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OPTIMIZING EXPANSION OF LOW VARIABLE COST CAPACITY IN COMPETITIVE ELECTRICITY MARKETS: VALUATION OF WIND GENERATION USING STOCHASTIC COMPLEMENTARITY PROGRAMMING

Paul G. Dabrowski

A thesis submitted in conformity with the requirements for the degree of

Master of Applied Science

Graduate Department of Mechanical and Industrial Engineering

University of Toronto

395 Wellington Street 395, rue Wellington Ottawa ON K1A0N4 Ottawa ON K1A0N4 Canada Canada

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OPTIMIZING EXPANSION OF LOW VARIABLE COST CAPACITY IN COMPETITIVE ELECTRICITY MARKETS: VALUATION OF WIND GENERATION USING STOCHASTIC COMPLEMENTARITY PROGRAMMING

Master of Applied Science 2008

Paul G. Dabrowski

Department of Mechanical and Industrial Engineering

University of Toronto

Abstract

We study capacity expansion in imperfectly competitive electricity markets by presenting two decision support models formulated as complementarity programs, with wind capacity as the expansion technology. WCAPCOMP 1.0 is a deterministic model with a single-price wind capacity market and separate electricity market, the latter approximating a year of competition. WCAPCOMP 2.WOM is stochastic with nuclear capacity uncertainty between the two markets, and an added "wind output market" (WOM) for contracts-for-difference, where conventional gencos can buy financial rights to wind output from a wind entrant competitor. All gencos behave strategically, anticipating how other gencos will react to their decisions, using an anticipation approach. Results show that conventional gencos with coal on the margin have incentive to purchase WOM contracts and reduce their coal output to the market, and that this increases the total amount of wind capacity constructed, the ratio of coal energy displaced to wind energy added, and profits for the wind entrant.

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Acknowledgements

To my supervisor, Dr. J. Scott Rogers, for perspective, advice, and accepting nothing less than my best work.

To the committee members, Dr. Roy Kwon & Dr. Daniel Frances, for their constructive comments.

To members of industry who provided helpful suggestions.

To Kevin Au, for document assembly advice, and crib.

To my wife Anne, for unconditional love.

To my mother, for her support, patience and understanding.

111 IV

Table of Contents 1.0 Introduction 1 1.1 Motivation 3 1.2 Purpose and Methodology 5 1.3 Literature Review 7 1.3.1 Electricity Market Equilibrium Models 7 1.3.2 Investment in Capacity Models 13 1.3.3 Comparison To Asl, Rogers 2004 15 1.3.4 Comparison to Yao et. al. 2007 16 1.3.5 Comparison to Murphy, Smeers 2005 17 1.4 Thesis Overview 18 1.4.1 Chapter Two Summary 18 1.4.2 Chapter Three Summary 19 1.4.3 Chapter Four Summary 21 1.4.4 Chapter Five Summary 22 2.0 Analytical Derivation of Value of Wind in Competitive Markets for Open-Loop and Closed-Loop Knowledge 24 2.1 WCM-EM Model: Wind Capacity Market and Electricity Market with Two Gencos 26 2.1.1 Cournot Wind Capacity Market and Open-Loop Electricity Spot Market 30 2.1.2 Cournot Wind Capacity Market with Closed-Loop Electricity Spot Market.... 3 5 2.1.3 Summary of WCM-EM Model Findings 42 2.2 WOM-EM Model: Wind Output Market and Electricity Spot Market with Two Gencos 43 2.2.1 Effect of WOM Contract from Wind-On-Margin Gw to Conventional-On- MarginGl 44 2.2.2 WTP of Gl For WOM Contract Using Total Derivatives 47 2.2.3 WTS of Gw For WOM Contract Using Total Derivatives 50 2.2.4 Verifying Existence of a Mutually-Acceptable Closed-Loop Price for Wind Output Contract in WOM 52 2.2.5 Pitfalls with Standard CP Formulation of WOM 53

IV V

2.2.6 Generalizing WTP and WTS for Imperfect Knowledge Cases 56 2.3 Conclusion 56 3.0 WCAPCOMP 1.0 - Deterministic Capacity Expansion Model for Ontario-like Market 59 3.1 WCAPCOMP 1.0 Model Structure 60 3.1.1 Description of WCAPCOMP 1.0 model 60 3.1.2 Electricity Spot Market 62 3.1.3 Use of Anticipation in Electricity Spot Market 64 3.1.4 Wind Capacity Market 65 3.1.5 Use of Anticipation in Wind Capacity Market 66 3.1.6 Representation of Wind Generation 68 3.2 WCAPCOMP 1.0 Model Notation 68 3.3 WCAPCOMP 1.0 Model Formulation 71 3.3.1 Conventional Genco Primal Formulation 71 3.3.2 Wind-Only Genco Primal Formulation 75 3.3.3 EMO Primal Formulation 75 3.3.4 WMO Primal Formulation 76 3.3.5 Primal Linking Constraints 77 3.3.6 WCAPCOMP 1.0 Dual Constraints 78 3.3.7 Complementarity Programming Formulation of WCAPCOMP 1.0 79 3.3.8 KKT Analysis of WCAPCOMP 1.0 80 3.4 Numerical Results 85 3.4.1 Wind Capacity Allocation and Willingness-to-Pay 86 3.4.2 Electricity Market Outputs 89 3.4.3 Coal Displacement 91 3.4.4 Electricity Market Price and Profits 93 3.5 Conclusion 96 4.0 WCAPCOMP 2.WOM - Stochastic Two-Stage Capacity Expansion Model for Ontario-like Market 98 4.1 WCAPCOMP 2.WOM Model Structure 99 4.1.1 Stage Two Wind Output Contract Market (WOM) 101

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4.1.2 Use of Anticipation for Effect of WOM Contracts on Electricity Market Equilibrium 103 4.1.3 Use of Anticipation of WOM Price 105 4.2 WCAPCOMP 2.WOM Model Notation 106 4.3 WCAPCOMP 2.WOM Model Formulation 108 4.3.1 Conventional Genco Primal Formulation 108 4.3.2 Wind-Only Genco Primal Formulation 110 4.3.3 EMO Primal Formulation 111 4.3.4 WMO Primal Formulation 112 4.3.5 Primal Linking Constraints 112 4.3.6 WCAPCOMP 2.WOM Dual Constraints 112 4.3.7 Complementarity Programming Formulation of WCAPCOMP 2.WOM 114 4.3.8 KKT Analysis of WCAPCOMP 2.WOM 114 4.4 Numerical Results 124 4.4.1 Wind Capacity Allocation and Wind Output Market Activity 125 4.4.2 Coal Displacement 128 4.4.3 Electricity Market and Wind Output Market Contract Prices 130 4.4.4 Genco Profits 132 4.5 Conclusion 134 5.0 Market Structure Sensitivity 136 5.1 Effect of Cross-Market WOM Knowledge and Fossil Generation Ownership 137 5.2 Numerical Results for Sensitivity to WOM Knowledge & Fossil Ownership 140 5.3 Effect of Demand Curve Slope 144 5.4 Numerical Results for Sensitivity to Demand Curve Slope 147 5.5 Note on Stability of the WCAPCOMP Models 150 5.6 Conclusion 151 5.7 Recommendations for Future Work 152 6.0 References 154 7.0 Appendices 158 7.1 Appendix A: Constructing Demand Curves 158 7.2 Appendix B: GAMS Implementation of WCAPCOMP 1.0 160

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7.3 Appendix C: Calculation of Seasonal Hydro Parameters 165 7.4 Appendix D: Parameters Used for Chapter Three Numerical Examples 166 7.5 Appendix E: Determining Basic and Advanced Knowledge Cross-Market WOM Anticipation Coefficients 169 7.6 Appendix F: GAMS Implementation of WCAPCOMP 2.WOM 170 7.7 Appendix G: Parameters Used for Chapter Four Numerical Examples 176 7.8 Appendix H: Parameters Used For Sensitivity Examples in Chapter Five 178

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List of Figures

Figure 1-1: Valuation of new capacity in a traditional cost-minimizing utility paradigm 4 Figure 2-1: Diagram of the analytical WCM-EM model 27 Figure 2-2 - Solution to the open-loop WCM-EM model using some simple test parameters, with WTP for Gl and Gw indicated 34 Figure 2-3: Diagram of the WOM-EM model 44 Figure 2-4: Comparison of electricity market equilibria, with and without a contract-for- difference on A units of wind output from Gwto Gl 46 Figure 3-1: Diagram of WCAPCOMP 1.0 model 61 Figure 3-2: Converting a seasonal LDC into a discretized load block, and fitting a demand curve 63 Figure 3-3: Illustration of electricity market anticipation and perception of residual demand curve slope 65 Figure 3-4: Illustration of wind capacity market anticipation and perception of residual supply curve slope 67 Figure 3-5: Total wind construction and allocation amongst spot market gencos for low, medium and high nuclear cases 87 Figure 3-6:: Hourly WTP (dual variable on wind output constraint) for wind output in selected blocks, for each genco, defined as the mariginal revenue expected from having an extra MWh of wind output 88 Figure 3-7: Electricity market outputs for each genco by generation type, for selected seasons & blocks, in the low, medium and high nuclear cases 90 Figure 3-8: Comparison of wind energy added to system, and coal energy displaced 91 Figure 3-9: Electricity market prices for selected blocks, in the low, medium and high nuclear cases 94 Figure 3-10: Total profits for each genco in each deterministic nuclear case 95 Figure 4-1: Diagram of WCAPCOMP 2.WOM, focusing on the addition of a market for wind output contracts (WOM) occurring after uncertainty is resolved 99 Figure 4-2: Definition of wind output market contract, where seller Gw sells financial rigkts to A units of wind output to buyer G(i), at a contract price of WOP*A 102

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Figure 4-3: Wind Capacity investment in stage 1, for deterministic medium nuclear case, uncertainty case with no WOM (2.0), and with a WOM (2.WOM) 125 Figure 4-4: WOM contracts for selected blocks, in the low, medium and high nuclear outcomes 127 Figure 4-5: Annual coal displacement for deterministic cases (chapter 3), base uncertainty case (2.0), and uncertainty case with WOM (2.WOM) 128 Figure 4-6: Additional wind energy due to WOM, and resulting additional coal energy displacement 129 Figure 4-7: Electricity market prices for the 2.0 and 2.WOM models, along with the wind output market contract price in the 2.WOM models 131 Figure 4-8: Total expected profits, and total profits for each potential outcome, for the no WOM (2.0) and WOM (2.WOM) model cases 133 Figure 4-9: Expected and outcome profits to each genco, with and without a WOM 134 Figure 5-1: Wind capacity construction and allocation, for basic & advanced cross-market WOM knowledge, for OLG+OSG and OIG cases 140 Figure 5-2: WOM contract purchases, for basic & advanced cross-market WOM knowledge, and OLG+OSG & OIG fossil ownership cases 141 Figure 5-3: Annual wind energy versus coal displacement, for basic and advanced WOM knowledge, and OLG+OSG vs OIG fossil ownership 143 Figure 5-4: Illustration of effect of using different point elasticity assumptions to compute demand curve slope, on electricity market anticipation coefficients ae,* for myopic, basic, and advanced knowledge levels 146 Figure 5-5: Wind construction & allocation in e=0.5, e=0.8 cases for low, medium and high nuclear scenarios using WCAPCOMP 1.0 model 147 Figure 5-6: Annual wind energy versus coal displacement, for e=0.5 & e=0.8 cases 148 Figure 5-7: Electricity market prices for selected blocks, for e=0.5 & e=0.8 149

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List of Tables

Table 2-1: Comparison of WCM-EM solutions for open-loop, and closed-loop cases 41 Table 3-1-WCAPCOMP 1.0 model notation 68 Table 3-2: Dual constraints for all agents in WCAPCOMP 1.0 78 Table 4-1: Definition of basic and advanced cross-market WOM knowledge in the WCAPCOMP 2.WOM model 103 Table 4-2 - Additional notation for WCAPCOMP 2.WOM 107 Table 4-3: Dual constraints for gencos & WMO agents in WCAPCOMP 2.WOM 112

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1.0 Introduction

Wind generation is emerging as the fastest growing renewable energy technology over the past decade. According to the Global Wind Energy Council's 2006 report (GWEC, 2006), at the end of 2006 there was 74 GW of worldwide installed wind generation capacity, an increase of 25% from the previous year. Europe is at the forefront of this trend, with 48.5 GW of installed capacity, with Asia and North America following as emerging markets.

In Canada, total installed wind capacity more than doubled in 2006 over the previous year, with 1.5 GW total installed wind capacity, according to (CanWEA 2007). The Ontario Power Authority announced a Request-for-Proposal (RFP) for 300 MW of renewable energy projects in 2004, and out of the 10 winning projects totalling 395 MW, 98% of that capacity was wind generation. This highlights the prominence of wind generation as the current leading renewable technology.

There are several driving factors behind this boom in the wind industry. Firstly, the production cost of one kilowatt-hour (kWh) of wind generation has declined to one-fifth of what it was 20 years ago, bringing it to the point where it is nearly able to compete with fossil fuel generation such as coal and gas on a cost basis alone (Sahin, 2004). Secondly, an increased political & social awareness of the ecological impact of conventional electricity generation technologies such as coal has resulted in a greater interest in renewable energy technologies, particularly in the context of the Kyoto Protocol, which mandates aggressive reductions in greenhouse emissions for all signing nations, including Canada (Leggett, 1998).

Along with the emergence of wind generation, the Ontario electricity system itself has undergone some important changes over the past few years. The Ontario government restructured the electricity sector in 2002, transforming it from a regulated vertically integrated monopoly, to a competitive market with several generating entities (gencos), with the intention of lowering rates for consumers, through increased competition. The electricity market is now administered by the Independent Electricity System Operator (IESO), a regulated entity that accepts offers of generation and price from gencos, and sets the spot market price at the price at which supply meets demand; the market-clearing price.

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One area of uncertainty in the current Ontario system is the amount of nuclear capacity which will be available in the medium- to long-term future. Several plants in Ontario are already beyond their designed life, and nuclear refurbishment projects have met with significant delays and cost overruns in the past (Pembina Institute & CELA, 2004). In addition, policy makers are currently reviewing proposals to build new nuclear capacity, something which has not been done in the province of Ontario since the 1960s. As gencos are evaluating the wind capacity investment option, they are doing so under uncertainty as to the total amount of nuclear capacity which will be operating in the system over the life of the wind project.

The focus of this thesis is to examine the value of wind capacity in a competitive market similar to the Ontario situation, by formulating an Ontario-like model consisting of a wind capacity market, where gencos purchase capacity from an aggregation of price-taking suppliers, and an electricity spot market where gencos sell output, consisting of conventional generation types and hydro as well as wind generation, to price-taking consumers. We then extend this model to a stochastic case, where wind investments must be made under uncertainty of the level of nuclear capacity in the system. In this stochastic Ontario-like model, we introduce the concept of a wind output contract-for-difference, where one genco can sell financial rights to wind output to another genco competing in the spot market, as recourse to the wind investment decisions, once nuclear uncertainty has been resolved.

Our interest lies in identifying the effect of market knowledge, nuclear uncertainty, and the market structure (number of gencos and their generation portfolios) on the amount of wind capacity constructed, it's allocation amongst the gencos, the perceived value of wind capacity to each genco, and how much coal generation is displaced by the addition of wind into the system.

Finally, we note that the approach and results of this thesis can be applied to any low-cost generation type which has an operating cost sufficiently low that it is always sold to market. We use wind generation in this thesis, because it is a renewable generation technology with a low operating cost, which is receiving a lot of attention as one candidate to displace

2 3 generation types with higher emissions such as coal, whenever the intermittent wind resource is available.

1.1 Motivation

A typical approach to the valuation of new capacity, in the traditional cost-minimizing utility paradigm, is to construct a model of the generation assets of the utility, along with the loads requirements it must meet, and the cost of constructing new capacity of a given type, with an objective of meeting the required load for a given time horizon at a minimum cost, potentially with some stochastic element such as demand uncertainty (for example, (Bienstock & Shapiro, 1988)).

For a given load duration curve (LDC), which represents the number of hours of the year in which load exceeds a given level and is assumed fixed, a ballpark approach to capacity valuation would be to determine the cost savings associated with having the new capacity type available to the utility.

To illustrate this ballpark approach graphically, we present an LDC which is reflective of the cost structure in our Ontario-like models from the later chapters. To determine the value of 1 GW of wind output to the utility, we add this wind output to the LDC, and observe how it affects the utilization of the incumbent generation types.

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Resource Adequacy Approach To Valuation of

Capacity For Cost Minimizing Utility

Load (GW) 15

10 _

5 _

Nuclear

ROR Hydro I i I r 2190 h 4380 h 6570 h 8760 h Duration (hrs)

Figure 1-1: Valuation of new capacity in a traditional cost-minimizing utility paradigm.

Assuming for the moment that the shaded region represents average wind output across the year (neglecting seasonal and diurnal variation, which are accounted for in our later models), the addition of wind output serves to displace coal generation for the majority of the year, and so for a unit of wind output made available to the utility, it would displace coal output on an almost 1:1 basis in terms of energy. Therefore, the value of a unit of wind output in this paradigm is approximately equal to the value of the coal output displaced in this paradigm. Computing the value of the coal energy displaced over the year by multiplying the operating cost of coal by the duration shown above, and assuming some wind capacity factor, which converts the maximum rated wind capacity to an average power output of a turbine after accounting for intermittent wind availability, the utility would obtain the value of an extra unit of installed wind capacity.

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By comparing this value of wind capacity to the cost of investing in the capacity, the utility would choose a level of wind capacity such that the marginal cost of an extra unit of capacity is equal to the marginal value of the output it produces.

However, the trend of liberalizing electricity markets in North America has provided a new paradigm for which the above approach does not apply. In a survey paper of stochastic models in the energy sector, (Fleten & Wallace 2003) make the following observation:

"In a deregulated and well-functioning market, capacity expansion decisions should be analyzed in view of their profit and market value adding potential, and not their ability to serve growing demand at minimum cost. "

The authors further point out that although much work has been done on stochastic programming models of capacity expansion under the minimum cost utility, there has been little academic focus on stochastic capacity expansion models in a competitive market context, and this is an emerging field that has not yet matured.

It is to this end that we present in this thesis a model of capacity expansion in which competing gencos have the objective of maximizing profits, rather than meeting demand at minimum cost. In our models, gencos have differing perceptions of the marginal value of wind capacity, as well as the marginal expenditure required to invest in this capacity, which yields different results than the traditional cost minimizing utility approach described at the beginning of this section.

1.2 Purpose and Methodology

This thesis presents several models of a competitive electricity market, starting with WCAPCOMP 1.0, where several gencos compete to purchase wind capacity in a single-price capacity market from a separately-owned price-taking supplier who does not have access to the electricity market, and then sell generation to the electricity market, with price-taking consumers. The useful outputs of this model are the total wind construction, it's allocation amongst the gencos, the value of wind capacity to each genco, how much coal output is displaced by this generation, as well as the profits to each genco. These outputs are obtained for a given set of market parameters including the amount of existing generation in the 5 6 system, costs, demands, and perceptions of market behaviour for each participant, as supplied by the analyst.

We also extend this model to a stochastic version WCAPCOMP 2.WOM, where the amount of nuclear capacity in the system is uncertain at the time of wind capacity investment, and is known prior to the electricity market stage. In this model, because of the presence of uncertainty, we introduce a wind output market, where gencos can trade financial rights to wind output via contracts-for-difference, where the purchasing genco pays a contract price for the financial rights to a quantity of wind output from another genco (who physically delivers the generation to the market), and receives the electricity spot market price for this quantity. This allows recourse to the wind investment decisions following the resolution of uncertainty.

The WCAPCOMP models are both decision support models which utilize an anticipation framework, in which gencos make assumptions about the response of their competitors in the various markets to their decisions (eg. Asl & Rogers, 2004). This approach allows for emulation of a range of market knowledge, from open-loop knowledge, where gencos assume that their competitors will not alter their decisions, to closed-loop knowledge, where, have knowledge of the response of their competitor's reactions to their decisions (these terms are more properly defined at the beginning of Chapter Two). The use of anticipation coefficients in these markets for each participant allows the analyst to vary knowledge levels between these theoretical bounds.

The models are all equilibrium models, which yield as the solution a Nash equilibrium, a concept which is defined in the literature review in the next section.

Prior to presenting the larger WCAPCOMP models, we first analytically explore a small, simplified equilibrium model of capacity expansion and spot market competition (WCM- EM), and obtain solutions for open-loop and closed-loop levels of knowledge. These provide a foundation to demonstrate how knowledge affects the value of wind capacity for gencos in a competitive market setting, which is used to foreshadow the results obtained from the larger WCAPCOMP 1.0 model. We then perform a similar analysis for a simplified wind output market and electricity market model (WOM-EM), exploring the value of wind output

6 7 contracts-for-difference to gencos under open-loop and closed-loop assumptions, which sets up the WCAPCOMP 2.WOM model which includes a wind output market for such contracts.

The models of this thesis are meant to represent a medium- to long-term market equilibrium, with the electricity spot market representing one year of electricity competition, and average market behaviour over hundreds of hours, taking a resource adequacy approach at a strategic level of detail. As such, we do not consider operational details such as ramping constraints, or imbalance penalties for a genco not meeting it's scheduled generation commitments due to short-term variability in wind. Finally, we do not consider transmission constraints, instead assuming that all supplies and demands of electricity exist at the same node.

1.3 Literature Review

The published literature that is relevant to this thesis can be divided into two general categories: electricity market equilibrium models, and capacity expansion models, which can be stochastic or deterministic. We will review the literature for these two categories in this section, followed by a more detailed comparison of our work to three relevant papers.

1.3.1 Electricity Market Equilibrium Models

Modeling of electricity markets has been a relatively active research area over the past ten years. (Ventosa et. al., 2005) presents a useful survey of electricity market modeling approaches, classifying them generally into three categories: optimization models, equilibrium models, and simulation models. Optimization models focus on a single firm's decision, while equilibrium models capture the behaviour of all firms in the market, in the context of competition. Simulation models are used when the situation to be modeled is too complex to be represented as an equilibrium model. Ventosa et. al. also state that equilibrium models are most useful for studying long-term planning and market power analysis, and less so for short-term, detailed operational analysis.

The concept of an equilibrium model is most attractive for this thesis, because competitive markets are comprised of several decision-making agents, each with their own optimization problem and objective function of maximizing profits. The goal of an equilibrium model is to identify Nash equilibria (Ventosa et. al. 2005). 7 8

(Fudenberg & Tirole, 1991) define the concept of a Nash equilibrium as follows:

"A Nash equilibrium is a profile of strategies such that each player's strategy is an optimal response to the other player's strategies. "

In other words, each decision-making entity believes that it cannot further improve it's objective function unilaterally by altering it's decisions. It is important to note that this equilibrium depends on the perceptions each genco has of it's counterpart's behaviour.

When discussing models of imperfectly competitive markets, it is important to understand the information held by each decision-making agent in a given model. This knowledge manifests itself in assumptions made by an agent as to the reaction of other agent's decision variables with respect to it's own decision variables.

Within equilibrium models, there are three main approaches which consider imperfect information held by agents: quantity-based competition (commonly referred to as Cournot, or open-loop Cournot competition), and supply function equilibrium (SFE), as well as imperfect knowledge approaches, which are more flexible in that they can represent varying levels of competitiveness between agents. Finally, leader-follower games are increasingly being used to model situations where one or more agents have perfect knowledge about other agents. The models we present in Chapters Three and Four can be classified as imperfect knowledge models, which allow us to generate results for various levels of knowledge held by each agent, as to the expected response of it's counterparts to it's decisions.

Some relevant papers illustrating these approaches to electricity market modeling will now be reviewed.

1.3.1.1 Quantity-based Competition Approaches

In quantity -based competition, each genco chooses it's generation output quantity, and the market price is determined by a demand curve representing consumers, which relates the total quantity of all gencos to a market-clearing price. In quantity competition under Cournot, or open-loop assumptions, gencos know the demand curve, each genco forecasts the output quantities of it's competitors, and takes them to be fixed, with the Cournot equilibrium being

8 9 the point where each genco's forecast is accurate, according to (Varian, 1996, p. 469). By contrast, a perfectly competitive (PC) genco assumes that market price is unresponsive to it's output (ie. no demand curve exists). The simplest Cournot models are one-stage games, and can be solved analytically for small models, or via a complementarity programming (CP) approach for larger ones.

For example, in (Hobbs, 2001) gencos choose production quantities on a spatialized network, with each node having a spot price, and nodes being connected via transmission network. The key assumption here is that each agent views the decision variables of the other agents as unresponsive to it's own decisions. The model is solved by aggregating the Karush-Kuhn- Tucker (KKT) conditions of each agent, and solving the resulting mixed complementarity program (MCP), which is the solution approach we use to solve our models. The authors compare perfect competition results to Cournot results, with the finding (common to Cournot models) that the Cournot equilibrium price is higher than the perfectly competitive result. This acknowledges that oligopolistic firms may be able to influence the spot market price under Cournot assumptions, and therefore possess some market power. Another similar example of a Cournot model that is solved as an MCP can be found in (Ivanic et. al., 2005), with the main difference being the inclusion of resistance losses along transmission lines.

We consider quantity-based competition in this thesis, but with open-loop Cournot knowledge as merely one possible level of knowledge (see imperfect knowledge models, Section 1.3.1.3). The above models are concerned with transmission networks in an imperfect competition setting, something that we do not address. Our simplification is that all demand and supply exist at a single node, and therefore a single spot market price exists at a given time, which would be the case for a network with unlimited transmission capacity, or transmission capacity that is not binding for the majority of the time. We are focused on modeling the electricity system from a resource adequacy perspective, rather than a network planning perspective.

1.3.1.2 Supply Function Equilibrium Approaches

In the SFE approach, each firm submits a supply curve to an independent market operator (IMO) which is a function of price, and the EVIO chooses a price at which the market clears 9 10 by matching the total supply to demand, given each genco's supply curve. The assumption made in a supply function equilibrium model is typically that the supply functions offered by competitors are fixed.

The SFE approach was initially proposed by (Klemperer & Meyer, 1989) for a general commodity market, and adapted to electricity markets by (Green & Newbery, 1992). SFE models do not rely on an explicit demand curve assumption, instead assuming a fixed load at a given time, whereas Cournot models typically assume a linear demand curve, implying some level of demand elasticity. This point is cited by some authors (Green & Newbery, 1992), (Song et. al., 2003) to suggest that SFE models are more appropriate for modeling short-to-medium term electricity prices, where demand can be viewed as relatively inelastic. The equilibrium price of SFE models tends to lie between PC and Cournot results.

However, SFE models are more difficult to solve, given that in the general case, there can be an infinite number of equilibria (Rudkevich et. al, 1998), due to the fact that there are many supply functions that gencos could offer that would clear the market. Because of this, (Ventosa et. al., 2005) points out that most SFE models adopt a very limited level of detail in representing the production of gencos, to aid in maintaining a tractable model.

Because we are interested in long-term prices useful for investment decisions, and will need a more detailed production model in order to capture the unique aspects of wind generation, the SFE approach was not considered for this thesis.

1.3.1.3 Imperfect Knowledge Approaches

Some models seek to represent levels of competition that are neither PC nor Cournot. Such models allow for each agent to have it's own assumption as to the net reaction of all other agents to it's own decisions; in other words, the assumptions of each firm can be varied to represent various levels of competitive behaviour.

One example of a conjectural variation model can be found in (Day et. al., 2002), where gencos choose parameters for a supply curve, but adopt an assumption of the quantity response of the other gencos to the market price. This is referred to as a conjectured supply function (CSF) approach, and the assumption made by each genco is that the aggregate 10 11 supply of other gencos changes linearly, with either a fixed y-intercept or slope. Changing the parameters of the CSF allows for various levels of competition to be modeled, with Cournot as a special case if desired. The resulting model in this paper is formed by grouping the KKT conditions of gencos, an ISO, arbitragers and a set of market clearing conditions, and is solved as an MCP.

In a more advanced CSF model proposed in (Song et. al., 2003), a CV parameter is supplied for each genco to capture the assumed rate of change of all other competitor's supplied quantity, as the price changes. This allows for models where each genco potentially behaves differently with regards to competitiveness. By adjusting the CV parameter for each genco, results for PC, Cournot, and other cases are presented, with linear elastic and inelastic demand.

Finally, (Asl, Rogers, 2004) present an imperfect knowledge model which is based on quantity competition, where each genco makes an offer of price and quantity to the IMO. Each genco has an anticipation coefficient which captures the assumed rate of change of the market price, if an additional unit of quantity is offered and accepted by the IMO, representing the aggregate output response of the remaining gencos. The models presented in Chapters Three and Four are extensions of this approach, adding a wind capacity market, uncertainty, and a contract market for wind output, as well as the ability to account for seasonal and diurnal variation in with availability. This is one of the models with which we compare this thesis to in more detail in Section 1.3.3.

The flexibility of these models allows for the study of varying levels of market knowledge in an electricity market. However, as pointed out in (Day et. al., 2003), the solution of such a model is dependent on a given set of assumptions held by each firm, and such a solution may not be a long-term Nash equilibrium if the assumptions held are not correct. This is because agents would conceivably change their strategy, after observing the market outcome and realizing that their assumption was not correct.

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1.3.1.4 Leader - Follower Approaches

In the previous models, each decision-making agent had imperfect knowledge of it's counterparts, and instead adopted some assumption as to their behaviour. A leader-follower game arises in any situation where one or more agents (leaders) act with perfect information about the reactions other agents (followers). Conversely, the followers assume no reaction from the leaders (that is, they have open-loop knowledge). This means that a leader solves an optimization problem which includes the KKT conditions of the followers as constraints, leading to a mathematical program with equilibrium constraints (MPEC), which is a non linear program (NLP). If more than one leader exists, the problem becomes an equilibrium program with equilibrium constraints (EPEC), in which each leader solves an MPEC. These models are also referred to as bilevel games, and can be viewed as two-stage optimization problems.

MPECs and EPECs are generally non-convex, and such problems have been difficult to solve, until recently. In (Leyffer & Munson, 2005), the authors outline some formulation and solution approaches to leader-follower games. NLP and MCP formulations of MPECs and EPECs are discussed, which can be solved using conventional solvers such as CONOPT3 or PATH, in the GAMS environment.

The NLP approach outlined by Leyffer & Munson was used by (Anjos et. al., 2007) to solve a Cournot transmission model which featured several leader gencos with perfect information about the followers, which consisted of the ISO and a competitive fringe of generators, who acted as perfect competitors. The largest resulting EPEC consisted of 2799 variables and 1665 constraints, and was solved in a few minutes using the CONOPT3 NLP solver. However, the authors noted that the model was highly sensitive to the initial starting point given to the algorithm, highlighting the need for some prior expectation of the equilibrium point, in order to specify a starting point which will lead to a Nash equilibrium, and to verify that the solution is sensible.

An example of a two-stage model is provided in (Yao et. al., 2007), where firms choose forward market commitments in a Cournot fashion in stage 1, and spot market output in stage 2, resulting in an EPEC model. We discuss this paper in more detail in Section 1.3.4. 12 13

1.3.2 Investment in Capacity Models

The models discussed in this section are concerned with investment in capacity in a competitive market context, where one or more gencos are able to choose a capacity level, either prior to choosing spot market production levels, or simultaneously. This relates to our topic of investment in wind capacity, in an imperfectly competitive market setting. Such models can either be deterministic or stochastic.

(Fleten & Wallace, 2003) provide a useful summary of stochastic capacity expansion literature in the energy sector, noting that while much work has been done on the traditional cost-minimizing utility paradigm, the application to competitive markets is an emerging field. Stochastic capacity expansion models involving hydrothermal capacity in a non competitive context for a single decision maker, and typically view demand levels or other system parameters as stochastic in nature (e.g. (Pokharel & Ponnambalam, 1997), (Bienstock & Shapiro, 1988)), and seek an investment plan which minimizes the expected cost of meeting demand, subject to various operating parameters. Our WCAPCOMP models, by contrast, are concerned with the competitive market paradigm, with multiple decision making gencos, all of whom may invest in capacity, and compete for that capacity in a single-price market.

In terms of stochastic models related to wind generation, the current literature focuses on stochasticity in terms of wind output variability, rather than uncertainty in the state of the system, with a short-term operational focus. For example, (Meibom et al, 2007) discuss a multi-stage stochastic model on an hourly scale, and estimate the cost and price effects due to the integration of wind power into the system, assuming an efficient market with no market power, market prices set by marginal production costs, and exogenous demand. Our WCAPCOMP models represent a competitive market with endogenous demand and spot prices determined by the offers of gencos, rather than marginal production cost. Further, different levels of market knowledge held by each genco imply some level of market power exists in our model, which focuses on a strategic level of detail, with no consideration of imbalance penalties or other costs incurred by the inability to deliver wind generation at a given point in time, as is the focus of (Meibom et al, 2007).

13 14

A comparison of simultaneous and sequential choice of capacity is presented in (Ventosa et. al., 2002). In the simultaneous model, firms choose capacity and output in a Cournot fashion, assuming that the choices of capacity and output of other firms is fixed, and the resulting model forms an MCP, which is solved via PATH. This is an example of a open-loop model, in which the decision makers do not have information about how their counterpart's decisions will change with respect to their own. An alternative model presented in the same paper features a single leader firm, which makes a capacity decision before all others, having perfect knowledge of the reactions of the following firms. This type of model is referred to as a closed-loop model, and is solved using CONOPT2. The results showed that the amount of capacity built by the closed-loop player is greater than the amount built by the open-loop model, and the leader is able to earn higher profits. This difference in profit reflects the value to the leader firm of having perfect information about the competitor's reactions to it's capacity expansion decision.

The model contained in (Gabszewicz & Poddar, 1997) is a stochastic two-stage capacity expansion model with uncertainty between stage 1 and stage 2, and is similar in structure to the forward/spot model of (Yao et. al. 2007). In stage 1, firms choose capacity knowing only the probability of each state of nature, which relate to demand levels. Output levels are chosen in stage 2 under Cournot competition, after the demand state is known. The authors illustrate that the capacity built under uncertainly is greater than or equal to the capacity built in a "certainty equivalent" case which uses the expected value of demand. Furthermore, if the firms choose a level of capacity such that they are constrained in at least one of the possible demand states, then the solution to the two-stage uncertainty model has higher capacity built than in the "certainty equivalent" model. The model takes the form of an EPEC, but only symmetrical genco cases are considered, and gencos have no incumbent capacity, other than what they build in stage 1. In contrast, our model assumes that each genco has some existing capacity of a conventional nature, and has the option to purchase wind to supplement that capacity.

Finally, (Murphy & Smeers, 2005) is another example of a comparison of open- and closed- loop investment capacity decision models, which take a similar form to the small analytical

14 15 capacity expansion model WCM-EM we present in Chapter Two, with some key differences. This paper is compared to the thesis in more detail in Section 1.3.5.

1.3.3 Comparison To Asl, Rogers 2004

(Asl & Rogers, 2004) present an imperfect knowledge model ELFORSPOT which is based on quantity competition, with forward and spot markets for output. In each market, there were strategic gencos and customers (custcos). Gencos would make an offer of price and quantity to the IMO, with an anticipation coefficient which captures the assumed rate of change of the market price, if an additional unit of quantity is offered and accepted by the IMO, representing the aggregate output response of the remaining gencos in the market. The custcos made bids to the IMO, and the IMO set the single market-clearing price for electricity, with an FMO administering the forward market clearing price in a similar manner.

The spot market structure of our model is based on the structure of the spot market component of ELFORSPOT, with multiple load blocks representing a year of electricity market competition, and multiple generation technologies available to each genco. However, our model does not have strategic custcos, assuming instead price-takers in the electricity market.

Like ELFORSPOT, our WCAPCOMP model is a resource adequacy model, in the sense that it focuses on average amounts of output, prices, and genco behaviour over several hundred hours in each block, and what the resulting generation mix is in each block, rather than a more detailed short-term operational focus. The WCAPCOMP model in this thesis extends the spot market portion of ELFORSPOT to include multiple seasons as well as blocks, to allow for the study of wind capacity, which varies in availability seasonally and diuraally. Our model also departs from the work of (Asl & Rogers, 2004) as we incorporate a wind capacity market (in Chapter Three), and uncertainty along with a wind output market (in Chapter Four), neither of which are considered in ELFORSPOT.

Another mode of comparison is the forward markets of ELFORSPOT versus the wind output market of WCAPCOMP 2.WOM. The difference between markets is that in the former

15 16 model, the buyers of forward output contracts are end customers, while the buyers of wind output contracts in our model are other gencos. As we show in Chapter Four, each genco has a different perceived value of a wind output contract, depending on who the contract is signed with, and so a market with potential for bilateral pricing emerges. Furthermore, in our model each genco, depending on whether it has generation on the margin in the electricity market, will adjust it's electricity market outputs differently, and so the effect of selling a contract to genco A vs. Genco B in the wind output market on the electricity market price will be different. In ELFORSPOT, gencos had a single cross-market anticipation of the effect of selling forward contracts on the spot price, regardless of who the buyer was, while in our model, the seller has a different anticipation coefficient for each potential buyer.

1.3.4 Comparison to Yao et. al. 2007

In (Yao et. al., 2007), firms choose forward market commitments in stage 1, and spot market output in stage 2, in both stages behaving as Cournot players, in a spatialized network with a system operator. There are also several stage 2 contingencies, consisting of demand variation and line outages, which result in several potential spot market outcomes. Forward commitments are made based on the expected stage 2 outcome, and spot market decisions are made after the contingency is realized.

This model has the same agents making decisions in stages 1 and 2, but with different levels of information. For example, in stage 2 the firms know the forward commitments of all other firms, and the contingency outcome, and choose spot outputs as Cournot players. In stage 1, firms know the probability of each contingency occurring, assume that the forward contract decisions of the other firms are fixed, and know the KKT conditions of all stage 2 agents for each contingency. Another way of stating this is that each genco in stage 1 knows that the players in stage 2 behave as Cournot, at the time the forward contract decision is being made.

