Bi-level Model Research Articles

A similarity-based data-driven car-following model considering driver heterogeneity

Human drivers usually have distinct driving patterns and preferences. Driver heterogeneity is crucial for modeling driving behaviors. This paper incorporates driver heterogeneity with data-driven approaches to predict car-following behaviors. A bi-level similarity-based car-following model is proposed to predict the vehicle's moving distance. In the upper level, drivers with similar driving patterns as the ego vehicle are identified using k-nearest neighboring (kNN) search. In the lower level, leveraging kNN model, candidate records are selected from the identified vehicles’ trajectories and applied to predict the ego vehicle's moving distance, combining the driving pattern similarity measured in the upper level. By taking into account the driver heterogeneity, the proposed model is capable of identifying the most relevant driving situations, which leads to an improvement of prediction accuracy. Furthermore, the established bi-level structure largely shrinks the searching space of candidate records, which reduces the searching complexity and enhances computational efficiency. We quantitatively evaluate and compare the performance of the proposed model in terms of both prediction accuracy and computational efficiency using real-world vehicle trajectory data.

Bi-Level Model Management Strategy for Solving Expensive Multi-Objective Optimization Problems

Bi-Level Model Management Strategy for Solving Expensive Multi-Objective Optimization Problems

Optimising hierarchical emergency logistics network under ambiguous demand and transportation cost

This article studies the response problem of an emergency logistics network with a decision hierarchy relationship under uncertainty. To account for the partial distribution information about uncertain demand and transportation costs, we construct a moment-based ambiguity set based on limited historical data, where the pivot variable method is employed to determine the confidence interval of the mean value. Based on the constructed ambiguity set, we develop a novel distributionally robust bi-level post-disaster emergency logistics location-routeing model. By exploiting the structural characteristic, chance-constrained models under box-ellipsoid and budget perturbation sets are reformulated as bi-level mixed-integer conic programming models. To accelerate the solution procedure, the bi-level models are further converted into single-level ones via Karush-Kuhn-Tucker condition, which can be directly solved to optimality using CPLEX software. Supply risk value for each supplier is obtained by applying analytic hierarchy process. We conduct extensive experiments using the Iranian flood as the case study to address the computational performance of our proposed optimisation method.

Allocation and trading schemes of the maritime emissions trading system: Liner shipping route choice and carbon emissions

In September 2020, the European Union (EU) decided to include the maritime industry into the EU emissions trading system to effectively control carbon emissions from ships, namely the maritime emissions trading system (METS). Various allocation and trading schemes are adopted in practice. This paper investigates a comparison of allocation and trading schemes of the METS from the perspective of liner shipping carriers and regulators, respectively. We explore how to determine the optimal allocation and trading scheme of the METS and how the liner shipping route (LSR) choice decision (following or avoiding the METS) and carbon emissions will be affected. For liner shipping carriers, we use a mixed-integer linear programming model that aims to reduce carriers’ costs to formulate our problem. For regulators, we present a bi-level programming model that aims to reduce carbon emissions. In numerical experiments, we further analyze the impact of joining the METS on the scheme choice, LSR choice and emissions by considering the European Economic Area and the Red Sea being covered by the METS. Numerical experiments are provided to show the effectiveness of our proposed models.

Bidding optimization for cascade hydropower plants in multi-regional electricity markets

Efficient cross-regional allocation of hydropower is critical for the low-carbon energy transition. However, co-optimizing bidding curves for different regional markets presents significant challenges for hydropower producers, due to the complex interplay of bidding strategies, multiple factors that contribute to uncertainty, and the diversity of regional markets. This paper proposes a novel bi-level model to optimize the bidding curves of cascade hydropower plants for multi-regional electricity markets with detailed modeling of their spatio-temporal correlations. The proposed model can adapt to regional markets with varying information transparency by employing a flexible approach to market modeling. A modified interval programming method is proposed to address the multiple uncertainties in the bidding, which mitigates the excessive conservatism inherent in traditional interval programming while preserving its robustness to the distribution estimation error. To efficiently solve the mathematical model, the complex nonlinear dependencies among interval variables are first eliminated through Taylor expansion and a modified multi-dimensional interpolation method. Subsequently, the order relation of interval and Karush-Kuhn-Tucker conditions are utilized to reformulate the proposed model into a tractable mixed-integer linear programming model. Numerical tests conducted on a real cascade hydropower system located in China validate the effectiveness and advantages of the proposed method.

