Equilibrium Problem With Equilibrium Constraints Research Articles

Decarbonizing Maritime Transport through Green Fuel-Powered Vessel Retrofitting: A Game-Theoretic Approach

Addressing the urgent global challenge of man-made greenhouse gas emissions and climate change necessitates collaborative action between shipping lines and government regulatory agencies. Aligning with the International Maritime Organization’s emissions reduction strategy, this paper presents a novel bi-level programming model that unifies these stakeholders. On the upper level of the proposed bi-level model, a number of shipping lines optimize retrofitting plans for their vessels to maximize economic benefits. On the lower level, the regulatory agency responds to the carbon reduction efforts by setting retrofitting subsidies and emission penalty rates. This framework represents a multi-leader–single-follower game involving shipping lines and the regulatory agency, and its equilibrium is determined through an equilibrium problem with equilibrium constraints (EPEC). The EPEC comprises multiple single-leader–follower problems, each of which can be formulated as a mathematical program with equilibrium constraints (MPEC). The diagonalization algorithm (DM) is employed for its solution. Simulation studies performed based on a ten-year planning period show that the proposed approach can effectively promote vessel retrofitting and the use of green fuels, which leads to an annual emission reduction of over 50%.

Strategic trading equilibria in local community considering co-existence of pool-based and peer-to-peer energy markets with free-rider problem in network usage cost

The peer-to-peer (P2P) energy trading market is considered a promising method for allocating prosumers’ locally produced power in smart buildings. However, the existing literature overlooks the coexistence of P2P and pool-based energy markets. Therefore, this paper establishes a bi-level energy market model that incorporates both P2P and pool-based markets. The prosumers’ individual profit maximization model considering their network usage costs in the P2P energy trading network at the upper level is first developed. A hybrid market-clearing framework determining the optimal trading amounts with the prosumers’ strategic offers at the lower level is then exploited. The distribution network power flow model in the market clearing framework is then convexified. The lower-level problem is replaced with its strong duality conditions. This bi-level problem can be reformulated as an equilibrium problem with equilibrium constraints (EPEC). This EPEC is then solved after being transformed into an equivalent mixed integer linear programming (MILP) model. Finally, a fifteen-bus system is employed to illustrate the characteristics of the hybrid market equilibrium depicted by this EPEC model. The influences of the prosumers’ strategic offers on the distribution locational marginal prices (DLMPs) and network usage costs are highlighted. Since the prosumers can choose to transmit their energy in the pool-based or P2P markets, the emission and pollution costs caused by power transmission can be minimized. A cleaner energy production is thus guaranteed. The free-rider problem is verified by comparing the network usage charges in the strategic and perfect competition scenarios.

Electricity market equilibria analysis on the value of demand-side flexibility portfolios’ mix and the strategic demand aggregators’ market power

Electricity market equilibrium analysis is becoming increasingly important within the today’s liberalized electricity market and regulatory context for both market participants and policy makers. The former ones want to make informed business decisions to optimize their market position, while the latter need to promote efficient equilibria that satisfy both supply and demand-side requirements and optimize social welfare. In this paper, we focus on modeling the strategic demand aggregators’ (DA) behavior via an Equilibrium Problem with Equilibrium Constraints (EPEC) formulation. We model typical demand flexibility portfolios that comprise of several types of distributed flexibility assets and renewable energy resources. We then quantify the value of demand flexibility portfolios’ mix and respective value of market power that a strategic DA may exercise with respect to each DA’s cost decrease and social welfare as a function of time for a day-ahead market use case. Simulation results show that: (i) there is an optimal equilibrium point where overall DAs’ costs decrease, while social welfare’s deteriorates negligibly, (ii) there may be a trade-off between the rate of a certain DA’s cost decrease and the rate of market power (or else % share of total demand flexibility) that this DA possesses, so the strategic DA should take this into account in order to find its optimal CAPEX/OPEX balance for its portfolio, (iii) competition among strategic DAs at the demand-side can mitigate grid congestion problems and serve as a counter-balance to the strategic behavior of the supply-side market participants.

