Single-level Mathematical Program Research Articles

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.

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.

Personalized retail pricing design for smart metering consumers in electricity market

In the current deregulated electricity market, flexible consumers are more active in participating in market activities via the representation of electricity retailers. However, without an effective communication infrastructure, the connection between retailers and the consumers they serve is incomplete. Nowadays, smart meters are being rolled out worldwide to enhance the connection and data exchanges between retailers and consumers. Specifically, smart meters enable retailers to provide customers with detailed information about retail tariffs and their energy usage at different times of the day, which in turn enables customers to manage their energy use more proactively. This paper drops this assumption and makes use of data acquired from smart meters to design a personalized retail pricing scheme for different types of consumers. To formulate this problem, a bi-level optimization model is proposed, with the upper-level problem representing the pricing decision made by the retailer and two lower-level problems representing the demand response of consumers and the wholesale market clearing process, respectively. Afterward, we convert this bi-level optimization model into a single-level mathematical program with equilibrium constraints by using its Karush Kuhn Tucker optimality conditions and complementary conditions. The scope of the examined case studies is fourfold. First, consumers are classified based on their daily load profiles using the advanced clustering method. Second, the physical benefit of fully exploring the consumer’s demand flexibility as well as the economic benefits of increasing retailers’ profitability and reducing consumers’ energy bills are evaluated with respect to the traditional uniform retail pricing scheme. Third, the impacts of consumers’ demand flexibility on electricity market outcomes and business cases are investigated. Finally, the proposed personalized retail pricing scheme is verified to relieve the strategic retailer’s market power reduction caused by the flexibility of demand, which is beneficial to the retailer’s profitability.

Flexibility Clearing in Joint Energy and Flexibility Markets Considering TSO-DSO Coordination

Active distribution networks (ADNs) with distributed generators (DGs) can provide flexibility for the upstream grid by participating in energy and flexibility markets. In this paper, a novel market-clearing model is proposed to facilitate energy and flexibility transactions through coordinating the flexibility providers in both transmission and distribution networks. The energy and flexibility market-clearing problem is formulated as a bi-level optimization model. The upper-level models the transmission-level joint energy and flexibility market clearings while the lower-level represents the distribution-level ADN market clearings. In the proposed model, the locational marginal prices (LMP) at the coupling bus will affect the dispatch of DGs and power demands of the ADNs. In turn, the power demands of the ADNs and the offers submitted by DGs will affect energy and flexibility market-clearing results and the LMPs as well. Karush-Kuhn-Tucker (KKT) conditions and the duality theory are used to transform the proposed nonlinear bi-level model to a single-level mathematical program with equilibrium constraints (MPEC) model. Case studies verify the effectiveness of the proposed model.

Operator-as-a-Consumer: A Novel Energy Storage Sharing Approach Under Demand Charge.

Energy storage systems (ESSs)-based demand response (DR) is an appealing way to save electricity bills for consumers under demand charge and time-of-use (TOU) price. In order to counteract the high investment cost of ESS, a novel operator-enabled ESS sharing scheme, namely, the "operator-as-a-consumer (OaaC)," is proposed and investigated in this article. In this scheme, the users and the operator form a Stackelberg game. The users send ESS orders to the operator and apply their own ESS dispatching strategies for their own purposes. Meanwhile, the operator maximizes its profit through optimal ESS sizing and scheduling, as well as pricing for the users' ESS orders. The feasibility and economic performance of OaaC are further analyzed by solving a bilevel joint optimization problem of ESS pricing, sizing, and scheduling. To make the analysis tractable, the bilevel model is first transformed into its single-level mathematical program with equilibrium constraints (MPEC) formulation and is then linearized into a mixed-integer linear programming (MILP) problem using multiple linearization methods. Case studies with actual data are utilized to demonstrate the profitability for the operator and simultaneously the ability of bill saving for the users under the proposed OaaC scheme.