An Optimization Method of Flexible Manufacturing System Reliability Allocation Based on Two Dimension-Reduction Strategies

As increasingly extensive applications of flexible manufacturing systems (FMSs) arise, their reliability allocation has been a research hot spot. But, since FMSs are always composed of transfer and buffer devices, production machines, and complex control systems, the large number of basic elements makes the number of variables and constraints of reliability-allocation optimization increase greatly, which leads to the difficulty and inefficiency of optimization. To solve the above problem, two dimension-reduction strategies are proposed for the FMS reliability optimization with low cost and a high level of reliability as the objectives, and they are the reliability-weight double-threshold qualification strategy (RWTS) and the bi-level optimization strategy (BLOS), respectively. Based on these two strategies, an overall reliability-allocation optimization model and a bi-level reliability-allocation optimization model are established based on the FMS reliability evaluation presented in our previous work, and their algorithms based on particle swarm optimization (PSO) are presented. In terms of their contributions, for the RWTS, thresholds of reliability and the weight index of each basic element are set to dynamically reduce the number of variables in each iteration of the optimization; for the BLOS, the overall reliability-allocation optimization problem for transitioning from the FMS to basic elements can be transformed into simpler allocation optimizations from the FMS to subsystems and from subsystems to basic elements, which have fewer variables, and this can largely improve the optimization convergence performance. Through applying this to a box-parts finishing FMS, compared with the traditional optimization method, the high efficiency and the good allocation effect of the optimization based on these two strategies for improving convergence speed are verified by the simulation results. The proposed method has great significance for FMS design due to its limited cost but high-reliability requirement.

Bi-level optimal dispatching of distribution network considering friendly interaction with electric vehicle aggregators

The widespread application of electric vehicles (EVs) is a positive force driving green development. However, their widespread penetration also poses significant challenges and threats to the security and stable operation of the power grid. To address this urgent issue, this article constructs a bi-level optimal dispatching model fostering collaboration between electric vehicle aggregators and the distribution network. The upper-level optimization targets the minimization of peak-valley differences in the distribution network via considerably arranging power outputs of gas turbines, while the lower-level one focuses on reducing the charging expense of EV aggregators via efficient charging transfer. Note that the charging expense is not only composed of electric cost but also a dynamic carbon emission factor-based cost, which contributes to the electricity economy and carbon reduction concurrently. A geometric mean optimizer (GMO) is introduced to solve the mode. Its efficiency is evaluated against three typical algorithms, i.e., genetic algorithm, great-wall construction algorithm, and optimization algorithm based on an extended IEEE 33-bus system with different charging behaviors of EVs on both a typical weekday and weekend. Simulation results demonstrate that the GMO outperforms other competitive algorithms in accuracy and stability. The peak-valley difference between the distribution network and the total cost of EV aggregators can be decreased by over 98% and 76%, respectively.

Low carbon-oriented planning of shared energy storage station for multiple integrated energy systems considering energy-carbon flow and carbon emission reduction

—With the development of energy storage technology and sharing economy, the shared energy storage in integrated energy system provides potential benefit to reduce system operation costs and carbon emissions. This paper presents a bi-level carbon-oriented planning method of shared energy storage station for multiple integrated energy systems. Firstly, the energy-carbon relationship of the multiple integrated energy systems is established, and the node carbon intensity models of power grid, integrated energy system and shared energy storage station are established. Secondly, a bi-level planning model of shared energy storage station is developed. The upper layer model solves the optimal capacity planning problem of shared energy storage station to minimize average emission reduction cost in a long time scale. The lower layer model solves the optimal operation problem of multiple integrated energy systems with the goal of minimizing the operation cost in a short time scale. Furthermore, considering the uncertainty of wind and photovoltaic power in integrated energy system, the conditional value at risk method is adopted as the risk measurement index in the lower layer operation cost. Finally, the case study verifies the advantages of the proposed method in economy and environmental friendliness through the comparative analysis of three different energy storage planning cases. Simulation results show that, compared with the energy storage planned separately for each integrated energy system, it is more environmental friendly and economical to provide energy storage services for each integrated energy system through shared energy storage station, the carbon emission reduction rate has increased by 166.53 %, and the system operation cost decreases by 33.48 %.