Micro-grid tri-level EPEC based model for finding equilibria in the day-ahead and balancing markets

This paper presents a model that explores the strategic bidding equilibrium within a microgrid and its interactions with other strategic and non-strategic rivals in a joint energy and balancing market. The model utilizes a tri-level mathematical program with equilibrium constraints (MPEC) to represent the behavior of each strategic producer. Upper level maximizes the profit of each strategic producer, including microgrid and another strategic rival. The first lower-level problem involves maximizing social welfare through the day-ahead market clearing process, while the second lower-level problem deals with the balancing market clearing process. These objectives form the core of the proposed model. To simplify the tri-level problem, duality theory and the Karush-Kuhn-Tucker (KKT) optimally conditions are employed to transform it into a mixed integer linear programming (MILP) problem. By simultaneously solving all MPECs, an equilibrium problem with equilibrium constraints (EPEC) is formulated. The resulting EPEC is then addressed using a diagonalization algorithm and game theory to obtain a market Nash equilibrium, which constitutes the secondary objective of this paper. The effectiveness of the model is evaluated using the 6-bus test system as a case study. The results indicate that, at the equilibrium point, both the microgrid and rivals experience reduced profits compared to the initial state.

Market equilibrium with strategic pricing and strategic constraints in renewable energy: the role of private energy storage

With the increasing prevalence of renewable energy (RE) companies equipped with private energy storage (ES) systems, a dual capability emerges to offer strategic pricing and strategic constraints in market competition. Specifically, these RE companies can strategically leverage their own private ESs to modulate the variability of RE output limits and introduce modified constraints within the market. To examine these new strategic behaviors and the resulting market equilibria, we introduce an innovative bilevel strategic behavior model. The upper level of the model delineates the strategy for RE profit maximization through the imposition of strategic constraints and pricing schemes, while the lower level calculates the revenue outcomes for all entities in the day-ahead energy market clearing. The integration of the bilevel models from all strategic entities leads to the formulation of a new equilibrium problem with equilibrium constraints (EPEC), the solution of which indicates a novel market equilibrium. The impacts of these market equilibria on critical system operation metrics are then evaluated across two representative market mechanisms. Our numerical experiments reveal that RE exhibits low sensitivity to the private ES’s cost, suggesting that the behavior of imposing strategic constraints may be widespread among RE companies owning private ESs. Furthermore, the introduction of strategic constraints enhances the competitiveness of RE, significantly affecting social welfare, energy pricing, and RE integration rate. The study concludes with insights that could inform practical market transactions and system operations.

The impact of market design and clean energy incentives on strategic generation investments and resource adequacy in low-carbon electricity markets

Well-designed electricity markets play a crucial role in maintaining reliable electric power systems, which are critical in modern society. This study examines the impact of different electricity market designs and clean energy incentive schemes on supporting renewable energy integration and achieving clean energy goals. To this end, we utilize a game-theoretical generation expansion planning model where generation companies make investment and retirement decisions to maximize their expected profit. The model is structured as an equilibrium problem with equilibrium constraints (EPEC) and solved using a diagonalization approach combined with progressive hedging. We analyze three types of electricity market designs: an energy-only market, a capacity market, and a clean energy market, and consider a wide range of market parameters resulting in 14 total scenarios. Wind and solar capacity comprise the majority of new investments in all considered scenarios, but the resultant system planning reserve margin (PRM) can differ significantly depending on market parameters. We also find that profit-driven investments lead to lower PRMs than a traditional system cost minimization approach. These individual scenario results further demonstrate how different market designs and clean energy incentive schemes may influence investor decision-making and impact resource adequacy throughout the clean energy transition.

Stochastic multi‐stage joint expansion planning of transmission system and energy hubs in the presence of correlated uncertainties

AbstractDeployment of Energy Hubs (EHs) across the power grid can alleviate the Transmission System (TS) capacity and substitute the conventional fossil fuel‐based thermal units. Therefore, this paper presents a tri‐level multi‐stage Joint Expansion Planning of the Transmission system and EHs (JEPT&EHs). In this approach, the Cholesky decomposition technique combined with the Nataf transformation is applied to make the uncertain input parameters correlated. Then, the k‐means data‐clustering method is employed to reduce the initial correlated samples. In the first level, the Transmission System Operator (TSO) optimizes the planning and scheduling strategies associated with the TS capacity requirements and operation costs of the conventional generators. In the second level, the financers specify the expansion of the EHs based on the Locational Marginal Prices (LMPs). In the third level, the Direct Current Optimal Power Flow (DCOPF) is determined to update the LMPs by the Independent System Operator (ISO). The optimization problem is an Equilibrium Problem with Equilibrium Constraints (EPEC) since there are multiple financers across the TS. The proposed model is implemented on the IEEE standard 30 bus TS to present the effectiveness of the EHs' deployment and the impact of the correlations in the total costs of the TSO and financers.

