Single-level Mixed-integer Linear Program Research Articles

Planning of distributed energy storage with the coordination of transmission and distribution systems considering extreme weather

As the penetration level of renewable energy is continuously growing, it is essential for transmission and distribution system operators to collaborate on optimizing the siting and sizing of distributed energy storage to enhance the operational flexibility and economic efficiency. Given the frequent occurrence of extreme weather in recent years, the planning should also account for such factors. Hence, a planning method of distributed energy storage with the coordination of transmission and distribution systems considering extreme weather is proposed. Firstly, a Gaussian mixture model-based chance constraint is established to describe the uncertainty of wind and solar power, ensuring high confidence that the bus voltage of the distribution system is within a safe range. Secondly, aiming to maximize the social welfare, a bi-level planning model for distributed energy storage is developed. The upper-level addresses the siting and sizing issues of distributed energy storage, while the lower-level characterizes the day-ahead clearing problem of power market. By leveraging Karush-Kuhn-Tucker (KKT) conditions and linearization techniques, the bi-level model is transformed into a single-level mixed integer linear programming model that is easier to solve. Finally, numerical analysis is conducted on a modified IEEE 24-node system combined with two IEEE 33-node systems. The case study verifies the effectiveness of the proposed model.

A logistics provider’s profit maximization facility location problem with random utility maximizing followers

We introduce a strategic decision-making problem faced by logistics providers (LPs) seeking facility location decisions that lead to profitable operations. The profitability depends on the revenue generated through agreements with shippers, and the costs arising when satisfying these agreements. The latter depends, in turn, on the service levels and characteristics of the shippers’ customers. However, at a strategic level, the LP has imperfect information thereof.We propose a stochastic bilevel formulation where a given LP (leader) anticipates the decisions of shippers (followers) arising from a random utility maximization model. Using a sample average approximation and properties of the associated optimal solutions, we propose a heuristic that can compute high-quality solutions. It is based on a reduced single-level mixed integer linear programming formulation that can be solved by a general-purpose solver after a preprocessing phase. We can quickly identify situations that lead to zero expected profit for the LP. Experimental results show that expected profit highly depends on shippers’ price sensitivities. Underestimating price sensitivities can lead to an overestimation of the expected profit.

Optimal configuration of energy storage considering flexibility requirements and operational risks in a power system

The integration of renewable energy units into power systems brings a huge challenge to the flexible regulation ability. As an efficient and convenient flexible resource, energy storage systems (ESSs) have the advantages of fast-response characteristics and bi-directional power conversion, which can provide flexible support for the power system. This paper establishes an optimization model for the ESS based on a bi-level programming model. The upper-level model optimizes the decision strategy of ESS configuration planning. The lower-level model is based on scenario analysis theory to simulate the operation of typical daily scenarios. Flexibility requirement constraints are added to characterize the required flexibility resources of the power system. In addition, the conditional value-at-risk (CVaR) is applied to characterize the risk of wind curtailment and load shedding during operation. To simplify the model, a set of association constraints is introduced to convert the original bi-level programming model into a direct-solvable single-level mixed-integer linear programming (MILP) model. Finally, the effectiveness of the proposed model is verified through case studies.

A Bilevel EV Charging Station and DC Fast Charger Planning Model for Highway Network Considering Dynamic Traffic Demand and User Equilibrium

