Generalized Benders Decomposition Research Articles

A generalized Benders decomposition approach for the optimal design of a local multi-energy system

A local multi-energy system (LMES) is a decentralized energy system producing energy under multiple forms to satisfy the energy demand of a set of buildings located in its neighborhood. We study the problem of optimally designing an LMES over a multi-phase horizon. This problem is formulated as a large-size mixed-integer linear program with a block-decomposable structure involving mixed-integer sub-problems. We propose a new way to adapt a recently published framework for generalized Benders decomposition to our problem. This is done by exploiting the fact that the constraint matrix appearing in front of the first-stage variables in the coupling constraints is non-negative. The obtained generalized Benders decomposition algorithm relies on the use of a new type of non-convex Benders cuts involving indicator functions. We first prove that, under the assumption that all first-stage decision variables are integer and bounded, the finite and optimal convergence of our algorithm is guaranteed in theory. We then investigate how to obtain a good numerical performance in practice. Finally, we report the results of a computational study carried out on a real-life case study. These results show that the proposed algorithm clearly outperforms both a mathematical programming solver directly solving the problem as a whole and a state-of-the art hierarchical decomposition algorithm.

Design and operation of modular biorefinery supply chain under uncertainty using generalized Benders decomposition

AbstractBiomass supply chain performance is heavily affected by uncertainties stemming from supply, demand, or unexpected disruptions. Unlike petrochemical plants that use crude oil, biorefineries often have to deal with the uneven spatial‐temporal distribution of feedstock supply. The modular production strategy provides more flexibility in chemical manufacturing by allowing fast capacity expansion and unit movement. However, modeling and optimizing modular biomass supply chain under uncertainties becomes challenging due to high dimensionality and the existence of discrete decisions. This work optimizes the multiperiod biomass supply chain using the rolling horizon planning and two‐stage stochastic programming framework. We then applied generalized Benders decomposition to reduce the computational complexity of the stochastic mixed integer nonlinear programming supply chain optimization. Furthermore, the solution of the stochastic programming could be used to quantitatively describe the life‐cycle assessment uncertainties of the biomass supply chain performance, demonstrating seasonality and random variability.

Taking the human out of decomposition-based optimization via artificial intelligence, Part II: Learning to initialize

The repeated solution of large-scale optimization problems arises frequently in process systems engineering tasks. Decomposition-based solution methods have been widely used to reduce the corresponding computational time, yet their implementation has multiple steps that are difficult to configure. We propose a machine learning approach to learn the optimal initialization of such algorithms which minimizes the computational time. Active and supervised learning is used to learn a surrogate model that predicts the computational performance for a given initialization. We apply this approach to the initialization of Generalized Benders Decomposition for the solution of mixed-integer model predictive control problems. The surrogate models are used to find the optimal initialization, which corresponds to the number of initial cuts that should be added in the master problem. The results show that the proposed approach can lead to a significant reduction in solution time, and active learning can reduce the data required for learning.

Computationally efficient solution of mixed integer model predictive control problems via machine learning aided Benders Decomposition

Mixed integer Model Predictive Control (MPC) problems arise in the operation of systems where discrete and continuous decisions must be taken simultaneously to compensate for disturbances. The efficient solution of mixed integer MPC problems requires the computationally efficient online solution of mixed integer optimization problems, which are generally difficult to solve. In this paper, we propose a machine learning-based branch and check Generalized Benders Decomposition algorithm for the efficient solution of such problems. We use machine learning to approximate the effect of the complicating variables on the subproblem by approximating the Benders cuts without solving the subproblem, therefore, alleviating the need to solve the subproblem multiple times. The proposed approach is applied to a mixed integer economic MPC case study on the operation of chemical processes. We show that the proposed algorithm always finds feasible solutions to the optimization problem, given that the mixed integer MPC problem is feasible, and leads to a significant reduction in solution time (up to 97% or 50×) while incurring small error (in the order of 1%) compared to the application of standard and accelerated Generalized Benders Decomposition.

Look-Ahead Rolling Economic Dispatch Approach for Wind-Thermal-Bundled Power System Considering Dynamic Ramping and Flexible Load Transfer Strategy

With a wind-thermal-bundled power system (WTBPS) under high wind penetration levels, the sharp power fluctuations of tie-lines for interconnected grids trigger a significant challenge of security-constrained power system operation. Smoothing power fluctuations with economic dispatch is widely concerned against this challenge. In this paper, an appropriate look-ahead economic dispatch framework for WTBPS considering the variable ramp rate of retrofitted coal-fired units and the flexible load transfer strategy (LTS) in high voltage distribution networks (HVDNs) is proposed. To cope with this issue, <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">i</i> ) we derive a new family of tightened ramping constraints of retrofitted coal-fired units in the linear and second-order conic (SOC) forms, respectively. <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">ii</i> ) Using graph characterization of typical HVDNs, the simplified voltage-constrained LTS via topological structures can be developed. This proposed look-ahead economic dispatch model is cast as a mixed-integer second-order cone programming (MISOCP) problem, which can be reformulated by Multi-cut Generalized Benders Decomposition (MGBD). This MGBD enables multiple sub-problems can be solved in parallel and accelerates the solving process. Finally, simulation results of a large-scale practical system validate the computational accuracy and efficiency of the proposed approach.