Multi-Microgrids Operation With Interruptible Loads in Local Energy and Reserve Markets

In the evolution of the power systems, a particular case is the presence of a number of microgrids (MGs) operated with mutual interconnection, but without connection to the main distribution system. The interconnected MGs form a structure in which the overall system operation and resource scheduling can be determined by considering centralized or decentralized approaches. This article introduces local energy and reserve markets (LERMs) in which the MG managers (MGMs) can meet their required energy and reserve with optimal scheduling of their resources, besides competing with the other MGs. To model such decision-making framework for MGMs, a bilevel optimization approach is developed in which the MGMs’ problem is modeled as the upper level problem and the LERMs clearing problem is modeled as the lower level problem. This model is transformed into a mathematical programming with equilibrium constraints (MPEC) using the primal-dual transformation. Then, the resulting MPEC for each MG is replaced with its Karush–Kuhn–Tucker conditions, obtaining an equilibrium problem with equilibrium constraints (EPEC) model. The nonlinear terms of the model are linearized through different approaches. Finally, the EPEC model is transformed into a mixed-integer linear problem considering the objective function of all MGMs. The model is applied to a test system with three interconnected MGs. Moreover, the sensitivity of the results to the probability of calling reserve is investigated.

A bilevel game-theoretic decision-making framework for strategic retailers in both local and wholesale electricity markets

This paper proposes a bilevel game-theoretic model for multiple strategic retailers participating in both wholesale and local electricity markets while considering customers’ switching behaviors. At the upper level, each retailer maximizes its own profit by making optimal pricing decisions in the retail market and bidding decisions in the day-ahead wholesale (DAW) and local power exchange (LPE) markets. The interaction among multiple strategic retailers is formulated using the Bertrand competition model. For the lower level, there are three optimization problems. First, the welfare maximization problem is formulated for customers to model their switching behaviors among different retailers. Second, a market-clearing problem is formulated for the independent system operator (ISO) in the DAW market. Third, a novel LPE market is developed for retailers to facilitate their power balancing. In addition, the bilevel multi-leader multi-follower Stackelberg game forms an equilibrium problem with equilibrium constraints (EPEC) problem, which is solved by the diagonalization algorithm. Numerical results demonstrate the feasibility and effectiveness of the EPEC model and the importance of modeling customers’ switching behaviors. We corroborate that incentivizing customers’ switching behaviors and increasing the number of retailers facilitates retail competition, which results in reducing strategic retailers’ retail prices and profits. Moreover, the relationship between customers’ switching behaviors and welfare is reflected by a balance between the electricity purchasing cost (i.e., electricity price) and the electricity consumption level.

Strategic investment decisions in an oligopoly with a competitive fringe: An equilibrium problem with equilibrium constraints approach

Modern wholesale electricity markets often have producers who exercise market power. The standard way to model market power in an oligopoly with a competitive fringe is by using Mixed Complementarity Problems (MCPs) and conjectural variations. However, such models can lead to myopic (sub-optimal) behaviour for oligopolists. We first build on existing literature to show that an oligopoly with a competitive fringe where all firms have investment decisions will also lead to myopic behaviour when modelled using MCPs with conjectural variations. To overcome this issue, we develop an Equilibrium Problem with Equilibrium Constraints (EPEC) to model such an electricity market structure. The EPEC models two types of players: price-making firms, who have market power, and price-taking firms, who do not. Our model is the first in the literature to consider an oligopoly with a competitive fringe where all firms have investment decisions. Our results indicate that an EPEC can model investment decisions in an oligopoly with a competitive fringe more credibly and thus overcome the myopic behaviour observed in MCPs. The EPEC found multiple equilibria for investment decisions and firms’ profits. Despite this, market prices and consumer costs were found to remain relatively constant across the equilibria. A further contribution of the modelling approach is that it shows how it may be optimal for price-making firms to accept losses in some time periods in order to disincentivize price-taking firms from investing further into the market. We conclude our paper with a discussion of the computational limitations of our approach.