A Joint Optimization Model for Transmission Capacity and Wind Power Investment Considering System Security

Due to continuous growth in electric demand and increasing connection of renewable energy, the power systems are being operated with smaller stability margins. Therefore, a sufficient loading margin is essential to maintain the system secure and ensure voltage stability. To this end, this paper studies the joint optimization of transmission capacity and wind power investment problem, the unique feature of which is incorporating voltage stability margin (VSM) in the planning model. A bi-level model has been formulated whose upper level minimizes the total investment and operation cost minus the weighted VSM. The lower level evaluates the VSM given the optimal expansion plan from the upper level. In addition, the stochastic nature of wind power and load can impact voltage stability. Thus, uncertainties related to intermittent wind generation and demand must be modeled. We use an approximated linear representation to model the AC power system at both levels of the problem. The duality theory (primal-dual formulation) is utilized to transform bi-level programming into single-level mathematical programming. The validity of the constructed methodology is demonstrated on the IEEE 24-bus RTS, which indicates the efficacy and feasibility of the presented model.

Bi-level Energy Trading Model Incorporating Large-scale Biogas Plant and Demand Response Aggregator

Increasing intermittent renewable energy sources (RESs) intensifies the imbalance between demand and generation, entailing the diversification of the deployment of electrical energy storage systems (ESSs). A large-scale biogas plant (LBP) installed with heating devices and biogas energy storage (BES) usually exhibits a storage-like characteristic of accommodating an increasing penetration level of RES in rural areas, which is addressed in this paper. By utilizing the temperature-sensitive characteristic of anaerobic digestion that enables the LBP to exhibit a storage-like characteristic, this paper proposes a bi-level energy trading model incorporating LBP and demand response aggregator (DRA) simultaneously. In this model, social welfare is maximized at the upper level while the profit of DRA is maximized at the lower level. Compared with cases only with DRA, the results show that the proposed model with the LBP improves the on-site accommodation capacity of photovoltaic (PV) generation up to 6.3%, 18.1%, and 18.9% at 30%, 40%, and 50% PV penetration levels, respectively, with a better economic performance. This nonlinear bi-level problem is finally recast by a single-level mathematical program with equilibrium constraints (MPEC) using Karush-Kuhn-Tucker (KKT) conditions and solved by the Cplex solver. The effectiveness of the proposed model is validated using a 33-bus test system and a sensitivity analysis is provided for analyzing what parameter influences the accommodation capacity most.

Strategic retail pricing and demand bidding of retailers in electricity market: A data-driven chance-constrained programming

• A hybrid deep learning forecasting method is proposed. • A bi-level model is proposed for retailer, consumers, and market clearing process. • A chance-constrained programming is proposed accounting for uncertainties. • The impact of demand flexibility on retailer’s strategies is investigated. This paper proposes a novel bi-level optimization model to study the strategic retail pricing and demand bidding problems of an electricity retailer that considers the interactions between demand response and market clearing process. In order to accurately forecast the day-ahead demand bids submitted by the retailer, a novel deep learning framework based on convolutional neural networks and long short-term memory is proposed that can capture both local trends and long-term dependency of the forecasting data. In addition, uncertainties about the retailer’s served demand, rivals’ demand bids, and wind power generation are incorporated using the data-driven uncertainty set constructed from data. We further propose chance-constrained programming that introduces a set of chance constraints to represent the operational risk associated with the market uncertainties. To solve this problem, we first reformulate chance-constrained programming as a tractable second-order conic programming and then convert it into a single-level mathematical program with equilibrium constraints by using its Karush Kuhn Tucker conditions. The scope of the examined case studies is four-fold. First, they evaluate the benefits of the proposed forecasting framework in terms of higher accuracy and expected profit compared to the conventional forecasting methods. Second, they demonstrate how demand flexibility affects the retailer’s strategies and its business cases. Third, they highlight the added value of the proposed bi-level model capturing the market clearing process by comparing its outcomes against the state-of-the-art bi-level model with exogenous market prices. Finally, they analyze the retailer’s strategies and business cases at different confidence levels regarding the imposed chance constraints.

