Bi-level Mixed Integer Linear Programming Research Articles

The Efficacy of a Cellular Approach to Resilience-Oriented Operation of Distribution Grids Based on Eigenvectors of the Laplacian Matrix

This study addresses the critical question for distribution system operators: how to maintain power supply during an impending disaster that could disrupt the grid? To address this challenge, we propose a cost-efficient cellular model for enhancing the resilience of smart distribution grids. This model prioritizes resilience in the face of natural disasters or other disruptions that could impact service delivery. This approach benefits both grid operators and consumers by ensuring reliable power supply while minimizing energy costs. Furthermore, the model's scalability allows it to be applied to distribution grids of varying sizes. The proposed method utilizes an innovative approach to form optimal cellular network configurations within the grid. As the first step in the formation of cellular topology for the grid, the eigenvectors of the Laplacian matrix of the grid will be used to decide on the optimal configurations. Subsequently, a bi-level mixed-integer linear programming model is proposed to decrease the network costs while simultaneously consider potential power transfer scenarios between the cells and the upstream network during both normal and emergency conditions. The researchers validated the effectiveness of the proposed method through simulations on an IEEE 33-bus test system. The results demonstrate outstanding performance, with a significant increase in the resilience index (around 96%) and a substantial reduction in load-shedding costs (around 80%), making the network considerably more robust.

A Competitive Bilevel Programming Model for Green, CLSCs in Light of Government Incentives

The growth of world population has fueled environmental, legal, and social concerns, making governments and companies attempt to mitigate the environmental and social implications stemming from supply chain operations. The state-run Environmental Protection Agency has initially offered financial incentives (subsidies) meant to encourage supply chain managers to use cleaner technologies in order to minimize pollution. In today’s competitive markets, using green technologies remains vital. In the present project, we have examined a class of closed-loop supply chain competitive facility location-routing problems. According to the framework of the competition, one of the players, called the Leader, opens its facilities first. The second player, called the Follower, makes its decision when Leader’s location is known. Afterwards, each customer chooses an open facility based on some preference huff rules before returning the benefits to one of the two companies. The follower, under the influence of the leader’s decisions, performs the best reaction in order to obtain the maximum capture of the market. So, a bilevel mixed-integer linear programming model is formulated. The objective function at both levels includes market capture profit, fixed and operating costs, and financial incentives. A metaheuristic quantum binary particle swarm optimization (PSO) is developed via Benders decomposition algorithm to solve the proposed model. To evaluate the convergence rate and solution quality, the method is applied to some random test instances generated in the literature. The computational results indicate that the proposed method is capable of efficiently solving the model.

Markov decision process design: A framework for integrating strategic and operational decisions

We consider the problem of optimally designing a system for repeated use under uncertainty. We develop a modeling framework that integrates the design and operational phases, which are represented by a mixed-integer program and discounted-cost infinite-horizon Markov decision processes, respectively. We seek to simultaneously minimize the design costs and the subsequent expected operational costs. This problem setting arises naturally in several application areas, as we illustrate through examples. We derive a bilevel mixed-integer linear programming formulation for the problem and perform a computational study to demonstrate that realistic instances can be solved numerically.

A Bilevel Model for Carbon Pricing in a Green Supply Chain Considering Price and Carbon-Sensitive Demand

In today’s industrial landscape, there is a mounting urgency to mitigate the adverse environmental impacts of emissions stemming from supply chain operations. On one front, policy-makers impose increasingly stringent emission reduction targets for supply chains, while on another front, consumers express a heightened preference for products and services with reduced carbon footprints. This study addresses the challenge of determining an optimal carbon pricing strategy by integrating the imperatives of a green supply chain with carbon taxation policies. To this end, we introduce a bi-level mixed-integer linear programming model for supply chain network planning, encompassing considerations of carbon taxation policies and the responsiveness of demand to the final product’s price and associated carbon emissions. Findings from a case study underscore that an escalation in carbon pricing prompts the supply chain to prioritize emissions reduction through the utilization of environmentally conscious approaches. The results reveal the need for a USD 0.9/kg carbon price to achieve a 10% emission reduction, resulting in an 80% profit decline. Notably, a 10% reduction has profound impacts, which leads to the suggestion of a gradual approach. Furthermore, as carbon prices reach higher levels, the supply chain tends toward curtailing production, thereby fostering an environment conducive to emission abatement. Consequently, policy formulators must judiciously calibrate a fitting carbon pricing mechanism to strike a harmonious equilibrium between emission reduction targets and the financial outlays of the supply chain.