Heterogeneous traffic information provision on road networks with competitive or cooperative information providers

This paper studies the long-term effects of heterogeneous information provision on road networks with multiple information providers (IPs). To capture the information heterogeneity resulting from various IPs, we consider that users subscribing to different IPs receive non-identical information sets about the available routes and make route choices within their information sets. We investigate both IPs’ strategic interactions and users’ route choices in a bi-level “leader–follower” game. At the upper level, IPs act as leaders who compete or cooperate to determine the optimal information sets provided for their users to maximize their objectives. Three models are developed to investigate IPs’ strategic interactions, i.e., Nash game, Stackelberg game, and cooperation, in which IPs are non-cooperative, leader and followers, and cooperative, respectively. At the lower level, users act as followers to choose their user-optimal routes based on their received information sets. Wardrop Equilibrium with Multiple Information Classes (WEMIC) is defined to capture users’ route choices. The WEMIC problem is formulated as an equivalent variational inequality (VI) problem. The existence and uniqueness of the VI solution are proved. When IPs play the Nash game, the problem is formulated as an Equilibrium Problem with Equilibrium Constraints (EPEC) with discrete variables and non-continuous payoff functions. To solve the EPEC efficiently, we propose a Binary Coded Nash Dominance based Evolutionary (BCNDE) algorithm embedded with a heuristic method for action space reduction. Finally, numerical experiments with two IPs (a big IP and a small IP) are conducted to analyze the effects of unit information cost in the Nash game and the impact of the leader’s strategy in the Stackelberg game, respectively. We find that the small IP may benefit from higher information costs in the Nash game. Moreover, the effects of IPs’ competition and cooperation are compared from both the perspectives of IPs and the system. The results indicate that cooperation can reduce congestion and make both IPs better off. On the Nguyen–Dupuis network, we demonstrate the effectiveness of the BCNDE algorithm and analyze the effects of market share distribution in the Nash game. We also extend this study by considering elastic demand. The extended models, theoretical proofs, and corresponding numerical experiments are provided.

Investigating How the Equilibria of the Electricity Market are Affected by Modeling the Strategic Behavior of Consumers

The paper proposes a formulation for a generalized Nash equilibrium model which incorporates the strategic biddings of some consumers capable of providing reserve and balancing services in the day-ahead and balancing markets, respectively. The strategic bidding in the electricity market is modelled using a bilevel optimization programming, where the electricity cost of consumers is minimized at the Upper Level (UL), and the energy and reserve costs in the market clearing process are co-optimized at the lower level (LL). By means of the strong duality theorem, the original bilevel model is transformed into an equivalent single-level Mathematical Problem with Equilibrium Constraints (MPEC). The joint problem of all MPECs, one per consumer, constitutes an Equilibrium Problem with Equilibrium Constraints (EPEC). The resulting EPEC is finally formulated as an auxiliary Mixed-Integer Linear Programming (MILP) problem. To this end, an exact linearization technique and Fortuny-Amat transformation are adopted to substitute the nonlinear terms and the complementarity conditions. In addition, a parametrization technique is used to replace the dual variable associated with the strong duality equation which appears in the EPEC problem. The diagonalization method is also adopted in this part as an ex-post analysis to verify the obtained solutions of the resulted MILP. Finally, a 3-bus illustrative example and the IEEE RTS 24-Bus System and 118-Bus System are considered to investigate the performance of the proposed approach.

Optimal Equilibrium Selection of Price-maker Agents in Performance-based Regulation Market

This paper analyzes the oligopolistic equilibria of multiple price-maker agents in performance-based regulation (PBR) markets. In these markets, there are price-maker agents representing some frequency regulation (FR) providers and a number of independent price-taker FR providers. A model of equilibrium problem with equilibrium constraints (EPECs) is employed in this paper to study the equilibria of a PBR market in the presence of price-maker agents and price-taker FR providers. Due to the incorporation of the FR providers' dynamics, the proposed model is reformulated as a mixed-integer linear programming (MILP) problem over innovative mathematical techniques. An optimal equilibrium point is also selected for the market, where none of the agents is the unique deviator and the dynamic performance of power system is improved simultaneously. The effectiveness of the proposed optimal equilibrium point is evaluated by comparing the outputs with the conventional optimal dispatches of the FR providers.

