Complementarity models can represent the simultaneous optimization problems of one or several int... more Complementarity models can represent the simultaneous optimization problems of one or several interacting decision-makers, and thus they have become an increasingly important and powerful tool for formulating and solving bottom-up energy market models. This paper provides an overview of the full range of complementarity-based formulations and how these can be applied to assist the different market participants and organizations with their decision-making processes. To this end, the first part of the paper introduces the mathematical formulation of some basic complementarity models, which are illustrated by highly simplified but illustrative energy market applications. Considering these models, the second part of the paper is devoted to describing in broad terms four areas of their potential application: electricity markets, emission markets, natural gas markets and economies comprising several interacting markets.
International Series in Operations Research & Management Science, 2012
In this chapter, we explore the notions of equilibria and optimization and show how in some cases... more In this chapter, we explore the notions of equilibria and optimization and show how in some cases they are related. The notion of an equilibrium is a fundamental concept that has been used in a variety of disciplines such as economics, engineering, and science to name just a few. At its core, an equilibrium is a state of the system being modeled for which the system has no “incentive” to change. These incentives can be monetary in the case of economics or based on natural forces and scientific laws such as total input equals total output. Some well-known engineering examples include: conservation of energy, conservation of mass, conservation of momentum [8], steady-state probabilities in Markov chains such as birth-and-death processes [53] to name a few. These and other engineering examples are typified by a balancing of forces or conditions so that the state once reached will not easily (if at all) be left.
International Series in Operations Research & Management Science, 2012
In this chapter, we present several advanced algorithms that can be useful for the solution of so... more In this chapter, we present several advanced algorithms that can be useful for the solution of some of the models discussed in this book.
International Series in Operations Research & Management Science, 2012
The purpose of this chapter is to explain variational inequality (VI) formulations of equilibrium... more The purpose of this chapter is to explain variational inequality (VI) formulations of equilibrium problems, and the close connection of a VI problem to an equivalent complementarity problem. There are sometimes advantages to a VI formulation compared to a complementarity formulation: the complementarity formulation has primal decision variables, and dual variables that arise, e.g., when specifying the KKT conditions of individual agents; but a VI formulation has the same primal variables, with few, or no dual variables, which can considerably ease the coding of the model in GAMS. This coding advantage is particularly evident when implementing large complex models, or decomposition algorithms as discussed in Chapter 9. However, the derivation of a complementarity model formulation is usually easier than the derivation of an equivalent VI model: e.g., many complementarity models in this book are derived by writing down the KKT conditions of the agents, together with market-clearing conditions, but the procedure to write down a VI model with few or no dual variables is not as easily stated. In this chapter, we alleviate this difficulty by showing how to arrive quickly at the formulation of a VI model for a large class of Nash equilibrium and generalized Nash equilibrium settings.
This chapter presents some of the key ideas in various algorithms that are used to solve some of ... more This chapter presents some of the key ideas in various algorithms that are used to solve some of the basic equilibrium models, namely, models in the form of the linear complementarity problem (LCP), the nonlinear complementarity problem (NCP) and the variational inequality (VI) problem. We explain the main ideas of the algorithms, but avoid detailed discussions or proofs of important mathematical issues such as the conditions under which an algorithm is guaranteed to converge.
International Series in Operations Research & Management Science, 2012
Natural gas is a key fuel in energy markets worldwide. It is produced from either onshore or offs... more Natural gas is a key fuel in energy markets worldwide. It is produced from either onshore or offshore wells, processed to remove impurities, and then transported by either pipeline in gaseous form or cooled to about -260 degrees F (about -160 degrees C) and then transported as liquefied natural gas (LNG) to destinations around the world. The main consuming sectors that use it are residential, commercial, industrial, electric power, and to some extent transportation. At present, the world has abundant gas supplies. According to [52], the global mean projected remaining recoverable resources is 16,200 trillion cubic feet (Tcf) or 150 times the current annual global consumption. About 9,000 Tcf is gauged to be economically available at less than or equal to $4 per million British Thermal Units (Btu) [52].