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.

An analysis of distribution planning under a regulatory regime: An integrated framework

AbstractDistribution system planning is a multifaceted topic involving financial, regulatory, and system level analysis. The wide nature of the topic warrants a holistic study considering all aspects of analysis. The distribution utility is a natural monopoly that is subjected to utility regulation. The regulator can impact customer experience by strategically influencing the planning decisions of the utility. Hence, this paper reviews the existing utility regulation methods in the context of the distribution system and their efficacy in improving certain reliability and efficiency objectives. A two‐bus system is used to demonstrate the impact of classical models in alleviating reliability and efficiency issues through demand response. Further, a review is conducted on distribution system planning models without a regulatory regime, and suitable models for holistic analysis are identified. A two‐person complete information regulator and utility game with a comprehensive distribution system model at the lower level is proposed. A framework based on the Mixed Integer Bilevel Linear Program (MIBLP) is discussed to find the equilibrium point of the proposed game.

On SOCP-based disjunctive cuts for solving a class of integer bilevel nonlinear programs

We study a class of integer bilevel programs with second-order cone constraints at the upper-level and a convex-quadratic objective function and linear constraints at the lower-level. We develop disjunctive cuts (DCs) to separate bilevel-infeasible solutions using a second-order-cone-based cut-generating procedure. We propose DC separation strategies and consider several approaches for removing redundant disjunctions and normalization. Using these DCs, we propose a branch-and-cut algorithm for the problem class we study, and a cutting-plane method for the problem variant with only binary variables. We present an extensive computational study on a diverse set of instances, including instances with binary and with integer variables, and instances with a single and with multiple linking constraints. Our computational study demonstrates that the proposed enhancements of our solution approaches are effective for improving the performance. Moreover, both of our approaches outperform a state-of-the-art generic solver for mixed-integer bilevel linear programs that is able to solve a linearized version of our binary instances.

Integrated stochastic transmission network and wind farm investment considering maximum allowable capacity

• A stochastic model is presented for transmission and wind power investment. • The model considers demand and wind generation uncertainty. • The model considers maximum allowable capacity. • The model is formulated as a stochastic bilevel optimization problem. • Multi-parametric programming is used to solve the problem. This research presents a novel mathematical methodology for integrated transmission network and wind farm investment (ITWI) considering maximum allowable capacity (MAC). The joint problem of transmission and wind farm investment planning is carried out under a central planner perspective, where load and wind power uncertainties are managed using scenario-based stochastic programming. Distinct from the existing models in which the installation capacity of wind farms is restricted only by the available investment budget, the wind power capacity is limited by MAC considering strength measures. In this regard, the investment problem is addressed as a bi-level programming model where the upper level seeks to minimize the investment cost associated with transmission lines and wind farms plus the operation cost while the lower level determines the MAC of wind farms. The existence of integer decision variables in the lower level renders Karush-Kuhn-Tucker (KKT) conditions invalid. Thus, the multi-parametric programming (MPP) method is utilized to solve the mixed-integer bi-level linear programming (MIBLP). Numerical tests on two different power systems corroborate the efficiency of the proposed model. The comparable results demonstrate that ignoring the MAC of wind farms leads to inefficient solutions due to additional unnecessary investment in the wind energy sector.