Optimal time-differentiated pricing for a competitive mixed traditional and crowdsourced event parking market

An event-based parking pricing problem, the Crowdsourced Event Parking Market Pricing Problem, is proposed wherein parking lot owners and others who are willing to rent out privately owned spaces compete to attract drivers who are looking for available parking spaces. Each parking location owner’s problem is modeled as a bi-level program, where the upper-level parking garages, individuals and consolidators (the players) compete for customers, setting their prices to maximize revenue given the response of the lower-level followers. The lower-level followers choose their parking locations based on the utilities they derive from the spaces, which is a function of the proximity of the spaces to their destinations, parking fees and crowdedness. Prices are set as a function of the time of reservation. Reservation-time based pricing enables price differentiation for late comers either in the form of lower prices to attract last-minute customers when excess spaces are anticipated or higher-pricing if few spaces are expected to remain empty. A multi-period, stochastic Equilibrium Problem with Equilibrium Constraints (EPEC) formulation of the Crowdsourced Event Parking Market Pricing Problem is presented, and a diagonalization method with embedded gradient ascent approach for solution of individual player Mathematical Programs with Equilibrium Constraints (MPECs) is proposed for its solution. Both supply- and demand-side uncertainties are explicitly modeled. Solutions provide competitive parking prices set by reservation time for each parking facility, whether the facility involves a large parking garage or single parking space owner. Numerical experiments were conducted to illustrate the proposed concepts and assess the potential impact of crowdsourced parking spaces on the parking market. The results show that social welfare increases by more than 5% when crowdsourced parking locations account for 7% of the parking market. Results from additional numerical experiments show that ignoring stochasticity results in revenue loss for all parking owners. The developed techniques aim to facilitate existing and new parking facility owners to participate in crowdsourced event-parking markets.

Equilibrium models of coordinated electricity and natural gas markets with different coupling information exchanging channels

In electricity market (EM) and natural gas market (GM), increasingly interacted energy flows and economic flows call for an appropriate market coordination mechanism to improve the overall market efficiency. In this paper, two equilibrium models of coordinated electricity and natural gas markets are proposed with difference in the coupling information exchanging channels between EM and GM. After fully considering the participants, especially the supply-side integrated strategic firm (ISF) and the demand-side regional integrated energy system (RIES), this paper numerically analyzes the social welfare, market power, participants’ profits/costs and market price in two equilibrium models. In addition, the overall social welfare of coordinated markets is compared with traditional uncoordinated markets to illustrate the improvement in market efficiency. The bi-level market equilibrium models are converted into a linearized equilibrium problem with equilibrium constraints (EPEC) which is a mixed-integer linear program (MILP) and can be efficiently calculated by commercial solvers.

Integrated Demand Response programs and energy hubs retail energy market modelling

The present research aims to formulate competition in a retail energy market in the presence of an Integrated Demand Response (IDR) program to reduce prosumer costs and increase retailer profits. This gives prosumers more degrees of freedom to reduce their energy costs. The retail energy market includes retailers and prosumers equipped with an energy hub containing a boiler for producing heat and combined heat and power (CHP). Retailers aim to maximize profit, whereas prosumers seek to minimize their costs. Hence, a multi-leader-follower game with a bi-level program emerges in which the upper level deals with the profit maximization of each retailer while the lower level considers the cost minimization of each prosumer. The strategic behaviour of each retailer is modelled as a Mathematical Program with Equilibrium Constraints (MPEC) problem. Simultaneously solving all MPECs, which leads to an Equilibrium Problem with Equilibrium Constraints (EPEC), determines the market equilibrium point. The equilibrium point is achieved using mathematical, analytical methods and linearization of nonlinear constraints by accurate techniques. Two different case studies are developed to investigate how the number of retailers influences the market equilibrium point. The first case includes two retailers, while the second case considers an increase in the number of retailers. The results demonstrate that with an increase in retailers' number, their competition increases, causing the prosumers costs to reduce. Furthermore, our results suggest the IDR impact on reduced prosumers cost and increased retailers profit.

