, we take into consideration these constraints so that our analysis presents more realistic results. Moreover, introducing transmission congestion essentially means our model allows different levels of granularity of locational prices, i.e., nodal prices in the spot market and zonal aggregation in the forward market. In addition, the model to be developed in this thesis considers system contingencies resulting from transmission and generator outages so as to simulate these unpredictable events. Price caps. Regulatory price/offer caps and competitive entry are additional means of market power mitigation. A regulatory price cap is imposed by regulators to restrain spot prices for energy consumption and generation. On the other hand, potential entry to the forward market implicitly limits forward prices because, if forward prices exceed the long-run amortized cost, additional capacity will be invested by competitive entry.

9 Studying how a spot price cap and competitive entry reduce market power is also a task of this thesis. Special-purpose algorithm. From a computational viewpoint, the models developed in this thesis are known as equilibrium problems with equilibrium constraints (EPECs), which are composed of mathematical programs with equilibrium constraints (MPECs, see [40]). Such an EPEC problem, when applied to realistic systems, presents a computational challenge. A major part of this thesis is devoted to developing specialpurpose algorithms for solving these EPEC problems. To summarize: this research considers centrally-dispatched energy systems with flow congestion, system contingencies and demand uncertainty, and studies the Cournot equilibria under two settlements. Subjects of this thesis also include the computational perspective and price caps resulting from regulation and competitive entry.

1.3.2

Outline of this Thesis We start in chapter 2 with two Nash-Cournot models of spot markets, Model S1

and Model S2. In these two models, the ISO’s strategic variables are postulated to be the redispatch quantities and the locational price differences, respectively. Also described in this chapter is the stylized Belgian electricity network, which shall be used as a benchmark system for comparing the models and illustrating economic insights. In chapters 3 and 5, we add a forward market to the two spot market models, and develop corresponding two-settlement models, Model T1 and Model T2, respectively. In both models, firms’ optimization problems in the forward market are constrained by a set of equilibrium conditions, which includes the optimality conditions for the firms and the ISO’s programs in the spot market, and a no-arbitrage relationship equating the forward prices to the expected spot zonal settlement prices. Numerical examples show that introducing a forward market increases consumer surplus and reduces spot energy prices. Chapter 4 develops a variant of Model T1 by introducing two types of price caps:

10 a forward price cap and a spot price cap. The spot price cap is a regulatory tool to limit spot price spikes; the forward price cap replicates the long-rum amortized cost, beyond which entry will occur. We introduce a reformulation for removing the non-smoothness in suppliers’ objectives due to the spot price cap. Numerical simulation demonstrates that a forward price cap catalyzes producers’ incentives for forward contracting, whereas a spot price cap reduces such incentives. The two-settlement models that will be developed in this thesis are the so-called EPEC problems, for which one has to solve simultaneously the MPEC problems therein. In chapter 6, we develop an MPEC algorithm via piece-wise quadratic programming and parametric LCP pivoting. We further construct an EPEC scheme by iteratively applying the MPEC algorithm.

11

Chapter 2

The Spot Market Models 2.1

Introduction Different markets have been following different blueprints for electricity restruc-

turing. In the US, one prevailing design is the so-called centrally-dispatched market. This type of market usually consists of a pool run by an independent system operator (ISO) that serves as a broker, or auctioneer, for the wholesale spot electricity market transactions. The ISO controls the transmission system and redispatches generator outputs so as to maximize social welfare while meeting all the network and security constraints. The ISO also sets locational energy prices and transmission charges for bilateral energy transactions. In this chapter, we consider a centrally-dispatched wholesale spot market with demand uncertainty, flow constraints and system contingencies, and develop two NashCournot models. The next section reviews literature on spot market models. In section 2.3, we introduce the ISO’s decision making. Sections 2.4 and 2.5 develop the two models, respectively. An example is given in section 2.6. Numerical results of the two models are reported in section 2.7.

12

2.2

Literature Review We review single-settlement models with transmission constraints. Schweppe et

al. [51] originated the theory of locational prices. Given the generator costs, the demand and the network topology, locational prices can be calculated using an optimal power flow model by minimizing the total cost of generation. In a decentralized system, these prices can elicit the optimal quantities from competitive agents, and reflect equilibrium marginal costs/benefits at various locations. The differences in these prices determine the transmission congestion rents for bilateral contracts. Subsequent studies have considered the effect of market power in electricity markets. Most models with transmission constraints assume the Cournot conjectural variation. An important modeling choice in these models is the assumption on how agents act in the transmission markets. This choice impacts locational marginal prices and the congestion rent paid to transmission rights holders. Assuming the agents game the transmission market leads to an equilibrium problem in which each firm solves a non-convex problem subject to shared constraints. Cardell, Hitt and Hogan [9] consider a network where Cournot generators own plants at multiple locations, and a competitive fringe acts as a price taker in the spot market. A complicating feature of their model is that, with the introduction of a competitive fringe, a given strategy profile of the generators may not lead to a unique outcome in the market. Another feature is that both the strategy sets and the objective values of a generator depend on the actions chosen by other generators. Their study shows that a firm owning multiple units gains a strategic advantage by forcing out some competition on a part of the network, and, due to transmission constraints, earns more profits on its remaining capacity. Borenstein, Bushnell and Stoft [5] demonstrate that interactions among Cournot generators in the presence of a transmission constraint can be quite complex. In a spatially separated market with a symmetric duopoly and a small interconnection, there might exist

13 no pure strategy Cournot equilibria due to the discontinuities in the response functions of the generators. Following the best response, one generator produces a constant amount, congesting the transmission line in the outgoing direction until the other generator’s production reaches a threshold, at which the former increases its output proportionally. At another threshold, the first generator abandons this strategy and reverts to a defensive strategy with a smaller production to congest the line in the incoming direction. The authors also analyze the cases with multiple equilibria. Gilbert, Neuhoff and Newbery [29] examine the role of transmission contracts on market power. They show that the settlement process of transmission rights significantly affects the results. In a uniform price auction, arbitrageurs will force generators to sign contracts, which, in turn, will mitigate their market power. On the other hand, if contracts are auctioned with discriminatory prices, they can enhance the producers’ market power. The authors also analyze the impact of the network topology on market power mitigation. The preceding models solve generators’ optimization problems by taking into account the effect of their actions on transmission prices; these approaches introduce significant computational intractability. Some other models avoid this complication by assuming generators act as price takers in the transmission market. This removes the non-convexity from generators’ optimization problems. Thus, the first-order optimality conditions for all generators’ programs can now be assembled along with those for the transmission owner’s program, and the equilibrium can be computed as a complementarity problem or variational inequality. Wei and Smeers [54] consider a Cournot model with regulated transmission prices. Variational inequalities are solved to determine unique long-run equilibria. Smeers and Wei [52] consider a separated energy and transmission market, where the system operator conducts a transmission capacity auction, and power marketers purchase transmission contracts to support bilateral transactions. They show that such a market converges to the optimal dispatch with many marketers.

14 Hobbs [32] assumes linear demand and cost functions, and establishes a mixed linear complementarity problem for solving Cournot market equilibria. In a bilateral market, he analyzes two types of markets, with and without arbitrageurs. In the market without arbitrageurs, non-cost-based price differences can arise because bilateral transactions endow generators more degrees of freedom to discriminate electricity demands at various locations. This result is equivalent to a separated market in [52]. In the market with arbitrageurs, non-cost differences are eliminated by arbitrageurs who trade electricity at nodal prices. This equilibrium is shown equivalent to a Cournot-Nash equilibrium in a POOLCO-type market.

2.3

The ISO’s Problem We consider an electricity network that consists of a set N of nodes, and a set L

of transmission lines. There are a set G of generation firms competing in the market, each operating the units at a subset of locations Ng ⊆ N . For simplicity, we assume that at most one generation firm operates at a node, and we shall introduce artificial nodes, if necessary, to meet this assumption. (Complete collection of notations can be found in Appendix A.1.) Given the firms’ production decisions ({qi }i∈N ), the ISO determines the import/export ric at each node i ∈ N (using the convention that positive quantities represent imports). These quantities must satisfy the network feasibility constraints, that is, the resulting power flows should not exceed the thermal limits ({Kl }l∈L ) of the transmission lines in both directions: −Kl ≤

X

Dl,i ri ≤ Kl ,

l ∈ L.

i∈N

We model the transmission network as a lossless DC approximation of Kirchhoff’s laws (see [10]). Specifically, flows on lines can be calculated using power transfer distribution factors (PTDFs) Dl,i which specify the proportion of flow on a line l ∈ L resulting from an injection of one-unit electricity at a node i ∈ N and a corresponding one-unit withdrawal at some

15 fixed reference bus (also known as the slack bus). Moreover, because electricity is not economically storable, the load and generation must be balanced. This leads to the energy balancing constraint. In a lossless grid, this constraint enforces the total import/export across all nodes to be zero: X

ri = 0.

i∈N

The ISO’s objective is to maximize social welfare of the entire system, which is given by the total consumer willingness-to-pay (the aggregated area under the nodal inverse demand curves) less the total generation cost. Mathematically, the ISO solves the following problem parametric on the firms’ production decisions ({qi }i∈N ): ¶ X µZ ri +qi max Pi (τi )dτi − Ci (qi ) ri :i∈N

i∈N

0

subject to: X ri = 0

(2.1)

i∈N

X

Dl,i ri ≥ −Kl ,

l∈L

(2.2)

i∈N

X

Dl,i ri ≤ Kl ,

l∈L

(2.3)

i∈N

Here, terms Pi (·) and Ci (·) are the inverse demand function (IDF) and generation cost function at node i, respectively. Note that, since the firms’ decision variables {qi }i∈N are treated as constants in this program, we can drop term Ci (qi ) from the objective without affecting the optimal solution. Let p, λl− and λl+ be the Lagrangian multipliers corresponding to constraints (2.1)(2.3), then the first order necessary conditions (the Karush-Kuhn-Tucker, KKT conditions [38, 39]) for the ISO’s problem are: Pi (qi + ri ) − p +

X

(λm− Dm,i − λm+ Dm,i ) = 0

i∈N

(2.4)

m∈L

X j∈N

rj = 0

(2.5)

16

0 ≤ λl−



X

Dl,j rj + Kl ≥ 0

l∈L

(2.6)

l∈L

(2.7)

j∈N

0 ≤ λl+

⊥ Kl −

X

Dl,j rj ≥ 0

j∈N

Condition (2.4) implies that the nodal price is Pi (qi + ri ) = p + ϕi ,

i ∈ N,

(2.8)

where X

ϕi = −

(λm− Dm,i − λm+ Dm,i ).

m∈L

Thus, qi + ri = Pi−1 (p + ϕi ) . and consequently, due to (2.5), X

qj =

j∈N

X

Pj−1 (p + ϕj ) .

(2.9)

j∈N

We can interpret p as the energy price at the reference bus and {ϕi }i∈N as node specific premiums which determine the relative nodal prices. Hence, the congestion charge for the transmission from node i to node j that will prevent arbitrage between nodes is ϕj − ϕi , P and the total congestion rent in the network is i∈N ϕi ri . Note the load and the generation are settled at the prices of their respective nodes; the total payment from load does not necessarily match the total charge from generation. Their difference is computed in the following proposition. Proposition 1. The difference between the total payment from load and the total charge from generation is nonnegative and equal to the total congestion rent of the network. Proof. The total payment from load is the net consumption paying at nodal prices: X i∈N

Pi (qi + ri )(qi + ri ),

17 and the total charge from generation is the production paid at nodal prices: X

Pi (qi + ri )qi .

i∈N

Their difference, denoted by ∆, is ∆=

X

Pi (qi + ri )ri

i∈N

By condition (2.4), we have ∆=

X

Ã

X pri + ri (λm+ − λm− )Dm,i

i∈N

=p

l∈L

X

ri +

i∈N

=

!

XX

(λm+ − λm− )Dm,i ri

i∈N m∈L

XX

(λm+ − λm− )Dm,i ri

i∈N m∈L

=

X

ϕi ri

i∈N

Hence, this difference is equal to the total congestion rent. Furthermore, conditions (2.6) and (2.7) imply ∆=

X (λl+ + λl− )Kl . l∈L

Since both λl+ and λl− are nonnegative, the total congestion rent is nonnegative. Moreover, it is zero if and only if λl+ = λl− = 0,

l ∈ L,

i.e., no transmission line is congested.

2.4 2.4.1

The First Spot Market Model: Model S1 The Firms’ Problems In the spot market, each firm g ∈ G determines the output quantities ({qi }i∈Ng )

from its units at Ng . These quantities must be nonnegative and not exceed the correspond-

18 ing upper capacity bounds ({q i }i∈Ng ): 0 ≤ qi ≤ q i ,

i ∈ Ng .

A number of modeling approaches have been proposed for characterizing generation firms’ decision making (see, for example, [32, 41, 52, 54]). One modeling consideration regarding the suppliers’ strategic behavior in these models is whether or not they game the transmission market. This choice is referred in [32] as whether or not the firms behave a la Bertrand with respect to transmission prices, and in [41] as where the market design is integrated or separated. As a result, we classify spot market models into two approaches. The first approach assumes that the energy and transmission markets are sequentially cleared, and the generation firms anticipate the impact of their production decisions on the transmission prices set by the ISO. The resulting formulation of the spot market is a multi-leader one-follower Stackelberg game. Each producer solves the following parametric mathematical program with equilibrium constraints (MPEC, see [40]), in which the optimality conditions for the ISO’s program are the constraints shared by all firms: max

qi :i∈Ng

X

Pi (ri + qi )qi −

i∈Ng

X

Ci (qi )

i∈Ng

subject to: qi ≥ 0,

i ∈ Ng

qi ≤ q i ,

i ∈ Ng

Pi (qi + ri ) − p +

X

(λm− Dm,i − λm+ Dm,i ) = 0,

m∈L

X

rj = 0

j∈N

0 ≤ λl−



X

Dl,j rj + Kl ≥ 0,

l∈L

j∈N

0 ≤ λl+

⊥ Kl −

X j∈N

Dl,j rj ≥ 0,

l∈L

i∈N

19 The above problem represents a so-called “generalized Nash game” (see [30]), and it may have none or multiple solutions (see [5]). Furthermore, even if a solution is found, it is usually degenerate; that is, firms will find it optimal to barely congest some transmission lines so as to avoid the congestion rent (see [43]). Moreover, finding equilibrium solutions of this formulation for a realistic size network is computationally intractable. The second approach is to assume that the firms do not anticipate the impact of their production decisions on congestion charges, or alternatively to assume that the energy and transmission markets are simultaneously cleared [32, 52, 54]. Empirical evidence suggests that ignoring potential gaming of transmission prices by generators is quite realistic since it is practically impossible for multiple generation firms to coordinate their production so as to avoid congestion charges by barely decongesting transmission lines. Indeed, the initial design of the California market that attempted to control congestion by relying on advisory congestion charges is proved to be unworkable. In the second approach, the ISO is a Nash player that acts simultaneously with the generation firms. The firms determine their supply quantities so as to maximize their profits; they are however price takers in the transmission market. The market equilibrium is then determined by aggregating the optimality conditions for both the firms and the ISO’s problems, resulting in a mixed complementarity problem or a variational inequality problem. This approach avoids the computational intractability of the first approach and, under a non-degeneracy assumption, leads to a unique equilibrium solution. There are still two modeling options within this simultaneous-move framework. Consequently, this chapter will develop two corresponding models of the spot market. In the first option, it is assumed that the ISO’s strategic variables are the redispatch quantities. Mathematically, each firm g ∈ G solves the following profit-maximization problem parametric on ({ri }i∈N ): max

qi :i∈Ng ,

X i∈Ng

Pi (qi + ri )qi −

X i∈Ng

Ci (qi )

20

subject to: qi ≥ 0,

i ∈ Ng

(2.10)

qi ≤ q i ,

i ∈ Ng

(2.11)

Note that, in this formulation, the residual demand functions faced by the firms are the original demand shifted by the import/export quantities, but retaining the original slopes. Let ρi− and ρi+ be the Lagrangian multipliers corresponding to (2.10) and (2.11); the KKT conditions for firm g’s program are: Pi (qi + ri ) +

2.4.2

dCi (qi ) ∂Pi (qi + ri ) qi − + ρi− − ρi+ = 0 ∂qi dqi

i ∈ Ng

(2.12)

0 ≤ ρi−

⊥ qi ≥ 0

i ∈ Ng

(2.13)

0 ≤ ρi+

⊥ q i − qi ≥ 0

i ∈ Ng

(2.14)

Market Equilibrium Conditions Aggregating the KKT conditions of the ISO and the firms’ programs, (2.4)-(2.7)

and (2.12)-(2.14), leads to the market equilibrium conditions which, in general, form a mixed nonlinear complementarity problem. When the nodal demand functions are linear: Pi (q) = a − bi q,

i ∈ N,

and the cost functions are quadratic 1 Ci (q) = di q + si q 2 , 2

i ∈ N,

these conditions become the following mixed linear complementarity problem (mixed LCP, see [12]): a − (2qi + ri )bi − di − si qi + ρi− − ρi+ = 0

i∈N

(2.15)

0 ≤ ρi−

⊥ qi ≥ 0

i∈N

(2.16)

0 ≤ ρi+

⊥ q i − qi ≥ 0

i∈N

(2.17)

21 X

rj = 0

(2.18)

j∈N

a − (qi + ri )bi − p +

X

(λm− Dm,i − λm+ Dm,i ) = 0

i∈N

(2.19)

l∈L

(2.20)

l∈L

(2.21)

m∈L

0 ≤ λl−



X

Dl,j rj + Kl ≥ 0

j∈N

0 ≤ λl+

⊥ Kl −

X

Dl,j rj ≥ 0

j∈N

The preceding conditions can be simplified to a linear complementarity problem (LCP) with a particular structure as shown below. Proposition 2. Conditions (2.15)-(2.21) are equivalent to an LCP with a bisymmetric PSD matrix. Proof. We group and relabel the parameters and variables as follows: • B (∈ R|N |×|N | ): A diagonal matrix where the (i, i)-th element is bi . • S (∈ R|N |×|N | ): A diagonal matrix where the (i, i)-th element is si . • q (∈ R|N | ): The vector of the generator capacity bounds. q = [q i

i ∈ N]

• d (∈ R|N | ): The vector of the first-order marginal generation costs. d = [di

i ∈ N]

• D (∈ R|L|×|N | ): The PTDF matrix where the (l, i)-th element is Dl,i . • k (∈ R|L| ): The vector of the flow capacities of the transmission lines. k = [Kl

l ∈ L]

22 • r (∈ R|N | ): The vector of the ISO’s import/export quantities. r = [ri

i ∈ N]

• q (∈ R|N | ): The vector of the firms’ generation quantities. q = [qi

i ∈ N]

• ρ− , ρ+ (∈ R|N | ): The vectors of the Lagrangian multipliers corresponding to the generation capacity constraints. ρ− = [ρi−

i ∈ N]

ρ+ = [ρi+

i ∈ N]

• λ− , λ+ (∈ R|L| ): The vectors of the Lagrangian multipliers corresponding to the flow capacity constraints. λ− = [λl−

l ∈ L]

λ+ = [λl+

l ∈ L]

Now, let e ∈ R|N | be a vector with all 1’s, then constraints (2.18) and (2.19) become 

            T T ae B B e r D D             0   − q −    +   λ− −   λ+ =   . 0 0 eT 0 p 0 0 0

23 Solving r and p yields    −1          T T  r   B e   ae   B   D   D    =   − q +   λ− −   λ+  p eT 0 0 0 0 0            B −1 e T T Q eT B −1 e   ae   B   D   D    q+ λ− −  − =    λ+      eT B −1 1 0 0 0 0 eT B −1 e eT B −1 e where Q = B −1 −

B −1 eeT B −1 . eT B −1 e

Hence, ¡ ¢ r = − QBq + Q DT λ− − DT λ+ p =a −

(2.22)

¢ eT eT B −1 ¡ T T . q + D λ − D λ − + eT B −1 e eT B −1 e

Substituting (2.22) into (2.15) and solving ρ− leads to ¢ ¡ ρ− = −ae + d + (2B − BQB + S)q + BQ DT λ− − DT λ+ + ρ+ . Next, let w and y be two variable  q¯ − q    ρ−  w=   k + Dr  k − Dr 

 q¯

   −ae + d  t=  k   k

    ,   

vectors, t and M be constants, such that    ρ+         q     , y =  ,      λ−     λ+ 

 0

−I

0

   I 2B − BQB + S BQDT  M =  −DQB DQDT  0  0 DQB −DQDT

then conditions (2.15)-(2.21) are represented as an LCP w = t + M y,

0≤w

⊥ y ≥ 0.

