Master Problem Research Articles

A non-cooperative decentralized model for Volt-VAr optimization of active distribution networks with multiple AC and DC microgrids

Volt-VAr optimization (VVO) is an important issue for distribution network operators in active distribution networks with a number of AC and DC microgrids. This paper introduces a coordinated decentralized framework with application to Volt-VAr optimization in distribution networks with multiple AC and DC microgrids in the presence of distributed energy resources (DERs). To this end, a non-cooperative coordination model is proposed so that micro-grids operators (MGOs) and distribution system operator (DSO) could minimize their own active power losses, separately. In fact, the DSO and MGOs interact together to achieve an equilibrium that satisfies their expected active power loss. In order to solve the proposed coordinated non-cooperative problem, a novel decentralized optimization approach is suggested so that the master problem is decomposed into two sub-problems from DSO and MGOs point of views. These two sub-problems are linked to each other through sharing boundary information. On this basis, the communicational and computational loads are decreased while information secrecy is guaranteed. The modified IEEE 69-bus distribution network considering a number of AC and DC MGs has been chosen with the aim of conducting several numerical analyses.

A Successive Linear Relaxation Method for MINLPs with Multivariate Lipschitz Continuous Nonlinearities

We present a novel method for mixed-integer optimization problems with multivariate and Lipschitz continuous nonlinearities. In particular, we do not assume that the nonlinear constraints are explicitly given but that we can only evaluate them and that we know their global Lipschitz constants. The algorithm is a successive linear relaxation method in which we alternate between solving a master problem, which is a mixed-integer linear relaxation of the original problem, and a subproblem, which is designed to tighten the linear relaxation of the next master problem by using the Lipschitz information about the respective functions. By doing so, we follow the ideas of Schmidt, Sirvent, and Wollner (Math Program 178(1):449–483 (2019) and Optim Lett 16(5):1355-1372 (2022)) and improve the tackling of multivariate constraints. Although multivariate nonlinearities obviously increase modeling capabilities, their incorporation also significantly increases the computational burden of the proposed algorithm. We prove the correctness of our method and also derive a worst-case iteration bound. Finally, we show the generality of the addressed problem class and the proposed method by illustrating that both bilevel optimization problems with nonconvex and quadratic lower levels as well as nonlinear and mixed-integer models of gas transport can be tackled by our method. We provide the necessary theory for both applications and briefly illustrate the outcomes of the new method when applied to these two problems.

Inverse optimization of integer programming games for parameter estimation arising from competitive retail location selection

When determining store locations, competing retailers must take customers’ store choice into consideration. Customers predominantly select which store to visit based on price, accessibility, and convenience. Incumbent retailers can estimate the weight of these factors (customer attraction parameters) using granular historical data. Their location decision under full information and simultaneous competition translates into an integer programming game. Unlike incumbents, new entrants lack this detailed information; however, they can observe the resulting location structure of incumbents. Assuming the observed location structure is (near-)optimal for all incumbent retailers, a new entrant can use these observations to estimate customer attraction parameters. To facilitate this estimation, we propose an “inverse optimization approach” for integer programming games (IPGs), enabling a new entrant to identify parameters that lead to the observed equilibrium solutions. We solve this “inverse IPG” via decomposition by solving a master problem and a subproblem. The master problem identifies parameter combinations for which the observations represent (approximate) Nash equilibria compared with optimal solutions enumerated in the subproblem. This row-generation approach extends prior methods for inverse integer optimization to competitive settings with (approximate) equilibria.We compare the decision-making of new entrants selecting locations based on scenarios, or information about the underlying distribution of customer attraction parameters (expected values), with new entrants using inversely estimated parameters for their location decisions. New entrants who rely on inversely optimized parameters can improve their profits. On average over a large set of synthetic numerical experiments, we observe improvements of 4–11%. This benefit can be realized with as little as one or two observations, yet additional observations help to increase prediction reliability significantly.