In the Yao et. al. model, each stage 1 firm solves an MPEC consisting of the KKT conditions of all stage 2 agents, and the resulting equilibrium is solved for as an EPEC. The stage 2 KKT conditions completely define the stage 2 behavior of the agents as Cournot in that stage, and including them as primal constraints in each genco's MPEC gives the gencos perfect information with regards to the impact of their forward decisions on the spot market 16 17 decisions of the stage 2 agents, while ensuring that their stage 2 decisions are made in a Cournot manner. A solution to the EPEC is obtained using an iterative approach where each MPEC is solved, taking the stage 1 decisions of the other firms as fixed, and iterating among MPECs until the difference between iteration solutions is arbitrarily small.

Our WCAPCOMP models are able to approximate EPEC results, but can also represent more realistic competitive behaviour paradigms within each stage than just Cournot, by choosing assumptions for each agent which approximate levels of knowledge that are somewhere between open-loop and closed-loop in each market, while avoiding the computational and theoretical difficulties associated with EPECs. Furthermore, our approach allows us to model gencos as having different levels of knowledge in each market, and examine the effect of varying the level of knowledge for one or several of the participants.

1.3.5 Comparison to Murphy, Smeers 2005

(Murphy & Smeers, 2005) provides a comparison of three investment capacity decision models: a perfectly competitive model, an open-loop Cournot model where decisions are made simultaneously similarly to the model of (Ventosa et. al., 2002), and a "closed-loop Cournot" model where investment in capacity is made by all firms prior to output decisions, which is an EPEC model. Output is sold in the spot market to a price-taking consumer, in a Cournot-like fashion. In the open-loop formulation, each genco assumes that the spot output of it's competitor remains fixed, while in the closed-loop formulation, each genco is assumed to have perfect knowledge of how the spot output of it's competitor will change as it increases its output. The models are intended to be simple enough to facilitate analytical analysis.

Murphy & Smeers discuss a vertically-integrated capacity construction case, where each genco can build it's own type of generating capacity (base or peak). The gencos are vertically integrated, in the sense that their capacity construction is done independently, and the cost of building capacity is not related to the amount of capacity that the other genco builds. The authors note that the value of a plant in the open-loop model is equal to the sum of the shadow prices on the capacity constraints in each time segment, but in the "closed-loop Cournot" model the value of building a plant is greater than the sum of these shadow prices.

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The additional value in the latter model comes from the benefit of being able to make investment decisions that affect the state of the spot market.

The WCAPCOMP models differ from (Murphy & Smeers, 2005) in two key areas: firstly, we include some incumbent capacity owned by the gencos (which we demonstrate has an effect on the willingness-to-pay for new wind capacity), and secondly, the gencos in WCAPCOMP compete for capacity in a single-price market, which means that the capacity decisions of one genco affect the others, rather than construct capacity as a vertically- integrated firm, where the capacity investment costs are not affected by the capacity construction of the competitors. This makes sense if gencos are competing for wind capacity, with a finite number of suitable sites in a geographical area, implying that the capacity decisions of the gencos affect one another.

1.4 Thesis Overview

The following section provides a summary of the important results of each subsequent chapter, to give the reader a preview of the general topics covered in each chapter.

1.4.1 Chapter Two Summary

In Chapter Two, we present two small analytical models, which are used to explain the effects observed in the larger Ontario-like models of Chapters Three and Four. The first is WCM-EM, and contains a wind capacity market (WCM) along with an electricity spot market (EM), with two gencos: Gl with conventional capacity, and Gw with no conventional capacity. Both gencos purchase wind capacity from a price taking supplier in the wind capacity market as Cournot competitors, and can sell generation to price-taking consumers in the electricity market. Gencos use their perceptions of marginal expenditure in the WCM, and marginal revenue in the EM to determine how much wind capacity to purchase, and how much output to sell.

In the electricity market, we compare the willingness-to-pay (WTP) for an extra unit of wind output for each genco in the electricity market, assuming either open-loop or closed-loop levels of knowledge. Our conclusion is that Gl has a constant WTP for wind capacity equal to the value of displacing conventional generation, regardless of whether it has open- or 18 19 closed-loop knowledge, as long as it is selling some conventional generation to the market. For Gw, however, the WTP is a decreasing function of not only the electricity market price, but also the amount of total output it has in the electricity market. In doing so we establish that the willingness-to-pay for wind generation in a competitive market context is dependent on the generation portfolios of the gencos, their perceptions of the other gencos behaviour in the electricity market, and their level of output to the market. This model is useful to set up the more realistic WCAPCOMP 1.0 model of Chapter Three.

The second analytical model, WOM-EM, looks at the concept of a contract-for-difference for wind output, whereby Gl can purchase financial rights to wind output from Gw in the electricity market, assuming wind capacity investments are fixed. We show that if Gl purchases such a contract, it has an incentive to reduce it's conventional output, and increase the electricity market price it receives over it's (now larger) financial output1 to the market. We demonstrate that with closed-loop levels of knowledge, there exists a contract price such that both gencos are better off after signing such a contract, and compare the WTP for contracts-for-difference for the two-genco case for both open-loop and closed-loop levels of knowledge. This foreshadows that the purchasers of such contracts will be conventional gencos with coal generation on the margin , which we observe in Chapter Four.

1.4.2 Chapter Three Summary

In Chapter Three, we formulate WCAPCOMP 1.0, which is a deterministic Ontario-like decision support market model representing a wind capacity market and one year of electricity market competition, including gencos with different generation assets (hydro, nuclear, coal, oil/gas), and incorporating seasonal and diurnal variation in wind availability,

We use the term 'financial output' to refer to the total output for which a genco is receiving revenues for in the electricity market, consisting of it's physical outputs, adjusted by the amount of contracts-for-difference it has purchased (or sold).

We use the term 'on the margin' to refer to a generation technology which is operating at an unconstrained level (ie. operating at a non-zero level, and not fully utilized).

19 20 seasonal hydro allocation, several load blocks, wind capacity and electricity market operators, and maintenance. Both markets in WCAPCOMP 1.0 are settled simultaneously. WCAPCOMP 1.0 utilizes an anticipation approach, where each genco has an anticipation of the net effect of buying wind capacity in the WCM, or selling extra output in the EM, given some level of knowledge as to how it's competitors will react. This gives the model analyst the ability to determine the Nash equilibrium for a given set of market parameters and knowledge assumptions for each genco, in each market, and the model outputs of interest are the wind construction and allocation, coal displacement due to wind, and profits to the gencos. The solution to WCAPCOMP 1.0 is a Nash equilibrium, given each genco's level of market knowledge.

After presenting the model structure, we exploit the properties of complementary slackness in the KKT conditions to analyze the model equations, and derive expressions for the WTP for wind output for gencos with conventional generation on or off the margin. We also define levels of market knowledge in the wind capacity and electricity markets as being either basic (gencos do not anticipate their competitors will respond to their actions), or advanced (gencos anticipate that competitors will respond to their actions); an approximation of open- and closed-loop imperfect knowledge as defined in the literature review.

We then present some numerical results for an Ontario-like market with fossil generation ownership split between two gencos, as per the Market Design Committee recommendation (Trebilcock & Hrab, 2005), with 4 gencos total, including a wind-only new entrant. The largest genco is modeled as having advanced knowledge in the electricity market, while the wind-only genco is assumed to have advanced knowledge in the wind capacity market, with all other gencos in each market having basic knowledge. We compare three deterministic nuclear capacity cases, with the medium case having 11.5 GW of total nuclear capacity in the system, and low and high nuclear cases with +/- 2.5 GW.

We show that low nuclear capacity results in a higher value of wind capacity to the new entrant due to higher electricity market prices, with approximately 1 GW (15%) more total wind capacity being constructed in the low nuclear case than the high case. We also compare the ratio of wind energy in the system to coal energy displaced, both annually and within

20 21 given blocks, and show that while annual ratios range from 0.48-0.75 depending on the case, the ratio within blocks can range from 0.3 to a value of 2. In general, coal displacement ratios below 1 are observed because the majority of the wind output is owned by the new entrant, rather than the coal genco(s) who would use it to displace coal output (as seen in Chapter Two), and the coal genco(s) are only reducing their coal output in response to lower electricity prices due to additional wind output in the system. Finally, we note that the wind entrant faces the largest financial downside risk due to nuclear uncertainty, with 56% lower profits in the high nuclear case than the low nuclear case.

1.4.3 Chapter Four Summary

In Chapter Four, we extend the WCAPCOMP 1.0 model to a stochastic version WCAPCOMP 2.WOM which incorporates nuclear uncertainty, with a forward wind capacity market pre-uncertainty, and the electricity market occurring post-uncertainty. We also allow the wind-only genco to sell wind output contracts-for-difference to the conventional gencos in a wind output market (WOM), occurring on the same timescale as the electricity market, as a recourse to the wind investment decisions in stage 1. To fulfill a WOM contract, the wind-only genco seller delivers the output to the electricity market, but gives the buying genco the revenue equal to the equilibrium price multiplied by the quantity covered by the contract, in return for which the buyer pays the seller a wind output contract price. An anticipation approach is used to represent each WOM participant's level of knowledge as to how the buying or selling of WOM contracts affect the optimal electricity market output, and resulting electricity market price. We present similar definitions of basic and advanced knowledge in the WOM as in the WCM and EM of Chapter Three, which mimic degrees of imperfect information ranging from open- to closed-loop like levels in the wind output market for contracts.

We again exploit the complementary slackness properties of the KKT conditions to demonstrate that conventional gencos with generation on the margin have incentive to purchase WOM contracts from the wind-only genco, and then reduce their conventional outputs to increase the price received for their (now larger) financial outputs to the electricity market. We also show that no mutually acceptable contract price exists between the wind

21 22 entrant and a conventional genco with no generation on the margin. However, a genco that has conventional generation on the margin is willing to pay a contract price which is an increasing function of three factors: the operating cost of it's marginal conventional capacity, it's total (financial) output to the electricity market, and the electricity market price. All three of these components depend on the genco's perception as to how these factors will change as it purchases more wind output in WOM contracts. Particularly, the larger the output quantity of a genco, the higher the price it is willing to pay for a WOM contract-for-difference if it has coal on the margin.

We conclude by presenting some numerical results of the WCAPCOMP 2.WOM model, as well as a 2.0 model with no WOM, and use the three nuclear outcomes from Chapter Three as the possible stage 2 scenarios. In the 2.WOM model example, we assume that the gencos have advanced WOM knowledge. The results show that the presence of the WOM yields 7% higher total wind construction in stage 1, and 13% higher expected profits for the wind entrant. Furthermore, the WOM contract buyers in stage 2 are the coal gencos, who purchase wind output contracts and then reduce their coal outputs, which increases the ratio of wind energy to coal energy displaced. In the low nuclear outcome, the large coal genco (who also owns nuclear and hydro generation) has the highest WTP for WOM contracts because it has a large offer quantity in the market, and is able to pay a contract price higher than the electricity price to the wind entrant, which the smaller coal genco cannot afford. In the medium and high nuclear outcomes, the smaller coal genco is the purchaser of WOM contracts, as the large genco does not have coal on the margin in these blocks, because it has more nuclear capacity available.

1.4.4 Chapter Five Summary

Finally, in Chapter Five we examine the sensitivity of the WCAPCOMP models to three dimensions: fossil generation ownership (split ownership vs single ownership), level of market knowledge in the WOM, and the slope of the demand curve used to describe electricity consumers. In the first half of the chapter, using the WCAPCOMP 2. WOM model, we run cases with unevenly split fossil ownership (from Chapter Four) and single fossil

22 23 ownership (combining the assets of the two fossil gencos), and comparing the effect of basic versus advanced WOM knowledge for each case, for a total of 4 comparison cases.

We observe that with basic knowledge in the WOM, the wind entrant will only accept contracts for a price greater than the electricity price. As a result, only the large coal genco (in the split ownership case) or the integrated genco (in the single ownership case) are able to buy wind output contracts at such a price, because they can then reduce their coal output, and increase the electricity price over their (now larger) amount of generation output. Therefore, in the split ownership case we only observe WOM contract purchases in the low nuclear outcome, when the large genco has coal on the margin. In the integrated genco case, the genco has a very large output quantity, and has coal on the margin in almost every block, and therefore purchases larger amounts of wind output under WOM contracts-for-difference than the split-ownership gencos do. The integrated fossil genco case yields the highest coal displacement ratios, and it is found that assuming basic or advanced knowledge in this case has little effect on contracts.

In the second half of the chapter, we look at the effect of shallower demand curve slopes in the electricity market (representing longer-term demand response by consumers) on the value of wind capacity, wind construction, and coal displacement, using the deterministic WCAPCOMP 1.0 model. Increasing the point elasticity used to compute the linear demand curves from e=0.5 (used in the Chapter Three and Four models) to e=0.8 results in 4-7% higher wind construction depending on the nuclear case, and increases the annual coal displacement ratios by 0.05 - 0.12 (for the medium and high nuclear cases, where coal is on the margin for the majority of the blocks). This increased wind construction occurs even as the electricity market prices in the e=0.8 model are lower by 4-11%, which is a counter intuitive result, and is explained by noting that gencos use expectation of marginal revenue to determine their capacity purchase and output decisions, and a shallower demand curve slope gives a higher perceived marginal revenue associated with selling additional wind output to the system.

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2.0 Analytical Derivation of Value of Wind in Competitive Markets for Open-Loop and Closed-Loop Knowledge

In Chapter Two, we will discuss open- and closed-loop versions of two simplified illustrative competitive market models involving wind capacity and electricity competition. Each analytical model is used to derive the value of wind capacity, or willingness-to-pay (WTP), for open-loop and closed-loop levels of knowledge, for two different gencos: Gl, which owns some incumbent conventional capacity, and Gw, who is a wind-only entrant with no conventional capacity. These results will be useful in interpreting the results of the larger Ontario-like models of Chapters Three and Four.

(Fudenberg & Tirole, 1991) define the terms "open-loop" and "closed-loop" in the context of multi-stage games as follows:

"The terms closed-loop and open-loop are used to distinguish between two different information structures in multi-stage games. Our definition of a multi-stage game with observed actions corresponds to the closed-loop information structure, where players can condition their play at time t on the history of play until that date. [...] the corresponding strategies are called closed-loop strategies or feedback strategies, while open-loop strategies are functions of calendar time alone. [...] The term closed-loop equilibrium usually means a subgame-perfect equilibrium of the game where players can observe and respond to their opponents' actions at the end of each period".

While the small models we will present in Chapter Two are not multi-stage models (we assume markets are settled simultaneously), the notions of open- and closed-loop knowledge still apply, if we recognize that the rationale for having closed-loop knowledge in a multi stage game as per (Fudenberg & Tirole, 1991) is that gencos know the history of their competitors' decisions, and therefore have some knowledge of how they will react to the genco's own decisions in a given market. In a real electricity spot market, which is settled thousands of times per year, this level of knowledge is not unreasonable.

From the above definition, an agent with open-loop knowledge assumes no reaction from it's competitors, while an agent with closed-loop knowledge recognizes that it's decisions will

24 25 affect the optimal decisions of it's competitors, which is the definition we use in Chapter Two. In Chapter Three onward, when we present our anticipation model structure, we use the terms "basic knowledge" and "advanced knowledge" to represent approximations of open- loop and closed-loop knowledge, respectively. The latter assumes imperfect knowledge of competitors reactions, rather than having perfect information, which is generally implied by the term closed-loop knowledge.

The first model, WCM-EM, features a wind capacity market (WCM), and a spot market for electricity output (EM). The two gencos compete in a Cournot fashion in the WCM, purchasing wind capacity from a price-taking supplier who constructs wind capacity, but is not an electricity spot market participant, and each genco pays a single market-clearing price equal to the marginal cost of constructing the last unit of wind capacity. The gencos also compete in the electricity market, selling their output to price-taking consumers and receiving the market-clearing price equal to the consumer's marginal willingness-to-pay of the last unit of generation purchased.

The WCM-EM model has similarities to (Murphy & Smeers 2005), which featured vertically integrated gencos who construct their own capacity prior to electricity market competition, but do not own any incumbent generation capacity. The WCM-EM model considers gencos competing for new capacity in a single-price capacity market, rather than constructing their own, and assumes that one of the gencos already has some conventional capacity.

From this model, we derive the WTP for wind output for Gl and Gw, and show how these values differ under open- and closed-loop knowledge assumptions in the electricity market. We will illustrate that conventional genco Gl is only willing to pay the value of it's conventional generation displaced in the electricity market, while Gw has a higher WTP equal to the marginal revenue of selling an extra unit of output to the electricity market. This model serves as an introduction to the more complex Ontario-like model of Chapter Three, which features the same market structure, but several competing gencos, different types of generation, more realistic representation of demand and wind availability, and varying levels of market knowledge for each participant, in each market.

25 26

The second model, WOM-EM, assumes some fixed level of wind endowment for each genco (ie. wind investment decisions already have been made). In this model, Gl has the option of purchasing financial rights to wind output from it's electricity market competitor Gw via a contract-for-difference, prior to electricity market competition, in a market which we refer to as a wind output market (WOM). We will contrast the open- and closed-loop WTP for Gl, and willingness-to-sell (WTS) for Gw, and show that in the closed-loop case, the WTP for Gl is higher than the WTP for wind in the WCM-EM model, and that there exists a price such that Gl and Gw would both have incentive to sign contracts for wind output in this manner. In doing so, we observe that Gl has incentive to reduce it's conventional output to the electricity market, after purchasing a WOM contract from Gw.

This model is useful to set up intuition for the Chapter Four model, which incorporates uncertainty in nuclear generation capacity, and adds a WOM between the wind capacity market and electricity spot market, after uncertainty is resolved, at which point the total wind investments are assumed to be fixed.

These examples motivate our use of an anticipation model in the later chapters, whereby we observe the optimal investments in wind capacity for levels of knowledge somewhere between the theoretically pure, but less realistic open-loop and closed-loop bounds. We use such a model to study the value of wind capacity in Chapters Three and Four.

2.1 WCM-EM Model: Wind Capacity Market and Electricity Market with Two Gencos

In the first illustrative model WCM-EM, we consider two gencos: Gl with an infinite amount of conventional capacity, and Gw as a new entrant with no conventional capacity. Both gencos sell their output to a price-taking consumer in the electricity market, and can purchase wind capacity in a wind capacity market from a price-taking supplier. We assume that both Gw and Gl can invest in wind generation in this manner.

This is a three-tier model, with the wind capacity supplier in the upstream tier, the gencos Gl and Gw in the middle tier, and the consumer in the downstream tier. The wind capacity supplier and gencos participate in a market for wind capacity, with gencos paying the same

26 27 market-clearing price for all wind capacity constructed. The wind capacity supplier can be thought of as a 'turn-key' wind project developer, who is capable of site evaluation, and turbine installation, or an aggregate of many such firms. It can equivalently be interpreted as a wind construction firm that builds wind plants, and signs a fixed-price contract for it's output to gencos who have access to the electricity market. The price paid for all capacity by both gencos is equal to the marginal cost to the supplier of producing an extra unit of new capacity. In the wind capacity market, we assume that both gencos are Cournot in all cases, so when purchasing wind in the WCM, they assume that the decision of their rival is fixed.

The gencos and consumers also participate in a spot market for electricity generation, where Gl can sell conventional output, and both gencos can sell any wind output coming from the capacity they have invested in. For the spot market, we will present results for both open- loop and closed-loop levels of market knowledge, deriving the WTP for wind capacity for each genco for each of these levels of knowledge.

For this model, all markets are settled simultaneously, and each market is a one-shot game. A diagram of this model showing each market is found below:

WCM-EM Model Diagram

Wind capacity G1 Generation sold to $ purchased by ^>* spot market by G1 $ G1 ^^ . (conventional +wind) y—-^^ %^ Wind WCM -—^_ 7 EM A ( )4 ^=-~~-^'^ Consumer(s) Capacity $ Supplier(s) f \ $

Total spot Total wind Generation sold to market capacity Wind capacity Gw spot market by Gw generation construction purchased by (wind only) Gw

Figure 2-1: Diagram of the analytical WCM-EM model.

27 28

A price-taking consumer is defined by a demand curve, which relates the total output sold to a willingness to pay, per unit, for that output. The price-taking consumer is not a strategic agent, in that there is no recognition of the effect of the customer's purchase on the market price. No decision is made by such an agent; it simply 'takes' a price related to the quantity sold in the market. Many price-taking consumers can be aggregated into a single demand curve, which means that a demand curve can be used to represent all consumers in a market that behave in this non-strategic manner.

The demand curve takes the following linear form:

EP{vt) = Po-P,v, (2.1)

, where EP is the price of electricity (in $/MWh), which is a function of Vt, the total volume sold (MWh), and/?0 and/>/ are the demand curve y-intercept and slope, respectively. We use the notation V\ to represent the amount of conventional output sold by Gl to the spot market, and Wi and Ww to denote the amount of wind output sold by Gl and Gw to the spot market, respectively. Therefore, V, = Vi +W/ +WW.

For the wind capacity market, we assume that there are many price-taking suppliers, which can be aggregated into a supply curve analogous to the spot market demand curve, which yields the market-clearing price for wind capacity in this market .

The supply curve takes the following linear form:

WCP(Ut) = y0+yxUt (2.2)

, where WCP is the price of wind capacity (in $/MW), Ut is the total amount of wind capacity constructed (MW), and yo and yi are the supply curve y-intercept and slope, respectively. We

3 It is justifiable to suggest that the cost of building new capacity of a type such as wind, where costs are site- dependent, would increase with each additional plant constructed. For example, sites with close proximity to transmission corridors, easy accessibility for construction, or on cheaper land would cost the least, and would be constructed first. Subsequent construction of capacity would eventually spread out to less desirable locations, with a higher price tag, suggesting an increasing marginal cost of adding new capacity.

28 29

refer to the amount of wind capacity purchased by Gl and Gw as Uj and Uw respectively, and therefore Ut = U/ +UW.

Each genco's objective is to maximize revenue in the electricity market minus cost in the wind capacity market, by choosing a level of wind capacity investment, and an output quantity to sell to consumers, subject to capacity constraints on wind output, and assuming infinite conventional capacity for Gl. The gencos in this case are strategic players in the electricity market, because they are aware that their output will affect the market price of electricity EP, and so each genco has a perception of marginal revenue of selling additional output to the market.

The gencos also behave strategically in the wind capacity market, being aware that their capacity purchases will affect the wind capacity price WCP, and so they also have a perception of the marginal expenditure associated with purchasing an extra unit of wind capacity.

We will use cj as the operating cost of conventional and new capacity, making the simplifying assumption that there is no operating cost for wind generation.

The WCM-EM model will be formulated in the following sections, first assuming open-loop knowledge for both gencos (ie. Cournot behaviour), and then assuming closed-loop knowledge for Gl (while assuming that Gw remains Cournot), and then for Gw (assuming that Gl remains Cournot). In all cases, we assume that the gencos behave in a Cournot manner in the wind capacity market - therefore we are only focusing on the effect of open versus closed loop knowledge in the electricity spot market. All of these are small complementarity programming models and will be solved analytically.

In the closed-loop case, the decision maker would have some sense of how the competitor's total output would change in response to it's decision, and this would modify each genco's perception of marginal revenue, and therefore it's WTP, for an extra unit of wind capacity, as we will show in this section. We will illustrate that the open- and closed-loop values for wind capacity are identical for Gl, but not for Gw, for reasons which will become clear.

29 30

2.1.1 Cournot Wind Capacity Market and Open-Loop Electricity Spot Market

We begin with presenting the open-loop optimization problem for Gl and Gw in the WCM- EM model, and derive the open-loop WTP for wind output for Gl and Gw in this context.

2.1.1.1 Open-Loop WTP for Wind Output for Conventional Genco Gl

Gl's open-loop optimization problem is to maximize revenue from the electricity market, minus electricity market operating costs and wind capacity purchase costs from the wind capacity market, assuming that Gw will not change it's decisions, which is the open-loop knowledge assumption. In this section, we will derive the solution to Gl's optimization model as a function of the decisions of Gw, and show the WTP for wind output in the open- loop case, which is the marginal revenue to Gl if it sold an extra unit of wind output in the electricity market, assuming open-loop knowledge as defined above.

Using the notation described in the previous section, Gl's optimization problem is as follows:

Gl:max[/70 -px{vx + Wx+Ww)\vx+Wx)-cxVx -\y0 + yx{Ux + UW)]-UX (2.3)

subject to:

The dual variable YWh orthogonal to the primal constraint on wind output, represents the open-loop shadow price, or WTP, for new wind capacity in the model to Gl.

For the output quantities of Gl to be optimal, assuming the non-trivial case where Vi, Wi, Uj > 0, the following first-order conditions (FOCs) must be satisfied in the optimal solution:

— :0 = Po-Px(vx + w1 + tVw)-Px(vl+W1)-cx =EP-Px{vx+Wx)-cx (2.5)

^:YWx=pQ-px{vx+Wx+Ww)-px{vx+Wx) =EP-PX(VX+WX) (2.6)

^:-YWx=-y0-yx{ux+Uw)-yxUx ^YWx=WP+yxUx (2.7)

30 31

Solving the above system of equations, and recognizing that in any optimal equilibrium W;=Ui, we can obtain the following solution to Gl 's model analytically:

Po- •~ci Vx = -Wx (2.8) 2p{

Po~ PiWw -^o- -y\P w Wx--= [/, (2.9) 2Pl + 2y{

YWX= ci (2.10)

Equations (2.8-2.9) yield the optimal electricity market outputs and optimal capacity investments for Gl in this open-loop model. Equation (2.10) yields the open-loop WTP for new wind capacity, which is the marginal revenue Gl expects for an extra unit of wind output.

Looking at the expression for the optimal Vj, we can see that it is a decreasing function of the output of competitor Gw. Furthermore, we can see that the optimal conventional output V/ decreases on a 1:1 basis as Wi increases. This means that if Gl were to receive an extra unit of wind capacity, Gl would increase it's wind output by 1 unit, and decrease it's conventional output by a corresponding amount, leaving it's total output to the electricity market unchanged, in order to satisfy all of the FOCs.

The optimal wind capacity and wind output U/ and W/ are also a decreasing function of the output decision of Gw, which affects the electricity market price that Gl receives, and a decreasing function of the wind capacity decision Uw of Gw, which affects the wind capacity price paid by Gl for it's own wind capacity purchases.

The open-loop WTP for wind capacity YWi is equal to the operating cost savings c/ of displacing a unit of conventional output, which we can see by looking at equation (2.10). Interestingly, this WTP is constant as long as Vj > 0, meaning that as long as Gl has some

31 32 conventional output on the margin , it will have a constant WTP for wind capacity, equal to

Cj.

Finally, we point out that it is clear from equations (2.5) and (2.6) that a high p\ , which implies a low price elasticity of demand (see Section 5.3 for more details), the total optimal output of Gl (Vi+Wj) would be lower, which would result in a higher electricity market price EP, by following the demand curve (2.1) . This illustrates that the use of inelastic demand curves in electricity market models result in less output sold by gencos, and a higher equilibrium market price than more elastic demand curve cases, which have a lower value of pi. We examine the effect of different values for p\ in our larger WCAPCOMP models in Chapter Five.

2.1.1.2 Open-loop WTP for Wind Output for Wind-Only Genco Gw

We now present wind-only Gw's open-loop optimization problem, which takes a similar form to Gl's in the previous section.

Gw's optimization problem is as follows:

Gw:m^[pQ-Px{Vx+Wx+Ww)\Ww-\y0+yx{ux+Uw)\uw (2.11)

subject to

YWW represents the WTP for new wind capacity in this open-loop Cournot model to Gw.

For the output quantities of Gw to be optimal, again assuming the non-trivial case where Ww,

Uw > 0, the following first-order conditions must be satisfied in the optimal solution:

^:YWw=p0-px{vx+Wx+Ww)-pxWw = EP-pxWw (2.13)

Recall that "on the margin" refers to a generation type which is operating at a non-zero value, but not at full capacity. Because we assume infinite conventional capacity in the WCM-EM model, this is always true as long as V,> 0.

32 33

^.:-YWw=-y0-y1(Ul+Uw)-yxUw ->YWw=WP + yxUw (2.14) dUw

Solving the above system of equation yields the optimal open-loop values of Ww and Uw, along with the WTP for wind output to Gw, given the decisions of Gl:

Ww=Uw = Po-pM^hy0-y^ (215) 2Pl+2y{

YWw=EP-PxWw (2.16)

We find from (2.15) that Gw's optimal wind output is a decreasing function of the total output Vj+Wi of Gl, as well as the capacity decision Uj of Gl. Note that from Gw's perspective, the marginal revenue received from the electricity market is a function of the total output of Gl, but in the electricity market Gw is not concerned with the individual outputs Vi or Wj, only the combined effect of Gl's total output Vi+Wi on the electricity market price.

For Gw, the open-loop WTP for wind capacity, found in (2.16), is not constant because Gw has no conventional output to displace if it increases it's wind output, and is therefore a decreasing function of Gw's output Ww to the electricity market. This is because, unlike Gl, if Gw were to receive an extra unit of wind capacity, it would be increasing it's total output, and therefore moving the electricity market price down along the demand curve, and earning a lower price for all existing generation Ww currently being sold. Therefore for Gw, the WTP for extra wind capacity is equal to the marginal revenue of selling that unit to the electricity market, which in contrast to Gl's WTP, is not constant but rather decreases as Ww (or Uw) is increased.

If we consider an initial equilibrium point at which no wind capacity has yet been constructed, meaning Wj = Ww = 0, then the WTP of Gw for the first unit of wind capacity will be higher than the WTP of Gl, because if conventional output is on the margin for Gl, then it must be true that EP > cj, and so YWW > YWi for the first unit of wind capacity purchased. Therefore, in this model we would expect the wind-only genco Gw to purchase more wind capacity in the wind capacity market than conventional genco Gl, at equilibrium.

33 34

Note that it is also possible that if Gw's WTP for wind output is sufficiently higher than Gl's, that Gl may not be able to purchase any wind capacity in the WCM. This is a result we observe in the Chapter Three models for some of the gencos, but for illustrative purposes, we discuss only the non-trivial case where Gl is active in the WCM.

2.1.1.3 Numerical Example of Open-Loop Solution to WCM-EM Model

Figure 2-2 shows an illustrative example using some simple test parameters: po= 10, pi=\, yo=l,yi=0.5, c/=2.5:

WCM-EM Open-Loop Solution

A: revenue to G1 from conventional output Vi B: revenue to G1 from wind output Wi C + D: revenue to Gw from wind output Ww YW, = F: G1's WTP for extra unit of wind YWw = D -E: Gw's WTP for extra unit of wind Electricity price

EP = 5.25

EP'ifGwsoldWw+1

Ci=2.5

Electricity market output V,= 2.25 Wi=0.5 W„ = 2

Figure 2-2 - Solution to the open-loop WCM-EM model using some simple test parameters, with WTP for Gl and Gw indicated.

In this solution, Gw purchases 2 units of wind capacity, with Gl purchasing 0.5 units. At equilibrium, the shadow price to Gw for new wind capacity YWW = 3.25, while for Gl, YWi =

34 35

2.5 = ci. Because Gl assumes that if it were to sell an extra unit of wind output, Gw will not change Ww, Gl will only use any new wind capacity to displace it's conventional generation to satisfy it's FOCs, and earn a surplus equal to the operating cost savings of 1 unit of Vj, which is equal to the region 'F'. In Gw's case, if it believes that Vj+Wi is fixed, then if it received an extra unit of wind output, it would be increasing it's total output and anticipate pushing the market price down to EP\ and it's marginal revenue would be equal to region 'E' - region 'D' in the above figure, which is larger than 'F' (note the diagram is not to scale, to improve clarity).

In the next sections, we will discuss how a closed-loop spot market would potentially yield a different solution than the open-loop case. We will show that the closed-loop WTP of the conventional genco Gl is equal to the open-loop value as long as it has conventional generation on the margin, while for the wind-only genco Gw, which does not have any conventional generation, the closed-loop WTP for wind output in the spot market is higher than the open-loop value.

2.1.2 Cournot Wind Capacity Market with Closed-Loop Electricity Spot Market

In the following sections, we will investigate how the WTP for wind output in the above model would change for each genco if it were to possess closed-loop knowledge in the electricity spot market. We maintain the assumption that both gencos are Cournot in the wind capacity market, and so they assume that the wind capacity investments of their counterparts are fixed.

To begin, we will discuss the knowledge assumptions of a closed-loop genco versus an open- loop genco, and then derive the closed-loop WTP for wind output if each genco, in turn, had closed-loop knowledge in the spot market.

2.1.2.1 Electricity Spot Market Knowledge Assumptions of Open-Loop versus Closed- Loop Gencos

We saw in the preceding model that for Gw, it's optimal wind output Ww was a decreasing function of the total output VJ + WJ of Gl, meaning if Gl were to increase it's total output,

35 36

Gw would respond by decreasing it's output, and so the no-response assumption is not necessarily an accurate one.

We will now examine the total derivative of revenue with respect to Gl's total output, to illustrate the difference between open- and closed-loop knowledge further. To generalize the discussion, we will use the notation OQi = Vi + Wi to represent Gl's total spot market outputs, which in the WCM-EM model is composed of conventional plus wind output.

The revenue portion of Gl's objective function from equation (2.3), if we substitute EP = po

-pi(Vi +Wj +WW) can be written simply as:

Gl: max EPOQi (2.17)

Taking the total derivative of (2.17) with respect to OQi gives the closed-loop WTP for an extra unit of total output OQi to the electricity market, recognizing that EP is a multivariable function, dependent on the parameters OQi and Ww :

dEPOQ ^EP+OQ{.^, where J^ = ^L+^.-^ (2.18) V dOQi *' dOQx dOQx BOQx dWw dOQx ' An open-loop genco assumes that other gencos will not change their output decision in dW response to its own. This means that an open-loop Gl believes that —— = 0, and so if it dOQx dEP increases output, = -P\, which is the demand curve slope. This reduces (2.18) to:

This illustrates that Gl, if it possesses open-loop knowledge, assumes the electricity market price will only change with respect to it's own output, and takes the output of it's competitor as given. Comparing (2.19) to (2.6), which was the FOC for wind output in the open-loop model, we see that they are the same, if we substitute Vi+Wi=OQi.

36 37

However, if Gl possesses closed-loop knowledge, then Gl knows that — > 0, which dOQx therefore affects its perception of marginal revenue from adjusting total electricity market output.

dW 1 For instance, if Gl believed that —— = —, meaning it expected Gw to reduce it's wind dOQx 2 output by V2 for every extra unit of OQi sold to the market (which would be true if Gl had perfect information about Gw), then the closed-loop perception of marginal revenue to Gl from selling extra generation to the electricity market would be:

dEP OQ - ^EP + OQx\^ + ^.^) = EP+0QAP,-ELVEP„ELoQx (2.20) l dOQx * [dOQx dWw dOQx) "" P 2 J 2 This marginal revenue is higher than the open-loop value, because Gl does not anticipate the market price to decrease as quickly if it knows that Gw will reduce it's output as Gl increases OQi, therefore raising the expected market price EP' after Gl increases total output.

We will now use this approach to derive the closed-loop WTP for wind output specifically, rather than total output, for Gl and Gw in the WCM-EM model, and show that while the WTP for wind of Gw would increase if it possessed closed-loop knowledge, Gl's WTP for wind output would actually be the same as the open-loop value.

2.1.2.2 Closed-Loop WTP for Wind Output for Conventional Genco Gl

Because we are now assuming that Gl has closed-loop knowledge, the FOC for Gl will be different than the open-loop model, because when taking the derivatives of Gl's objective function, we must take into account that Gl knows the rate of change of Gw's output with respect to it's own total output.

Gl's optimization problem in the closed-loop is functionally identical to the open-loop case, but we present a slightly different formulation, to support the necessary derivations of WTP which will follow. Specifically, we define OQi as the total output quantity of Gl sold to the

37 38 spot market as we did in the previous section, and add a primal constraint to the model which ensures that this value is equal to the sum of conventional and wind outputs.

The revised formulation of Gl's optimization problem for the closed-loop model is as follows:

Gl:maxEP• OQx -cxVx-\y0 + yx(ux +UW)\Ux (2.21)

subject to

OQx

Wx < Ux 1YWX (2.23)

YOQi in the above model represents the marginal profit associated with increasing total output, while YW/ remains the marginal profit associated with specifically increasing wind output. Both of these shadow prices are interpreted as the increase in the objective function associated with increasing the RHS of their orthogonal constraints by 1 unit, and assuming that the values of the other primal variables in the basis (Vi, OQi) are adjusted to maintain feasibility, without changing the set of basic variables. This is the definition of a shadow price as found in (Winston 1993).

Before presenting the FOCs for this model, we must determine what Gl's estimate of dOQx would be if it had closed-loop knowledge. If we return to equation (2.13), which gives Gw's optimal output of Ww as a function of the outputs of Gl, then we can see that the rate of

dWw 1 change of Ww with respect to OQj, assuming that Uj and Uw remain fixed, is dOQx

The FOCs for the electricity market decision variables of Gl (V/, W/ and OQi), assuming closed-loop information, are:

r dEP dEP dW„ ^ -^-:YOQ=EP+OQ = EP OQ\p -^-) = EP-^OQ (2.24) M M + + x dOQ 80Q 8WW dOQ

^-:-YOQx=-cx ^YOQx=cx (2.25) dvx

38 39

—: -YOQx + YWl=0 -> YOQl = YWX = c, (2.26)

Firstly, we note that in the closed-loop model, the amount of total output OQi sold to the electricity market by Gl will be higher than the open-loop model, as can be illustrated by solving for OQi in equations (2.24) & (2.25), and comparing it to the value of (Vi +Wi) which is the total output in the open model, obtained from equation 2.5:

06i (dosedloop) = -i ^ > ^- = OQi {openloop) (2.27) Pi Pi However, looking at the value of YWi in the closed-loop case in (2.26), it is still equal to ci, which was the open-loop WTP for wind output. This is because if Gl were to increase wind output Wi by 1 unit, this would not affect it's optimal OQi, and so Gl would displace 1 unit of Vi for every unit of Wi it receives, to maintain OQi at it's optimal level. Because OQi does not change, Gw's optimal output would be unaffected, and so the market price would be identical to the value before Gl had an extra unit of wind output. Therefore, the marginal profit associated with receiving more wind to Gl is equal to the displacement value of Vi in the closed-loop model, just as it is in the open-loop model. To summarize,

YWX [dosedloop) = YWX (openloop )=cl (2.28)

The above result is true as long as Gl has conventional generation on the margin, which it can displace as Wi increases.