Coordinated Planning of Integrated Electric and Heating System Considering the Optimal Reconfiguration of District Heating Network

The tight coupling of electric and heating systems would make it necessary to consider the coordinated planning of the energy facilities. The seasonal district heating network (DHN) reconfiguration can notably impact the planning results and offer a potential approach for better wind penetration. To address this issue, a coordinated planning of electric and heating (CPEH) model is proposed in this paper by considering the seasonally DHN reconfiguration, which is formulated as a mixed-integer linear programming problem. Considering the computation difficulties brought in by numerous operation scenarios involved in the CPEH problem, a scenario-oriented generalized Benders decomposition (S-BD) algorithm is proposed to solve the CPEH model with higher efficiency by generating feasibility and optimality cuts with a unified form. The numerical simulations and a practical case in Jilin, China, are simulated to demonstrate the efficiency of the S-BD and illustrate the potential benefits of DHN reconfiguration in enhancing wind energy penetration.

A two-stage stochastic programming approach for dynamic OD estimation using LBSN data

Estimating origin–destination (OD) demand is essential for urban transport management and traffic control systems. With the ubiquity of smartphones, location based social networks (LBSN) data has emerged as a new rich data source with broad urban spatial and temporal coverage highly suitable for OD estimation. Its nature of confirmed trip purpose (activity) and activity chain host makes it more advantageous than other data (e.g., household travel surveys and traffic network detection). On the other hand, LBSN data is a more direct and accurate representation of demand patterns and can remove the significant burden of developing traffic models and estimating simulation-based objective functions. However, thus far, most LBSN-based estimation models only focus on static (day-level) OD estimation, making less use of those characteristics. To this end, this paper establishes a two-stage stochastic programming (TSSP) framework integrating the activity chains to model activity-level dynamic mobility flows using LBSN data. The first-stage model aims to minimize the errors introduced by the inter-zone OD flows alongside the expected errors of the check-in patterns. The second-stage model attempts to minimize the errors produced by the considered check-in pattern scenarios. Markov chain Monte Carlo (MCMC) sampling is used to generate plausible check-in scenarios. A generalized Benders decomposition (GBD) algorithm is presented to solve the two-stage stochastic programming model. We conduct the experiments on the case study of Tokyo, Japan, under the employment of the generalized least squares (GLS) estimator. The results show that algorithm convergence can be guaranteed within several iterations. The approach can provide satisfactory estimations of check-in patterns, zonal production and attraction, and OD flows. Furthermore, multiple objective function states are tested for evaluating the completeness of the proposed framework and exploring its potential for simplification and extension. Incorporating specific penalty terms into the objective function also provides a way to verify the reliability of the two-stage structure and validate the effectiveness of the model. Finally, we discuss the model enhancement from the perspectives of online OD estimation by integrating with LBSN simulations, network-wide OD extrapolation using appropriate scaling methods, and removing the user-side data requirement by leveraging activity chain modeling. The proposed framework provides a novel and effective approach to OD demand estimation using LBSN data.

Joint Optimization of Security Strength and Resource Allocation for Computation Offloading in Vehicular Edge Computing

Vehicular Edge Computing (VEC) is a promising new paradigm that has attracted much attention in recent years, which can enhance the storage and computing capabilities of vehicular networks to provide users with low latency and high-quality services. Due to the open access and unreliable wireless channels, some appropriate security measures should be implemented in the VEC to ensure information security. However, the operation of the security mechanism dominates supererogatory computing resources, thus affecting the performance of VEC systems. The scarcity of computation and energy resources of the vehicles conflicts with the requirement of tasks for time delay and information security. In this paper, taking the driving velocity and position of the vehicles, the number of lanes, the model and density of the attackers, and security strength into consideration, we formulate a max-min optimization problem to jointly optimize offloading decision, transmit power, task computation frequency, encryption computation frequency, edge computation frequency, and block length to obtain optimal secure information capacity and local computation delay. The formulated optimization problem is a mixed integer nonlinear programming (MINLP), which is intractable. We apply the generalized benders decomposition (GBD)-based method to solve it. The simulation results show that our proposed algorithms have convergence and effectiveness and achieve fairness among vehicles on the road.