Edge Trusted Sharing: Task-Driven Decentralized Resources Collaborate in IoT

6G is committed to providing a fully connected world. The deployment and application of 6G technology in the Internet of Things (IoT) can efficiently collaborate IoT resources and realize resource sharing, which mainly encourages IoT development. However, due to the lack of trust between IoT resources, security and privacy become the main challenges. As an emerging technology, blockchain can solve the trust-absence issues and provide more benefits, but the introduction of blockchain also brings problems for IoT resource collaboration and sharing. This article first proposes a blockchain-enabled edge resource sharing (BEERS) architecture by combining blockchain and edge computing technology. Then, based on a typical resource sharing and collaboration scenario, the joint optimization problem of resource scheduling and task assignment (JRSTA) is constructed. We decompose JRSTA into a primal problem and a master problem and design a greedy-based task assignment (GBTA) algorithm to solve the primal problem. Based on the GBTA algorithm, the resource scheduling and task assignment (RSTA) algorithm is developed. Next, we design a layered parallel edge resource scheduling and task assignment (LPRSTA) mechanism to improve practicability. Finally, we analyze the security and the performance of our proposed architecture and algorithms. The results show that the proposed architecture can support the secure collaboration of IoT resources, and the proposed algorithm can achieve effective resource scheduling and task assignment.

Pickup capacitated vehicle routing problem with three-dimensional loading constraints: Model and algorithms

This study addresses the pickup capacitated vehicle routing problem with three-dimensional loading constraints (3L-PCVRP), which combined the container loading problem (CLP) and the capacitated vehicle routing problem (CVRP). A mixed-integer linear programming model for 3L-PCVRP is designed with the objective of minimizing the total transportation cost. The basic constraints of CLP and CVRP as well as the practical constraints imposed by the pickup operations are incorporated in 3L-PCVRP. An improved branch-and-price-based (B&P) algorithm is proposed to solve 3L-PCVRP. The 3L-PCVRP model is decomposed into the restricted master problem (RMP) for selecting routes and the subproblem (SP) for generating routes that satisfy the loading and routing constraints. A label-correcting-based algorithm (LCA) is presented to solve the SP, which employs a two-stage method and two enhanced loading algorithms to ensure the loading feasibility of the routes. The two loading algorithms (tree search and greedy heuristic algorithms) are improved by a new evaluation function and an improved space-merging method. The two-stage method is designed to call the two loading algorithms at different frequencies to investigate the route feasibility accurately and efficiently. Numerical experiments are designed to test the performance of the proposed algorithms. The efficiency and effectiveness of the improved loading algorithms are demonstrated in hundreds randomly generated instances. For small-scale 3L-PCVRP instances, the B&P algorithm can generate solutions that are not worse than those of the solver within 0.14% of the solver’s running time. The efficiency and effectiveness of the improved B&P algorithm are also validated in large-scale benchmark instances.

A column generation-based framework for ATFM incorporating a user-driven prioritization process

A column generation-based framework is proposed for air traffic flow management (ATFM) incorporating a user-driven prioritization process (UDPP). Airspace Users' (AUs') preferences and priorities can be explicitly reflected in the framework. AUs are allowed to propose their preferred routes as initial options and AUs' delay cost structure can be customized and not revealed to other AUs. Major UDPP features, including Fleet Delay Reordering, Selective Flight Protection, and Margins, are fully incorporated in the proposed framework. In the master problem, the model selects 4D-routes for flights subject to sector capacity constraints. In the subproblems, new 4D-routes are generated. Numerical experiments demonstrate that the proposed framework can efficiently solve the ATFM problem considering UDPP features with often zero or small optimality gaps. The effectiveness of UDPP is also verified using case studies. Parallelism of subproblems considerably further reduces the runtime and the large-scale scenario computational time can be reduced to around 30 min.