These findings are an extension of the findings of (Murphy, Smeers 2005), who compared the open- and closed-loop value for new capacity for the case with two gencos with no incumbent capacity, and concluded that the closed-loop WTP for new capacity was higher than the open-loop value.

In our analysis above, we demonstrate that their conclusion is not the case for an incumbent genco who has conventional generation on the margin, but still applies to a new entrant with no incumbent capacity, as we will show next.

39 40

2.1.2.3 Closed-Loop WTP for Wind Output for Wind-Only Genco Gw

Gw's optimization problem in the closed-loop case is again functionally identical to the open-loop model, but we write EP in place of the actual demand curve p0 -px{OQ{ +Ww)iox the purposes of clarity in the following analysis. The closed-loop model for Gw is as follows:

Gw.maxEP-Ww-\y0+yl{U1+Uw)lul (2.29)

subject to

Before presenting the FOCs for Gw in the closed-loop case, we must first determine the closed-loop assumption of Gw, which is the value of ——, the expected change in Gl's total output if Gw were to sell an extra unit of Ww to the electricity market. Using Gl's FOC from equation (2.5) in the open-loop model, we can determine that —— = —.

The FOC for the electricity market decision variable Ww in Gw's closed-loop model, by taking total derivatives and using the above closed-loop knowledge, is:

( r „ \ dEP dEP dOQx ^ P\ P\ ^-:YWX=EP-W„ = EP-W„ EP-^WW (2.31) 3WW dOQx dWw

Clearly, for Gw the closed-loop WTP for an extra unit of wind output is greater than the open-loop value of the previous model:

YWW {closedloop) =EP-Q-Ww>EP-PlWw= YWW (openloop) (2.32)

From this we can conclude that if Gw were a closed-loop genco in the spot market, it would have a higher perceived value for new wind capacity than in the open-loop case, because it would recognize that by increasing it's wind output, it would cause a withdrawal of output by Gl, and this would result in a higher expected electricity market price, and higher marginal profit.

40 41

2.1.2.4 Numerical Example of Closed-Loop Solution to WCM-EM Model

Using the same test parameters as the open-loop example, we now compare the closed-loop solution to the open-loop solution in the table below, for the cases where either Gl, or Gw has closed-loop knowledge in the electricity market. The genco that does not have closed- loop knowledge in these runs is assumed to have open-loop knowledge.

Table 2-1: Comparison of WCM-EM solutions for open-loop, and closed-loop cases

Open-Loop Gl Gl Closed-Loop, Gw Closed-Loop, & Gw Gw Open-Loop Gl Open-Loop v, 2.25 3.24 2.25 w,,u, 0.5 0.72 0.214 w u 2 1.56 2.571 EP $5.25 $4.48 $4.96

WP $2.25 $2.14 $2.39

YW, (WTP for wind output) 2.5 2.5 2.5

YWW (WTP for wind output) 3.25 2.92 3.679

As we can see from these results, the WTP of Gl is unaffected by open- or closed-loop knowledge, as long as Gl has conventional generation on the margin, which it does in all of these examples. However, the WTP for Gw is higher in the case where it has closed-loop knowledge than the cases where it has open-loop knowledge. The total amount of wind constructed is highest in the case where Gw has closed-loop knowledge, and lowest in the case where Gl has closed-loop knowledge.

Interestingly, the WTP for Gw is lower in the case where Gl has closed-loop knowledge, and therefore Gw purchases less wind in this case. This is because when Gl has closed-loop knowledge, it's total output quantity is increased, and the resulting electricity market price is lower than the open-loop case. Gw's WTP for wind is a function of EP, and so with a lower EP, Gw purchases less wind capacity in the WCM. As a result, the WCM price WP is lower, 41 42 and Gl is able to purchase more wind in the closed-loop case than in the open-loop case, even though it's perceived marginal revenue is the same, because the marginal expenditure to purchase wind is lower.

The above numerical example demonstrates that the WTP for wind for Gl is unaffected by open- or closed-loop assumptions in the electricity market, but the WTP for Gw depends on what level of knowledge it possesses in the electricity market. These findings apply to the WTP for wind capacity purchased from a separately-owned supplier in the WCM, but we will see in the WOM-EM model that the level of knowledge makes a difference to the perceived value of wind for both gencos.

2.1.3 Summary of WCM-EM Model Findings

In summary, we have shown here that for a genco with conventional generation on the margin, the closed- and open-loop values for new wind output are identical, and equal to the displacement value of conventional generation. Contrasting with this is the result that for a genco that does not have conventional generation on the margin, such as a wind-only entrant, a closed-loop level of knowledge yields a higher value for new wind output than an open- loop level of knowledge in the electricity market.

A corollary of the above conclusion is that if new capacity were made available to a number of gencos in a competitive setting who all had conventional capacity on the margin, then each genco would displace conventional output on a one-for-one basis, and the total output sold to the market, and therefore the market price, would be unaffected, even though a new type of capacity with a lower operating cost has been introduced to the system. This is a counter intuitive result which emerges from strategic models of electricity markets such as the simple one shown here.

In the next section, we will examine a slightly different model, in which we introduce the possibility of Gl purchasing financial rights to wind output from Gw in a wind output market, which occurs in a simultaneous time frame to electricity market competition, under conditions where the total wind capacity in the system is fixed, with gencos having made their investment decisions already. We will examine the closed-loop and open-loop WTP for

42 43 the buyer, and WTS of the seller in this new market, for some fixed level of wind endowments, and show that the WTP for Gl in this market is in fact higher than the WTP derived in the previous WCM-EM model, where wind capacity was purchased from a supplier who was not also an electricity market participant. From this analysis, we demonstrate that there exists incentive for Gl to purchase financial rights to wind output from Gw in the WOM.

2.2 WOM-EM Model: Wind Output Market and Electricity Spot Market with Two Gencos

The preceding WCM-EM model assumed that the capacity and electricity markets took place simultaneously, and all new capacity was purchased from a price-taking seller who was not an electricity market participant. Now, we turn our attention to a situation where rather than purchasing wind capacity from a supplier who is not in the electricity market, genco Gl has the option of purchasing financial rights to wind output from it's competitor Gw, via a contract-for-difference, where Gw physically delivers the wind output to the electricity market, but Gl receives the revenue received from this output. This contract is available to gencos in the model of Chapter Four.

In this paradigm, we will show that with closed-loop information (meaning the genco knows how the electricity market equilibrium will be affected by it's contract decision), a genco with conventional generation has incentive to purchase financial rights to wind output from a wind-only genco in this manner, and that such a purchase can be mutually beneficial in terms of profit for the buyer and seller. Furthermore, the WTP for wind output in the WOM to Gl with closed-loop information is higher than the WTP for wind capacity in the wind capacity market of the WCM-EM model.

In the WOM-EM model, if Gl purchases financial rights to wind output from Gw, we will show that Gl has incentive to reduce it's conventional output, to increases the electricity market price received over all of it's output, including the wind output for which it has purchased financial rights. The existence of such contracts in the WOM, therefore, yields added environmental benefits if the marginal conventional technology is of a type that produces harmful emissions, such as coal.

43 44

A diagram of the WOM-EM model is presented below:

WOM-EM Model Diagram

EP * (V1 +Ww) WOM contract for A wind output Consumers

Wind capacity endowments U1 &Uw V1 +Ww Ww (includes A)

Figure 2-3: Diagram of the WOM-EM model.

The following section will derive the effect of a WOM contract-for-difference on the optimal conventional output of Gl, and the resulting equilibrium electricity market price, for the simple two-genco case. We will derive the closed-loop WTP of the buyer Gl, and the closed- loop willingness-to-sell (WTS) of the seller Gw, and contrast these values to the open-loop equivalents, where gencos do not anticipate the effect of the contract on the electricity market equilibrium.

2.2.1 Effect of WOM Contract from Wind-On-Margin Gw to Conventional-On- Margin Gl

We will begin by defining Gl and Gw as in the previous WCM-EM model. However, in this model, we assume that the wind capacity owned by each genco is fixed, at some values of £// and Uw, and that the conventional output of Gl is on the margin, so V/ > 0. For the analytical derivations below, we assume that Ui=0 and UW>0, so Gw is the only participant with wind capacity in the model to simplify the analysis. Because Gl starts with no wind capacity, prior to the contract the electricity market wind output of Gl is Wj=0, and we assume that Gw is

44 45

selling all of it's wind output to the spot market, so Ww = Uw. Therefore we are deriving the WTP for the first unit of wind output purchased by Gl from Gw.

For the derivations in the remainder of the chapter, we will use the notation Wi' and Ww' to represent the amount of wind output Gl and Gw have financial rights to, incorporating any WOM contracts, rather than their physical outputs to the electricity market. Regardless of the amount of wind output of each genco prior to the WOM contract, since all wind output ends up sold to the electricity market, Wi + Ww=Wi'+ Ww'

The optimal value of Vi, if Gl is an open-loop genco, is derived in the previous model in equation 2.8, and repeated here:

j?0 f, Cl Vx = ~ '^" -^1 (2.8)

If a WOM contract was formed such that Gl were to receive the electricity market price for A units of wind output from Gw, then for the purposes of determining the revenue to each genco, the new wind outputs by each agent would be Ww' = Uw -A, and Wi'= A.

Using (2.8), and substituting the new wind outputs Wi' and Ww', we can solve for the new optimal value of VV , if this contract existed:

F|.= /'o-^^._yi.= /'o-/'.(^-A)-c,_(A)= A (2J3) 2P\ ?-P\ 2

Gl has incentive to reduce it's output, because it is now receiving the electricity market price over a larger amount of output, and so by withdrawing some conventional output from the market, it is able to increase it's revenue by increasing the price along the demand curve. This increased conventional displacement which is gained when the income from wind output is transferred from a wind-only genco to a conventional genco is one of the key benefits of allowing such contracts in a wind output market, a point that will be illustrated in more detail in Chapter Four.

The resulting electricity market price after this contract is:

45 46

EP=PO-PI{V1'+W1'+IVW')=P0-P1\V1--+W1+A+WW-AUEP+^-A (2.34)

Therefore, we can see analytically that if the financial rights to A units of wind output is sold to Gl, assuming each genco behaves Cournot in the electricity market, the resulting reduction in Gl's conventional output increases the electricity market price.

The following diagram compares the two equilibrium points, prior to and following the WOM contract for A units of wind output from Gw to Gl in the WOM:

Effect of WOM contract between Gw and G1

WTP(G1) = WTS(Gw) Electricity price

Electricity market output

Figure 2-4: Comparison of electricity market equilibria, with and without a contract-for-difference on A units of wind output from Gw to Gl.

Figure 2-4 shows the electricity market equilibria, with and without a WOM contract for A units of wind output from Gw to Gl. The total amount of wind output in the market remains

46 47 the same, but Gl is now receiving the electricity spot price for a larger amount of total output, and so by decreasing the total output by reducing Vi to Vi', it is able to earn higher profits.

The total surplus to Gl, which is it's WTP, is defined as the maximum price Gl would be willing to pay for the WOM contract, and is indicated by the horizontally-hatched region, plus the vertically-hatched shaded region, minus the black region in Figure 2-4. These areas represent the additional revenue from the WOM contract, plus the gains from an increased EP' across Gl's total output (including the contract output), minus the lost revenue from reducing V/ to Vi', including operating cost savings, respectively. These areas correspond to the equation for WTP derived in the next section.

To the seller Gw, we define it's willingness-to-sell as the minimum price that Gw would be willing to accept for the WOM contract. This is equal to the horizontally-hatched region, which represents the opportunity cost of not receiving revenue for A wind output, minus the grey region, which represents gains to Gw across it's remaining wind output Ww', because the electricity price has increased from EP to EP'.

We will now derive equations for the WTP of the buyer and WTS of the seller analytically, having presented conceptually how the electricity market equilibrium is affected by a WOM contract, when the total amount of wind in the system is fixed. In order to compute these values from the model, we require the participants to have some closed-loop knowledge, reflecting their anticipation of the electricity market equilibrium changing as a result of this type of contract.

2.2.2 WTP of Gl For WOM Contract Using Total Derivatives

In this section, we will compare the WTP of Gl for financial rights to wind output from Gw for closed- and open-loop levels of knowledge, using the total derivative of Gl's objective function, which is an analytical approach, independent of the specifics of the formulation of the model.

As we will show further on in the chapter in Section 2.2.5, taking a traditional approach to formulating this model as a complementarity program, which appears on the surface to 47 48 properly characterize the wind output market and electricity markets, can actually yield nonsensical results where the WOM contract is improperly valued, which motivates the use of a total derivative approach here.

2.2.2.1 Closed-Loop WTP of Gl in WOM

The closed-loop WTP of Gl for an extra unit of wind output via a WOM contract from Gw is defined as the perceived marginal revenue from such a purchase, given Gl knows how the electricity market equilibrium will respond. This represents the maximum price that Gl is willing to pay for the contract.

From the derivations of the previous section, we know that for small values of A:

^L^Ei. 4h.-zL dWl -1 dW™ = i dA ~ 2 ' dA ~ 2 ' dA ~ ' dA

To determine the closed-loop WTP of Gl, we will compute the total derivative of only the revenue portion of each genco's objective function, because we are interested in the WTP associated with a WOM contract, not the marginal profit, which includes the cost of purchasing the contract. The WTP is the incremental revenue associated with buying an extra unit of wind output in a WOM contract from Gw incorporating closed-loop knowledge, by assuming that Gl knows the value of the derivatives shown above.

For the buyer Gl, we compute the closed-loop WTP as follows, assuming for the purposes of the derivation that the cost of the contract is zero:

Gl: max EP^+W^-c^

WTP(Gl,closed,WOM)=^- = EP^ + V^ + EP^- + Wl^-^-cl dA dA dA dA dA dA (2.35)

= \EP+\PA^+W{)+UX

In (2.35), the first and third terms, when added together, represent the horizontally-hatched shaded region minus the black region in Figure 2-4, which is the marginal revenue gained from receiving revenue for the WOM contract wind quantity A, along with the operating cost savings of reducing conventional output. The middle term corresponds to the vertically- 48 49 hatched shaded region of Figure 2-4, which is the surplus gained from increasing the electricity price from EP to EP'.

We can see from this expression that the middle term, which represents the gains to the buyer from an increased market price across it's output, would be larger for a genco who had a large amount of total output being sold to the spot market. Furthermore, the third term shows that the higher the operating cost ci, the higher the WTP for this type of contract will be.

Taken together, we would expect that if there were several possible buyers, as we have in the Chapter Four model, and all potential buyers have conventional capacity on the margin, then a genco with a large amount of total output in the spot market, and/or a high operating cost for it's marginal technology, would have the highest closed-loop WTP for wind output in the WOM.

Finally, we note that Gl's closed-loop WTP for a WOM contract from Gw (2.35) is higher than Gl's WTP in the wind capacity market (2.26), which was shown in the first half of the chapter to be equal to operating cost c/, for both closed- and open-loop knowledge levels. This can easily be proven by noting that EP>ci (since we assume that Gl is not perfectly competitive, and V/ > 0, this must be true), and so WTP(Gi,dosed,woM) > ci.

2.2.2.2 Open-Loop WTP for Gl in WOM

We define open-loop knowledge in the WOM as a genco assuming that the decision variables of it's counterpart remain fixed. In the open-loop case, Gl assumes that WW'=WW. Since this is fixed, we can use the WCM-EM model open-loop case equation (2.8) for optimal Vi

(which was derived under the assumption that Gw would not change Ww) to conclude that Gl will reduce it's conventional output 1:1 as it obtains more wind output, so Vi'=Vt-A and Wi '=Wi+A. and therefore Gl does not expect the electricity market price to change, because total output by both gencos to the market remains constant.

Therefore, an open-loop Gl would have the following perceptions of the relevant dEP dW, dW dV, derivatives: - = 0,—L = l,—- = 0,—- = -1. Substituting these into (2.35) we see that the dA dA dA dA & V '

49 50 open-loop WTP for Gl in the WOM is equal to the WTP from the WCM-EM model for an extra unit of wind output in the WCM:

WTP(Gl, open, WOM) = = -EP + EP + cx =c, (2.36) dA The closed-loop WTP for wind in the WOM (2.35) is clearly greater than the open-loop WTP in the WOM (2.36).

Therefore, we have demonstrated analytically that a conventional genco with conventional capacity on the margin has a higher closed-loop willingness-to-pay for wind output if it purchases it from a wind-only competitor with closed-loop knowledge incorporated into the model. This higher WTP comes from Gl having sufficient knowledge of the electricity market equilibrium to forecast how it's conventional output, and thus the electricity market price, will change if it purchases a WOM contract.

2.2.3 WTS of Gw For WOM Contract Using Total Derivatives

We now perform a similar analysis for the seller Gw, computing the closed- and open-loop WTS, which represents the minimum price at which Gw would be willing to sell a WOM contract.

2.2.3.1 Closed-Loop WTP of Gw in WOM

The total derivative of the revenue portion of Gw's objective function, assuming closed-loop knowledge of the derivatives of EP and Ww as shown in Section 2.2.2.1, is computed as follows:

Gw: max EP(fVw)

WTS(Gw,closed,WOM) = EP^^ + WW ~=-EP + ^-Ww (2.37) dA dA 2 Equation (2.37) shows the net loss in profit to Gw when it gives financial rights to 1 unit of wind output to Gl via a WOM contract, with a price of zero, assuming closed-loop knowledge. Therefore, multiplying (2.37) by (-1) yields the minimum price that Gw would be willing to accept for the contract.

50 51

The willingness-to-sell price comes from the opportunity cost of not selling the wind output to the spot market, and therefore losing EP, but accounts for the gains to Gw that come from knowing that Gl will reduce it's conventional output, and therefore increase the electricity market price across the wind output Gw retains financial rights to, after selling a portion to Gl under the WOM contract.

2.2.3.2 Open-Loop WTS of Gw in WOM

For the open-loop WTS, Gw would assume that the total output to the electricity market would remain fixed, meaning that Gl will not re-optimize it's conventional output. We compute the total derivative of Gw's objective function for the open-loop case, by dEP dW substituting = 0 , and —- = -1 into (2.37) , which yields: dA dA

dW WTS(Gw, open) = EP ^ = -EP (2.38) dA

Therefore, in the open-loop case for Gw, it would only sell wind output to Gl if it were offered a WOM contract price higher than the electricity market price EP, which it is currently receiving.

It is clear that for non-zero values of Gw's wind output Ww, the closed-loop WTS price is lower than the open-loop WTS price, and so if Gw had closed-loop knowledge, it would be more willing to sell financial rights to wind output to conventional genco Gl, for a price below EP. This increased willingness-to-sell comes from Gw having knowledge that Gl will reduce it's conventional output, and therefore increase the electricity market price across Gw's remaining wind output.

Therefore, we have established the minimum price that a genco with no conventional output would be willing to accept, and the price which a conventional-on-margin genco would be willing to pay, when trading wind output in the WOM, for open- and closed-loop levels of knowledge.

If both gencos had open-loop knowledge, then the two gencos would not be able to agree on a mutually acceptable contract price, because Gl would only value wind at ci, while Gw

51 52 would value it at EP and no trading would occur. This is proven more rigorously following the presentation of the WCAPCOMP 2.WOM model of Chapter Four.

However, if Gl had closed-loop knowledge, and Gw had open-loop knowledge, then if ci or {Vi+Wi) was sufficiently high, by looking at (2.35) Gl may be able to negotiate a WOM contract Gw if Gl's WTP > WTS = EP, meaning both would perceive some value from the trade.

In all, we have shown that there is definitely mutual incentive to sign a contract if both gencos possess closed-loop knowledge, it is possible that a contract would be signed if Gl had closed-loop knowledge but Gw did not, and that a mutually acceptable contract does not exist if Gl and Gw both had open-loop knowledge. In the next section, we will demonstrate that a mutually beneficial contract price exists in the mutual closed-loop knowledge case, such that both the buyer and seller have incentive to trade, and would earn a profit by doing so.

2.2.4 Verifying Existence of a Mutually-Acceptable Closed-Loop Price for Wind Output Contract in WOM

We will now demonstrate that if both gencos possess closed-loop knowledge of the effect of WOM contracts on the electricity market equilibrium, then there is a feasible contract price which is mutually acceptable to both the buyer Gl and seller Gw. We must prove that:

WTP(Gl,closed) = -EP + -pl{Vl+Wl)+-cl >WTS(Gw,closed) = EP-^-Ww (2.39)

To show that this is true, we substitute the optimal values of Vi, Wi, and Ww into the above expression, to compare the WTP and WTS as functions of the model parameters po, pi, and

Ut (total wind in the system):

f u c u c ( Po ~P\ t ~ \ 'Po-P\ t~ \^ WTP(G\, closed) = - Po--P\ -PV, + P\ \ V 2/7, 2PI f2.4m

= -{Pa-P\U,+cx}

52 53

„ ,Po-p\ut~cA WTS{Gw, closed) = •PV, +EI£L Po (2.41)

-\PO-P\U,+CX] ~-

Comparing these two expressions, it is clear that for any value of total wind capacity in the system Ut, the WTP for the first unit of wind output purchased by Gl from Gw is higher than the minimum WTS price that Gw would need to receive to make a profit.

Therefore, we have established that for two gencos who behave as open-loop Cournot competitors in the spot market, and have closed-loop knowledge in the WOM, a mutually acceptable contract price does exist, such that Gl and Gw would both do better by participating in a trade of wind output from Gw to Gl via a WOM contract.

In Chapter Four, we will use the results of Section 2.2 as a guide to explain the results of our stochastic Ontario-like model which features a wind output market of a similar nature. The WOM in that model occurs on the same timescale as the electricity market, after uncertainty is resolved, and we assume that by this second stage, the total amount of wind capacity in the system is fixed, which aligns with the assumptions made during the preceding analyses.

2.2.5 Pitfalls with Standard CP Formulation of WOM

In Section 2.2.2, it was mentioned that taking a typical approach of formulating a CP model to represent the WOM-EM paradigm can return erroneous results. We will demonstrate this issue now, which motivates our use of total derivatives in the previous sections, and is one of the reasons why the Chapter Four model utilizes anticipation coefficients to properly characterize the wind output market.

We define Xb andXs as the desired contract amounts of the buyer and seller, respectively, and

WOP as the contract price, and assume that Ui and Uw are parameters representing the wind capacity endowment of Gl and Gw.

The following CP would appear to represent the WOM-EM paradigm accurately, and one might try to use the dual variables to determine the open-loop WTP or WTS for WOM contracts (although we will soon demonstrate this approach does not work):

53 54

G\:max[p0 -px(vx + Wx + Ww)]• fa +Wl)-crVl-WOP-Xb

subject to FF, < C/, + Xb 1 IW,

subject to WW

Xs > Xb _L WOP

The FOCs for both gencos in this model are as follows (taking care not to double-count WOP, which appears in the objective function of each genco and as a dual variable on the link constraint, and should only be counted once):

dG\ : -YWX > -WOP 1 Xb dXb

^-:0>p0 -Pl(Vx +W, +Ww)-Pl(Vi +Wx)-cx IV,

— •.YWx>pQ-px{Vx+Wx+Ww)-px{Vx+Wx) ±WX oWx dGw — :-YWw>-WOP ±XS

L ^ .YWw>pa-px{vx+Wx+Ww)-pxWw ±WW

Note that this model assumes implicitly that, when taking the partial derivative of the objective function with respect to a unit increase in the orthogonal primal variable, the other variables of the decision maker are adjusted to maintain feasibility, the decision maker's basis does not change, nor do the decision variables of the other genco. Therefore, this model should, in theory, yield the open-loop WTP and WTS for WOM contracts derived in the previous section, but we now show that this is not the case.

By looking at the dual equations above, if some non-zero contract amount is traded (ie. Xb=Xs

>0), then the model requires that YWX = YWW = WOP = cx by complementary slackness, which means that the seller Gw, according to this model, would accept a price equal to the operating 54 55 cost of Gl's conventional technology. However, in the previous section, using the total derivative approach, we derived that for the open-loop case, YWW=EP, which is greater than

YWw-ci in this model, and so the model shows a contradiction.

In fact, the above CP formulation under-values the WOM contract from Gw's perspective. When taking the partial derivative to form the FOCs as per the standard CP approach, the dW dual equation implicitly assumes that —- = 0, meaning Gw assumes that when selling 1 unit dXs of wind output under a WOM contract to Gl, this wind output does not end up being sold to the market by way of Wi increasing, which is a wholly myopic view that no decision maker, open-loop or otherwise, would presumably have.

According to (Winston 1993), we interpret the dual variable YWW as the change in the objective function after increasing the RHS of Gw's wind output constraint by 1 unit, after adjusting the value of the basic variables (but not the basic set) to maintain feasibility, and assuming that the decision variables of Gl remain fixed. -YWW is therefore interpreted as the change in the objective function from decreasing the RHS of the wind output constraint, which would require reducing Ww by 1 unit. Therefore, Gw, when valuing the WOM contract in this incorrect model, is assigning a value equal to the value of withdrawing one unit of wind output from the spot market, ignoring the fact that it is giving the rights to this output to Gl, and therefore the wind output is not removed from the market at all, but is merely "owned financially" by Gl.

Therefore, if one were to assume that the dual variable YWW properly reflects the value of a WOM contract sold to Gl, and applied this typical CP approach to formulate a WOM-EM model, then they would obtain an erroneous model result, and this basic CP model approach would not be useful.

This is the main reason for using a total derivatives approach to valuing the WOM contract in Chapter Two for open- and closed-loop knowledge assumptions. In the Chapter Four model, we formulate the WOM using an anticipation framework, whereby each genco forecasts how it's own output to the electricity market, along with the electricity market price (reflecting the

55 56 expectation of how the other genco's outputs will adjust to the WOM trade) will be affected by signing a WOM contract.

2.2.6 Generalizing WTP and WTS for Imperfect Knowledge Cases

As we have seen in the previous two sections, for a genco to forecast the value of a WOM contract, even for open-loop levels of knowledge, it must have some perception of how it's outputs will change, along with what effect this will have on the electricity market price, if any. In our Chapter Four model, we use an anticipation approach, whereby each genco in the WOM anticipates how it's own offer quantity to the spot market OQ„ along with the spot price EP will change, as it signs financial contracts for wind output in the WOM.

This anticipation approach involves defining, for each genco, the expected change in the spot market price and it's own spot market output, so that the model properly reflects the perceived value of a WOM contract for a given level of knowledge, which may be open or closed loop, or somewhere in between. We saw in Section 2.2.5 that a non-anticipation model does not yield the correct open-loop valuation of the WOM contract.

The approach of the previous two sections which used total derivatives to compute the closed-loop perfect information WTP and WTS values can be applied to situations with imperfect knowledge as well, simply by substituting a genco's knowledge of the derivatives

,—-,—-,—- into equations (2.35) and (2.37), using a similar approach to Section dA dA dA dA 2.1.2.1. In the Chapter Four model formulation, we will examine the dual equations of our Ontario like WCAPCOMP 2.WOM model with respect to the WOM contract quantity decision variables, and illustrate the use of anticipation coefficients to characterize each genco's perception of the value of a WOM contract, given imperfect knowledge.

2.3 Conclusion

The preceding chapter has laid an analytical framework which demonstrates that the willingness-to-pay for new capacity in a competitive market setting varies depends on the

56 57 generation portfolio of the purchasing genco, whether that genco has conventional capacity on the margin, and the level of knowledge that the genco has with respect to the response of it's competitors reactions to it's decisions.

We have shown that if a genco has conventional generation capacity, and that capacity is on the margin, then the WTP for a new capacity type such as wind is constant, and equal to the operating cost of the displaced technology. This means that a genco with a higher operating cost technology, such as gas-fired generation, would be willing to pay more for new capacity than a genco with a lower operating cost technology, such as coal generation, and this is true in both open-loop and closed-loop contexts. This WTP does not depend on the outputs of the other gencos, in both the open- and closed-loop contexts. Furthermore, for a genco with conventional capacity on the margin, any new capacity received will be used to displace the marginal generation on a one-for-one basis.

However, for a genco without conventional capacity on the margin, such as a wind-only new entrant, then the willingness-to-pay is a function of the outputs of the other gencos. In this case, we observed that a closed-loop model will yield a higher value for new capacity than an open-loop model, because a genco with higher knowledge of it's competitors will recognize that if it sells new generation to the market, the outputs of it's competitors will decrease, which will affect the market price.

We then showed that if the total amount of wind capacity is fixed, as was the case in our wind output market model WOM-EM, then with closed-loop levels of information, a conventional-on-margin may have incentive to purchase wind capacity contracts-for- difference from a wind-only entrant. If both gencos have closed-loop knowledge in the WOM, then a mutually acceptable price for the WOM contract exists, while if Gl has closed- loop knowledge and Gw has only open-loop knowledge, then a mutually acceptable price may not exist, and it is unclear a priori whether the two gencos have incentive to trade wind output. The WOM also had the added benefit of reducing the amount of conventional output sold by the conventional genco if it purchases wind from the new entrant, which can have environmental benefits in the case of coal displacement.

57 58

In the next chapter, we will move to a more realistic anticipation model of wind capacity and electricity spot markets with several oligopolistic gencos with portfolios similar to the Ontario electricity system, along with a wind-only new entrant. In this larger Ontario-like model, we have the ability to simulate various levels of electricity market knowledge, as well as wind capacity market knowledge, by specifying a parameter which represents a genco's anticipation of how the electricity market or wind capacity market price will change, given what assumptions it has as to the response of it's rivals in the market. This information can be perfect (closed-loop), or open-loop, or any value in between, as will be illustrated in the coming chapter.

In Chapter Four, we will extend the Chapter Three model, and introduce uncertainty in nuclear capacity which is resolved between the wind capacity and spot market stages, as well as a wind output market which occurs in the second stage, and supports various knowledge assumptions held by each genco as to the effect of WOM contracts on the electricity market equilibrium.

58 59

3.0 WCAPCOMP 1.0 - Deterministic Capacity Expansion Model for Ontario-like Market

In Chapter Three, we construct a more detailed Ontario-like market model, referred to as WCAPCOMP 1.0. This model consists of two markets: a wind capacity market and a spot market for electricity, with both markets being settled simultaneously. WCAPCOMP 1.0 is a decision support model which provides insight into the value of wind capacity in a competitive setting, with several gencos competing for wind capacity and competing in the spot market for electricity. This model is useful to determine the amount of wind purchased by each genco, the amount of coal generation displaced by investment in wind, as well as total profits to each genco in the model, for different levels of nuclear capacity. In Chapter Four, we take nuclear capacity as an element of uncertainty, and compare those results to the results of the deterministic model presented in Chapter Three.

The wind capacity market features a price-taking supplier, which can be imagined as an amalgamation of smaller firms, who acts as wind market operator (WMO), creating a single- price market where all gencos can make offers to purchase wind capacity built by the supplier(s), with the price set at the cost of constructing the marginal unit of wind capacity.

The spot market for electricity is more detailed than the Chapter Two version, by dividing load into four seasons with three blocks in each season. This electricity market represents one year of competition, and is administered by an electricity market operator (EMO), who accepts offers of price and quantity from each genco in each block, and sets the electricity market price equal to the marginal willingness-to-pay of the price-taking consumers. The electricity market loads and operating costs used in the numerical examples are approximate to year 2000 levels.

Each genco has a different generation portfolio, consisting of hydro, nuclear, coal and/or oil generation types, as well as any wind which is purchased in the wind capacity market. We model one genco as a wind-only new entrant, with no conventional generation capacity. Each genco has a different level of market knowledge in the each market, representing how the

59 60 genco anticipates it's competitors will respond to their wind capacity decisions in the WCM, and spot market output decisions in the EM.

The spot market structure is inspired by the ELFORSPOT model of (Asl & Rogers, 2004), but with new improvements to account for seasonal and diurnal variation in wind generation and hydro output, seasonal scheduled maintenance, and the addition of a wind capacity market. The model is formulated as a complementarity program, and solved using the PATH algorithm in GAMS.

The solution to the WCAPCOMP 1.0 model is a Nash equilibrium, where each decision maker believes it cannot unilaterally change it's decisions and increase it's profit (Fudenberg &Tirole, 1991).

In Chapter Three, we will describe the general model structure, after which the notation used in the WCAPCOMP 1.0 model will be presented. Following these descriptions, we will formulate the model as a CP. Lastly, we present some numerical examples, in which we vary the amount of nuclear capacity available to the nuclear gencos, representing potential contingencies related to the success or failure of nuclear rehabilitation projects similar to those currently being undertaken in Ontario. In these results, we will focus on the amount of wind capacity constructed in each scenario, the amount of coal displaced by wind, and the profits to the various gencos.

3.1 WCAPCOMP 1.0 Model Structure

In the first section, we will describe the structure of the WCAPCOMP 1.0 model, using a model diagram and an explanation of the markets and decision making agents in each market.

3.1.1 Description of WCAPCOMP 1.0 model

The WCAPCOMP 1.0 model features a wind capacity market where gencos can purchase wind capacity from a price-taking supplier, and compete in a spot market for electricity divided into seasons and load blocks, using any wind capacity they acquire, in addition to any conventional generation capacity which may be available to the gencos.

60 61

A diagram of the model, showing the product and cash flows between agents, is presented below:

WCAPCOMP 1.0 Model Diagram i i i Wind Capacity Market (WCM) i

Figure 3-1: Diagram of WCAPCOMP 1.0 model.

In Figure 3-1, the wind capacity market and electricity markets are shown, along with the product and financial flows between each entity in each market. In WCAPCOMP 1.0, both markets are settled simultaneously. Within each market, gencos have a given level of knowledge as to how the other gencos will respond to their output or purchasing decisions, which is facilitated via the use of anticipation coefficients, as will be explained further on.

For the electricity spot market portion of WCAPCOMP 1.0, we use a model structure similar to ELFORSPOT (Asl & Rogers, 2004), which features gencos, an EMO, and price-taking customers. The WCAPCOMP 1.0 models also includes a wind capacity market, and seasonal and diurnal variation in available wind output into the spot market stage, to more accurately represent wind generation on an annual basis. Seasonal hydro availability and seasonal

61 62 scheduled maintenance are also additions found in the WCAPCOMP 1.0 model, but not in the ELFORSPOT model. These will also be described in subsequent sections in more detail.

3.1.2 Electricity Spot Market

The electricity spot market portion of WCAPCOMP 1.0 represents a year of competition, with load broken into four seasons, with three load blocks per season. Each season and block has a demand curve, which assumes that load varies with price, and customers are represented as price-takers. The use of a demand curve implies that there is some elasticity in electricity demand over the medium-to-long term, which manifests itself in several ways, for example larger customers such as aluminium smelters or manufacturing plants reducing their consumption during peak hours or moving to a different location in response to high electricity prices, or smaller customers conserving electricity by switching from incandescent to fluorescent bulbs over the long-term.

To derive the electricity market demand curves, we start with a load duration curve for each season, and split it into discretized blocks, with each block representing the average hourly demand and price for a number of hours, and the total energy in each block adding up to the area under the LDC, as can be seen below in Figure 3-2 for a simple 1 block example. Taking this load level, along with a given electricity market price and an assumed elasticity of demand at this point, we fit a linear demand curve to each block. Each block is meant to represent hourly average levels of load, price, and genco behaviour over several hundred hours.

More details on the equations used to compute the demand curves for the numerical examples can be found in Appendix A.

62 63

Simple 1-Block Illustration of Approximating LDC

Average Hourly Block demand curve Demand (GW)

Actual seasonal Block height LDC (average hourly load)

Duration (khrs)

Figure 3-2: Converting a seasonal LDC into a discretized load block, and fitting a demand curve.

Within each season and block, gencos choose the amount of generation to offer to the EMO, along with an offer price. The EMO takes an amount less than or equal to the offer quantity, in order to satisfy demand. The market price is set as the marginal cost to the EMO of satisfying an additional unit of demand, and the EMO ensures that each market price must lie on the demand curve for that particular season and block.

Also in the spot market, the EMO pays each genco a fixed annual peak payment, for the maximum amount of conventional capacity (nuclear, coal, or oil) and hydro capacity made available during the year by that genco. Gencos pay a fixed annual capital cost to operate a given generation plant, as well as a variable operating cost which is related to the cost of operating a given generation type on a per hour basis. These will be described in more detail in the model formulation. 63 64

3.1.3 Use of Anticipation in Electricity Spot Market

Each genco in the electricity market has a perception of how it's competitors will react if it offers an extra unit of output to the EMO (assuming the EMO accepts this output). By using different levels of electricity market anticipation for each genco, the model can represent different levels of competitive behaviour, or market knowledge. This anticipation takes the same functional form as in the ELFORSPOT model of (Asl & Rogers, 2004), and will be briefly described below.

If a genco assumes that none of it's competitors will react to it's output decision, then the anticipation coefficient is set equal to the demand curve slope, which we refer to in this thesis as a "basic" level of knowledge. A genco with an anticipation coefficient close to zero does not expect the electricity market price to change significantly if it offers more output, which is a "myopic" level of knowledge. Meanwhile, a genco with "advanced" knowledge recognizes that as it increases it's output, other gencos will decrease their outputs (as we saw in the closed-loop WCM-EM model of Chapter Two), and so it perceives the residual demand curve slope to be less than the actual demand curve slope, and expects that the electricity market price will not decrease to the same degree as a basic knowledge genco would expect, if it increases it's output.

The perceived slope of the residual demand curve, given an assumed level of knowledge, is shown in Figure 3-3 below.

64 65

Genco Anticipation of Residual Demand Curve Slope

Electricity price EP after competitors react, for various knowledge levels

- -Myopic knowledge

Advanced knowledge

Electricity market (current total spot output) (extra output A) output

Figure 3-3: Illustration of electricity market anticipation and perception of residual demand curve slope.

In Figure 3-3, we illustrate how each genco will anticipate the new electricity market price after offering A extra units of output to the market, given a level of knowledge of how it's competitors will adjust their outputs. A genco with advanced knowledge will have a higher perceived marginal revenue for selling extra output to the electricity market than a basic knowledge genco, because it recognizes that the other gencos will reduce their outputs if the advanced genco increases it's output, and so the residual demand curve slope to the advanced genco has a shallower slope.