International Series in Operations Research & Management Science, 2012
Several of the models previously introduced in this book have focused on the market for a single ... more Several of the models previously introduced in this book have focused on the market for a single commodity with a single price, such as power at a particular location in a particular hour. However, many of this book’s models instead considered several markets simultaneously, recognizing that linkages among them imply that equilibrium prices in one market cannot be calculated without considering how they affect, and are affected by, prices in other markets. In the earlier chapters, linkages among markets were mainly through the supply-side, in which the cost of providing commodity in one market depends in part on prices in other markets. For instance, a power generator with only a small amount of capacity with low running costs might experience a rise in its marginal cost of serving one part of the network if it also sells a lot of power elsewhere, thereby exhausting its cheap capacity. The purpose of this chapter is to introduce the modeling of multiple energy markets in which it is instead the behavior of consumers that links the markets. In particular, the amount that final consumers buy of one commodity affects how much they are willing to pay for other commodities.
International Series in Operations Research & Management Science, 2012
The purpose of this chapter is to provide a more in-depth exploration of applications of compleme... more The purpose of this chapter is to provide a more in-depth exploration of applications of complementarity models to electricity markets. In doing so, we introduce two crucial features of energy markets. The first is transportation networks with capacity limits on links between different markets. The second is environmental restrictions, such as emissions markets. We address these in turn by building, analyzing, and solving models for electric power markets that incorporate these features.
International Series in Operations Research & Management Science, 2012
This chapter provides a friendly introduction to several mathematical structures used in the foll... more This chapter provides a friendly introduction to several mathematical structures used in the following chapters. These structures are useful to describe the functioning of markets and the behavior of market agents. Throughout the chapter clarity and simplicity are emphasized.
International Series in Operations Research & Management Science, 2012
In this chapter, we explain some useful principles of microeconomics for those readers with littl... more In this chapter, we explain some useful principles of microeconomics for those readers with little or no background in the subject. Readers who have studied microeconomics may also benefit from this chapter, as we show how to construct several different kinds of models of markets, using optimization and complementarity techniques.
The paper concerns a new class of optimization-related problems called Equilibrium Problems with ... more The paper concerns a new class of optimization-related problems called Equilibrium Problems with Equilibrium Constraints (EPECs). One may treat them as two level hierarchical problems, which involve equilibria at both lower and upper levels. Such problems naturally appear in various applications providing an equilibrium counterpart (at the upper level) of Mathematical Programs with Equilibrium Constraints (MPECs). We develop a unified approach to both EPECs and MPECs from the viewpoint of multiobjective optimization subject to equilibrium constraints. The problems of this type are intrinsically nonsmooth and require the use of generalized differentiation for their analysis and applications. This paper presents necessary optimality conditions for EPECs in finite-dimensional spaces based an advanced generalized variational tools of variational analysis. The optimality conditions are derived in normal form under certain qualification requirements, which can be regarded as proper analogs of the classical Mangasarian-Fromovitz constraint qualification in the general settings under consideration.
We discuss a petroleum discovery model that greatly simplifies the approach initiated by Barouch ... more We discuss a petroleum discovery model that greatly simplifies the approach initiated by Barouch and Kaufman (1976) in which exploration is viewed as a sampling without replacement process, and the probability of discovery of a pool is proportional to its size. Calculations that formerly required lengthy Monte Carlo simulations have been reduced to compact formulas.
Abstract This paper proposes a novel schematic approach for coordinating the selection of distrib... more Abstract This paper proposes a novel schematic approach for coordinating the selection of distributed generation unit investment proposals submitted by multiple, competing, private investors to achieve maximum investor participation while complying with the technical ...
Abstract In deregulated electricity sector climates, such as in Ontario, the production of clean ... more Abstract In deregulated electricity sector climates, such as in Ontario, the production of clean or renewable energy by small power producers through distributed generation (DG) is encouraged. This paper examines the policies that can be used to encourage DG ...