Capacity Allocation Optimization Framework for Hydrogen Integrated Energy System Considering Hydrogen Trading and Long-Term Hydrogen Storage

As an essential part of the future national energy system, hydrogen energy has the advantages of clean, long-time scale energy storage and good complementary characteristics to electrical power, making it an important player in the low-carbon transformation of energies. However, the long timescale storage characteristics and tradability of hydrogen energy are rarely considered in existing capacity allocation optimization methods for hydrogen integrated energy systems(HIES). Meanwhile, the impact of the physical characteristics of the hydrogen storage equipment is rarely considered in the construction of HIES. Therefore, this paper proposes a new capacity allocation method for HIES in industrial park considering hydrogen trading and long-term hydrogen storage. The proposed capacity allocation optimization method is a bilevel mixed-integer linear programming model, which is solved by the reconfiguration decomposition algorithm. The nonlinear constraint problem due to the physical characteristics of the hydrogen storage device is solved by the Big-M method and the binary method. The proposed method can effectively improve the economy of HIES and reduce the cost of hydrogen production. Meanwhile, the reconstruction decomposition algorithm can effectively solve the bilevel mixed integer programing model. Case studies demonstrate that the proposed method can reduce hydrogen production economics by 28%. Considering hydrogen trading, the total investment and operating cost of HIES is reduced by 25%, while long-term hydrogen storage can reduce the cost of hydrogen production for HIES and reduce the impact of hydrogen trading fluctuations.

A Bilevel Mixed-Integer Linear Programming Model for Emissions Reduction

A Bilevel Mixed-Integer Linear Programming Model for Emissions Reduction

Interdicting restructuring networks with applications in illicit trafficking

We consider a new class of max flow network interdiction problems, where the defender is able to introduce new arcs to the network after the attacker has made their interdiction decisions. We prove properties of when this restructuring will not increase the value of the minimum cut, which has important practical interpretations for problems of disrupting drug trafficking networks. In particular, it demonstrates that disrupting lower levels of these networks will not impact their operations when replacing the disrupted participants is easy. For the bilevel mixed integer linear programming formulation of this problem, we devise a column-and-constraint generation (C&CG) algorithm to solve it. Our approach uses partial information on the feasibility of restructuring plans and is shown to be orders of magnitude faster than previous C&CG methods. We demonstrate that applying decisions from standard max flow network interdiction problems can result in significantly higher flows than interdictions that account for the restructuring.

A possibilistic-robust-fuzzy programming model for designing a game theory based blood supply chain network

This paper presents a bi-level blood supply chain network under uncertainty during the COVID-19 pandemic outbreak using a Stackelberg game theory technique. A new two-phase bi-level mixed-integer linear programming model is developed in which the total costs are minimized and the utility of donors is maximized. To cope with the uncertain nature of some of the input parameters, a novel mixed possibilistic-robust-fuzzy programming approach is developed. The data from a real case study is utilized to show the applicability and efficiency of the proposed model. Finally, some sensitivity analyses are performed on the important parameters and some managerial insights are suggested.

Preventing Outages Under Coordinated Cyber–Physical Attack With Secured PMUs

Due to the potentially severe consequences of coordinated cyber-physical attacks (CCPA), the design of defenses has gained significant attention. A popular approach is to eliminate the existence of attacks by either securing existing sensors or deploying secured PMUs. In this work, we improve this approach by lowering the defense target from <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">eliminating attacks</i> to <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">preventing outages</i> and reducing the required number of PMUs. To this end, we formulate the problem of <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">PMU Placement for Outage Prevention (PPOP)</i> under DC power flow model as a tri-level non-linear optimization problem and transform it into a bi-level mixed-integer linear programming (MILP) problem. Then, we propose an alternating optimization framework to solve PPOP by iteratively adding constraints, for which we develop two constraint generation algorithms. In addition, for large-scale grids, we propose a polynomial-time heuristic algorithm to obtain suboptimal solutions. Next, we extend our solution to achieve the defense goal under AC power flow model. Finally, we evaluate our algorithm on IEEE 30-bus, 57-bus, 118-bus, and 300-bus systems, which demonstrates the potential of the proposed approach in greatly reducing the required number of PMUs.