3.1.4 Wind Capacity Market

The wind capacity market is formulated as a single-price supply market, where gencos can make offers to purchase an amount of wind capacity from the WMO, along with an offer

65 66 price. The WMO represents an aggregation of many price-taking wind capacity suppliers. The willingness-to-sell of the suppliers is defined by a supply curve, which reflects the marginal cost of constructing an additional unit of wind capacity. The WMO takes offers from each genco, and chooses it's corresponding accepted quantities, setting the market- clearing price for wind capacity such that it lies on the supply curve. Essentially, this market operates in a similar manner to the electricity market, but replacing a demand curve with a supply curve, and with the strategic gencos as purchasers, rather than sellers as they are in the electricity market, with the WMO playing the same role as the EMO does in the electricity market.

By contrast to the electricity market, for which we have enough data to replicate an approximate demand curve for a given block, we do not have sufficient data to create a supply curve for wind capacity, given that this is an emerging technology. Our approach, given this deficiency, is to assume a linear supply curve, with the y-intercept and slope as parameters which can be altered by the model analyst.

We note also that our model reflects one year of electricity market competition, and so the wind capacity market prices need to reflect approximately one year's share of the total investment cost of wind capacity, given that wind turbines can last up to 20 years or more.

3.1.5 Use of Anticipation in Wind Capacity Market

Each genco in the wind capacity market has a perception of how its competitors will adjust their purchase quantities, if it purchases an extra unit of wind capacity from the WMO. This is the same concept as used for electricity market anticipation. By varying the level of the anticipation coefficient for a given genco, we can observe results for a variety of different levels of competitive behaviour in the wind capacity market.

The definition of myopic, basic and advanced levels of wind capacity market knowledge are analogous to those described for the electricity market in the previous section. The perception of the residual supply curve slope is shown below in Figure 3-4:

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Genco Anticipation of Residual Supply Curve Slope

WCP after competitors react

Initial WCP Basic knowledge

Wind capacity price Advanced knowledge

Myopic knowledge

Wind Capacity Constructed (current total capacity (extra purchase A) purchased)

Figure 3-4: Illustration of wind capacity market anticipation and perception of residual supply curve slope.

An advanced genco recognizes that if it buys more capacity in the wind market, other gencos will react to the higher WCP by decreasing their purchase quantities, and therefore an advanced genco perceives a shallower residual supply curve slope than a genco with basic knowledge, who assumes that the other gencos purchases will remain fixed.

The benefit of this approach in both the wind capacity and electricity markets is that we can model a given genco to have a high amount of electricity market knowledge, but a low amount of wind capacity market knowledge, or vice versa, as we do in the numerical examples at the end of this chapter.

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3.1.6 Representation of Wind Generation

The use of seasonality facilitates varying levels of wind availability in each season and block, to capture seasonal and diurnal variation in wind energy. For example, wind availability is typically higher at night than during the day, and higher in the winter than in the summer. In the numerical examples at the end of the chapter, the wind capacity factor in the winter base block (representing overnight demand periods) is 3 times higher than the capacity factor in the summer peak block (representing mid-afternoon demand periods). As a result, the model will show higher levels of wind output in winter base blocks than summer peak blocks, for a given amount of wind capacity.

The capacity factor for wind is an average measure of the amount of power a turbine can provide, as a fraction of it's rated maximum capacity. We assume that gencos purchase wind capacity in units of rated turbine capacity, but can only sell output which is adjusted by the wind capacity factor cfkb in a particular season and block.

As a simplification, we do not represent imbalance charges for wind output imposed by the EMO, which would occur in a real electricity spot market if a genco cannot meet it's output commitment due to a decrease in wind energy available at the scheduled time.

3.2 WCAPCOMP 1.0 Model Notation

The notation used in the WCAPCOMP 1.0 model presented in this chapter is now presented below in Table 3-1. We distinguish between the incumbent conventional gencos using set notation /, and formulate the optimization problem for the wind-only genco separately, for which many of the conventional, maintenance and hydro constraints do not apply. Variables which apply to the wind-only genco are denoted by using the suffix 'w'.

Table 3-1 - WCAPCOMP 1.0 model notation

Notation Description

SETS: / Conventional gencos

/ Conventional generation technologies (non-hydro)

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k Seasons

b Load blocks within each season

PARAMETERS:

Electricity Pokb Electricity market demand curve y-intercept ($/MWh) Market:

Plkb Electricity market demand curve slope ($/MW*MWh)

aetkb Conventional genco anticipation of electricity market price change if additional GW of generation is offered and accepted ($/MW*MWh)

aewkb Wind-only genco anticipation of electricity market price change if additional MW of generation is offered and accepted ($MW*MWh)

wbkb Duration of seasonal demand block (khrs)

oc, Operating cost of conventional technology ($/MWh)

fct Fixed annual capital cost of conventional technology ($M/GW)

oh Operating cost of hydro technology ($/MWh)

Jh Fixed annual capital cost of hydro technology ($M/GW)

Cfkb Wind capacity factor (%)

pcapu Hourly production capacity limit for conventional technology (GW)

hcapik Seasonal hourly production capacity limit for hydro technology (GW)

hrrik Seasonal hourly hydro run-of-river limit (GW)

hengik Total seasonal hydro energy available (TWh)

mrct Annual scheduled maintenance requirement for conventional technology (%)

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foutt Forced outage rate for conventional technology (%)

PP Fixed peak payment for highest annual capacity made available ($M/GW)

Wind Capacity act Conventional genco anticipation of wind capacity price Market: change if additional GW is purchased ($M/GW2)

acw Wind-only genco anticipation of wind capacity price change if additional GW is purchased ($M/GW2)

yo Wind capacity marginal cost curve y-intercept ($M/GW)

2 yi Wind capacity marginal cost curve slope ($M/GW )

DECISION VARIABLES:

Electricity Market:

EMO: Dkb Hourly average total demand in each block (GW)

BQikb Generation quantity accepted by EMO from conventional genco (GW)

BQwkb Generation quantity accepted by EMO from wind-only genco (GW)

Conventional OQm Generation offer quantity to EMO (GW) Gencos:

OEPikb Electricity market offer price by Genco to EMO ($/MWh)

VCukb Hourly average conventional generation output (GW)

VHM Hourly average hydro generation output (GW)

vwikb Hourly average wind generation output (GW)

PVCit Annual peak conventional capacity made available (GW)

PVHt Annual peak hydro capacity made available (GW)

MCitk Amount of conventional capacity taken down for seasonal maintenance (GW)

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Wind-only OQwkb Generation quantity to EMO (GW) Genco:

OEPwkb Electricity market offer price by wind genco to EMO ($/MWh)

VWwkb Hourly average wind generation output (GW)

Wind Capacity Market: WMO: u, Total amount of wind capacity constructed (GW) BUi Amount of wind capacity accepted by WMO from conventional genco (GW)

BUw Amount of wind capacity accepted by WMO from wind- only genco (GW)

Conventional OUi Amount of wind capacity bid by genco from WMO (GW) Gencos:

OWCPi Wind capacity supply market offer price by genco to WMO (SM/GW)

Wind-only OUw Amount of wind capacity offered by genco to buy from Genco: WMO (GW)

OWCPw Wind capacity supply market offer price by genco to WMO ($M/GW)

3.3 WCAPCOMP 1.0 Model Formulation

In the following section, we define the WCAPCOMP 1.0 model, using the notation presented in the previous section, defining in turn the optimization problem for the conventional gencos, the wind-only genco, the EMO and the WMO, as well as some necessary linking constraints for each market which are visible to all decision makers in that market.

3.3.1 Conventional Genco Primal Formulation

In the electricity market stage, each genco maximizes profit from sales to the electricity market. Each genco receives revenue for generation sold of all types, for which it receives 71 72 the electricity market price as determined by the EMO. Gencos also receive a fixed annual peak payment pp from the EMO for the highest amount of capacity of each conventional technology, and hydro, made available to the electricity market. We assume that no peak payments are made for wind capacity, because it is currently considered non-dispatchable in the Ontario system.

In terms of costs, each conventional genco must pay a fixed capital cost related to the highest amount of capacity operated of each conventional and hydro plant type, which reflects the capital cost of making capacity from each plant available for a year. In addition, gencos pay the variable operating cost for each unit of generation produced by each generation technology.

We also account for annual maintenance of conventional plants, by requiring that each plant is taken down for mrct % of the total hours it operates in the year. Gencos may choose to spread maintenance across several seasons, if desired. We also include forced outage for conventional generation types, which reduces the amount of available capacity in each block to reflect unexpected outages.

In the wind capacity market, each genco chooses how much wind capacity to purchase. The output from this wind capacity is subject to seasonal and diurnal variation in the electricity market, with a capacity factor cfkb for each season and block.

3.3.1.1 Conventional Genco Objective Function

The objective function for a given conventional genco maximizes the total electricity market revenue from sales of generation, weighted by each block and season, minus variable operating costs weighted by block and season, plus the difference between the peak payments from the EMO and fixed annual operating costs for the maximum available capacity of each conventional and hydro plant., minus the total cost of wind capacity purchases.

The objective function for each conventional genco is as follows:

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G(i): max ^jT block duration *[ {EM offer price * EM offer quantity} -]T {variable k,b t operating cost of plant t * plant t output} -{variable hydro operating cost * hydro output}] -^ {fixed capital costs of plant t- peak payment price}* highest plant j capacity available -{fixed capital costs of hydro plant - peak payment price}*highest hydro capacity available -{wind capacity offer price * wind capacity offer quantity}

\ ' EPt G(i): max £ wbkb • —^ - aeikb {0Qikb - BQikb) • OQikb -Yocr VCitkb - oh • VH'ikb U J (3.1) PVC +X(PP - f°< )• -> + (PP - Jh)-PVH, -[WCP + ac, {OUi -BU, )]-OU, i

3.3.1.2 Conventional Genco Primal Constraints

Each genco must produce at least as much output from all of it's generation types as it offers to the electricity market:

£ VCm + VHikb + VWikb > OQikb 1 YOQikb (3.2)

Each genco must choose the maximum annual capacity made available for each conventional plant, for purposes of peak payments and fixed operating costs, and the average hourly output plus scheduled maintenance must not be greater than this level, after adjusting for the forced outage rate:

PVCit

VC m + MCjtk < (l - font, )PVCit J. YVCm (3.4)

We have similar constraints for hydro, except that we use the spring hydro capacity for purposes of determining the peak level of hydro available for peak payments, but with no forced outage rate assumed for hydro:

PVH, < hcaPi,spring, 1YPVH, (3.5)

VHikb

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The total seasonal hydro output must be less than the total hydro energy available in that season. Total hydro output is also constrained on the lower bound by run-of-river constraints:

£ wbkb • VHm < hengik 1 YHENGik (3.7) k,b

VHikb>hrrik ±YHRRjkb (3.8)

The total scheduled maintenance across all seasons much amount to the annual maintenance requirement for each conventional plant type:

]T wbkb • MCilk > 8.76 • mrc, • PVCU 1 YMCU (3.9) k,b

The wind output must be less than the amount of capacity purchased in stage 1, adjusted by the capacity factor for the given season and block:

VWikb

The spot market and wind capacity market offer prices by the genco to the market operators are set at the current equilibrium price in the market, adjusted by the anticipated change in price if an extra unit is sold or purchased in each market:

EP OEPikb=—^--aeikb{OQikb-BQikb) ±YOEPikh (3.11)

OWCP^WCP-aCiiOUj-BUi) LYOWCPi (3.12)

The market prices EPikb and WCP come from the EMO and WMO formulations, which are presented following the formulation of the wind-only genco problem. The anticipation

coefficients aeaa, and act are multiplied by the difference between the genco's desired quantity in a given market, minus the amount that the EMO / WMO accepts, such that at equilibrium the anticipation term is equal to zero, and so the offer prices are equal to the market-clearing prices determined by each market operator. At equilibrium, the bids and offers in a given market will be equal.

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3.3.2 Wind-Only Genco Primal Formulation

The wind-only genco's formulation takes a similar form to that of the conventional gencos, but omitting any constraints or terms which deal with conventional or hydro output.

The wind-only genco formulation is relatively straightforward, being analogous to the previous conventional genco formulation, and is presented below for completeness:

EPkbt Gw: max \\ wbkb -aewkb(OQwkb-BQwkb) •OQwkb-[WCP + acw(OUw-BUw)]-OUw (3.13) k.h wbkb subject to

VWwkb>OQwkb ±YOQwkb (3.14)

VWwkb

FP OEPwkb = —2- - aewkb {OQwkb - BQwkb ) ±YOEPwkb (3.16) wbkb

OWCPw = WCP+acw(OUw-BUw) lYOWCPw (3.17)

3.3.3 EMO Primal Formulation

The EMO serves as a link between the price-taking customers and the gencos who participate in the electricity market. The role of the EMO in this model is to choose bid quantities of output from each genco, taking a quantity up to but not exceeding the amount offered, minimizing the total cost of meeting demand. The EMO also pays each genco a fixed peak payment for the highest amount of capacity operated during the year for conventional and hydro plants, but has no optimization decisions related to peak payments, and so they are omitted from the EMO's objective function.

3.3.3.1 EMO Objective Function

The EMO's objective function is as follows:

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EMO: min V block width * {conv genco offer price * quantity accepted from conv i,k,b genco} + T block width * {wind genco offer price * quantity accepted from wind k,b genco}

EMO:xnm^wbkb -{OEPikb • BQikb)+^wbkb -{OEPwkb • BQwkb) (3.18) ;,*,* k,b

3.3.3.2 EMO Primal Constraints

The EMO must satisfy average hourly total demand of customers in each season and block by choosing bids from the gencos.

B + B w Dkb < X Qm Q kb -L EPkb (3.19) i

The EMO also sets the spot market clearing price on the demand curve for each season and block, with the market price taken as the dual variable of constraint (3.19):

L ^ = Pm-PiktDkb ±YDkb (3.20) w bkb Note that there is no anticipation on the part of the EMO as to the change in the offer price if the demand shifts, which means that the EMO is a price-taker with respect to offers of the gencos. Also, in this equation the electricity market price is in units of $M/MWblock, and must be divided by the block duration wbkb (which is in khrs) to ensure that the LHS and RHS are in the same units of $/MWh.

3.3.4 WMO Primal Formulation

The WMO, who is an aggregate price-taking supplier, views offers from gencos to purchase a quantity of wind capacity along with an offer price. The WMO chooses wind capacity offer quantities, and sets the market-clearing price in the wind capacity market equal to the marginal cost of producing the last unit of wind capacity.

Mechanically, this capacity market operates analogously to the electricity spot market, with the cost of marginal supply curve replacing the consumer demand curve, and gencos

76 77 competing for supply of wind capacity, rather than sales of electricity output. The WMO is also a price-taker with respect to the gencos, as the EMO is in the electricity market.

3.3.4.1 WMO Objective Function

The objective of the WMO is to maximize revenues from selling wind capacity to the gencos, minus the total construction cost of wind capacity.

The objective function for the WMO is as follows:

WMO: max V {conv genco offer price * conv genco accepted quantity}

+{wind genco offer price * wind genco accepted quantity}-{total cost of wind construction}

2 WMO:max.^(OWCP, •BUj)+OWCPw-BUw-y0Ul --y{Ut (3.21)

3.3.4.2 WMO Primal Constraints

For the WMO, it must construct enough wind capacity to meet it's offers of wind capacity to the conventional gencos and wind-only genco:

U^Y^BUi+BUw 1WCP (3.22)

The wind capacity market price is the dual variable on the capacity balance constraint above.

In addition, the WMO sets the market-clearing price WCP to be on the supply curve:

WCP = y0+ylUl ±YU, (3.23)

These constraints are both analogous to the constraints of the EMO in the electricity market.

3.3.5 Primal Linking Constraints

The following linking constraints are primal constraints which link the market operators and gencos in the wind capacity and electricity markets.

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For the electricity market, at equilibrium, the amount offered by each genco is equal to the amount accepted by the EMO, and the EMO must not accept more output from a genco than is offered. Therefore, we include the following primal constraint which links the offers and bids together, for each conventional genco and the wind-only genco to the EMO:

BQikb

BQwkb

A similar linking constraint is required for the wind capacity market, ensuring that each genco cannot purchase more wind capacity than is being offered by the WMO:

BU^OU, 1YBU, (3.26)

BUw

3.3.6 WCAPCOMP 1.0 Dual Constraints

The dual constraints for all agents in the WCAPCOMP 1.0 model are presented below in Table 3-2, as the dual equations are necessary in order to formulate the WCAPCOMP 1.0 model as a complementarity program.

Table 3-2: Dual constraints for all agents in WCAPCOMP 1.0.

Agent Dual Constraints:

Conventional -YBQikb +YOQikb + aemPxkbYOEPikb > Genco G(i) y (3.28) EPkb -aeikhwbkb(20Qikb -BQikb) ±OQikb

-YOQikb+YVCukb>-wbkboc, ±VCitkb(3.29)

-YOQikb+YVWikb>0 lVWikb(3.30)

-YOQikh+YVHikb -YHRRikb + wbkbYHENGik > -wbkboh lVHikb(3.3l)

YPVCit - Y, (l - font, )YVCitkb + 8.76/nrc, YMCit >pp-fc, ±.PVCit (3.32) k,b

YPVHi >pp-Jh ±PVH,(3.33)

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wb Ya kb YMCit > 0 IMC,, (3.34) b K b J

-Y,cfkbYVWikb -aciYOWCPi-YBUi > k,b (3.35) -WP-aCjilOUj-BUj) 10t/,-

YOWCPt > 0 ±OWCPt(336)

YOEPikb>0 ±OEPikb(331)

Wind-only -YBQwkb + YOQwkb + aewkbYOEPwkb > Genco Gw (3.38) EPkb -aewkbwbkb(20Qwkb -BQwkb K ) ±OQwkb

-YOQwkb + YVWwkb > 0 ±VWwkb(339)

-^cfkbYVWwkb -acw-YOWPw-YBUw> k,b (3.40) -WP-acw(20Uw- -BUw) ±OUw

YOWCPw>0 1 OWCPw (3.41)

YOEPwkb > 0 ±OEPwkb (3.42)

EMO YBQikb-EPkb>-wbkbOEPikb ±fiO,tt(3.43)

YBQwkb -EPkb >-wbkbOEPwkb ±BQwkb(3AA)

PmYDkb+EPkb>0 ±/>tt(3.45)

WMO YBUi + WCP > OWCPt ±BU,(3.46)

YBUwi + WCP > OWCPw ± BUw (3.47)

-WP-yxYUt>-y0-yxUt ± [7,(3.48)

3.3.7 Complementarity Programming Formulation of WCAPCOMP 1.0

WCAPCOMP 1.0 is solved as a complementarity program using the PATH algorithm in GAMS. The resulting CP is defined by the set of complementarity pairs defined by all primal and dual equations for the gencos, EMO, WMO, and linking constraints, paired with their

79 80 respective orthogonal variables, as shown in the previous sections. These pairings form the KKT conditions for optimality for each agent. The GAMS implementation of the model can be found in Appendix B.

3.3.8 KKT Analysis of WCAPCOMP 1.0

We will now present an analysis of the primal and dual problems of the WCAPCOMP 1.0 model, exploiting the complementary slackness properties of the KKT conditions of the model to derive some results which can be used to describe genco behaviour in the model. We will refer back to the propositions of this section when discussing and interpreting the numerical results of the WCAPCOMP 1.0 model in Section 3.4.

We divide this analysis into two sections, with the first focusing on properties of the electricity market equilibrium, and the second focusing on the wind capacity market equilibrium. In the derivations, we sometimes introduce the subscript * in place of the season and block k and b, to improve readability.

3.3.8.1 Analysis of Electricity Market KKT Conditions

The first two propositions describe the relationship between the offer quantities of the gencos, and the electricity market price.

Proposition 3.1: An increase in any genco i's offer quantity OQi*, results in a decrease in the FP electricity market price —- (assuming the EMO accepts this quantity), and vice versa. wb.

Proof. To maintain feasibility, total demand D* must increase to satisfy EMO demand balance constraint Z), = y]BQit+BQw,(3.\9), which must be tight if EP*>0 by i complementary slackness. Noting that increasing D* requires market price to decrease due to EP, the EMO price constraint = p , - /?,»£>» completes the proof. wb* 0

A corollary of Proposition 3.1 is that an increase in the amount of nuclear capacity pcapi/nuc- available to genco / would result in an increased offer quantity to the electricity market, if

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YVCitnUC*>0, which is the case in our model because nuclear generation is a base load technology, and so this would result in a lower electricity market price.

FP Proposition 3.2: Decreasing electricity market price —- results in a decrease in the offer wb* quantity of any genco i with generation on the margin (ie. with VCit, > 0, YVCit* = Ofor some t), by decreasing VQ,*, and this decrease in output quantity is proportional to it's level of electricity market knowledge aej*, and vice versa.

Proof: If a genco has generation type t on the margin, then VClt*, OQi* > 0, and by complementary slackness dual equations (3.28) and (3.29) must be tight, and so we can solve both as equalities to obtain the following identity:

EP, ae , OQr = oc (3.49) ( wb* t

, which implies that a decrease in the electricity market price requires a decrease in OQi* to maintain feasibility, with the decrease proportional to ae,*. Conventional output VCu* will be reduced to accomplish this, to satisfy the offer quantity balance constraint

OQit < £ VCit, + VWit + VH-,n (3.2). This completes the proof.

In the WCAPCOMP 1.0 model, we interpret the dual variables YVWikb (for conventional genco i) and YVWwkb (for the wind genco) as the willingness-to-pay for an additional unit of wind output of each genco in season k, block b. Proposition 3.3 below derives the WTP for a genco with no conventional generation on the margin, using the notation of the conventional genco, although the result applies equally to the wind-only genco, which has no conventional generation on the margin by definition.

Proposition 3.3: For a genco i with no conventional generation on the margin, the WTP for wind output YVWf, is a decreasing function of the electricity market price, the electricity market anticipation coefficient ae/*, and the total output quantity OQi* of the genco.

Proof. Assuming that OQi*>0, the dual equation with respect to OQi* (3.28) must be tight:

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- YBQi. + YOQj, + aeit YOEPt, = —- - ae,„ (200,., - BQ,.) (3.50) wb.

To reduce this expression, we first recognize that if VWi*>0, then the dual equation (3.30) is tight, and so YOQ,, = YVWt,. Secondly, if OQi*=BQi* this reduces the RHS term to -ae,,0g,»,

FP and genco offer price primal equality constraint (3.11) reduces to OEPt, =—-. This implies wb, that YBQt* = 0, by looking at EMO dual constraint YBQ,.-EP. = wb,OEPl, (3.43), which is tight because BQi*>0. Finally, if we recognize that since primal variable OEPj* > 0, then by complementary slackness we require by dual equation (3.37) that YOEPt, = 0. Substituting these three results into (3.50), we obtain the expression:

FP YVW^^—^—aepOQp. (3.51) wb.

This expression relates the dual variable YVW,*, which is interpreted as the WTP for an extra unit of wind output in the market, to the electricity market price, the genco's total output quantity, and anticipation coefficient aem, and completes the proof.

As we can see from (3.51), a lower market price results in a lower WTP for wind output for a genco with no conventional generation on the margin. Furthermore, this genco would have a lower WTP if it has a high offer quantity in the electricity market, or a larger anticipation coefficient ae*.

This result is in line with the derivation of WTP for wind output in the Chapter Two WCM- EM model, which showed that Gw's WTP for wind was a function of the market price and it's output.

Proposition 3.4 below presents the derivation of WTP for wind output of a conventional genco with generation technology t on the margin.

Proposition 3.4: For a genco i with conventional generation t on the margin, the WTP for wind output is constant, and equal to the operating cost of the marginal generation technology t.

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Proof: If VCjf, VWt> 0, then the dual equations with respect to these variables (3.29) & (3.30) must be tight by complementary slackness. Solving the two equations as equalities

yields the identity YVWit = oct, which completes the proof.

Therefore, in contrast to the result in Proposition 3.3, the WTP for an extra unit of wind output for a conventional-on-margin genco is not a function of the market price, nor the genco's total output, nor it's level of market knowledge. This is because a genco who has marginal generation will reduce it's generation of that type, to exactly offset it's increased wind output, and therefore perceives the value of wind output as the value of reducing generation output of type / by 1 unit.

In the WCAPCOMP 1.0 numerical examples, the only generation technology on the margin is coal, with coal ownership split unevenly between two gencos. For one genco, coal is on the margin for some blocks of the year, while coal is always on the margin for the other. As the preceding derivations demonstrate, this is important in evaluating the perceived total value of wind output to a given genco.

3.3.8.2 Analysis of Wind Capacity Market KKT Conditions

We now present some similar propositions which relate to the wind capacity market equilibrium. Again, we use the notation of the conventional genco problem for convenience, but the results apply equally to the wind-only genco.

Proposition 3.5: Purchasing an extra unit of wind capacity OU, (assuming the WMO accepts the offer) increases the wind capacity price WCP.

Proof: Increasing OUt requires that the WMO increase total wind capacity construction Ut to

maintain feasibility of primal constraint Vst/,+5t/w

market price constraint WCP = y0 + yl(u,)(3.23) requires WCP to increase to remain feasible completes the proof.

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Proposition 3.6: If one genco increases it's capacity purchase quantity 0U„ other purchasing gencos will reduce their purchased capacity, and this response depends on their level of wind capacity market knowledge ac{.

Proofs For the other gencos who are purchasing in the wind capacity market, OUi =BUt > 0, and so the dual equation with respect to OU (3.35) is tight, and can be written as: acflU^Y^cfuYVW^-WCP (3.52) k,b

From Proposition 3.5, if WCP increases, other purchasing gencos will reduce their OU to maintain feasibility of that dual equation, and this response will be proportional to the anticipation coefficient acj, which completes the proof.

Finally, we point out that in equation (3.52), if genco / has a high YVWM, it will purchase more wind capacity in the WCM than a genco with lower YVWikb- We can also infer that a genco i with conventional generation on the margin will have a lower WTP for wind capacity than genco/ with conventional generation off the margin under the following conditions:

EP EP YVWi*=oct=—^--aei*OQi* < YVWjt =—--ae.-.Og,-. trueif ae^OQ,* > ae:*OQ•* (3.53) wh, wb» Therefore, for equal ae* between the gencos, the conventional-on-margin genco / would only have a higher WTP than conventional-on-margin genco/ if OQj* > OQt*. This demonstrates, as the analysis of the Chapter Two WCM-EM model did (see Section 2.1.1.2), that the wind- only genco will purchase more wind capacity than the conventional gencos in the WCAPCOMP 1.0 model, where the conventional gencos have higher output in the market than the wind entrant.

Some numerical results of the WCAPCOMP 1.0 model will now be presented in the next section, as a demonstration of the model and to provide motivation for the WCAPCOMP 2.WOM model of the next chapter.

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3.4 Numerical Results

To demonstrate the usefulness of the WCAPCOMP 1.0 model formulated above, we now present a few comparative numerical examples.

Our numerical examples feature three incumbent gencos, a single new entrant wind-only genco, and a genco with the option of building combined-cycle generation as a fixed investment, representing prospective investors who would take advantage of high market prices should they arise. The largest genco, OLG, controls approximately half of the installed nuclear capacity, all of the hydro, and 1/3 of the fossil fuel generation, including coal and oil- fired generators. The second genco is ONG, which is a nuclear-only genco with the remaining nuclear capacity in the system. A third smaller genco, OSG, controls the remaining 2/3 of the fossil fuel generation, while the wind entrant OWE and cc investor OCE have no incumbent generation types. OCE does not participate in the wind capacity market, and can only build cc generation by paying a fixed capital cost in the spot market stage to construct and operate such capacity.

Load is broken into 12 segments, consisting of winter, spring, summer and fall seasons with three demand blocks each, representing peak, off-peak and base load. The demand curves for each season were calibrated using data published online at the IESO website, scaled back to year 2000 levels, and using the 2006 LDC shape to divide demand into each season and block. A point demand elasticity of e=0.5 is used to compute the linear demand curves assumed in all blocks for Chapter Three (we present results for a lower elasticity of e=0.8 in Chapter Five). See Appendix A for more detail on how the demand parameters were calculated.

Wind capacity factors in the off-peak (middle) load block range from 40% in the winter, to 20% in the summer, with +/- 5% in the base (typically overnight) and peak (daytime) blocks, respectively. Therefore the range of wind capacity factors used is from 0.15-0.45. This represents wind generation more accurately, accounting for seasonal and diurnal variation in available output from wind capacity.

85 86

For hydro availability by season, we take the annual hydro energy sold in Ontario, and divide it into 4 seasons, assuming 20% higher energy available in the spring blocks, and 20% lower energy available in the summer blocks. See Appendix C for more detail on how hydro generation parameters were selected for these runs.

We assign each genco a different level of electricity market and wind capacity market knowledge. In the wind capacity market, the incumbent gencos are assumed to possess basic market knowledge (as defined in Section 3.1.4), while the wind entrant is assumed to have an advanced level of knowledge. In the electricity market, we model the largest genco OLG as an advanced genco, and all other electricity market participants having basic market knowledge (as defined in Section 3.1.3).

A complete list of the level of parameters used in the examples of Chapter Three, with regards to the generation capacities of each genco, the fixed and variable operating costs, and other model inputs can be found in Appendix D.

We show results for three potential cases, in which we vary the amount of nuclear capacity available to the nuclear gencos OLG and ONG. In the medium nuclear case, we assume capacities of 6.5 GW and 5 GW, respectively. In the low and high nuclear cases, we vary these capacities by +/- 1 GW for OLG, and by +/- 1.5 GW for ONG. The total difference in nuclear capacity in the system between the low and high nuclear scenarios is therefore 5 GW.

We are interested in the wind construction and allocation amongst the gencos, the resulting spot market outputs, amount of coal generation displaced by the addition of wind (when compared to a base model run with no wind capacity in the system), electricity market prices, and genco profits, and how the level of nuclear capacity available affects all of these aspects.

3.4.1 Wind Capacity Allocation and Willingness-to-Pay

We find in all cases that OWE is the largest purchaser of wind capacity, and that as more nuclear capacity is available in the system, OSG purchases an increased amount of wind capacity as OWE decreases it's purchases. The total wind capacity purchased in the wind capacity market for each nuclear case, along with the allocation of wind capacity between the spot market gencos, is presented below in Figure 3-5: 86 87

Wind Construction & Allocation

8-i-

7 --

6 -- s o 8 5" o EOWE 3 • OSG ft. SONG £ 4 - I • OLG u ? 3- s

1 --

o-L L M H Nuclear Case Figure 3-5: Total wind construction and allocation amongst spot market gencos for low, medium and high nuclear cases.

What is interesting here is that in the low nuclear case, OLG and ONG each purchase small amounts of wind capacity, while they are not willing to buy in the medium or high nuclear cases. This is because coal is not on the margin in those cases for either genco (market outputs are shown in Section 3.4.2), and so by Proposition 3.3, each genco's WTP is a decreasing function of their output (which is larger with more nuclear available) and the electricity price (which is lower for the same reason, see Section 3.4.4 for these prices).

In the high nuclear case, OSG purchases 56% more wind output than in the medium case, while OWE builds 19% less. This is because OSG's WTP is not affected by the lower electricity market price because it has coal on the margin (from Proposition 3.4), while OWE's WTP is lower because it does not have conventional on the margin, and is therefore affected by low electricity prices (see Proposition 3.3). The fact that OWE purchases less leads to a lower wind capacity price WP, and so OSG is able to purchase more (see Proposition 3.6)

We find that removing 5 GW of nuclear capacity (from the H to L case) only results in 1 GW of extra wind capacity construction. The remaining difference in output is made up of 87 88 demand reduction due to higher prices, and increased coal output by the coal gencos in response to those higher prices, results which are explained by Propositions 3.1 and 3.2.

We now present the WTP5 for 1 MWh of extra wind output for each genco, for some selected blocks in Figure 3-6:

Hourly Willingness-To-Pay for Wind Output (selected blocks)

• OLG • ONG • OSG El OWE

Season/Block/Nuclear Case

Figure 3-6:: Hourly WTP (dual variable on wind output constraint) for wind output in selected blocks, for each genco, defined as the mariginal revenue expected from having an extra MWh of wind output.

The above results are very interesting for several reasons. We see that in the low scenario, because coal is on the margin6 for both OLG and OSG in the shown blocks, their WTP is identical and equal to the displacement value of coal (Proposition 3.4). This explains why OLG is able to purchase some wind capacity in stage 1 of the low nuclear case, when it has

5 Recall the WTP for wind output in WCAPCOMP 1.0 is defined by the dual variable YVWikb for the conventional gencos, and YVWwkb for the wind genco.

6 Recall that our definition of "on-the-margin" generation means that the generation type is being utilized at a non-zero level, but not at full capacity.

88 89 coal on the margin and therefore the same WTP as OSG in all blocks where both have coal on the margin.

For ONG and OWE, the WTP is different in each block, as both gencos have no conventional generation on the margin (ONG is fully utilizing it's nuclear capacity). This means that if they were increase their wind output, this would change their total outputs, and lower the electricity market price (see Proposition 3.1). Thus the WTP for these gencos depends on the electricity price in a given block, along with the amount of total output each genco has in the market. For OWE, the WTP for wind is higher than that of ONG in all blocks, because OWE has a smaller amount of output in the electricity market, and therefore is less affected if it reduces the market price by increasing wind output than ONG would be. These results are explained by Proposition 3.3.

In the high nuclear case, we see that again, OSG has WTP equal to the displacement value of coal in all blocks, because it has coal on the margin. Therefore, OSG has an identical WTP in all blocks, and this is the same in the low, medium and high nuclear cases. In the winter-base block for the high nuclear case, OSG actually has a higher WTP for wind output than OWE does, because the lower electricity price in that block decreases the WTP of OWE, but not that of OSG (from Propositions 3.3 and 3.4). OLG does not operate coal in the high nuclear case, and so it's WTP becomes a function of the electricity price and it's offer quantity, and varies in each block, which is not true in the low nuclear case when it is operating coal.

All of these results were foreshadowed in the Chapter Two WCM-SM model, which showed that WTP was constant and equal to displacement value if a genco has conventional generation on the margin, and is a decreasing function of the spot market price and total output offered, if conventional generation is fully-utilized, or not utilized at all.

3.4.2 Electricity Market Outputs

Because we showed in the previous section that the WTP of a genco for wind capacity depends on whether it has coal generation on the margin in a given block, we now present the electricity market outputs for each genco for selected blocks in Figure 3-7:

89 90

Electricity Market Output (selected blocks)

Figure 3-7: Electricity market outputs for each genco by generation type, for selected seasons & blocks, in the low, medium and high nuclear cases.

We note that OCE, the combined-cycle turbine investor genco, does not invest in combined- cycle capacity in any outcome, and therefore is not mentioned further in the results.

OLG, the largest genco in terms of generation capacity, only utilizes its coal capacity in the low nuclear case, and in the summer peak of the medium nuclear case, while OSG, the genco with 2/3 of the system's coal capacity, sells coal in all blocks in all cases. This is significant, because as we have seen from Propositions 3.3 and 3.4, the WTP for wind output for a genco is determined by what type of conventional generation it has on the margin, if any.

As expected, the amount of coal in the system is lowered in the high nuclear case compared to the low case. When nuclear capacity is low, there is higher coal output, reduced consumer demand due to higher electricity prices, and slightly higher wind construction which account for the loss of nuclear capacity in the model, when compared to the medium and high nuclear cases (see Propositions 3.1 and 3.2.)

90 91

3.4.3 Coal Displacement

Figure 3-8 below highlights the total coal output displaced by the addition of wind to the model. What is interesting about coal displacement is that we see various ratios of coal energy displaced to wind energy added, ranging from blocks where very little coal is displaced (ratio = 0.3), and some blocks where the total coal displacement is in fact higher than the amount of wind energy output in that block (ratio > 1).

These results are obtained by first running a version of the WCAPCOMP 1.0 model with the wind capacity market removed, and therefore no wind in the system, and observing the amount of coal output in each block. By then running the WCAPCOMP 1.0 model with the wind market added and observing the change in coal energy output, we obtain the results presented below in Figure 3-8.

Wind Energy vs Coal Energy Displaced - annual & selected blocks

ESWind Energy •^•Coal Displaced -*-Ratio

Block/Nuclear Case

Figure 3-8: Comparison of wind energy added to system, and coal energy displaced.

The ratio of annual total wind energy versus coal energy displaced ranges from 0.48-0.75 amongst the cases, although within specific blocks, this ratio varies from 0.3, to approximately 2.

91 92

Looking at the summer peak block in the low nuclear case, we see that even with 0.75 TWh of wind energy added to the system, there is less than 0.24 TWh coal displacement in that block. This is because OSG is fully utilizing coal in this block, and so by complementary slackness, the dual variable on the primal coal output constraint (3.4) YVCitkb >0, suggesting that OSG will lose profit if it reduces coal output. Thus no coal output is removed from the model when wind is added, in that particular block.

However, in the medium nuclear case, summer peak block, we see the opposite: the amount of coal displacement is actually greater than the amount of wind energy added. This occurs because for a given wind capacity, the amount of wind output is higher in the winter than the summer (due to seasonal variation), and this causes OLG and ONG to shift some of their nuclear maintenance from the summer season to the winter season, resulting in higher nuclear output in the summer blocks, which causes the coal-on-margin OSG to reduce it's coal output further (see Proposition 3.2). We will now explain this result in more detail, using the model equations.

