Mathematical Program With Equilibrium Constraints Research Articles

Congestion-aware Stackelberg pricing game in urban Internet-of-Things networks: A case study

We consider a one-leader multi-follower Stackelberg pricing game in urban Internet-of-Things networks. The leader manages multiple services and sets service prices to maximize profit, while followers seek the most cost-effective services, taking into account service prices, service congestion, and individual locations. Existing approaches often simplify the game process by neglecting followers’ locations, but they are not applicable to real-world complex spatial distributions. In this paper, we focus on the congestion-aware Stackelberg pricing game in urban Internet-of-Things networks, using electric vehicle charging as a case study. The game is reformulated as a mathematical program with equilibrium constraints (MPEC) that considers both congestion effects and followers’ spatial distributions. To solve the MPEC, we introduce the Segmentation-based Pricing with ITERative Optimization (SPITER) algorithm, which converges to a local maximum. Additionally, optimization techniques are developed to improve the performance of solving SPITER in urban environments. We evaluate the performance of SPITER using extensive experiments with two real-life urban datasets, demonstrating the advantages of our model and illustrating SPITER’s effectiveness and convergence.

A hierarchical structure for harnessing the flexibility of residential microgrids within active distribution networks: Advancing toward smart cities

The increasing integration of Renewable Energy Sources (RESs) and the replacement of fossil fuel vehicles with electric types in distribution systems, while enhancing economic and environmental aspects, have introduced new challenges to network production-consumption balance due to the inherent randomness of these resources. This factor has magnified the significance of balancing markets in renewable-based distribution networks, as these markets address uncertainties and are vital for operational stability. Consequently, this paper introduces a two-stage multi-layer methodology for organizing local energy and balancing markets among Residential Microgrids (RMGs) within an Active Distribution Network (ADN). The AND operator organizes both energy and balancing markets.In the first stage, the intra-day energy market operates, engaging non-profit-seeking aggregators, RMGs, and ADN. Moving to the second stage, the intra-hour balancing market is conducted involving profit-seeking Flexible Load (FL) and Electric Vehicle (EV) aggregators, RMGs, and ADN. The ADN operator organizes both markets and can alter the network topology online within the energy market. The planning of profit-seeking aggregators is formulated using Karush-Kuhn-Tucker (KKT) conditions and Strong Duality Theory (SDT) in the form of Mathematical Programming with Equilibrium Constraints (MPEC), considering the correlation of aggregators' profit with the satisfaction index of end-users. The simulation results of the proposed methodology on a modified 118-bus IEEE distribution test system using GAMS indicate that this approach not only provides a significant share of balancing capacities through FLs and EVs but also ensures system stability against intra-hour fluctuations across various scenarios. The results also validate that the proposed methodology achieves a reduction of 24.8 % in power losses and 5.06 % in market costs, while simultaneously ensuring profitability for both aggregators and RMGs.

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.

A coordinated green hydrogen and blue hydrogen trading strategy between virtual hydrogen plant and electro‐hydrogen energy system

AbstractIn the hydrogen‐based integrated energy system (HIES), there exists a hydrogen trading market where hydrogen producers and consumers are distinct stakeholders. Current research in hydrogen trading predominantly focuses on high‐cost green hydrogen (GH), which is not aligned with the current trend of utilizing hydrogen from multiple sources. To address this, this paper proposes a hydrogen trading strategy between the virtual hydrogen plant (VHP) and electro‐hydrogen energy system (EHES) based on a bi‐level model, considering the synergy of GH produced from electrolyzers and blue hydrogen (BH) derived from natural gas in the HIES. In the VHP level, the objective is to maximize profit from hydrogen sales, allowing for the determination of hydrogen prices. In the EHES level, the goal is to minimize the cost of energy supply, leading to the formulation of GH and BH purchasing plans based on hydrogen prices. Additionally, this paper incorporates a risk‐averse model from the information gap decision theory (IGDT) to account for the impact of wind power output uncertainties in the VHP level. Subsequently, leveraging the Karush–Kuhn–Tucker (KKT) conditions of the EHES level, the bi‐level problem is transformed into a solvable single‐level mathematical program with equilibrium constraints (MPEC), with the non‐linear equilibrium constraints linearized. The proposed bi‐level optimization model is validated through case studies encompassing industrial and residential hydrogen utilization within the HIES. The outcomes confirm the rationality of the proposed model, demonstrating that, in comparison to exclusively trading GH, the coordinated GH and BH trading can increase the profit of the VHP by 2.7% and reduce the costs of the EHES by 8.5%.