Alliance or Independence? Optimal Operation of Commuter Buses by Suburban Employers Based on Cooperative Game Theory

To reduce commuting expenses incurred by suburban employers and their urban employees, this paper proposes the operation of commuter buses by an employer alliance and builds a bi-level programming model to optimize their operation. The upper model determines the composition of the alliance with the objective of minimizing total commuting expenses. The lower model optimizes the commuter bus routes so as to minimize the total commuting expenses of the alliance through a consideration of differences in the value of employees’ time. We fairly allocate total operating costs among alliance members based on cooperative game theory, which reinforces the optimized alliance’s stability. This paper then takes Meishan District in Ningbo as a numerical case study. The analysis shows the following: (1) under the alliance model, the total commuting expenses of three allied employers will be reduced by 8.09% per day; (2) the total commuting time of employees can be reduced by 3.75% per day; moreover, (3) the spatial distribution of the bus stops has a significant impact on employers’ willingness to participate in an alliance as opposed to operating an independent commuter bus service. Specifically, the farther dispersed the bus stops and the fewer the employees at each stop, the greater the likelihood that the employer will participate in an alliance.

EMD-based model with cooperative training mechanism for tourism demand forecasting

A novel EMD-based model with the cooperative training mechanism is proposed to enhance the tourism demand forecasting performance of tourist attraction. Our proposed model develops a cooperative training mechanism integrating a bi-level optimization model, a differential evolution algorithm and interactive fuzzy programming to optimize the neural networks of sub-series decomposed by EMD, which overcomes the difficulty of capturing the relationship among the sub-series and acquires the forecasting accuracy. To verify the performance of the model proposed in this study, extensive experiments are conducted. The experimental results illustrate the superiority of our model among all benchmark models and indicate its effectiveness in incorporating the cooperative training mechanism into the decomposition-ensemble model.

Joint optimal pricing strategy of shared autonomous vehicles and road congestion pricing: A regional accessibility perspective

Current studies on shared autonomous vehicles (SAVs) and road congestion pricing (RCP) mainly focus on their independent effect on mobility, and rarely consider the joint impact of the two policies on accessibility. In light of this incentive, this paper establishes a bi-level programming model with multi-objective functions and multi-decision variables to solve the joint optimal pricing problem of SAVs and RCP considering accessibility. Because of the complex relationship between the decision variables and the objective functions, the proposed bi-level programming model belongs to a black-box problem. A machine learning algorithm based on multi-objective Bayesian optimization was developed to solve the proposed model, by taking Jiangyin City as an empirical case study. The results show that the combined implementation of SAVs and RCP plays an essential role in enhancing the performance of land-use and transportation systems. However, this advantageous outcome relies on an appropriately appropriate pricing strategy. Failing to account for the influence of the pricing strategy on implementation efficacy, SAVs and RCP could result in a 22.2 % and 5.0 % degradation in regional accessibility and total flow time, respectively. In contrast, the joint optimal pricing strategy of SAVs and RCP can not only mitigate traffic congestion and improve transportation network efficiency but also optimize urban spatial distribution and promote compact urban development, ultimately achieving a 14.0 % improvement in regional accessibility and 18.5 % reduction in total flow time. Sensitivity analysis reveals that with the increase in SAV market penetration rate, regional accessibility sees a steady rise. Conversely, the total flow time initially escalates before experiencing a decline. The method proposed in this paper serves not only in determining the joint optimal pricing strategy of SAVs and RCP to enhance accessibility, but also in analyzing and optimizing the accessibility impact of other land-use and transportation policies.

Whether the off-design characteristics of equipment should be considered in integrated energy system optimization design models – A case study

The operational performance of integrated energy systems (IESs) varies with their configuration. The equipment model directly influences the capacity configuration of IESs. The conventional static model assumes that equipment efficiency is constant, ignoring its off-design characteristics. For this reason, this paper establishes a dynamic model considering the off-design characteristics of the equipment. Based on the dynamic and static models, bi-level optimization models with the minimum total cost as the optimization objective are established to optimize the configuration of IESs, and the optimization results are System 1 and System 2, respectively. Then, the operational performance of the two systems under the dynamic optimal scheduling model is compared. The results demonstrate that the capacity of the absorption chiller (ABC), combined heating and power (CHP) unit in System 1 are larger than in System 2 owing to the difference in equipment models, whereas the capacity of ground source heat pumps (GSHPs) shows the opposite trend. The larger capacity of the GSHP increases the grid power consumption of System 2 but also reduces its operating efficiency. The greater operation energy consumption and grid power consumption make the operating cost, carbon tax, and total cost of System 2 under the design condition higher than that of System 1, with an increased rate of 1.38%, 3.47%, and 1.14%, respectively. As with the design condition, the carbon tax, operating cost, and total cost of System 2 are higher than those of System 1 under both summer and winter operating conditions, particularly under summer operating conditions, where the carbon tax, operating cost, and total cost increase by 2.27%, 1.24%, and 0.88% respectively. Even if the energy price changes, the carbon tax, operating cost and total cost of System 2 in a typical year are still greater than those of System 1. Therefore, it is necessary to consider the off-design characteristics of equipment in the bi-level optimization design model.