The dual equation with respect to the maintenance decision MCi/nuc'k is:

/ -\ YMCit lWC/>rt (3.34) b V b J

Comparing the values of YVCj'nucu- for OLG (which represent the value of an extra unit of nuclear capacity in each block) between the no-wind model run, and the model run with wind s added, we observed that in the summer' £_YVCOLGnudearsummerb decreased by 13% with the b presence of wind, while in the winter ^YVC0LGnuclearwinterb decreased by 22%. Recall that b wind capacity factors are higher in the winter season than summer, which means that for a given level of wind capacity, there is more wind output in the winter blocks than summer blocks. This lowers the electricity price, and hence value of nuclear capacity in the winter more than it does in the summer, as shown by the disproportionate effect on

2_j OLG,nuclear,w\nter,b '

92 93

Therefore, nuclear maintenance is increased in the winter (where the value of extra nuclear output has decreased the most), and reduced in the summer, such that the dual equation with respect to MC-oLG,"nuc\Summer-becomes slack in the summer (requiring that MCVLCrnucysummer' = 0), but remains tight in the winter, resulting in higher winter maintenance. The net result is more coal displacement in the summer blocks because of increased nuclear capacity available in the summer blocks, which was caused by the addition of wind and subsequent nuclear maintenance re-allocation.

All of these results are somewhat counter-intuitive to the belief that if one unit of wind output is introduced into the system, then one unit of coal will be removed. The actual coal displacement, as we have seen above, is dependent on the block, who owns the wind capacity in the market, and who operates the coal generation.

3.4.4 Electricity Market Price and Profits

We find that with high nuclear availability, OWE earns significantly lower profits, which suggests there is financial downside risk to the wind-entrant, if nuclear capacity available is higher in the model, because of the downward effect of nuclear output on electricity market prices. The electricity prices obtained from each of these cases, for selected blocks, are presented below in Figure 3-9:

93 94

Electricity Market Price (selected blocks)

£" 45 1 i j c Marke t P ;it y s srag e El 3 Hourl y A v n c

sm-peak wt-base sm-peak wt-base sm-peak wt-base

L M H Season/Block/Nuclear Case

Figure 3-9: Electricity market prices for selected blocks, in the low, medium and high nuclear cases.

From Figure 3-9, as one would expect, we see that the more nuclear capacity is available in the system, the lower the electricity market price becomes (from Proposition 3.1). We can also see that the difference in the summer peak prices are smaller than the differences in the winter base prices, due to a higher wind capacity factor in the winter base block.

It should be noted that the electricity prices in the WCAPCOMP 1.0 model represent hourly average prices for a given block, and do not reflect short-term spikes in price that are sometimes observed in many competitive electricity systems with hourly spot markets. They are also somewhat lower than the current Ontario market prices, as we are using operating costs from 2000 levels, which in the case of coal are currently approximately twice the costs used in this model. The reason for this choice lies in being internally consistent with using 2000 load levels to derive the demand curve in each scenario.

The total profits for each genco in each nuclear case are presented below in Figure 3-10:

94 95

Total Profits

3500 -i — - - -

L M H Nuclear Case

Figure 3-10: Total profits for each genco in each deterministic nuclear case.

As expected, we see that OSG and OWE have lower profits as OLG and ONG gain market share from having increased nuclear capacity (comparing the low and high cases).

For OWE in particular, the difference in profits for each nuclear case is substantial, with OWE earning approximately 75% higher profits in the low nuclear case than the high nuclear case. Lower profits in the high nuclear case are a combination of a lower market price received for it's wind, due to more nuclear capacity in the system (see Proposition 3.1), and OWE constructing less wind capacity to begin with due to a lower WTP in the high nuclear case (see Proposition 3.6). Therefore, we see that OWE would face financial downside risk if the wind capacity investment decision occurred under nuclear uncertainty, as the value of wind decreases as a result of increased nuclear capacity in the high scenario. In Chapter Four, we extend this model to look at the impact of nuclear uncertainty on the value of wind.

One would expect the total profits of both OLG and ONG would increase as more nuclear capacity is available, because they benefit from having more cheap nuclear capacity available. However, the nuclear-only genco ONG is the only genco to actually earn increased

95 96 profits when nuclear capacity increases, while OLG's profits slightly decrease as it receives more nuclear capacity.

The reason for this surprising result is that when competitor ONG increases it's nuclear output to the electricity market due to having more nuclear capacity available, this has the effect of lowering the electricity price in the high nuclear cases across all of OLG's existing output (from Proposition 3.1). From OLG's perspective, it has a much larger output to the market than ONG (2 to 3 times higher depending on the block), and so it loses more market revenue (EP*OQ) than ONG does as the market prices decreases. In the end, ONG benefits more from having additional nuclear capacity than OLG does, for this reason.

3.5 Conclusion

Chapter Three has presented a decision support model WCAPCOMP 1.0 that simulates a wind capacity market, followed by one year of electricity market competition, along with some numerical results for an Ontario-like system.

Several key results emerged from the numerical examples, which varied the amount of nuclear capacity available to the two largest gencos. We noted that the WTP for wind output for coal-on-margin genco OSG is constant in all blocks and seasons, and was equal to the operating cost of coal, while the WTP for OLG, ONG and OWE varied with the season and block. Interestingly, while in the majority of blocks the WTP of OWE was higher than that of the other gencos, in some blocks this was not the case, with OSG having a higher WTP in the winter base block for the medium and high nuclear cases.

The total wind constructed was highest in the low nuclear case, with 6.8 GW wind capacity construction, while the high nuclear case had approximately 1 GW less. The majority of wind investment came from wind-only genco OWE, with OSG purchasing a maximum of 22% of the total wind construction in the high nuclear case, and with OSG, OLG and ONG together purchasing a total of 13% of wind capacity in the low nuclear example.

In terms of coal displacement due to wind output, we observed that the amount of coal displaced in the system is not simply a function of the amount of wind available due to seasonal and diurnal variation, and that knowing which genco(s) are operating coal and/or 96 97 wind plays a role. We saw that in some blocks, coal displacement due to the addition of wind was 0.3 (ie. low nuclear, summer peak block), while in other blocks, the coal energy displaced was actually greater than the amount of wind energy added in that block (ie. medium nuclear, summer peak block, ratio of 2). The annual ratios of coal-out/wind-in ranged from 0.48 to 0.75.

We also saw that the high nuclear case results in 75% lower profits to OWE than the low nuclear case, due to lower electricity market prices and less wind capacity investment by OWE in the high nuclear case. In the next chapter, we will extend the deterministic WCAPCOMP 1.0 model to include uncertainty in the amount of nuclear capacity available, which is resolved between the wind capacity market and electricity market stages. The resulting model WCAPCOMP 2.WOM is a stochastic complementarity programming model, and can be used to look at the effect of nuclear uncertainty on wind investments, coal displacement due to wind, as well as profits to the gencos.

In Chapter Four, we will also discuss the impact of a wind output market, which occurs in stage 2, after uncertainty is resolved and on the same timescale as the electricity market, and provides recourse on the wind investment decisions in stage 1. In the WOM, the wind-only entrant has the option of selling wind output contracts-for-difference to other electricity market gencos, assuming a willing buyer exists.

97 98

4.0 WCAPCOMP 2.WOM - Stochastic Two-Stage Capacity Expansion Model for Ontario-like Market

In Chapter Three we presented a deterministic model WCAPCOMP 1.0, which featured a market for wind capacity, along with electricity spot market competition. Now, we extend this model to WCAPCOMP 2.WOM, which incorporates uncertainty in the form of nuclear capacity available, between the wind capacity and electricity market stages, and a wind output market (WOM) where the wind entrant genco can sell the financial rights to a quantity of wind output to another conventional genco via a contract-for-difference in each season and block. This market is assumed to occur post-uncertainty, on the same timescale as the spot market for electricity. The wind capacity market in WCAPCOMP 2.WOM is a forward capacity market.

The wind output market allows participants to adjust their wind outputs via contracts-for- difference, as an additional recourse to the resolution of this uncertainty. The concept of a market for wind output contracts, under conditions of fixed total wind capacity, was explored analytically in Chapter Two using the WOM-EM model (see Section 2.2).

We will show that the presence of a market for wind output contracts yields higher levels of total wind construction under nuclear uncertainty, as well as increased coal displacement in all outcomes, and higher expected profits and outcome profits for the wind entrant.

In Chapter Four, we will first present a high-level description of the WCAPCOMP 2.WOM model, emphasizing the extensions on the WCAPCOMP 1.0 model from Chapter Three. Next, we introduce the notation and formulation of the WCAPCOMP 2.WOM model, and explain how we incorporate anticipation (market knowledge) into the WOM contract decisions of the participants. Finally, we present some numerical results of the WCAPCOMP 2.WOM model, which focus on the amount of wind capacity constructed, coal displacement due to wind, and profits to the gencos. We show results for a base uncertainty model without a WOM (referred to as WCAPCOMP 2.0), and then with a WOM (WCAPCOMP 2.WOM) with advanced levels of market knowledge for the WOM participants, as will be defined shortly in Section 4.1.2. The effect of WOM market knowledge on the results will be

98 99 explored in more detail in Chapter Five, where we contrast these results with a case where gencos have basic knowledge in the WOM.

4.1 WCAPCOMP 2.WOM Model Structure

WCAPCOMP 2.WOM is an extension of the WCAPCOMP 1.0 strategic planning model which is a representation of a forward wind capacity market operating pre-uncertainty, followed by a wind output market for contracts, and spot market for electricity, post- uncertainty. We refer to the pre-uncertainty market as epoch 1, and the post-uncertainty markets collectively as epoch 2. The total amount of wind capacity in the system in epoch 2 is assumed fixed by all gencos, and both epoch 2 markets are settled simultaneously.

WCAPCOMP 2.WOM Model Diagram

High nuclear outcome p(H) I WOM(H) | ! EM(H) |

Wind Output Market (WOM)

G1 X1 ^—" •

Wind capacity - i Elec market i •— m m / m • \ ! WCM ^v Med nuclear outcome p(M) purchases ^ ( Gw ! ou,pute j, EM(M) \ J

Xn ^ -_^ Gn

Xi- wind output purchased in RWM by t enco i WOPi - price in RWM to genco i

Low nuclear outcome p{L) [ WOM (L) ! [ EM (L) ! Pi

i J

Figure 4-1: Diagram of WCAPCOMP 2. WOM, focusing on the addition of a market for wind output contracts (WOM) occurring after uncertainty is resolved.

The wind capacity and electricity market structure in the WCAPCOMP 2.WOM model is the same as the WCAPCOMP 1.0 model of Chapter Three. However, in WCAPCOMP 2.WOM,

99 100 gencos choose their forward wind capacity purchases under uncertainty, knowing the probability of each potential nuclear outcome, rather than simultaneously with their electricity market outputs. Gencos use the expected value of wind capacity from each of the scenarios to make their forward wind capacity decisions.

Following the wind capacity market, uncertainty is resolved, meaning that the remaining epoch 2 decisions in the wind output and electricity markets are made under deterministic conditions. Gencos do not anticipate a change in either of the epoch 2 markets as a result of their epoch 1 purchases, which is a reasonable assumption if the epochs are separated in time by uncertainty.

In epoch 2, each genco also has the option of participating in the WOM, which occurs in each season and block, where the conventional gencos can buy financial rights to wind output from the wind-only entrant for that block. We use an anticipation framework in the WOM to model different levels of market knowledge in the WOM for the genco participants, as will be explained in the following section. The WOM provides an avenue of recourse to uncertainty by allowing gencos to buy wind output contracts, after their investment decisions are fixed, and opens up an alternate revenue stream for the wind-only entrant, who is the seller of wind output contracts in this market.

With the presence of nuclear uncertainty, as we saw in the deterministic Chapter Three results for low, medium and high nuclear cases, there exists financial and emissions risk to the decision makers and customers. In terms of financial risk, the gencos, particularly the wind-only genco, face the possibility of lower electricity market prices if nuclear availability is high, and so there is downside risk present. In terms of emissions risk, the presence of uncertainty affects the wind capacity investment decisions of each genco, and if the low nuclear outcome is realized, the amount of coal output sold to the electricity market increases, and so customers face the possibility of increased carbon, S02 and NOx emissions. Our model demonstrates that the presence of a WOM helps to increase the total expected and actual profits of the wind-only entrant in all outcomes, and improves coal displacement by transferring financial ownership of wind output to gencos who are operating

100 101 coal in the electricity market, who then have incentive to reduce their coal outputs, as we will show in this chapter.

Finally, we point out that the framework of the WCAPCOMP 2.WOM model allows for other types of uncertainty to be examined, such as demand, operating costs, or other generation availability, with some slight formulation adjustments, although we do not explore these areas in this thesis.

In summary, WCAPCOMP 2.WOM is a stochastic complementarity programming decision support model, which extends WCAPCOMP 1.0 by adding uncertainty, as well as a recourse action on wind investments in the form of a wind output market, which can be used to evaluate the effectiveness of various market structures or policies on wind investment, coal displacement, and profits to the gencos.

4.1.1 Stage Two Wind Output Contract Market (WOM)

Because we are adding uncertainty to our model, we must also consider an avenue of recourse which could potentially be available to gencos in the electricity market. Specifically, we propose that the wind-only genco, with available wind capacity built in stage 1, is able to sign contracts to sell the financial rights to wind output to another genco in a given block, as long as there is a genco who is willing to buy, at a mutually acceptable contract price. These transactions open up an alternative for the wind-only genco to selling wind to the electricity market, and receiving the electricity market price, instead receiving a contract price from another genco in the market.

Unlike the wind capacity and electricity markets, there is no market operator in the WOM, because there are no price-taking market participants, and so trades are negotiated bilaterally between the wind-only seller, and each conventional genco. In essence, this market gives the gencos the ability to obtain 'virtual ownership' of wind output in a given season and block.

These contracts for wind output in the WOM are structured as follows. For a contract quantity A, the purchasing genco pays a fixed wind output price multiplied by the purchase quantity (WOP*A) to the seller. The seller still delivers the A units of wind output to the

101 102 electricity market physically, but gives the revenue it receives (EP*A) to the purchaser of the WOM contract, as is shown below in Figure 4-2:

Wind Output Market Contract-for-Difference

EP*A

Net revenue to Gw: Net revenue to Gi: = EP*(OQw -A) = EP*(OQi +A) - +WOP*A -EP*A Gw WOP*A G(i) WOP*A (seller) (buyer)

OQw OQi EP*OQw EP*OQi

EMO

Figure 4-2: Definition of wind output market contract, where seller Gw sells financial rigkts to A units of wind output to buyer G(i), at a contract price of WOP*A.

This is similar to a contract-for-difference, whereby the purchasing party pays a fixed price for financial rights to output, while the physical output is delivered to the market by the seller, and the revenue from this output is given to the purchaser.

We will show in our results that if G(i) purchases in the WOM, then it has incentive to reduce it's conventional output, and increase EP, thereby increasing the revenue it receives for the wind output purchased in the WOM, as well as the other output it sells to the electricity market. This was also foreshadowed in Chapter Two Section 2.2.1, using the WOM-EM model to derive how the buyer reduces conventional output as a result of a WOM contract purchase.

Similar to the wind capacity and electricity markets of the Chapter Three model, each genco in may have a different perception of how the epoch 2 markets will respond to it's buy or sell decisions in the WOM, and therefore we use an anticipation approach in the WCAPCOMP

102 103

2.WOM model to simulate varying levels of market knowledge, and thus competitive behaviour in the WOM, as will be discussed in the next two sections.

4.1.2 Use of Anticipation for Effect of WOM Contracts on Electricity Market Equilibrium

In the wind output market of WCAPCOMP 2.WOM, we assume that each genco anticipates that buying or selling in the WOM may cause changes in the electricity market equilibrium. It was pointed out in Chapter Two that in order for a genco to forecast it's WTP or WTS in the WOM, it must have some perception of these effects. Collectively we refer to these assumptions as cross-market anticipation, reflecting the fact that gencos are forecasting the effect of their decisions in the wind output market on the electricity market equilibrium.

In the formulation of the WOM for the WCAPCOMP 2.WOM model, we introduce two anticipation coefficients, which capture each genco's level of cross-market knowledge as to how the electricity market price, and it's electricity market output (in the case of the conventional gencos only) will change if the genco buys or sells an extra unit of wind output in the WOM. Having this information in the model allows a genco to compute it's WTP if it is the buyer, or WTS if it is the seller, when determining how much wind to purchase or sell under a contract-for-difference in the WOM.

We define levels of WOM cross-market knowledge as being either advanced, or basic, which line up loosely with the concepts of closed-loop and open-loop knowledge as discussed in the analytical WOM-EM model of Chapter Two. Table 4-1 below describes, in words, the assumptions made by a buyer or seller in the WOM, if the genco increases the amount purchased or sold in the WOM by 1 unit:

Table 4-1: Definition of basic and advanced cross-market WOM knowledge in the WCAPCOMP 2.WOM model

103 104

Basic Advanced

Buyer (conventional) Buyer will re-optimize conventional All gencos (including the buyer) gencos) outputs, but other genco's outputs will will re-optimize conventional remain fixed. outputs in response.

Seller (wind-only Other gencos conventional outputs will All gencos will re-optimize genco) remain fixed (seller cannot re-optimize total conventional outputs in response output as it has no conventional generation (seller cannot re-optimize as it has to adjust) no conventional generation to adjust)

In the WOM-EM model of Chapter Two, Cournot electricity market behaviour was assumed, and so we were able to derive closed- and open-loop values for the anticipation coefficients given that assumed level of electricity market behaviour. In the larger WCAPCOMP 2.WOM model, each genco may have a different level of competitive behaviour in the electricity market, and so analytically deriving values for the cross-market anticipation coefficients in the WOM becomes more difficult.

Instead, we apply a simple calibration technique to obtain values for the basic and advanced levels of WOM cross-market knowledge. This approach involves first solving the WCAPCOMP 2.WOM model with no wind output market, and then fixing the wind capacity investments at the levels observed from that case. Then, we simulate a contract of 1 unit of wind output between each conventional genco, in turn, and the wind-only genco, and observe the change in the electricity market price and buyer & seller electricity market outputs. By either fixing the outputs of the non-trading gencos, or letting them respond by allowing them to re-optimize, we can obtain the correct anticipation values for basic or advanced knowledge levels, respectively, in the neighbourhood of the no-WOM equilibrium. This calibration procedure is discussed in more detail in Appendix E for the interested reader.

It is necessary to point out that when applying this model to a real market, the model analyst would be required to supply values for these anticipation coefficients, using market data and experience, rather than using the model itself to generate the anticipation coefficients

104 105 endogenously. However, for the purposes of demonstrating the model, and having internally consistent results, we use this calibration approach to derive the values of cross-market anticipation in the WOM for basic and advanced knowledge paradigms. In Chapter Five, we will look at the difference between having basic versus advanced knowledge in the WOM, and how it affects the results for some sensitivity cases.

4.1.3 Use of Anticipation of WOM Price

In the analytical WOM-EM model of Chapter Two, we verified that a mutually-beneficial price exists between a conventional-on-margin buyer, and a wind-only seller, by showing that the WTP of the buyer was higher than the WTS of the seller, for the first unit of wind output traded. Theoretically, as long as WTP > WTS, any price between WTP and WTS would be acceptable to both gencos, and so there exist infinite potential prices in the WOM where each genco would earn a surplus. In reality, the buyer and seller would negotiate a contract price somewhere between these upper and lower bounds.

The wind output market contract price is related to the WTP and WTS of the participating gencos. The acceptable WOM price for the buyer and seller will change in response to the amount traded in the WOM, because as we saw in Chapter Two, the electricity market price and outputs are affected by WOM transactions, as gencos re-optimize their electricity market decisions.

Unlike the wind capacity or electricity markets, we do not have an explicit demand or supply curve in this market, because we have strategic decision makers on both sides of the WOM, rather than a price-taker engaging with strategic gencos as we do in the WCM or EM. The WCAPCOMP 2.WOM model addresses this issue, by allowing each genco to anticipate the rate of change of the negotiated WOM price for wind output, if it offers to buy or sell another unit in the market. The buyer is made to perceive an increasing WOM contract price as it buys more, while the seller is made to perceive a decreasing WOM contract price as it sells more. In this manner, we avoid the problem of having many feasible WOM prices, and are able to obtain a solution in the WOM which has a single negotiated price for each buyer- seller pair.

105 106

This WOM price anticipation takes a functional form similar to the wind capacity and electricity markets of WCAPCOMP 1.0. This anticipation affects the perceived marginal expenditure to a buyer, or the perceived marginal revenue of the seller, for purchases or sales of contracts in the WOM.

We do not distinguish between basic or advanced levels of knowledge with respect to this anticipation term. We simply note that the lower the value assumed for this anticipation, the higher the quantity that a genco will choose to buy or sell in the WOM, because the genco will not expect the WOM price to increase or decrease by a significant amount, for the buyer or seller, respectively. This anticipation is necessary merely to guarantee that the buyer and seller in the RWM can agree on a price in the model, and simulates the negotiation of that price.

4.2 WCAPCOMP 2.WOM Model Notation

We now present the notation used in WCAPCOMP 2.WOM, emphasizing the changes from the WCAPCOMP 1.0 model notation found in Table 3-1.

The notation used for the stage 1 forward wind capacity market in WCAPCOMP 2.WOM is unchanged from Chapter Three. In the stage 2 electricity market, we add a scenario index V to all decision variables of the gencos and EMO, as well as the parameter pcapsnuct, which represents the amount of nuclear capacity, and is different in each outcome to reflect the amount of nuclear capacity available.

In the WCAPCOMP 2.WOM model, we interpret the electricity market offer quantities

OQSikb and OQwsj± as the conventional and wind genco's physical outputs to the spot markets. For purposes of determining revenue to each genco in consideration of WOM contracts for wind output, we define new variables Xbsikb and Xssikb to represent the WOM contract quantities, such that the total financial output to the spot market for genco / is

OQsikb + Xbsikb, and OQwskb -^Xssikb for the wind genco.

We now present the following new notation, which is used in the WCAPCOMP 2.WOM model but not found in the WCAPCOMP 1.0 model:

106 107

Table 4-2 - Additional notation for WCAPCOMP 2.WOM

Notation Description

SETS: s Epoch 2 scenarios

PARAMETERS: Ps Probability of scenario s occurring in epoch 2

EPOCH TWO:

Wind Output QXsikb Conventional genco (buyer) anticipated change in Market: electricity market price, if additional unit of wind output is purchased in WOM ($/MWh*MW)

axwsikb Wind genco (seller) anticipated change in electricity market price, if additional unit of wind output is sold to genco i in WOM ($/MWh*MW)

aoqSikb Conventional genco (buyer) anticipated change in it's physical output OQSjkb , if additional unit of wind output is purchased in WOM (GW)

awpsikb Conventional genco (buyer) anticipated change in WOM price, if additional unit of wind output is purchased in WOM ($/MWh*MW)

awpwsikb Wind genco (seller) anticipated change in WOM price, if additional unit of wind output is sold to genco i ($/MWh*MW)

VARIABLES:

EPOCH TWO:

Wind Output Market:

Conventional Xbsikb Amount of wind output contracts bought from seller in Gencos: WOM (GW output)

Wind-Only -Xssikb Amount of wind output contracts sold to genco i in Genco: WOM (GW output)

107 108

4.3 WCAPCOMP 2.WOM Model Formulation

We now present the formulation of the WCAPCOMP 2.WOM model, for the conventional gencos, wind-only genco, EMO and WMO, along with the linking constraints. In this section, we emphasize the differences between this model, and the deterministic WCAPCOMP 1.0 model of Chapter Three.

4.3.1 Conventional Genco Primal Formulation

We now present the formulation of each genco's primal problem, consisting of the objective function and primal constraints.

4.3.1.1 Conventional Genco Objective Function

Each conventional genco maximizes it's expected epoch 2 profits, weighted by the probability of each outcome, minus stage 1 wind investment costs. Epoch 2 profits are equal to the revenue from selling generation to the electricity market (including revenue from WOM contracts) plus peak payments for capacity, minus operating costs, fixed capital costs, and costs incurred from purchasing WOM contracts from the wind-only genco.

Each conventional genco's objective function is as follows:

FP M aeikb {OQsikb - BQsikb )+ axsikb {xbsikb -Xssikh) wb ™bkb G(i):maxYJPsYJ kb k,b aoc Xs C VC h VH (°Qsikb ~ tsikb i^sikb ~ sikb ) + Xbsikb ) "Z ° ' ' »kb - ° • iklikb

WOP (4.1) 1 kb awp (Xb -Xs ) ' Xbsikb -Z^Z*^ wb sikb sjkb sikb k,b _Ps- kb C PVC + h PVH +YJP^PP~f ')- sit iPP-J )- si s I

-[fVCP + aCi (OU, - BU, )]• OUj

The first line captures expected spot market revenue from financial outputs to spot market (physical outputs plus WOM contract output), along with variable operating costs. The second line captures the expected purchasing cost of contracts in the WOM. The third line captures the expected annual fixed operating costs and peak payments from the EMO. Finally, the fourth line captures the cost of purchasing wind capacity in stage 1. 108 109

Note the appearance of the cross-market WOM anticipation coefficients axslkb and aoqSikb, representing the expected change in the electricity market price, and the genco's offer quantity to the electricity market, respectively, if they were to buy an extra unit of wind XbSikb in the WOM. At equilibrium, since Xbsikb=XsSikb, these anticipation terms net to zero in the objective function, but appear in the dual equation with respect to WOM purchase Xbsikb, as we will show further on. This allows the genco to forecast it's WTP for a WOM contract, by anticipating how it's market outputs will adjust if it signs the contract, and how the electricity market price will be affected, as we will illustrate via an analysis of the KKT conditions of the WCAPCOMP 2.WOM model in Section 4.3.8.

4.3.1.2 Conventional Genco Primal Constraints

In terms of primal constraints, these are very similar to the WCAPCOMP 1.0 model, but with the expansion to a set of electricity market constraints for each scenario s, and with modifications to the total electricity output constraint to incorporate anticipation of the effect of WOM contracts on the genco's offer quantity to the electricity market.

The total electricity market output constraint is modified to facilitate anticipation by a genco of how it's physical output to the market will change if it purchases more wind in the WOM:

X VCsm + VHsikb + VWsikb > OQsikb - aoqsikb (Xbsikh - Xssikb ) 1 YOQsikb (4.2)

Again, the anticipation term on the RHS will be zero at equilibrium, but allows for the dual equation with respect to WOM purchases Xbsikb to properly capture the expected operating cost savings from displacing conventional output with wind output, as we will discuss later in Section 4.3.8.2.

The remaining primal constraints from the WCAPCOMP 1.0 model, handling the conventional and hydro outputs, conventional seasonal maintenance, and offer prices to each market, are functionally unchanged in WCAPCOMP 2.WOM. The only additions involve inserting subscript ^ to the decision variables, and production capacity parameter pcapsi-nuc; which varies by epoch 2 outcome. We present the remaining primal constraints for the conventional genco below for completeness:

109 110

PVCsit ^ PcaPsti 1 YPVC, (4.3)

VCsitkb+MCsitk<(\-foutt)PVCSl ±YVCsilkb (4.4) ywsikb

PVHsi < heap, spring 1 YPVH„ (4.6)

VHsikb ^ hcaPik ±YVHsikb (4.7)

^wbkb-VHsjkb

VHsikb * hrrik ± YHRRsikb (4.9)

^wbu-MC^ZSJG-mrcrPVCs, 1 YMC., (4.10) k,b

EPskb ae B (4.11) OEP,sikb - sikb{°Qsikb- QSikb) ±YOEPsikb wbkb

OWCP, = WCP-aci(Of/,. -BU,) 1 YOWCP; (4.12)

4.3.2 Wind-Only Genco Primal Formulation

For the wind-only genco, one additional constraint required, which ensures that wind-only genco cannot sell more WOM contracts than it's total physical wind output in a given season and block:

^Xssikb

The wind-only genco's formulation is otherwise similar to the conventional genco in terms of the objective function, and is presented below along with the remaining constraints:

EP, skb - aewkb {oQyoskb - BQwskb)- X axw^ (^<** ~~ ^«*) k.b wbkb Gw: max Zp* Xs OQwskb-Yj sikb

wb WOP sikb •Xs sikb (4.14) ^P'H »> Z - - awpwsikb (Xssikb - Xbsikb) k,b Ps-wb,kb -[WCP + acw(OUw-BUw)]-OUw 110 Ill subject to

(4.13) above

VWwskb ^OQw: skb ±YOQwskb (4.15)

VWwskb

Y,Xssikb

EP. OEPwskb = -f- - aewkb {OQwskh - BQwskb) 1 YOEPwskb (4.18) wbkb

OWCPw= WCP+acw(OUw-BUw) 1 YOWCPw (4.19)

The only significant difference between the wind-only genco and the conventional gencos is that the wind-only genco can sell wind output to multiple buyers, and requires a separate anticipation coefficient to represent the effect of sales to each potential buyer on the electricity market price. The wind genco does not require anticipation of the effect of selling a WOM contract to the gencos on it's physical market outputs, because these will remain unchanged as the genco has no conventional generation which it may adjust, which is not the case for the conventional gencos.

4.3.3 EMO Primal Formulation

The EMO is formulated once for each potential scenario s, but is otherwise functionally the same as Chapter Three, and is presented below for completeness:

EMO{s): min £ wbkb • {OEPsikb • BQsikb )+ J] wbkb • {OEPwskb • BQwskb) (4.20) i,k,b k,b subject to.

Dskb^BQsM+BQWsu ±EP,skb (4.21)

EP,skb - Pokb " Plkb ' Askb ±YD skb (4.22) wb,kb

111 112

4.3.4 WMO Primal Formulation

The WMO formulation in the WCAPCOMP 2.WOM model is identical to that of WCAPCOMP 1.0, and can therefore be found in Section 3.3.4.

4.3.5 Primal Linking Constraints

The link constraints for the wind capacity and electricity markets from the Chapter Three model appear again in WPCAPCOMP 2.WOM, with the wind capacity market links being identical to those found in Section 3.3.5, and with the electricity market links being expanded to cover all outcomes, by adding a scenario s index to those constraints.

In addition, the WCAPCOMP 2.WOM model also has a set of WOM linking constraints, which ensure that the buyers in the WOM cannot purchase more wind output than the seller is willing to offer them:

Xssikb>Xbsikh ±WOPsikb (4.23)

The dual variable orthogonal to this linking constraint is the equilibrium WOM price for the contract between conventional genco / and the wind-only genco, which appears in the genco's objective functions in the preceding sections. In order for WOPSikb to be positive, by complementary slackness, the WOM linking constraint must be tight, and so we ensure that at equilibrium, the price of the WOM contract will be positive only if the buyer and seller agree on a WOM contract quantity.

4.3.6 WCAPCOMP 2.WOM Dual Constraints

The dual constraints of the WCAPCOMP 2. WOM model for all gencos, along with the EMO below in Table 4-3 (the WMO dual formulation is the same as WCAPCOMP 1.0 and is found in Table 3-2):

Table 4-3: Dual constraints for gencos & WMO agents in WCAPCOMP 2.WOM.

Agent Dual Constraints

112 113

Conventional -YBQsikb +YOQsikb +aeikbYOEPsikb > psEPskb K(4.24 ) Genco G(i) ±OQsikb ~Ps wbkb [aeikb (20Qsikb - BQsikb + Xbsikb)]

lVCsitkb(4.25) -YOQM + YVCsitkb > -Pswbkboc,

±VWsikb(4.26) -YOQsikb+YVWsikb>0

±VHsikb(4.27) -YOQsikb + YVHsikb - YHRRsikb + wbkbYHENGsik > -pswbkboh

±PVCsil (4.28) YPVCsit -^(l-foutt)YVCM +8.76mrctYMCsit>ps(pp-fct) k,b ±PVH„(4.29) YPVHsi>Ps{pp-Jh)

lMCsitk(4.30) Y,YVCsm-Y.wbaYMCsi,^ b b ao( YO ax Y0EP +W0P l a0( EP ~ isikb Qsikb - sikb Sikb Sikb ^ Ps( - Isikb) skb ax sikb \PQsikb + 2Xbs!kb - Xssikb ) (4.31) +pswb: kb LXb sikb -awpsikb(2Xbsikb-Xsslkb)

c -Y, fkbYVWsikb -aciYOWCPi -YBU, > -WCP'- ac t{lOU) -BU,) LOU ,{432) s,k,b

YOWCPj > 0 ± OWCPi (4.33)

YOEPsikb>0 LOEPsikb{434)

Wind-only -YBQwskb + YOQwskb + aewkbYOEPM > PsEPskbs Genco Gw 2 y (4.35) nu OQw -BQw -^Xs -pswbs kb aewkb skb M sikb ^OQwskb

-YOQwskb+YVWwM>0 LVWwskb{436)

YWOMskb -axwsikbYOEPskb -WOPsikb>-PsEPsk

axwsikb °Qy^skb-Yu^ sikb (4.37) wb IXs -Ps , kb \ i sikb

_- awpwsikb {2Xssikb - Xbsikb )

-^cfkbYVWwskb -acw-YOWCPw-YBUw> s,k,b (4.38) ±OUw -fVCP-acw{20Uw-BUw) LOWCPw (4.39) YOWCPw>0

113 114

YOEPwskb>0 lOEPwskb(AA0)

EMO YBQslkb -EPskb >-pswbkbOEPsikb ±BQsikb(4Al)

YBQwskb -EPskb >-pswbkbOEPwskb ±BQwsikb(4A2)

pmYDskb+EPM>0 1/^(4.43)

4.3.7 Complementarity Programming Formulation of WCAPCOMP 2.WOM

The WCAPCOMP 2.WOM CP is defined by the set of complementary pairs defined by all primal and dual equations for the gencos, EMO, WMO, and linking constraints as presented above, which define the KKT conditions for optimality for all decision-making agents in the model. The GAMS implementation of the WCAPCOMP 2.WOM model as a CP can be found in Appendix F.

4.3.8 KKT Analysis of WCAPCOMP 2.WOM

As we did for the WCAPCOMP 1.0 model, we will now present an analysis of the KKT conditions of the WCAPCOMP 2.WOM model, exploiting complementary slackness to derive some relationships which will help to explain the numerical results which will follow in Section 4.4, focusing on the WOM and wind capacity markets. The insights derived from the WCAPCOMP 1.0 model in Section 3.3.8.1 with regards to the electricity spot market apply to this model as well.

4.3.8.1 Analysis of Wind Output Contract Effect on Genco Outputs

In this section we will examine the effect of WOM trades on the optimal electricity market decisions of the gencos, via an important Proposition 4.1, which will be referred to often to explain the numerical results which conclude Chapter Four.

Proposition 4.1: If genco i has conventional generation type t on the margin, and purchases some wind output in a WOM contract in a particular block *, then genco i will reduce it's

114 115

conventional output to the electricity market in that block such that it's new optimal OQsi*' < EP.' EP, OQsi*, and the resulting equilibrium electricity market price —— > —— . wb, wb»

Proof: If technology t is on the margin, and so VCsit*, OQsi*>0, then their orthogonal dual equations (4.24) will be tight by complementary slackness. Since coal output primal constraint (4.4) is slack for technology t since VCsjt*, it's orthogonal dual variable YVCsit* =

0. Assuming OQiskb=BQSjkb and Xbsikb=Xssikb, and recognizing that YBQsikb= YOEPsikb (as demonstrated in the proof of Proposition 3.3), we can solve the dual equations with respect to

VCsu* (4.25) and OQsl* (4.24) as equalities to obtain the following expression:

FP aeit (OQsi, + Xbsi, ) = —f-oc, (4.44) wb,

EP * From the EMO market balance constraint, ——-P0* ~P\*DS* (4.22), and recognizing that in wb* order to be an equilibrium solution, Ds* = Vog, +OQw, we can substitute these into (4.44) i to remove EPS* and obtain the following identity which must be satisfied in the equilibrium solution:

a«i* (OQsi* + XbSi*) = Pa* - P\* X OQsi* + OQw\-oc,s* (4.45)

If we solve this equation first assuming Xbsi* = 0, and then for a value Xbsi* = A (A being sufficiently small to not change the set of basic variables), we can compare the optimal values of OQsl* (output quantity prior to the WOM contract) and OQsi*' (output quantity following the WOM contract for A units) and show the result:

OQsi.'=OQsi.-A-^^ (4.46) aejt + px* This shows that the optimal output quantity of genco / following a purchase of a WOM contract is lower than the optimal output had no contract been formed. Genco / would therefore reduce it's marginal output VCsit* to satisfy it's new optimal total output. It follows

115 116

EP ' EP from Proposition 3.1 that —— >—— because of this output reduction by genco i, which wb* wb» completes the proof.

Another way of stating Proposition 4.1 is that for a given level of wind capacity in the model, if some non-zero amount of this wind output is purchased via a contract-for-difference from the wind entrant by a genco with coal on the margin, the total amount of coal output in the market will be lower than if no such WOM contract existed, and the wind-only genco simply sold the output to the electricity market

4.3.8.2 Analysis of Wind Output Market Dual Equation for Conventional Genco Buyers

We will now explain how the various WOM anticipation coefficients appear in the dual equations of the conventional genco buyers, as an illustration of what factors contribute to each buyer's willingness-to-pay for a wind output contract from the wind-only genco seller in the WOM. From this we can derive two useful Propositions 4.3 and 4.4, which provide insight into the maximum price a conventional genco would be willing to pay for a WOM contract, for cases with conventional generation on or off the margin for the buyer.

Recall from the WOM-EM model of Chapter Two (Section 2.2.2.1), we showed that the WTP of a conventional genco buyer with generation on the margin for a WOM contract from a wind-only seller was made up of three components: revenue from selling additional output to the electricity market, revenue from an increased electricity market price, and operating cost savings due to the displacement of the marginal conventional technology. Depending on the cross-market WOM knowledge held by the buyer (ie. basic or advanced) and whether it has conventional generation on the margin or not, it will have a different perception of each component, as we will now illustrate.

7 Recall that cross-market knowledge in the WOM refers to each genco's anticipation of how it's physical output, and the electricity market price, will change if it purchases additional WOM contracts, as defined in Section 4.1.2.

116 117

The dual equation with respect to Xbsi* (4.31) for some block *, rearranged so that marginal costs are on the LHS, and marginal revenues are on the RHS, and recognizing that at equilibrium Xbsi*-Xssi*, is as follows:

WOP.,, ,„ _ EP* EP * YOQ . + awp *Xb *> ^ x *(OQ * Xb ,.)-aoq *\—^-^f\S si (4.47) J IT SIsi SIsi J + a si si + s si wb* pswb* pswb» wb* MR from increase in EP marginal expenditure MR from WOM contract output jost revenue from decrease in OQ

On the LHS, the first term is the hourly equilibrium wind output contract price in a scenario 5 (which must be adjusted by the probability of scenario s and the number of hours in block *).