Capturing Electricity Market Dynamics in Strategic Market Participation Using Neural Network Constrained Optimization

In competitive electricity markets, the optimal bid or offer problem of a strategic agent is commonly formulated as a bi-level program and solved as a mathematical program with equilibrium constraints (MPEC). If the lower-level part of the problem can be well approximated as a convex problem, this approach leads to a global optimum. However, electricity markets are governed by non-convex (partially known) constraints and reward functions of the participating agents. In this paper, an alternative data-driven paradigm, labeled as a mathematical program with neural network constraint (MPNNC), is developed. The method uses a neural network to represent the mapping between the upper-level (agent) decisions and the lower-level (market) outcomes, i.e., it replaces the lower-level problem with a surrogate model. In the presented case studies, the proposed model is used to find the optimal load shedding strategy of a strategic load-serving entity. First, the MPNNC performance is compared to the MPEC approach, both in convex and non-convex environments, showing that the proposed MPNNC achieves similar performance to an ideal MPEC that has perfect knowledge of the simulated market environment. Then, aggregated supply curves from the Belgian spot exchange are used to assess the potential gains of using the developed model in real-life applications.

BilevelJuMP.jl: Modeling and Solving Bilevel Optimization Problems in Julia

In this paper, we present BilevelJuMP.jl, a new Julia package to support bilevel optimization within the JuMP framework. The package is a Julia library that enables the user to describe both upper and lower-level optimization problems using the JuMP algebraic syntax. Because of the generality and flexibility that our library inherits from JuMP’s syntax, our package allows users to model bilevel optimization problems with conic constraints in the lower level and all constraints supported by JuMP in the upper level including conic, quadratic, and nonlinear constraints. Moreover, the models defined with the syntax from BilevelJuMP.jl can be solved by multiple techniques that are based on reformulations as mathematical programs with equilibrium constraints (MPEC). Manipulations on the original problem data are possible due to MathOptInterface.jl’s structures and Dualization.jl features. Hence, the proposed package allows quick modeling, deployment, and thereby experimenting with bilevel models based on off-the-shelf mixed-integer linear programming and nonlinear solvers. History: Accepted by Ted Ralphs, Area Editor for Software Tools. Funding: The authors were partially supported by the Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - Brasil (CAPES) - Finance Code 001. The work of A. Street was also partially supported by Fundação Carlos Chagas Filho de Amparo à Pesquisa do Estado do Rio de Janeiro (FAPERJ) and Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq). The work was partially funded by the project P&D ANEEL PD-00403-0050/2020 sponsored by ENGIE BRASIL ENERGIA S.A. Supplemental Material: The software that supports the findings of this study is available within the paper and its Supplemental Information ( https://pubsonline.informs.org/doi/suppl/10.1287/ijoc.2022.0135 ) as well as from the IJOC GitHub software repository ( https://github.com/INFORMSJoC/2022.0135 ). The complete IJOC Software and Data Repository is available at https://informsjoc.github.io/ .

Unlocking Flexible Electric Vehicle Charging via New Rate Design

A high penetration of electric vehicles (EVs) will deeply impact the management of electric power systems. The risk of not providing adapted EV pricing signals can lead to inefficient investments in grid infrastructure. To avoid costly grid reinforcements and to ensure proper guidance for EV charging, a solution allowing customers to access EV-only rates without installing a separate meter, which we refer to as submetering, is an attractive option for EV owners and grid operators. We develop a game-theoretical model expressed and treated as a mathematical program with equilibrium constraints (MPEC) to capture the interaction between a national regulatory authority (NRA) designing these tariffs and heterogeneous agents. This framework represents a stylized regulatory setup applicable to several European countries. First, we analyse the conditions in which EV-only tariffs can be applied for domestic charging by comparing different energy profiles. Second, we study the impact of EV charging on different tariff structures to identify the most efficient way of recovering network costs. We found that the adoption of submetering under a pure volumetric tariff can bring yearly gains varying from $64 to $125 to consumers with EV. Finally, we derive policy implications from the results.