A Stackelberg Game-based planning approach for integrated community energy system considering multiple participants

The game relationship between the Energy Server (ES) and the Energy Consumer (EC) in the Integrated Community Energy System (ICES) can change the energy supply and consumption behaviors of both participants, and affect the economy of the ICES planning scheme. This paper proposes a bi-level model to address a joint master-slave planning-operation problem of the ICES. Firstly, the revenue models of the ES and EC in the ICES is established. Secondly, a bi-level planning model based on Stackelberg Game for the ICES is proposed. In the upper level, a joint optimal planning model of energy conversion devices and energy prices is built and used by the ES (game leader) to maximize its profit. In the lower level, an optimal energy consuming strategy is proposed for the EC (game follower) to minimize its energy bill. The Karush-Kuhn-Tucker (KKT) conditions are utilized to convert the bi-level planning problem to a single-level mathematical program with equilibrium constraints (MPEC). A typical ICES is employed as a test system to illustrate the effectiveness of the proposed bi-level planning approach, and the results show that the proposed approach can reduce the investment cost of ES while ensuring the interests of EC.

Optimal energy storage planning for stacked benefits in power distribution network

Energy storage system (ESS) is regarded as an effective tool to promote energy utilization efficiency and deal with the operational risk of the power distribution network (PDN), which is caused by the inherent uncertainties of distributed energy resources and the surging of new loads from emerging energy sectors. Multiple benefits could be accrued by ESSs when providing various services to the PDN. However, investing in ESSs without a comprehensive cost-benefit analysis will underestimate the ESSs’ profitability and result in a suboptimal planning scheme. To this end, this paper proposes a bi-level ESS planning method considering the stacked benefits of ESSs. First, to facilitate the comprehensive cost-benefit analysis in the investment decision making process, systematic definitions and quantitative methods of ESS benefits are proposed to evaluate both the investment and operational benefits of ESSs properly. Novel ESS reserve model and load hosting capacity evaluation model are also developed to improve the accuracy of the benefit assessment. Second, a game theory based bi-level programming method is proposed to precisely capture the two-layer dynamic interaction structure of ESS planning using an easy-to-implement mathematical optimization-based formulation with global optimum guarantee. Specifically, the upper level optimizes the sites and sizes of ESSs for maximizing the stacked benefits, while the lower level economic dispatch model determines the optimal scheduling of the installed ESSs. Finally, to make the bi-level ESS planning problem tractable, it is reformulated as a single-level mathematical programming with equilibrium constraints by the duality-based reformulation. The linearization method is applied to further convert it into a mixed-integer programming problem that could be directly solved by commercial solvers. Numerical results verify that the proposed planning strategy could yield cost-effective results under different settings. Comprehensive cost-benefit analysis also indicates that transmission and distribution system upgrade deferral, reserve/regulation provision and energy arbitrage are the top-three applications for maximizing ESS benefits, and it is not always reasonable to keep investing in ESSs as their marginal benefits will decrease to zero with the increasing ESS investment.

Strategic Biddings of a Consumer demand in both DA and Balancing Markets in Response to Renewable Energy Integration

This paper proposes strategic biddings for a consumer demand that participates in both the day-ahead and balancing markets. The strategic behavior of the consumer is represented by the bilevel optimization programming with minimization of consumer costs at the upper level (UL) subject to the co-optimization of energy and reserve in the market clearing process at the lower level problem (LL). Using the Karush-Kuhn-Tucker (KKT) optimality constraints to replace the LL problem, the bilevel model is recast into a single-level mathematical program with equilibrium constraints (MPEC). The resulted model is finally formulated as a mixed-integer linear programming (MILP) problem using the exact linearization technique and Fortuny-Amat transformation to replace bilinear terms and the complementarity constraints, respectively. The results demonstrate a reduction in the electricity consumption payment for the strategic consumer and a decline in social welfare.© 2017 Elsevier Inc. All rights reserved.

