Bilevel Programming Research Articles

Carbon emission allocation policy making in liner shipping: A novel approach toward equitable and efficient maritime sustainability

Maritime shipping, an integral component of global trade, accounts for nearly 3% of global carbon dioxide emissions and faces significant challenges in mitigating its carbon footprints. This study addresses the shortcomings of current carbon emission surcharges levied by shipping companies and the deficiencies of allocation methods utilized by the government, such as the inequity of uniform carbon emission charges across diverse origin–destination (od) pairs and the reliance on potentially inaccurate self-reported emissions data to assign carbon emission allowances. We propose a novel approach to carbon emission allocation per twenty-foot equivalent unit (TEU) container per od pair in liner shipping from the perspective of the government, aiming to enhance government oversight and minimize social costs. Leveraging a bi-level programming model, our research navigates the complex interplay between government regulatory objectives and shipping companies’ operational decisions, identifying a socially optimal carbon allocation policy. Our approach introduces three innovative carbon allocation methods that advance beyond the traditional metric that typically focuses solely on emissions and cargo volume. Our first two innovative methods integrate shipping distances into the carbon allocation process, specifically considering both the actual traveling distance and the shortest sailing distance. Furthermore, we propose the third innovative “harmonic distance,” which is a convex combination of the actual traveling distance and the shortest sailing distance. This measure effectively reflects both the operational realities and equity concerns associated with shipping services. Our findings reveal the effectiveness of utilizing the shortest sailing distance for carbon allocation per TEU per od pair, offering a straightforward, route-independent, and equitable solution. This approach diverges from current industrial practices, favoring a fairer emission allocation basis and balancing the interests of governments, shipping companies, and consumers. Through comprehensive experiments, we demonstrate the superiority of this method in minimizing total social costs and providing a viable path toward environmental sustainability and economic efficiency in liner shipping.

Optimization of Charging Station Capacity Based on Energy Storage Scheduling and Bi-Level Planning Model

With the government’s strong promotion of the transformation of new and old driving forces, the electrification of buses has developed rapidly. In order to improve resource utilization, many cities have decided to open bus charging stations (CSs) to private vehicles, thus leading to the problems of high electricity costs, long waiting times, and increased grid load during peak hours. To address these issues, a dual-layer optimization model was constructed and solved using the Golden Sine Algorithm, balancing the construction cost of CSs and user costs. In addition, the problem was alleviated by combining energy storage scheduling and the M/M/c queue model to reduce grid pressure and shorten waiting times. The study shows that energy storage scheduling effectively reduces grid load, and the electricity cost is reduced by 6.0007%. The average waiting time is reduced to 2.1 min through the queue model, reducing the electric vehicles user’s time cost. The bi-level programming model and energy storage scheduling strategy have positive implications for the operation and development of bus CSs.

Improving capacity configuration and load scheduling for an integrated multi-sourced cogeneration system

The integration of renewable energy sources and fossil fuel e.g. coal in a heat and power cogeneration system has been proven to be an effective way to reduce CO2 emissions. An integrated multi-sourced cogeneration (IMC) system consisting of wind, solar photovoltaic (PV), and coal-fired combined heat and power (CHP) plant on the energy supply side, is studied in this paper. A bi-level collaborative optimization model for capacity configuration and load scheduling of the wind-PV-coal IMC system is developed. In the upper level, the capacity of wind farm and PV farm are optimized with the enumeration method, while the load scheduling under typical annual conditions is optimized with adaptive particle swarm optimization algorithm in the lower level. The results show that the optimal capacities of wind and PV are 120 MW and 280 MW for the IMC system, respectively. The annual and typical day load scheduling with optimal wind and PV capacities are analyzed. The result of the load scheduling shows that heating and power load distribution unevenly between the two CHP units contributes to reducing total standard coal consumption, which saves 34.96 t in a typical day during the heating season. This study can provide a reference for the capacity configuration and operation scheduling of the IMC system.