Flexibility enhancement of multi-district DISCOs considering a trade-off between congestion and extractable reserve capacity from virtual energy storage systems

One of the fundamental concerns of distribution companies (DISCOs) is how to satisfy security requirements under uncertain conditions and real-time contingencies. To address this concern, it is vital to employ operational flexibility levers aimed at mitigating congestion and retaining optimal reserve capacity, particularly in critical feeders. Increased interaction between DISCOs and virtual energy storage systems (VESSs) offers an unparalleled opportunity to unlock the energy flexibility of distribution grids at the local level without increasing operational complexity. According to the state-of-the-art, models that tap the economic performance of VESSs while respecting the flexibility and security setpoints adopted by DISCOs in a distributed manner have remained scarce. This study goes beyond by developing a flexibility-oriented decision-making strategy as a single-leader multi-followers stochastic model; in which a multi-district DISCO involves the flexibility setpoints and congestion management strategy in the market participation plan of each VESS locally to enhance the energy flexibility of each district. In the upper level, DISCO aims to deal with stressful situations in real-time sessions by (i) determining the minimum reserve capacity requirements in each district to be provided by VESSs with respect to the existing uncertainty sources and the importance of each district, and (ii) alleviating the congestion in critical feeders to release the feeder capacity for deployment of the available reserve power promptly. In the lower level, each VESS makes optimal decisions on power and reserve bidding in day-ahead, reserve, and real-time markets that are affected by the flexibility services required by the DISCO. Inspired by the activity schedule of different users, it also examines how demand response programs can affect DISCO’s ability to unlock consumer-side flexibility in realizing congestion management strategies. The developed bi-level scheduling model is first reformulated as single-level mathematical program with equilibrium constraints (MPEC) by employing Karush–Kuhn–Tucker (KKT) optimality conditions and then linearized into a mixed-integer linear program (MILP). The proposed strategy is validated through extensive simulations conducted on the modified IEEE 33-bus test system consisting of three districts with different users. Furthermore, DIgSILENT PowerFactory is used to verify the feasibility of the proposed model under contingency cases and to provide insights for DISCOs to control multiple VESSs and capacity-correlated uncertainties.

Risk‐constrained expansion planning of wind integrated networks using innovative MPEC primal‐dual formulation for directly involving price‐based demand response in MILP problem

AbstractPrice‐based demand response (PBDR) programs can push consumers to reconsider their electricity demand regarding the electricity price in the market. The load profile of the whole network can be reshaped in response, which can directly affect the network investment decisions. The decision‐maker had to consider this effect in order to reach an optimal plan for the network. Here, a mixed‐integer linear programming (MILP) model considering a price‐based demand response (PBDR) program is developed for transmission expansion planning (TEP) of wind‐integrated networks and the problem is constrained by the conditional value at risk (CVaR) measure to model the risk of planning and investments for both sides. The proposed model is an originally bi‐level problem with different objective functions in both layers. These objectives are as follows, minimizing the total cost of TEP, consumer payments, and wind curtailment in the first layer, and minimizing the network operational costs in the second layer. Then, using an innovative formulation to overcome the non‐linearities, and using KKT conditions of the second layer problem, the problem recast into a single‐layer mixed integer non‐linear program (MINLP) problem which is called a mathematical program with equilibrium constraints (MPEC) with primal‐dual formulation. The proposed model had been applied to IEEE standard 24‐bus RTS and IEEE standard 118‐bus test systems to show its efficiency.