Strategic investment for district heating systems participating in energy and reserve markets using heat flexibility

The flexibility of district heating systems can be utilized to provide ancillary services for electric power systems and hence the investors can make profits by taking advantage of the arbitrage in multiple markets. This paper proposes a strategic investment scheme for a district heating system participating in both electricity energy and reserve markets, which enables the enhanced investment decisions for the installation of combined heat and power units, heat pumps, and thermal energy storage. A new business model is proposed for the district heating system that acts as a prosumer participating in energy and reserve markets, where the flexibility region of the district heating system is particularly modeled to ensure the feasibility of deployed reserves. The strategic behavior is modeled as a bi-level optimization problem. The upper-level problem determines the capacities and optimal offering/bidding strategies of district heating system, while the lower-level problem represents the energy and reserve market clearing over a set of representative days. The bi-level model is converted into a single-level mathematical program with equilibrium constraints using Karush-Kuhn-Tucker optimality conditions, and then reformulated as a mixed-integer linear problem by linearizing the nonlinear terms. Simulation results demonstrate that the total investment and operational costs of the district heating system are reduced by strategically offering in reserve market and the investor prefers to invest in thermal energy storage to shave peak heat load and combined heat and power units to increase the district heating system’s flexibility other than heat pumps. The effectiveness of the flexibility model of the district heating system for feasible reserve deployment is also verified.

Modeling the α-max capacity of transportation networks: a single-level mathematical programming formulation

Network capacity, defined as the largest sum of origin–destination (O–D) flows that can be accommodated by the network based on link performance function and traffic equilibrium assignment, is a critical indicator of network-wide performance assessment in transportation planning and management. The typical modeling rationale of estimating network capacity is to formulate it as a mathematical programming (MP), and there are two main approaches: single-level MP formulation and bi-level programming (BLP) formulation. Although single-level MP is readily solvable, it treats the transportation network as a physical network without considering level of service (LOS). Albeit BLP explicitly models the capacity and link LOS, solving BLP in large-scale networks is challenging due to its non-convexity. Moreover, the inconsideration of trip LOS makes the existing models difficult to differentiate network capacity under various traffic states and to capture the impact of emerging trip-oriented technologies. Therefore, this paper proposes the α-max capacity model to estimate the maximum network capacity under trip or O–D LOS requirement α. The proposed model improves the existing models on three aspects: (a) it considers trip LOS, which can flexibly estimate the network capacity ranging from zero to the physical capacity including reserve, practical and ultimate capacities; (b) trip LOS can intuitively reflect users’ maximum acceptable O–D travel time or planners’ requirement of O–D travel time; and (c) it is a convex and tractable single-level MP. For practical use, we develop a modified gradient projection solution algorithm with soft constraint technique, and provide methods to obtain discrete trip LOS and network capacity under representative traffic states. Numerical examples are presented to demonstrate the features of the proposed model as well as the solution algorithm.

Distribution LMP-Based Demand Management in Industrial Park via a Bi-Level Programming Approach

To enhance industrial park's economic gains and effectively allocate its electricity bill among industrial users with combined heat and power (CHP) units and photovoltaic (PV) panels, this paper proposes a distribution locational marginal price (DLMP)-based bi-level demand management approach. The upper level optimizes dispatching decisions of industrial users with the objective of minimizing their energy bills, and the lower level is a DLMP-based market clearing problem to minimize the two-part tariff cost of the industrial park operator. In order to solve the proposed bi-level model efficiently, it is first equivalently converted into a single-level mathematical programming with equilibrium constraints (MPEC), and then reformulated as a mixed-integer second-order conic programming (MISOCP) model by linearizing bilinear terms. Numerical results demonstrate the effectiveness of our proposed bi-level method in lowering industrial park's electricity bill and achieving effective allocation among users.