Determining an Optimal Government Subsidy Scheme for Shared Parking Management via a Bi-Level Programming Approach

A bi-level programming model is formulated to determine a government subsidy scheme for shared parking management. The government, as the upper-level decision-maker, seeks to maximize the social benefit by providing subsidy to the shared parking platform, incentivizing the platform to rent more shared parking slots, which can increase the utilization rate of idle parking slots and reduce the curbside parking cruising time of parking demanders. At the lower-level formulation, the shared parking platform, as a reseller, not only matches shared parking slot demanders but also determines which shared parking slots to rent, which is approached by a two-sided decision-making process. A “one-to-many” matching principle is adopted for the platform to maximize its revenue. A modified genetic algorithm is designed to solve the proposed model. Results indicate that the government subsidy has a positive impact on stimulating the shared parking market. Specifically, from the perspective of the government, the subsidy can increase the number of matched shared parking demanders, improve the supply of parking slots, and reduce curbside parking cruising time. From the perspective of the shared parking platform, the subsidy can lead to higher revenue.

Financial arbitrage in electricity markets via virtual bidding

The concept of virtual bidding serves as a pure financial tool in electricity markets, aiming at enhancing their performance. Although virtual bidding helps reduce price gaps between different trading floors, there is evidence suggesting that this does not necessarily lead to improved welfare for market consumers. To gain a deeper understanding of virtual bidding and its impact on electricity market prices, this work delves into the strategic behavior of a financial player engaged in an electricity pool-based market. To this end, a bi-level model is formulated. The upper level optimizes the expected revenues of the virtual bidder while the lower-level represents the clearing of the market under network-constrained economic dispatch through a two-stage stochastic programming. The proposed model derives optimal arbitrage strategies in both scenarios of explicit and implicit virtual bidding. It also assesses the performance of virtual bidding under stochastic generation and transmission contingencies, including considerations of up-to-congestion virtual bidding.

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/ .

Bi-level planning approach for incorporating the demand-side flexibility of cloud data centers under electricity-carbon markets

With the prevalence of cloud computing applications, cloud data centers (CDCs) are proliferating around the world. However, in the context of hybrid energy‑carbon markets environment, the high energy costs and environmental expenses caused by CDC operation are pressing CDC owners to restructure the future development of CDC in a more economical and low-carbon manner. The potential spatio-temporal transferability and reducibility of workloads provides CDCs with significant flexibility in their operations and thus may interact with the power grid as active demand users. Nonetheless, other than private data centers, public CDCs have no direct control over the workloads submitted by terminal cloud users. As such, this paper presents a bi-level model for CDC allocation planning, so as to incorporate cloud service-demand response (CS-DR) from the perspective of a hybrid electricity-carbon market. The upper level pertains to multi-domain resource collaborative planning model, which determines the optimum siting and sizing of CDCs as well as the incentive design for CS-DR program, with the objective of maximizing the total expected benefits of CDC. The lower level models correspond to the market clearing (electricity-carbon tariff model) of Independent system operator (ISO) and the decision-making of cloud users regarding CS-DR participation. The proposed model belongs to a bi-level mixed integer nonlinear programming problem with two non-convex lower levels, which can be intractable in mathematics. To solve such difficult problem, a hybrid solution method combining multiple linearization techniques and reformulation and decomposition (R&D) strategy based on column-and-constraint generation (C&CG) algorithm is developed. The proposed model is demonstrated on a modified IEEE 30-bus test case, and the simulation results verified the effectiveness of the proposed approach.

