Generalized Benders Decomposition Algorithm Research Articles

Coordinated Planning Strategy for Integrated Energy Systems in a District Energy Sector

With the ever-growing integration of diverse distributed energy resources, modern district energy sectors are transitioning into integrated energy systems (IESs), which generally consist of various energy carriers such as electric power, natural gas, and heat. Instead of modeling individual energy carriers, the emergence of IESs requires comprehensive consideration of all involved energy systems in both planning and operation phases. This paper proposes a comprehensive planning strategy for a district energy sector to address the challenges of IES planning considering the coupling of power, gas, and heat systems. The proposed planning model contains an operational module that develops a steady-state optimal multi-energy flow (OMEF) for the IES considered and a multi-stage expansion module that optimizes the investment decisions. To efficiently solve the proposed planning model, which is formulated as a mixed-integer nonlinear programming problem, an improved generalized Benders decomposition algorithm that utilizes dynamic dual multipliers to improve the convergence speed is employed. The effectiveness of the proposed planning model and the feasibility of the improved Benders decomposition algorithm are verified in case studies.

A stochastic programming model for emergency supply planning considering traffic congestion

Traffic congestion is one key factor that delays emergency supply logistics after disasters, but it is seldom explicitly considered in previous emergency supply planning models. To fill the gap, we incorporate traffic congestion effects and propose a two-stage location-allocation model that facilitates the planning of emergency supplies pre-positioning and post-disaster transportation. The formulated mixed-integer nonlinear programming model is solved by applying the generalized Benders decomposition algorithm, and the suggested approach outperforms the direct solving strategy. With a case study on a hurricane threat in the southeastern USA, we illustrate that our traffic congestion incorporated model is a meaningful generalization of a previous emergency supply planning model in the literature. Finally, managerial insights about the supplies pre-positioning plan and traffic control policy are discussed.

Optimal Scheduling of Battery Charging Station Serving Electric Vehicles Based on Battery Swapping

A battery charging station (BCS) is a charging facility that supplies electric energy for recharging electric vehicles’ depleted batteries (DBs). A BCS has a certain number of charging bays and maintains a dynamic inventory of fully charged batteries (FBs). This paper studies a BCS scheduling (BCSS) problem whose target is to schedule the charging processes of the charging bays such that the charging cost is minimized while satisfying the FB demand. Specifically, the BCSS problem has two types of operations: 1) loading DBs into the charging bays and then unloading them to the FB inventory when they are fully charged and 2) controlling the charging rate of each charging bay. We formulate the BCSS problem as a mixed-integer program with quadratic battery degradation cost. A generalized benders decomposition algorithm is then developed to solve the problem efficiently. The salience of the developed algorithm is that: 1) each charging bay can solve its own subproblem separately and 2) each subproblem can be further partitioned into multiple independent and identically structured quadratic programming problems, and thus the algorithm facilitates an efficient parallel implementation. We perform extensive real data simulation to validate the optimization model and demonstrate the efficiency of the proposed algorithm.

Adaptive Traffic Management and Energy Cooperation in Renewable-Energy-Powered Cellular Networks

In this paper, we consider a cellular system, in which base stations (BSs) are powered by both on-grid and renewable energy sources. To efficiently utilize the harvested energy of the BSs, we study adaptive traffic management (TM) and energy cooperation (EC) that aim at minimizing the on-grid energy consumption, while guaranteeing minimum average throughputs. To achieve this, we develop an adaptive TM and EC algorithm that jointly decides the energy sharing among BSs, the user association to BSs, and the sub-channel and power allocation in BSs. Within the algorithm, a network scheduling problem, which is mixed-integer non-linear programming (MINLP), should be solved in each timeslot. To efficiently solve it, we develop a network scheduling algorithm applying generalized Benders decomposition (GBD) that optimally solves the MINLP problem. In addition, we also develop a heuristic network scheduling algorithm that has a much lower computational complexity than the GBD algorithm, while providing comparable performance. Through the numerical results, we show that our algorithms always outperform the algorithms that use only one of TM or EC regardless of the system conditions.