0 −BQDT −DQDT DQDT

    ,   

24 We notice that H is symmetric positive-definite. Moreover, v T B −1 eeT B −1 v eT B −1 e 1 1 1 − 12 − kB vk2 kB 2 ek2 − kv T B − 2 B − 2 ek2

v T Qv = v T B −1 v − =

1

kB − 2 ek2 v ∈ R|N | .

≥ 0,

Hence, Q is symmetric PSD. Now, since 

 0

M+ 2

MT

0

0 0

   0 H 0 0  =   0 0 0 0  0 0 0 0



        +      

 0



T 0

     0  0   Q   D   D   −D −D

     ,   

we conclude that M is bisymmetric PSD. Proposition 3. Conditions (2.15)-(2.21) are also equivalent to an LCP with a symmetric PSD matrix. Proof. To show that conditions (2.15)-(2.21) are also equivalent to a symmetric-PSD-matrix LCP, we substitute (2.22) into (2.15) and solve q: ¢ ¡ q = H ae − d + ρ− − ρ+ − BQDT λ− + BQDT λ+ ,

(2.23)

where H = (2B − BQB + S)−1 . Substituting (2.23) into (2.22), we obtain ¡ ¢ r = − QBH (ae − d) − QBHρ− + QBHρ+ + QBHBQDT + QDT (λ− − λ+ ) Now, let w and y be two variable vectors, t be a constant vector, and M be a

25 constant matrix, such that  q¯ − q    q  w=   k + Dr  k − Dr





 ρ+

    ,   

   ρ−  y=   λ−  λ+

H

H(ae − d)

    ,   

 −H



    q − H(ae − d)  t=   k − QBH(ae − d)  k + QBH(ae − d)

−HBQDT

   −H H HBQDT  M =  DQBH DQBHBQDT  −DQBH  DQBH −DQBH −DQBHBQDT   0 0 0 0      0 0 0 0    + ,   DQDT −DQDT   0 0   T T 0 0 −DQD DQD

    ,   

HBQDT −HBQDT −DQBHBQDT

        

DQBHBQDT

then conditions (2.15)-(2.21) become the following LCP problem w = t + M y,

w ≥ 0,

y ≥ 0,

wT y = 0.

Here, because H and Q are symmetric PSD, and 





T



I I 0              −I  −I  0      M = H   +       −DQB   −DQB   D      DQB DQB −D





        Q      

T 0

  0    ,  D   −D

M is symmetric PSD.

2.4.3

Difficulties of Model S1 Notice that each firm g’s program is parametric on {ric }i∈N ; one can decompose it

into |Ng | subproblems, each corresponding to the production decision at one node. There-

26 fore, this model will yield a spot market equilibrium that is invariant to the generator ownership structure, i.e, it doesn’t matter whether one firm owns one or multiple generators. Moreover, under this formulation, when the network constraints (5.2)-(5.3) are relaxed, the equilibrium solution predicts uniform nodal prices that are systematically higher than the Cournot equilibrium price corresponding to a single market with the aggregated system demand function. This phenomenon is consequence of the fact that, even in the absence of congestion, the residual demand function faced by each generator has the local demand slope rather than the aggregated demand slope.

2.5

The Second Spot Market Model: Model S2 The second spot market model assumes that the ISO’s strategic variables are the

locational price premiums, as opposed to the nodal imports/exports in the first model.

2.5.1

The Firms’ Problems Equation (2.9) implicitly defines the aggregated demand function at the reference

bus. Thus, each firm g ∈ G determines its own production so as to affect the energy price at the reference bus via (2.9) while anticipating the rivals’ production and the nodal price premiums. It solves the following profit-maximization problem: X

max

qi :i∈Ng ,p

(p + ϕi ) qi −

i∈Ng

X

Ci (qi )

i∈Ng

subject to: qi ≥ 0,

i ∈ Ng

(2.24)

qi ≤ q i , i ∈ Ng X X qj = Pj−1 (p + ϕj )

(2.25)

j∈N

(2.26)

j∈N

Let ρi− , ρi+ and βg be the Lagrangian multipliers corresponding to (2.24)-(2.26),

27 then the KKT conditions for firm g’s program are: dCi (qi ) + ρi− − ρi+ = 0 dqi X X dPj−1 (p + ϕj ) βg + qj = 0 dp p + ϕi − βg −

j∈N

X

qj =

i ∈ Ng

(2.27) (2.28)

j∈N g

X

Pj−1 (p + ϕj )

(2.29)

j∈N

j∈N

0 ≤ ρi−

⊥ qi ≥ 0

i ∈ Ng

(2.30)

0 ≤ ρi+

⊥ q i − qi ≥ 0

i ∈ Ng

(2.31)

Here, the first two equations are the derivatives of the Lagrangian function with respect to qi and p, respectively.

2.5.2

Market Equilibrium Conditions Combining conditions (2.4)-(2.7) with (2.27)-(2.31) leads to the market equilibrium

conditions as a mixed nonlinear complementarity problem. Furthermore, when both the nodal demand functions and the marginal cost functions are linear: Pi (q) = a − bi q,

i∈N

1 Ci (q) = di q + si q 2 , 2

i ∈ N,

the equation (2.9) becomes P p=a−

ϕj j∈N bj P 1 j∈N bj

P −P

j∈N

qj

1 j∈N bj

,

and the market equilibrium conditions become the following mixed LCP. p + ϕi − βg − di − si qi + ρi− − ρi+ = 0 X 1 X − βg + qj = 0 bj j∈N j∈N g P P ϕj j∈N bj j∈N qj p=a− P 1 − P 1 j∈N bj

j∈N bj

i ∈ Ng , g ∈ G

(2.32)

g∈G

(2.33)

g∈G

(2.34)

28 X

ϕi = −

(λm− Dm,i − λm+ Dm,i )

i∈N

(2.35)

i∈N

(2.36)

i∈N

(2.37)

m∈L

0 ≤ ρi−

⊥ qi ≥ 0

0 ≤ ρi+ ⊥ q i − qi ≥ 0 X rj = 0

(2.38)

j∈N

a − (qi + ri )bi − p +

X

(λm− Dm,i − λm+ Dm,i ) = 0

i∈N

(2.39)

l∈L

(2.40)

l∈L

(2.41)

m∈L

0 ≤ λl−

X



Dl,j rj + Kl ≥ 0

j∈N

0 ≤ λl+

⊥ Kl −

X

Dl,j rj ≥ 0

j∈N

Proposition 4. Conditions (2.32)-(2.41) are equivalent to a bisymmetric-PSD-matrix LCP problem. Proof. We group the variables and parameters as follows: • B (∈ R|N |×|N | ): A diagonal matrix where the (i, i)-th element is bi . • q = [q i

i ∈ N]

• d = [di

i ∈ N]

• D (∈ R|L|×|N | ): The PTDF matrix where the (l, i)-th element is Dl,i . • k = [Kl

l ∈ L]

• r = [ri

i ∈ N]

• q = [qi

i ∈ N]

• ρ− = [ρi−

i ∈ N]

• ρ+ = [ρi+

i ∈ N]

• λ− = [λl−

l ∈ L]

29 • λ+ = [λl+

l ∈ L]

We derive r and p from (2.38) and (2.39) ¡ ¢ r = − QBq + Q DT λ− − DT λ+ p =a −

eT eT B −1 e

q+

¢ eT B −1 ¡ T D λ− − DT λ+ T −1 e B e

where Q = B −1 −

B −1 eeT B −1 . eT B −1 e

Next, merging conditions (2.32)-(2.35) leads to ¢ eeT B −1 ¡ T D λ− − DT λ+ + DT λ− − DT λ+ + ρ+ , T −1 e B e ¢ ¡ = −ae + d + Hq + BQ DT λ− − DT λ+ + ρ+ ,

ρ− = −ae + d + Hq −

where H is a symmetric PSD matrix such that   2+si  if i = j  T −1   e B e 2 hij = if i 6= j, and the units at nodes i and j belong to the same firm eT B −1 e     1  otherwise eT B −1 e Finally, let w and y be two variable vectors, t be a constant vector, and M be a constant matrix, such that 

 q¯ − q

   ρ−  w=   k + Dr  k − Dr 

 q¯

   −ae + d  t=  k   k

    ,   

    ,   



 ρ+

   q  y=   λ−  λ+

    ,    

 0

−I

0

   I H BQDT  M =  DQDT  0 −DQB  0 DQB −DQDT

0 −BQDT −DQDT DQDT

    ,   

30 then conditions (2.32)-(2.40) become w = t + M y,

0≤w

⊥ y ≥ 0.

Notice M is bisymmetric PSD.

2.5.3

Difficulties of Model S2 The second model overcomes the shortcomings of Model S1. It takes account of the

resource ownership structure and, when the network constraints are relaxed, the locational price premiums go to zero so that the model produces the same equilibrium solution as the Cournot equilibrium applied to the aggregate system demand. Unfortunately, this model has a different shortcoming which manifests itself if we reduce the capacity of a radial transmission line to zero or if it is common knowledge that a radial line is constantly congested. In such situations, the sub networks connected by the radial line are effectively decoupled from a competitive interaction point of view. For instance, in the case of a symmetric two-node one-line network, reducing the line capacity to zero creates two local monopolies. However, in this situation, our model will produce a symmetric duopoly equilibrium with prices that are systematically lower than the monopoly prices (this phenomenon is similar to that in [5]). Recognizing this problem, we can address it by identifying decoupled sub-networks and systematically congested links in the system, and account for such aspects by introducing “common knowledge constraints” on the corresponding flows which will appear in the ISO and the firms’ problems. Alternatively, we can treat a decoupled network as separate sub-networks whose equilibria are calculated separately. In this thesis, we will, however, assume that the network is connected physically and strategically, i.e., it represents a single congested market.

31 51

38 39

50

40 33 34 35

53

48

41

46

36 42 44

43

52

22

45 29 28

19 18

49

47

37

30 31

27

21 10 9 8 7 24 11 6 25 32 26 23 20

14 13

12 5 4 3

17 1

2

15 16

Figure 2.1: Belgian high voltage network

2.6

A Numerical Example: the Stylized Belgian Network Throughout this thesis, we use a stylized version of the Belgian electricity net-

work (see figure 2.1) to exhibit numerical results and economic insights. This system is originally composed of 92 380kV and 220kv transmission lines including some lines in neighboring countries for the effect of loop flows.

Parallel lines between the same

pairs of nodes have been collapsed into single lines with equivalent electric characteristics. In total, the stylized network comprises 71 transmission lines (see table 2.4) and 53 nodes. The generation units in this system are located, respectively, at the nodes {7, 9, 10, 11, 14, 22, 24, 31, 33, 35, 37, 40, 41, 42, 44, 47, 48, 52, 53}. The ownership structure, zonal aggregation in the forward market and contingency states are fictitious and so are the nodal demand functions, although they are calibrated to actual demand information. We assume six independent contingency states in the spot market. In the first state, the demands are at peak, all generation plants operate at full capacities, and all transmission lines are rated at the full thermal limits. The second state is the same as the

32

Table 2.1: States of the Belgian spot market State 1

Probability 0.20

2

0.50

3

0.03

4

0.03

5

0.04

6

0.20

Type and description On-peak state: Demands are at peak. Shoulder state: Demands are at shoulder. Shoulder demands with line breakdown: Line 31-52 goes down. Shoulder demands with generation outage: Plant at node 10 goes down. Shoulder demands with generation outage: Plant at node 41 goes down. Off-peak state: Demands are off-peak.

first state except that it has shoulder demands. The next three states have also shoulder demands, but comprise system contingencies. In particular, state 3 denotes the transmission breakdown of line 31-52. State 4 and 5 capture the unavailability of two plants at nodes 10 and 41, respectively. Off-peak state 6 differs from states 1 and 2 with very low demand levels. The hypothetical intercepts of the inverse demand functions in these six states are assumed to be 1000, 500, 500, 500, 500 and 250 $/MWh, respectively. The probabilities of these states are given in table 2.1. Tables 2.2 and 2.3 list the nodal information, including the inverse demand function slopes, the marginal generation costs and the full capacities of generation plants. In this example, we assume that the demand at each node shifts inward and outward in different states, but that the slope remains unchanged. For flow congestion, we focus on transmission lines 22-49, 29-45, 30-43 and 31-52, which are assumed to be the so-called flowgates. We consider five hypothetical ownership structures with two, three, four, five and six firms, respectively. Table 2.5 reports these five structures.

33

Table 2.2: Node information in the Belgian network node 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

IDF slope ($/MW2 ) 1 0.82 1.13 1 0.93 0.85 1 1 0.88 0.9 1 0.73 1 0.85 1 1.3 1 0.79 0.68 1.05 1 1.1 1 0.75 1 0.8 1.13 1 0.93 0.85 ∗

marginal cost ($/MW) 45 18 18 20 13 19 10 -

This capacity is zero in state 4.

capacity (MW) 0 0 0 0 0 0 70 0 460 121* 124 0 0 1164 0 0 0 0 0 0 0 602 0 2985 0 0 0 0 0 0

34

Table 2.3: Node information in the Belgian network (continued) node 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53

IDF slope ($/MW2 ) 1 1 0.88 0.5 1 0.73 1 0.85 1 1.15 1 0.79 0.68 1.03 1 1 1 0.73 1.2 1.5 1 1 0.7 ∗

marginal cost ($/MW) 18 20 25 10 10 21 18 20 22 20 58

This capacity is zero in state 5.

capacity (MW) 712 0 496 0 1053 0 1399 0 0 1378* 522 385 0 538 0 0 0 258 0 0 0 879 95

35

Table 2.4: Transmission lines in the Belgian network from 1 1 2 3 3 4 4 4 5 6 6 7 7 8 8 8 9 9 11 12 13 13 13 15 16 17 17 17 18 19 20 21 22 22 23 23

to 2 15 15 4 15 5 12 15 13 7 8 21 32 9 10 32 11 32 32 32 14 15 23 16 17 18 19 20 19 52 23 22 23 49 24 25

impedance 23716 6269 8534 5339 11686 6994 5887 3644 6462 23987 9138 14885 5963 45360 26541 11467 20157 10012 18398 4567 121410 5094 5481 8839 2633 4236 1939 8071 1465 11321 13165 47621 11391 9138 41559 16982

capacity (MW) 345 345 345 240 240 510 405 240 510 300 400 541 410 400 800 400 410 375 375 405 2700 790 2770 400 5154 1715 5140 1179 13710 1179 1316 1420 1350 1350 5540 1420

from 23 23 25 25 27 28 29 29 30 30 31 33 34 34 35 35 36 36 36 37 37 38 38 39 40 41 41 43 44 46 47 48 49 50 52

to 28 32 26 30 28 29 31 45 31 43 52 34 37 52 41 52 41 42 43 39 41 39 51 51 41 46 47 45 45 47 48 49 50 51 53

impedance 8610 33255 134987 11991 64753 38569 284443 14534 269973 10268 1453 40429 7048 12234 14204 9026 15777 11186 15408 66471 21295 10931 17168 8596 11113 11509 13797 34468 47128 34441 14942 6998 5943 2746 1279

capacity (MW) 1350 1350 1420 1420 1420 1350 1350 1350 1420 1420 400 1420 1350 1350 1350 1420 2770 2840 2770 1420 1350 1650 946 1650 2770 2840 1420 1350 1420 1420 1420 1420 3784 5676 2840

36

Table 2.5: Resource ownership structures node 7 9 10 11 14 22 24 31 33 35 37 40 41 42 44 47 48 52 53

2 firms firm #1 firm #1 firm #2 firm #1 firm #2 firm #2 firm #2 firm #1 firm #1 firm #1 firm #1 firm #2 firm #1 firm #2 firm #2 firm #1 firm #2 firm #1 firm #1

3 firms firm #1 firm #3 firm #1 firm #2 firm #2 firm #1 firm #3 firm #1 firm #1 firm #3 firm #2 firm #2 firm #3 firm #1 firm #3 firm #3 firm #1 firm #2 firm #1

4 firms firm #3 firm #1 firm #2 firm #3 firm #2 firm #2 firm #4 firm #3 firm #1 firm #3 firm #1 firm #4 firm #1 firm #2 firm #4 firm #3 firm #4 firm #4 firm #1

5 firms firm #2 firm #4 firm #5 firm #1 firm #4 firm #2 firm #4 firm #4 firm #1 firm #3 firm #5 firm #3 firm #4 firm #5 firm #2 firm #5 firm #1 firm #5 firm #1

6 firms firm #1 firm #3 firm #4 firm #5 firm #2 firm #4 firm #6 firm #1 firm #1 firm #3 firm #5 firm #2 firm #3 firm #4 firm #6 firm #3 firm #4 firm #1 firm #1

37 950 model S1 model S2 900

850

Price

800

750

700

650

600

550

0

5

10

15

20

25 30 Node

35

40

45

50

Figure 2.2: Duopoly nodal prices with on-peak demands ($/MWh)

2.7

Numerical Results We test the two spot market models on the stylized Belgian network with the

preceding five resource structures. Thus, there are ten test cases in total. Note that Model S1 results in the identical equilibrium outcomes for different resource structures; the actual number of test cases is six. We study states 1, 2 and 6 of the spot market, in which the demand varies from the highest to the lowest level. States 3, 4, and 5 yield prices similar to those in state 2 (except for the impact of generator and transmission line outages); we exclude these three states from the analysis in this chapter. We first examine the Duopoly structure. We notice that, under the different demand levels, the nodal prices resulting from Model S2 are consistently lower than those from Model S1 (see figure 2.2, 2.3 and 2.4). This is because the first model does not reflect

38 500 model S1 model S2

450

Price

400

350

300

250

200

0

5

10

15

20

25

30

35

40

45

50

Node

Figure 2.3: Duopoly nodal prices with shoulder demands ($/MWh) the generator ownerships, and each unit behaves as a locational monopoly. Next, we focus on the second state and compare the nodal prices resulting from different resource structures. In figure 2.5, we plot from left to right the prices at the selected nodes {5, 10, 15, 20, 25, 30, 35, 40, 45, 50} for Model S1 (the same prices for all resource structures) and for the two-, three-, four-, five-, and six-firm structures of Model S2, respectively. As expected, the nodal prices in Model S2 at most nodes decrease as the resources become more diversified. However, due to the network constraints, there might exist some nodes at which the prices do not follow this monotone relationship. For example, the price at node 45 increases when the number of firms increases from three to four.