A new bi-level model for the false data injection attack on real-time electricity market considering uncertainties

State Estimation (SE) plays a critical role in various applications of power systems including the electricity market. False Data Injection Attacks (FDIAs) are capable of manipulating the power system SE process aiming to gain financial profits by the players. In this paper, the potential economic influences of FDIA on SE and consequently the Real-Time (RT) electricity market are examined. Therefore, a new bi-level optimization framework is proposed to realistically model the FDIAs from the attacker perspective. At the upper-level, the attacker intends to intelligently choose the most critical congested transmission lines and manipulate their associated meter readings to maximize the expected financial profitability. At the lower-level, the RT market-clearing model is formulated under the influence of FDIA. Considering the incomplete attacker's prior knowledge of the network topology i.e., connection/disconnection of transmission lines, the angles of the phase-shifting transformers, and the real-time load uncertainty, the proposed formulation is extended to a new bi-level scenario-driven stochastic model. The developed attack model, which must remain undetectable by the system operator in all scenarios, is solved using the Karush–Kuhn–Tucker (KKT) approach. The correctness and effectiveness of the proposed optimization scheme have been validated in GAMS software using CONOPT and OQNLP solvers based on the IEEE 14-bus test system and also comparative results are presented to demonstrate the merits of the proposed formulation.

Aggregator-supported strategy for electric bus fleet charging: A hierarchical optimisation approach

The increasing adoption of electric buses in cities presents an opportunity for these vehicles to serve as mobile energy storage systems, providing valuable grid services through their substantial battery capacity. However, the complex interplay of fleet management and energy market participation requires that public transportation operators (PTOs) use adequate optimisation models and algorithms to make the best operational decisions. To address these issues, our research proposes an aggregator-supported charging strategy using a bi-level optimisation framework for electric bus fleets. The main goal is to achieve a balance between reducing PTO operational costs and maximising aggregator revenues, while ensuring reliable bus system operation. The results indicate that a coordinated charging strategy, including energy arbitrage and reserve provision, can significantly reduce operational costs, although potential limitations arise from associated battery degradation costs. Furthermore, managing fleets for reserve services appears to be more lucrative than engaging in energy arbitrage schemes. In this context, this study emphasizes the role of aggregator entities in managing the energy flexibility of electric bus fleets and enhancing the sustainability and cost-effectiveness of urban public transportation.

Research on Risk-Averse Procurement Optimization of Emergency Supplies for Mine Thermodynamic Disasters

Reducing procurement risks to ensure the supply of emergency supplies is crucial for mitigating the losses caused by mine thermodynamic disasters. The risk preference of decision-makers and supply chain collaboration are the important aspects for this reductiom. In this study, a novel P-CVaR (Piecewise conditional risk value) distributionally robust optimization model is proposed to accurately assist the decision-makers’ decision of risk preference for reducing procurement risks. Meanwhile, the role of cooperation between procurement and reserves are considered for the weakening procurement risks. A risk-averse bi-level optimization model is proposed to obtain the optimal procurement strategy. Furthermore, by applying the Lagrange duality theorem, the complexity of the bi-level optimization model is simplified then solved using a PSO algorithm. Using empirical analysis, it has been verified that the model presented in this paper serves as a valuable guideline for mine thermodynamic pre-disaster emergency material procurement strategies for the prevention of thermodynamic disasters.

A bilevel optimization approach of energy transition in freight transport: SOS1 method and application to the Ecuadorian case

A bilevel optimization approach of energy transition in freight transport: SOS1 method and application to the Ecuadorian case

Bilevel Optimization of the Kantorovich Problem and Its Quadratic Regularization

This paper is concerned with an optimization problem which is governed by the Kantorovich problem of optimal transport. More precisely, we consider a bilevel optimization problem with the underlying problem being the Kantorovich problem. This task can be reformulated as a mathematical problem with complementarity constraints in the space of regular Borel measures. Because of the non-smoothness that is induced by the complementarity constraints, problems of this type are often regularized, e.g., by an entropic regularization. However, in this paper we apply a quadratic regularization to the Kantorovich problem. By doing so, we are able to drastically reduce its dimension while preserving the sparsity structure of the optimal transportation plan as much as possible. As the title indicates, this is the second part in a series of three papers. While the existence of optimal solutions to both the bilevel Kantorovich problem and its regularized counterpart were shown in the first part, this paper deals with the (weak-∗\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$*$$\\end{document}) convergence of solutions to the regularized bilevel problem to solutions of the original bilevel Kantorovich problem for vanishing regularization parameters.