Two-stage distributionally robust optimization-based coordinated scheduling of integrated energy system with electricity-hydrogen hybrid energy storage

A coordinated scheduling model based on two-stage distributionally robust optimization (TSDRO) is proposed for integrated energy systems (IESs) with electricity-hydrogen hybrid energy storage. The scheduling problem of the IES is divided into two stages in the TSDRO-based coordinated scheduling model. The first stage addresses the day-ahead optimal scheduling problem of the IES under deterministic forecasting information, while the second stage uses a distributionally robust optimization method to determine the intraday rescheduling problem under high-order uncertainties, building upon the results of the first stage. The scheduling model also considers collaboration among the electricity, thermal, and gas networks, focusing on economic operation and carbon emissions. The flexibility of these networks and the energy gradient utilization of hydrogen units during operation are also incorporated into the model. To improve computational efficiency, the nonlinear formulations in the TSDRO-based coordinated scheduling model are properly linearized to obtain a Mixed-Integer Linear Programming model. The Column-Constraint Generation (C&CG) algorithm is then employed to decompose the scheduling model into a master problem and subproblems. Through the iterative solution of the master problem and subproblems, an efficient analysis of the coordinated scheduling model is achieved. Finally, the effectiveness of the proposed TSDRO-based coordinated scheduling model is verified through case studies. The simulation results demonstrate that the proposed TSDRO-based coordinated scheduling model can effectively accomplish the optimal scheduling task while considering the uncertainty and flexibility of the system. Compared with traditional methods, the proposed TSDRO-based coordinated scheduling model can better balance conservativeness and robustness.

Logic-Based Benders Decomposition in Answer Set Programming for Chronic Outpatients Scheduling

AbstractIn answer set programming (ASP), the user can define declaratively a problem and solve it with efficient solvers; practical applications of ASP are countless and several constraint problems have been successfully solved with ASP. On the other hand, solution time usually grows in a superlinear way (often, exponential) with respect to the size of the instance, which is impractical for large instances. A widely used approach is to split the optimization problem into subproblems (SPs) that are solved in sequence, some committing to the values assigned by others, and reconstructing a valid assignment for the whole problem by juxtaposing the solutions of the single SPs. On the one hand, this approach is much faster due to the superlinear behavior; on the other hand, it does not provide any guarantee of optimality: committing to the assignment of one SP can rule out the optimal solution from the search space. In other research areas, logic-Based Benders decomposition (LBBD) proved effective; in LBBD, the problem is decomposed into a master problem (MP) and one or several SPs. The solution of the MP is passed to the SPs that can possibly fail. In case of failure, a no-good is returned to the MP that is solved again with the addition of the new constraint. The solution process is iterated until a valid solution is obtained for all the SPs or the MP is proven infeasible. The obtained solution is provably optimal under very mild conditions. In this paper, we apply for the first time LBBD to ASP, exploiting an application in health care as case study. Experimental results show the effectiveness of the approach. We believe that the availability of LBBD can further increase the practical applicability of ASP technologies.

Optimal Planning of Internet Data Centers Decarbonized by Hydrogen-Water-Based Energy Systems