Joint PEV Charging Network and Distributed PV Generation Planning Based on Accelerated Generalized Benders Decomposition

Integration of plug-in electric vehicles (PEVs) with distributed renewable power sources will reduce PEVs’ well-to-wheels greenhouse gas emissions, promote renewable power adoption, and defer power system investments. This paper proposes a multidisciplinary approach to jointly plan PEV charging stations and distributed photovoltaic (PV) power plants on a coupled transportation and power network. We formulate a two-stage stochastic programming model to determine the sites and sizes of: 1) PEV charging stations and 2) PV power plants. This proposed method incorporates comprehensive models of: 1) transportation networks with explicit PEV driving range constraints; 2) PEV charging stations with probabilistic quality of service constraints; 3) PV power generation with reactive power control; and 4) alternating current distribution power flow. The formulation results in a mixed-integer second-order cone program. We then design a generalized Benders decomposition algorithm to efficiently solve it. Numerical experiments show that investing in distributed PV power plants with PEV charging stations has multiple benefits, e.g., reducing social costs, promoting renewable power integration, and alleviating power congestion. The benefits become more prominent when utilizing PV generation with reactive power control, which can also help to enhance power supply quality.

Optimal User Scheduling in Energy Harvesting Wireless Networks

We consider a wireless network where multiple energy harvesting transmitters communicate with the common receiver in a time-sharing manner. In each slot, a transmitter can either harvest energy or send its data to the receiver. Given a time deadline, the goal is to maximize the sum rate of transmitters under random energy arrivals with both perfect and imperfect channel state information at the receiver. The original sum-rate maximization (SRM) problem is a non-convex mixed integer non-linear program (MINLP). To obtain the optimal scheduling policy, we first reduce the original optimization problem to a convex MINLP and solve it using the generalized Benders decomposition algorithm. We observe that the SRM problem results in an unfair rate allocation among transmitters, i.e., the transmitter closer to the receiver achieves a higher rate than that by the transmitter farther from the receiver. Hence, to induce fairness among transmitters, we consider the minimum-rate maximization (MRM) problem. For the bounded channel estimation error, we obtain a robust scheduling policy by solving the worst-case SRM and MRM problems. Finally, we compare the proposed policies with myopic policies studied in the literature and show that the former outperform the latter in terms of achievable rates.

Lossless convexification of nonconvex MINLP on the UAV path‐planning problem

SummaryMost unmanned aerial vehicle path‐planning problems have been modeled as linear optimal control problems, ie, as mixed‐integer linear programming problems. However, most constraints cannot be described accurately in linear form in practical engineering applications. In this paper, the traditional unmanned aerial vehicle path‐planning problem is modified as a nonconvex mixed‐integer nonlinear programming problem, whose continuous relaxation is a nonconvex programming problem. A lossless convexification method is introduced into the generalized Benders decomposition algorithm framework. Thus, an optimal solution can be obtained without directly solving the nonconvex programming problem. The output of the proposed algorithm has been rigorously proved to be the optimal solution to the original problem. Meanwhile, the simulation results verify the validity of the theoretical analysis and demonstrate the superior efficiency of the proposed algorithm.

Solving multi-period interdiction via generalized Bender’s decomposition

This paper considers a novel formulation of the multi-period network interdiction problem. In this model, delivery of the maximum flow as well as the act of interdiction happens over several periods, while the budget of resource for interdiction is limit. It is assumed that when an edge is interdicted in a period, the evader considers a rate of risk of detection at consequent periods. Application of the generalized Benders decomposition algorithm considers solving the resulting mixed-integer nonlinear programming problem. Computational experiences denote reasonable consistency with expectations.

Covariance‐based hardware selection part IV: Solution using the generalized benders decomposition

Recently the covariance based hardware selection problem has been shown to be of the mixed integer convex programming (MICP) class. While such a formulation provides a route to global optimality, use of the branch and bound search procedure has limited application to fairly small systems. The particular bottleneck is that during each iteration of the branch and bound search, a fairly slow semi‐definite programming (SDP) problem must be solved to its global optimum. In this work, we illustrate that a simple reformulation of the MICP and subsequent application of the generalized Benders decomposition algorithm will result in massive reductions in computational effort. While the resulting algorithm must solve multiple mixed integer linear programs, this increase in computational effort is significantly outweighed by the reduction in the number of SDP problems that must be solved. © 2016 American Institute of Chemical Engineers AIChE J, 62: 3628–3638, 2016

MINLP optimization of a heterogeneous azeotropic distillation process: Separation of ethanol and water with cyclohexane as an entrainer

Heterogeneous distillation processes are widely used in industry for the separation of azeotropic and close-boiling mixtures. This paper addresses the optimization of a heterogeneous distillation process for the separation of an azeotropic ethanol/water mixture using cyclohexane as an entrainer. Starting from a given process superstructure a MINLP problem is set up to consider continuous as well as discrete decision variables such as the feed locations and the number of stages of the distillation columns. A modified Generalized Benders Decomposition algorithm to account for non-convexities of the model equations solves the MINLP problem. The algorithm can be attached via Visual Basic for Applications (VBA) to any commercial process simulator with NLP and VBA capabilities. Various optimization runs show that the algorithm is easily applicable and returns solutions independent of the initial values.

