Equilibrium Problem With Equilibrium Constraints Research Articles

Pool equilibria including strategic storage

With a rapid increase in capacity, independently-invested energy storage systems (ESSs) might take up a significant share in the generation mix, and would be required to participate in the electricity market. Considering the arbitrage capability of ESSs, their behaviors would be remarkably different from those of conventional generators, impacting the market equilibrium. These impacts would vary with the type of ESSs and the generation mix. Therefore, this paper formulates a multi-period market equilibrium problem with equilibrium constraints (EPEC) to study the strategic behaviors of different types of ESSs and their impacts on the market outcomes, assuming that ESSs behave as price-makers. The EPEC model is established within a general framework, which considers the individual profit-maximization behaviors of different ESSs and generators, including thermal units, hydro units and renewable units. Interactions between different generators and the market operator are also represented. Finally, numerical studies based on a modified IEEE 57-node system with different wind generation curves are performed for illustration and validation.

A bi-level approach for optimal contract pricing of independent dispatchable DG units in distribution networks

Distributed Generation (DG) units are increasingly installed in the power systems. Distribution Companies (DisCo) can opt to purchase the electricity from DG in an energy purchase contract to supply the customer demand and reduce energy loss. This paper proposes a framework for optimal contract pricing of independent dispatchable DG units considering competition among them. While DG units tend to increase their profit from the energy purchase contract, DisCo minimizes the demand supply cost. Multi-leader follower game theory concept is used to analyze the situation in which competing DG units offer the energy price to DisCo and DisCo determines the DG generation. A bi-level approach is used to formulate the competition in which each DG problem is the upper-level problem and the DisCo problem is considered as the lower-level one. Combining the optimality conditions ofall upper-level problems with the lower level problem results in a multi-DG equilibrium problem formulated as an equilibrium problem with equilibrium constraints (EPEC). Using a nonlinear approach, the EPEC problem is reformulated as a single nonlinear optimization model which is simultaneously solved for all independent DG units. The proposed framework was applied to the Modified IEEE 34-Bus Distribution Test System. Performance and robustness of the proposed framework in determining econo-technically fare DG contract price has been demonstrated through a series of analyses.

Pricing Bilateral Electricity Trade between Smart Grids and Hybrid Green Datacenters

Datacenter demand response is envisioned as a promising approach for mitigating operational instability faced by smart grids. It enables significant potentials in peak load shedding and facilitates the incorporation of distributed generation and intermittent energy sources. This work considers two key aspects towards realtime electricity pricing for eliciting demand response: (i) Two-way electricity flow between smart grids and large datacenters with hybrid green generation capabilities. (ii) The geo-distributed nature of large cloud systems, and hence the potential competition among smart grids that serve different datacenters of the cloud. We propose a pricing scheme tailored for geo-distributed green datacenters, from a multi-leader single-follower game point of view. At the cloud side, in quest for performance, scalability and robustness, the energy cost is minimized in a distributed manner, based on the technique of alternating direction of multipliers (ADMM). At the smart grid side, a practical equilibrium of the pricing game is desired. To this end, we employ mathematical programming with equilibrium constraints (MPEC), equilibrium problem with equilibrium constraints (EPEC) and exact linearization, to transform the multi-leader single-follower pricing game into a mixed integer linear program (MILP) that can be readily solved. The effectiveness of the proposed solutions is evaluated based on trace-driven simulations.

Cordon toll competition in a network of two cities: Formulation and sensitivity to traveller route and demand responses

While there exists extensive literature on the first- and second-best tolling of congested transportation networks, much of it presumes the existence of a single agent responsible for toll-setting. The present paper extends the small but growing body of work studying the impact of several agents independently regulating tolls on different parts of a network. Specifically we consider the problem of a network consisting of two ‘cities’, each city independently regulated by a city ‘authority’ able to set a single cordon toll for entry to the city. It is supposed that each authority aims to maximise the social welfare of its own residents, anticipating the impact of its toll on travellers’ route and demand decisions, while reacting to the toll level levied by the other authority. In addition, we model the possibility of the cities entering into a ‘tax-exporting agreement’, in which city A agrees to share with city B the toll revenues it collects from city B residents using city A’s network. It is assumed that the sensitivity of travellers, in terms of their route and demand responses, is captured by an elastic demand, Stochastic User Equilibrium (SUE) model. Conditions for a Nash Equilibrium (NE) between cities are set out as an Equilibrium Problem with Equilibrium Constraints (EPEC). It is shown that weaker, ‘local’ solutions to the EPEC (which we term LNE for local NE) satisfy a single variational inequality, using the smooth implicit function of the SUE map. Standard variational algorithms may then be used to identify such LNE solutions, allowing NE solutions to be identified from this candidate set; we test the use of a Sequential Linear Complementarity Problem algorithm. Numerical results are reported in which we see that the sensitivity of travellers may affect many factors, including: the number of LNE solutions, the initial conditions for which algorithms might determine such solutions, the gap between LNE and a global regulator solution, and the incentive for cities to cooperate in terms of tax-exporting.