This paper presents a stochastic linear programming model that can be used for pricing in electri... more This paper presents a stochastic linear programming model that can be used for pricing in electrical energy and reserve markets. It addresses capacity, energy, and reserve dispatch problems that may arise from n-1 contingency scenarios. Possible market solutions focusing on generator compensation using realtime, day-ahead, and hybrid schemes are enumerated, along with opportunities for consumer pricing and transmission costing. This model is illustrated on a 6-bus test system as well as a larger 66-bus system representing the Ontario network. A key difference among schemes is the degree of risk to the generators, measured by variance in profit. Index Terms-Electricity markets, energy and reserve pricing, 1 contingency criterion, operating reserves, stochastic optimization. NOMENCLATURE The symbols used here follow the economic tradition of using for price and for quantity. A. Sets and Indices Scenario. Base scenario (most probable/ zero contingency). Actual (real-time) scenario. Set of scenarios. Node. Set of nodes in network.
In this paper we present a new Benders decomposition method for solving stochastic complementarit... more In this paper we present a new Benders decomposition method for solving stochastic complementarity problems based on the work by Fuller and Chung
We present a modification to Dantzig-Wolfe decomposition of variational inequality (VI) problems ... more We present a modification to Dantzig-Wolfe decomposition of variational inequality (VI) problems that allows for approximation of the VI mapping in the subproblem. The approximation is parameterized by the most recent master problem solution, and it must satisfy two simple requirements. In an electronic companion (online appendix), we show that the proofs of convergence and other important properties go through with subproblem approximation. The approximation procedure is illustrated by an application to a class of multicommodity economic equilibrium models (MCEEMs): the standard Dantzig-Wolfe decomposition by commodity does not allow the subproblem to be decomposed into separate subproblems for each commodity, but we show two ways to approximate the subproblem's inverse demand function, and both ways allow the subproblem to be broken into separate single-commodity problems. A further approximation is combined with each of the inverse demand approximations; in effect, an approximate supply or demand curve is introduced into each commodity's subproblem for transfers of commodities between different subproblems, thus allowing the subproblems to produce better proposals. An illustration is included for an MCEEM that represents energy markets in Canada.
We present a modification to Dantzig-Wolfe decomposition of variational inequality (VI) problems ... more We present a modification to Dantzig-Wolfe decomposition of variational inequality (VI) problems that allows for approximation of the VI mapping in the subproblem. The approximation is parameterized by the most recent master problem solution, and it must satisfy two simple requirements. In an electronic companion (online appendix), we show that the proofs of convergence and other important properties go through with subproblem approximation. The approximation procedure is illustrated by an application to a class of multicommodity economic equilibrium models (MCEEMs): the standard Dantzig-Wolfe decomposition by commodity does not allow the subproblem to be decomposed into separate subproblems for each commodity, but we show two ways to approximate the subproblem's inverse demand function, and both ways allow the subproblem to be broken into separate single-commodity problems. A further approximation is combined with each of the inverse demand approximations; in effect, an approximate supply or demand curve is introduced into each commodity's subproblem for transfers of commodities between different subproblems, thus allowing the subproblems to produce better proposals. An illustration is included for an MCEEM that represents energy markets in Canada.
ABSTRACT An analysis of the policy implications of Ontario's FITs on overall societal wel... more ABSTRACT An analysis of the policy implications of Ontario's FITs on overall societal welfare suggests that, if unbounded, existing FIT tariffs would have a large negative impact on consumer welfare, with an overall net loss on total social welfare. Negative impacts could be minimized by controlling the quantities.
ABSTRACT Integration of large-scale energy storage systems (ESSs) is desirable nowadays to achiev... more ABSTRACT Integration of large-scale energy storage systems (ESSs) is desirable nowadays to achieve higher reliability and efficiency for smart grids. Controlling ESS operation usually depends on electricity market prices so as to charge when the price is low and discharge when the price is high. On the other hand, the market-clearing price itself is determined based on the net demand, i.e., including energy storage output, at every hour. Therefore, it is crucial to develop a mathematical model to determine the optimal ESS operation as well as the market-clearing prices. The problem is formulated as a mixed complementarity problem (MCP) that allows the representation of special (incentive) prices, which cannot be represented in a single optimization model. The proposed model is useful for power system operators to determine the optimal storage dispatch simultaneously with the market-clearing price in addition to the conventional generation dispatch. The impact of energy storage size and location on market price, total generation cost, energy storage arbitrage benefit, and total consumer payment is further investigated in this paper. The latter analysis provides some guidelines for power system planners to identify the optimal size and location for installing large-scale ESSs.