39 300 model S1 model S2 280

260

240

Price

220

200

180

160

140

120

100

0

5

10

15

20

25 30 Node

35

40

45

50

Figure 2.4: Duopoly nodal prices with low demands($/MWh)

2.8

Summary In this chapter, we introduce two Nash-Cournot models of spot energy markets by

following two assumptions regarding the agents’ strategic sophistication. In the first model (Model S1), the locational imports/exports are the ISO’s strategic variables, whereas in the second model (Model S2), the firms expect the location price premiums set by the ISO. Both models lead to the formulations of mixed LCP under linear demand and marginal cost functions. Numerical results from the stylized Belgian network demonstrate the differences between these two models.

40

450 S1 S2: 2 firms S2: 3 firms S2: 4 firms S2: 5 firms S2: 6 firms

400

Price

350

300

250

200

5

10

15

20

25

30

35

40

45

50

Nodel

Figure 2.5: ($/MWh)

Nodal prices under different resource structures with shoulder demands

41

Chapter 3

The First Two-settlement Model: Model T1 3.1

Introduction We shall now describe the first two-settlement model (denoted as Model T1) for

computing the Oligopolistic equilibrium over a given network. We view the two settlements as a complete-information two-period Nash-Cournot game in which the forward market is period zero, and the spot market is period one. In the forward market, generation firms determine their forward commitments in order to maximize their expected utilities while anticipating each other’s forward quantities and the spot market outcomes. In the spot market, all firms’ forward commitments become public knowledge. The spot market is a subgame with two stages. In stage one, Nature picks a state of the world to realize the actual capacities of the generation units, the flow capacities of the transmission lines and the shapes of the locational demand functions. In stage two, the firms recognize the realized contingency state and determine their spot productions so as to maximize their total profits including the financial settlements of their forward commitments and the payments to their generation units. In doing so, they take

42

Figure 3.1: The model dynamics as constants both the revealed forward commitments and the conjectured spot production decisions of all other firms, as well as the redispatch decisions of the independent system operator (ISO) specifying the import/export quantity at each node. Simultaneously with the generators’ decision making, the ISO recognizes the realized network topology and determines the imports/exports at the nodes so as to maximize social welfare. The spot market equilibrium is characterized by the KKT conditions for the firms and the ISO’s programs. Note that each individual firm’s forward commitments affect the spot market outcomes and, in turn, its total profits; a natural question for a firm is how to choose its best forward commitments, given its rational expectation of its rivals’ forward commitments. Hence, we characterize the equilibrium in two-settlement markets as a sub-game perfect Nash equilibrium (SPNE, see [23]). Figure 3.1 illustrates the dynamics of this model. In this figure, the solid directed lines show the time progress whereas the dashed lines represent the rational expectation of the spot equilibrium outcomes.

43 From a mathematical perspective, the model is presented as an equilibrium problem with equilibrium constraints (EPEC), which comprises a set of mathematical programs with equilibrium constraints (MPECs, see [40]) characterizing the decisions of the individual competing firms. In each MPEC, the upper level is the firm’s profit-maximization decisions in the forward market, and the lower level, which is shared by all MPECs, consists of the spot market equilibrium conditions. This chapter is organized as follows. We review literature on two-settlement markets in the next section. Section 3.3 proposes three pricing schemes for settling forward and spot energy transactions. In section 3.4, we develop the firms and the ISO’s programs in the spot market. The forward market EPEC problem is given in section 3.5. Numerical results are reported at the end.

3.2

Literature Review Multi-settlement models are employed to simulate generation firms’ inter-temporal

decision making in both the forward and the spot markets. Allaz [1] pioneered the research on forward incentives. He identifies three types of motives for forward commitments: speculative, hedging and strategic motives. Speculative motives aim at exploiting the price differences between the forward and the spot market. Hedging motives help risk-averse firms handle the uncertainty in the spot market. Strategic motives allow the holders of forward contracts to influence the spot market equilibrium. A subsequent model by Allaz [2] assumes that producers meet in a two-period market with uncertain demand in the second period. In the first period, producers write contracts and speculators take opposite positions. In the second period, a competitive market based on Cournot conjectures is modeled. A no-arbitrage relation between forward and expected spot prices determines the forward price. Allaz shows that generators have a strategic incentive to contract forward even if other producers do not. Hence, a producer

44 with access to the forward market can use its forward commitment as the detriment to its competitors while improving its own profitability. Allaz shows that, if the forward market is accessible to all producers, it leads to a prisoners’ dilemma effect and reduces profits of all producers. Social welfare as the sum of consumer and producer surpluses is higher than that without a forward market. Allaz points out that the results are very sensitive to the type of conjectural variation assumed, and shows that Cournot and market-sharing conjectural variations in the forward market lead to very different results. Allaz and Vila [3] extend this model to the case with multiple periods for forward trading. They show that, as the number of periods for forward trading tends to infinity, producers lose their ability to raise market prices above marginal cost. Kamat and Oren [37] analyze the welfare and distributional properties of a twosettlement system, which consists of a nodal spot market over two- and three-node networks and a single forward energy market. The system is subject to congestion with uncertain transmission capacities in the spot market. Willems [55] studies the effect of forward call options with exogenous strike prices. He shows that, if the strike price is medium, firms will write many options in the forward market, and flood the spot market until the spot price is equal to the strike price. On the other hand, when the strike price is high enough, the options cannot be in-the-money, and they fail to impact the spot market. When the strike price is too low, forward options yield the same strategic effect as forward contracting. Researches on the impact of contracts on the UK pool are conducted in [18, 46]. von de Fehr and Harbord [18] focus on the price competition in the spot market with capacity constraints and multiple demand scenarios. They find that contracts put downward pressure on spot prices. Although this provides disincentive to generators to offer such contracts, a countervailing force is that selling a large number of contracts makes a firm more aggressive in the spot market, and ensures a full dispatch of its capacity in more scenarios. Powell [46] models the re-contracting by Regional Electricity Companies (RECs)

45 after the maturation of the initial portfolio of contracts. He adds risk aversion to the RECs, and assumes generators to be price setters in the contract market. He shows that the degree of coordination affects the hedge cover of the RECs, and illustrates a “free rider” problem that leads to a lower hedge cover chosen by the RECs. Newbery [42] considers nonlinear supply functions, and analyzes the role of contracts as a barrier to entry in the England and Wales electricity market. He assumes constant marginal costs so as to derive analytical solutions to the spot supply functions. Generation firms make “take-it-or-leave-it” offers with a fixed contract quantity at a specified price to consumers. He analyzes equilibria with the forward contract price being an unbiased estimation of the spot price. Newbery shows that, if entrants can sign base load contracts and incumbents have enough capacity, the latter can sell enough contacts to drive down the spot price below the entry deterring level. This could result in more volatile spot pries if producers coordinate on the highest profitable equilibrium. Capacity limit, however, prevents incumbents from playing a low enough equilibrium in the spot market to deter entry. Green [28] adds linear marginal costs to Newbery’s model. In the forward market, each duopoly generator chooses a quantity of contracts while anticipating the rivals’ forward contracts; in the spot market, each firm bids a supply function. An interesting result is that, when generators compete in supply functions in the spot market, the assumption of Cournot conjectural variations in the forward market implies that no contracting will take place unless buyers are risk averse and are willing to have a hedge premium in the forward market. The author shows that forward sales can deter excess entry, increase economic efficiency and raise the long-run profit of a large incumbent firm facing potential entrants.

46

3.3

Pricing Schemes We consider a market with a finite set C of states. Its network is composed of

a set N of nodes connected by a set L of transmission lines. In addition, the network is divided into a set Z of zones for settling forward contracts. A set G of Oligopolistic firms behave a la Cournot in both the forward and the spot markets; each firm g ∈ G operates the units at the node set Ng ⊆ N . (More notations are available in Appendix A.2.) For the sake of generality, our model permits different levels of granularity in financial settlements, with equal granularity being a limiting case. This is motivated by the fact that, in real markets, we observe different granularity levels in the spot and the long-term forward markets — spot markets are mostly organized at the nodal level, whereas forward markets typically involve zonal aggregation and trading hubs. For example, in the PJM system, the western hub representing the weighted average price over nearly 100 nodes is the most liquid electricity forward market in the US. Different granularity levels suggest different prices for spot generations and forward contracts. We introduce three pricing schemes: spot nodal prices, spot zonal settlement prices and forward zonal prices. Energy production in the spot market is paid for at nodal prices through the recognition of the transmission constraints. Forward contracts are settled at spot zonal prices in the spot market and traded at forward zonal prices in the forward market. Intra-zonal transmission congestion is ignored in the forward market — although such congestion implicitly affects the forward and spot zonal prices through a weighted average of nodal prices that are based on historical load shares. The spot nodal prices {pci }i∈N settle the loads and the generations at their respective nodes. In each state c ∈ C of the spot market, the spot nodal price at a node i ∈ N is given by the inverse demand function Pic (qic + ric ), where the net local consumption is the sum of the production (qic ) from the local generation unit and the redispatch (ric , using the convention that positive quantities represent imports) by the ISO.

47 In each state c ∈ C, the spot zonal settlement price (or spot zonal price) ucz of a zone z ∈ Z is used to settle zonal forward contracts financially. In this research, we define it as the weighted average of the nodal prices in the zone with predetermined weights δi . These weights are assumed to be the historical load shares, so they are constants in our model, and not endogenously determined by the actual load ratios in the spot market. Mathematically, the spot zonal settlement prices are given by: ucz =

X

δi Pic (ric + qic ),

z ∈ Z,

i:z(i)=z

where z(i) denotes the zonal index of node i. We assume that the forward prices are collected in the spot market. The forward zonal prices, or the forward prices, {hz }z∈Z are the prices at which the forward contracts are traded in respective zones. In this model, we assume that the forward market is a standardized market with uniform prices or an over-the-counter market where risk-neutral speculators take opposite positions to generation firms and exploit any arbitrage opportunities, so that, in the equilibrium, no profitable arbitrage is possible between the forward and the spot zonal settlement prices. This is the so-called “no-arbitrage”, or the “perfect-arbitrage” assumption. Hence, the forward zonal price of a zone is equal to the expected spot zonal settlement prices: hz = E c [ucz ] X = P r(c)ucz ,

z ∈ Z,

c∈C

where P r(c) denotes the probability of state c.

3.4

The Spot Market In the spot market, Nature first selects a state c ∈ C so as to realize the uncer-

c } tainties, such as the power transfer distribution factors (PTDFs) {Dl,i l∈L,i∈N , the trans-

48 mission thermal limits {klc }l∈L , the generation capacities {¯ qic }i∈N and the nodal demand curves {Pic (·)}i∈N .

3.4.1

The ISO’s Problem In observance of the firms’ outputs {qic }i∈N and the realized uncertainty of the

spot market, the ISO redispatches electricity within the grid. Mathematically, it solves a problem for maximizing social surplus, i.e., the consumer willingness-to-pay minus the generation cost: c

S :

max

ric :i∈N

X µZ i∈N

ric +qic

0

¶ Pic (τi )dτi

subject to: X ric = 0



Ci (qic )

(3.1)

i∈N

X

c c Dl,i ri ≥ −Klc ,

l∈L

(3.2)

i∈N

X

c c Dl,i ri ≤ Klc ,

l∈L

(3.3)

i∈N

In this program, constraint (3.1) enforces the energy balance, and constraints (3.2) and (3.3) impose the flow feasibility.

3.4.2

The Firms’ Problems Given a realized state c, each firm g ∈ G chooses the output {qic }i∈Ng for its units.

These outputs must be nonnegative and not exceed the units’ capacity bounds in this state: 0 ≤ qic ≤ q ci

i ∈ Ng .

Firm g’s revenue in a state c is the sum of the financial settlements of its forward commitments {xgz }z∈Z at the spot zonal prices and the payment for its production quantities {qic }i∈Ng at the spot nodal prices. Mathematically, firm g’s profit in the spot market

49 is πgc =

X

Pic (ric + qic )qic −

i∈Ng

X

(hz − ucz ) xgz −

z∈Z

X

Ci (qic )

i∈Ng

For convenience, we treat the forward contracts as (financial) Contracts For Difference (CDFs). However, the mathematical formulation for financial forward contracts is equivalent to that for physical forward contracts. When making production decisions, firm g treats as constants its own forward commitments {xgz }z∈Z , forward prices {hz }z∈Z and the ISO’s decisions {ric }i∈N . It solves the following problem in the spot market: Ggc :

πgc

max

qic :i∈Ng

subject to: X X X πgc = Pic (ric + qic )qic + (hz − ucz ) xgz − Ci (qic ) i∈Ng

qic ≥ 0,

i∈Ng

i ∈ Ng

qic ≤ q ci ,

3.4.3

z∈Z

(3.4)

i ∈ Ng .

(3.5)

Spot Market Equilibrium Conditions If we let pc , λcl− and λcl+ be the Lagrangian multipliers corresponding to constraints

(3.1)-(3.3), the first order optimality conditions, or the Karush-Kuhn-Tucker (KKT) conditions for problem S c are: Pic (qic + ric ) − pc +

X

c c (λct− Dt,i − λct+ Dt,i ) = 0,

t∈L

X

rjc = 0

j∈N

0 ≤ λcl−



X

c c Dl,j rj + Klc ≥ 0,

l∈L

j∈N

0 ≤ λcl+

⊥ Klc −

X j∈N

c c Dl,j rj ≥ 0,

l∈L

i∈N

50 Similarly, if we let ρci− and ρci+ be the Lagrange multipliers for constraints (3.4) and (3.5), the KKT conditions for problem Ggc are: Pic (qic + ric ) +

∂Pic (qic + ric ) dCi (qic ) − + δi bci xgz(i) + ρci− − ρci+ = 0, ∂qic dqic

0 ≤ ρci−

⊥ qic ≥ 0,

0 ≤ ρci+

⊥ q ci − qic ≥ 0,

i ∈ Ng

i ∈ Ng i ∈ Ng

Aggregating the preceding KKT conditions for the firms’ and the ISO’s problems leads to the spot market equilibrium conditions, which, in general, form a mixed nonlinear complementarity problem. When both the nodal inverse demand functions and the marginal generation costs are linear; i.e., Pic (q) = ac −bci q and Ci (q) = di q+ 12 si q 2 , problems Ggc and S c are both concave-maximization programs, which implies that the spot market equilibrium conditions are sufficient. Moreover, these conditions become a mixed linear complementarity problem (LCP, see [12]) parametric to the firms’ forward commitments {xgz }g∈G,z∈Z : ac − (qic + ric )bci − pc +

X

c c (λct− Dt,i − λct+ Dt,i )=0

i∈N

(3.6)

t∈L

X

rjc = 0

(3.7)

j∈N

0 ≤ λcl−



X

c c Dl,j rj + Klc ≥ 0

l∈L

(3.8)

l∈L

(3.9)

i ∈ Ng , g ∈ G

(3.10)

j∈N

0 ≤ λcl+

⊥ Klc −

X

c c Dl,j rj ≥ 0

j∈N

ac − 2bci qic − bci ric − di + δi bci xgz(i) + ρci− − ρci+ = 0 0 ≤ ρci−

⊥ qic ≥ 0

i∈N

(3.11)

0 ≤ ρci+

⊥ q ci − qic ≥ 0

i∈N

(3.12)

Proposition 5. If we restrict {xgz = 0}z∈Z,g∈G , the solutions to conditions (3.6)-(3.12) lead to the outcome of a single-settlement market, i.e. there exists no forward market, and all firms act only in the spot market.

51

3.5

The Forward Market The forward market is organized at the zonal level with the network constraints

ignored. Each firm forms rational expectations of the rivals’ forward quantities and determines its own forward commitments. A set of risk-neutral arbitrageurs or energy retailers take opposite positions to the producers in order to exploit any profitable arbitrage opportunity arising from the discrepancy between the forward and the spot prices. When trading forward contracts, the firms, who may be risk verse, aim to maximize their total utility in both the forward and spot markets. In this thesis, however, we assume that the firms are risk neutral, so that their forward objectives are to maximize the expected profits from forward contracting and spot generation. The constraints for the producers’ decision making are the “no-arbitrage” condition relating the forward and spot prices and conditions {(3.6) − (3.12)}c∈C characterizing the anticipated spot market outcomes. Therefore, in the forward market, each firm g ∈ G solves the following stochastic mathematical program with equilibrium constraints (MPEC, see [40]), which is parameterized by the rest of the firms’ forward decisions: X

max

xgz :z∈Z

P r(c)πgc

c∈C

subject to: X X X πgc = Pic (ric + qic )qic + (hz − ucz ) xgz − Ci (qic ) i∈Ng

hz =

X

z∈Z

c∈C

i∈Ng

P r(c)ucz

z∈Z

c∈C

ucz =

X

δi Pic (ric + qic ),

z ∈ Z, c ∈ C

i:z(i)=z

ac − (qic + ric )bci − pc +

X c c (λct− Dt,i − λct+ Dt,i )=0 t∈L

X j∈N

rjc = 0

i ∈ N, c ∈ C

52 0 ≤ λcl−



X

c c Dl,j rj + Klc ≥ 0

l ∈ L, c ∈ C

j∈N

0 ≤ λcl+

⊥ Klc −

X

c c Dl,j rj ≥ 0

l ∈ L, c ∈ C

j∈N

ac − 2bci qic − bci ric − di + δi bci xgz(i) + ρci− − ρci+ = 0

i ∈ Ng , g ∈ G, c ∈ C

0 ≤ ρci−

⊥ qic ≥ 0

i ∈ N, c ∈ C

0 ≤ ρci+

⊥ q ci − qic ≥ 0

i ∈ N, c ∈ C

Combining all firms’ MPEC programs leads to an equilibrium problem with equilibrium constraints (EPEC). A solution to this EPEC is a set of the variables, including the firms’ forward and spot decisions, the ISO’s redispatch decisions, as well as the corresponding Lagrangian multiplies, at which all firms’ MPEC problems are simultaneously solved, and no market participant wants to unilaterally change its decisions in either market. In the following numerical example, we have employed the MPEC/EPEC algorithms that we will develop in chapter 6 by exploiting the special structure of the MPECs.