Green and sustainable development of Internet data centers (IDCs) has attracted more attention in both academia and industry. Full utilization of renewable energy sources is widely known as the most effective way to supply electrical and thermal energy while reducing carbon emission. However, the integration of renewable energy into IDCs is still challenging due to the mismatch between uncertain renewable supply and time-varying demand requirements, and high requirement of operation reliability against IDC failures. Therefore, in this paper a hydrogen-water-based energy (HWBE) system is developed and its integrated planning-and-operation problem is formulated as a mixed-integer linear programming problem to determine the optimal capacity of energy facilities in the HWBE system with considering IDC operation reliability. A hybrid physics-based and data-driven method is developed to accurately capture the electrical and thermal energy consumption characteristics and their coupling which are the basis for the optimal planning of the HWBE system. Furthermore, a Benders decomposition-based reliability improvement algorithm is developed to enhance the operation reliability, which decomposes the problem into the planning problem with normal operation as the master problem and the operation problem with IDC failure as the subproblem. The reliability can be enhanced using the solution obtained by the master problem with the feasibility cut obtained from the subproblem. Numerical results show that the developed HWBE system is energy-efficient with low carbon emission, since the power usage efficiency of IDCs could be as low as 1.09 and the carbon emission could be reduced by 74.9% as compared by the electricity-driven IDC energy system. <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Note to Practitioners</i> —This paper focuses on the integrated planning-and-operation optimization of an HWBE system for the application in IDCs. We improve the energy consumption model of IDCs based on a hybrid physics-based and data-driven method, which can describe the interaction between the dynamic thermal process and electricity consumption of IDCs. In this way, both the high accuracy of the physics-based model and the lower computational effort of the data-driven method could be simultaneously achieved in the energy consumption model. Furthermore, in practice, the optimal planning problem of IDCs is necessary to take into account the operation reliability against data center failures, since the capital expenditure of the backup energy devices is generally significant. This means that a trade-off between the solution accuracy of the planning problem and the computational complexity caused by the operation problem should be considered. Therefore, we develop a Benders decomposition-based reliability improvement algorithm to address the trade-off mentioned above. This technique can integrate the feasibility cut obtained from the operation problem with IDC failure into the planning problem, in order to improve the operation reliability against the supply-demand mismatching and IDC failures while reducing the capital cost, as compared to the system designed by the conventional redundancy standard. Numerical results show the effectiveness of the developed method which can make full use of renewable energy sources and support the green and sustainable development of IDCs.

Parallel Batch Processing for the Coating Problem

We solve a challenging scheduling problem with parallel batch processing and two-dimensional shelf strip packing constraints that arises in the tool coating field. Tools are assembled on so-called planetaries (batches) before they are loaded into coating machines to get coated. The assembling is not trivial and must fulfil specific constraints, which we refer to as shelf strip packing constraints. Further, each tool is associated with a starting time window s.t. tools can only be put on the same planetary if their time window overlap. The objective is to minimise the makespan and the number of required planetaries. Since the problem naturally decomposes into scheduling and packing parts, we tackle the problem with a two-phase logic-based Benders decomposition approach. The master problem assigns items to batches. The first phase solves as subproblem the packing problem by checking if the assignment is feasible, whereas the second phase solves the scheduling subproblem. The approach is compared with a monolithic mixed integer linear programming approach as well as a monolithic constraint programming approach. Experimental evaluation shows that our proposed approach outperforms the state-of-the-art benchmarks by solving more instances to optimality in a shorter time.

Modular Bus Unit Scheduling for an Autonomous Transit System under Range and Charging Constraints

Recent advances in vehicle technology offer new opportunities for an electric, automated, modular bus (MB) unit with an adjustable capacity to be applied to transit systems, promising to tackle the resource allocation challenges of traditional buses in coping with uneven travel demand. Drawing on the concept of modular vehicles, this paper introduces a novel scheduling system in which MB units can be combined/separated from fulfilling imbalanced trip demands through capacity adjustments. We develop an optimization model for determining the optimal formation and trip sequence of MB units. In particular, given that the vehicles are electrically powered, battery range limits and charging plans are considered in the system scheduling process. A column-generation-based heuristic algorithm is designed to efficiently solve this model, with constraints related to travel demand and charging station capacity incorporated into the master problem and the trip sequence for modular units with limited energy solved by the subproblem. Taking real data from transit operations for numerical examples, the proposed model performs well in terms of both algorithmic performance and practical applications. The generated optimal MB dispatching scheme can significantly reduce the operating cost from $1534.31 to $1144.26, a decrease of approximately 25% compared to conventional electric buses. The sensitivity analysis on the MB dispatch cost and battery capacity provides some insights for both the scenario configuration and the battery selection for MB system implementation.