Coordinative dispatching of shared and public transportation under passenger flow outburst

In the passenger flow outburst area (e.g., metro emergency, dispersal of sporting events, and concerts), passengers put forward large demand for ride-sourcing services. Insufficient ride-sourcing supply of the area induces increasing matching time, and the ride-sourcing influx from outside picking up passengers leads to heavier road traffic congestion in the flow outburst area. To solve this problem, we propose a novel coordinative dispatching method of ride-sourcing and multi-modal transit, in which subsidies and buses are provided for passengers in flow outburst areas to encourage taking other public transportation to low passenger flow areas first and then taking ride-sourcing to their destinations. In this paper, there are two options for passengers: (a) wait for ride-sourcing to pick them up; (b) take other public transportation first and then take ride-sourcing. To obtain the optimal dispatching and subsidy schemes, we develop a bi-level mixed integer programming model based on network flow theory and design the corresponding iterative algorithm to solve it. Considering the high uncertainty of ride-sourcing demand in the area, we further develop a robust optimization model to obtain more reliable schemes. Case studies based on a real-problem-scale dataset are conducted. The results demonstrate that our approach can be carried out in real time, and encouraging passengers to take multi-modal transit shows great potential in reducing affected users’ delays. The robust optimization model offers a more reliable and competitive solution when demand varies widely. Our method offers a win–win–win way for passengers, ride-sourcing service providers, and public transportation systems.

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

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

Low-carbon oriented planning of shared photovoltaics and energy storage systems in distribution networks via carbon emission flow tracing

With the targets of carbon peaking and carbon neutrality, carbon emission reduction measures have become significant issues in new-type power systems. Distribution networks (DNs), which multiple power sources and flexible loads access to, play an essential role in exploiting the potential of reducing carbon emissions on the demand side. To achieve a global carbon emission reduction considering the carbon quota of each customer, shared photovoltaics (PVs) and energy storage systems (ESSs) are allocated with a centralized calculation and optimization conducted by DNs. To solve two key points in demand-side planning of shared PVs and ESSs in distribution networks, i.e., the accuracy of carbon emission flow (CEF) calculation and carbon quota-oriented optimization planning, this paper proposes a low-carbon oriented planning method for shared PVs and ESSs via CEF tracing. Firstly, a novel CEF calculation method is proposed based on the forward current sweep and backward voltage sweep algorithm, which considers the tracing of reactive power and power losses. Next, carbon emission index and economic index of allocating shared PVs and ESSs are defined. Subsequently, a bi-level optimization model is presented, considering the excess carbon emissions of each load based on carbon accounting and carbon quotas. Finally, the superiority of the proposed method is verified through a modified IEEE 33-node distribution network, and the effects of various investment constraints and carbon quotas are discussed.

On the complexity of inverse bivariate multi-unit assignment valuation problems

Inverse and bilevel optimization problems play a central role in both theory and applications. These two classes are known to be closely related due to the pioneering work of Dempe and Lohse (2006), and thus have often been discussed together ever since. In this paper, we consider inverse problems for multi-unit assignment valuations. Multi-unit assignment valuations form a subclass of strong-substitutes valuations that can be represented by edge-weighted complete bipartite graphs. These valuations play a key role in auction theory as the strong substitutes condition implies the existence of a Walrasian equilibrium. A recent line of research concentrated on the problem of deciding whether a bivariate valuation function is an assignment valuation or not. In this paper, we consider an inverse variant of the problem: we are given a bivariate function g, and our goal is to find a bivariate multi-unit assignment valuation function f that is as close to g as possible. The difference between f and g can be measured either in ℓ 1 - or ℓ ∞ -norm. Using tools from discrete convex analysis, we show that the problem is strongly NP -hard. On the other hand, we derive linear programming formulations that solve relaxed versions of the problem.

Momentum recursive DARTS

DARTS has emerged as a popular method for neural architecture search (NAS) owing to its efficiency and simplicity. It employs gradient-based bi-level optimization to iteratively optimize the upper-level architecture parameters and lower-level super-network weights. The key challenge in DARTS is the accurate estimation of gradients for two-level object functions, leading to significant errors in gradient approximation. To address this issue, we propose a new approach, MR-DARTS, that incorporates a momentum term and a recursive scheme to improve gradient estimation. Specifically, we leverage historical information by using a running average of past observed gradients to enhance the quality of current gradient estimation in both upper-level and lower-level functions. Our theoretical analysis shows that the variance of our estimated gradient decreases with each iteration. By utilizing momentum and a recursive scheme, MR-DARTS effectively controls the error in stochastic gradient updates that result from inaccurate gradient estimation. Furthermore, we utilize the Neumann series approximation and Hessian Vector Product scheme to reduce computational requirements and memory usage. We evaluate our proposed method on several benchmarks and demonstrate its effectiveness through comprehensive experiments.