3.6

Numerical Results We apply the model to the stylized Belgian network described in the previous

chapter, and study economic implications of two settlements under a hypothetical two-zone oligopoly structure. In this test, nodes 1 to 32 are clustered into zone #1, and the rest of the nodes into zone #2. We compute the equilibrium outcomes for the Belgian system with a single settlement and with two settlements, and observe the firms’ forward commitment values, the behavior of the units’ output, the spot prices and social welfare, as well as the likelihood of flow congestion.

3.6.1

Forward Commitment Values We observe that, at the equilibrium of two settlements, the firms have strategic

incentives to enter the forward market, and the total forward quantities in both zones are

53

Figure 3.2: Total forward commitment with different numbers of firms positive (see tables 3.1 and 3.2 for the cases of two and four firms, respectively). However, there are some firms taking negative forward positions in a zone. This is because the transmission constraints in the spot market create spatial price variations among the nodes, and the weights used to determining the spot zonal prices are imperfect; this creates an incentive for firms operating the units at the nodes with higher prices have incentives to short forward contracts anticipating cash inflows. Of course, such gains are not sustainable in the long run. Table 3.1: Forward commitments with two firms (MW) zone zone 1 zone 2

firm 1 920.1 716.1

firm 2 2741.4 1003.2

total 3661.5 1719.3

Table 3.2: Forward commitments with four firms (MW) zone zone 1 zone 2

firm 1 -245.6 2653.9

firm 2 848.5 -304.6

firm 3 379.4 459.4

firm 4 8105.3 217.4

total 9087.6 3026.1

Next, we perform the sensitivity analysis of the total forward commitment vs.

54 market concentration. This is accomplished by dividing the units into two, three, four, five and six firms, respectively (see table 2.5). Figure 3.2 depicts the total forward commitment (relative to total market capacity) as a function of the number of firms. We find that the overall incentives for forward contracting are strengthened by a decreased market concentration.

3.6.2

Spot Prices and Social Welfare Firms’ forward contracts affect spot market outcomes. In this subsection, we focus

on the Duopoly structure to explore such effects. Two settlements alter the firms’ production patterns. We report in figure (3.3) the expected generation quantities from the units. The dark bars denote the outputs under two settlements, and the bright bars show those under a single settlement. Comparing the output of two settlements to that of a single settlement, we find that the overall output in all states of the spot market increases under two settlements. However, it is still possible for some generation units to curtail the output in some, or all, states (see figure 3.3). This is because the firms can take advantage of the network constraints so as to exercise local market power by restricting the outputs from these units while increasing the outputs from other plants. For example, the plants at nodes 14 and 24 decrease outputs in all states, whereas the plant at node 22 increases output only in the on-peak state (state 1), and decreases in all other five states. The altered production pattern in turn affects the spot energy prices. Figure 3.4 compares the expected spot nodal prices under two settlements to those under a single settlement. The prices of two settlements are drawn dark, and the overall heights show the expected spot nodal prices under a single settlement; the bright portions capture the price differences. As shown in this figure, the expected spot nodal prices under two settlements are consistently lower than those under a single settlement. This suggests that two settlements reduce the firms’ market power. For illustrative purpose, we further plot in figure 3.5 the

55 500

450

400

350

Quantity

300

250

200

150

100

50

0

7

9

10

11

14

22

24

31

32

33 35 Nodes

37

40

41

42

44

47

48

53

Figure 3.3: Expected spot generation (MW) spot nodal prices for all six states. Notice that the nodal prices are uniform in the off-peak state, but vary dramatically in the on-peak state. In table 3.3, we report the spot zonal prices under a single settlement in columns 2 and 3, and those under two settlements in column 4 and 5. The last row of this table shows the expected spot zonal prices, which, in the two-settlement system, are equal to the forward prices (due to the “no-arbitrage” relationship). Consistent with the trend of the nodal prices, the zonal prices under two settlements are lower than those under a single settlement. Two settlements increase social and consumer surpluses. For example, the expected social welfare increases from 227 k$/h under a single settlement to 357 k$/h under two settlements. This follows because the total output increases under two settlements. Note that the forward prices are equal to the expected spot zonal prices; consumers do not owe outstanding lump-sum balances for financial forward contracts, and they are better off

56

500

490

480

470

Price

460

450

440

430

420

410

400

5

10

15

20

25 30 Nodes

35

40

45

50

Figure 3.4: Expected spot nodal prices ($/MWh)

900 800 700

Price

600 500 400 300 200 100 0 1 2

50 3 State

40 4

30 20

5 6

10

Node ID

Figure 3.5: Spot nodal prices with two settlements ($/MWh)

57

Table 3.3: Spot zonal prices ($/MWh)

state 1 state 2 state 3 state 4 state 5 state 6 Expected

single settlement zone 1 zone 2 849.75 844.72 428.29 429.04 428.20 429.14 429.96 430.78 431.35 432.31 232.30 232.30 473.55 473.01

two settlements zone 1 zone 2 845.18 841.02 423.95 424.77 419.75 420.97 421.55 422.53 423.34 424.49 224.23 224.24 468.03 467.73

with two settlements. Such a welfare enhancement effect of two settlements is qualitatively consistent with that in [2]. Table 3.4: Surpluses ($/h) consumer surplus social surplus

single settlement 226640 2072200

two settlements 356710 2551700

Finally, we compare the forward quantities to the spot zonal generation for the Duopoly structure (see figure 3.6). In this figure, the left-most bars are the zonal forward commitments, and the rest are the zonal aggregated production in the states of the spot market. It shows that the forward contracts are below the firms’ outputs in the peak states and above their outputs in the off-peak states. Therefore, the firms are net suppliers in the peak states, and net buyers in the off-peak states.

3.6.3

Flow Congestion Transmission lines not congested in the single-settlement system might be con-

gested in the two-settlement system, or vice versa. This follows because the firms adjust their outputs responding to their forward commitments, and consequently, affect the ISO’s redispatch decisions, which in turn alter the flows on the transmission lines. For example, line 22-49, which is not congested under a single settlement, becomes congested in state 2

Quantity

58

3500 3000 2500 2000 1500 1000 500 0 firm 2 to zone 1 firm 2 to zone 2 firm 1 to zone 2 firm 1 to zone 1 forward

state 1

state 2

state 3

state 4

state 5

state 6

Figure 3.6: Forward commitments and spot zonal outputs (MW) through 6 under two settlements, whereas a single settlement causes the congestion on line 31-52 in states 2, 4 and 5, which is decongested under two settlements. Such effects, which may have a profound impact on prices, are complex and difficult to predict; this highlights the need for a computational approach as that are developed in this thesis.

3.7

Summary In this chapter, we model a two-settlement electricity system as a two-period

Nash-Cournot game with probabilistic contingency states in the spot market. The subgame perfect Nash equilibrium for this model is characterized as an EPEC. We assume linear demand and linear marginal generation cost functions, so the spot market equilibrium can be computed as a mixed LCP. In the forward market, firms solve MPECs subject to the “no-arbitrage” relationship and the equilibrium outcomes of the spot market. Numerical examples show that, in the presence of both flow congestion and system contingencies, the firms do have strategic incentives to enter forward contracts, which decreases spot energy

59

Table 3.5: Flow congestion state 1 1 1 1 2 2 2 2 3 3 3 3 4 4 4 4 5 5 5 5 6 6 6 6

line 22-49 29-45 30-43 31-52 22-49 29-45 30-43 31-52 22-49 29-45 30-43 31-52 22-49 29-45 30-43 31-52 22-49 29-45 30-43 31-52 22-49 29-45 30-43 31-52

single settlement uncongested congested congested congested uncongested uncongested congested congested uncongested uncongested congested breakdown uncongested uncongested congested congested uncongested uncongested congested congested uncongested uncongested uncongested uncongested

two settlements uncongested congested congested congested congested uncongested congested uncongested congested uncongested congested breakdown congested uncongested congested uncongested congested uncongested congested uncongested congested uncongested uncongested uncongested

60 prices and increases social and consumer surplus.

61

Chapter 4

Price Caps 4.1

Introduction Concentrated electricity markets endow producers with substantial locational mar-

ket power. It is clear that, without some sort of price ceiling, on-peak prices in most markets could rise too high. Moreover, this is extremely severe in the areas where the capability of importing electricity is limited. Hence, local unregulated wholesale suppliers can raise prices to an artificially high level, and consumers have to pay for the unreasonably high charge. As a result, regulators in many restructured electricity markets have imposed price or bid caps in the spot markets in an attempt to restrain price spikes. Price caps can also rectify the market imperfections due to demand inelasticity and imperfect capital markets. Determining the “right” level of the spot price cap, if any, has been a subject of debate among economists, policy makers and steak holders in electricity markets. If the price cap is set too low, in the short term it will discourage the production from high-cost plants and demand response, and in the long run it will lead to disinvestments and shortages in the industry as producers are unable to cover their capital cost. On the other hand, if the price cap is set too high, the exercise of market power will cause significant wealth transfers from consumers to producers. Furthermore, artificially high prices raised by the exercise of

62 market power, rather than actual scarcities, can stimulate inefficient entry. Competitive entry in the forward market is another mean of market power mitigation. For long-term forward contracts, potential competitive entry imposes an implicit price cap on forward contract prices because new investment in generation capacity will occur when forward prices rise above the amortized long-run cost. (In contrast, the models in Chapters 3 and 5 assume a fixed generation stock which is for a two-settlement system over a short or medium time span.) The threat of entry encourages the competitive behavior of the incumbents, and actual entry reduces market concentration – although the possibility of entry alone cannot alleviate all concerns about market power abuse. In this chapter, we study the impact of forward and spot price caps on energy prices and consumer benefits. In particular, we will address the following questions. To what extent do the generators commit forward contracts under price caps? How do caps affect the spot and forward prices? What is the relative impact of spot caps versus competitive entry in mitigating market power? We explore these questions via the two-settlement Cournot equilibrium framework developed in the previous chapter, and extend it with the forward and spot price caps. We present the model as an equilibrium problem with equilibrium constraints (EPEC), where each generation firm solves a mathematical program with equilibrium constraints (MPEC, see [40]). The remainder of this chapter is organized as follows. The next section presents the formulation for the two-settlement markets with price caps. In section 4.3, we investigate the implications of hypothetical price caps in the stylized Belgian electricity system. Some remarks are drawn to conclude this chapter.

4.2

The Model We introduce a generic two-period Nash-Cournot model of the two-settlement

electricity system with both spot and forward price caps. The system with either a spot or

63 a forward price cap can be treated as a special case of this generic model with the other cap relaxed.

4.2.1

Pricing Schemes with Price Caps With price caps, the three locational price schemes, spot nodal prices, spot zonal

(settlement) prices and forward prices, should be also be capped. When no spot price cap exists, the nodal consumption qic +ric at node i ∈ N should be priced at Pic (qic + ric ) according to the inverse demand function (IDF) Pic (·) at node i. Whenever a spot price cap does exist, the actual spot nodal prices pˆci are expressed as pˆci = min{¯ u, Pic (qic + ric )},

i ∈ N,

where u ¯ is the spot price cap. Consequently, the spot zonal price ucz of a zone z ∈ Z is given by ucz =

X

δi pˆci ,

z ∈ Z.

i:z(i)=z

Here, {δi }i∈N are the predetermined weights, such as historical load ratios, to settle the spot zonal prices, and z(i) denotes the zone containing node i. Similar to Model T1, the current model assumes that, in an equilibrium, profitable arbitrage is impossible between the forward and the spot zonal prices. Thus, the forward zonal price hz in zone z ∈ Z is equal to the expected spot zonal price, that is, hz =

X

P r(c)ucz ,

z ∈ Z,

c∈C

¯ where P r(c) is the probability of state c in the spot market. With a forward price cap h, an upper bound is imposed on these forward prices: ¯ hz ≤ h,

z ∈ Z.

As discussed earlier, this upper bound typically reflects a backstop price at which new entry becomes ex ante profitable.

64

4.2.2

The Spot Market We shall now be ready to derive our model starting from the spot market formu-

lation. The Generic Spot Model In each state c ∈ C of the spot market, each generation firm g ∈ G determines the output quantities {qic }i∈Ng of its generation units. These quantities are confined within the range determined by the minimal and maximal possible output of the plants in that state: 0 ≤ qic ≤ q ci

i ∈ Ng .

Firm g ∈ G receives the payments for its generation at the spot nodal prices, possibly capped, and settles its forward contracts at the spot zonal prices. Its spot market profit πgc is given by πgc =

X

qic pˆci −

i∈Ng

X

ucz xg,z −

z∈Z

X

Ci (qic ),

i∈Ng

where term Ci (·) stands for the generation cost of the unit at node i. Thus, it solves the following program in each state c of the spot market: max

qic :i∈Ng

πgc

subject to: X X X πgc = qic pˆci − ucz xg,z − Ci (qic ). i∈Ng

z∈Z

i∈Ng

pˆci = min{¯ u, Pic (qic + ric )}, X ucz = δi pˆci , z ∈ Z i:z(i)=z

hz =

X

P r(c)ucz ,

c∈C

¯ z∈Z hz ≤ h

z∈Z

i ∈ Ng

65 0 ≤ qic ≤ q ci ,

i ∈ Ng

Given the production decisions by the generators, the independent system operator (ISO) determines the adjustment {ric }i∈N at the nodes for each state c ∈ C in the spot market. These adjustments are subject to the thermal constraints on the power flow, which is computed in terms of the power transfer distribution factors based on a Direct Current (DC) approximation of the Kirchhoff’s law (see [10]). Mathematically, the network feasibility constraints are −Klc ≤

X

c c Dl,i ri ≤ Klc ,

l ∈ L.

i∈N

The ISO also maintains the real time balance of loads and generations, that is, X

ric = 0.

i∈N

Note that price caps do not affect consumers’ willingness-to-pay; in state c, the ISO’s social-welfare-maximization problem should be the same as that in the previous chapter: max

ric :i∈N

X µZ 0

i∈N

ric +qic

¶ Pic (τi )dτi



Ci (qic )

subject to: X ric = 0 i∈N

− Klc ≤

X

c c Dl,i ri ≤ Klc ,

l∈L

i∈N

Spot Market Smooth Formulation The firms and the ISO’s decision problems in the spot market do not have straightforward optimality conditions due to the non-smooth functions characterizing the spot prices. In this section, we reformulate these problems by removing the “min” operator for the capped spot nodal prices. It is accomplished by considering two separate cases.

66 In the first case, the spot price cap exceeds the price intercepts of the IDFs, so that the spot price cap cannot be binding. Thus, we have the spot nodal prices as pˆci = Pic (qic + ric ),

i ∈ N.

This implies the spot zonal prices are X

ucz =

δi Pic (qic + ric ),

z ∈ Z.

i:z(i)=z

Consequently, the firms’ spot decision problems become Ggc :

πgc

max

qic :i∈Ng

subject to: X X X πgc = Pic (qic + ric )qic − uzc xg,z − Ci (qic ). i∈Ng

z∈Z

X

ucz =

δi Pic (qic + ric ),

i∈Ng

z∈Z

i:z(i)=z

hz =

X

P r(c)ucz ,

z∈Z

c∈C

¯ hz ≤ h,

z∈Z

0 ≤ qic ≤ q ci ,

i ∈ Ng

and the ISO’s decision problem is Sc :

max c

ri :i∈N

X µZ i∈N

0

ric +qic

¶ Pic (τi )dτi

subject to: X ric = 0 i∈N

− Klc ≤

X

c c Dl,i ri ≤ Klc ,

l∈L

i∈N

When the spot price cap is below the price intercepts of IDFs, it could be binding (see figure 4.1).

Note that the standard technique of relaxing the equality pˆci =

67 Price pc Inverse demand function

Spot price cap

u

Marginal cost di 0

vic

Quantity

Figure 4.1: Spot price cap min{¯ u, Pic (qic + ric )} with two inequality constraints, pˆci ≤ u ¯ and pˆci ≤ Pic (qic + ric ), may not work here because the sign of the terms containing pˆci in the firms’ objective functions may be positive or negative, depending on the relative magnitudes of xg,z and qic . To overcome this complication, we introduce auxiliary variables vic to represent min{qic + ric , v¯ic }, where 0 ≤ vic ≤ qic + ric , and vic ≤ v¯ic =

ac − u ¯ , bi

and rewrite the spot nodal prices (considering the linearity of the demand functions) as pˆci = u ¯ − (qic + ric − vic )bi . However, expressing the spot nodal prices in such a way is correct only if vic = min{qic + ric , v¯ic }. To guarantee this, we reformulate the ISO’s problem as: Ã ! Z v¯c +rc +qc −vc X Z vic i i i i c c c Sˆ : max P (τi )dτi + P (τi )dτi ric ,vic

i∈N

subject to:

0

i

v¯ic

i

68

pc

Price Inverse demand function

Price cap u

vic qic + ric vic vic + qic + ric − vic Load

Figure 4.2: The ISO’s objective when qic + ric ≤ v¯ic X

ric = 0

(4.1)

i∈N

X

c c Dl,i ri , ≥ −Klc ,

l∈L

(4.2)

i∈N

X

c c Dl,i ri ≤ Klc ,

l∈L

(4.3)

i∈N

vic ≥ 0, vic ≤ v¯ic ,

i∈N

(4.4)

i∈N

ric + qic − vic ≥ 0,

(4.5) i∈N

(4.6)

The two components of the ISO’s objective function are illustrated in figures 4.2 and 4.3. Notice that, due to the difference of the heights of the two intervals, the maximization of the ISO’s objective function will assign a positive load (qic + ric − vic ) in the second interval (beyond v¯ic ) only if the load in the first interval reaches its limit of v¯ic . Thus, at the optimal solution, vic = v¯ic if qic + ric ≥ v¯ic , and vic = qic + ric if qic + ric < v¯ic . In other word, the optimal solution successfully sets vic = min{¯ vic , qic + ric }.

69

Price

pc

Inverse demand function Price cap

u

Load vic

vic qic + ric

vic + qic + ric − vic

Figure 4.3: The ISO’s objective when qic + ric > v¯ic Each firm g’s spot profit-maximization program can now be written as Gˆgc :

πgc

max

qic :i∈Ng

subject to: X X X πgc = (¯ u − (qic + ric − vic )bi )qic − ucz xg,z − Ci (qic ) i∈Ng

z∈Z

X

ucz =

(¯ u − (qic + ric − vic )bi )δi ,

i∈Ng

z∈Z

i:z(i)=z

hz =

X

P r(c)ucz ,

z∈Z

c∈C

¯ hz ≤ h,

z∈Z

(4.7)

qic ≥ 0,

i ∈ Ng

(4.8)

qic ≤ q ci ,

i ∈ Ng

(4.9)

It is worth noting that, if {¯ vic }i∈N are set to zero, programs Gˆgc and Sˆc also capture the generation firms and the ISO’s programs when the spot price cap is higher than the vic }i∈N fully price intercepts of IDFs. Therefore, problems Gˆgc and Sˆc along with suitable {¯ represent the agents’ spot market decision problems.