Demand Management for Peak-to-Average Ratio Minimization via Intraday Block Pricing

Price based demand response schemes may significantly improve power system efficiency. Additionally, it is desired that such schemes yield improved power operation, by reducing the peak consumption. This paper proposes the Intraday Block Pricing (IBP) scheme, aiming to promote effective demand response among consumers by charging their electricity usage based on intraday time-slots. To design the prices associated with the proposed scheme, we formulate a bilevel optimization problem that aims to minimize the Peak-to-Average Ratio (PAR) and simultaneously benefit the consumers and the utility company. The bilevel problem is converted into a single-level Mathematical Program with Equilibrium Constraints (MPEC). The resulting MPEC is non-convex and includes nonlinear constraints. Hence, to obtain a solution, it is relaxed into a Mixed Integer Linear Program by dealing with all nonlinearities. To evaluate the conservativeness of the proposed approach, a lower bound to the cost of the original bilevel problem is obtained. The applicability of the proposed scheme is demonstrated with simulations on various case studies, which exhibit a significant reduction in PAR and economic gains for the utility company and consumers. Moreover, simulation results show that the solutions of the original and relaxed problems are equivalent, demonstrating the effectiveness of the proposed solution approach. Further simulation results demonstrate significant advantages in the performance of the IBP scheme when compared to existing schemes in the literature.

Iterative Backpropagation Method for Efficient Gradient Estimation in Bilevel Network Equilibrium Optimization Problems

Network optimization or network design with an embedded traffic assignment (TA) to model user equilibrium principle, sometimes expressed as bilevel problems or mathematical programs with equilibrium constraints (MPEC), is at the heart of transportation planning and operations. For applications to large-scale multimodal networks with high dimensional decision variables, the problem is nontrivial, to say the least. General-purpose algorithms and problem-specific bilevel formulations have been proposed in the past to solve small problems for demonstration purposes. Research gap, however, exists in developing efficient solution methods for large-scale problems in both static and dynamic contexts. This paper proposes an efficient gradient estimation method called Iterative Backpropagation (IB) for network optimization problems with an embedded static TA model. IB exploits the iterative structure of the TA solution procedure and simultaneously calculates the gradients while the TA process converges. IB does not require any additional function evaluation and consequently scales very well with higher dimensions. We apply the proposed approach to origin-destination (OD) estimation, an MPEC problem, of the Hong Kong multimodal network with 49,806 decision variables, 8,797 nodes, 18,207 links, 2,684 transit routes, and 165,509 OD pairs. The calibrated model performs well in matching the link counts. Specifically, the IB-gradient based optimization technique reduces the link volume squared error by 98%, mean absolute percentage error (MAPE) from 95.29% to 21.23%, and the average GEH statistics from 24.18 to 6.09 compared with the noncalibrated case. The framework, even though applied to OD estimation in this paper, is applicable to a wide variety of optimization problems with an embedded TA model, opening up an efficient way to solve large-scale MPEC or bilevel problems. Funding: The study is supported by IVADO Postdoctoral Fellowship scheme 2021, HSBC 150th Anniversary Charity Programme HKBF17RG01, National Science Foundation of China (71890970, 71890974), General Research Fund (16212819, 16207920) of the HKSAR Government, and the Hong Kong PhD Fellowship.

Mathematical program with equilibrium constraints approach with genetic algorithm for joint optimization of charging station location and discrete transport network design

ABSTRACT This paper focuses on the joint optimization of the charging station location problem (CSLP) and discrete network design problem (DNDP) in a transportation network. We present a variational inequality (VI) formulation to describe the user equilibrium (UE) state of gasoline vehicles (GVs) and electric vehicles (EVs). Based on the mixed-UE model, a mathematical program with equilibrium constraints (MPEC) model is formulated for integrating the decisions of deploying EV charging stations (EVCSs) and adding new links to minimize the total travel cost (TTC) of all vehicles. A modified genetic algorithm is developed to tackle the MPEC model with an adaptive path generation procedure to address the mixed-UE model. Finally, we conduct numerical experiments to identify the efficacy of the proposed models and algorithms. Specifically, we propose a two-step optimization model and explore a performance comparison between the joint and two-step optimization approaches, while the joint optimization exhibits superiority in minimizing the TTC.