Designing a collaborative maintenance planning mechanism between airline and maintenance company using a bilevel model – Considering flexible maintenance schedule and service charges

Periodic aircraft Maintenance Repair and Overhaul (MRO) activities are compulsory. To capture the features of MRO operations between airline and maintenance company under MRO outsourcing mode, we consider a collaborative aircraft maintenance planning problem, in which the airline outsources its self-operated maintenance checks to an independent maintenance company. Under such context, MRO planning conflicts between two collaborates exists especially during maintenance peak period, as airline aims to maximize flight operations revenues by receiving timely MRO service. On the other hand, maintenance company strives to accomplish heterogenous demands without causing tardiness and involving excessive MRO resources cost. Therefore, finding an effective mechanism to eliminate the conflicts between maintenance demands (from airline company) and maintenance service supply capability (from maintenance company) is indispensable for collaboration. A bi-level optimization model is formulated to describe the conflicting objectives and maintenance planning features of two independent stakeholders. To facilitates a synchronization of demand and supply and reconcile MRO conflicts, flexible maintenance schedules and service charges decisions from two independent collaborators is involved in the model, enabling a maintenance planning negotiation for finding mutual beneficial maintenance plan. An iterative algorithm deploying bounding updating strategy is developed, which generates a series of convergent-oriented bounding constraints along the iterative negotiation and guides the adjustment of maintenance demand and supply towards equilibrium. Based on the computational results in solving various maintenance planning scenarios, the impacts of different features (e.g. resource availability, resource demands level, and penalty cost of service tardiness) on airline and service company’s planning performance are analysed, revealing that maintenance service company’s own interest can be assured and maintained though it is a follower in the collaboration framework Afterwards, the managerial insights for facilitating the maintenance collaboration are further discussed.

Collaborative Planning and Comprehensive Evaluation Method for Energy Storage System and Flexible Interconnection in Distribution Networks Considering Island Operation

Introduction: When the superior power grid fails and loses power, due to the traditional single power supply radial grid structure of low-voltage substation areas, all loads in the lowvoltage substation area will lose power, causing a poor electricity consumption experience for low-voltage users. To enhance the resilience of low-voltage distribution areas against faults and power outages from higher-tier power grids, a collaborative planning and comprehensive evaluation method for grid-forming energy storage systems (GFM-ESS) and flexible interconnection in distribution networks, considering island operation, is proposed. Method: This method integrates GFM-ESS and flexible interconnections, taking islanding operations into account. Firstly, the structure of GFM-ESS and voltage source converters is analyzed. The control modes of GFM-ESS in parallel or off-grid scenarios are also scrutinized. Secondly, minimizing the annual comprehensive cost and the annual power outage load are the objective functions. A bilevel programming model for GFM-ESS and flexible interconnection of lowvoltage distribution networks, considering island operation, has been established. Moreover, a comprehensive evaluation method based on the analytic hierarchy process for proposed lowvoltage distribution network structural schemes is introduced. An evaluation index system is established to assess the rationality of these proposed network structural planning schemes. Finally, the superiority of the proposed collaborative planning method is verified through a comparative analysis of three planning methods. Result: The proposed collaborative planning method can effectively ensure the continuous power supply of low-voltage distribution networks under the fault scenario of the upper power network. Conclusion: Moreover, the rationality of the results obtained from the proposed collaborative planning method is verified through a comprehensive evaluation method of grid structure schemes based on the analytic hierarchy process. The proposed comprehensive evaluation method can determine which two areas are optimal for grid planning. result: The superiority of the proposed collaborative planning method was verified by comparative analysis of three planning methods. And the rationality of the proposed collaborative planning method results was verified by the comprehensive evaluation method of grid structure schemes based on analytic hierarchy process.