The second term is the anticipated increase in the WOM contract price awpsi* multiplied by the current amount of wind output being purchased by genco i, which is perceived as an additional expenditure associated with increasing the contract quantity. Combined, the LHS captures the perceived marginal expenditure of purchasing an extra unit of wind output in the WOM.

On the RHS, in the first term, —— is the revenue received from an additional unit of WOM wb* contract output, for which the genco receives the electricity market price.

The second term on the RHS multiples the anticipated electricity market price change axsi*

(axsi*>0 if genco / has generation on the margin because of Proposition 4.1, and caSj*-0 if it does not) by the total financial output of genco i (OQsi* + Xbsi*). This term represents the perceived marginal revenue from the electricity market price by axsi* across it's financial output to the electricity market in block *, for a unit increase mXbsi*.

( EP * YOO * 1 The final term on the RHS -aoqsi* —£- ^>L is the anticipated reduction in conventional ^ wb* pswb* J output, multiplied by the difference between the electricity market price and the operating cost associated with the marginal technology. Recall that for gencos with marginal conventional generation, aoqSj* > 0 as a result of Proposition 4.1, while for gencos with no generation on the margin, aoqSi*=0. Together, this term represents the loss in revenue associated with selling aoqSj* fewer units of output to the electricity market. To illustrate that

117 118

YOQsj, is the expected operating cost of the marginal technology / of genco /, we present

Proposition 4.2:

Proposition 4.2: If a genco has conventional generation type t on the margin, the dual

variable YOQSi* is equal to the expected operating cost of I unit of output of generation type t across block *.

Proof. VCsit*>0 because technology / is on the margin by definition, and so dual variable

orthogonal to the coal output constraint (4.4) YVCsit*=0 by complementary slackness, and

dual equation with respect to VCsi*, - YOQsi, + YVCslt* > -pswb*oct (4.25) must be tight, which

reduces to the identity YOQsi, = pswb*oct. This the expected cost of operating 1 unit of generation type t in scenario s for a duration wb* of block *, which completes the proof.

The appearance of YOQSj* in the dual equation with respect to XbSi* (4.31) is a direct result of

including the anticipation term -aoqsi»(xbsi.-Xssi*)or\. the RHS of the primal offer quantity

balance constraint ^VCsit, + VHsi. + VWsi*>OQsl.-aoqst*(Xbsi*-Xssi,)(4.2). Without this term,

gencos would not factor in any operating cost savings which could arise as a result of purchasing a WOM contract and reducing conventional output (which we saw in Proposition 4.1), and would therefore undervalue the WOM contract, which is the reason this term was included in the model.

Having explained the dual equation with respect to XbM, we now present two useful Propositions 4.3 and 4.4, which describe the maximum WTP for a WOM contract for gencos with or without conventional generation on the margin.

Proposition 4.3: A genco with no conventional generation on the margin in a particular

block * (ie. VCsu*=0 or VCsit*-PVCSit for all t) would only be willing to purchase a WOM EP, contract for a price lower than the electricity price ——. wb*

Proof: Note that if genco / does not have conventional generation on the margin, then it

cannot adjust it's output, so it would have anticipation coefficient aoqsi*=0. The electricity market price will be unaffected because total output to the market does not change, and so

118 119

there will be no reaction from the other gencos in the market, and as a result axsi*=0.

Substituting these into the dual equation with respect to Xbsikb (4.47) yields the following identity which would have to be true if genco i purchases a WOM contract such that Xbsi^O, for an amount sufficiently small that it does not affect the set of basic variables:

WOP*. „ EPS* ,. . f-+awPsi*Xbsi*=—^- (4.48) pswb* wb*

Therefore, for any XbSi*>0, the KKT conditions for the conventional-off-margin genco WOP •* EP * require that —-^- < ——, which completes the proof. pswb* wb*

Proposition 4.4: For a conventional genco with technology t on the margin, the maximum

WOP * acceptable WOM contract price in block * , —, is an increasing function of it's total pswb* conventional output in the electricity market, the operating cost of technology t, and the current electricity market price.

Proof. The dual equation with respect to Xbsi* (4.47) must be an equality if Xbsi*>0 by complementary slackness, and by using Proposition 4.2 to substitute for YOQsi*:

WOP,, ,„ EPt EP* + awpsi*Xbsi*=—^ + axsi*(OQsi*+Xbsj*)-aoqsj*\ —^-oct (4.47) j * si si j wb* pswb> wb* For conventional-on-margin genco /', we know from Proposition 4.1 that there will be some output response by genco /, and so \>aoqSi*>Q. Because there is an output response, there will also be an electricity market price response (from Proposition 3.1), and so axsi*>0. Therefore, first, second, and third terms are all non-zero, and are each an increasing function of the spot price, output quantity, and technology t operating cost, respectively, which completes the proof.

An important observation from Proposition 4.4 is that while we cannot determine a priori whether the maximum acceptable WOM contract price WOP^t •* o the conventional-on-margin pswb* genco is greater than or less than the electricity market price—EP— * in general, we can wb*

119 120

conclude that for some given set of market cost and demand parameters, a larger OQsi* or oc, would make it more possible that the genco would be able to offer a price higher than the electricity price for a WOM contract.

4.3.8.3 Analysis of Wind Output Market Dual Equation for Wind-Only Genco Seller

We will now perform a similar analysis for the wind-only genco in the WOM, by breaking down the dual equation with respect to Xssikb and then presenting three useful Propositions 4.5- 4.7 which characterize the minimum price that the contract seller is willing to accept for WOM contracts with gencos with conventional generation on or off the margin, given basic or advanced levels of WOM knowledge.

From Chapter Two, we saw that the WTS of a wind-only genco to sell wind output to another genco via a contract-for-difference comes from two components: the opportunity cost of losing revenue from that unit in the electricity market, and increased profits on the remaining output sold to the electricity market, if the market price increases as a result of the contract.

Re-arranging the dual equation with respect to Xssncb (4.37) so that marginal revenue is on the RHS, and marginal costs on the LHS, for some block * and a buyer i we have:

r EPS, WOPsi, OQw^-^XSsi* (4.49) > awpwsi tXs sj* +axw wb, ps wb* V MC from losing spot output MR from selling extra WOM contract MR from spot price increase

The LHS is simply the current electricity market price, and represents the opportunity cost of losing revenue from 1 unit of wind output in the electricity market, and is the marginal cost to the wind-only genco of signing the contract.

The first bracketed term on the RHS are the current wind output contract price, minus the anticipated decrease in the contract price if it sells more wind output to genco / across it's total quantity sold to genco i, which taken together represent the marginal revenue associated with selling more wind output under a WOM contract.

The remaining RHS term is the anticipated increase in the electricity market price, multiplied by the wind genco's total financial outputs to the spot market (it's physical output minus the

120 121 total output it has sold under a contract-for-difference with the other gencos). Together, this represents the perceived marginal revenue associated with an increasing electricity market price across it's financial outputs to the market.

We are now ready to present some propositions which provide insight into the minimum acceptable WOM contract price to the wind-only genco, for various situations.

Proposition 4.5: Ifthe buyer of a WOM contract is a genco with conventional generation off the margin, and/or the wind-only genco has only basic cross-market WOM knowledge (ie. does not anticipate that other gencos will adjust their outputs), the wind-only genco will only sell a contract-for-difference if the WOM contract price is greater than the electricity market price, for some block *.

Proof. We know that if a conventional genco with no generation on the margin received wind output via a WOM contract, it cannot adjust it's outputs because all primal output constraints (4.4) are tight by definition, thus the electricity market price would not change, and so the wind genco cross-market anticipation coefficient axwsi*=0. IfXsSj*>0, meaning the wind genco was offering to sell a contract to genco /, then dual equation (4.49) must be tight by complementary slackness, and so we have the following identity:

WOPsi, v EPs S, .. „. f- - awpwsitXssi. = — - (4.50) pswb* wo*

This requires that for any Xssi*>0, for the wind genco to satisfy it's KKT conditions must have

WOP• EP — >——, if the seller is a strategic negotiator with awpwsi*>0, which proves the first pswb* wbm half of the proposition. Note that if the wind genco had only basic knowledge, then by definition (see Section 4.1.2) it would not expect an output response from the buyer

(regardless of whether it has conventional generation on the margin or not), and so axwi/*=0 in this case as well. Thus the same result follows if the wind-only contract seller has basic knowledge, which completes the second half of the proposition, and the proof.

121 122

Proposition 4.6: If the wind-only genco has advanced cross-market WOM knowledge (ie. it anticipates the other gencos with conventional generation on the margin will reduce their outputs), then it may be willing to accept a WOM contract price lower than the electricity market price, from a genco with conventional generation on the margin in some block *.

Proof: From Proposition 4.1, we know that the electricity market price will increase if a conventional-on-margin buys a WOM contract for wind output, because it will reduce it's conventional output to satisfy it's KKT conditions. Because the definition of advanced knowledge means that the wind-only genco would have some perception of this change by it's counterpart, the wind genco's anticipation coefficient axwsi*>0. Looking at the dual equation with respect to Xssi* (4.49) which must be an equality if Xssi*>0, we have the following equation:

WOPsl> v ,. c1. -axw„ OQws*-^Xssit wb* = awpwsi* Xssi* (4. j 1) pswb,

EP» WOP •» One can see readily that if axwsi*>0, then it is possible that —— > — while still

wb. pswbkb satisfying the above equality which must be true if the KKT conditions are to hold, which completes the proof. The reason that the wind genco may be willing to accept a WOM contract price lower than the equilibrium electricity price (if it has advanced knowledge) is that it recognizes that selling the contract will increase the electricity market price, such that the wind genco receives higher electricity market revenues for the remaining wind output (OQws, - VXssit ) i under it's control.

Proposition 4.7: No mutually acceptable WOM contract price exists between a conventional- off-margin genco, and the wind genco in some block * regardless of the level of WOM market knowledge.

Proof. This is a direct result of Propositions 4.3 and 4.5, which showed that the conventional- off-margin genco would only accept a contract price lower than the electricity price, while

122 123 the inverse is true for the wind genco, and so no mutually acceptable price exists for any

XbSi*=XsSi*=0 in a block * where the buyer / has no marginal generation technology, which completes the proof.

The propositions presented in this section are particularly useful for explaining the WOM contract decisions of the gencos in the WCAPCOMP 2. WOM model, and are used in Chapter Five as well to explain the results of the 2.WOM model with basic WOM knowledge assumptions.

4.3.8.4 Analysis of Willingness-to-Pay for Wind Capacity in the Forward Wind Capacity Market

In the WCAPCOMP 2.WOM model, gencos must purchase wind output under conditions of nuclear uncertainty. We will briefly illustrate via that each genco values wind capacity in the WCM at the total expected value of an additional unit of capacity, by looking at the dual equation with respect to wind capacity investment OUt (4.24) for the conventional gencos. The following analysis applies equally to the wind-only genco.

If a genco is purchasing wind capacity in the forward wind capacity market such that OUi>0, the purchased quantity is an increasing function of the total expected value of extra wind capacity in each outcome. By looking at the dual equation with respect to OUj (4.32), which must be tight if OUi>0, if we are at an equilibrium OUj=BUj > 0, then it can trivially be shown that YOWCPi=YBUi=0 by complementary slackness, and so we are left with the following equality which must hold if a genco is a purchaser of wind capacity in the forward WCM: acfiU^Y^cf^YVW^-WCP (4.52) s,k,b

Recall from (Winston, 1993), the dual variable YVWsikb is interpreted as the marginal increase in the objective function if the RHS of the wind output primal constraint (4.5) is increased by 1 unit (assuming the basis does not change, and the basic variables are adjusted to maintain feasibility). Since the objective function is in units of total expected profit, by definition YVWsikb is weighted by the probability of scenario s occurring, and the duration wbkb of block

123 124 k,b. Therefore, multiplying this by the wind capacity factor cfkb and summing across all scenarios, seasons and blocks yields the total expected value of an extra unit of wind capacity, which is on the RHS of (4.52) and is thus used by the genco to determine how much forward wind capacity OU( to purchase.

As in WCAPCOMP 1.0, YVWsikb depends on whether a genco has conventional generation on the margin or not, as was illustrated in 3.3 and 3.4 in the previous chapter.

Therefore, in the WCAPCOMP 2.WOM model, each genco decides how much wind capacity to invest in by comparing the total expected value of an extra unit of wind capacity to it's perception of marginal expenditure in the WCM, which is captured on the RHS of (4.52).

Having explored the KKT conditions of the WCAPCOMP 2.WOM model and derived some insightful conclusions from the various propositions, we are now ready to present some numerical results.

4.4 Numerical Results

We now illustrate the usefulness of the WCAPCOMP 2.WOM model by showing some numerical examples, for the WCAPCOMP 2.0 model (with no WOM), and the WCAPCOMP 2.WOM model.

We consider the same electricity system as the WCAPCOMP 1.0 models, with the same gencos, costs, demand, and wind parameters in those examples being used again here. The probability of the low, medium and high nuclear outcomes are each set at p=l/3. The level of electricity market and wind capacity market competitive behaviour is also assumed to be the same as the Chapter Three model numerical examples.

In terms of genco cross-market knowledge in the WOM (anticipation of electricity price and output changes from WOM contract decisions), we assume an advanced level of knowledge for all WOM market participants, as defined in

124 125

Table 4-1 in terms of how WOM trades affect the electricity market equilibrium. This level of knowledge is akin to closed-loop like levels, which assume that each participant anticipates that the electricity market price, and output quantity will adjust in response to the genco's actions in the WOM.

A complete list of the level of parameters used in the examples of Chapter Four can be found in Appendix G.

4.4.1 Wind Capacity Allocation and Wind Output Market Activity

We find that the presence of a WOM in the 2.WOM results increases the amount of wind capacity constructed in stage 1, when compared to the base uncertainty 2.0 results, and the WCAPCOMP 1.0 deterministic medium nuclear results. Below, we present the wind allocation in the 2.0 and 2.WOM cases, along with deterministic medium nuclear case from the Chapter Three for comparison.

Wind Capacity Investment • WCAPCOMP 1.0 (medium case), 2.0,2.WOM

deterministic med nuclear stochastic, no WOM (2.0) stochastic, WOM (2.WOM) Case

Figure 4-3: Wind Capacity investment in stage 1, for deterministic medium nuclear case, uncertainty case with no WOM (2.0), and with a WOM (2.WOM).

125 126

We can see that there is less than 1% difference in the wind investments when comparing the deterministic medium nuclear case of Chapter Three and the uncertainty case. This is because the expected value of wind output to the gencos across the low, medium and high nuclear scenarios is within 1% of the value of wind in the medium deterministic case, so y]cfkbYVWsikb s y\YVWsikb and therefore the stage 1 investments are similar (see Section k,b s,k,b 4.3.8.4).

However, when comparing the 2.0 and 2.WOM models, we can see that the total wind capacity constructed increases by 0.4 GW, with OWE purchasing 1.2 GW more wind, which drives up the price in the WCM (see Proposition 3.5) and therefore the amount purchased by OSG reduces. OWE invests in 19% more wind capacity in the WCM when a WOM is present, because WOM contracts sold to OSG and OLG by OWE increase the electricity market price (from Proposition 4.1). This in turn increases OWE's valuation of additional wind output YVWWM (an increasing function of electricity market price, see Proposition 3.4), and therefore the presence of the WOM increase the amount of wind capacity OWE invests in.

The trades which occur in the 2.WOM model are presented below, for selected blocks, along with who the purchasing genco is in each block. Keep in mind that these trades are in units of average hourly GW output, and have been adjusted by the capacity factor for the given season and block, which range from 0.15 in the summer peak, to 0.45 in the winter base.

126 127

Wind Output Market Activity (selected blocks)

a 0.6

a OLG • OSG * 0.4

Season/Block/Nuclear Outcome

Figure 4-4: WOM contracts for selected blocks, in the low, medium and high nuclear outcomes.

In the low nuclear outcome, OLG is the buyer, in all blocks. In these blocks, OLG has coal capacity on the margin, and is therefore earning a surplus by buying in the WOM, by displacing some of it's coal output, and increasing the electricity price (from Proposition 4.1). OLG's willingness to pay for wind output in the WOM is higher than OSG's in these blocks, even though both gencos have the same technology on the margin, because OLG benefits more from increasing the electricity market price, since it has a higher output quantity in the spot market (see Proposition 4.4).

However, in the non-summer blocks of the medium nuclear case, and all blocks of the high nuclear case, OSG has coal on the margin, while OLG is operating coal at full capacity and therefore a mutually beneficial WOM contract does not exist between OLG and OWE (a result shown in Proposition 4.7). As a result, OSG becomes the only buyer of wind in these remaining blocks.

Overall, the presence of a WOM for contracts-for-difference serves to increase the amount of wind capacity purchased by the wind entrant OWE, and drives total wind construction up by approximately 8% in our model when compare to the base 2.0 model with no WOM.

127 128

The prices at which the WOM transactions occur are presented in Section 4.4.3, along with the electricity market prices for comparison.

4.4.2 Coal Displacement

We find that the presence of a WOM reduces coal emissions by transferring financial control of wind output from the new entrant to the coal-on-margin gencos, who then have incentive to withdraw coal output (an effect shown in Proposition 4.1). The WOM therefore increases the annual amount of coal displacement due to wind in all three nuclear outcomes.

Figure 4-5 below compares the amount of annual coal displacement observed in the WCAPCOMP 1.0 deterministic nuclear cases, the 2.0 case, and the 2.WOM case, when compared to running the equivalent models with no wind capacity in the system.

Annual Coal Displacement - WCAPCOMP 1.0, 2.0, 2.WOM

• OSG BOLG

stochastic, no WOM (2.0) Case

Figure 4-5: Annual coal displacement for deterministic cases (chapter 3), base uncertainty case (2.0), and uncertainty case with WOM (2.WOM).

Comparing the deterministic 1.0 and stochastic 2.0 models, we see that the annual coal displacement in the deterministic model is higher than the stochastic model, for the low nuclear outcome, while the coal displacement in the medium and high nuclear outcomes is

128 129 almost the same. The reason for lower coal displacement in the stochastic 2.0 model (compared to the deterministic low nuclear model) is that the amount of wind constructed is lower by 3% because the possibility of a high nuclear outcome reduces the amount of wind capacity purchased by OWE in stage 1, due to a lower expected value of wind capacity ^cfkbYVWyvsikb (see Section 4.3.8.4). s,k,b

Therefore, the presence of nuclear uncertainty introduces an element of emissions risk, with a chance that there will be approximately 1 TWh more annual coal output if nuclear uncertainty is present during the wind investment decision stage, and the low nuclear outcome is realized.

However, we see that the presence of a WOM increases the annual coal displacement in all nuclear outcomes, due to WOM contracts bought by OLG and OSG and their subsequent withdrawal of coal output (from Proposition 4.1) compared to the base 2.0 model, and even out-performs the coal displacement of the deterministic results.

Wind Energy vs Coal Displacement due to WOM

ElWind energy added by WOM • Coal energy displaced by WOM 2 1

Season/Block/Outcome

Figure 4-6: Additional wind energy due to WOM, and resulting additional coal energy displacement.

129 130

As we can see above in Figure 4-6, in almost all outcomes and blocks in the low and medium nuclear cases, the amount of additional coal energy displaced when a WOM is added to the model is greater than the amount of additional wind energy built. The extra coal displacement beyond the 2.0 model levels is a direct result of the WOM contract purchases by the coal gencos OLG and OSG, which displace coal output when they re-optimize their conventional outputs after the purchase (from Proposition 4.1).

The WOM has the least effect on coal displacement in the high nuclear scenario, where the amount of WOM contract activity is lowest (see Figure 4-4), but the 2.WOM model still yields approximately 0.5 TWh more annual coal displacement than the base 2.0 model.

Overall, the WOM serves to reduce coal emissions in all cases, not simply the low nuclear case, by transferring financial control of wind output from the wind-only entrant OWE, to gencos OLG and OSG, who have coal on the margin and therefore displace some of it after purchasing contracts-for-difference from OWE in the WOM.

4.4.3 Electricity Market and Wind Output Market Contract Prices

We now compare the electricity market and wind output contract prices, and show that when OLG is the buyer (in all low nuclear and summer medium nuclear blocks), the WOM contract price is greater than the electricity price, while when OSG is the buyer in the remaining blocks, the WOM contract price is lower than the electricity market price.

130 131

Electricity Market & Wind Output Market Contract Prices (selected blocks)

HEP-2.0 • EP-2.WOM QWOP-2.WOM

sm-peak | wt-base sm-peak

M Season/Block/Nuclear Outcome

Figure 4-7: Electricity market prices for the 2.0 and 2.WOM models, along with the wind output market contract price in the 2.WOM models.

From Figure 4-7, we observe that in the majority of blocks, there is not more than a 1.5% difference in the electricity market prices when comparing the 2.WOM model to the base 2.0 model.

Most interestingly, the WOM price in the 2.WOM model is actually higher than the electricity market price in the low nuclear and medium nuclear summer-peak blocks, where OLG is the purchaser (refer to Figure 4-4). To explain, recall from Proposition 4.4, we know that the WTP for a wind contract-for-difference for OLG when it has coal on the margin is an increasing function of it's total output in the electricity market. Because OLG has such a high output quantity, consisting of nuclear and hydro as well as coal output, we observe that the WTP is sufficiently high that OLG agrees to a contract price which is up to 4% higher than the electricity market price (eg. in the medium nuclear summer peak block), to purchase a WOM contract from OWE.

In all of the other outcomes and blocks of the 2.WOM model, where OSG is the buyer, OSG is unable to offer a contract price as high as the electricity market price, because it has much lower coal output to the market (see Proposition 4.4, noting that OSG has smaller spot 131 132 market outputs than OLG). However, OWE is still willing to sell to OSG at a contract price lower than the electricity price, because it receives a higher electricity market price across all remaining units of wind output in the market, due to OSG withdrawing some of it's coal output (a result predicted by Proposition 4.6).

These results were also foreshadowed in the Chapter Two WOM-EM model, which showed that for two gencos with identical marginal generation technologies, the genco with the higher output to the market would have a higher WTP, because it benefits more from increasing the market price across all of it's existing output.

4.4.4 Genco Profits

We saw in Chapter Three that some gencos, particularly OWE, faced the risk of lower profits in the low nuclear outcome, due to a lower electricity market price. The presence of a WOM, as we will now see, increases the expected profits of OWE by giving it recourse against uncertainty, as well as increasing OWE's total outcome profits for each potential outcome.

Below we compare the total expected profits, and total profits if each of the three outcomes was realized, in the 2.0 and 2.WOM model cases, for the wind-only entrant:

132 133

Wind Entrant Total Expected & Outcome Profits

Expected Profit Low Nuclear Outcome Med Nuclear Outcome High Nuclear Outcome Expected / Nuclear Outcome

Figure 4-8: Total expected profits, and total profits for each potential outcome, for the no WOM (2.0) and WOM (2.WOM) model cases.

Above we can see that the difference in total profits to the wind entrant OWE from the low outcome realization to the high outcome realization is not reduced by the presence of the WOM, and therefore the WOM does not necessarily mitigate financial risk to OWE, in terms of the variance of profits between outcomes.

However, the profits to OWE, both expected and actual for all three outcomes, dominate the profits of the base 2.0 model, suggesting that OWE would benefit from the presence of a wind output market. OWE earns more than 13% higher expected profits when a WOM is present, with the outcome profits for each scenario realization increasing from 19% (low outcome) to 4% (high outcome).

We now present the expected profits and outcome profits for the remaining conventional gencos, along with the wind-only entrant:

133 134

Total Expected & Outcome Profits

g 2000 EOLG BONG • OSG DOWE

stochastic, no stochastic, WOM stochastic, no stochastic, WOM stochastic, no stochastic, WOM stochastic, no stochastic, WOM WOM(2.0) j (2.WOM) WOM (2.0) | (2.WOM) WOM (2.0) | (2.WOM) WOM (2.0) j (2.WOM)

Expected Profit Low Outcome Medium Outcome High Outcome

Figure 4-9: Expected and outcome profits to each genco, with and without a WOM.

Figure 4-9 shows that the profits of the incumbent conventional gencos are not affected by as large a degree by the addition of a WOM, varying by no more than 2% from the base 2.0 case with no WOM. It is OWE that earns the highest % gains in profit from the presence of a WOM, which is understandable given that OWE is most sensitive to the electricity market price, by virtue of not having other generation types which can be adjusted to react to the nuclear outcome realization, as the conventional gencos do.

From these results, it appears that the strategy for OWE which yields the highest profits is to purchase more wind capacity in stage 1, and sell some of the wind output under contracts- for-difference to either OLG or OSG, whenever they have coal on the margin, who then have incentive to buy such contracts and withdraw coal output to increase their profits.

4.5 Conclusion

In Chapter Four, we extended the WCAPCOMP 1.0 model to a stochastic version with uncertainty in nuclear capacity, and introduced a wind output market as a recourse action for gencos to adjust their wind purchases post uncertainty.

134 135

We found that the presence of a WOM increases the total amount of wind constructed, along with the total expected profits to OWE. Therefore the WOM is useful to guarantee a higher expected profit to the wind entrant when nuclear uncertainty is present, although the WOM does not reduce financial risk, defined as the variation in potential profits across the nuclear outcomes.

Furthermore, we also observed that the WOM results in higher coal displacement in all nuclear outcomes, by transferring financial ownership of wind output from the wind-only entrant OWE to the coal-on-margin gencos OLG or OSG, depending on who has coal on the margin and is therefore willing to buy. The price in the WOM was found to be close to the electricity market price, with OLG offering a price higher than the electricity price in some blocks, demonstrating a high willingness-to-pay for wind output when it has coal on the margin, and large amounts of output sold to the market. Overall, the WOM was found to have financial benefits to the wind entrant, and emissions benefits to society.

In the final chapter of the thesis, we will explore the sensitivity of our Ontario-like model results to some alternate market configurations, by using the WCAPCOMP 1.0 and 2.WOM models to look at the effect of basic versus advanced WOM knowledge, the number of gencos who own fossil generation in the market, and the slope of the demand curves used to describe consumers. In all of these examples, we focus on how each of these different market configurations affects the amount of wind capacity constructed, and the amount of coal displacement. We provide these additional results as a further demonstration of the usefulness of the WCAPCOMP models.

135 136

5.0 Market Structure Sensitivity

In the previous chapters of this thesis, we have presented models of wind capacity expansion in a competitive market setting. These models have allowed us to examine the value of wind, specifically the amount of wind capacity constructed, the allocation of wind capacity amongst the gencos, the amount of coal displaced by wind generation, and the resulting profits to each of the gencos. We have also looked at the effect of nuclear uncertainty on the value of new capacity, and explored the concept of a market for wind output contracts-for- difference post-uncertainty, where the wind entrant can sell financial rights to wind output to other competitors, rather than receive payment from the electricity market.

In the last chapter, we will present some case studies, which focus on the effect of various market structures or policies on the amount of wind constructed, and the amount of coal energy displaced by wind. We will focus on three areas: the effect of WOM market knowledge on the amount of contracts traded, number of gencos in the market with coal generation, and sensitivity to the assumed demand curve slopes describing the consumers. These results will provide an illustration of the sensitivity of the results of the previous two chapters to these aspects of market structure.

In the first half of the chapter, we will explore the interaction between two of the above dimensions: ownership of fossil generation (single ownership versus split ownership) and level of wind output cross-market knowledge (basic versus advanced), using the WCAPCOMP 2.WOM model, for a total of 4 comparison cases. For ownership of fossil generation, we will compare the Chapter Three & Four case of split ownership 1:2 between OLG and OSG, versus a case where OLG and OSG are combined into a single integrated genco named "OIG", who then has a monopoly on fossil generation. For each of these ownership cases, we will also contrast results obtained assuming either basic knowledge in the WOM (genco does not expect other gencos to adjust outputs in response to a WOM contract purchase) or advanced knowledge (when gencos anticipate their counterparts will adjust outputs, which was assumed in the Chapter Four results).

136 137

In the second half of the chapter, we will look at the effect of using a higher point elasticity of demand to derive the demand curves for each block in the electricity market. The effect of using a higher point elasticity of demand results in a shallow demand curve slope, which impacts the electricity market price, and therefore the model results, as will be illustrated in the subsequent sections. In the WCAPCOMP 1.0 model of Chapters Three, we presented results using e=0.5 to compute the demand curves. We will contrast those results with a case where we use e=0.8 to calibrate the demand curves, which results in a shallower demand curve slope, and illustrate the effect this has on electricity market prices, wind construction & coal displacement, using the WCAPCOMP 1.0 deterministic model as our comparison tool.

We close the chapter with some notes on the stability of our model, along with some recommendations for future work.

5.1 Effect of Cross-Market WOM Knowledge and Fossil Generation Ownership

In the first two sections, we will look at the sensitivity of the WCAPCOMP 2.WOM results to two dimensions of market structure: wind output market knowledge, and fossil generation ownership. We look at the effect of basic knowledge versus advanced cross-market WOM knowledge (which was assumed in Chapter Four, as defined in Section 4.1.2), and also the effect of combining the two fossil gencos OLG & OSG from Chapters Three and Four into a single entity OIG, for a total of 4 comparison cases. The OIG case represents a similar situation to that currently in Ontario, where a single genco controls the marginal capacity type (coal), while the former represents a recommendation from the Market Design Committee to split up ownership of the price-setting generation units (Trebilcock & Hrab, 2005), which are coal in Ontario.

For the OIG case, we assume that the integrated genco behaves reasonably competitively, and ensure this by giving OLG a lower anticipation coefficient ae^oiG1* in the electricity market than OLG had in Chapters Three and Four, to prevent OIG, who controls more than % of the generation assets in the market for that case, from abusing market power and withholding generation to increase electricity prices.

137 138

We will begin by recalling from Section 4.1.2 the definitions of basic and advanced cross- market WOM knowledge. A basic genco assumes that if it purchases a WOM contract, the other gencos in the electricity market will not adjust their outputs in response, which corresponds to an open-loop level of knowledge. An advanced knowledge genco, by contract, recognizes that if there are other gencos with conventional generation on the margin, then they will adjust their outputs in response to the buying genco's output re- optimization, and this will affect their perceived marginal revenue associated with buying WOM contracts.

Given these definitions, we can make two propositions, which will set up the numerical results to follow.

Proposition 5.1: For a WOM contract buyer i with no conventional generation on the margin in some block *, there is no difference between basic and advanced cross-market WOM knowledge from that genco's perspective in that block.

Proof. If a conventional genco / with no marginal technology t purchases a wind output contract, all of it's conventional generation types are either not utilized or are fully-utilized,

and therefore either the non-negativity constraint VCsit* > 0 is tight, or the conventional output

primal constraint VCsit.+MCsil <(\-fout,)PVCsil (4.4) is tight by definition, for all generation types t. This means that if genco i were to purchase a WOM contract for some sufficiently small quantity A of wind output (such that the set of basic variables does not change) from the wind-only genco seller, it would not be able to adjust it's conventional outputs in response as they are primally constrained, and so OQ* would remain unchanged, and the genco would anticipate aoqi*=0. Furthermore, this implies that the other conventional gencos would not change their outputs in response (because the electricity market price remains the same), and so ax*=0. Therefore, the basic knowledge assumption that other gencos will not react is in fact correct when genco i has no conventional generation on the margin, and if the genco is advanced, then it knows this with confidence. In both cases, aoq*=ax*::=0, and so the genco behaves identically, which completes the proof.

Proposition 5.2: For a WOM contract buyer i with conventional generation on the margin in some block *, there is a difference in behaviour between basic and advanced knowledge in 138 139 the WOM in that block, unless there are no other competitors in the electricity market with conventional generation on the margin, in which case level of knowledge makes no difference in that block.

Proof: First, we know from Proposition 4.1 that since the buyer / has conventional generation on the margin, it will reduce it's OQsi* if it purchases a WOM contract from the wind-only genco seller, and as a result the electricity market price will increase (by Proposition 3.1). If there are no competitors with conventional generation on the margin, then following the logic of Proposition 5.1, there will be no other output response by the other gencos, and so the perception of the genco as to the net electricity market effect of the trade on price & output is the same for a basic genco (ie. assuming no response) as it is for an advanced genco (ie. knowing that there is no response), and the genco will behave identically given either level of knowledge in the WOM.

However, if there is another genco j with conventional generation on the margin, then since

EPS* increases as genco / reduces it's output (by Proposition 4.1), genco j will increase it's output in response (by Proposition 3.2), which will decrease the equilibrium electricity market price to partly counteract the effect of genco / reducing output. If genco /' is advanced, then it recognizes this reaction by definition, and so it would perceive a smaller net increase in EPS* than if it were basic (and therefore does not anticipate the output increase of genco j, which lowers EPS*). Since anticipation coefficient axsi* is defined as genco /'s anticipated change in the electricity price as a result of a WOM contract purchase, we have shown that axSi* (basic) > axsi*(advanced), and since this term appears in the dual equation with respect to XbSi*

(4.31), the level of cross-market WOM knowledge will affect the optimal Xbsi* decision of genco i, which completes the proof.

The implication of Proposition 5.2 is that from the buying genco's perspective, the only blocks in which the level of cross-market WOM knowledge makes a difference are the blocks in which there are more than one genco with conventional generation on the margin. In the OLG+OSG case used in Chapter Four, this was only the case in the low scenario, where OLG and OSG were both operating coal, and hence assuming advanced knowledge in the WOM would yield different perceptions for the contract buyers than assuming basic

139 140 knowledge. Note also that in the OIG cases which we will present next, since OIG is the only genco to own coal capacity, which is the marginal technology in the system, then in the Ontario-like WCAPCOMP 2.WOM model, then by the preceding discussion we do not expect a significant difference in results if we assume basic versus advanced knowledge in the OIG case.

We will use these propositions, along with some of the derived results from Chapters Three and Four, to interpret the numerical results which we now present.

5.2 Numerical Results for Sensitivity to WOM Knowledge & Fossil Ownership

We now present some numerical results to illustrate how WOM knowledge and the number of fossil gencos affect the amount of wind constructed, WOM contract decisions, and coal displacement due to wind generation. A complete list of the parameters used in these examples can be found in Appendix H.

Wind Construction & Allocation - OIG vs OLG+OSG, basic vs advanced WOM knowledge

(3. BOWE • OSG n u a OIG •ca i 3

basic WOM advanced WOM basic WOM advanced WOM

OLG+OSG case OIG case WOM Knowledge/Case

Figure 5-1: Wind capacity construction and allocation, for basic & advanced cross-market WOM knowledge, for OLG+OSG and OIG cases.

140 141

Interestingly, within each fossil ownership case, the amount of WOM knowledge does not materially affect the total wind construction, or allocation, which suggests that the total expected value of wind capacity is the same for the gencos within each fossil ownership paradigm, causing their stage 1 wind investment decisions to remain the same. We do not find this surprising in the OIG case, as we saw from Proposition 5.2 that since OIG has a monopoly on the marginal technology (coal), there is no difference between basic and advanced WOM knowledge.

In the OLG+OSG case, while total expected value of wind capacity is the same, within outcomes there is a difference. In the basic knowledge results, the low nuclear outcome yields higher value for capacity due to more WOM contract sales to OLG, but lower WOM contract sales in the medium and high nuclear outcomes give a lower value for wind capacity in those outcomes, as we will now see in Figure 5-2, which shows the WOM contract purchases by each genco:

WOM Purchases (selected blocks) - OIG vs OLG+OSG, basic vs advanced WOM knowledge

4 n Fl 4 4 4 o * 1 * 4 4 S 1.2 * * t 4 4 4 * 4 > 4 1 1 : 1 \ 4 4 4 a OIG * 4 4 * 4 > 4 DOLG 1 : 4 > 4 * ! * • OSG I 4 * 4 * 1 * 4 ! ^ 4 * 4 • 4 1 1 1 4 4 4 4 * 4 1 4 * 4 ; 4 • 4 * 4 * 4 ^ 4 I: 4 : * 0.2 * 4 4 * 4 4 * 4 \ 4 i o o a -a O TJ to t/i a en S o 9 8 o 1 8 OI G basi c basi c 4 basi c basi c basi c 4 S basi c 6 9 advance d advance d advance d advance d advance d advance d OLG+OS G OLG+OS G OLG+OS G OLG+OS G d O * OLG+OS G OLG+OS G !' d d ^ 3 advance d 8 £ • OI G basi c ? • advance d m OI G basi c „ OLG+OS G B advance d £ • OI G basi c S OI G basi c ? r OI G basi c wt-ba

• a wt-b sm-pe

Case/Block/Nuclear Outcome

Figure 5-2: WOM contract purchases, for basic & advanced cross-market WOM knowledge, and OLG+OSG & OIG fossil ownership cases.

We will begin by discussing the results of the OLG+OSG fossil ownership case, for basic and advanced WOM knowledge. We see that in the basic knowledge case, OLG purchases 141 142

WOM contracts in the same blocks that it does in the advanced knowledge case (ie. whenever it has coal on the margin), but the purchased quantities are higher. This is because OLG anticipates a greater electricity market price increase due to it's WOM purchase when it has basic knowledge and does not take into account the reaction of the other coal genco OSG

(see Proposition 5.2). This means that OXS-OLG-* in the basic knowledge case is greater than the advanced case, and OLG's WTP for WOM contracts is an increasing function of OX^OLG-* (shown in Proposition 4.4). As a result, there are more WOM contracts sold to OLG in the low scenario, which increases the market price (by Proposition 4.1), and therefore the value of wind to OWE in the low scenario (by Proposition 3.3).

However, in the medium nuclear non-summer blocks, and all high nuclear blocks, there are no WOM contracts purchased by either genco when they have basic knowledge, while in the advanced knowledge case, OSG is purchasing wind output contracts. Recall from Proposition 4.5 that if the wind-only genco is basic, it will only accept a WOM contract price greater than the electricity price. OSG does not have a sufficiently high output quantity in the spot market to be able to offer a price greater than the electricity price for the WOM contract, and so

OWE is unwilling to sell. We noted after 4.4 that a genco with a large OQsi* may be able to offer a contract price greater than the electricity price and still benefit from the contract (which turns out to be the case for OLG, but not for OSG).