A flexibility-based multi-objective model for contingency-constrained transmission expansion planning incorporating large-scale hydrogen/compressed-air energy storage systems and wind/solar farms

The remarkable arrival of uncertainties given by the fluctuations in the output of renewable energy sources (RESs) calls for a more flexible power grid. In confronting this issue, large-scale energy storage systems play a vital role in achieving higher flexibility. In this regard, the authors of this study present here a new multi-objective model for contingency-constrained transmission expansion planning that incorporates large-scale hydrogen/compressed-air energy storage systems and wind/solar farms to simultaneously boost both supply-demand-related flexibility (SDFX) and grid-related flexibility (GDFX). The proposed model is formulated as a non-convex mixed-integer nonlinear master-slave optimization problem. The planning objectives, modeled through the master problem, are intended to minimize the investment and operation costs, while maximizing the SDFX enhancement metric minus the GDFX degradation metric. The slave problem, however, aims to maximize the community welfare function (CIWF) and microgrid welfare function (MIWF) under both normal and N-1 operating conditions. To solve the resulting non-convex mixed-integer nonlinear master-slave optimization problem, a potent symphony orchestra search algorithm (SOSA)—empowered with a multi-computational-step, multi-dimensional, multiple-homogeneous structure—is employed to allow better diversification and exploration. Following the determination of optimal solutions, a conservative-based fuzzy satisfying approach is applied to characterize the best trade-off solution that suits cost-effectiveness and flexibility requirements. The newly developed model was implemented on the IEEE 24-bus and IEEE 118-bus test grids, and the simulation results proved its effectiveness and practicality.

Security-constrained unit commitment: A decomposition approach embodying Kron reduction

We address the day-ahead scheduling of electricity production units throughout a network imposing N-1 security constraints, which ensures uneventful operation under any single-branch failure. For realistic electric energy systems, this optimization problem, which is mixed-integer linear or nonlinear but convex, involves millions of continuous variables, millions of constraints, and thousands of binary variables. This problem is intractable if state-of-the-art branch-and-cut solvers are used. As a solution methodology, we propose a Benders-type decomposition technique with a dynamically enriched master problem. Such master problem incorporates scheduling (binary) decisions and decisions pertaining to under-contingency operating conditions. The subproblems represent the operation of the system under no failure and single-branch failure. As the algorithm progresses, the master problem incorporates additional under-contingency operating conditions, which increases its computational burden. We use Kron reduction to compact (reducing variables and constraints) the description of the under-contingency operating conditions in the master problem without losing accuracy, which renders major computational gains. The methodology proposed allows solving, within reasonable computing times, instances intractable with state-of-the-art branch-and-cut solvers and decomposition algorithms.

Small-signal stability constrained unit commitment based on decomposition and SQP-GS

This paper proposes a small-signal stability constrained unit commitment(SSSC-UC) model to ensure that the solution of SCUC can provide a sufficient small-signal stability margin. To solve the model, Benders decomposition (BD) is applied to separate the problem into a SCUC master problem, which is a mixed-integer linear programming problem, and a series of subproblems to check the AC power flow constraint and small-signal stability constraint. The small-signal stability check subproblems are non-smooth and non-linear programming problems, which are solved by Sequential Quadratic Programming with Gradient Sampling (SQP-GS). When the subproblem checks do not pass, cuts formed by the sensitivity of the spectral abscissa with respect to generator output are added to the master problem for the next iteration of the SCUC. The iteration process continues until all AC power flow and small-signal constraint violations are eliminated and a SCUC solution is found. Simulations on the New England 10-machine 39-bus system and the IEEE 54-machine 118-bus system show that our proposed SSSC-UC can guarantee small-signal stability for all time periods while SCUC cannot. Furthermore, the proposed parallel technique can speed up the computation up to 5 and 11 times respectively for the two test systems.