Robust model predictive control with embedded multi-scenario closed-loop prediction

Robust MPC seeks to mitigate the effects of uncertainty which can lead to suboptimal MPC performance. Previous work on robust MPC includes a stochastic multi-scenario approach to simulate multiple plant realizations and create a control scheme to optimize a performance metric based on the plant scenarios. This paper seeks to combine a scenario-based approach with embedded closed-loop prediction under future MPC control action by directly incorporating MPC subproblems into the overall robust MPC formulation. This allows the current MPC to predict closed-loop MPC responses to a range of uncertain future plant realizations. The resulting multilevel programming problem is solved by reformulating the inner MPC optimization subproblems as algebraic constraints corresponding to their first-order optimality conditions, resulting in a single level mathematical program with complementarity constraints (MPCC). The performance of the robust MPC scheme is evaluated against a standard MPC formulation in linear and nonlinear case studies.

Integrated demand response for congestion alleviation in coupled power and transportation networks

The increasing prevalence of electric vehicles (EVs) and emerging dynamic wireless charging (DWC) techniques have strengthened the interdependence between transportation networks (TNs) and power distribution networks (PDNs). Under the DWC mode, EV charging demand is shifted from residential plug-in charging to charging-while-driving during commuting hours, resulting in a simultaneous congestion in coupled networks. The present study addresses this issue by developing a bi-level integrated demand response (IDR) framework for alleviating congestion in coupled networks. At the upper level, an independent system operator aims to alleviate congestion by imposing the lowest possible traffic tolls and electricity tariffs. At the lower level, rational travelers in the TN schedule their routes and departure times according to traffic tolls and traffic conditions, yielding a multi-period user equilibrium state in which the generalized travel cost of users cannot be decreased by unilaterally changing routes or departure times. Simultaneously, load aggregators in the PDN schedule flexible power demands according to electricity tariffs to minimize total energy costs. The overall bi-level programming is reformulated into a single-level mathematical program with a complementarity constraint problem, which is efficiently solved as a sequence of relaxed non-linear programming problems by a specially designed algorithm. Numerical results demonstrated the effectiveness of the proposed IDR framework in alleviating congestion and reducing total procurement costs.

Elastic Traffic Engineering Subject to a Fair Bandwidth Allocation via Bilevel Programming

The ability of TCP’s congestion control scheme to adapt the rate of traffic flows and fairly use all the available resources is one of the Internet’s pillars. So far, however, the elasticity of traffic has been disregarded in traffic engineering (TE) methodologies mainly because, only recently, the increase in access capacity has moved the bottlenecks from the access network to the operator network and hungry cloud-based applications have begun to use all the available bandwidth. We propose a new approach to TE with elastic demands which models the interaction between the network operator and the end-to-end congestion control scheme as a Stackelberg game . Given a set of elastic traffic demands only specified by their origin-destination pairs, the network operator chooses a set of routing paths (leader’s problem) which, when coupled with the fair bandwidth allocation that the congestion control scheme would determine for the chosen routing (follower’s problem), maximizes a network utility function. We present bilevel programming formulations for the above TE problem with two widely-adopted bandwidth allocation models, namely, max-min fairness and proportional fairness, and derive corresponding exact and approximate single-level mathematical programming reformulations. After discussing some key properties, we report on computational results obtained for different network topologies and instance sizes. Interestingly, even feasible solutions to our bilevel TE problems with large optimality gaps yield substantially higher network utility values than those obtained by solving a standard single-level TE problem and then fairly reallocating the bandwidth a posteriori .

Computing a Pessimistic Stackelberg Equilibrium with Multiple Followers: The Mixed-Pure Case