70 Spot Market Equilibrium Conditions To obtain the simultaneous solutions to the firms’ and the ISO’s spot market problems (Gˆgc and Sˆc ), we combine their KKT conditions into one problem. In the case of linear IDFs (Pic (q) = ac − bci q), and quadratic generation costs (Ci (q) = di q + 12 si q 2 ), all the problems involved are strictly concave-maximization programs, and their global solutions provided by the KKT conditions characterize the spot market equilibrium outcomes. In the spot market equilibrium, for each firm, the shadow prices (the Lagrangian multipliers) corresponding to constraint (4.7) for each zone must be equal across all states for each firm. If not, the firm could increase its profit by curtailing its output in the states with lower shadow prices while increasing its output in the states with higher shadow prices. Moreover, all firms should face equal shadow prices of this constraint in the same zone because these shadow prices reflect the marginal profit of new entry via the forward price caps. Mathematically, a single Lagrangian multiplier ηz is assigned to (4.7) for a zone z ∈ Z. c , β c , µc , ρc and ρc be the Lagrangian multipliers correspondLet pc , λcl− , λcl+ , βi− i+ i i− i+

ing to constraints (4.1)-(4.6) and (4.8)-(4.9), respectively; we obtain the following equilibrium conditions in the spot market, which is a combination of the KKT conditions for the firms and the ISO’s program: X

rjc = 0

(4.10)

j∈N

ac − bi vic − u ¯ + (ric + qic − vic )bi − pc + µci +

X

c (λcm− − λcm+ )Dm,i =0

i∈N

(4.11)

i∈N

(4.12)

i∈N

(4.13)

l∈L

(4.14)

m∈L c c u ¯ − (ric + qic − vic )bi + βi− − βi+ − µci = 0

u ¯ − 2bi qic − (ric − vic )bi − di − si qic + δi bi xg,z(i) + ρci− − ρci+ + P r(c)δi bi ηz = 0 X c c 0 ≤ λcl− ⊥ Klc + Dl,i rj ≥ 0 j∈N

71 0 ≤ λcl+

⊥ Klc −

X

c c Dl,i rj ≥ 0

l∈L

(4.15)

j∈N c 0 ≤ βi−

⊥ vic ≥ 0

i∈N

(4.16)

c 0 ≤ βi+

⊥ v¯ic − vic ≥ 0

i∈N

(4.17)

0 ≤ µci

⊥ qic + ric − vic ≥ 0

i∈N

(4.18)

0 ≤ ηz

¯ − hz ≥ 0 ⊥ h

z∈Z

(4.19)

0 ≤ ρci−

⊥ qic ≥ 0

i∈N

(4.20)

0 ≤ ρci+

⊥ q ci − qic ≥ 0

i∈N

(4.21)

4.2.3

The Forward Market As discussed earlier, we assume that the forward contracts are settled financially.

Each firm g ∈ G takes all its competitors’ forward quantities as given, and determines its own forward quantities so as to maximize its expected utility. When the firms are risk neutral, their forward objectives are to maximize their profits from both the forward contracts and the spot generation, subject to the preceding spot market KKT conditions {(4.10) − (4.21)}c∈C . Thus, for firm g, its optimization problem in the forward market is the following MPEC problem: max xg,z

X

hz xgz +

z∈Z

X

P r(c)πgc

c∈C

subject to: xgz ∈ Xgz X X X πgc = (¯ u − (qic + ric − vic )bi )qic − ucz xg,z − Ci (qic ) i∈Ng

ucz =

X

z∈Z

(¯ u − (qic + ric − vic )bi )δi

z∈Z c∈C

i∈Ng

z ∈ Z, c ∈ C

i:z(i)=z

hz =

X c∈C

P r(c)ucz

z∈Z

72 X

rjc = 0

c∈C

j∈N

ac − bi vic − u ¯ + (ric + qic − vic )bi − pc + µci +

X

c (λcm− − λcm+ )Dm,i =0

i ∈ N, c ∈ C

m∈L c c u ¯ − (ric + qic − vic )bi + βi− − βi+ − µci = 0

i ∈ N, c ∈ C

u ¯ − 2bi qic − (ric − vic )bi − di − si qic + δi bi xg,z(i) + ρci− − ρci+ + P r(c)δi bi ηz = 0 X c c Dl,i rj ≥ 0 0 ≤ λcl− ⊥ Klc +

i ∈ N, c ∈ C l ∈ L, c ∈ C

j∈N

0 ≤ λcl+

⊥ Klc −

X

c c Dl,i rj ≥ 0

l ∈ L, c ∈ C

j∈N c 0 ≤ βi−

⊥ vic ≥ 0

i ∈ N, c ∈ C

c 0 ≤ βi+

⊥ v¯ic − vic ≥ 0

i ∈ N, c ∈ C

0 ≤ µci

⊥ qic + ric − vic ≥ 0

i ∈ N, c ∈ C

0 ≤ ηz

¯ − hz ≥ 0 ⊥ h

z ∈ Z, c ∈ C

0 ≤ ρci−

⊥ qic ≥ 0

i ∈ N, c ∈ C

0 ≤ ρci+

⊥ q ci − qic ≥ 0

i ∈ N, c ∈ C

Combining the firms’ MPEC problems, the equilibrium problem in the two-settlement markets is an EPEC, where the spot market equilibrium conditions form the lower-level problem that is shared by all MPECs.

4.3

Numerical Results We use the stylized Belgian electricity network to test the model and to illustrate

the numerical results (these results are obtained through the algorithm developed in Chapter 6). We consider five test cases: • Case 1: The price-uncapped single-settlement system, i.e. a single settlement without

73 a (spot) price cap. • Case 2: The price-capped single-settlement system, i.e. a single settlement with a (spot) price cap. • Case 3: The price-uncapped two-settlement system, i.e. two settlements without either a spot or a forward price cap. • Case 4: The forward-capped two-settlement system, i.e. two settlements with only a forward price cap. • Case 5: The spot-capped two-settlement system, i.e. two settlements with only a spot price cap. Note that cases 1 and 2 are indeed equivalent to a two-settlement system with the allowable forward commitments confined to zeros. We first run the test cases with a fixed number of firms and with fixed price caps. We assume a Duopoly structure with two zones, where nodes 1 through 32 are in the first zone and the remaining nodes in the second zone. We also assume the spot price cap in cases 2 and 5 is 600$/MWh, and the forward price cap in case 4 is 425$/MWh. Table 4.1: Spot zonal prices of zone 1 ($/MWh) state 1 state 2 state 3 state 4 state 5 state 6 Expected

Case 1 428.29 849.75 232.30 428.20 429.96 431.35 473.55

Case 2 428.29 600.00 232.30 428.20 429.96 431.35 423.56

Case 3 423.95 845.18 224.24 419.75 421.55 423.34 468.03

Case 4 363.30 823.53 189.45 405.46 407.19 409.34 425.00

Case 5 425.27 600.00 229.46 424.38 426.11 428.77 421.19

We find that, when both the spot and the forward prices are not capped, the generation firms have the incentives for forward contracting. Moreover, two settlements

74

Table 4.2: Spot zonal prices of zone 2 ($/MWh) state 1 state 2 state 3 state 4 state 5 state 6 Expected

Case 1 429.04 844.72 232.30 429.14 430.78 432.31 473.01

Case 2 429.04 600.00 232.30 429.14 430.78 432.31 424.07

Case 3 424.79 841.02 224.24 420.97 422.53 424.49 467.73

Case 4 363.03 823.75 189.45 406.32 407.93 410.47 425.00

Case 5 425.60 600.00 229.46 424.56 426.12 428.14 421.34

increase the spot generation, decrease the spot prices (see tables 4.1 and 4.2), and thus increase the social welfare. This result is consistent with those in chapter 3 and in [2, 3, 37]. Table 4.3: Total spot generation (MW) state state state state state state

1 2 3 4 5 6

Case 1 4153 8813 1049 4153 4051 3971

Case 2 4153 9694 1049 4153 4051 3971

Case 3 4440 9089 1549 4667 4568 4460

Case 4 7893 10173 3500 5443 5345 5211

Case 5 4406 11014 1524 4644 4545 4437

When the price caps are binding, they create opposite effects on the firms’ incentives for forward commitments. The spot price cap reduces the firms’ incentives to commit forward contracts when compared to the uncapped two-settlement system. In our test example, the spot price cap causes the firms to commit totally about 90% of the corresponding forward contract in the uncapped two-settlement markets. This observation can be explained in terms of the positive correlation between the expected spot output and the forward commitments. Spot price caps cause the firms to increase the productions in the on-peak states when the cap is binding, and to offset their profit losses by cutting off the output and raising the prices in the states when the caps are not binding. The net effect is a reduction in the expected output in the spot market and, consequently, a reduction in the forward commitments.

75 Forward price caps, on the other hand, increase the firms’ incentives for forward contracting. In our simulation, the firms’ commitments in the forward-capped twosettlement markets are about 80% higher than those in the same markets without caps. This observation is appealing because it is consistent with the intuition that competitive entry will trigger more forward contracting by incumbents in order to deter such entry. Thus, it validates the use of forward caps as a proxy for competitive entry. A more direct explanation, again, is based on the positive correlation between the expected output and the forward commitments. Unlike spot caps that confine the prices only in on-peak states, forward caps reduce the spot prices in all states through the no-arbitrage relationship. Such price reduction is accomplished only through an increase of the productions in all the states, which results in an increase in both the spot outputs and the forward commitments. Different incentives yielded from the forward and the spot caps further imply different results of the spot productions and prices as compared to the uncapped two-settlement system. Tables 4.1, 4.2 and 4.3 report the spot zonal prices and the spot generation quantities under different cases. Next, we run more tests to find the sensitivity of forward contracting on the number of firms. We keep as constants the price caps in cases 4 and 5, but vary the number of firms in the system from two to six. We find that the total forward contracting is greater with more firms. This is true for all types of two-settlement systems, whether or not the forward or the spot prices are capped. Such increase in forward contracting can be explained by the fact that the prisoners’ dilemma effect, as an incentive to drive forward contracting, is amplified as the number of players increases. Figure 4.4 shows the relationship between the total contracting quantity as a portion of the total market capacity and the number of firms in the market. Again, it is shown that the spot-capped system results in less forward contracting than the uncapped system, and the latter, in turn, causes less forward contracting than the forward-capped system.

76 1

0.9

0.8

0.7

0.6

0.5 uncapped spot capped forward capped 0.4

2

3

4

5

6

Figure 4.4: Comparison of total forward contracting (MW) Finally, we examine the forward contracting at different levels of the spot and forward price caps. Figures 4.5 and 4.6 plot the total forward contracting with respect to the price caps and the number of firms. With all other parameters fixed, we find that the lower the spot cap is, the less firms are willing to commit in the forward market. On the other hand, the total forward commitment grows as the forward price cap decreases. For a given spot or forward cap, the total forward contracting quantity raises as the number of firms increases. Note that the forward prices are decreasing functions of both the forward commitments and the spot generation quantities; the firms will succeed in deterring potential entry by playing an equilibrium with the forward prices just below the forward cap if they have sufficient capacities and the forward price cap is high enough. However, when the forward price cap is too low or the firms do not have enough capacity, the forward prices will attempt to rise above the forward price cap, and entry to the markets is inevitable as noted by [42].

77

1 0.9

contracting

0.8 0.7 0.6 0.5 0.4 0.3 6 900

5 800

4 number of firms 3

600 2

700 spot price cap

500

Figure 4.5: Total forward contract with spot caps (MW)

1 0.9

contracting

0.8 0.7 0.6 0.5 0.4 6 425

5 4 450 3 number of firms

2

475

forward price cap

Figure 4.6: Total forward contract with forward caps (MW)

78

4.4

Summary This chapter develops a two-period Nash-Cournot model for analyzing the capping

alternatives under a variety of scenarios in the framework of two-settlement markets. In doing so, we extend the two-settlement model in chapter 3 to the case in which either the forward prices or the spot prices are capped. We formulate the Cournot equilibrium in the price-capped two-settlement markets as a stochastic EPEC problem. We run test cases based on the stylized Belgian electricity market. The resulting equilibrium reveals fewer incentives for the firms to commit forward contracts due to spot price caps, but more incentives under forward price caps induced through competitive entry. However, the spot zonal prices, under both cap types, still decrease when a forward market is available. Sensitivity studies show that the results are robust and that forward contracting increases as the forward price cap decreases or as the spot cap increase. We also find that the total forward contracting and the sensitivities are amplified as the markets become less concentrated.

79

Chapter 5

The Second Two-settlement Model: Model T2 5.1

Introduction In this chapter, we extend the second spot market model, Model S2, to two set-

tlements. Similar to Model T1, the current model views two-settlement markets as a twoperiod Nash-Cournot game, in which the forward and the spot markets form the two periods. We present the Cournot equilibrium for this model as an equilibrium problem with equilibrium constraints (EPEC), which consists of all firms’ mathematical programs with equilibrium constraints (MPECs, see [40]). Because the current model follows a modeling approach similar to that in Model T1, we will discuss minimally its overlap with Model T1, but focus on the differences.

5.2

The Spot Market As in Model T1, the spot market equilibrium in this model is characterized by the

solutions to the combined optimality conditions for both the firms and the ISO’s programs.

80 However, unlike model T1 where the ISO’s strategic variables are its redispatches, the current model assumes these variables to be locational price premiums.

5.2.1

The ISO’s Problem Recall that, in each state c ∈ C, the ISO conducts the imports/exports {ric }i∈N

at the nodes so as to maximize social surplus while obeying the energy balance constraint (5.1) and the flow feasibility constraints (5.2)-(5.3): max

X µZ

ric :i∈N

ric +qic

0

i∈N

¶ Pic (τi )dτi



Ci (qic )

subject to: X ric = 0

(5.1)

i∈N

X

c c Dl,i ri ≥ −Klc ,

l∈L

(5.2)

i∈N

X

c c Dl,i ri ≤ Klc ,

l∈L

(5.3)

i∈N

Here, {qic }i∈N are the firms’ productions treated as parameters, {Pic (·)}i∈N are the nodal inverse demand functions, and {Ci (·)}i∈N are the generator cost functions. Let pc , λcl− and λcl+ be the Lagrangian multipliers corresponding to constraints (5.1)-(5.3), then the Karush-Kuhn-Tucker (KKT) conditions for the ISO’s problem are: Pic (qic + ric ) − pc +

X

c c (λcm− Dm,i − λcm+ Dm,i ) = 0,

i∈N

(5.4)

m∈L

X

rjc = 0

j∈N

0 ≤ λcl−



X

c c Dl,j rj + Klc ≥ 0,

l∈L

j∈N

0 ≤ λcl+

⊥ Klc −

X

c c Dl,j rj ≥ 0,

l∈L

j∈N

Let ϕci = −

P m∈L

c −λc D c ), condition (5.4) implies that the nodal prices (λcm− Dm,i m+ m,i

81 are Pic (qic + ric ) = pc + ϕci ,

i ∈ N.

Thus, qic + ric = (Pic )−1 (pc + ϕci ) ,

i ∈ N.

Aggregating this expression over all nodes, we have X

(qjc + rjc ) =

j∈N

X

¡ ¢ (Pic )−1 pc + ϕcj ,

j∈N

or X

X

qjc =

j∈N

¡ ¢ (Pic )−1 pc + ϕcj .

(5.5)

j∈N

Hence, pc can be viewed as the energy price at the reference bus, and {ϕci }i∈N as the locational price premiums. This equation denotes the demand of the entire network evaluated at the reference bus.

5.2.2

The Firms’ Problems In each state c ∈ C, the firms maximize their profits with respect to the residual

demand of (5.5). Each firm g ∈ G determines the output {qic }i∈N g of its units so as to affect the energy price pc at the reference bus (taking {ϕci }i∈N as given). Its generation at each node i ∈ Ng is paid at the nodal price pc + ϕci , whereas its forward contracts xgz in each zone z ∈ Z are settled at the spot zonal price ucz . Here, ucz =

X

(pc + ϕci ) δi ,

z ∈ Z,

i:z(i)=z

with given weights {δi }i∈N that are exogenous to our model. Mathematically, firm g solves the following problem parametric on its competitors’ decisions {qjc }j∈N \Ng and the locational price premiums {ϕci }i∈N : max c

qi :i∈Ng ,pc

X i∈Ng

(pc + ϕci ) qic −

X z∈Z

ucz xgz −

X i∈Ng

Ci (qic )

82

subject to: X ucz = (pc + ϕci ) δi ,

z∈Z

i:z(i)=z

qic ≥ 0,

i ∈ Ng

(5.6)

qic ≤ q ci , i ∈ Ng X X ¡ ¢ qjc = (Pic )−1 pc + ϕcj . j∈N

(5.7) (5.8)

j∈N

Let ρci− , ρci+ and βgc be the Lagrangian multipliers corresponding to (5.6)-(5.8), then the KKT conditions for this program are: dCi (qic ) + ρci− − ρci+ = 0 dqic ³ ´ −1 pc + ϕcj X d (Pic ) X X c βgc + q − δi xgz(i) = 0 j dpc pc + ϕci − βgc −

j∈N

X

qjc =

j∈N g

X

i ∈ Ng

i∈Ng

¡ ¢ (Pic )−1 pc + ϕcj

j∈N

j∈N

0 ≤ ρci−

⊥ qic ≥ 0

i ∈ Ng

0 ≤ ρci+

⊥ q ci − qic ≥ 0

i ∈ Ng

Here, the first two conditions are the derivatives of the Lagrangian function with respect to qic and pc , respectively.