Integrated Optimization of Production Scheduling and Haulage Route Planning in Open-Pit Mines

In mining, deposits are divided into blocks, forming the basis for open-pit mine planning, covering production and haulage route planning. Current studies often stage optimization and lack the consideration of road capacity, leading to suboptimal solutions. A novel approach integrates production scheduling and haulage route planning through a bilevel optimization model. The upper-level model integrates ore mining constraints to establish a mixed-integer production scheduling model, minimizing haulage costs. Spatiotemporal correlation constraints for block mining are determined using a two-stage algorithm. The lower-level model incorporates road capacity, forming a haulage route optimization model based on multicommodity network flow. A solution algorithm with a distance penalty strategy facilitates feedback between the upper and lower levels, achieving optimal solutions. Tested on a real open-pit coal mine with over 5 million blocks, this approach reduces haulage costs by 10.06% compared to stage optimization. Additionally, this approach allows for adjusting haulage demand in both temporal and spatial dimensions, effectively preventing road congestion. This study advances rational mining processes and enhances the efficiency of open-pit mining haulage systems.

A Knee Point Driven Evolutionary Algorithm for Multiobjective Bilevel Optimization.

Bilevel optimization is a special type of optimization in which one problem is embedded within another. The bilevel optimization problem (BLOP) of which both levels are multiobjective functions is usually called the multiobjective BLOP (MBLOP). The expensive computation and nested features make it challenging to solve. Most existing studies look for complete lower-level solutions for every upper-level variable. However, not every lower-level solution will participate in the bilevel Pareto-optimal front. Under a limited computational budget, instead of wasting resources to find complete lower-level solutions that may not be in the feasible region or inducible region of the MBLOP, it is better to concentrate on finding the solutions with better performance. Bearing these considerations in mind, we propose a multiobjective bilevel optimization solving routine combined with a knee point driven algorithm. Specifically, the proposed algorithm aims to quickly find feasible solutions considering the lower-level constraints in the first stage and then concentrates the computational resources on finding solutions with better performance. Besides, we develop several multiobjective bilevel test problems with different properties, such as scalable, deceptive, convexity, and (dis)continuous. Finally, the performance of the algorithm is validated on a practical petroleum refining bilevel problem, which involves a multiobjective environmental regulation problem and a petroleum refining operational problem. Comprehensive experiments fully demonstrate the effectiveness of our presented algorithm in solving MBLOPs.

A Robust, resilience and risk-aware solar energy farm location by Bi-Level programming approach

A Robust, resilience and risk-aware solar energy farm location by Bi-Level programming approach

Estimation of schedule preference and crowding perception in urban rail corridor commuting: An inverse optimization method

This paper introduces an inverse optimization method to uncover commuters’ schedule preference and crowding perception based on aggregated observations from smart card data for an urban rail corridor system. The assessment of time-of-use preferences typically involves the use of econometric models of discrete choice based on detailed travel survey data. However, discrete choice models often struggle with potential endogeneity issues in behavioral observations when estimating individual samples from massive transit data with limited exogenous identifying information. This motivates us to employ an equilibrium modeling approach to capture the dynamism hidden in commuters’ departure time decision-making from aggregations. Assuming user optimality in observed choices, an inverse optimization method is proposed to find a set of preference parameters in the stochastic user equilibrium-based morning commuting model with heterogeneous commuters so that the resulting equilibrium pattern best approximates the observed departure rate distribution over time. The proposed inverse optimization problem can be formulated by a bi-level programming model and a sensitivity analysis-based solution framework is further designed for model estimation. Lastly, the smart card data and train timetable data from the rail corridor along the Beijing Subway Batong Line are synthesized for a case study to estimate commuters’ departure time choice preferences during morning peak periods, as well as to validate the robustness and practicality of the proposed method.

Necessary optimality conditions for strictly robust bilevel optimization problems

The study of robust bilevel programming problems is a relatively new area of optimization theory. In this work, we investigate a bilevel optimization problem where the upper-level and the lower-level constraints incorporate uncertainty. Reducing the problem into a single-level nonlinear and nonsmooth program, necessary optimality conditions are then developed in terms of Clarke subdifferentials. Our approach consists of using the optimal value reformulation together with a partial calmness condition for the robust counterpart of the initial problem. To aid in the detection of Karush-Kuhn-Tucker (KKT) multipliers, an appropriate nonsmooth Mangasarian-Fromovitz constraint qualification is introduced. There are examples highlighting both our results and the limits of certain past studies.