Dynamic equilibrium analyses in a ride-sourcing market under travel time uncertainty

This paper proposes a dynamic equilibrium framework for the morning commute problem in the ride-sourcing market under travel time uncertainty. It is formulated as a simultaneous departure time and mode choice problem (SDTMCP) with random travel delay. Commuters are considered to choose among four travel modes: (1) driving alone, (2) taking non-shared service, (3) taking ride-sharing service, and (4) taking the metro. The equilibrium state is reached when no commuter can reduce their expected travel cost by unilaterally changing their mode or departure time. We derive the dynamic equilibrium queuing patterns analytically under certain assumptions, and develop an approach using nonlinear complementarity problem (NCP) to solve general problems where analytical results cannot be readily derived. We further explore the pricing strategy of the ride-sourcing platform, in which travel time uncertainty and the prices of ride-sourcing services jointly determine the profit performance. A mathematical program with equilibrium constraints (MPEC) is formulated to search for the optimal service prices to achieve the maximum profit. The framework is then extended to investigate the pricing strategy in a two-sided ride-sourcing market and examine the metro crowdedness effect in the numerical studies.

Fast-charging station location problem: A two-phase approach with mathematical program with equilibrium constraints considering charging choice behaviour

Electric vehicles (EVs) produce zero emissions, which makes them an essential solution for reducing air pollution in urban areas. Fast-charging station (FCS) planning is crucial for ensuring a successful transition to EVs. Therefore, to address the FCS location problem (FCSLP) for EVs in urban areas, a two-phase approach consisting of data processing and model optimization is proposed. During data processing, the spatiotemporal charging demand distribution, existing FCSs, and potential locations for new FCSs are sequentially identified. The results are then used in the model optimization phase, where a spatiotemporal mathematical program with equilibrium constraints (MPEC) model is developed to minimize the total social cost (TSC). The MPEC model integrates the maximal coverage location model and charging equilibrium model, which is formulated as a variational inequality to capture the charging choice behaviour of the EVs within the FCS service radius. To solve the MPEC model, a modified genetic algorithm is developed, in which the charging equilibrium model is iteratively solved. Finally, a case study is conducted in a practical area in Shenzhen, China. The results indicate that a trade-off exists between the TSC and the system service level concerning the optimal FCS locations under a specific FCS service radius. Moreover, the FCS service radius significantly influences EV drivers’ charging choice behaviour and FCS planning.

Optimal bidding strategy of a gas-fired power plant in interdependent low-carbon electricity and natural gas markets

This work presents a bi-level optimization framework to determine the optimal bidding strategies for a strategic gas-fired power plant, exerting market power in interdependent pool-based electricity and natural gas markets, under a carbon emission trading scheme (CETS). The upper-level problem aims at maximizing the profits of the strategic player, while at the lower-level problem, the day-ahead electricity and natural gas markets are cleared sequentially, considering the provision of carbon emission allowances for conventional power producers and high penetration of wind power generation. The bi-level formulation is initially recast into a mathematical program with equilibrium constraints (MPEC), using the Karush-Kuhn-Tucker optimality conditions and duality theory, and is further reformulated into a mixed integer linear program. The proposed algorithm is applied to a Pennsylvania-New Jersey-Maryland (PJM) 5-bus power grid, incurred by transmission constraints and a single node natural gas network. Numerical simulations provide CETS-embedded electricity clearing prices and optimal bidding decisions for the strategic gas-fired power plant, under plausible power transmission congestions and natural gas prices increment scenarios.

Value of distribution system information for DER deployment

AbstractIntegration of Distributed Energy Resources (DERs) in power systems exacerbates the existing information problems between power utilities and regulators. DER policies oblivious to the trilemma of information asymmetry between power utilities, DER aggregators, and regulators result in distorted price signals to DER investors, and socially inefficient DER roll‐out. Therefore, in this paper, a game‐theoretic approach is proposed for modeling information asymmetry in distribution network information and consumer data between the DER aggregators and the power utilities. The proposed framework is based on Single Leader Single Follower (SLSF) games, reformulated as Mathematical Programs with Equilibrium Constraints (MPECs), and solved using the Scholtes's relaxation technique. The results, based on the 7‐bus Manhattan power network, show that unless the DER aggregators have complete information about the distribution network characteristics, the welfare along with the realized installed capacity of DERs in the system decreases. Moreover, progressively decreasing DER investment costs alleviate the effects of information asymmetry, suggesting that early adopters face disproportionately high welfare losses attributed to incomplete information between the DER stakeholders. Hence, policy interventions to alleviate the rampant information problems are imperative to ensure an optimal DER roll‐out.