5.2.3

Spot Market Equilibrium Conditions Aggregating the KKT conditions for the ISO and the firms’ programs leads to the

market equilibrium conditions, which form a mixed nonlinear complementarity problem. When both the nodal demand functions and the marginal cost functions are linear: Pic (q) = ac − bci q,

i ∈ N,

1 Ci (q) = di q + si q 2 , 2

i ∈ N,

83 these conditions become the mixed linear complementarity problem (mixed LCP, see [12]) given below: pc + ϕci − βgc − di − si qic + ρci− − ρci+ = 0 P ϕj P c j∈N bcj j∈N qj c c p =a − P − P 1 1

i ∈ Ng , g ∈ G

(5.9)

g∈G

(5.10)

c c (λcm− Dm,i − λcm+ Dm,i )

i∈N

(5.11)

X 1 X X + qjc − δi xgz(i) = 0 c bj

g∈G

(5.12)

i∈N

(5.13)

i∈N

(5.14)

j∈N bcj

ϕci = −

X

j∈N bcj

m∈L

− βgc

j∈N

0 ≤ ρci−

j∈N g

i∈Ng

⊥ qic ≥ 0

0 ≤ ρci+ ⊥ q ci − qic ≥ 0 X rjc = 0

(5.15)

j∈N

X

ac − (qic + ric )bci − pc +

c c (λcm− Dm,i − λcm+ Dm,i )=0

i∈N

(5.16)

l∈L

(5.17)

l∈L

(5.18)

m∈L

0 ≤ λcl−



X

c c Dl,j rj + Klc ≥ 0

j∈N

0 ≤ λcl+

⊥ Klc −

X

c c Dl,j rj ≥ 0

j∈N

5.3

The Forward Market In the forward market, risk-neutral arbitrageurs eliminate any profitable opportu-

nity raised from the disparity between the forward zonal prices and the expected spot zonal settlement prices (the so-called “no-arbitrage” condition). Thus, the forward market price (hz ) in each zone z is determined by hz =

X

P r(c)ucz ,

z ∈ Z.

c∈C

The risk-neutral firms simultaneously determine their forward contracts {xgz }g∈G,z∈Z so as to maximize the total profits from both the forward contracts and the spot generations, while anticipating the forward commitments from all other firms as well as the spot

84 market equilibrium conditions {(5.9) − (5.18)}c∈C . The equilibrium problem in the forward market is an EPEC in which each firm g ∈ G solves the following MPEC: X

max

xgz :z∈Z

hz xgz +

z∈Z

X

P r(c)πgc

c∈C

subject to: X X X πgc = (pc + ϕci )qic − ucz xgz − Ci (qic ) i∈Ng

hz =

X

z∈Z

c∈C

i∈Ng

P r(c)ucz

z∈Z

c∈C

ucz =

X

(pc + ϕci )δi ,

z ∈ Z, c ∈ C

i:z(i)=z

pc + ϕci − βkc − ci − di qic + ρci− − ρci+ = 0 P ϕj P c j∈N bcj j∈N qj c c − p =a − P P 1 1 j∈N bcj

ϕci = −

X

i ∈ Nk , k ∈ G, c ∈ C c∈C

j∈N bcj

c c (λcm− Dm,i − λcm+ Dm,i )

i ∈ N, c ∈ C

m∈L

− βkc

X 1 X X c + q − δi xkz(i) = 0 j bcj

j∈N

0 ≤ ρci−

j∈Nk

k ∈ G, c ∈ C

i∈Nk

⊥ qic ≥ 0

i ∈ N, c ∈ C

0 ≤ ρci+ ⊥ q ci − qic ≥ 0 X rjc = 0

i ∈ N, c ∈ C

j∈N

X

ac − (qic + ric )bci − pc +

c (λcm− − λcm+ )Dm,i =0

i ∈ N, c ∈ C

m∈L

0 ≤ λcl−



X

c c Dl,j rj + Klc ≥ 0

l ∈ L, c ∈ C

j∈N

0 ≤ λcl+

⊥ Klc −

X j∈N

c c Dl,j rj ≥ 0

l ∈ L, c ∈ C

85

5.4

Numerical Results We apply the model to the stylized Belgian network. In particular, we consider two

different generator ownership structures with two zones: nodes 1..32 belong to zone #1, and the rest to zone #2. The first structure is composed of two firms, and the second structure of three firms (see Section 2.6). Tables 5.1 and 5.2 report the forward commitments at the two-settlement equilibrium of the two- and three-firm structures, respectively. Table 5.1: Forward commitments with two firms (MW) zone 1 2

firm 1 -1225.3 673.1

firm 2 837.1 910.6

Table 5.2: Forward commitments with three firms (MW) zone 1 2

firm 1 3900.7 3474.5

firm 2 701.4 772.3

firm 3 -513.6 480.9

We observe that, under both resource structures, firms have strategic incentives for forward contracting. While some firms in our example have taken short forward positions, the total forward commitment in the entire market is positive. We plot in figure 5.1 the expected spot nodal prices under two settlements and contrast them with the corresponding nodal prices in the equilibrium of a single-settlement market which is obtained by constraining all firms’ forward positions to zeros. We first notice that, whether or not there is a forward market, the three-firm structure yields lower spot prices than the Duopoly structure, as one would expect. Moreover, under both the two- and three-firm structures, with few exceptions, a two-settlement equilibrium results in lower spot prices at most nodes than a single settlement. However, nodes #29 and #31 do not follow this trend due to the transmission constraints. Consequently, two settlements increase social welfare and consumer surplus. These results suggest that the

86

Single Settlement

Two Settlements

Figure 5.1: Expected spot nodal prices ($/MWh) welfare-enhancing effect described in [2] and [3] generalizes to the case with flow congestion, system contingency and demand uncertainty.

5.5

Summary In this chapter, we extend the spot market model S2 to two settlements. We

present this two-settlement model as an EPEC, which comprises the individual firms’ MPECs. We apply the model to the stylized Belgian network and observe the welfareenhancement effect of forward contracting. Such effect is qualitatively consistent with, but quantitatively different from, that resulting from Model T1.

87

Chapter 6

The Algorithms The two-settlement models developed in the previous chapters are the so-called equilibrium problems with equilibrium constraints (EPECs). These problems comprise a set of mathematical programs with equilibrium constraints (MPECs), which share the lower-level linear complementarity constraints. Such EPECs are hard to solve because the non-convexities in both complementarity constraints and the MPECs’ objective functions. In this chapter, we will develop special-purpose MPEC/EPEC algorithms for solving these EPECs. The MPEC/EPEC algorithms that we will develop are associated with the concepts of the linear complementarity problem (LCP), MPEC and EPEC, and with complementarity problems in general (Detailed introductions and discussions of LCP and MPEC are available in [12] and [40], respectively). Below, we first briefly introduce some concepts related to complementarity problems, then compact the notions of our models, and finally describe the algorithms. Numerical examples are provided at the end.

88

6.1 6.1.1

Complementarity Problem Linear Complementarity Problem Given a matrix M ∈ Rn×n and a vector q ∈ Rn , an LCP problem LCP (q, M ) is

to find vector w, z ∈ Rn such that w = q + Mz 0≤w

⊥ z ≥ 0.

The LCP problem with a positive-semi-definite (PSD) matrix is closely related to convex quadratic programming. Consider the following program min x

1 cT x + xT Qx 2

subject to Ax ≥ b x ≥ 0. with given c ∈ Rn , Q ∈ Rn×n , A ∈ Rm×n and b ∈ Rm . If vector x ∈ Rn is a local minimizer for this problem, there must exit a vector y ∈ Rm such that the following Karush-KuhnTucker (KKT) conditions are satisfied: Ax ≥ b, x ≥ 0, c + Qx − AT y ≥ 0, y ≥ 0, y T (Ax − b) = 0, xT (c + Qx − AT y) = 0. If we define



 AT y

 c + Qx − w= Ax − b

 

89 and





 x  z =  , y then the preceding KKT conditions become an LCP problem LCP (q, M ) with respect to (w, z) where 



 c  q= , −b and



  Q M = A

−AT

 .

0

Moreover, the above KKT conditions are also sufficient for the (global) optimality when Q is PSD.

6.1.2

Mathematical Program with Equilibrium Constraints An MPEC is an optimization problem with two variable vectors, x ∈ Rl and

y ∈ Rm , in which some or all of its constraints are defined by a parametric variational inequality or complementarity system with y as the primary variables and x as the parameter vector. Specifically, suppose that f : Rl+m → R and F : Rl+m → Rm are given functions, Z ⊆ Rl+m is a non-empty closed set, and C : Rl → Rm is a set-valued map (possibly empty), the MPEC is defined as: min x,y

f (x, y)

subject to: (x, y) ∈ Z y ∈ S(x) where S(x) is the solution set of the variational inequality defined by the pair (F (x), C(x)), i.e. y ∈ S(x) if and only if y ∈ C(x) and (v − y)T F (x, y) ≥ 0 for all v ∈ C(x). The

90 monograph by Luo, Pang and Ralph [40] gives a comprehensive study of MPEC problems and presents theorems regarding first- and second-order optimality conditions. There has been growing literature on this type of important mathematical programs. In particular, [40] describes some iterative algorithms, such as the penalty interior point algorithm (PIPA) and the piecewise sequential quadratic programming (PSQP) algorithm. Other early contributions are contained in [20, 21, 22, 44, 45]. More recent advances in MPEC algorithms can be found, for example, in [11, 14, 24, 25, 26, 35, 47]. Facchinei, Jiang and Qi [14] introduce an algorithm for MPEC problems with strong monotone variational inequality constraints. This algorithm approximates the nonsmooth one-level minimization constraints of an MPEC problem by a series of smooth, regular problems. They show that every sequence of the global (stationary) solutions to the re-formulated problems converges to the global (stationary) solution to the MPEC problem. Fukushima, Luo and Pang [24] present a similar algorithm, sequential quadratic programming (SQP), through a reformulation of the complementarity condition as a system of semismooth equations by means of Fischer-Burmcister functional. This algorithm shares several common features with PIPA in terms of the computational steps and convergence properties; however, it differs from PIPA in the way it updates the penalty parameters and determines the step sizes. Chen and Fukushima [11] consider a type of MPECs whose lower constraints form a parametric P-matrix LCP. The complementarity constraints are smoothed through the Fischer-Burmcister functional, from which the state variables are viewed as implicit functions of the design variables. The MPECs with P-matrix LCP constraints can thus be solved through a sequence of convex programs. Fukushima and Tseng [25] propose an ²-active set algorithm to solve MPECs with linear complementarity constraints and establish the convergence to B-stationary points under the uniform linear independence constraint qualification on the feasible set. Without assuming the non-degeneracy on the lower-level strict complementarity or the upper-level strict complementarity, this algorithm generates a sequence of variable value sets such that

91 the objective value is almost decreasing, while maintaining the ²-feasibility of the complementarity constraints. Moreover, if the objective function is quadratic and ² is zero, the algorithm terminates finitely at a B-stationary point.

6.1.3

Equilibrium Problem with Equilibrium Constraints An EPEC problem is a set of MPEC problems each of which is parameterized by

other MPECs’ design variables. Specifically, suppose that fi : Rnl+m → R, i = 1..n and F : Rnl+m → Rm are given functions, Zi ⊆ Rl+m , i = 1..n are a non-empty closed sets, and C : Rnl → Rm is a set-valued map (possibly empty), then the EPEC is to find {x1 , .., xn , y} such that, for i = 1..n, (xi , y) solves the MPEC min xi ,y

fi (x1 , .., xn , y)

subject to: (xi , y) ∈ Zi y ∈ S (x1 , .., xn ) where y ∈ S (x1 , .., xn ) if and only if y ∈ C (x1 , .., xn ) and (v − y)T F (x1 , .., xn , y) ≥ 0 for all v ∈ C (x1 , .., xn ). One approach to solve EPEC problems is to derive the optimality conditions for the nonlinear-programming regularization scheme of the MPECs (see [19, 48, 50]), then either solve the nonlinear complementarity conditions of the EPEC as a whole [34, 53] or iteratively solve the nonlinear complementarity conditions of individual MPECs [34, 53]. Another more straightforward approach as demonstrated in [33, 34] is to iteratively solve MPECs using MPEC-based algorithms, such as those in [11, 14, 24, 25, 26, 35, 47].

92

6.2

A Compact Presentation of Model T2 In this section, we give a compact presentation of Model T2 (one can follow a

similar approach to simplify Model T1). Recall that, in the EPEC problem of Model T2, each firm solves an MPEC: X

max

xgz :z∈Z

hz xgz +

z∈Z

X

P r(c)πgc

c∈C

subject to: X X X πgc = (pc + ϕci )qic − ucz xgz − Ci (qic ) i∈Ng

hz =

X

z∈Z

c∈C

i∈Ng

P r(c)ucz

z∈Z

c∈C

ucz =

X

(pc + ϕci )δi ,

z ∈ Z, c ∈ C

i:z(i)=z

pc + ϕci − βkc − ci − di qic + ρci− − ρci+ = 0 P ϕj P c j∈N bcj j∈N qj c c p =a − P − P 1 1

i ∈ Nk , k ∈ G, c ∈ C

(6.1)

c∈C

(6.2)

c c (λcm− Dm,i − λcm+ Dm,i )

i ∈ N, c ∈ C

(6.3)

X 1 X X c + q − δi xkz(i) = 0 j bcj

k ∈ G, c ∈ C

(6.4)

i ∈ N, c ∈ C

(6.5)

i ∈ N, c ∈ C

(6.6)

j∈N bcj

ϕci = −

X

j∈N bcj

m∈L

− βkc

j∈N

0 ≤ ρci−

j∈Nk

i∈Nk

⊥ qic ≥ 0

0 ≤ ρci+ ⊥ q ci − qic ≥ 0 X rjc = 0

(6.7)

j∈N

X

ac − (qic + ric )bci − pc +

c (λcm− − λcm+ )Dm,i =0

i ∈ N, c ∈ C

(6.8)

l ∈ L, c ∈ C

(6.9)

l ∈ L, c ∈ C

(6.10)

m∈L

0 ≤ λcl−



X

c c Dl,j rj + Klc ≥ 0

j∈N

0 ≤ λcl+

⊥ Klc −

X j∈N

c c Dl,j rj ≥ 0

93 Here, the inner problem (6.1)-(6.10) is shared among all firms’ MPECs. We group and relabel the variables, including the dual variables, as follows: • xg (∈ R|Z| ): The vector of the forward commitments by firm g ∈ G. · xg =

¸ xgz z ∈ Z

• rc (∈ R|N | ): The vector of the ISO’s import/export quantities in state c ∈ C. · c

r =

¸ ric

i∈N

• q c (∈ R|N | ): The vector of the firms’ generation quantities in state c ∈ C. · c

q =

¸ qic i ∈ N

• ρc− , ρc+ (∈ R|N | ): The vectors of the Lagrangian multipliers associated with the generation capacity constraints in state c ∈ C. · ρc−

= ·

ρc+

=

¸ ρci− i ∈ N ρci+

¸

i∈N

• λc− , λc+ (∈ R|L| ): The vectors of the Lagrangian multipliers associated with the flow capacity constraints in state c ∈ C. · λc−

= ·

λc+ =

¸ λci−

l∈L

λcl+

l∈L

¸

94 In addition, the parameters are relabelled as • ∆ (∈ R|N |×|Z| ): A matrix where the (i, z)-th element is −1 if z(i) = z, and 0 otherwise. • q c (∈ R|N | ): The vector of the generator capacity bounds in state c ∈ C. q c = [q ci

i ∈ N]

• Bc (∈ R|N |×|N | ): A diagonal matrix for state c ∈ C where the (i, i)-th element is bci . • d (∈ R|N | ): The vector of the first-order marginal generation costs. d = [di

i ∈ N]

c . • Dc (∈ R|L|×|N | ): A PTDF matrix for state c ∈ C where the (l, i)-th element is Dl,i

• k c (∈ R|L| ): The vector of the flow capacities of the transmission lines in state c ∈ C. k c = [Klc

l ∈ L]

• Xg (∈ R|Z| ): The feasible region of xg for each firm g ∈ G. Xg = Xg1 × Xg2 × ... × Xg|z|

6.2.1

Compact Presentation of the Inner Problem in Model T2 Let e ∈ R|N | be a vector with all 1’s, constraints (6.7) and (6.8) for a state c ∈ C

become    Solving  c  r  pc

 ac e









 rc



 DcT







DcT



  0      Bc  c  Bc e      λ+ =    λ− −  − q −   + 0 0 0 0 0 eT 0 pc

rc and pc yields     −1        T c T   Bc e   a e   Bc  c  Dc  c  Dc  c   λ+  =    λ− −  − q +  0 eT 0 0 0 0

95

 =



 

 Qc

Bc−1 e eT Bc−1 e

eT Bc−1 eT Bc−1 e

−1 eT Bc−1 e

   









ac e

DcT







DcT

  Bc  c   c   c − q +   λ− −   λ+  0 0 0 0

where Qc = Bc−1 −

Bc−1 eeT Bc−1 eT Bc−1 e

Hence, ¡ ¢ rc = − Qc Bc q c + Qc DcT λc− − DcT λc+ pc =ac −

¢ eT eT Bc−1 ¡ T c c T c q + D λ − D λ c − c + −1 −1 eT B c e eT B c e

Now, consolidating conditions (6.1)-(6.4) for state c, we have ¡ ¢ ρc− = −ac e + d + Hc q c + Bc Qc DcT λc− − DcT λc+ + ρc+ +

X 1 ∆ xg eT Bc−1 e g∈G

where Hc is a matrix such that   2+si  if i = j  −1   eT Bc e 2 (hc )ij = if i 6= j, and the units at nodes i and j belong to the same firm  eT Bc−1 e    1  otherwise eT B −1 e c

Next, let wc and y c be two variable vectors, and tc , Ac and Mc be constants such that



q¯c

qc

−    ρc−  c w =  c  k + Dc rc  k c − Dc rc 

q¯c

   −ac e + d  tc =   kc   kc

     ,   





    ,   

   qc  c y =  c  λ−  λc+

ρc+

     ,    

     Ac =    

0 ∆ eT Bc−1 e

0 0

    ,   

96 



0 −I 0    I Hc Bc Qc DcT  Mc =    0 −Dc Qc Bc Dc Qc DcT  0 Dc Qc Bc −Dc Qc DcT

0 −Bc Qc DcT −Dc Qc DcT

       

Dc Qc DcT

The preceding applied to conditions (6.1)-(6.10) for the state c leads to wc = tc + Ac

X

xg + Mc y c ,

wc ≥ 0,

y c ≥ 0,

(y c )T wc = 0

(6.11)

g∈G

Finally, aggregating (6.11) for all states c ∈ C, we present the inner problem (6.1)-(6.10) as w =t+A

X

xg + M y,

w ≥ 0,

y ≥ 0,

yT w = 0

g∈G

where y, w are variables, and t, A and M are constants as follows: · y=

¸

, ¸ w = wc c ∈ C , · ¸ c t= t c∈C , · ¸ A = Ac c ∈ C ,  M 0  1   M2  M =  ...   0 M|C| yc

c∈C

·

6.2.2

     .   

Compact Presentation of the MPEC Problems in Model T2 In period zero, each firm g ∈ G solves the following MPEC problem: Fg (¯ x−g ) :

min

xg ,y,w

fg (xg , y, w, x ¯−g )

subject to :

97

xg ∈ Xg w = t + A¯ x−g + Axg + M y,

w ≥ 0,

y ≥ 0,

yT w = 0

(6.12)

In this program, xg is the design variable, (y, w) are the state variables, and x ¯−g = P ¯k is a parameter that denotes the sum of the rest of the firms’ forward conk∈G\{g} x tracts. An equilibrium of the EPEC problem in period zero is a set ({¯ xg }g∈G , y, w) that solves Fg (¯ x−g ) for all g ∈ G, i.e., (¯ xg , y, w) ∈ SOL (Fg (¯ x−g )) ,

g ∈ G,

where SOL (Fg (¯ x−g )) denotes the solution set of Fg (¯ x−g ).