The rapid electric vehicle (EV) adoption and relatively short EV driving range urge city planners to expand charging infrastructure (i.e., EV charging stations (EVCSs) and direct current fast chargers (DCFCs)) in the highway network for supporting long-distance intercity EV travels. To properly address distinct roles as well as interactions of the EVCS planner, traffic network, power distribution networks (PDNs), and drivers, this paper explores a bilevel planning model in which the upper level determines EVCS and DCFC construction plans and the two lower-level subproblems describe the user equilibriumbased traffic assignment model of the traffic network and the operation model of PDNs. Moreover, the statistical charging time is modeled in the traffic network subproblem as a function of statistical charging demand and the number of DCFCs, leveraging the macro traffic pattern and micro charging behavior of individual drivers with balanced accuracy and computational complexity. The proposed model also captures the impacts of social factors, charging infrastructure, and driver behaviors on dynamic EV traffic demands. Reformulation and linearization techniques are applied to convert the proposed bilevel problem into a tractable single-level mixed-integer linear programming model for effective computation. The effectiveness of the proposed approach is tested on an intercity highway network with multiple PDNs. The findings from numerical case studies highlight the important interplay among the construction plans of EVCSs and DCFCs, the PDN constraints, the user equilibrium traffic network, and the EV drivers. Case studies illustrate the feasibility and necessity of considering the number of DCFCs to calculate the macro-perspective charging time in the user equilibrium traffic network model. The dynamic traffic demand, induced by social factors and traffic demand elasticity, was also found to be a crucial factor in reshaping EV traffic. Thus, properly considering these important factors would lead to a more accurate quantification of traffic demands and ultimately result in a more reasonable charging facility construction plan.

A novel framework for the day-ahead market clearing process featuring the participation of distribution system operators and a hybrid pricing mechanism

The electricity market liberalization propelled by the increased operational flexibility observed in distribution systems urges the development of mathematical models for competitive electricity markets accounting for the participation of distribution systems operators. Most existing approaches model the interaction between the agents and the market as a bilevel programming problem, wherein the market is characterized at the lower level. Hence, the upper-level agent’s objective function is favored over minimizing the costs of supplying the system’s demand. In this paper, a new bilevel formulation is proposed. In the upper level, the market operator determines the optimal energy price that ensures the demand supply and maximizes the overall social welfare. In the lower level, the generation companies and distribution systems operators determine the amount of energy injected/extracted into/from the transmission system to maximize profits/minimize operational costs. A new pricing method is proposed in this paper as an alternative to the transmission system’s dual variables, which are inaccessible under the proposed framework. The resulting nonlinear mixed-integer bilevel programming formulation is transformed into an equivalent single-level mixed-integer linear program. Numerical results illustrate the proposed approach’s effectiveness in maintaining the minimum operational costs while distributing the payoff amongst the agents.

Optimal stochastic scheduling of a multi-carrier multi-microgrid system considering storages, demand responses, and thermal comfort

This paper focuses on the day-ahead scheduling of a multi-carrier multi-microgrid that delivers electricity, heat, and hydrogen to its customers. The microgrids are equipped with renewable sources, combined heat and power, gas boiler, electric boiler, multi-energy storage, and power-to-hydrogen facilities, and aim at minimizing their operational and environmental costs. Demand response for electrical and hydrogen loads as well as thermal comfort for heat demand have been considered to increase flexibility. The distribution company, which aims at maximizing its profit, imports electricity and natural gas from the main grid. Stochastic optimization is proposed, and the generated scenarios were reduced using the mixed-integer linear programming method. Considering that the problem possesses two conflicting objective functions, the problem was two-level with nonlinear terms. The Karush–Kuhn–Tucker conditions, duality theory, and Fortuny–Amat transformation were implemented to transfer the problem into a single-level mixed-integer linear programming problem. The effectiveness of the proposed framework in four cases has been tested on a hypothetical multi-carrier multi-microgrid. Energy storage, demand response, and thermal comfort resulted in a microgrid cost reduction of 12.31 % and a distribution company profit reduction of 17.25 %. According to the results, thermal comfort can reduce MGs costs remarkably since it adds more flexibility.

Linearized AC power flow model based interval total transfer capability evaluation with uncertain renewable energy integration

Total transfer capability (TTC) evaluation plays an essential role in guaranteeing reliable power transactions in a deregulated electricity market. However, the large-scale integration of wind power brings significant challenges to TTC calculations. The conventional probabilistic TTC evaluation methods are highly dependent on the pre-defined probabilistic distributions of the input random variables, which could not be accurately predicted in practical situations. This paper proposes an interval TTC evaluation method based on the linearized AC power flow (LACPF) model, which can provide intuitive TTC boundary information without any prior distributions. The adoption of LACPF also contributes to yielding more accurate TTC values than those DC power flow (DCPF) based methods. The proposed interval optimization model is equivalently translated into an optimistic model and a pessimistic model. The optimistic model is formulated as a max-max problem and can be directly converted to a single-level linear programming model. The pessimistic model is formulated as a min-max problem that cannot be solved directly. Hence, the strong duality theory and the big-M method are introduced to transfer the original bilevel model into a single-level mixed integer linear programming (MILP) model for an efficient solution. The proposed method is validated on the IEEE 118-bus system and a modified actual power grid in Northwest China. Numerical results demonstrate its computational efficiency.