Complementarity models can represent the simultaneous optimization problems of one or several int... more Complementarity models can represent the simultaneous optimization problems of one or several interacting decision-makers, and thus they have become an increasingly important and powerful tool for formulating and solving bottom-up energy market models. This paper provides an overview of the full range of complementarity-based formulations and how these can be applied to assist the different market participants and organizations with their decision-making processes. To this end, the first part of the paper introduces the mathematical formulation of some basic complementarity models, which are illustrated by highly simplified but illustrative energy market applications. Considering these models, the second part of the paper is devoted to describing in broad terms four areas of their potential application: electricity markets, emission markets, natural gas markets and economies comprising several interacting markets.
International Series in Operations Research & Management Science, 2012
In this chapter, we explore the notions of equilibria and optimization and show how in some cases... more In this chapter, we explore the notions of equilibria and optimization and show how in some cases they are related. The notion of an equilibrium is a fundamental concept that has been used in a variety of disciplines such as economics, engineering, and science to name just a few. At its core, an equilibrium is a state of the system being modeled for which the system has no “incentive” to change. These incentives can be monetary in the case of economics or based on natural forces and scientific laws such as total input equals total output. Some well-known engineering examples include: conservation of energy, conservation of mass, conservation of momentum [8], steady-state probabilities in Markov chains such as birth-and-death processes [53] to name a few. These and other engineering examples are typified by a balancing of forces or conditions so that the state once reached will not easily (if at all) be left.
International Series in Operations Research & Management Science, 2012
In this chapter, we present several advanced algorithms that can be useful for the solution of so... more In this chapter, we present several advanced algorithms that can be useful for the solution of some of the models discussed in this book.
International Series in Operations Research & Management Science, 2012
The purpose of this chapter is to explain variational inequality (VI) formulations of equilibrium... more The purpose of this chapter is to explain variational inequality (VI) formulations of equilibrium problems, and the close connection of a VI problem to an equivalent complementarity problem. There are sometimes advantages to a VI formulation compared to a complementarity formulation: the complementarity formulation has primal decision variables, and dual variables that arise, e.g., when specifying the KKT conditions of individual agents; but a VI formulation has the same primal variables, with few, or no dual variables, which can considerably ease the coding of the model in GAMS. This coding advantage is particularly evident when implementing large complex models, or decomposition algorithms as discussed in Chapter 9. However, the derivation of a complementarity model formulation is usually easier than the derivation of an equivalent VI model: e.g., many complementarity models in this book are derived by writing down the KKT conditions of the agents, together with market-clearing conditions, but the procedure to write down a VI model with few or no dual variables is not as easily stated. In this chapter, we alleviate this difficulty by showing how to arrive quickly at the formulation of a VI model for a large class of Nash equilibrium and generalized Nash equilibrium settings.
This chapter presents some of the key ideas in various algorithms that are used to solve some of ... more This chapter presents some of the key ideas in various algorithms that are used to solve some of the basic equilibrium models, namely, models in the form of the linear complementarity problem (LCP), the nonlinear complementarity problem (NCP) and the variational inequality (VI) problem. We explain the main ideas of the algorithms, but avoid detailed discussions or proofs of important mathematical issues such as the conditions under which an algorithm is guaranteed to converge.
International Series in Operations Research & Management Science, 2012
Natural gas is a key fuel in energy markets worldwide. It is produced from either onshore or offs... more Natural gas is a key fuel in energy markets worldwide. It is produced from either onshore or offshore wells, processed to remove impurities, and then transported by either pipeline in gaseous form or cooled to about -260 degrees F (about -160 degrees C) and then transported as liquefied natural gas (LNG) to destinations around the world. The main consuming sectors that use it are residential, commercial, industrial, electric power, and to some extent transportation. At present, the world has abundant gas supplies. According to [52], the global mean projected remaining recoverable resources is 16,200 trillion cubic feet (Tcf) or 150 times the current annual global consumption. About 9,000 Tcf is gauged to be economically available at less than or equal to $4 per million British Thermal Units (Btu) [52].