As a result of no WOM contracts in the majority of medium nuclear blocks, and all high nuclear blocks, the electricity price is lower in the basic knowledge case than the advanced knowledge case, and so the value of wind capacity to OWE is lower in those outcomes with basic knowledge.

Discussing the OIG cases for basic and advanced WOM knowledge, we see, as predicted by Proposition 5.2, that there is little difference in the WOM contract quantities purchased by OIG, because OIG is the only genco with coal on the margin. We do observe some increases in WOM contracts in the advanced versus the basic knowledge case - this is because the wind-only genco is willing to accept a lower WOM contract price when it has advanced knowledge, by anticipating an electricity price increase when OIG reduces it's output (see Proposition 4.6). Overall, these results show that when there is only one genco with

142 143 conventional generation on the margin, the level of cross-market WOM knowledge is not very important, as the buyer OIG is the only genco who can respond with an output adjustment (ONG is always operating nuclear at full capacity). An advanced level of knowledge merely lowers the acceptable WOM contract price to OWE, resulting in more purchases by OIG.

Finally, to conclude the numerical results, we present the annual wind energy added to the system, versus the amount of coal displaced, for all cases in Figure 5-3:

Annual Wind Energy vs Coal Displacement - OIG vs OLG+OSG, basic vs advanced WOM knowledge

a Wind Energy I Coal Energy Displaced -Ratio

Figure 5-3: Annual wind energy versus coal displacement, for basic and advanced WOM knowledge, and OLG+OSG vs OIG fossil ownership.

We observe that the coal displacement ratios in the OIG case for the medium and high nuclear scenarios are very high (> 0.85), compared to the OLG+OSG cases. This is a direct result of greater amounts of WOM contracts being purchased by OIG, than are purchased by OLG or OSG individually, and OIG correspondingly reducing it's coal output (from Proposition 4.1). The reason that OIG purchases more wind output contracts comes from Proposition 4.4, by noting that OIG has a much larger output quantity in the electricity market (because it owns all of the coal), and it's WTP for a WOM contract is an increasing function of OQSOLG*- 143 144

The reason that the coal displacement ratio for the OIG cases is lower than the OLG+OSG case in the low nuclear outcome is because OIG is operating coal at full capacity in the highest load blocks in that outcome, and so adding wind output to the system does not yield a decrease in the coal output of OIG for those particular blocks (because dual variable on coal output YVC-oiG'.coai^O, and so by definition OIG would lose profit by reducing coal in those blocks).

In conclusion, we have looked at the effect of cross-market WOM knowledge levels on the wind construction, and concluded that the level of WOM knowledge is important when there are multiple gencos with coal generation on the margin, and less so when all marginal generation is owned by a single genco. Furthermore, we showed that the single genco fossil generation ownership case yielded the highest amount of WOM contract purchases, and the highest coal displacement ratios for all scenarios where coal was on the margin for the majority of the year.

5.3 Effect of Demand Curve Slope

In the final two sections of the thesis, we will present results which examine the effect of assuming different demand curve slopes to describe consumer demand behaviour in the electricity market. We will show in the numerical results in the subsequent section that shallower demand curve slopes yield lower electricity market prices, but also increased wind capacity investment, which is a non-intuitive result. We limit our numerical examples to the OLG+OSG case for this section, and use the deterministic WCAPCOMP 1.0 model of Chapter Three to provide these examples.

In the thesis, we use an assumed point elasticity to compute linear demand curves. Price elasticity of demand in the electricity market is defined as the rate of change in demand for a given change in price, and for a price P and demand D, is defined as e = ; when measured at a given point (P, D), this is referred to as point elasticity of demand (Varian, 1996). In this thesis, we assume a linear inverse demand curve which gives the market price dP for a given level of demand, ie. EP = po* -pi*D*, with slope — = -px* for some block *. We

144 145 use an assumption of point elasticity at the measurement point (P,D), which was obtained for each block using year 2000 data, and the above formula to compute the linear demand curve slopes used in the electricity market (refer to Appendix A for an example of this calculation). Therefore, our demand curves approximate the assumed elasticity level around the given initial point, but do not have constant elasticity at all points on the demand curve, as this would require a hyperbolic curve (see Varian, 1996).

In the electricity market, a demand curve with a high pi reflects short-term demand response where consumers cannot react to price increases by reducing their demand. Using a low e value to compute/?/ yields a steeper demand curve, with a higher/)/. A lower/?/, however, reflects a more long-term demand response where consumers, having observed high prices over a long period of time, are able to lower their demand, for example by way of investments in more efficient technology. Using a higher e value yields a shallower demand curve which reflects this long-term situation.

In the WCAPCOMP models, gencos had a given level of electricity market knowledge, represented by anticipation coefficient ae,*, which was their estimate of the residual demand curve slope, either assuming that their competitors would not react (basic knowledge) or that they would (advanced knowledge). Decreasing the demand curve slope, by increasing the assumption of e used to derive the demand curve, reduces the ae,* coefficient for the gencos with basic knowledge, and allows the reduction of the same coefficient for gencos with advanced knowledge, as is shown in Figure 5-4 below:

145 146

High p1, low elasticity Low p1, high elasticity

Po Perceived residual

Electricity price demand curve slope

Electricity market output

Figure 5-4: Illustration of effect of using different point elasticity assumptions to compute demand curve

slope, on electricity market anticipation coefficients aer. for myopic, basic, and advanced knowledge levels.

8 In the WCAPCOMP models, a smaller aet* results in a higher perceived marginal revenue associated with selling more wind output to the market, for any genco with conventional generation not on the margin, as can be shown using equation (3.51) from Proposition 3.3:

EP, YVW *=—--ae *OQ * (3.51) t wo. i i

Recall from equation (3.52) in Proposition 3.6 that wind investments are an increasing function of YVWt*\

c YJ fkbYVWikb=WCP + aciOUi (3.52) k,b

Together, this makes it clear that having a lower/*/ in the electricity market (and hence lower ae,i) would yield higher amounts of wind capacity investment in the model. We will illustrate this effect, along with the resulting effect on coal displacement and electricity prices, in the

Recall from the Chapter Three WCAPCOMP 1.0 model that the dual variable YVW-,* is interpreted as the value of an extra unit of wind output in the electricity market to genco / in some block *.

146 147 following section, for numerical examples using the WCAPCOMP 1.0 model, assuming e=0.5 (as we did in Chapter Three) and e=0.8, to compute the linear demand curves.

5.4 Numerical Results for Sensitivity to Demand Curve Slope

We now present a numerical example, illustrating the effect of varying price elasticity of demand, using the WPCAPCOMP 1.0 model. We refer to the cases where we use point elasticities e= 0.5 or e=0.8 to compute the demand curves as the "e=0.5" or "e=0.8" cases, cognizant that the demand elasticity is not equal to this value at all points on the demand curve. A complete list of the parameters used for these examples can be found in Appendix H.

Wind Capacity Construction & Allocation for e=(0.5,0.8)

e=0.5 | e=0.8 Medium Nuclear Case

Figure 5-5: Wind construction & allocation in e=0.5, e=0.8 cases for low, medium and high nuclear scenarios using WCAPCOMP 1.0 model.

As is clearly shown in Figure 5-5, increasing demand elasticity increases the total amount of wind construction, by approximately 4-7%, depending on the nuclear case. One interesting side observation is that the coal-on-margin genco OSG does not purchase as much wind capacity in the e=0.8 results as in the e=0.5 results. This is because OSG's value of wind

1 output (as captured by YVW^OSG *) is constant and equal to occoai, and is not a function of

147 148 ae^osG-* while OSG has coal on the margin (a result shown in Proposition 3.4), in virtually all blocks, in all nuclear cases. This results in OWE purchasing a larger share of the wind output in the e=0.8 case, because it's WTP for wind output goes up due to it's electricity market anticipation coefficient aew* being smaller (since/?/* is smaller), and so it increases it's wind capacity purchase OU, (see equations (3.51) & (3.52) from the previous section), and this decreases the amount purchased by OSG (from Proposition 3.6).

Annual Wind Energy vs Coal Displacement for e=(0.5, 0.8)

KIWind Energy ••Coal Energy Displaced -*- Ratio

Figure 5-6: Annual wind energy versus coal displacement, for e=0.5 & e=0.8 cases.

In the medium and high nuclear cases, decreasing the demand curve slope in the e=0.8 case results in a higher coal displacement ratio. To see why this occurs, note that for the coal genco OSG, who has coal on the margin in all blocks in the medium and high nuclear scenarios, the following identity applies, from Proposition 3.2:

EP* >OQ, (3.49) 'OSG' OSG'" wb. 'coal

The above equation shows that if EP* is decreased, due to the addition of wind output to the system which lowers the electricity market price (from Proposition 3.1), if ae-osG'* is small due to a low demand curve slope (the e=0.8 case) , then OSG must decrease OQOSG* by a

148 149 larger degree to satisfy it's KKT conditions, which require that (3.49) remain tight. This explains why there is a larger reduction in coal for a unit increase in wind output in the medium and high scenarios, in the e=0.8 (lowpi) case.

However, in the low nuclear case, we see that a lower demand curve slope in the e=0.8 case actually yields a lower coal displacement ratio. The reason for this is that in many of the blocks in the low scenario, due to decreased nuclear capacity, coal is operating at full capacity for one or both of the coal gencos (and is therefore not on the margin), and so the addition of wind to the system does not cause these gencos to decrease their coal output. This is because dual variable YVCvcoav* > 0 (ie. primal constraint VCit+MCitt <(l-fout,)PVCit (3.4) is tight) when coal is operating at full capacity, meaning that decreasing VC-coar* will decrease the profits, as per the definition of the shadow price YVC-coai* according to

(Winston, 1993). Since in many blocks YVCVcoaV* > 0 (for both OLG and OSG) in the low nuclear case, adding wind to the system does not decrease wind in those blocks, and so the annual coal/wind displacement ratio is lower in the low nuclear, e=0.8 case.

Electricity Market Prices (selected blocks, e=0.5, 0.8)

^m L ^^^1 ^^^_ o. 30 • I 1 • El EP e-0.5 I I • • EP e=0.8 I• I • I• I •

sm-peak ..•wt-base _ _sm- )eaIk ,| wt-bas•e sm-peak L M H s eason/Block/Nucl Bar Case 1

1wt-base Figure 5-7: Electricity market prices for selected blocks, for e=0.5 & e=0.8.

149 150

Finally, we can see from Figure 5-7 that electricity market prices in the e=0.8 case are lower than the e=0.5 case across the nuclear scenarios (4-11% lower). This is interesting, as despite lower market prices, we still observed higher investment in wind capacity in the e-O.S case, from Figure 5-5. This counter-intuitive result comes from the fact that in competitive market models with strategic gencos, gencos use marginal revenue (as shown in equation (3.51)), which takes into consideration the slope of the demand curve, and possibly other genco's reactions, rather than simply the electricity market price, to determine the value of additional output, and hence the value of wind capacity.

In summary, we have illustrated that a shallower demand curve slope (obtained by using a higher point elasticity of demand to compute demand curves) results in higher wind construction in all nuclear scenarios, despite also yielding lower electricity market prices. We also showed that in the medium and high nuclear scenarios, the low pi case had a much higher coal/wind displacement ratio, although this was not the case in the low scenario, because coal was operating at full capacity in many of the blocks in that scenario, for the e=0.8 (shallow demand curve) case.

5.5 Note on Stability of the WCAPCOMP Models

For any equilibrium model, it is important that the model results be stable, in the sense that a small change in the magnitude of some model parameter does not produce a large change in the equilibrium solution.

Throughout the examples presented in this chapter, we have varied the demand curve parameters by testing different levels of e used to compute the linear demand curves, for the WCAPCOMP 1.0 model. The two test cases of e=0.5 & e=0.8 yielded y-intercepts po which differ by 25%, along with slopes pi which differed by 37.5% between the two cases. Despite these changes, the total output to the electricity market only varied by 4-11% within the blocks.

In addition, we also varied the nuclear generation capacity in the model by 5 GW between the low and high nuclear cases, in Chapters Three and Four, which is approximately 20% of

150 151 the utilized generation capacity in the model. However, the net change in the electricity market price was less than 10% in all blocks.

These results highlight the stability of the model, in the sense that large changes in the market parameters result in relatively small changes in the market equilibrium.

5.6 Conclusion

In Chapter Five, we explored the sensitivity of our WCAPCOMP models to some market structure parameters, namely the number of gencos with fossil generation, the effect of basic versus advanced knowledge in the wind output contract market, and the slope of the demand curves used to describe customers.

We have illustrated that a paradigm in which there is only a single coal genco yields the most WOM contracts, with the coal monopoly OIG having a high WTP for contracts in all nuclear outcomes, due to it's large total output quantity in the electricity market. OIG benefits from purchasing financial rights to wind output via a contract-for-difference, and then reducing it's marginal coal output to increase the electricity price which it receives over it's large offer quantity to the market. This correspondingly yields higher coal displacement due to wind, which has benefits to society. We also observed that OIG does not require advanced knowledge in the WOM to make such purchases, because there are no other competitors in the market with generation on the margin which may react to it's WOM contract purchase and subsequent output reduction.

In the OLG+OSG cases, when all gencos were assumed to have basic knowledge, we noted larger WOM contract purchases by OLG in the low nuclear outcome (due to OLG not anticipating OSG's output reaction to it's WOM contract purchases), but little to no WOM contract activity in the medium and high nuclear outcomes. OSG, which was a purchaser of contracts in the advanced knowledge case, does not purchase in the medium and high scenarios even though it has coal on the margin, because it is unable to offer OWE a price higher than the electricity price. OWE, being basic, does not perceive any electricity market effect from WOM contracts, and so is only willing to accept a contract price higher than what

151 152 it is currently receiving from the spot market, and OSG is unable to offer such a price, and still make a profit.

Finally, we found that using a smaller demand curve slope (by using a higher point elasticity e to compute the demand curves) results in larger amounts of wind construction by 4-7%, because the gencos perceive a higher marginal revenue from selling extra wind output to the electricity market with a shallow demand curve slope than in a more inelastic demand case with a higher pu This result persists even as the high pi case yields higher equilibrium electricity market prices, which is non-intuitive.

5.7 Recommendations for Future Work

In closing, we will now suggest some areas of potential future research which would expand upon the contributions made in this thesis.

One aspect of the Ontario electricity market that was not represented in the WCAPCOMP models was the presence of imports and exports of power to adjacent markets, such as Quebec or the Northeastern United States. In peak periods of demand, the market operator may need to import power from gencos in these markets to meet demand, and gencos may also offer generation to these markets as exports, or purchase rights to power from these markets as imports. This aspect may affect the market prices in the Ontario market, and may yield different strategies for the gencos. For instance, one plausible strategy that could be examined would be a genco with no generation on the margin purchasing WOM contracts for wind output, and simultaneously exporting conventional generation output to another market, which would potentially increase the electricity market price in Ontario, and therefore affect the value of the WOM contract. A potential expansion of the WCAPCOMP model would be the inclusion of some representation of these external markets, potentially with limits on the amount of generation that can be purchased or sold to an adjacent market, due to limited transmission capacity between markets, or regulatory limits.

Imbalance penalties for wind generation are another detail that was not represented in the WCAPCOMP. In the Ontario market, any genco that cannot fulfill it's offer quantity, which is typically made a day ahead of dispatch, is assessed a penalty by the market operator, which

152 153 would affect the profitability of an intermittent generation technology such as wind. In theory, gencos with wind generation on offer to the market could purchase rights to dispatchable capacity in a reserve market, such that they would be able to avoid imbalance charges. For gencos with hydro or oil/gas generation, they may be able to utilize these other assets to ensure that they can fulfill their offer quantity and avoid such charges, if it is economical to do so. This is another potential area for future work.

Finally, as EPEC models are receiving increasing attention in the literature (see (Anjos et. al., 2007), (Yao et. al. 2007), (Leyffer & Munson, 2005)), another area of future research would be to construct an EPEC model of the wind capacity and electricity markets of WCAPCOMP 1.0, such that wind capacity decisions are made by each genco subject to the KKT conditions of the electricity spot market, reflecting perfect knowledge of the latter market, similar to the closed-loop model of (Murphy & Smeers, 2005), rather than using imperfect knowledge via an anticipation approach as we have in this thesis. In an EPEC model, the gencos would potentially value wind capacity differently due to the influence of investments on the electricity market equilibrium. However, converting the stochastic WCAPCOMP 2.WOM model to an EPEC structure would be substantially more difficult, as there are three markets, rather than two as most current EPECs in the literature consider. Such an equilibrium model would require that the WOM decisions of each genco are subject to the KKT conditions of the electricity market, and furthermore that the stage 1 capacity decisions of each genco must solve the EPEC representing the subsequent two markets. To the best of our knowledge, there are no EPEC models that consider three consecutive decision stages, and the successful formulation and solution of such a model would represent a substantial contribution to the mathematical programming literature.

153 6.0 References

M. Anjos et. al, "Finding Nash Equilibria in Electricity Markets: an AC Network Approach", presentation to CORS 2007 conference in London, ON, May 14, 2007.

L. Asl, J. S. Rogers, "ELFORSPOT: An Equilibrium Model of the Combined Forward Market and Spot Market for Electricity, Using Complementarity Programming", MASc thesis, University of Toronto, 2004.

D. Bienstock, J. Shapiro, "Optimizing Resource Acquisition Decisions by Stochastic Programming", Management Science, v. 34, no. 2, Feb 1988, pp. 215-229.

C. Day, B. Hobbs, J. Pang, "Oligopolistic competition in power networks: a conjectured supply function approach", IEEE Transactions on Power Systems, v. 17, no. 3, 2002, pp. 597-607.

Canadian Wind Energy Association (CanWEA), "2006 A Record Breaking Year for the Global Wind Energy Industry With Canada Now Ranking 12th In The World For Total Installed Wind Energy Capacity", press release Feb. 8, 2007, URL: http://www.canwea.ca/images/uploads/File/PressRelease- Februarv82007 FINAL rh 2.pdf. last accessed June 1, 2007.

S. Fleten, S. W. Wallace, "Stochastic programming in energy", published in "Stochastic Programming" (A. Ruszcynski & A. Shapiro (eds)), v. 10 in series Handbooks in Operations Research and Management Science, 2003, pp. 637-677.

D. Fudenberg, J. Tirole, "Game Theory", 7th printing, MIT Press, 2000, pp. 130-131.

J. Gabszewicz, S. Poddar, "Demand fluctuations and capacity utilization under duopoly", Economic Theory, v. 10, 1997, pp. 131-146.

Global Wind Energy Council (GWEC), "Global Wind 2006 Report" URL : http://www.gwec.net/index.php?id=78. last accessed June 1, 2007.

154 155

R. Green, D. Newbery, "Competition in the British Electricity Spot Market", The Journal of Political Economy, v. 100, no. 5, October 1992, pp. 929-953.

B. Hobbs, "Linear Complementarity Models of Nash-Cournot Competition in Bilateral and POOLCO Power Markets", IEEE Transactions on Power Systems, v. 16, no. 2, May 2001, pp. 194-202.

Ivanic, Preckel, Yu, "Market power and welfare effects in DC power flow electricity models with thermal line losses", Decision Support Systems, v. 40, issue 3-4, October 2005, pp. 529-542.

P. Klemperer, M. Meyer, "Supply Function Equilibria in Oligopoly under Uncertainty", Econometrica, v. 57, no. 6, November 1989, pp. 1243-1277.

J. Leggett, "A guide to the Kyoto protocol: a treaty with potentially vital strategic implications for the renewables industry", Renewable and Sustainable Energy Reviews, v. 2, 1998, pp.345-351.

S. Leyffer, T. Munson, "Solving Multi-Leader-Follower Games", draft, April 26, 2005.

P. Meibom, R. Barth, H. Brand, C. Weber, "Wind power integration studies using a multi stage stochastic electricity system model", work done within the WILMAR (Wind Power Integration in Liberalised Electricy Markets) project, URL: http://www.risoe.dk/rispubl/art/2007 222 paper.pdf, last accessed Aug 14, 2008.

F. Murphy, Y. Smeers, "Generation Capacity Expansion in Imperfectly Competitive Restructured Electricity Markets", Operations Research, v. 53, no. 4, July-Aug 2005, pp. 646-661.

Ontario Ministry of Energy, "McGuinty Government Gives Green Light to Renewable Energy Projects", press release Nov. 24, 2004. URL: http://www.energy.gov.on.ca/index.cfm?fuseaction=english.news&body=yes&news_ id=82 , last accessed June 1, 2007.

155 156

Pembina Institute for Appropriate Development & Canadian Environmental Law Association, "Power for the Future: Towards a Sustainable Electricity System for Ontario", published May 2004, URL: http://pubs.pembina.org/reports/energvreport- fullreport a.pdf, last accessed August 9, 2008.

S. Pokharel, K. Ponnambalam, "Investment Planning For Electricity Generation Expansion", International Journal of Energy Research, v. 21, 1997, pp. 185-194.

A. Rudkevich, A. Duckworth, R. Rosen, "Modeling electricity pricing in a deregulated generation industry: the potential for oligopoly pricing in a Poolco", Energy Journal, v.l9,no. 3, 1998, pp. 19-48.

A. Sahin, "Progress and recent trends in wind energy", Energy and Combustion Science, v. 30, 2004, pp. 501-543.

Y. Song et. al., "Analysis of Strategic Interactions among Generation Companies Using Conjectured Supply Function Equilibrium Model", Power Engineering Society General Meeting, 2003, IEEE, v. 2. Published July 13-17 2003.

M. Trebilcock, R. Hrab, "Electricity Restructuring in Ontario", Energy Journal, v. 26, no. 1, 2005, pp. 123-146.

H. Varian, "Intermediate Mircoeconomics: a modern approach (4' ed)", W.W. Norton & Company, NY, 1996, (p. 469).

M. Ventosa et. al., "Electricity market modeling trends", Energy Policy, v. 33, no. 7, 2005, pp. 897-913.

M. Ventosa, R. Denis, C. Redondo, "Expansion Planning in Electricity Markets: Two different approaches", proceedings of the 14th Power Systems Computation Conference (PSCC) Conference, Seville, June 2002.

W. Winston, "Operations Research: Applications and algorithms, 3rd edition", Duxbury Press, 1993, pp. 236-237, 288-289.

156 157

J. Yao, I. Adler, S. Oren, "Cournot Equilibrium in Two-Settlement Electricity Markets with System Contingencies", InternationalJournal of Critical Infrastructures, v. 3, no. 1- 2, 2007, pp. 142-160.

157 158

7.0 Appendices

7.1 Appendix A: Constructing Demand Curves

To construct demand curves for the WCAPCOMP 1.0 and 2.WOM models, we first divide a year of demand into 4 seasons (winter Dec-Feb, spring Mar-May, summer Jun-Aug, fall Sept-Nov), with 3 blocks per season (bi=peak, b2=offpeak, and b3=base), for a total of 12 blocks. The blocks represent the upper, median and lower 1/3 of load levels during a given season. Fitting a linear demand curve for each block requires a demand & price point, and an assumption of price elasticity of demand e. We use year 2000 load levels from the Ontario system, because these prices were regulated and therefore stable, and consumers had time to adjust their consumption accordingly, meaning the system was at a relatively stable medium- term equilibrium from a demand perspective. Regulated prices during this period are assumed to be rP(k,b) = $48/MWh for all blocks. We use the peak load from the year 2000 (23.5 GW), but load shapes from Sept 2005-Aug 2006 (referred to as "2006" data), as year 2000 load shape data for each season is not available. To determine the initial load levels in each block, we determine the average % of peak load (using 2006 data), and scale it down to year 2000 load levels by multiplying the average % of peak load by the year 2000 peak load level. The algorithm for this procedure is as follows: 1. Compute year 2000 peak demand for each season, using following ratio: annual peak 2006 _ annual peak 2000 season peak 2006 season peak 2000 where annual peak 2000 = 23.5 GW, and we define model "season peak 2000" as rQ(k) for season k. 2. Compute height of each load block in each season L(k,b), by averaging height of LDC (using 2006 data) in a given block of hours as a fraction of peak demand in that season: L(k,b) = sum(h, AvgHrlyDemand/PeakDemand) / block duration 3. Compute initial demand point rD(k,b) point for each block as: rD(k,b) = rQ(k)*L(k,b) 4. Now we have rD(k,b), rP(k,b), and assume a point elasticity of demand e, and demand curve parameters p0(k,b) and pl(k,b) can be computed as: pl(k,b) = P(k,b) / (rD(k,b)*e) where rP(k,b) = $48/MWh, e = 0.5 p0(k,b) = P(k,b) +pl(k,b)*rD(k,b)

158 159

An example of the above procedure for the fall season demand curve calculations follows: Fall Season (Sept - Nov 2005) - duration 2184 hrs

LDC - Fall 2005 (Sept - Nov) - 2184 hrs

30000 T , , 1 , 1 1

10000

5000

0.| 1 1 1 1 1 1 1 365 729 1093 1457 1821 Hours

Step 1: Compute seasonal peak load for model scaled to year 2000 levels: Actual season peak demand in 2006 = 23.9 GW scaled season peak demand iQ(fl) = (23.5)*23.9 / 27 = 20.8 GW

Step 2: Compute height of each load block as % of season peak: L(fl,b) = [t>! 0.821, b2 0.717, b3 0.597] (%)

Step 3: Compute scaled height of each load block: rD(fl,b) = rQ(fl)*L(fl,b) = [bi 17.1, b2 14.9, b3 12.4] GW

Step 4: Compute po, pi: pi(fl,b) = P(fl,b) / (rD(fl,b)*e) = [bi 5.61, b2 6.44, b3 7.74] po(fl,b) = P(fl,b)+pl(fl,b)*rD(fl,b) = [b, 144, b2 144, b3 144]

Note: these calculations are performed for each block and season in GAMS prior to solving the model, by using the data inputs of rQ(k), L(k,b), rP(k,b), and e.

159 7.2 Appendix B: GAMS Implementation of WCAPCOMP 1.0

OPTION ITERLIM = 200000;

* SET NOTATION SET i conventional gencos /OLG, ONG, OSG, OCE/; ALIAS (i, ii); SET t conventional technologies /n, c, o, cc/; SET k seasons /fl, wt, sp, sm/; ALIAS (k, kk); SET b blocks /bl, b2, b3/; ALIAS (b, bb);

* DATA * DEMAND PARAMETERS PARAMETER rQ(k) peak demand in EM {GW} /fl 20.8, wt 20.8, sp 21.7, sm TABLE wb(k,b) duration of block in EM {khrs} bl b2 b3 fl 0.728 0.728 0.728 wt 0.720 0.720 0.720 sp 0.736 0.736 0.736 sm 0.736 0.736 0.736

SCALAR e price elasticity of demand in EM /0.5/; TABLE L(k,b) height of demand blocks as % of peak demand {%} bl b2 b3 fl 0.821 0.717 0.597 wt 0.879 0.785 0.668 sp 0.767 0.673 0.562 sm 0.789 0.668 0.542

PARAMETER rP initial marginal cost in EM {$ per MWh} /48/; PARAMETER rD(k,b) initial demand in EM (GW); rD(k,b) = rQ(k) *L(k,b) ; PARAMETER pl(k,b) demand curve slope in EM {$ per MWh*MW} pi (k,b) = rP/(rD(k,b)*e) ; PARAMETER p0(k,b) demand curve intercept in EM {$ per MWh}; p0(k,b) = rP +pl(k,b)*rD(k,b) ;

* WIND PARAMETERS TABLE cf(k,b) wind capacity factor {%) bl b2 b3 fl 0.25 0.3 0.35 wt 0.35 0.4 0.45 sp 0.25 0.3 0.35 sm 0.15 0.2 0.25

SCALAR yO supply curve intercept in WCM {$M per GW) /307; SCALAR yl supply curve slope in WCM {$M per GW*GW) /5/;

* GENCO PARAMETERS PARAMETER oc(t) operating cost of technology t {$ per MWh} /n 4, c 25, o 50, cc 40/; SCALAR oh operating cost of hydro {$ per MWh}; oh = 2.5; SCALAR pp peak payments for conventional generation ($M per GW}; pp = 60; PARAMETER fc(t) fixed operating cost of technology t {$M per GW} /n 100, c 25, o 15, cc 125/; SCALAR fh fixed operating cost of hydro {$M per GW}; fh = 5; TABLE pcap(i,t) conv generation capacity (GW} n c o cc OLG 6.5 2.3 0.7 0 ONG 5.0 0 0 0 OSG 0 4.7 1.3 0 OCE 0 0 0 5

PARAMETER mrc(t) annual maintenance requirement {%}

160 /n 0.15, c 0.15, o 0.15, cc 0.15/; PARAMETER fout(t) forced outage rate {%} /n 0.1, c 0.1, o 0.1, cc 0.1/; TABLE hcap(i,k) seasonal hrly hydro capacity {GW - includes ror) fl wt sp sm OLG 6 6 7.2 4.8

TABLE hrr(i,k) seasonal hrly hydro run of river (GW) fl wt sp sm OLG 2.4 2.4 2.88 1.92 ; TABLE heng(i,k) seasonal total hydro energy {TWh - includes ror} fl wt sp sm OLG 8.75 8.75 10.5 7

^ANTICIPATION COEFFICIENTS TABLE ae(i,k,b) ant of demand curve slope for conv gencos {% of pi} bl b2 b3 OLG.(fl,wt,sp,sm) 0.333 0.333 0.333 (ONG,0SG,OCE).(fl,wt,sp,sm) 111

TABLE aew(k,b) ant of demand curve slope for wind genco {% of pi} bl b2 b3 (fl,wt,sp,sm) 111

PARAMETER ac(i) ant by conv genco of WCM supply curve slope {% of yl} /OLG 1, ONG 1, OSG 1, OCE 1/

SCALAR acw ant by wind genco of WCM supply curve slope {% of yl) acw = 0.5;

* MODEL SPECIFICATION POSITIVE VARIABLES *Link YBQ(i,k,b) dual var on BQ <= OQ constraint ($M per GW} YBQw(k,b) wind genco analogue to YBQ YBU(i) dual var on BU <= OU constraint ($M per GW} YBUw wind genco analogue to YBU *EMO(s) BQ(i,k,b) spot market bid quantity from genco (GW output) BQw(k,b) wind genco analogue to BQ D(k,b) total spot market demand {GW output} EP(k,b) electricity spot market price {$ per MWh} *WMO BU(i) wind capacity market accepted quantity {GW} BUw wind genco analogue to BO {GW} Ut total wind constructed by FMO {GW} WCP wind capacity price {$M per GW} *CONVENTIONAL GENCO primals 0Q(i,k,b) electricity spot market generation offered to EMO {GW} OEP(i,k,b) electricity spot market offer price to EMO {$ per MWh} OU(i) wind capacity market offer quantity to WMO {GW} OWCP(i) wind capacity market offer price to WMO {$M per GW} VC(i,t,k,b) conventional spot market generation {GW} VW(i,k,b) wind spot market generation {GW} VH(i,k,b) hydro spot market generation {GW) PVC(i,t) annual peak conventional generation made available {GW} PVH(i) annual peak hydro generation made available {GW} MC(i,t,k) generation capacity taken down for seasonal maintenance {GW} *CONVENTOINAL GENCO duals YVC(i,t,k,b) dual var on hrly conv output constraint ($M per GW) YVW(i,k,b) dual var on hrly wind output constraint ($M per GW) YVH(i,k,b) dual var on peak hydro constraint ($M per GW) YHRR(i,k,b) dual var on ror hydro constraint ($M per GW) YHENG(i,k) dual var on seasonal hydro energy constraint ($M per TWh) YOQ(i,k,b) dual var on hryl total output constraint ($M per GW) YPVC(i,t) dual var on PVC <= pcap constraint ($M per GW) YPVH(i) dual var on PVH <= heap(spring) constraint ($M per GW) YMC(i,t) dual var on conv maintenance constraint ($M per GW) *WIND GENCO primals 161 162

OQw(k,b) wind genco analogue to OQ OEPw(k,b) wind genco analogue to OEP OUw wind genco analogue to OU OWCPw wind genco analogue to OWCP VWw(k,b) wind genco analogue to VW *WIND GENCO duals YVWw(k,b) wind genco analogue to YVW YOQw(k,b) wind genco analogue to YOQ YWOM(k,b) dual var on Xs <= VWw constraint {$ per GW)

FREE VARIABLES YD dual var on demand curve equality {units} YOEP(i,k,b) dual var on electricity spot offer price equality {units} YOEPw(k,b) wind genco analogue to YOEP YOWCP(i) dual var on wind capacity market offer price equality {units} YOWCPw wind genco analogue to YOWCP YUt dual var on wind capacity supply curve equality {units}

EQUATIONS *LINKS linkEM(i,k,b) linkEMw(k,b) linkWCM(i) linkWCMw *EMO EMOcurve(k,b) EMOdemsat(k,b) delBQ(i,k,b) delBQw(k,b) delD(k,b) *WMO WMOcurve WMOdemsat delBU(i) delBUw delUt *CONV GENCOS offerQ(i,k,b) peakconv (i,t) conv(i, t, k,b) maintconv(i, t) wind (i, k,b) peakhyd(i) hydout(i,k,b) hydrr (i, k,b) hydeng(i,k) offerEP(i,k,b) offerWCP(i) delOQ(i,k,b) delVC(i,t,k/b) delVW(i,k,b) delVH(i,k,b) delPVC(i,t) delPVH(i) delMC(i,t,k) delOU(i) delOWCP(i) delOEP(i,k,b) •WIND GENCO offerQw(k,b) windw(k,b) windWOM(k,b) offerEPw(k,b) offerWCPw delOQw(k,b) delVWw{k,b) delOUw delOWCPw delOEPw(k,b) 162 *MODEL DEFINITION *LINKS linkEM(i,k,b).. -BQ(i,k,b) =G= -OQ(i,k,b); linkEMw(k,b).. -BQw(k,b) =G= -OQw(k,b); linkWCM(i).. -BO(i) =G= -OU(i) ; linkWCMw.. -BUw =G= -OUw; *EMO EMOcurve(k,b).. EP(k,b)/wb(k,b) =E=pO(k,b) pl(k,b)*D(k, EMOdemsat(k,b).. sum(i, BQ(i,k,b)) +BQw(k,b) D(k,b) =G= 0 delBQ(i,k,b) . . YBQ(i,k,b) -EP(k,b) =G -wb(k,b)*OEP(i,k, delBQw(k,b).. YBQw(k,b) -EP(k,b) =G= wb(k,b) *OEPw(k,b) delD(k,b) . . pl(k,b)*YD(k,b) +EP(k,b) =G= 0; *WMO WMOcurve. . WCP =E= yO +yl*Ut; WMOdemsat. -sum(i, BU(i)) -BUw +Ut =G= 0; delBO(i).. YBU(i) +WCP =G= OWCP(i); delBUw.. YBOw +WCP =G= OWCPw; delUt.. -WCP -yl*YUt =G= -yO -yl*Ut; CONVENTIONAL GENCOS offerQ(i,k,b) . . sum(t, VC(i,t,k,b) ) +VW(i,k,b) +VH(i,k,b) =G= OQ(i,k,b); peakconv(i,t) . . -PVC(i,t) =G= -pcap(i,t); conv(i,t,k,b).. -VC(i,t,k,b) -MC(i,t,k) =G= - (1-fout (t) ) *PVC (i, t) maintconvfi,t) . . sum(k, sum(b, wb (k,b) *MC (i, t, k) ) ) =G= 8.76*mrc(t*PVC(i,t) ) ; wind(i, k,b) -VW(i,k,b) =G= -cf (k,b)*OU(i); peakhyd(i).. -PVH(i) =G= -hcap(i,'sp'); hydout(i,k,b).. -VH(i,k,b) =G= -hcap(i,k); hydrr(i,k,b).. VH(i,k,b) =G= hrr (i,k) ; hydeng (i, k) . . -sum(b, wb(k,b)*VH(i,k,b) ) =G= -heng(i,k); offerEP(i,k,b) . . OEP(i,k,b) =E= EP(k,b)/wb(k,b) -ae(i,k,b)*pl(k,b)*{OQ(i,k,b) -BQ(i,k,b) }; offerWCP(i) .. OWCP(i) =E= WCP +ac(i)*yl*{OU (i) -BU(i)}; delOQ(i,k,b) -YBQ(i,k,b) +YOQ(i,k,b) +ae (i, k, b) *pl (k, b) *YOEP (i, k, b) =G= EP(k,b) -wb(k,b)*[ae(i,k,b)*pl(k,b)*{2*OQ(i,k,b) -BQ( i,k,b) }]; delVC(i,t,,k, b .. -YOQ(i,k,b) +YVC(i,t,k,b) =G= -wb(k,b)*oc(t) ; delVW(i,k,,b ) . -YOQ(i,k,b) +YVW(i,k,b) =G= 0; delVH(i,kWb, ) . -YOQ(i,k,b) +YVH(i,k,b) -YHRR(i,k,b) +wb (k,b) *YHENG (i, k) =G= wb(k,b)*oh; delPVC(i,t) . . YPVC(i,t) -sum(k, sum(b, (1-fout (t) ) *YVC(i,t,k,b) ) ) + 8.7 6*mrc(t) *YMC(i,t) =G= (pp -fc (t) ) ; delPVH(i).. YPVH(i) =G= (pp -fh) ; delMC(i,t ,k). sum(b, YVC(i,t,k,b) ) -sum(b, wb (k,b) ) *YMC (i, t) =G=0; delOU(i). -sum(k, sum(b, cf (k,b) *YVW (i, k,b) ) ) ~ac (i)*YOWCP(i ) -YBU(i) =G= -WCP -ac (i ) *yl*{2*OU(i) -BU(i)}; delOWCP(i I . . YOWCP(i) =G= 0; delOEPd, k,b) YOEP(i,k,b) =G= 0;