Flower Greenhouse Energy Management to Offer Local Flexibility Markets

Electricity access is strongly linked to human growth. Despite this, a portion of the world’s population remains without access to energy. In Colombia, rural communities have energy challenges due to the National Interconnected System’s (NIS) lack of quality and stability. It is common to find that energy services in such locations are twice as costly as in cities and are only accessible for a few hours every day due to grid overload. Implementing market mechanisms that enable handling imbalances through the flexible load management of main loads within the grid is vital for improving the rural power grid’s quality. In this research, the energy from the rural grid is primarily employed to power a heating, ventilation, and air-conditioning (HVAC) system that chills flowers for future commerce. This load has significant consumption within the rural grid, so handling HVAC consumption in a suitable form can support the grid to avoid imbalances and improve the end-user access to energy. The primary responsibilities of the flower greenhouse operator are to reduce energy costs, maximize flexibility, and maintain a proper indoor temperature. Accordingly, this research proposes a flexible energy market based on the bi-level mixed-integer linear programming problem (Bi-MILP), involving the Agricultural Demand Response Aggregator (ADRA) and the flower greenhouse. ADRA is responsible for assuring the grid’s stability and quality and developing pricing plans that promote flexibility. A flower greenhouse in Colombia’s Boyacá department is used as an application for this research. This study looked at the HVAC’s flexibility under three different pricing schemes (fixed, time-of-use, and hourly) and graded the flower greenhouse’s flexibility as a reliable system.

Coordinating aircraft maintenance routing and integrated maintenance staffing and rostering: a Stackelberg game theoretical model

There is an interdependence between operational-based flight delay aircraft maintenance routing problem (AMRPD) and maintenance staffing problem (MSP). Furthermore, MSP and maintenance rostering problem (MRP) are interdependent. In the literature, most of the studies only consider a single interdependence and neglect the other one. This results in difficulties in implementing the optimal plan constructed by each problem, leading finally to an increase in the operational cost of airlines and maintenance providers. Therefore, the main objective of this study is to investigate these interdependences simultaneously. This is accomplished by developing a Stackelberg game model (LFSGM), in which leader and follower are the airlines and maintenance providers, respectively. This LFSGM is modelled as a bi-level mixed integer linear programming (MILP) model. The upper-level is represented by the AMRPD, while the integrated maintenance staffing and rostering (IMSR) functions as the lower-level. The solution of the LFSGM is obtained from an iterative algorithm. The viability and superiority of the LFSGM are demonstrated by presenting a case study of a leading airline in the Middle East serviced by four maintenance providers. The results reveal a significant cost saving for airline and maintenance providers.

On the Bilevel Optimization for Remapping Virtual Networks in an HOE-DCN

Hybrid optical/electrical datacenter network (HOE-DCN) uses the inter-rack networks that consist of both electrical Ethernet switches and optical cross-connects (OXCs), for better cost-efficiency and scalability. Meanwhile, to provision dynamic network services well, the operator of an HOE-DCN needs to deploy virtual networks (VNTs) and remap them adaptively. Therefore, this work studies the problem of VNT remapping in an HOE-DCN from a novel perspective, i.e., the remapping schemes should be optimized for not only the network status after the remapping but also the transition to realize it. Specifically, we model this problem as a bilevel optimization, where the upper-level optimization aims at selecting proper virtual machines (VMs) to migrate such that the estimated latency of VM migration can be minimized, and the lower-level optimization determines the actual scheme of VNT remapping for minimizing the number of resource hot-spots. We first formulate a bilevel mixed integer linear programming (BMILP) model for the bilevel optimization, and then propose a polynomial time algorithm based on enumeration to solve it approximately. Extensive simulations verify the effectiveness of our proposal.