Designing a reliable bio-fuel supply chain network considering link failure probabilities

This study presents a pre-disaster planning model that seeks to strengthen a bio-fuel supply chain system’s multi-modal facility links while accounting for limited budget availability. The model presented here determines which set of facilities and links to select that will maximize post-disaster connectivity and minimize bio-fuel supply chain related costs. The failure probability of the links between the multi-modal facilities is estimated using a spatial statistic model, which is developed from real world data. This paper develops a generalized Benders decomposition algorithm to solve this challenging NP-hard problem. The proposed algorithm is validated via a real-world case study with data from Mississippi and Alabama. Computational results show that the proposed solution approach is capable of solving the problem efficiently. Several experiments are conducted to demonstrate the applicability of this model by testing various model parameters on bio-fuel supply chain network performance, including reliability improvement cost, availability of budget, biomass supply changes, and the risk averseness degree for decision makers. Numerical analysis indicates that, under normal conditions, the minimum cost model determines a unit bio-fuel delivery cost of $3.56/gallon. However, in case of a disaster, the unit bio-fuel delivery cost provided by the minimum cost model increases to $3.96/gallon, compared to $3.69/gallon provided by the reliable model solution.

Integration of Scheduling and Dynamic Optimization of Batch Processes under Uncertainty: Two-Stage Stochastic Programming Approach and Enhanced Generalized Benders Decomposition Algorithm

Integration of scheduling and dynamic optimization significantly improves the overall performance of a production process compared to the traditional sequential method. However, most integrated methods focus on solving deterministic problems without explicitly taking process uncertainty into account. We propose a novel integrated method for sequential batch processes under uncertainty. The integrated problem is formulated into a two-stage stochastic program. The first-stage decisions are modeled with binary variables for assignment and sequencing while the second-stage decisions are the remaining ones. To solve the resulting complicated integrated problem, we develop two efficient algorithms based on the framework of generalized Benders decomposition. The first algorithm decomposes the integrated problem according to the scenarios so that the subproblems can be optimized independently over each scenario. Besides the scenario decomposition, the second algorithm further decomposes dynamic models from the sc...

Single allocation hub location problem under congestion: Network owner and user perspectives

The single allocation hub location problem under congestion is addressed in this article. This mixed integer non-linear programming problem is referential in discrete location research having many real applications. Two different network design perspectives are proposed: the network owner and the network user. These perspectives can be translated into mathematical programming problems that are very hard to solve due to their inherently high combinatorial nature combined to the nonlinearities associated to congestion. A very efficient and effective generalized Benders decomposition algorithm is then deployed, enabling the solution of large scale instances in reasonable time.

A facility location model with safety stock costs: analysis of the cost of single-sourcing requirements

We consider a supply chain setting where multiple uncapacitated facilities serve a set of customers with a single product. The majority of literature on such problems requires assigning all of any given customer's demand to a single facility. While this single-sourcing strategy is optimal under linear (or concave) cost structures, it will often be suboptimal under the nonlinear costs that arise in the presence of safety stock costs. Our primary goal is to characterize the incremental costs that result from a single-sourcing strategy. We propose a general model that uses a cardinality constraint on the number of supply facilities that may serve a customer. The result is a complex mixed-integer nonlinear programming problem. We provide a generalized Benders decomposition algorithm for the case in which a customer's demand may be split among an arbitrary number of supply facilities. The Benders subproblem takes the form of an uncapacitated, nonlinear transportation problem, a relevant and interesting problem in its own right. We provide analysis and insight on this subproblem, which allows us to devise a hybrid algorithm based on an outer approximation of this subproblem to accelerate the generalized Benders decomposition algorithm. We also provide computational results for the general model that permit characterizing the costs that arise from a single-sourcing strategy.