Impact of customer portrait information superiority on competitive pricing of EV fast-charging stations

The rapid development of electric vehicles (EVs) has increased the demand for public fast-charging stations (FCSs). Multiple self-interested charging service operators (CSOs) in an urban transportation network compete with each other through pricing signals to maximize their respective profits, where the information about customer demands and heterogeneous preferences work as important decision criteria. With the prevailing data transaction, the information superiority differences between CSOs may significantly affect the competitive pattern in the charging market. In this study, a Nash-Stackelberg-Nash game framework is established to investigate the strategic pricing of CSOs under different information distribution scenarios. The customer response pattern, namely, the routing and charging choices of EV drivers to pricing signals, is described via a computationally efficient extended stochastic user equilibrium model comprehensively considering heterogeneous customer preferences and decision stochasticity. Then, aiming at different data forms, both parameter-driven and sample-driven CSO pricing models are proposed. The pricing models are reformulated as a tractable single-level mixed-integer linear program, and the game equilibrium is solved alternately via Gauss-Seidel iteration. Numerical simulations are conducted to analyze the impact of single CSO information superiority and double CSO information distribution on CSO profitability and customer experience in both data forms. The influences of endogenous factors like CSO data amount and processing ability, as well as exogenous scenario parameters like customer preference distribution and perception error distribution, are discussed.

A Risk-Averse Adaptive Stochastic Optimization Method for Transactive Energy Management of a Multi-Energy Microgrid

This paper proposes a new energy management method for a multi-energy microgrid (MEMG) which supplies both electrical and thermal energies. Based on the transactive energy (TE) concept, the problem is formulated as a Stackelberg game-theoretic bi-level optimization model. The MEMG operator optimizes the energy scheduling and pricing strategies at the upper level, and the industrial, commercial and residential agents optimize their energy trading strategies at the lower level. The equivalent single-level mixed-integer linear program (MILP) reformulation is then derived for computational tractability. To coordinate the strategies made in the day-ahead and intra-day energy markets, an adaptive stochastic optimization (SO) approach is adopted, by which a day-ahead stochastic MILP and an intra-day deterministic model are formed. Furthermore, the conditional value-at-risk (CVaR) measure is incorporated in the day-ahead stage for risk aversion of the MEMG operator towards uncertainties. To solve the models, an adaptive Progressive Hedging (PH) algorithm is developed to decompose the day-ahead stochastic MILP into multiple scenario-based subproblems which can be solved in parallel, and an outer approximation (OA) algorithm is adopted in the intra-day stage to linearize the bilinear objective function. Finally, the simulation results verify the effectiveness of the proposed energy management method and the efficiency of the solution algorithms.

A data-driven Stackelberg game approach applied to analysis of strategic bidding for distributed energy resource aggregator in electricity markets

In the context of the participation of distributed energy resource (DER) aggregators in day-ahead energy and reserve markets, most of the existing analysis models of strategic bidding behavior of DER aggregators ignore the consideration of the safe operation of distribution systems. To this end, this paper builds a one-leader multi-follower Stackelberg game model considering the safety checking of distribution systems with the discrete features of capacitor banks and tap ratio of the substation transformer in the lower level. Mathematically, this model constitutes a bi-level mixed-integer nonlinear programming (BMINLP) problem. We convert the BMINLP model to a bi-level mixed-integer linear programming (BMILP) model based on KKT conditions, strong duality theorem, and big-M approach, which is still difficult to be solved directly. As a result, we propose a customized data-driven algorithm based on sampling without tuning hyper-parameters and an improved linear surrogate function method, to accurately fit the corresponding constraints of the lower-level model of BMILP problem, and then transform the BMILP problem into a single-level mixed-integer linear programming (MILP) problem to solve. The simulation of data-driven algorithm and comparison with the particle swarm optimization algorithm shows the high practicability of the bi-level Stackelberg game model and the effectiveness of the data-driven algorithm.