Low-Carbon economic scheduling with Demand-Side response uncertainty in regional integrated energy system

Effective mitigation and reduction of carbon emissions is consistently underscored in Regional Integrated Energy Systems (RIES), aligning with the goals of carbon-neutralization. However, the inherent trade-off between carbon emissions reduction and economic operation is an obvious challenge not only in power systems but especially in the coupled electricity-gas-heat energy systems. This paper develops a bi-level low-carbon optimal scheduling model, highlighting reliable operation and the balance of supply and demand of RIESs with demand-response uncertainty. In this paper we firstly redefined the carbon emission of a RIES as a carbon flow and proposed a reward and punishment carbon trading mechanism based on the ladder carbon price. Secondly, we designed an incentive-based demand-side response policy within the subsystem encompassing electricity, gas, and heat. Concurrently, the probability density distribution functions of transferable, replaceable, and interruptible loads are computed separately. Based on the above, the deviation of the load reduction and the actual load is recalculated to delimit the constraint of the demand-side response. Finally, the mechanism of carbon trading and the demand-response is leveraged to address the economic scheduling incorporating operation cost, carbon transaction cost, imbalance costs of supply and demand, and load loss. Simulations demonstrated in the paper prove that the optimal solution with demand-response uncertain may effectively increase the economic performance and reduce carbon emissions of a RIES.

Day-ahead optimal dispatch of a virtual power plant in the joint energy-reserve-carbon market

As an efficient technique to manage massive distributed energy resources (DER), the virtual power plant (VPP) can leverage the aggregated flexibility to enhance the economy and security of the system. However, few efforts have been reported on the carbon emission reduction benefit of the VPP. In this paper, a novel incentive mechanism for low-carbon demand response is designed, which is based on dynamic pricing and carbon intensity. Considering the uncertainty of renewable energy output characterized by the chance-constrained approach, a coordinated optimization model in the joint day-ahead energy, reserve and carbon market for the VPP operator and flexible consumers is proposed, including the pre-dispatch model, the energy storage systems (ESSs) incorporated carbon emission flow calculation model, and the bi-level final-dispatch model. Numerical simulations of both small-scale and large-scale cases demonstrate that the proposed low-carbon dispatch strategy can boost the payoff of the VPP operator, reduce carbon emissions in the joint energy-reserve-carbon market, and improve the robustness of the power system against uncertainty.

Node location of bi-level urban metro-based ground-underground logistics distribution

Metro-based underground logistics (M-ULS) can relieve the pressure of urban traffic and avert the road congestion through coordinating with ground logistics system. A bi-level node location model is proposed based on the ground-underground logistics distribution mode. The cost of delivery and pickup from the perspective of service companies and customers are minimized as the upper and lower objectives. Since the model is belonged to NP-hard problem, the simulated annealing algorithm (SA algorithm) whose new solution generation part was improved according to the characteristics of the model is used to solve the problem. In the case analysis part, the distribution nodes and customer points close to Nanjing Metro Line 3 were selected to verify the validity of the model and algorithm. The result shows that the improved SA algorithm outperformed in taking less iteration times and obtaining better solutions. The total transportation cost of the system can be reduced by 21.7 % with the optimized node location plan, and 63.6 % of the ground transportation distance can be replaced by the underground transportation, such results are also obtained when the cargo demand increases by 1.5 times and 2 times, which verifies the advantages of M-ULS node location in reducing truck emission and total delivery cost.

Resilience-economy-environment equilibrium based configuration interaction approach towards distributed energy system in energy intensive industry parks

Because unexpected, high-impact, extreme events and disasters can seriously threaten power system safety and resilience, there has been an increased research focus on power system resilience. This study proposes a bi-level multi-objective model for industrial park distributed energy configuration optimization to deal with extreme events, which considers the interactions between the authority and the industrial park in a leader–follower decision process and seeks to trade-off between economic cost, environmental protection, and power system resilience. An interactive algorithm is designed and the ɛ-constraint method is used to convert the model into a bi-level single objective model to solve the proposed complex model. A case example from an industrial park in Wuxi, Jiangsu Province, China, is given to demonstrate the practicality and efficiency of the proposed optimization method. The joint analysis of the power supply and financial losses found that a distributed energy system with an optimal resilience configuration could guarantee 95.85% of demand for three consecutive hours after an extreme event, maintain more than 50% of demand at any time, and reduce power outage financial losses at any time to a minimum of 0.232 million CNY. Based on the analyses and discussion, this model was proven to provide a reasonable and practical strategy for a resilience-economy-environment trade-off in industrial park distributed energy systems.