6.3

Solution Approach To solve the EPEC as stated above, we propose an iterative scheme that solves in

turn the MPEC problems Fg (¯ x−g ) by holding fixed the design variables of the rest MPEC problems. To avoid ambiguity, we refer to the iterations for solving the MPECs as the inner iterations, while the iterations for solving the EPEC problem are referred as outer iterations. Below, we first observe some properties of the MPEC problems; then exploiting these properties, we propose an MPEC algorithm; and finally develop the EPEC scheme.

6.3.1

Properties of the MPEC Problems We observe the following properties of Fg (¯ x−g ).

1. fg (xg , y, w, x ¯−g ) is quadratic with respect to (xg , y, w). 2. M is positive semi-definite. To show this, we first notice that Hc is symmetric positivedefinite. Secondly, v T Qc v = v T Bc−1 v −

v T Bc−1 eeT Bc−1 v eT Bc−1 e

98 −1

=

−1

−1

−1

kBc 2 vk2 kBc 2 ek2 − kv T Bc 2 Bc 2 ek2 −1

kBc 2 ek2 ≥ 0,

v ∈ R|N | .

Hence, Qc is symmetric positive semi-definite. Now, since 

 0

Mc + 2

McT

0

0 0



 0

       0 Hc 0 0   0    = +     0 0 0 0   Dc    −Dc 0 0 0 0

T



         Qc       

0

  0    ,  Dc   −Dc

we conclude that Mc is positive semi-definite. ¯−g , the constraint set (6.12) is an LCP parameterized by xg . Moreover, for 3. Given x any xg ,    ³   −ac e + d + wc =     

 q¯c

 ´+   ∆x−g +∆xg  eT Bc−1 e ,  c k    c k

 ³ c  −a e + d +    yc =     

∆x−g +∆xg eT Bc−1 e

´−     0  ,  0    0

c∈C

satisfy the linear constraints of this LCP. By Theorem 3.1.2 in [12], the LCP problem (6.12) is always solvable. From the economic perspective, it is not unreasonable to assume that, for each state in period one, there is a unique market equilibrium, that is, (6.12) has unique solution (w, y) for all xg ∈ Xg . By Theorem 3.1.7 in [12], this is equivalent to assuming that the active constraints at the optimal solutions to the period-one problems are linearly independent.

6.3.2

The MPEC Algorithm The uniqueness of solution (w, y) in (6.12) implies its solution (y, w) is an implicit

functions of xg . Consequently, Fg (¯ x−g ) can be reduced to an optimization problem with

99 respect only to xg . This feature motivates us to develop an algorithm for solving Fg (¯ x−g ) via a divide-and-conquer approach. Specifically, we partition Xg into a set of polyhedra according to feasible complementary bases of (6.12). In each polyhedron, we derive explicitly the affine functions for the state variables in terms of xg , and solve a quadratic program regarding only xg . Through parametric LCP pivoting, the proposed MPEC algorithm searches in the space of xg for a B-stationary point of Fg (¯ x−g ) along adjacent polyhedra. Partition of Xg The partition of Xg is determined by the feasible complementary bases of the LCP problem (6.12) (see Definition 1.3.2 in [12]). Let n and m be the dimensions of xg and y (and w), respectively. Consider LCP (6.12), given a partition (α, α ¯ ) of {1, 2, ..., m}, we define matrix CM (α) ∈ Rm×m as

CM (α)•i =

   −M•i if i ∈ α  

I•i

,

if i ∈ α ¯

CM (α) is called a complementary matrix of [−M, I] with respect to α; it is a complementary basis if nonsingular; it is a feasible complementary basis with respect to xg if −1 CM (α)(q + Axg ) ≥ 0,

where q = t + A¯ x−g . Now, given the partition (α, α ¯ ), let wα = 0 and yα¯ = 0, then (6.12) is reduced to −1 v α = CM (α)(q + Ag xg ) ≥ 0,

where viα =

   yi

if i ∈ α

  wi if i 6∈ α

.

100 The preceding is equivalent to the following two conditions −1 yα = Mαα (qα + Aα• xg ) ≥ 0 −1 wα¯ = −Mαα ¯ Mαα (qα + Aα• xg ) + qα ¯ + Aα• ¯ xg ≥ 0.

Note that the orthogonal condition between w and y is guaranteed for wα = 0 and yα¯ = 0. When CM (α) is a feasible complementary basis with respect to xg , the above nonnegative constraints for yα and wα¯ are satisfied. Thus, the polyhedron © ª −1 P˜g (α) = xg ∈ Rn : CM (α)(q + Ag xg ) ≥ 0 defines the set of xg with respect to which CM (α) is feasible. Moreover, the state variables y and w are linear functions of xg ∈ P˜g (α): −1 yα = Mαα (qα + Aα• xg )

(6.13)

yα¯ = 0

(6.14)

wα = 0

(6.15)

−1 wα¯ = −Mαα ¯ Mαα (qα + Aα• xg ) + qα ¯ + Aα• ¯ xg

(6.16)

m Equations (6.13)-(6.16) imply that xg ∈ Int(P˜g (α)) if and only if (yα , wα¯ ) ∈ R++

(y and w are non-degenerate), and that xg ∈ Bd(P˜g (α)) if and only if there exists some i ∈ α such that yi = wi = 0 (y and w are degenerate). Enumerating all feasible complementary bases CM (α), one can partition Xg into T a set of polyhedra Pg (α) = Xg P˜g (α). The uniqueness of the solution (y, w) to (6.12) guarantees a unique partition of Xg (but with respect to a fixed x ¯−g ). Figure 6.1 illustrates a typical partition for the case of n = 2. Stationary Point Equations (6.13)-(6.16) imply that, whenever CM (α) is a feasible complementary basis, fg (xg , y, w, x ¯−g ) is reduced to a quadratic function with respect to xg ∈ Pg (α). We

101

Figure 6.1: A typical partition of Xg denote this function as fg,α (xg , x ¯−g ). Now, limiting Fg (¯ x−g ) to xg ∈ Pg (α) leads to the following program parameterized by x ¯−g : QPg (α) :

min xg

fg,α (xg , x ¯−g )

subject to: xg ∈ Xg −1 Mαα (qα + Aα• xg ) ≥ 0 −1 − Mαα ¯ Mαα (qα + Aα• xg ) + qα ¯ + Aα• ¯ xg ≥ 0

Because this problem does not involve y and w, it has a smaller dimension than Fg (¯ x−g ). We call α the associated (index) basis of polyhedron Pg (α). Let xg ∈ Xg be given, and (y, w) be the corresponding solution to (6.12), equations (6.13)-(6.16) hold for all associated bases in the set Bg (xg , x ¯−g ) = {α ⊆ {1, 2, ..., m} : {i : yi > 0} ⊆ α ⊆ {i : wi = 0}} . We refer to this set as the association (basis) set at xg . Clearly, xg ∈ P (α) for all α ∈ Bg (xg , x ¯−g ). We are now ready to characterize the B-stationary points of Fg (¯ x−g ). Following [40], a vector (¯ xg , y, w) is called a B-stationary point of Fg (¯ x−g ) if, for all feasible

102 (with respect to (6.12)) directions u ∈ Rn+2m at (¯ xg , y, w), the directional derivative ∇u fg (xg , y, w, x ¯−g ) ≥ 0. Thus, a point x ¯g is a B-stationary point of Fg (¯ x−g ) if and only if, for all α ∈ Bg (¯ xg , x ¯−g ), either of the following two cases holds 1. Pg (α) is a singleton containing only the point x ¯g , i.e., Pg (α) = {¯ xg }; 2. for any unit-vector direction u ∈ Rn such that there exists a sufficiently small scaler ² > 0 satisfying x ¯g + ²u ∈ Pg (α), the directional derivative of fg (xg , y, w, x ¯−g ) at x ¯g with respect to u is non-negative, i.e., ∇u fg (xg , y, w, x ¯−g ) =

∂fg ∂fg dy ∂fg dw u+ u+ u ≥ 0, ∂xg ∂y dxg ∂w dxg

where y and w are as in (6.13)-(6.16). (When the derivative is zero for all directions, x ¯g is a stationary point of QPg (α).) The above B-stationary conditions suggest that, if a local minimum or stationary point x ¯g of QPg (α) yields non-degenerate (y, w) in (6.12), it is a B-stationary point of Fg (¯ x−g ); otherwise, one should identify whether this point is a B-stationary point of Fg (¯ x−g ) by checking whether it is a local minimum or stationary point with respect to all polyhedra associated with Bg (¯ xg , x ¯−g ). The MPEC Algorithm Let x ¯g ∈ Xg be a given starting point. If there exists an α ∈ Bg (¯ xg , x ¯−g ) such that QPg (α) is unbounded, then Fg (¯ x−g ) is also unbounded. If x ¯g is a local minimum or stationary point of problems QPg (α) for all α ∈ Bg (¯ xg , x ¯−g ), it is a B-stationary point of Fg (¯ x−g ). Otherwise, there exists an α∗ ∈ Bg (¯ xg , x ¯−g ) for which QPg (α∗ ) leads to a solution different from x ¯g . Let this point be x∗g , then its corresponding state variables y ∗

103 and w∗ are as in (6.13)-(6.16) with α replaced by α∗ . If y ∗ and w∗ are non-degenerate, x∗g ∈ Int(P˜g (α∗ )), and hence a B-stationary point of Fg (¯ x−g ); otherwise, it serves as the starting point for the next (inner) iteration. The dashed lines in figure 6.1 illustrate such a sample path. The MPEC algorithm Input: x ¯g , x ¯−g 0. (Initialization) Set α∗ := ∅. 1. (Subroutine call) Call the search subroutine. 2. (Termination check) If the subroutine reports unboundedness, report the problem Fg (¯ x−g ) as unbounded, stop. else if the subroutine reports x ¯g as a B-stationary point, x ¯g is a B-stationary point of Fg (¯ x−g ), stop. else let x∗g and α∗ be the returned point and the associated basis, respectively. let y ∗ and w∗ solve (6.12) with x∗g . m , If (yα∗ ∗ , wα∗¯ ∗ ) ∈ R++

x−g ), x∗g is a B-stationary point of Fg (¯ stop. else set x ¯g := x∗g , go to step 1. end

104 end

The search subroutine Input: α∗ , x ¯g , x ¯−g 0. (Initial pivoting) Pivot to an associated basis α ∈ Bg (¯ xg , x ¯−g )\{α∗ } at x ¯g . 1. (Search) Call a quadratic programming subroutine to solve QPg (α). If QPg (α) has an unbounded direction, report unboundedness. end If QPg (α) yields a point x∗g 6= x ¯g with a decreased objective value, set α∗ := α, return x∗g and α∗ . end 2. (Termination Check) If all bases in Bg (¯ xg , x ¯−g )\{α∗ } have been visited, return x ¯g as a B-stationary point. else pivot (at x ¯g ) to the next α ∈ Bg (¯ xg , x ¯−g )\{α∗ }, go to step 1. end Here, one can use any available quadratic programming solver as the quadratic programming subroutine.

105 Remarks 1. Because the number of zones is typically much smaller than the number of nodes, the dimension of QPg (α), |xg |, is usually much smaller than that of Fg (¯ x−g ). Therefore, the proposed MPEC algorithm advances the PSQP algorithm [40] in the ability of solving relatively larger problems. 2. The MPEC algorithm maintains the satisfaction of all constraints (including the complementarity constraint) in (6.12). 3. If |Bg (¯ xg , x ¯−g )| ≤ 2 throughout the course of the MPEC algorithm, then no basis will be repeated. This, combined with the fact that there exists a finite number of partition of Xg (bounded by the number of feasible complementary bases in (6.12)), establishes the finite global convergence of the MPEC algorithm. If the preceding condition is violated (that is, if |Bg (¯ xg , x ¯−g )| > 2 for some x ¯g ), one can use any of the standard lexicographic schemes (in the context of LCP pivoting; see, for example, [12]) to avoid cycling. It should be noted that different lexicographic schemes might lead to different search paths of the MPEC algorithm and thus possibly to different B-stationary points. For example, if the lexicographic scheme selects the basis α2 , instead of α1 , at point A in figure 6.1, the algorithm terminates immediately. −1 and possibly, depending on the 4. Note that, to solve QPg (α), we need to compute Mαα −1 can be computed quadratic programming subroutine in use, a starting point. Mαα

efficiently from the corresponding matrix of the previously-visited basis, which differs from α by one index. The solution to the quadratic program with respect to the previous-visited basis can be used as the starting point.

106

6.3.3

The EPEC Scheme

B-stationary equilibrium To define the B-stationary equilibrium for {Fg (·)}g∈G , we extend the definition of association set for the MPECs to the EPEC. Because the association set relies on the signs of the components of (y, w), which are determined through (6.12) jointly by all MPECs’ design variables, we conclude the following equivalence property. Let (y, w) solve (6.12) with given {¯ xg ∈ Xg }g∈G and consider any two firms g and g 0 , then the association set at x ¯g for Fg (¯ x−g ) and the association set at x ¯g0 for Fg0 (¯ x−g0 ) are equivalent, i.e., Bg (¯ xg , x ¯−g ) = Bg0 (¯ xg0 , x ¯−g0 ),

g, g 0 ∈ G.

The above equivalence of the association sets among all MPECs implies that ¯g ∈ Int(P˜g (α)) for some α, then x ¯g0 ∈ Int(P˜g0 (α)); • if x • if x ¯g is in the boundaries of a set of polyhedra P˜g (α1 ), P˜g (α2 ), ..., P˜g (αk ), x ¯g0 is also in the boundaries of polyhedra P˜g0 (α1 ), P˜g0 (α2 ), ..., P˜g0 (αk ). We define the association set of the EPEC {Fg (·)}g∈G as follows. Given {¯ xg }g∈G , let (y, w) solve (6.12), then the association set for {Fg (·)}g∈G is B({¯ xg }g∈G ) = {α ⊆ {1, 2, ..., m} : {i : yi > 0} ⊆ α ⊆ {i : wi = 0}}. The association set of {Fg (·)}g∈G allows us to characterize the B-stationary equilibria of {Fg (·)}g∈G as follows. A set {¯ xg }g∈G is a B-stationary equilibrium of {Fg (·)}g∈G if x ¯g is a B-stationary point of Fg (¯ x−g ) for all g ∈ G, i.e., x ¯g is a local minimum or stationary point of QPg (α) for all α ∈ B({¯ xg }g∈G ). The Scheme To solve {Fg (·)}g∈G , we start with an arbitrary set {¯ x0g ∈ Xg }g∈G . At each outer ¯k−g as given. The xk−g ) for each g ∈ G while taking x iteration k, we compute x ¯kg from Fg (¯

107 algorithm terminates when the improvement of the design variables in two consecutive iterations is reduced to a predetermined limit, or when the number of iterations reaches a predetermined upper bound. The EPEC Scheme 0. (Initialization) Select an arbitrary {¯ x0g ∈ Xg }g∈G . Let k := 1. 1. (Loop) Let {¯ xkg }g∈G := {¯ xk−1 g }g∈G . For each g ∈ G, apply the MPEC algorithm to Fg (¯ xk−g ). if Fg (¯ xk−g ) is unbounded, report the failure of finding an equilibrium, stop. else let x ¯kg and (y, w) be the returned design and state variables. end 2. (Termination check) If k{¯ xkg − x ¯k−1 g }g∈G k is within a given error bound, report ({¯ xkg }g∈G , y, w) as a B-stationary equilibrium, stop. else if the predetermined bound of the number of iterations is reached, stop. else go to step 1 with k := k + 1. end

108 Remarks 1. The termination basis of an MPEC problem can be used as the starting basis for the next MPEC problem. 2. The termination point for an MPEC problem can be used as the starting point for solving the next MPEC problem.

6.4

Computational Results We implemented in MATLAB the MPEC and EPEC algorithms which utilize the

optimization toolbox for solving quadratic programs. In the implementation, we treat any number below 10−16 as zero to account for roundoff errors. Tests of the algorithms are performed on both randomly generated problems and practical problems specific to the context of electricity markets.

6.4.1

Tests of the MPEC Algorithm The main computational effort involved in the EPEC scheme is to solve the

MPECs. While Our MPEC algorithm is guaranteed to terminate in finite number of steps (see Section 6.3.2), it is not known whether it can be solved in polynomial time. In this section, we test the actual performance of the algorithm on a randomly generated set of generic MPEC problems with quadratic objective functions. Specifically, these MPEC problems are of the form: min x,y

1 2

·

¸ x y

   x    x  P   + cT   y y

subject to: Ax + a ≤ 0 w = N x + M y + q,



109

w ≥ 0,

y≥0

wT y = 0 where P , A, B, M (a positive-semidefinite matrix), N , c, a and q are constant matrices and vectors with suitable dimensions. We use the “QPECgen” package by Jiang and Ralph [36] to generate these MPEC programs. In the tests, we launch the MPEC algorithm from random starting point. Table 6.1 summarizes the test results. The first three columns list the dimensions of the decision and state variables, and columns 5 to 7 report the minimum, maximum and average numbers of iterations, respectively. We observe that 1. The average number of iterations increases moderately as the dimension of the MPEC problems grows (expect the case of n = 150 and m = 100); but, there does not exist such a trend for the minimum and maximum numbers of iterations. 2. The algorithm is able to effectively solve MPEC problems with relatively large dimensions. Note that all instances in table 6.1 have greater dimensions than those reported in [36].

Table 6.1: Test results of the MPEC algorithm Dim(x) 25 50 50 100 150 100 200 200

Dim(y) 50 50 100 100 100 200 200 500

Dim(w) 50 50 100 100 100 200 200 500

Total dimension 125 150 250 300 350 500 600 1200

min 3 7 2 10 2 3 2 2

Iterations max average 34 16 35 18 49 22 43 23 30 14 38 23 88 29 76 43

110

6.4.2

Test the EPEC algorithm To test the EPEC algorithm, we use the numerical examples derived from the

stylized Belgian electricity network (see chapter 2). It is worth a mention that the stylized system has the dimension m = 2|C| × (|N | + |L|) = 684 of y (and w), and the total number of possible partitions is 2684 . In this implementation, we terminate the EPEC algorithm at an outer iteration k if the relative improvement of the MPECs’ design variables (forward commitments) is no greater than 10−8 , i.e., −8 k{¯ xkg − x ¯k−1 xk−1 g }g∈G k ≤ 10 k{¯ g }g∈G k.