Hub Network Design Problem with Capacity, Congestion, and Stochastic Demand Considerations

Our study introduces the hub network design problem with congestion, capacity, and stochastic demand considerations (HNDC), which generalizes the classical hub location problem in several directions. In particular, we extend state-of-the-art by integrating capacity acquisition decisions and congestion cost effect into the problem and allowing dynamic routing for origin-destination (OD) pairs. Connecting strategic and operational level decisions, HNDC jointly decides hub locations and capacity acquisitions by considering the expected routing and congestion costs. A path-based mixed-integer second-order cone programming (SOCP) formulation of the HNDC is proposed. We exploit SOCP duality results and propose an exact algorithm based on Benders decomposition and column generation to solve this challenging problem. We use a specific characterization of the capacity-feasible solutions to speed up the solution procedure and develop an efficient branch-and-cut algorithm to solve the master problem. We conduct extensive computational experiments to test the proposed approach’s performance and derive managerial insights based on realistic problem instances adapted from the literature. In particular, we found that including hub congestion costs, accounting for the uncertainty in demand, and whether the underlying network is complete or incomplete have a significant impact on hub network design and the resulting performance of the system. Funding: This work was supported by Türkiye Bilimsel ve Teknolojik Araştırma Kurumu [Grant 218M520]. Supplemental Material: The online appendices are available at https://doi.org/10.1287/trsc.2022.0112 .

Market-based coordination of price-responsive demand using Dantzig-Wolfe decomposition method

The increasing penetration of Distributed Generation (DG) and Demand Responsive (DR) loads in power systems has necessitated the development of novel approaches to address the coordination problem of Price Responsive Devices (PRD). These PRDs are treated as self-interested players aiming to optimize their consumption patterns based on prevailing prices. In this paper, we propose a new algorithm based on the Dantzig-Wolfe (DW) Decomposition method, which tackles the coordination problem of self-interested PRDs in a distributed manner. Leveraging the distributed nature of the DW approach, we model the self-interested algorithms of PRDs as sub-problems within the DW framework. The coordinator, or grid operator, responsible for collecting the energy consumption information (energy bids) of PRDs, solves the master problem of the DW and determines the price signal accordingly.The proposed algorithm exhibits fast convergence as the sub-problems within DW, which could involve a large number of PRDs (potentially millions), can be solved simultaneously. Additionally, based on the DW theory, if the PRDs' subproblems are convex, reaching the optimal point (equivalent to Nash Equilibrium) is guaranteed within a limited number of iterations. To evaluate the proposed model, we conducted a simulation involving 200 participant households, each equipped with two types of loads: Electric Vehicles (EVs) as examples of interruptible loads, and Electric Water Heaters (EWHs) as examples of Thermostatically Controlled Loads (TCLs). The results demonstrate that when the algorithm converges to the optimal point, both the generation cost and user payment (based on the marginal cost of generation) decrease. Furthermore, there is a significant reduction in the Peak to Average Ratio (PAR) of the aggregate load

A benders decomposition algorithm for a bid generation problem in the procurement of Three-echelon transportation services

In this paper, we address a bid generation problem in a combinatorial auction for the procurement of transportation services with time windows in a three-echelon transportation network. By considering the interconnectivity of transportation operations in different echelons, a path-based mathematical programming model is proposed for this problem. To solve the model with complicated linking constraints, a Benders decomposition algorithm in a heuristic manner is designed. The algorithm iteratively and alternatively solves a master problem that is a selective routing problem with time windows for the second echelon and two subproblems that are routing problems with time windows for the first and third echelon respectively. The basic Benders decomposition algorithm is enhanced by applying the acceleration techniques of multiple cuts, Pareto-optimal cuts and multiple cuts combined with Pareto-optimal cuts. Extensive computational experiments were performed to evaluate the effectiveness of the acceleration techniques and the efficiency of the algorithm. The computational results indicate that the enhanced algorithm significantly outperforms CPLEX in terms of solution quality and CPU time. Moreover, our sensitivity analysis on some key parameters of the model show that the lower the proportion of reserved requests, the larger of the capacity of each vehicle, and the higher the percentage of profit increase of a carrier after bidding.