Optimal bidding strategy of multi-carrier systems in electricity markets using information gap decision theory

This paper proposes a novel optimization framework based on information gap decision theory (IGDT) for strategic participation of multi-carrier systems (MCS) in the electricity market under price uncertainty. The proposed framework helps the operator of MCS to take advantage of flexible generation units such as combined heat and power (CHP) systems, boilers, and battery energy storage systems (BESS) and participate in the electricity market considering the strategies presented by IGDT against the uncertainty in the market clearing price. The proposed optimization framework is developed as a bilevel problem, in which the risk-based bidding strategy of the MCS operator is modeled in the upper-level problem, and the energy market is cleared by the bulk power system operator in the lower-level problem. Karush–Kuhn–Tucker conditions are used to transform the bilevel problem into a single-level optimization problem. Duality theory is later used to approximate the product of power and market clearing price in the MCS operation problem, and the non-linearity enforced by the implementation of IGDT is handled by a linearization method called Triangle method. The studied optimization framework is modeled as a single-level mixed-integer linear programming problem, and CPLEX is used to solve it. The simulations are carried out on the modified IEEE 6-bus transmission network, and the results are analyzed to verify the economic and computation efficiency of the studied model. According to the simulation results, by taking a risk-averse bidding strategy provided by the IGDT, the operation cost of MCS increases 16.6% in order to increase the robustness level of MCS up to 36%, indicating that MCS becomes robust against 36% of the increase in the electricity market price, compared to the case with a risk-neutral bidding strategy being taken.

RETRACTED: Robust bidding strategy of interconnected multi-carrier systems in the electricity markets under the uncertainty in electricity load

This paper presents a robust bi-level model for optimal bidding of interconnected multi-carrier systems (MCSs) in the wholesale electricity markets under the uncertainty in electricity load. The proposed model implements a robust optimization method (ROM) to model the risk-based bidding of interconnected MCSs under the uncertainty in load in the upper-level problem while including the energy and reserve markets in the lower-level problem. The studied MSCs include combined heat and power (CHP) units, boilers, central air conditioning (CAC) systems, and battery storage units to provide local energy and make the MCSs’ operator capable of participating in the reserve market to gain profit. ROM models the electricity demand and provides different bidding strategies (i.e., risk-seeking, risk-neutral, and risk-averse) for the participation of MCSs’ operator in the energy and reserve markets. The studied bi-level model is converted to a single-level mixed-integer linear programming problem using the Karush–Kuhn–Tucker conditions and strong duality and then solved using the CPLEX solver of the General Algebraic Modeling System. Numerical experiments reveal that the interconnection of MSCs, along with the integration of storage units, enables the MSCs’ operator to control the uncertainty in the electricity load and increase the energy efficiency in order to optimally bid in the day-ahead energy and reserve markets. Results show that by enabling the MCSs to locally trade energy, the total operation cost of MCSs in the risk-seeking, risk-neutral, and risk-averse modes decreases by 5.65%, 4.28%, and 3.49%. Moreover, with the integration of battery storage systems into the interconnected operation of MCSs, the total operation cost of systems in the risk-seeking, risk-neutral, and risk-averse modes decreases by 7.65%, 6.99%, and 6.40%, respectively.

Multi-stage robust optimal economic accommodation method for renewable energy considering carbon emission cost

The integration of renewable energy contributes to the target of energy saving and emission reduction, which helps to realize “2030 carbon peak, 2060 carbon neutral” strategy of China. Renewable outputs, on the other hand, exhibit uncertainties, intermittentness, and volatility. To accommodate renewable energy, traditional generator assets such as thermal units must incur higher operating costs. To that end, optimizing power system scheduling requires balancing the benefits and costs of renewable integration. In addition, with the establishment of the carbon trading market, carbon emission cost has become one of the essential costs of power generation. As a result, this paper develops a multi-stage robust optimal economic accommodation model that accounts for both operation and carbon emission costs. Besides, the solution nonanticipativity and robustness can be guaranteed by the established. The optimal economic accommodation model is then equivalently reformed into a single-level mixed integer linear programming (MILP) structure using a multi-stage robust explicit decision method with affine policies. Furthermore, a fast evaluation method for economic accommodation rate is proposed based on the solved affine coefficients. Numerical testing confirms that the proposed method outperforms traditional methods. Furthermore, the proposed method avoids paying 52.93% more cost to increase the accommodation rates by 5.74%, 0.74%, 2.28%, and 1.14% for four farms.