International Series in Operations Research & Management Science, 2012
Several of the models previously introduced in this book have focused on the market for a single ... more Several of the models previously introduced in this book have focused on the market for a single commodity with a single price, such as power at a particular location in a particular hour. However, many of this book’s models instead considered several markets simultaneously, recognizing that linkages among them imply that equilibrium prices in one market cannot be calculated without considering how they affect, and are affected by, prices in other markets. In the earlier chapters, linkages among markets were mainly through the supply-side, in which the cost of providing commodity in one market depends in part on prices in other markets. For instance, a power generator with only a small amount of capacity with low running costs might experience a rise in its marginal cost of serving one part of the network if it also sells a lot of power elsewhere, thereby exhausting its cheap capacity. The purpose of this chapter is to introduce the modeling of multiple energy markets in which it is instead the behavior of consumers that links the markets. In particular, the amount that final consumers buy of one commodity affects how much they are willing to pay for other commodities.
International Series in Operations Research & Management Science, 2012
The purpose of this chapter is to provide a more in-depth exploration of applications of compleme... more The purpose of this chapter is to provide a more in-depth exploration of applications of complementarity models to electricity markets. In doing so, we introduce two crucial features of energy markets. The first is transportation networks with capacity limits on links between different markets. The second is environmental restrictions, such as emissions markets. We address these in turn by building, analyzing, and solving models for electric power markets that incorporate these features.
International Series in Operations Research & Management Science, 2012
This chapter provides a friendly introduction to several mathematical structures used in the foll... more This chapter provides a friendly introduction to several mathematical structures used in the following chapters. These structures are useful to describe the functioning of markets and the behavior of market agents. Throughout the chapter clarity and simplicity are emphasized.
International Series in Operations Research & Management Science, 2012
In this chapter, we explain some useful principles of microeconomics for those readers with littl... more In this chapter, we explain some useful principles of microeconomics for those readers with little or no background in the subject. Readers who have studied microeconomics may also benefit from this chapter, as we show how to construct several different kinds of models of markets, using optimization and complementarity techniques.
The paper concerns a new class of optimization-related problems called Equilibrium Problems with ... more The paper concerns a new class of optimization-related problems called Equilibrium Problems with Equilibrium Constraints (EPECs). One may treat them as two level hierarchical problems, which involve equilibria at both lower and upper levels. Such problems naturally appear in various applications providing an equilibrium counterpart (at the upper level) of Mathematical Programs with Equilibrium Constraints (MPECs). We develop a unified approach to both EPECs and MPECs from the viewpoint of multiobjective optimization subject to equilibrium constraints. The problems of this type are intrinsically nonsmooth and require the use of generalized differentiation for their analysis and applications. This paper presents necessary optimality conditions for EPECs in finite-dimensional spaces based an advanced generalized variational tools of variational analysis. The optimality conditions are derived in normal form under certain qualification requirements, which can be regarded as proper analogs of the classical Mangasarian-Fromovitz constraint qualification in the general settings under consideration.
We discuss a petroleum discovery model that greatly simplifies the approach initiated by Barouch ... more We discuss a petroleum discovery model that greatly simplifies the approach initiated by Barouch and Kaufman (1976) in which exploration is viewed as a sampling without replacement process, and the probability of discovery of a pool is proportional to its size. Calculations that formerly required lengthy Monte Carlo simulations have been reduced to compact formulas.
Abstract This paper proposes a novel schematic approach for coordinating the selection of distrib... more Abstract This paper proposes a novel schematic approach for coordinating the selection of distributed generation unit investment proposals submitted by multiple, competing, private investors to achieve maximum investor participation while complying with the technical ...
Abstract In deregulated electricity sector climates, such as in Ontario, the production of clean ... more Abstract In deregulated electricity sector climates, such as in Ontario, the production of clean or renewable energy by small power producers through distributed generation (DG) is encouraged. This paper examines the policies that can be used to encourage DG ...