A stochastic-robust optimization model for inter-regional power system planning

This study develops a stochastic-robust optimization model for inter-regional power system planning. The model is formulated as Mathematical Programs with Equilibrium Constraints (MPEC) with two levels of decision makers. In the bottom-level, various individual regions separately make generator investment, operating, and trading plans for the minimization of their own costs in a perfectly competitive market. In the top-level, a central system operator makes the investment plan of cross-border transmission lines for the minimization of total system cost, anticipating how generators in various regions respond to those investments. Two different levels of data uncertainties are considered in the problem, namely scenario and local uncertainties. The uncertainties are handled by combining the advantages of stochastic programming and robust optimization methods. The proposed model is applied for power system planning in the Association of Southeast Asian Nations (ASEAN) between 2020 and 2040. The modelling results show that the integration of a cross-border power grid results in a substantial shift in the generation portfolio, and can potentially reduce total costs by up to 6.0%. The cost savings primarily result from the increased utilization of renewable energy resources, facilitated by cross-border electricity transmission. We assessed the performance of our proposed approach in handling data uncertainties by comparing it with existing practices. The comparison results demonstrate that our stochastic-robust approach not only enhances the capacity expansion plan’s robustness to meet power demand and CO2 emissions targets, but also significantly reduces investment and operational costs.

Continuous Network Design Using Partial Linearized Subgradient Methods

Although demand management has shown to be a vital tool in managing congestion, many metropolitan planning organizations (MPOs) and departments of transportation (DOTs) still pursue network modification and capacity increase as a congestion relief and a mitigation measure. The network design problem (NDP) still has, therefore, an essential and significant role in shaping and sizing urban transportation networks. The literature has traditionally treated the NDP as a bi-level mathematical programming problem or a mathematical program with equilibrium constraints (MPEC). In the bi-level optimization setting, the problem is approached as a leader-follower problem in which the lower level is a user equilibrium (UE) assignment problem as the follower, and the upper level is the network sizing problem as the leader problem. NDP has long been known as a challenging problem, and many solution algorithms have been proposed to solve it. This study proposed an efficient solution algorithm for the continuous network design problem (CNDP). The solution algorithm has been shown empirically to solve the CNDP in a shorter time using partial linearized subgradient methods. The proposed method was applied to a small network that was traditionally used to evaluate the performance of solution algorithms versus Braess’s paradox. It was then applied to the Sioux Falls network as a well-known benchmark network to compare the results with previous studies. The proposed method has been shown to run much faster than all the previous studies reviewed in this paper with minimal degradation of accuracy (0.52% lower than the best solution).

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.

Stochastic Strategic Participation of Active Distribution Networks With High-Penetration DERs in Wholesale Electricity Markets

With the increasing penetration of distributed energy resources (DERs), traditional distribution networks as load-serving entities in wholesale electricity markets, now evolve towards active distribution networks (ADNs) which can proactively participate in wholesale markets by optimally controlling the DERs in their networks. A stochastic bilevel optimization model is proposed in this paper for the strategic participation of ADNs and DERs to provide energy and grid services in wholesale electricity markets. The bilevel optimization model can capture the interactions between the ADN and the wholesale energy and ancillary service markets, considering the uncertainties of DERs in the ADN. In the upper-level model, the ADN makes optimal decisions on energy and reserve bidding considering the availability, uncertainties, and flexibility of DERs. The joint energy and reserve market-clearing of the independent system operator (ISO) is modeled as the lower-level problem. Using strong duality theory and Karush-Kuhn Tucker (KKT) conditions, the proposed bilevel optimization problem is reformulated as mathematical programming with equilibrium constraints (MPEC) problem and further converted into a computationally-solvable mixed-integer second-order-cone programming (MISOCP) model. The simulation results demonstrate the effectiveness of the model and the interactions between an ADN and wholesale electricity markets.