Low-carbon planning for park-level integrated energy system considering optimal construction time sequence and hydrogen energy facility

With the increasing concern about global energy crisis and environmental pollution, the integrated renewable energy system has gradually become one of the most important ways to achieve energy transition. In the context of the rapid development of hydrogen energy industry, the proportion of hydrogen energy in the energy system has gradually increased. The conversion between various energy sources has also become more complicated, which poses challenges to the planning and construction of park-level integrated energy systems (PIES). To solve this problem, we propose a bi-level planning model for an integrated energy system with hydrogen energy, considering multi-stage investment and carbon trading mechanism. First, the mathematical models of each energy source and energy storage in the park are established respectively, and the independent operation of the equipment is analyzed. Second, considering the operation state of multi-energy coordination, a bi-level planning optimization model is established. The upper level is the capacity configuration model considering the variable installation time of energy facilities, while the lower level is the operation optimization model considering several typical daily operations. Third, considering the coupling relationship between upper and lower models, the bi-level model is transformed into a solvable single-level mixed integer linear programming (MILP) model by using Karush–Kuhn–Tucker (KKT) condition and big-M method. Finally, the proposed model and solution methods are verified by comprehensive case studies. Simulation results show that the proposed model can reduce the operational cost and carbon emission of PIES in the planning horizon, and provide insights for the multi-stage investment of PIES.

Active Model Discrimination for Piecewise Affine Inclusion Systems

This paper proposes a novel set-membership active model discrimination (AMD) algorithm for actively separating/discriminating among a set of piecewise Affine inclusion systems with bounded noise. Specifically, to overcome the difficulties of dealing the integer variables in the lower/inner level of the associated bilevel optimization problem that stem from the mapping of the piecewise inclusions and subregions, we propose an alternative reformulation that moves the integer variables into the higher/outer level. This reformulation allows us to leverage Karush-Kuhn-Tucker (KKT) conditions to obtain an equivalent single level mixed-integer linear programming (MILP) problem. Moreover, in contrast to standard AMD algorithms that strictly enforces model separation, we propose a slight modification/extension that separates as many models as possible when a strict separation of all models is not possible.

Co-Operative Optimization Framework for Energy Management Considering CVaR Assessment and Game Theory

In this paper, a bi-level energy management framework based on Conditional Value at Risk (CVaR) and game theory is presented in the context of different ownership of multiple microgrid systems (MMGS) and microgrid aggregators (MAs). The energy interaction between MMGS and MAs can be regarded as a master–slave game, where microgrid aggregators as the leaders set the differentiated tariff for each MG to maximize its benefits, and MMGS as the follower responds to the tariff decision specified by the leader through peer-to-peer (P2P) energy sharing. The P2P energy sharing of MMGS can be regarded as a co-operative game, employing asymmetric Nash bargaining theory to allocate the co-operative surplus. The Conditional Value at Risk model was used to characterize the expected losses by microgrid aggregators due to the uncertainties of renewable energy resources. The Karush–Kuhn–Tucker conditions, Big-M method, and strong duality theory were employed to transform the bi-level nonlinear model of energy management into a single-level mixed integer linear programming model. The simulation results show that when MGs adopt the P2P energy-sharing operation mode, the total operating cost of MMGS can be reduced by 7.82%. The simulation results show that the proposed co-operative optimization framework can make the multiple microgrid systems obtain extra benefits and improve the risk resistance of microgrid aggregators.