Exercise of Market Power on Ramp Rate in Wind-Integrated Power Systems

With an increasing penetration of wind power, there is likely to be an increasing need for fast-ramping generating units. These generators ensure that no load is lost if supply drops due to the uncertainties in wind power generation. However, it is observed in practice that, in a presence of network constraints, fast-ramping generating units are prone to act strategically and exercise market power by withholding their ramp rates. In this paper we model this gaming behavior on ramp rates. We assume a market operator who collects bids in form of marginal costs, quantities, and ramp rates. He runs a ramp-constrained economic dispatch given the generators' bids, forecasted demand, and contingencies. Following the game-theoretic concepts, we set up a multi-level optimization problem. The lower-level problem is the ramp-constrained economic dispatch and the higher-level represents the profit maximization problems solved by strategic generators. The whole problem is formulated as an equilibrium problem with equilibrium constraints (EPEC). The outcome of the EPEC problem is a set of Nash equilibria. To tackle the multiple Nash equilibria problem, the concept of the extremal-Nash equilibria is defined and formulated. We model the concept of extremal-Nash equilibria as a single-stage mixed-integer linear programming problem (MILP) and demonstrate the application of this mathematical framework on an illustrative case and on a more realistic case study with tractable results.

Equilibria in an Oligopolistic Market With Wind Power Production

This paper proposes an approach for analyzing the impacts of large-scale wind power integration on electricity market equilibria. A pool-based oligopolistic electricity market is considered including a day-ahead market and a number of real-time markets. Wind power is considered within the generation portfolio of the strategic producers, and the uncertainty of wind power production is modeled through a set of plausible scenarios. The strategic behavior of each producer is modeled through a stochastic bilevel model. The resulting nonlinear equilibrium problem with equilibrium constraints (EPEC) is linearized and then solved. Numerical results for a test case with increasing levels of the wind power penetration is provided.

A Tri-Level Model of Centralized Transmission and Decentralized Generation Expansion Planning for an Electricity Market—Part I

We develop a tri-level model of transmission and generation expansion planning in a deregulated power market environment. Due to long planning/construction lead times and concerns for network reliability, transmission expansion is considered in the top level as a centralized decision. In the second level, multiple decentralized GENCOs make their own capacity expansion decisions while anticipating a wholesale electricity market equilibrium in the third level. The collection of bi-level games in the lower two levels forms an equilibrium problem with equilibrium constraints (EPEC) that can be approached by either the diagonalization method (DM) or a complementarity problem (CP) reformulation. We propose a hybrid iterative solution algorithm that combines a CP reformulation of the tri-level problem and DM solutions of the EPEC sub-problem.

A Tri-Level Model of Centralized Transmission and Decentralized Generation Expansion Planning for an Electricity Market—Part II

We study a tri-level integrated transmission and generation expansion planning problem in a deregulated power market environment. The collection of bi-level sub-problems in the lower two levels is an equilibrium problem with equilibrium constraints (EPEC) that can be approached by either the diagonalization method (DM) or a complementarity problem (CP) reformulation. This paper is a continuation of its Part I, in which a hybrid iterative algorithm is proposed to solve the tri-level problem by iteratively applying the CP reformulation of the tri-level problem to propose solutions and evaluating them in the EPEC sub-problem by DM. It focuses on the numerical results obtained by the hybrid algorithm for a 6-bus system, a modified IEEE 30-bus system, and an IEEE 118-bus system. In the numerical instances, the (approximate) Nash equilibrium point for the sub-problem can be verified by examining local concavity.

Competition between two cities using cordon tolls: an exploration of response surfaces and possibilities for collusion

Extending the literature on competition in the presence of cordon tolls, this article explores the implications of competition between two cities. The city authorities are assumed to maximise the welfare of their own residents whilst taking advantage of tax export mechanisms by charging traffic from the competing authority. The problem is posed as an Equilibrium Problem with Equilibrium Constraints (EPEC) which is a special form of a Nash game with a hierarchical structure. Employing a grid search to explore the response surfaces and to determine the Nash Equilibrium toll levels, we demonstrate the possibility that there may exist multiple local Nash solutions and that competition may lead to a sub-optimal outcome for one or both authorities depending on whether there exists a stronger player. We then consider the impact of elasticity of demand and other parameter assumptions on the potential number of Nash solutions. Finally incentives for collusion are studied.