*WIND GENCO offerQw(k,b) . . VWw(k,b) =G=OQw(k,b); windw(k,b).. -VWw(k,b) =G= -cf(k,b)*OUw; offerEPw(k,b) . . OEPw(k,b) =E= EP (k,b)/wb (k,b) -aew(k,b)*pl(k,b)*{OQw(k,b) -BQw(k,b)}; offerWCPw.. OWCPw =E= WCP +acw*yl*(OUw -BUw}; delOQw(k,b).. -YBQw(k,b) +YOQw(k,b) +aew(k,b)*pl(k,b)*YOEPw(k,b) =G= EP(k,b) -wb(k,b)*[aew(k,b)*pl(k,b)'t{2*OQw(k,b) -BQw(k,b)}]; delVWw(k,b).. -YOQw(k,b) +YVWw(k,b) =G= 0; delOUw.. -sum(k, sum(b, cf(k,b)*YVWw(k,b))) -acw*YOWCPw -YBUw =G= -WCP -acw*yl*{2*OUw -BUw}; delOWCPw.. YOWCPw =G= 0; delOEPw(k,b).. YOEPw(k,b) =G=0;

OU.upCOCE' ) = 0;

MODEL WCAPCOMP1 / *link constraints linkEM.YBQ, linkEMw.YBQw, linkWCM.YBU, linkWCMw.YBUw

163 *EMO primal and dual EMOcurve.YD, EMOdemsat.EP, delBQ.BQ, delBQw.BQw, delD.D, *WMO primal and dual WMOcurve.YUt, WMOdemsat.WCP, delBU.BU, delBUw.BUw, delUt.Ut, *conventional genco primal offerQ.YOQ, peakconv.YPVC, conv.YVC, maintconv.YMC, wind.YVW, peakhyd.YPVH, hydout.YVH, hydrr.YHRR, hydeng.YHENG, offerEP.YOEP, offerWCP.YOWCP, *conventional genco dual delOQ.OQ, delVC.VC, delVW.VW, delVH.VH, delPVC.PVC, delPVH.PVH, delMC.MC, delOU.OU, delOWCP.OWCP, delOEP.OEP, *wind genco primal offerQw.YOQw, windw.YVWw, offerEPw.YOEPw, offerWCPw.YOWCPw, *wind genco dual delOQw.OQw, delVWw.VWw, delOUw.OOw, delOWCPw.OWCPw, delOEPw.OEPw /;

SOLVE WCAPCOMP1 USING MCP;

164 165

7.3 Appendix C: Calculation of Seasonal Hydro Parameters

In a typical year, the total annual hydro energy sold in the province of Ontario is approximately 35 TWh, and is operated by a single genco, represented in our model by OLG. This hydro comes from two types of hydro plants: run-of-river plants, and peaking hydro reservoirs. Run-of-river hydro generates a relatively constant output in a given season, while peaking hydro energy can be dispatched whenever desirable. In Ontario, annually, the highest total hydro dispatch (run of river plus peaking hydro) is approximately 6 GW, and the run-of-river hydro capacity is 2.4 GW. In the WCAPCOMP models, we have only a single output variable for hydro, VHjkb which represents the average hourly hydro output operated in a given block, and includes both run- of-river and peaking hydro. To obtain the required hydro parameters of maximum hourly hydro capacity, run-or-river output (representing a lower bound), and hydro energy in each season, we do the following: 1. Divide the annual hydro energy into 4 seasons, and then assume that in the spring, the hydro energy available is 20% higher than the average, and in the summer is 20% lower, which incorporates seasonal variation in hydro availability. This results in hydro energy in available in each season, heng-oiG'k as: heng(OLG,k) = [fall 8.75, winter 8.75, spring 10.5, summer 7] TWh 2. The base Ontario annual hourly hydro capacity 6GW is likewise adjusted in the spring and summer by +/- 20%, to yield seasonal values of: hcap(OLG,k) = [fall 6, winter 6, spring 7.2, summer 4.8] GW 3. The base Ontario annual hourly run-of-river hydro lower limit of 2.4 GW is adjusted by +/- 20% in the spring/summer to yield seasonal values of: hror(OLG,k) = [fall 2.4, winter 2.4, spring 2.88, summer 1.92] GW

165 7.4 Appendix D: Parameters Used for Chapter Three Numerical Examples

DEMAND PARAMETERS (used to compute demand curve parameters pokb and pikb) Block duration wbkb (khrs): Bl B2 B3 Fall 0.728 0.728 0.728 Winter 0.72 0.72 0.72 Spring 0.736 0.736 0.736 Summer 0.736 0.736 0.736

Price elasticity of demand e = 0.5 Block height Lkb (% of peak): Bl B2 B3 Fall 0.821 0.717 0.597 Winter 0.879 0.785 0.668 Spring 0.767 0.673 0.562 Summer 0.789 0.668 0.542

Initial price rPkb = $48/MWh Seasonal peak loads rQkb = [fall 20.8, winter 20.8, spring 21.7, summer 23.5] {GW} WIND PARAMETERS: Average wind capacity factor cfkb {% of installed wind capacity}: Bl B2 B3 Fall 0.25 0.3 0.35 Winter 0.35 0.4 0.45 Spring 0.25 0.3 0.35 Summer 0.15 0.2 0.25

Wind capacity supply curve y-intercept yo = 30 {$M/ GW} Wind capacity supply curve slope yi = 5 {$M/ GW2}

166 167

GENCO PARAMETERS: Generation non-nuclear capacities {GW}: Coal Oil/Gas Combined- Cycle OLG 2.3 0.7 0 ONG 0 0 0 OSG 4.7 1.3 0 OWE 0 0 0 OCE* 0 0 5 *OCE must pay lien fixed cost to operate cc generat ion, simulating inv Fixed & Variable operating costs, below)

Nuclear capacities (for L, M, H nuclear scenarios) {GW} : Low Nuclear Medium High Nuclear Nuclear OLG 5.5 6.5 7.5 ONG 3.5 5.0 6.5 OSG, OWE, 0 0 0 OCE

Hydro hourly peak capacity, hourly run-of-river, and total energy in each season (for OLG only, all other i set to zero) {GW}: hcapik hrorik hengik Fall 6.0 2.4 8.75 Winter 6.0 2.4 8.75 Spring 7.2 2.88 10.5 Summer 4.8 1.92 7

Fixed & Variable operating costs:

Fixed operating cost fct Variable operating cost oct

167 168

(Jh for hydro){$M/GW} (oh for hydro) {$/MWh} Nuclear 100 4 Hydro 5 2.5 Coal 25 25 Oil/Gas 15 50 Combined-Cycle* 125 40 * note that high fixed capital cost simulates investment cost in new cc capacity, for genco OCE who is presumed to be a new entrant.

Annual fixed peak payments for making capacity available pp = 60 $M/GW

Scheduled maintenance and fixed outage rates {%}: Nuclear Coal Oil/Gas Combined- Cycle

Scheduled maintenance (mrct) 0.15 0.15 0.15 0.15 Forced outage rate (foutt) 0.1 0.1 0.1 0.1

ANTICIPATION PARAMETERS: Electricity market anticipation aeikb Wind capacity market anticipation (all blocks) {% of p]} aci {%ofyi} OLG 0.333** 1 ONG 1 1 OSG 1 1 OWE 1 0.5* OCE 1 1 ** reflects advanced market knowledge for gencos in given market, all others basic knowledge

168 169

7.5 Appendix E: Determining Basic and Advanced Knowledge Cross-Market WOM Anticipation Coefficients

To obtain the cross-market WOM anticipation coefficients axsikb (axwsikb for wind entrant) and aoqsikb (for conventional gencos only), we use the following step-by-step techniques, for basic and advanced cross-market WOM knowledge. The anticipation parameters, once obtained, are used as inputs into the WCAPCOMP 2.WOM model, and solve normally.

The approach is to simulate a small WOM contract in each block, and observe the effect on the buying genco's output, and the electricity market price in that block. To obtain the basic knowledge parameters, which assume that the genco believes that it's rivals will not adjust their outputs in response to the WOM contract, we fix the outputs of the rival gencos, and observe the resulting change in genco f s output and the electricity market price, which is what genco i would perceive if it held basic knowledge. To obtain advanced knowledge parameters, we allow the rival gencos to re-optimize their outputs by not fixing them, which yields the net effect of the WOM contract on the electricity market equilibrium, given that genco /' knows the responses of the rival gencos.

For basic knowledge for genco i: 1. Run WCAPCOMP 2.WOM, with Xbsikb=Xssikb = 0 (does not allow for WOM contracts), and observe wind investments OUi and OUw, electricity market prices EPskb, and outputs OQSikb of the conventional gencos. 2. Fix all wind investments OUi and OUw , and rival gencos -/ outputs OQs.ikb at levels observed in Step 1. = 3. For genco i, set Xbsikb XsSikb= 0.1 (leaving all others at zero), to simulate a WOM contract for 0.1 units of wind output (in all blocks) from the wind entrant to genco /. 4. Re-run WCAPCOMP 2.WOM, with rival genco outputs and all wind capacity investments fixed as per Step 2, and observe new genco i outputs OQsikb' and electricity market prices EPskb '• 5. Compute the anticipation coefficients for all scenarios and blocks, which are defined for a unit increase in Xb, and are hence divided by 0.1: 1 axsikb = (EPskb -EPskb)/0.\ aoqsikb= (OQSikb - OQsikbl/0.1

(for wind genco basic knowledge case, same procedure applies, but note that since all other = gencos OQSikb are fixed, EPskb' EPskb, and so axwSikb-0 for all gencos and blocks, in the basic knowledge case).

For advanced knowledge for genco i: (same procedure as basic knowledge case, but do not fix OQs-tkb in Step 2).

We further point out that regardless of the level of cross-market WOM knowledge, for any genco with conventional capacity off the margin for all types in a given block, = = axSikb cixWsikb~aoqsikb 0 (genco cannot adjust outputs in this case, and electricity market price will therefore remain constant).

169 7.6 Appendix F: GAMS Implementation of WCAPCOMP 2.WOM

OPTION ITERLIM = 200000;

* SET NOTATION SET s scenarios /L, M, H/; SET i conventional gencos /OLG, ONG, OSG, OCE/; ALIAS (i, ii); SET t conventional technologies /n, c, o, cc/; SET k seasons /fl/ wt, sp, sm/; ALIAS (k, kk); SET b blocks /bl, b2, b3/; ALIAS (b, bb);

* DATA * UNCERTAINTY PARAMETER p(s) probability of scenario s /L 0.333, M 0.333, H 0.333/; * DEMAND PARAMETERS PARAMETER rQ(k) peak demand in EM {GW} /fl 20.8, wt 20.8, sp 21.7, sm 23.5/; TABLE wb(k,b) duration of block in EM {khrs} bl b2 b3 fl 0.728 0.728 0.728 wt 0.720 0.720 0.720 sp 0.736 0.736 0.736 sm 0.736 0.736 0.736

SCALAR e price elasticity of demand in EM /0.5/; TABLE L(k,b) height of demand blocks as % of peak demand bl b2 b3 fl 0.821 0.717 0.597 wt 0.879 0.785 0.668 sp 0.767 0.673 0.562 sm 0.789 0.668 0.542

PARAMETER rP initial marginal cost in EM {$ per MWh} /48/; PARAMETER rD(k,b) initial demand in EM {GW}; rD(k,b) = rQ(k) *L(k,b) ; PARAMETER pl(k,b) demand curve slope in EM {$ per MWh*MW}; pl(k,b) = rP/(rD(k,b)*e) ; PARAMETER p0(k,b) demand curve intercept in EM {$ per MWh} p0(k,b) = rP +pl(k,b)*rD(k,b);

* WIND PARAMETERS TABLE cf(k,b) wind capacity factor {%} bl b2 b3 fl 0.25 0.3 0.35 wt 0.35 0.4 0.45 sp 0.25 0.3 0.35 sm 0.15 0.2 0.25

SCALAR yO supply curve intercept in WCM ($M per GW} /307; SCALAR yl supply curve slope in WCM {$M per GW*GW) /5/;

* GENCO PARAMETERS PARAMETER oc(t) operating cost of technology t {$ per MWh} /n 4, c 25, o 50, cc 40/; SCALAR oh operating cost of hydro {$ per MWh}; oh = 2.5; SCALAR pp peak payments for conventional generation {$M per GW}; pp = 60; PARAMETER fc(t) fixed operating cost of technology t {$M per GW} /n 100, c 25, o 15, cc 125/; SCALAR fh fixed operating cost of hydro {$M per GW}; fh = 5; TABLE pcap(s,i,t) conv generation capacity {GW} n c o cc L.OLG 5.5 2.3 0.7 0 L.ONG 3.5 0 0 0 M.OLG 6.5 2.3 0.7 0

170 171

M.ONG 5 0 0 0 H.OLG 7.5 2.3 0.7 0 H.ONG 6.5 0 0 0 (L,M,H) .OSG 0 4.7 1.3 0 (L,M,H).OCE 0 0 0 5

PARAMETER mrc(t) annual maintenance requirement (%} /n 0.15, c 0.15, o 0.15, cc 0.15/; PARAMETER fout(t) forced outage rate {%} /n 0.1, c 0.1, o 0.1, cc 0.1/; TABLE hcap(i,k) seasonal hrly hydro capacity {GW - includes ror} fl wt sp sm OLG 6 6 7.2 4.8

TABLE hrr(i,k) seasonal hrly hydro run of river !GW) fl wt sp sm OLG 2 . 4 2.4 2.88 1.92

TABLE heng(i,k) seasonal total hydro energy {TWh includes ror} fl wt sp sm OLG 8.75 8.75 10.5 7

•ANTICIPATION COEFFICIENTS TABLE ae(i,k,b) ant of demand curve slope for conv gencos of pi} bl b2 b3 OLG.(fl,wt,sp,sm) 0.333 0.333 0.333 (ONG,OSG,OCE).(fl,wt,sp,sm) 111

TABLE aew(k,b) ant of demand curve slope for wind genco of pi} bl b2 b3 (f 1, wt, sp, sm) 1 1 1

TABLE ax(s,i,k,b) ant by conv genco of EP change if buying extra unit of WOM contract {% of pi} bl b2 b3 L.OLG.(fl,wt,sp,sm) 0.2 0.2 0.2 M.OLG.(fl,wt,sp) 0 0 0 M.OLG.sm 0.2 0 0 H.OLG.(fl,wt,sp,sm) 0 0 0 (L,M,H).ONG.(fl,wt,sp,sm) 0 0 0 L.OSG.(fl,wt,sp) 0.2 0.2 0.2 L.OSG.sm 0.35 0.2 0.2 M.OSG.(fl,wt,sp) 0.5 0.5 0.5 M.OSG.sm 0.2 0.2 0.5 H.OSG.(fl,wt,sp,sm) 0.35 0.35 0.35

TABLE axw(s,i,k,b) ant by wind genco of EP change if selling extra unit of WOM contract {% of pi} bl b2 b3 L.OLG.(fl,wt,sp,sm) 0.2 0.2 0.2 M.OLG.(fl,wt,sp) 0 0 0 M.OLG.sm 0.2 0 0 H.OLG.(fl,wt,sp,sm) 0 0 0 (L,M,H).ONG.(fl,wt,sp,sm) 0 0 0 L.OSG.(fl,wt,sp) 0.2 0.2 0.2 L.OSG.sm 0.35 0.2 0.2 M.OSG.(fl,wt,sp) 0.5 0.5 0.5 M.OSG.sm 0.2 0.2 0.5 H.OSG.(fl,wt,sp,sm) 0.35 0.35 0.35

TABLE aoq(s,i,k,b) ant by conv genco of OQ(i) decrease if buying extra unit of WOM contract {GW} bl b2 b3 L.OLG.(fl,wt,sp,sm) 0.4 0.4 0.4 M.OLG.(fl,wt,sp) 0 0 0 M.OLG.sm 0.4 0 0 H.OLG.(fl,wt,sp,sm) 0 0 0 (L,M,H).ONG.(fl,wt,sp,sm) 0 0 0 L.OSG.(fl,wt,sp) 0.8 0.8 0.8 L.OSG.sm 0.65 0.8 0.8 171 M.OSG.(fl,wt,sp) 0.5 0.5 0.5 M.OSG.sm 0.8 0.8 0.5 H.OSG. (fl,wt,sp,sm) 0.65 0.65 0.65

TABLE awp(s,i,k,b) ant by conv genco of WOP increase if buying extra contract {% of pi} bl b2 b3 (L,M,H). (OLG,ONG,OSG). (f1,wt,sp,sm) 0.5 0.5 0.5

TABLE awpw(s,i, k,b) ant by wind genco of WOP decrease if selling extra contract {% of pi) bl b2 b3 (L,M,H).(OLG,ONG,OSG).(f1,wt,sp,sm) 0.5 0.5 0.5

PARAMETER ac(i) ant by conv genco of WCM supply curve slope {% of yl} /OLG 1, ONG 1, OSG 1, OCE 1/ / SCALAR acw ant by wind genco of WCM supply curve slope {% of yl}; acw = 0.5;

* MODEL SPECIFICATION POSITIVE VARIABLES *Link YBQ(s,i,k,b) dual var on BQ <= OQ constraint {$M per GW) YBQw(s,k,b) wind genco analogue to YBQ YBU(i) dual var on BU <= OU constraint ($M per GW} YBUw wind genco analogue to YBU WOP(s,i,k,b) WOM contract price for contract with buyer i {$ per MWh) *EMO(s) BQ(s,i,k,b) spot market bid quantity from genco {GW output} BQw(s,k,b) wind genco analogue to BQ D(s,k,b) total spot market demand {GW output} EP(s,k,b) electricity spot market price {$ per MWh} *WMO BU(i) wind capacity market accepted quantity {GW} BUw wind genco analogue to BU (GW) Ut total wind constructed by FMO {GW} WCP wind capacity price {$M per GW) *CONVENTOINAL GENCO primals OQ(s,i,k,b) electricity spot market generation offered to EMO {GW} OEP(s,i,k,b) electricity spot market offer price to EMO {$ per MWh} Xb(s,i,k,b) WOM contracts purchased from wind genco {GW} OU(i) wind capacity market offer quantity to WMO {GW} OWCP(i) wind capacity market offer price to WMO {$M per GW} VC (s,i,t,k,b) conventional spot market generation {GW} VW(s,i,k,b) wind spot market generation {GW} VH(s,l,k,b) hydro spot market generation {GW} PVC(s,i,t) annual peak conventional generation made available {GW} PVH(s,i) annual peak hydro generation made available {GW} MC(s,i,t,k) generation capacity taken down for seasonal maintenance {GW} *CONVENTOINAL GENCO duals YVC(s,i,t, k,b) dual var on hrly conv output constraint ($M per GW) YVW(s,i,k,b) dual var on hrly wind output constraint ($M per GW) YVH(s,i,k,b) dual var on peak hydro constraint ($M per GW) YHRR(s,i,k,b) dual var on ror hydro constraint ($M per GW) YHENG(s,i,k) dual var on seasonal hydro energy constraint ($M per TWh) YOQ(s,i,k,b) dual var on hryl total output constraint ($M per GW) YPVC(s,i,t) dual var on PVC <= pcap constraint ($M per GW) YPVH(s,i) dual var on PVH <= heap(spring) constraint ($M per GW) YMC(s,i,t) dual var on conv maintenance constraint ($M per GW) *WIND GENCO primals OQw(s,k,b) wind genco analogue to OQ OEPw(s,k,b) wind genco analogue to OEP Xs(s,i,k,b) WOM contracts sold to genco i {GW} OOw wind genco analogue to OU OWCPw wind genco analogue to OWCP VWw(s,k,b) wind genco analogue to VW *WIND GENCO duals YVWw(s,k,b) wind genco analogue to YVW YOQw(s,k,b) wind genco analogue to YOQ YWOM(s,k,b) dual var on Xs <= VWw constraint {$ per GW} 172 FREE VARIABLES YD dual var on demand curve equality {units} YOEP(s,i,k,b) dual var on electricity spot offer price equality {units} YOEPw(s,k,b) wind genco analogue to YOEP YOWCP(i) dual var on wind capacity market offer price equality {units} YOWCPw wind genco analogue to YOWCP YUt dual var on wind capacity supply curve equality {units}

EQUATIONS *LINKS linkEM(s,i,k,b) linkEMw(s,k,b) linkWCM(i) linkWCMw linkWOM(s,i,k,b) *EMO EMOcurve(s,k,b) EMOdemsat(s,k,b) delBQ(s,i,k,b) delBQw(s,k,b) delD(s,k,b) *WMO WMOcurve WMOdemsat delBU(i) delBUw delUt *CONV GENCOS of ferQ(s, i, k,b) peakconv(s,i,t) conv(s, i, t, k,b) maintconv(s,i,t) wind (s, i, k,b) peakhyd(s,i) hydout(s,i,k,b) hydrr (s, i, k,b) hydeng(s, i,k) offerEP(s,i,k,b) offerWCP(i) delOQ(s,i,k,b) delVC(s,i,t,k,b) delVW(s,i,k,b) delVH(s,i,k,b) delPVC(s,i,t) delPVH(s,i) delMC(s,i,t,k) delXb(s,i,k,b) delOO(i) delOWCP(i) delOEP(s,i,k,b) *WIND GENCO offerQw(s,k,b) windw (s, k,b) windWOM(s,k,b) offerEPw(s,k,b) offerWCPw delOQw(s,k,b) delVWw(s,k,b) delXs(s,i,k,b) delOUw delOWCPw delOEPw(s,k,b)

*MODEL DEFINITION *LINKS linkEM(s,i,k,b) . . -BQ(s,i,k,b) =G= -OQ(s,i,k,b) , linkEMw(s,k/b) . . -BQw(s,k,b) =G= -OQw(s,k-b) ; 173 linkWCM(i).. -BU(i) =G=-OU(i); linkWCMw.. -BUw =G= -OOw; linkWOM(s,i,k,b) . . Xs(s,i,k,b) =G= Xb(s,i,k,b); *EMO EMOcurve(s,k,b).. EP(s,k,b)/wb(k,b) =E= pO(k,b) -pi (k,b)*D(s,k,b) ; EMOdemsat (s,k,b) . . sum(i, BQ(s,i,k,b)) +BQw(s,k,b) -D(s,k,b) =G= 0; delBQ(s,i,k,b) . . YBQ(s,i,k,b) -EP(s,k,b) =G= -wb(k,b) *OEP(s,i,k,b) ; delBQw(s,k,b) . . YBQw(s,k,b) -EP(s,k,b) =G= -wb(k,b) *OEPw(s,k,b) ; delD(s,k,b) . . pi (k,b) *YD (s, k,b) +EP(s,k,b) =G= 0; *WMO WMOcurve.. WCP =E= yO +yl*Ut; WMOdemsat.. -sum(i, BU(i)) -BUw +Ut =G= 0; delBU(i).. YBO(i) +WCP =G= OWCP(i); delBUw.. YBUw +WCP =G= OWCPw; delUt.. -WCP -yl*YUt =G= -yO -yl*Ut; 'CONVENTIONAL GENCOS offerQ(s,i,k,b) . . sum(t, VC (s, i, t, k,b) ) +VW(s,i,k,b) +VH(s,i,k,b) =G= OQ(s,i,k,b) -aoq (s, i, k,b) * {Xb (s, i, k,b) -Xs (s, i, k,b) }; peakconv (s,i,t).. -PVC(s,i,t) =G= -pcap(s,i,t); conv(s,i,t,k,b) . . -VC (s, i, t, k,b) -MC(s,i,t,k) =G= -(l-fout(t) ) *PVC(s,i,t); maintconv(s,i,t) . . sum(k, sum(b, wb(k,b) *MC (s, i, t, k) )) =G= 8.76*mrc(t)*PVC(s,i,t); wind(s,i,k,b) . . -VW(s,i,k,b) =G= -cf (k,b) *OU (i) ; peakhyd (s,i).. -PVH(s,i) =G= -heap(i,'sp'); hydout(s,i,k,b) . . -VH(s,i,k,b) =G= -heap (i, k) ; hydrr(s,i,k,b) . . VH(s,i,k,b) =G= hrr (i, k) ; hydeng (s, i, k) . . -sum(b, wb (k,b) *VH (s, i, k, b) ) =G= -heng(i,k); offerEP(s,i,k,b) . . OEP(s,i,k,b) =E= EP (s, k,b)/wb (k,b) -ae(i,k,b) *pl (k,b) * {OQ (s, i, k,b) -BQ(s,i,k,b) } +ax(s,i,k,b) *pl (k,b) *{Xb(s,i,k,b) -Xs (s, i, k,b) } ; offerWCP(i).. OWCP(i) =E=WCP +ac(i)*yl*{OU(i) -BU(i)}; delOQ(s,i,k,b) . . -YBQ(s, i, k,b) +YOQ (s, i, k,b) +ae(i,k,b) *pl (k,b)*YOEP(s,i,k,b) =G= p (s) *EP (s, k,b) -p(s)*wb(k,b) *[ae(i,k,b)*pl(k,b) * (2*OQ (s, i, k,b) -BQ(s,i,k,b) +Xb(s,i,k,b) }] ; delVC(s,i,t,k,b).. -YOQ(s,i,k,b) +YVC(s,i,t,k,b) =G= -p(s)*wb(k,b) *oc(t) ; delVW(s,i,k,b).. -YOQ(s,i,k,b) +YVW(s,i,k,b) =G= 0; delVH(s,i,k,b) . . -YOQ(s, i, k,b) +YVH (s, i, k,b) -YHRR(s,i, k,b) +wb(k,b) *YHENG(s,i,k) =G= -p (s) *wb (k,b) *oh; delPVC(s,i,t) . . YPVC(s,i,t) -sum(k, sum(b, (l-fout(t) ) *YVC(s,i,t,k,b) ) ) + 8.76*mrc(t)*YMC(s,i,t) =G= p(s)*(pp -fc(t)); delPVH(s,i) . . YPVH(s,i) =G= p(s)*(pp -fh) ; delMC(s,i, t,k) . . sum(b, YVC (s, i, t, k, b) ) -sum(b, wb(k,b))*YMC(s,i,t) =G= 0; delXb(s,i,k,b) . . -aoq(s, i, k,b) *YOQ (s, i, k,b) -ax(s,i,k,b)*pl(k,b)*YOEP(s,i,k,b) +WOP (s, i, k,b) =G=p(s)*(l -aoq(s,i,k,b) )*EP(s,k,b) +p (s) *wb (k,b) * [ ax(s,i,k,b)*pl(k,b)*(OQ(s,i,k,b) +2*Xb(s, i, k,b) -Xs (s,i, k,b) } -awp(s,i,k,b) *pl (k,b) * {2*Xb (s, i, k,b) -Xs (s, i, k,b) } ] ; delOU(i).. -sura(s, sum(k, sum(b, cf (k,b) *YVW (s, i, k,b) ) ) ) -ac(i)*YOWCP(i) -YBU(i) =G= -WCP -ac(i)*yl*{2*OU(i) -BU(i)}; delOWCP(i).. YOWCP(i) =G= 0; delOEP(s,i,k,b).. YOEP(s,i,k,b) =G= 0;

*WIND GENCO offerQw(s,k,b) . . VWw(s,k,b) =G= OQw (s, k,b) ; windw(s,k,b) . . -VWw(s,k,b) =G= -cf (k,b) *OUw; windWOM(s,k,b) . . -sum(i, Xs(s,i,k,b)) =G= -VWw (s, k,b) ; offerEPw(s,k,b) . . OEPw(s,k,b) =E= EP (s, k,b) /wb (k,b) -aew(k/b)*pl(k,b)*{OQw(s,k,b) -BQw(s,k,b) } +sum(i, axw(s,i,k,b) *pl (k,b) *(Xs (s,i,k,b) -Xb (s, i, k,b) }) ; offerWCPw.. OWCPw =E= WCP +acw*yl*{OUw -BUw); 174 delOQw(s,k,b) . . -YBQw (s, k,b) +YOQw(s,k,b) +aew(k,b)*pl(k,b)*YOEPw(s,k,b) =G= p (s) *EP (s, k,b) -p(s)*wb(k,b)*[aew(k,b)*pl (k,b) *{2*0Qw(s,k,b) -BQw(s,k,b) -sum(i, Xs (s, i, k,b) ) ) ] ; delVWw(s,k,b) . . -YOQw(s,k,b) -YWOM(s,k,b) +YVWw(s,k,b) =G= 0; delXs(s,i,k,b) . . YWOM(s,k,b) -axw (s, i, k,b) *pl (k,b) *YOEPw (s, k,b) -WOP(s,i,k,b) =G= -p(s)*EP(s,k,b) +p (s) *wb (k,b) * [ axw(s,i,k,b)*pl (k,b) * {OQw (s, k,b) -sum(ii, Xs (s, ii, k,b) ) } -awpw(s,i,k,b)*pl (k,b) *{2*Xs(s,i,k,b) -Xb(s,i,k,b) }] ; delOUw.. -sum(s, sum(k, sum(b, cf(k,b)*YVWw(s,k,b)))) -acw*YOWCPw -YBUw =G= -WCP -acw*yl*{2*OUw -BUw}; delOWCPw.. YOWCPw =G= 0; delOEPw(s,k,b).. YOEPw(s,k,b) =G= 0;

* OCE cannot build wind OU.up ('OCE') = 0;

MODEL WCAPCOMP2WOM / *link constraints linkEM.YBQ, linkEMw.YBQw, linkWCM.YBU, linkWCMw.YBUw, linkWOM.WOP, *EMO primal and dual EMOcurve.YD, EMOdemsat.EP, delBQ.BQ, delBQw.BQw, delD.D, *WMO primal and dual WMOcurve.YUt, WMOdemsat.WCP, delBU.BU, delBUw.BUw, delUt.Ut, * conventional genco primal offerQ.YOQ, peakconv.YPVC, conv.YVC, maintconv.YMC, wind.YVW, peakhyd.YPVH, hydout.YVH, hydrr.YHRR, hydeng.YHENG, offerEP.YOEP, offerWCP.YOWCP, Conventional genco dual delOQ.OQ, delVC.VC, delVW.VW, delVH.VH, delPVC.PVC, delPVH.PVH, delMC.MC, delXb.Xb, delOU.OU, delOWCP.OWCP, delOEP.OEP, *wind genco primal offerQw.YOQw, windw.YVWw, windWOM.YWOM, offerEPw.YOEPw, offerWCPw.YOWCPw, *wind genco dual delOQw.OQw, delVWw.VWw, delXs.Xs, delOUw.OUw, delOWCPw.OWCPw, delOEPw.OEPw /;

SOLVE WCAPCOMP2WOM USING MCP;

175 176

7.7 Appendix G: Parameters Used for Chapter Four Numerical Examples

For WCAPCOMP 2.0 (base stochastic model with no WOM), we fix all Xbsikb=Xssikb=0 so that no WOM contracts can be formed.

All parameters from WCAPCOMP 1.0 are used (see Appendix D), with the following additions:

probability of nuclear outcome^ : PL=PM=PH= 1/3.

For WCAPCOMP 2.WOM, as above, but with Xbsikb,Xssikb>^ 0 (ie. not fixed), and with the following anticipation coefficients in the WOM (obtained using the procedure outlined in Appendix E):

TABLE ax (s,i k,b) ant by conv genco of EP change if buying extra unit of WOM contract {% of Pl> bl b2 b3 L.OLG . (fl,wt,sp , sm) 0.2 0. 0.2 M.OLG .(fl,wt, sp) 0 0 0 M.OLG . sm 0.2 0 0 H.OLG .(fl,wt, sp, sm) 0 0 0 (L,M, H) .ONG. fl( , wt,sp,sm) 0 0 0 L.OSG (fl,wt, sp) 0.2 0 2 0.2 L.OSG sm 0.35 0 2 0.2 M.OSG (fl,wt, sp) 0.5 0 5 0.5 M.OSG sm 0.2 0 2 0.5 H.OSG (fl,wt. sp,sm) 0.35 0 35 0.35

TABLE axw(s,i,k,b ) ant by wind genco of EP change if selling extra unit of WOM contract {% of pl) bl b2 b3 L.OLG . (fl,wt,sp,sm ) 0.2 0.2 0.2 M.OLG . (fl,wt,sp) 0 0 0 M.OLG . sm 0.2 0 0 H.OLG . (fl,wt,sp,sm ) 0 0 0 (L,M, H).ONG.(fl, wt,sp,sm) 0 0 0 L.OSG . (fl,wt,sp) 0.2 0.2 0.2 L.OSG . sm 0.35 0.2 0.2 M.OSG . (fl,wt,sp) 0.5 0.5 0.5 M.OSG . sm 0.2 0.2 0.5 H.OSG . (f1,wt,sp,sm ) 0.35 0.35 0.35

TABLE aoq(s,i,k,b) ant by conv genco of OQ(i) decrease if buying extra unit of WOM contract (GW) bl b2 b3 L.OLG.(fl,wt,sp,sm) 0.4 0.4 0.4 M.OLG.(fl,wt,sp) 0 0 0 M.OLG.sm 0.4 0 0 H.OLG.(fl,wt,sp,sm) 0 0 0 (L,M,H).ONG.(fl,wt,sp,sm) 0 0 0 L.OSG.(fl,wt,sp) 0.8 0.8 0.8 L.OSG.sm 0.65 0.8 0.8 M.OSG.(fl,wt,sp) 0.5 0.5 0.5 M.OSG.sm 0.8 0.8 0.5 H.OSG.(fl,wt,sp,sm) 0.65 0.65 0.65

TABLE awp(s,i,k,b) ant by conv genco of WOP increase if buying extra unit of WOM contract {% of pl) bl b2 b3 (L,M,H).(OLG,ONG,OSG).(f1,wt,sp,sm) 0.5 0.5 0.5

176 177

TABLE awpw (s, i, k,b) ant by wind genco of WOP decrease if selling extra unit of WOM contract {% of pi} bl b2 b3 (L,M,H).(OLG,ONG,OSG).(f1,wt,sp,sm) 0.5 0.5 0.5

177 178

7.8 Appendix H: Parameters Used For Sensitivity Examples in Chapter Five

For OLG+OSG, advanced cross-market WOM knowledge case, model parameters are identical to the Chapter Four WCAPCOMP 2.WOM run (see Appendix G).

For OLG+OSG, basic cross-market WOM knowledge, use same model parameters as the preceding case, but replacing WOM cross-market anticipation coefficients ax*, axw* and aoq* with:

TABLE ax(s,i,k,b) ant by conv genco of EP change if buying extra unit of WOM contract {% of pi) bl b2 b3 L.OLG. (f1, wt, sp, sm) 0.25 0.25 0.25 M.OLG.(fl,wt,sp) 0 0 0 M.OLG.sm 0.25 0 0 H.OLG. (fl,wt, sp, sm) 0 0 0 (L,M,H).ONG.(fl,wt, sp, sm) 0 0 0 (L,M,H).OSG.(fl,wt. sp, sm) 0.5 0.5 0.5

TABLE axw(s,i,k,b) ant by wind ge nco of EP change if {% of pi} bl b2 b3 (L,M, H) . (OLG,ONG,OSG) .. (fl, wt,sp ,, sm) 0 0 0

TABLE aoq(s,i,k,b) ant by conv genco of OQ(i) dec contract {GW} bl b2 b3 L.OLG.(fl,wt,sp,sm) 0.25 0.25 0.25 M.OLG.(fl,wt,sp) 0 0 0 M.OLG.sm 0.25 0 0 H.OLG.(fl,wt,sp,sm) 0 0 0 (L,M,H).ONG.(fl,wt, sp, sm) 0 0 0 (L,M,H) .OSG. (fl,wt, sp, sm) 0.5 0.5 0.5

For OIG, advanced cross-market WOM knowledge case, model parameters are identical to the Chapter Four WCAPCOMP 2.WOM run, but with the following changes:

- Remove genco OSG from the model - Conventional capacities pcapsit of OIG = capacities OLG + OSG from the OLG+OSG case Set all OIG electricity market anticipation coefficients for aes-oiG'kb- 1/6 (reflects increased competitiveness, as would be required by policy makers due to OIG being near monopoly) Use the following cross-market WOM anticipation parameters:

TABLE ax(s,i,k,b) ant by conv genco of EP change if buying extra unit of WOM contract {% of pi} bl b2 b3 L.OIG.(fl,wt,sp) 0.15 0.15 0.15 L.OIG.sm 0 0.15 0.15 (M,H).OIG.(fl,wt,sp,sm) 0.15 0.15 0.15 (L,M,H).ONG.(fl,wt,sp,sm) 0 0 0

TABLE axw(s,i,k,b) ant by wind genco of EP change if selling extra unit of WOM contract {% of pi} bl b2 b3 178 179

L.OIG.(fl,wt,sp) 0.15 0.15 0.15 L.OIG.sm 0 0.15 0.15 (M,H).OIG.(fl,wt,sp,sm) 0.15 0.15 0.15 (L,M,H).ONG.(fl,wt,sp,sm) 0 0 0

TABLE aoq(s,i,k,b) ant by con? genco of OQ(i) decrease if buying extra unit of WOM contract {GW} bl b2 b3 L.OIG.(fl,wt,sp) 0.15 0.15 0.15 L.OIG.sm 0 0.15 0.15 (M,H).OIG.(fl,wt,sp,sm) 0.15 0.15 0.15 (L,M,H).ONG.(fl,wt,sp,sm) 0 0 0 /

For OIG, basic cross-market WOM knowledge case, same parameters as the preceding OIG advanced case but using the following WOM anticipation coefficients:

TABLE ax(s,i,k,b) ant by conv genco of EP change if buying extra unit of WOM contract {% of pi) bl b2 b3 L.OIG.(fl,wt,sp) 0.15 0.15 0.15 L.OIG.sm 0 0.15 0.15 (M,H).OIG.(fl,wt,sp,sm) 0.15 0.15 0.15 (L,M,H) .ONG. (fl,wt,sp,sm) 0 0 0

TABLE axw(s,i,k,b) ant by wind genco of EP change if selling extra unit of WOM contract {% of pi} bl b2 b3 (L,M,H).(OIG,ONG).(f1,wt,sp,sm) 0 0 0

TABLE aoq(s,i,k,b) ant by conv genco of OQ(i) decrease if buying extra unit of WOM contract {GW} bl b2 b3 L.OIG.(fl,wt,sp) 0.15 0.15 0.15 L.OIG.sm 0 0.15 0.15 (M,H).OIG.(fl,wt,sp,sm) 0.15 0.15 0.15 (L,M,H) .ONG. (fl,wt, sp,sm) 0 0 0

179