Distribution Systems Reconfiguration Based on OPF Using Benders Decomposition

This paper presents a new and efficient methodology for distribution network reconfiguration integrated with optimal power flow (OPF) based on a Benders decomposition approach. The objective minimizes power losses, load balancing among feeders, and is subject to constraints: capacity limit of branches, minimum and maximum power limits of substations or distributed generators, minimum deviation of bus voltages, and radial optimal operation of networks. A specific approach of the generalized Benders decomposition algorithm is applied to solve the problem. The formulation can be embedded under two stages: the first one is the master problem and is formulated as a mixed integer nonlinear programming problem. This stage determines the radial topology of the distribution network. The second stage is the slave problem and is formulated as a nonlinear programming problem. This stage is used to determine the feasibility of the Master problem solution by means of an OPF and provides information to formulate the linear Benders cuts that connect both problems. The model is programmed in the general algebraic modeling system. The effectiveness of the proposal is demonstrated through three examples extracted from the literature.

Optimal methodology for distribution systems reconfiguration based on OPF and solved by decomposition technique

AbstractThis paper presents a new and efficient methodology for distribution network reconfiguration integrated with optimal power flow (OPF) based on a Benders decomposition approach. The objective minimizes power losses, balancing load among feeders and subject to constraints: capacity limit of branches, minimum and maximum power limits of substations or distributed generators, minimum deviation of bus voltages and radial optimal operation of networks. The Generalized Benders decomposition algorithm is applied to solve the problem. The formulation can be embedded under two stages; the first one is the Master problem and is formulated as a mixed integer non‐linear programming problem. This stage determines the radial topology of the distribution network. The second stage is the Slave problem and is formulated as a non‐linear programming problem. This stage is used to determine the feasibility of the Master problem solution by means of an OPF and provides information to formulate the linear Benders cuts that connect both problems. The model is programmed in GAMS. The effectiveness of the proposal is demonstrated through two examples extracted from the literature. Copyright © 2009 John Wiley & Sons, Ltd.

Multiple allocation hub-and-spoke network design under hub congestion

The multiple allocation hub-and-spoke network design under hub congestion problem is addressed in this paper. A non-linear mixed integer programming formulation is proposed, modeling the congestion as a convex cost function. A generalized Benders decomposition algorithm has been deployed and has successfully solved standard data set instances up to 81 nodes. The proposed algorithm has also outperformed a commercial leading edge non-linear integer programming package. The main contribution of this work is to establish a compromise between the transportation cost savings induced by the economies of scale exploitation and the costs associated with the congestion effects.

Optimal production and scheduling of a process with decaying catalyst

AbstractA multiperiod mixed‐integer nonlinear programming (MINLP) optimization model is presented to schedule catalyst changeovers and determine the best operating policy for a chemical process with decaying performance. Two solution strategies are proposed to reduce the effect of nonconvexities and to reduce the solution time: (1) a search strategy that relies on partitioning the time horizon and (2) a strategy that makes use of the Generalized Benders Decomposition (GBD). Examples are presented for different time horizons to illustrate the performance of the proposed methods. The same objective value is reached with both strategies, which find better solutions than the full‐space MINLP. For small models, the GBD algorithm is faster than the partitioning search strategy, whereas the latter method is faster for larger models. Comparison with the operating strategy used in a real plant is also presented. © 2005 American Institute of Chemical Engineers AIChE J, 51:909–921, 2005

A DECOMPOSITION APPROACH FOR SOLVING KYP-SDPS

Semidefinite programs originating from the Kalman-Yakubovich-Popov lemma are convex optimization problems and there exist polynomial time algorithms that solve them. However, the number of variables is often very large making the computational time extremely long. Algorithms more efficient than general purpose solvers are thus needed. In this paper a generalized Benders decomposition algorithm is applied to the problem to improve efficiency.

Value-based transmission planning and the effects of network models

A value-based automated transmission system planning model is presented in this paper and the impact of the various network representations on the results of the model is discussed. Value-based integrated resource planning provides a societal cost (utility operation and investment costs plus customer outage costs) minimizing composite generation-transmission system expansion plan. The model trades-off and compares the costs and benefits of alternative generation and transmission resources using a consistent economic and engineering criterion in order to determine an optimal resource expansion plan. The generalized Benders decomposition algorithm with importance sampling used to solve the problem enables the model to estimate certain average information about scenarios or contingencies by examining only a very small fraction of all the scenarios or contingencies. The impacts of system resources upon both real and reactive power can be analyzed when using the AC power flow equations. It is concluded that for transmission system planning where reactive power flows and voltage constraints are important, it is imperative that an AC power flow model be used. The combination of AC power flow and linearized power flow models, with linearization about the solution of the AC power flow model, was found to be a promising compromise between accuracy and computation time.