Bi-level Optimization Model of PMU Fault Recovery Based on Observability

Aiming at the failure of PMU caused by cyber-attacks and extreme natural disasters, a bi-level optimization model of PMU fault recovery based on observability is proposed in this paper. By optimizing the sequential restoration of PMU, the control center can maximize the state observability of the system under the condition of limited recovery resources and adjust the power in real time according to the recovered observable nodes, so as to alleviate the power imbalance caused by a disturbance in the recovery process. The upper optimization model takes the maximum observable branch power flow as the optimization objective, and the lower optimization model is constructed as the DC optimal power flow model with the lowest power generation cost. The upper and lower models are coupled by the branch power flow and the node observation state. Then, the lower model is transformed into KKT conditions through Lagrange multiplication, and the nonlinear objectives and constraints are linearized so that the bi-level model can be transformed into a single-level mixed integer linear programming model for solving. Finally, the effectiveness of the proposed strategy is verified by simulation analysis of IEEE39 standard systems.

A local flexibility market framework for exploiting DERs’ flexibility capabilities by a technical virtual power plant

This paper presents a new intra-day intra-hourly local flexibility market (LFM) framework to exploit distributed energy resources’ (DERs) flexibility capabilities. A technical virtual power plant (TVPP) operates the LFM in which the DER aggregators participate. The TVPP offers the provided flexibility capabilities to wholesale flexibility market (WFM) as well as compensating the intra-hourly variability in power distribution network. In the proposed framework, the TVPP clears the LFM by considering hierarchical transactions with the aggregator agents to find the market equilibrium in which the DERs’ flexibility capabilities are optimally exploited while all participating agents make profit by trading flexibility capabilities in the LFM. A bilevel optimization model with multiple lower levels is considered to address different agents’ preferences and transactions in the LFM. In the upper-level problem, the TVPP aims at maximizing its profit while each lower-level problem represents an aggregator agent's optimization problem. The proposed model is reformulated into a single-level mixed integer linear programming problem and is implemented on the distribution network connected to Bus 5 of the Roy Billinton test system (RBTS) as well as a 119-bus test system. The results demonstrate the effectiveness of the model to utilize DERs’ flexibility and provide revenue opportunities for different agents.

Day-Ahead Strategic Operation of Hydrogen Energy Service Providers

The green hydrogen provides a pathway to the 100% sustainable energy future, rendering hydrogen energy service providers (HESPs) an important role in energy supply. Different from the conventional electricity service providers, HESP is not only responsible for the production, purchase and retailing but also the transportation of hydrogen, which makes its operation a challenging problem. To this end, this paper proposes a bi-level strategic operation model for HESPs. In the operation level, the bidding, hydrogen production and hydrogen transportation are coordinated to minimize the overall cost of HESPs. In the market level, the market clearing is simulated to estimate the influence of HESP behavior on electricity prices in the market. Then, a model reformulation technique is developed to connect the discrete-time based hydrogen production and continuous-time based hydrogen transportation. The bi-level optimization model is transformed into a single-level mixed-integer linear programming (MILP) problem with Karush-Kuhn-Tucker (KKT) conditions. The case studies with the modified IEEE-RTS-79 system were given to validate the proposed model and confirm the necessity to coordinate the hydrogen production and transportation in the strategic operation of HESPs. Sensitivity analysis is also conducted to study the potential influencing factors.

A game-theoretic model for wind farm planning problem: A bi-level stochastic optimization approach

As the electric network progressively shifts away from fossil fuel-fired generators to inverter-based generators, including most renewable energy sources (RESs), considering the system strength measure in the planning and operation problems is essential to maintaining the power system stability. To this end, this paper aims to introduce a new game-based bi-level framework for the wind farm planning problem (WFPP). The upper level seeks to maximize each investor's profit, and the lower level aims to clear the electricity market by considering the optimal investment decisions of investors. One innovative contribution of this paper is to incorporate the equivalent short circuit ratio (ESCR) as an indicator for the system strength measure in the WFPP. The bi-level model is converted into its equivalent single-level mixed-integer linear programming (MILP) employing the Karush-Kuhn-Tucker (KKT) conditions and strong duality theorem. This will yield a generalized Nash equilibrium problem (GNEP) containing binary decision variables, which is further transformed into a MILP. The intermittency of wind-demand scenarios is considered in the mathematical formulation of the presented bi-level model through scenario-based stochastic programming. Numerical studies are carried out to validate the effectiveness of the proposed model.