A Branch-and-Cut Algorithm for Submodular Interdiction Games

Many relevant applications from diverse areas such as marketing, wildlife conservation, and defending critical infrastructure can be modeled as interdiction games. In this work, we introduce interdiction games whose objective is a monotone and submodular set function. Given a ground set of items, the leader interdicts the usage of some of the items of the follower in order to minimize the objective value achievable by the follower, who seeks to maximize a submodular set function over the uninterdicted items subject to knapsack constraints. We propose an exact branch-and-cut algorithm for this kind of interdiction game. The algorithm is based on interdiction cuts, which allow the leader to capture the follower’s objective function value for a given interdiction decision of the leader and exploit the submodularity of the objective function. We also present extensions and liftings of these cuts and discuss additional preprocessing procedures. We test our solution framework on the weighted maximal covering interdiction game and the bipartite inference interdiction game. For both applications, the improved variants of our interdiction cut perform significantly better than the basic version. For the weighted maximal covering interdiction game for which a mixed-integer bilevel linear programming (MIBLP) formulation is available, we compare the results with those of a state-of-the-art MIBLP solver. Whereas the MIBLP solver yields a minimum of 54% optimality gap within one hour, our best branch-and-cut setting solves all but four of 108 instances to optimality with a maximum of 3% gap among unsolved ones.

A bi-level charging management approach for electric truck charging stationconsidering power losses

This article presents an optimized heavy duty electric truck charging station (ETCS) design based on bi-level mixed integer linear programming. Electric truck parameters are integrated with the grid model and charging sequences are firstly formulated to optimize charging stages. As the second level of the optimization stage, line losses are aimed to be minimized for the charging station. ETCS model is obtained from actual parameters of the Istanbul Muratbey Customs zone which is one of the busiest customs zone in Europe and an ideal location for ETCS application in the future. The ETCS is equipped with roof type photovoltaic (PV) modules and the capacity of the PV generation is determined in accordance with the actual data of the customs zone buildings. The PV generated power is integrated with electric truck demand which varies in daytime based on hourly intervals along with the seasonal impact. Ultimately, an optimized ETCS model is presented including PV generated power to reduce line losses and CO2 emission along with optimized charging sequences for ETCS with huge power consumption. The proposed model represents the reduction of line losses along with the obtained environmental and economic benefits.

Microgrids energy management in automated distribution networks by considering consumers’ comfort index

Modern distribution networks' scheduling faces many more challenges than traditional networks. Because on the one hand, it must satisfy the technical and security constraints of the distribution system, and on the other hand, it must observe the optimal scheduling of microgrids. Hence, in this paper, a bi-level mixed-integer linear programming (MILP) model is presented in which the scheduling of the distribution network is done by considering the optimal scheduling of microgrids and smart homes. In the first level, each microgrid performs its optimal scheduling in a decentralized manner and delivers it to the distribution system operator (DSO). Then, in the second level, DSO plans the main grid according to the optimal schedule of microgrids and by implementing distribution feeder reconfiguration (DFR). Finally, the problem is solved in the form of four case studies by the CPLEX solver in GAMS software and the results show that the proposed model not only reduces the total operating cost by 27.05% but also significantly increases the consumers’ comfort index. The results also illustrate that DFR implementation and participation in the DR program reduce total operating costs by 19.11% and 14.02%, respectively.

Production planning method for seru production systems under demand uncertainty

The seru production system is a new-type manufacturing system arising from Japanese production practices that can simultaneously achieve high efficiency, flexibility, and responsiveness. In this paper, a worker and production quantity allocation optimization method for multi-product seru production systems under demand uncertainty is proposed. To handle such demand uncertainty, the Monte Carlo technique is employed and the optimization problem is formulated as a bi-level mixed-integer linear programming problem. At the first step, a worker allocation problem is solved before demand realization is observed. As the second optimization step, the production quantity allocation is then conducted based on the observed demand realization. The proposed method is applied to several numerical problems to illustrate its effectiveness.