We run the tests with different numbers of zones and firms, and, for each test, we start with randomly generated design variables of the MPECs. In the implementation, the MPEC algorithm terminates after one inner iteration. (Note that, if the MPEC algorithm terminates before it reaches a B-stationary point, the LCP problem (6.12) is still satisfied; this allow us to trade off the accuracy of the MPEC algorithm for the speed of the EPEC scheme. We also tried some other rules for terminating the MPEC algorithm; however, they don’t provide comparable results.) The test results are summarized in table 6.2. Columns 4 to 9 show the minimum, maximum and average numbers of outer iterations and quadratic programs, respectively. In addition, tables 6.3 and 6.4 report the outer iterations of the firm’s total forward commitments for the cases of two and three firms, respectively. We find that 1. For all test problems, the EPEC scheme converges promptly. 2. There exists no clear relationship between the problem dimensions and the number of iterations. However, the total number of quadratic programs grows as the number of firms increases. 3. In the tests, the EPEC scheme quickly reaches the proximity of the B-stationary equilibrium, after which it only improves the decimal significant digits (see, for example,

111 tables 6.3 and 6.4).

Table 6.2: Test results of the EPEC algorithm |Z|

|G|

Dim(y)

2 2 2 3 3 3 4 4 4

2 3 4 2 3 4 2 3 4

684 684 684 684 684 684 684 684 684

Outer iterations min max average 4 8 6 8 11 9 7 12 10 2 4 3 4 6 5 4 10 8 3 9 7 7 21 13 9 25 15

Quadratic programs min max average 15 29 24 47 65 55 55 94 77 7 15 11 23 34 29 32 79 62 11 36 27 41 126 80 70 197 122

Table 6.3: Iterations of firms’ total forward commitments (two firms) Outer iteration 0 1 2 3 4 5

6.5

Firm 1 0.000000 -513.063752 -331.223467 -545.254227 -552.287608 -552.287608

Firm 2 0.000000 575.219726 1546.721883 1747.692181 1747.692181 1747.692181

Summary This chapter studies the computational aspect of the EPEC models of two-settlement

electricity markets. In these models, each firm solves an MPEC problem parameterized on the design variables of the other MPECs. We propose an MPEC algorithm by taking advantage of the special properties of the MPECs. This algorithm partitions the feasible region of the design variables into a set of polyhedra, and projects the state variables to the space of the design variables. The algorithm solves a quadratic program for a stationary point in each polyhedron, and pivots through adjacent polyhedra while maintaining the feasibility

112

Table 6.4: Iterations of firms’ total forward commitments (three firms) Outer iteration 0 1 2 3 4 5 6 7 8 9 10 11

Firm 1 0.000000 6739.889190 6739.889190 6851.687937 7001.487699 7154.268773 7237.416442 7239.775870 7239.859233 7239.862110 7239.862110 7239.862110

Firm 2 0.000000 -16.249658 246.601419 556.357457 849.405693 1001.093059 1006.167745 1006.342137 1006.348165 1006.348374 1006.348382 1006.348382

Firm 3 0.000000 -288.471837 -103.536223 71.319790 154.719273 149.846951 149.619740 149.611431 149.611140 149.611129 149.611130 149.611130

and complementarity of the linear complementarity constraints. We establish the finitely global convergence of this MPEC algorithm. An EPEC algorithm is further constructed by deploying the MPEC problems through an iterative approach. Numerical examples of randomly generated quadratic MPECs and the EPECs derived from the stylized Belgian electricity network demonstrate the effectiveness of the algorithms.

113

Chapter 7

Concluding Remarks In this thesis, we consider electricity markets with demand uncertainty, network constraints and system contingencies, and study the effect of forward contracts on consumer benefits. In particular, we follow two game-theoretical approaches and model two settlements as an equilibrium problem with equilibrium constraints. We apply our models to the stylized Belgian network, and observe the strategic incentives of the firms for forward contracting, the likelihood of congestion, increased social surplus and decreased spot prices. We also notice that these effects are amplified when the market is less concentrated. When imposing price caps, we find a spot price cap reduces the firms’ incentives for committing forward contracts whereas a forward price cap creates the opposite effect. We develop special-purpose EPEC/MPEC algorithms for solving such EPEC problems. The MPEC algorithm converts the state variables into piece-wise linear functions of the design variables, and search for a B-stationary point through parametric LCP pivoting and finite quadratic programs. The EPEC algorithm deploys the MPEC problems through an iterative approach. We point out that, although the MPEC and EPEC algorithms are presented in the context of two-settlement electricity markets, they can be applied to quadratic EPEC

114 problems provided that the linear complementarity constraints yield unique values of the state variables. It should also be pointed out that our numerical tests are limited, and that their objective is aimed primarily at validating our modelling methodology rather than reaching conclusive economic results, which would require far more extensive simulation studies. As a result, one should expect quantitatively dissimilar results if the models are applied to different settings or different networks. We plan to relax the “no-arbitrage” assumption between the forward and spot prices with a market-clearing condition that sets the forward prices based on the expected demands in the spot market. Such analysis will attempt to capture how lack of liquidity (or high risk aversion) on the buyers side might be reflected in a high risk premium embedded in the forward prices. We expect that such a condition enhances firms’ market power and enables them to raise forward prices above the expected spot prices while increasing their profits. Another extension of the modeling perspective is to introduce the forward and the spot price caps to Model T2. On the computational aspect, we plan to challenge the algorithms by applying them to the California network. We will also continue the theoretical analysis on the convergence of the EPEC algorithm. In the case that the firms are risk averse and their MPECs in the forward market have non-quadratic objectives, we will develop new algorithms to solve such EPEC problems.

115

Bibliography [1] Allaz, B. 1987. Strategic Forward Transactions under Imperfect Competition: The Duopoly Case. Ph.D. dissertation, Department of Economics, Princeton University. [2] Allaz, B. 1992. Oligopoly, Uncertainty and Strategic Forward Transactions. Internal Journal of Industrial Organization. 10 297-308. [3] Allaz, B. and J.-L. Vila. 1993. Cournot Competition, Forward Markets and Efficiency. Journal of Economic Theory. 59 1-16. [4] Anderson, S. 2004. Analyzing Strategic Interaction in Multi-settlement Electricity Markets: A Closed-loop Supply Function Equilibrium Model. Ph.D. Thesis, Harvard University. [5] Borenstein, S., J. Bushnell and S. Stoft. The Competitive Effects of Transmission Capacity in A Deregulated Electricity Industry. RAND Journal of Economics. 31 2 294-325. [6] Bower, J. and D. W. Bunn. 2000. A Model-based Comparison of Pool and Bilateral Market Mechanisms for Electricity Trading. Energy Journal. 21 3 [7] Bushnell, J. A Mixed Complementarity Model of Hydrothermal Electricity Competition in the Western United States. Operations Research. 51 80-95. [8] California Public Utilities Commission. 1993. Californias Electric Services Industry: Perspectives on the Past, Strategies for the Future. [http://www.cpuc.ca.gov/Published/Report/3822.htm] [9] Cardell, J., C. Hitt and W. W. Hogan. Market Power and Strategic Interaction in Electricity Networks. Resource and Energy Economics. 19 109-137. [10] Chao, H.-P. and S. C. Peck. 1996. A Market Mechanism for Electric Power Transmission. Journal of Regulatory Economics. 10 1 25-60. [11] Chen, X. and M. Fukushima. 2004. A Smoothing Method for a Mathematilcal Program with P-matrix Linear Complementarity Constraints. Computational Optimization and Applications. 27 223-246. [12] Cottle, R. W., J.S. Pang and R. E. Stone. 1992. the Linear Complementarity Problem. Academic Press, Boston, MA.

116 [13] Fabra, N., N.-H. von de Fehr and D. Harbord. 2004. Designing Electricity Auctions. CSEM working paper 122, University of California Energy Institute, Berkeley, California [14] Facchinei, F., H. Jiang and L. Qi. 1999. A smoothing method for mathematical programs with equilibrium constraints. Mathematical programming. 85 107-134. [15] Federal Energy Regulatory Commission. 1996a. Promoting Wholesale Competition Through Open Access Non-discriminatory Transmission Services by Public Utilities, Recovery of Stranded Costs by Public Utilities and Transmitting Utilities (Order No. 888). Washington, DC. [16] Federal Energy Regulatory Commission. 1996b. Open Access Same-Time Information System (formerly Real-Time Information Networks) and Standards of Conduct (order 889). Washington, DC. [17] Federal Energy Regulatory Commission. 1999. Order 2000: Final Rule. Docket No. RM99-2-000 (89 FERC 61,285), Regional Transmission Organizations. Washington, DC. [18] von der Fehr, N.-H. M. and D. Harbord. 1992. Long-term Contracts and Imperfectly Competitive Spot Markets: A Study of UK Eletricity Industry. Memorandum no. 14, Department of Economics, Univeristy of Oslo, Oslo, Sweden. [19] Fletcher, R. and S. Leyffer. 2004. Solving Mathematical Programs with Complementarity Constraints as Nonlinear Programs. Optimization Methods and Software. 19 15-40 [20] Kocvara, M. and J. V. Outrata. 1994. On optimization of systems governed by implicit complementarity problems. Numerical Functional Analysis and Optimization. 15 869–887 [21] Kocvara, M. and J. V. Outrata. 1995a. On the solution of optimum design problems with variational inequalities. In: Recent Advances in Nonsmooth Optimization (D.-Z. Du, L. Qi and R.L. Womersley, eds.), World Scientific Publishers, Singapore. 171–191. [22] Kocvara, M. and J. V. Outrata. 1995b. A nonsmooth approach to optimization problems with eqilibrium constraints. Proceedings of the International Conference on Complementarity Constraints, M.C. Ferris and J.S. Pang (Eds.). Baltimore, Maryland, SIAM publications. 148-164 [23] Fudenberg, D. and J. Tirole. 1991. Game Theory. the MIT Press, Cambridge, MA. [24] Fukushima, M., Z.-Q. Luo and J.-S. Pang. 1998. A Globally Convergent Sequential Quadratic Programming Algorithm for Mathematical Programs with Linear Complementarity Constraints. Computational Optimization and Applications. 10 1 5-34 [25] Fukushima, M. and P. Tseng. 2002. An Implementable Active-set Algorithm for Computing a B-stationary Point of a Mathematical Problem with Linear Complementarity Constraints. SIAM Journal of Optimization. 12 3 724-739 [26] Fukushima, M. and G.-H. Lin. 2004. Smoothing Methods for Mathematical Programs with Equilibrium Constraints. International Conference on Informatics Research for Development of Knowledge Society Infrastructure (ICKS’04). 206-213

117 [27] Green, R. J. and D. M. Newbery. 1992. Competition in the British Electricity Spot Market. Journal of Political Economy 100 92953 [28] Green, R. J. 1999. the Electricity Contract Market in England and Wales. Jounal of Industrial Economics. 47 1 107-124 [29] R. Gilbert, K. Neuhoff and D. Newbery. 2004. Allocating Transmission to Mitigate Market Power in Electricity Networks. RAND Journal, Winter. 35 4 691-711 [30] P.T. Harker. 1991. Generalized Nash Games and Quasivariational inequalities. European Journal of Operations Research. 54 8194 [31] Harvey, S. M. and W. W. Hogan. 2000. Nodal and Zonal Congestion Management and the Exercise of Market Power, John F. Kennedy School of Government, Harvard University, Working paper [32] Hobbs, B.F. 2001. Linear Complementarity Models of Nash-Cournot Competition in Bilateral and POOLCO Power Markets. IEEE Transactions on Power Systems. 16 2 194-202. [33] Hobbs, B.F., C.B. Metzler and J.-S. Pang (2000): Strategic Gaming Analysis for Electric Power Networks: an MPEC Approach. IEEE Transactions on Power Systems. 15 638645. [34] Hu, X. 2002. Mathematical Programs with Complementarity Constraints and Game Theory Models in Electricity Markets, Ph.D. Thesis, Department of Mathematics and Statistics, University of Melbourne. [35] Hu, X.M. and D. Ralph. 2004. Convergence of a penalty method for mathematical programming with complementarity constraints. Journal of Optimization Theory and Applications. 123 2 365-390. [36] Jiang, H. and D. Ralph. 1999. QPECgen: A MATLAB Generator for Mathematical Programs with Quadratic Objectives and Affine Variational Inequality Cnostraints. Computational Optimization and Applications. 13 25-49. [37] Kamat, R. and S. S. Oren. 2004. Multi-Settlement Systems for Electricity Markets: Zonal Aggregation under Network Uncertainty and Market Power. Journal of Regulatory Economics. 25 1 5-37 [38] Karush, W. 1939. Minima of Functions of Several Variables with Inequalities as Side Constraints, M.S. Thesis, Department of Mathematics, University of Chicago, Chicago, IL [39] Kuhn, H. W. and A. W. Tucker. 1951. Nonlinear Programming, Proceedings of the Second Berkeley Symposium, J Neyman (Ed.), University of California Press, Berkeley, CA [40] Luo, Z.Q., J.S. Pang and D. Ralph. 1996. Mathematical Programs with Equilibrium Constraints, Cambridge University press, Cambridge, UK.

118 [41] Neuhoff, K., J. Barquin, M. G. Boots, A. Ehrenmann, B. F. Hobbs, F. A.M. Rijkers and M. Vzquez. 2005. Network-constrained models of liberalized electricity markets: the devil is in the details. Energy Ecomnomics 27 3 495-525 [42] Newbery, D. M. 1998. Competition, Contracts, and Entry in the Electricity Spot Market. Rand Journal of Economics. 29 4 726-749. [43] Oren, S. S. 1997. Economic Inefficiency of Passive Transmission Rights in Congested Electricity Systems with Competitive Generation. the Energy Journal . 18 63-83. [44] Outrata, J.V. 1990. On optimiztion problems with variabional inequality constraints. SIAM Journal on Optimization. 4 340-357. [45] Outrata, J.V. and J. Zowe. 1995. A numerical approach to optimization problems with variabional inequality constraints. Mathematial Programming. 68 105-130. [46] Powell, A. 1993. Trading Forward in an Imperfect Market: the Case of Electricity in Britain. the Economic Journal. 103 444-453. [47] Ralph, D. and S.J. Wright. 2004. Some properties of regularization and penalization schemes for MPECs. Optimization Methods and Software. 19 5 527-556. [48] Scheel, H. and S. Scholtes. 2000. Mathmatical Programs with Equilibrium Constraints: Stationarity, Optimality, and Sensitivity. Mathematics of Operations Research. 25 1-22 [49] Schmalensee R, B. W. Golub. 1984. Estimating Effective Concentration in Deregulated Wholesale Electricity Markets. RAND Journal of Economics. 15 1 12-26. [50] Scholtes, S. 2001. Convergence Properties of a Regularization Scheme for Mathmatical Programs with Complementarity Constraints, SIAM J. on Optimization. 11 918-936 [51] Schweppe, F.C., M.C. Caramanis, R.E. Tabors and R.E. Bohn. 1988. Spot Pricing of Electricity. Kluwer Academic Publishers. [52] Smeers, Y., J.-Y. Wei. 1997. Spatial Oligopolistic Electricity Models with Cournot Firms and Opportunity Cost Transmission Prices. Center for Operations Research and Econometrics, Universite Catholique de Louvain, Louvain-la-newve, Belgium. [53] Su, C.-L. 2005. Equilibrium Problem with Equilibrium Constraints: Stationarities, Algorithms and Applications. Ph.D. thesis, Department of Management Science and Engineering, Standford University, Standford, CA [54] Wei, J.-Y., Y. Smeers. 1999. Spatial Oligopolistic Electricity Models with Cournot Firms and Regulated Transmission Prices. Operations Research. 47 1 102-112. [55] Willems, B. 2004. Cournot Competition, Financial Option Markets and Efficiency. CSEM working paper 139, University of California Energy Institute, Berkeley, California [56] Wilson, R. 2002. Architecture of Power Markets. Econometrica. 70 4 1299-1340

119 [57] World Energy Council. 1998. The Benefits and Deficiencies of Energy Sector Liberalisation: Current Liberalisation Status. Volume II. London. http://www.worldenergy.org/wec-geis/publications/reports/current cls/ClsTOC.asp [58] Wolak, F. and R. H. Patrick. 1996 The Impact of Market Rules and Market Structure on the Price Determination Process in the England and Wales Electricity Market. POWER working paper, University of California Energy Institute, Berkeley, CA [59] Wolfram, C. 1999. Strategic Bidding in a Multiunit Auction: An Empirical Analysis of Bids to Supply Electricity in England and Wales. RAND Journal of Economics. 29 703-725. [60] Wolfram, C. 1999. Measuring Duopoly Power in the British Electricity Spot Market. American Economic Review. 89 805-826.

120

Appendix A

Model Notations A.1

Notations of Single-settlement Models The following notations are included in Model S1 and Model S2. Sets:

• N : The set of nodes. • L: The set of transmission lines. • G: The set of generation firms. Ng denotes the set of nodes at which firm g ∈ G owns generation units. Parameters: • q i : The upper capacity bound of the generation unit at node i ∈ N . • Pi (·): The inverse demand function at node i ∈ N . The corresponding demand function is denoted by Pi−1 (·). In the case of linear demand, Pi (q) = a − bi q and Pi−1 (p) = (a − p)/bi , where a denotes the uniform price intercept across the nodes, and bi stands for the slope. • Ci (·): The generation cost function at node i ∈ N . We assume it takes the quadratic form Ci (q) = di q + 12 si q 2 . • Kl : The thermal limit of line l ∈ L. • Dl,i : The power transfer distribution factor on line l ∈ L with respect to node i ∈ N . Decision variables: • qi : The output from the unit at node i ∈ N . • ri : The import/export quantity at each node i ∈ N .

121

A.2

Notations of Two-settlement Models

Sets: • N : The set of nodes. • Z: The set of zones. Moreover, z(i) represents the zone where node i resides. • L: The set of transmission lines. • C: The finite set of states in the spot market. • G: The set of generation firms. Ng denotes the set of nodes where generation facilities of firm g ∈ G are located. Parameters: • q ci : The upper capacity bound of generation facility at node i ∈ N in state c ∈ C. • Pic (·): The inverse demand function at node i ∈ N in state c ∈ C. In the case of linear demand, Pic (q) = ac − bci q,

i ∈ N, c ∈ C,

where the price intercepts of the inverse demand curves are uniform across the nodes in each state. • Ci (·): The cost function at node i ∈ N . We assume quadratic cost functions 1 Ci (q) = di q + si q 2 , 2

i ∈ N.

• Klc : The flow capacity of line l ∈ L in state c ∈ C. c : The power transfer distribution factor in state c ∈ C on line l ∈ L with respect • Dl,i to node i ∈ N .

• P r(c): The probability of state c ∈ C of the spot market. • δi : The nonnegative weights to settle the spot zonal prices. It holds that 1.

P

• u ¯: The spot price cap. ¯ The forward price cap. • h: Decision variables: • xgz : The forward commitment from firm g ∈ G to zone z ∈ Z. • qic : The output from the unit at node i ∈ N in state c ∈ C. • ric : The import/export quantity at node i ∈ N by the ISO in state c ∈ C.

i:z(i)=z δi

=

Suggest Documents