The search problem of computing a Stackelberg (or leader-follower)equilibrium (also referred to as an optimal strategy to commit to) has been widely investigated in the scientific literature in, almost exclusively, the single-follower setting. Although the optimistic and pessimistic versions of the problem, i.e., those where the single follower breaks any ties among multiple equilibria either in favour or against the leader, are solved with different methodologies, both cases allow for efficient, polynomial-time algorithms based on linear programming. The situation is different with multiple followers, where results are only sporadic and depend strictly on the nature of the followers’ game. In this paper, we investigate the setting of a normal-form game with a single leader and multiple followers who, after observing the leader’s commitment, play a Nash equilibrium. When both leader and followers are allowed to play mixed strategies, the corresponding search problem, both in the optimistic and pessimistic versions, is known to be inapproximable in polynomial time to within any multiplicative polynomial factor unless textsf {P}=textsf {NP}. Exact algorithms are known only for the optimistic case. We focus on the case where the followers play pure strategies—a restriction that applies to a number of real-world scenarios and which, in principle, makes the problem easier—under the assumption of pessimism (the optimistic version of the problem can be straightforwardly solved in polynomial time). After casting this search problem (with followers playing pure strategies) as a pessimistic bilevel programming problem, we show that, with two followers, the problem is NP-hard and, with three or more followers, it cannot be approximated in polynomial time to within any multiplicative factor which is polynomial in the size of the normal-form game, nor, assuming utilities in [0, 1], to within any constant additive loss stricly smaller than 1 unless textsf {P}=textsf {NP}. This shows that, differently from what happens in the optimistic version, hardness and inapproximability in the pessimistic problem are not due to the adoption of mixed strategies. We then show that the problem admits, in the general case, a supremum but not a maximum, and we propose a single-level mathematical programming reformulation which asks for the maximization of a nonconcave quadratic function over an unbounded nonconvex feasible region defined by linear and quadratic constraints. Since, due to admitting a supremum but not a maximum, only a restricted version of this formulation can be solved to optimality with state-of-the-art methods, we propose an exact ad hoc algorithm (which we also embed within a branch-and-bound scheme) capable of computing the supremum of the problem and, for cases where there is no leader’s strategy where such value is attained, also an alpha -approximate strategy where alpha > 0 is an arbitrary additive loss (at most as large as the supremum). We conclude the paper by evaluating the scalability of our algorithms via computational experiments on a well-established testbed of game instances.

Risk-based optimal power portfolio methodology for generation companies considering cross-region generation right trade

The generation right trades (GRTs) are of great significance for the improvement of unit utilization hours and reduction of carbon emission, which plays an important role in the cross-regional electricity transactions. In order to hedge the risk in spot markets and ensure the incomes, it is crucial for generation companies (Gencos) to determine the optimal GRT scheme at their risk preferences. In this paper, a power portfolio optimization methodology is proposed considering the bidding behaviours in spot markets, independent system operator (ISO) centralized dispatching, and the cross-region GRTs. Specifically, various risks for Gencos are modelled including spot price fluctuation and system component failures. Then, a bi-level optimal portfolio (BLOP) model is established where the Gencos maximize their total incomes with subject to the ISOs which minimize the local dispatching costs simultaneously. The BLOP model is transformed into a solvable single-level mathematical program with equilibrium constraints (MPCE) through the Karush-Kuhn-Tucker (KKT) conditions. The numerical results on a realistic Chinese testing system illustrate the effect of GRTs for income assurance of Gencos and the necessity to consider the stochastic contingencies in portfolio decisions. Specifically, it is a preferred option for Gencos to exploit approximately 77% of spot power to GRT at the normal state, which can increase the total income by 266.21%. At contingency states, the optimal GRT ratios are 70–80%, the corresponding income grows nearly twice more than pure spot trades.

Sufficient optimality conditions and duality results for a bilevel multiobjective optimization problem via a Ψ reformulation

ABSTRACTIn this paper, we are concerned with a bilevel multiobjective optimization problem . Using the function Ψ introduced by Gadhi and Dempe [Necessary optimality conditions and a new approach to multiobjective bilevel optimization problems. J Optim Theory Appl. 2012;155:100–114], we reformulate as a single level mathematical programming problem and establish/exhibit the global equivalence between the two problems and . Using a generalized convexity introduced by Dutta and Chandra [Convexificator, generalized convexity and vector optimization. Optimization. 2004;53:77–94], we derive sufficient optimality conditions for the problem and establish Mond-Weir duality results. To illustrate the obtained results some examples are given.