This paper presents a stochastic linear programming model that can be used for pricing in electri... more This paper presents a stochastic linear programming model that can be used for pricing in electrical energy and reserve markets. It addresses capacity, energy, and reserve dispatch problems that may arise from n-1 contingency scenarios. Possible market solutions focusing on generator compensation using realtime, day-ahead, and hybrid schemes are enumerated, along with opportunities for consumer pricing and transmission costing. This model is illustrated on a 6-bus test system as well as a larger 66-bus system representing the Ontario network. A key difference among schemes is the degree of risk to the generators, measured by variance in profit. Index Terms-Electricity markets, energy and reserve pricing, 1 contingency criterion, operating reserves, stochastic optimization. NOMENCLATURE The symbols used here follow the economic tradition of using for price and for quantity. A. Sets and Indices Scenario. Base scenario (most probable/ zero contingency). Actual (real-time) scenario. Set of scenarios. Node. Set of nodes in network.
In this paper we present a new Benders decomposition method for solving stochastic complementarit... more In this paper we present a new Benders decomposition method for solving stochastic complementarity problems based on the work by Fuller and Chung
We present a modification to Dantzig-Wolfe decomposition of variational inequality (VI) problems ... more We present a modification to Dantzig-Wolfe decomposition of variational inequality (VI) problems that allows for approximation of the VI mapping in the subproblem. The approximation is parameterized by the most recent master problem solution, and it must satisfy two simple requirements. In an electronic companion (online appendix), we show that the proofs of convergence and other important properties go through with subproblem approximation. The approximation procedure is illustrated by an application to a class of multicommodity economic equilibrium models (MCEEMs): the standard Dantzig-Wolfe decomposition by commodity does not allow the subproblem to be decomposed into separate subproblems for each commodity, but we show two ways to approximate the subproblem's inverse demand function, and both ways allow the subproblem to be broken into separate single-commodity problems. A further approximation is combined with each of the inverse demand approximations; in effect, an approximate supply or demand curve is introduced into each commodity's subproblem for transfers of commodities between different subproblems, thus allowing the subproblems to produce better proposals. An illustration is included for an MCEEM that represents energy markets in Canada.
We present a modification to Dantzig-Wolfe decomposition of variational inequality (VI) problems ... more We present a modification to Dantzig-Wolfe decomposition of variational inequality (VI) problems that allows for approximation of the VI mapping in the subproblem. The approximation is parameterized by the most recent master problem solution, and it must satisfy two simple requirements. In an electronic companion (online appendix), we show that the proofs of convergence and other important properties go through with subproblem approximation. The approximation procedure is illustrated by an application to a class of multicommodity economic equilibrium models (MCEEMs): the standard Dantzig-Wolfe decomposition by commodity does not allow the subproblem to be decomposed into separate subproblems for each commodity, but we show two ways to approximate the subproblem's inverse demand function, and both ways allow the subproblem to be broken into separate single-commodity problems. A further approximation is combined with each of the inverse demand approximations; in effect, an approximate supply or demand curve is introduced into each commodity's subproblem for transfers of commodities between different subproblems, thus allowing the subproblems to produce better proposals. An illustration is included for an MCEEM that represents energy markets in Canada.
ABSTRACT An analysis of the policy implications of Ontario's FITs on overall societal wel... more ABSTRACT An analysis of the policy implications of Ontario's FITs on overall societal welfare suggests that, if unbounded, existing FIT tariffs would have a large negative impact on consumer welfare, with an overall net loss on total social welfare. Negative impacts could be minimized by controlling the quantities.
ABSTRACT Integration of large-scale energy storage systems (ESSs) is desirable nowadays to achiev... more ABSTRACT Integration of large-scale energy storage systems (ESSs) is desirable nowadays to achieve higher reliability and efficiency for smart grids. Controlling ESS operation usually depends on electricity market prices so as to charge when the price is low and discharge when the price is high. On the other hand, the market-clearing price itself is determined based on the net demand, i.e., including energy storage output, at every hour. Therefore, it is crucial to develop a mathematical model to determine the optimal ESS operation as well as the market-clearing prices. The problem is formulated as a mixed complementarity problem (MCP) that allows the representation of special (incentive) prices, which cannot be represented in a single optimization model. The proposed model is useful for power system operators to determine the optimal storage dispatch simultaneously with the market-clearing price in addition to the conventional generation dispatch. The impact of energy storage size and location on market price, total generation cost, energy storage arbitrage benefit, and total consumer payment is further investigated in this paper. The latter analysis provides some guidelines for power system planners to identify the optimal size and location for installing large-scale ESSs.
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Papers by David Fuller