Strategic Generation Capacity Choice Under Demand Uncertainty: Analysis of Nash Equilibria in Electricity Markets

Abstract: We analyze a two-stage game of strategic firms facing uncertain demand and exerting market power in decentralized electricity markets. These firms choose their generation capacities at the first stage while anticipating a perfectly competitive future electricity spot market outcome at the second stage; thus it is a closed loop game. In general, such games can be formulated as an equilibrium problem with equilibrium constraints (EPEC) and examples have been posed in the literature that have multiple or no equilibria. Therefore, it is of interest to define general sets of conditions under which solutions exist and are unique, which would enhance the value of such models for policy andmarket intelligence purposes. In this paper, we consider various types of such a closed loop model regarding the underlying price-demand relations (elastic and inelastic demand), the assumed demand uncertainty with a broad class of continuous distributions, and any finite number of players with symmetric or asymmetric costs. We establish sufficient conditions for the random demand’s probability distribution which guarantee existence and uniqueness of equilibria in most of the cases of this closed loop model. We identify a broad class of commonly used continuous probability distributions satisfying these conditions.

Generation Investment Equilibria With Strategic Producers—Part I: Formulation

The first of this two-paper series proposes a methodology to characterize generation investment equilibria in a pool-based network-constrained electricity market, where the producers behave strategically. To this end, the investment problem of each strategic producer is represented using a bilevel model, whose upper-level problem determines the optimal investment and the supply offering curves to maximize its profit, and whose several lower-level problems represent different market clearing scenarios. This model is transformed into a mathematical program with equilibrium constraint (MPEC) through replacing the lower-level problems by their optimality conditions. The joint consideration of all producer MPECs, one per producer, constitutes an equilibrium problem with equilibrium constraints (EPEC). To identify the solutions of this EPEC, each MPEC problem is replaced by its Karush-Kuhn-Tucker (KKT) conditions, which are in turn linearized. The resulting mixed-integer linear system of equalities and inequalities allows determining the EPEC equilibria through an auxiliary MILP problem.

Medium-Term Power Dispatch in Predominantly Hydro Systems: An Equilibrium Approach

This paper proposes an alternative formulation for the problem of dispatching generation units in a predominantly hydro system considering a medium-term horizon. Hydro-power dispatch is expressed as a equilibrium problem with equilibrium constraints (EPEC) and its solutions are compared to the solutions of a classical optimal hydro dispatch problem. A stochastic formulation is used to represent uncertainties in water inflows. Results are presented for realistic test systems with interconnected reservoirs.

An EPEC approach to the yearly maintenance scheduling of generating units

This paper considers the yearly maintenance scheduling of generating units within a market environment. Each producer schedules its units' maintenance periods to maximize its revenue using a bilevel approach. The upper-level problem of this bilevel model seeks maximum revenue and contains unit scheduling constraints, while the lower-level problems represent the market clearing process under different operating conditions. This single producer maintenance problem can be recast as a mathematical program with equilibrium constraints (MPEC). Since the MPECs of all producers have to be considered simultaneously and the market clearing process is common to all of them, the proposed formulation for maintenance scheduling is an equilibrium problem with equilibrium constraints (EPEC) corresponding to a multiple-leader-common-follower game. The solution of this EPEC is a set of equilibria, in which none of the producers is able to increase its revenue unilaterally by changing the maintenance periods of its generating units.

A Bilevel Approach to Operational Decision Making of a Distribution Company in Competitive Environments

This paper presents a model for operational decision making of a distribution company (disco) with distributed generation (DG) and interruptible loads (IL) in a competitive market. The disco objective, modeled through the upper-level problem, is to minimize the cost of market purchases and DG unit dispatch while taking into account the responses of the other discos. This upper-level problem is constrained by a lower-level market clearing problem whose objective corresponds to maximization of social welfare. Such a setting results in a multi-disco equilibrium problem formulated as an equilibrium problem with equilibrium constraints (EPEC) by combining the optimality conditions of all upper-level problems. Using a nonlinear approach, the EPEC problem is reformulated as a single nonlinear optimization model which is simultaneously solved for all discos. The proposed approach is applied to an ac model of a power system to account for voltage and reactive power constraints of the transmission and distribution networks. Numerical examples based on two test systems are presented to illustrate the application of the proposed method.