Distribution Market-Clearing and Pricing Considering Coordination of DSOs and ISO: An EPEC Approach

Distribution-level electricity market provides a platform for trading energy and grid services from large-amount of small-scale distributed energy resources (DERs) located on distribution grids. The behavior of the DERs in the distribution electricity market may ultimately impact the market-clearing and locational marginal prices (LMPs) in the wholesale market. This paper proposes a bi-level optimization model for distribution market clearing and distribution locational marginal pricing (DLMP) considering the interactions between distribution and transmission wholesale markets. In the proposed model, the upper-level model represents the distribution system operator (DSO) market-clearing and the lower-level model represents the wholesale market-clearing by the independent system operator (ISO). The LMP at the substation will impact the DER dispatch and power demands of the DSO as well as the DLMP. In turn, the power demands of the DSO will further impact the ISO market and its LMPs. The equilibrium problem with equilibrium constraints (EPEC) approach is applied to find the equilibria of multiple DSOs and the ISO. The EPEC problem is transformed into a single-level mixed-integer convex problem in order to allow for efficient solving. The effectiveness of the proposed model and solution method are demonstrated through case studies.

Pricing and Resource Allocation Optimization for IoT Fog Computing and NFV: An EPEC and Matching Based Perspective

The number of devices connected to the Internet of Things (IoT) is growing at an enormous rate globally. In the next generation networks, distributed fog computing deployments at the network edge can provide computing resources to the users, especially for latency-sensitive applications. Further, the heterogeneous needs of the fifth generation (5G) networks demand the virtualization of network functions, termed as network function virtualization (NFV). Therefore, an integrated NFV and fog computing resource allocation framework for IoT is of prime importance. Accordingly, in this paper, we model the interactions between the data service operators (DSOs) and the authorized data service subscribers (ADSSs) as an equilibrium problem with equilibrium constraints (EPEC), and utilize the alternating direction method of multipliers (ADMM) as a large-scale optimization tool to obtain solutions. This results in the optimization of resource pricing for the DSOs and the amount of resources to be purchased by the ADSSs. Moreover, we propose a many-to-many matching based model to allocate the fog node (FN) resources according to the VNF resource requirements of the ADSSs. Simulation results show the effectiveness of our proposed approach in achieving efficient resource allocation in NFV enabled IoT fog computing.

Dynamic pricing in electricity and natural gas distribution networks: An EPEC model

Abstract Independent clearing of coupled retail electricity and natural gas markets generally results in a loss of market efficiency. This study investigates the problem of dynamic pricing in retail electricity and natural gas markets in the presence of network constraints by developing a two-leader multiple-follower bi-level model. Here, upper-level electricity and gas utility companies serve as leaders in the model, and determine their optimal dynamic retail prices in anticipation of the demand response (DR) provided by multiple lower-level integrated third-party DR providers or aggregators. Moreover, the model considers the dynamic gas prices provided by gas utility companies to electric utility companies for distributed gas-fired power units, which coordinates the operations of the separate utilities. The resulting model is recast as an equilibrium problem with equilibrium constraints (EPEC). Numerical results for a test system composed of an electric grid with distributed gas-fired power units, a gas distribution network, and multiple integrated DR aggregators indicate that the economic value of the proposed dynamic pricing scheme is up to 5.5% compared with existing regular pricing schemes.

Hierarchical Game for Networked Electric Vehicle Public Charging Under Time-Based Billing Model

Electric Vehicle (EV) public charging is important to meet the exploding charging demand and to address the range anxiety issue. In this paper, we focus on the EV public charging market with heterogeneous charging stations (CSs) under the time-based billing model. We jointly consider the charging time optimization for EVs, the EV-CS pairing, and the pricing mechanism for CSs. A hierarchical game, which mathematically corresponds to an equilibrium problem with equilibrium constraints (EPEC), is then developed to formulate the three coupled problems. In the proposed hierarchical game, each CS sets the charging price to maximize its own revenue first, then the EVs choose their desired CSs and determine the charging time. We analyze the optimal charging time strategies for EVs, and a many-to-one matching algorithm is applied to solve the EV-CS pairing problem. Besides, a block coordinate descent (BCD) based algorithm is applied for each CS to solve the pricing problem. Simulation results show that our proposed schemes can achieve the performance improvement of the charging system.