Capacity reservation for humanitarian relief: A logic-based Benders decomposition method with subgradient cut

Prepositioning of relief supplies has been widely addressed to cover the demands of humanitarian emergencies. However, cost inefficiency, item type limitation, and damage risk make solely relying on prepositioning unrealistic. We develop two-stage stochastic models that incorporate prepositioning, physical capacity reservation, and production capacity reservation for reactive procurement. As an alternative to the traditional physical capacity reservation, the production capacity reservation is inspired by the practice of the automotive industry, garment industry, etc. for manufacturing personal protective equipment during the pandemic. Our models minimize the supply-side monetary costs and the demand-side social impacts, i.e., deprivation costs. The discretized deprivation cost function is introduced to handle the nonlinear deprivation cost function. The hierarchical nature of our stochastic models motivates us to utilize the logic-based Benders decomposition (LBBD). Our Benders master problem contains one set of nonbinary integer variables for prepositioning inventory and another set of continuous variables for estimating the second-stage costs, which differs from the existing LBBD works that typically contain binary and continuous variables in the master problem. Hence, a new type of logic-based Benders optimality cut, namely logic-based subgradient cut, is first introduced. To compute the cutting coefficients efficiently, heuristics that can lead to almost optimal solutions is developed. We also develop warm-start cutting planes, namely wait-and-see cuts and expected-value cuts, to help with better upper bounds and lower bounds. Extensive numerical results followed by a case study validate the efficiency of the solution method, the value of incorporating stochasticity, and the superiority of the capacity reservation.

Formation and Routing of Worker Teams for Airport Ground Handling Operations: A Branch-and-Price-and-Check Approach

We address workforce optimization for ground handling operations at the airport, focusing on baggage loading and unloading. Teams of skilled workers have to be formed and routed across the apron to unload the baggage from the aircraft after a landing and to load it before takeoff. Such tasks must be performed within time windows and require a team of workers with different skill levels. The goal is to find a feasible plan that minimizes the sum of the tasks completion times. We formalize a variation of the workforce scheduling and routing problem, integrating team formation, hierarchical skills with downgrading, multiple trips, and different execution modes. We propose a solution approach based on branch-and-price-and-check and test it on real-world instances from a major European hub airport. We propose a model based on the Dantzig–Wolfe decomposition. In the pricing problem, we generate tours of teams as shortest paths with constrained resources in a network. In the master problem, we select an optimal set of tours that do not exceed the workforce availability. Our experiments show that the proposed algorithm can produce optimal solutions for small- and medium-sized instances and good or optimal solutions for large instances. The results also show that our approach outperforms the current airport dispatching policy. Funding: G. Dall’Olio was funded by the Deutsche Forschungsgemeinschaft [Grant Advanced Optimization in a Networked Economy Graduiertenkolleg 2201, Project 277991500]. Supplemental Material: The online appendix is available at https://doi.org/10.1287/trsc.2022.0110 .

A Method for Calculating Aggregate Flexibility Parameters of Virtual Power Plant Based on Improved Column and Constraint Generation Algorithm

The virtual power plant (VPP) can aggregate massive, dispersed, and heterogeneous distributed energy resources (DERs) and provide an effective energy management way for an active distribution system (ADS). In this paper, three parameters of upper and lower limits of active power output, ramping, and discharging energy is used to describe the aggregate flexibility of VPP. Based on this, a two-stage robust optimization (RO) model with dispatch instructions (DI) from the superior power grid (SPG) as uncertainty variables was constructed. Considering that the range of DI from the SPG changes during the iteration of the column and constraint generation (C&CG) algorithm, a position set is proposed to record the boundary information of the uncertain set of DI issued by the SPG when the sub-problem (SP) finds the worst scenario in each iteration, and the corresponding constraints are added to the master problem (MP). For the solution of SP, the bilevel max-min model is firstly transformed into a single-level bilinear programming (BLP) model by the duality theory and then solved by the concave-convex process (CCP) iterative method. Compared with transforming the BLP problem to a mixed integer programming (MILP) problem based on Karush-Kuhn-Tucker (KKT) condition, it can not only meet the engineering accuracy but also greatly save the solution time. Finally, the effectiveness of the proposed model and algorithm was verified on a modified IEEE 33-bus system.