Equilibria in an Oligopolistic Electricity Pool With Stepwise Offer Curves

We study the equilibria reached by strategic producers in a pool-based network-constrained electricity market. The behavior of each producer is modeled by a mathematical program with equilibrium constraints (MPEC) whose objective is maximizing profit and whose complementarity constraints describe market clearing. The joint solution of all these MPECs constitutes an equilibrium problem with equilibrium constraints (EPEC). The equilibria associated with the EPEC are analyzed by solving the strong stationarity conditions of all MPECs, which can be linearized without approximation by mixed-integer linear programming (MILP) techniques. The resulting mixed-integer linear conditions can be reformulated as an optimization problem that allows establishing diverse objectives to differentiate among alternative equilibria.

Smoothing approach to Nash equilibrium formulations for a class of equilibrium problems with shared complementarity constraints

The equilibrium problem with equilibrium constraints (EPEC) can be looked on as a generalization of Nash equilibrium problem (NEP) and the mathematical program with equilibrium constraints (MPEC) whose constraints contain a parametric variational inequality or complementarity system. In this paper, we particularly consider a special class of EPECs where a common parametric P-matrix linear complementarity system is contained in all players' strategy sets. After reformulating the EPEC as an equivalent nonsmooth NEP, we use a smoothing method to construct a sequence of smoothed NEPs that approximate the original problem. We consider two solution concepts, global Nash equilibrium and stationary Nash equilibrium, and establish some results about the convergence of approximate Nash equilibria. Moreover we show some illustrative numerical examples.

Analysis of M-stationary points to an EPEC modeling oligopolistic competition in an electricity spot market

We consider an equilibrium problem with equilibrium constraints (EPEC) arising from the modeling of competition in an electricity spot market (under ISO regulation). For a characterization of equilibrium solutions, so-called M-stationarity conditions are derived. This first requires a structural analysis of the problem, e.g., verifying constraint qualifications. Second, the calmness property of a certain multifunction has to be verified in order to justify using M-stationarity conditions. Third, for stating the stationarity conditions, the coderivative of a normal cone mapping has to be calculated. Finally, the obtained necessary conditions are made fully explicit in terms of the problem data for one typical constellation. A simple two-settlement example serves as an illustration.

Modeling and analysis of strategic forward contracting in transmission constrained power markets

Taking the effects of transmission network into account, strategic forward contracting induced by the interaction of generation firms’ strategies in the spot and forward markets is investigated. A two-stage game model is proposed to describe generation firms’ strategic forward contracting and spot market competition. In the spot market, generation firms behave strategically by submitting bids at their nodes in a form of linear supply function (LSF) and there are arbitrageurs who buy and resell power at different nodes where price differences exceed the costs of transmission. The owner of the grid is assumed to ration limited transmission line capacity to maximize the value of the transmission services in the spot market. The Cournot-type competition is assumed for the strategic forward contract market. This two-stage model is formulated as an equilibrium problem with equilibrium constraints (EPEC); in which each firm's optimization problem in the forward market is a mathematical program with equilibrium constraints (MPEC) and parameter-dependent spot market equilibrium as the inner problem. A nonlinear complementarity method is employed to solve this EPEC model.

Two-settlement electricity markets with price caps and Cournot generation firms

We compare two alternative mechanisms for capping prices in two-settlement electricity markets. With sufficient lead time, forward market prices are implicitly capped by competitive pressure of potential entry that will occur when forward prices rise above some backstop price. Another more direct approach is to cap spot prices through a regulatory intervention. In this paper we explore the implications of these two alternative mechanisms in a two-settlement Cournot equilibrium framework. We formulate the market equilibrium as a stochastic equilibrium problem with equilibrium constraints (EPEC) capturing congestion effects, probabilistic contingencies and horizontal market power. As an illustrative test case, we use the 53-bus Belgian electricity network with representative generator costs but hypothetical demand and ownership structure. Compared to a price-uncapped two-settlement system, a forward cap increases firms’ incentives for forward contracting, whereas a spot cap reduces such incentives. Moreover, in both cases, more forward contracts are committed as the generation resource ownership structure becomes more diversified.

Modeling and Computing Two-Settlement Oligopolistic Equilibrium in a Congested Electricity Network

A model of two-settlement electricity markets is introduced, which accounts for flow congestion, demand uncertainty, system contingencies, and market power. We formulate the subgame perfect Nash equilibrium for this model as an equilibrium problem with equilibrium constraints (EPEC), in which each firm solves a mathematical program with equilibrium constraints (MPEC). The model assumes linear demand functions, quadratic generation cost functions, and a lossless DC network, resulting in equilibrium constraints as a parametric linear complementarity problem (LCP). We introduce an iterative procedure for solving this EPEC through repeated application of an MPEC algorithm. This MPEC algorithm is based on solving quadratic programming subproblems and on parametric LCP pivoting. Numerical examples demonstrate the effectiveness of the MPEC and EPEC algorithms and the tractability of the model for realistic-size power systems.