Process Scheduling Analysis and Dynamic Optimization Maintaining the Operation Margin for the Acetylene Hydrogenation Fixed-Bed Reactor

The full-cycle operation optimization of the acetylene hydrogenation reactor should strictly adhere to the operation optimization scheme within the operation cycle, regardless of scheduling changes. However, in actual industrial processes, in order to meet temporary process scheduling requirements, the acetylene hydrogenation reactor needs to adjust its operation strategy temporarily within the remaining operation cycle based on the results of dynamic optimization for a certain period. It brings additional challenges and a research gap to the operational optimization problem. To make up for this research gap, this paper focuses on researching a type of full-cycle dynamic optimization problem where the operation optimization scheme is temporarily adjusted during the operation cycle. The methods employed for changing the operation optimization scheme include modifying the operation cycle, maximizing economic benefits, and altering the optimization goal to maximize the operation cycle. A novelty full-cycle scheduling optimization framework based on surplus margin estimate is proposed to build a platform for these methods. The paper analyzes the impact of process scheduling changes on full-cycle optimization using a dynamic optimization model that maintains the operation margin. It establishes a full-cycle scheduling optimization model and obtains the optimal scheduling strategy by a novelty method NSGBD (non-convex sensitivity-based generalized Benders decomposition). In this process, an adaptive CVP (control vector parameterization) based on a decomposition optimization algorithm is proposed, which tackles the challenge of optimizing complex acetylene hydrogenation reactor models on a large time scale. Scheduling optimization can be realized as an annualized benefit of 1.56 × 106 and 1.57 × 106 ¥ separately within two scheduling optimization constraints, and the computational time required is much less than previous operational optimizations.

An exact decomposition technique for the deadline-constrained discrete time/cost trade-off problem with discounted cash flows

This paper addresses the deadline-constrained discrete time/cost trade-off problem with discounted cash flows in project scheduling optimization. Our objective is to optimize the net present value (NPV) of cash flows by selecting execution modes for activities while considering project deadlines. To address this problem, we propose an adapted model that combines the activity-on-node representation of the project network with the milestone activity payment system commonly used in real-world scenarios. This enhanced model accurately depicts project dynamics, enabling improved decision-making in NPV optimization. To solve the problem, we employ the generalized benders decomposition technique, which provides an exact solution approach to handle the computational complexity of the problem. Our approach introduces three types of feasibility cuts. The first type eliminates infeasible solutions by analyzing subproblem infeasibility. The second type enhances the generation of high-quality optimality cuts by iteratively introducing constraints to the master problem, utilizing dominated paths in the project network graph. Lastly, the third type targets the elimination of inferior infeasible solutions. Additionally, we generate a comprehensive dataset encompassing various scenarios and parameter ranges from existing literature. This dataset serves as a valuable resource for further research, with online accessibility and the latest solutions available for approximately 32,400 generated instances. Through our research, we demonstrate the superiority of our proposed solution procedure compared to existing methods in the field. By considering the specific requirements of project scheduling with NPV optimization and project deadlines, our approach contributes to more effective project planning and decision-making in real-world contexts.

Efficient iterative optimization to real-time train regulation in urban rail transit networks combined with Benders decomposition method

Considering the rapid expansion of urban rail transit networks and increasing transfer demands, this paper investigates the design of real-time train regulation strategies for urban rail networks under disturbances. Specifically, a mixed-integer nonlinear programming (MINLP) model is constructed in a rolling horizon (RH) framework to reduce train deviations and passenger waiting times, which involves complex coupling relationships among network-wide non-transfer passengers, transfer passengers and train traffics of different lines. To effectively address the model complexity and adapt to the real-time nature of this issue, we proposed an efficient iterative optimization (IO) approach to split the original problem into a smaller-scale regulation issue and a computationally cheap passenger loading procedure, which can efficiently treat the introduced nonconvexity of the exact modelling scheme. Besides, tailored to the mixed-integer property of the model, the generalized Benders decomposition (GBD) technique is incorporated into our algorithm. Furthermore, to verify the effectiveness of our method, we describe the detailed implementation of a series of numerical experiments on a small-scale and a real-world case. Computational results show that our regulation method can effectively contribute to the reliability of train systems, improving operational efficiency and service level. Besides, it performs promising computational efficiency in dealing with large-scale real-world cases, applicable for real-time applications.

Multi-IRS-Aided Millimeter-Wave Multi-User MISO Systems for Power Minimization Using Generalized Benders Decomposition

Difficulties in controlling IRSs to form the optimized passive beamforming have rarely been considered in intelligent reflecting surface (IRS)-aided systems, which are summarized as follows: 1) sending the optimized passive precoding vectors to the IRS controller incurs significant control overheads; 2) implementing the optimized passive precoding needs to set massive modes in the IRS control circuit. To address these issues, we investigate codebook-based passive beamforming for multi-IRS-aided millimeter-wave (mmWave) multi-user multiple-input single-output (MU-MISO) systems, where the control overheads are reduced to several scalars and the number of modes set in the IRS control circuit is reduced to that of codewords. Moreover, we formulate a joint passive and active precoding problem in the multi-IRS-aided mmWave MU-MISO system as a mixed-integer nonlinear programming (MINLP) problem, and then develop a generalized Benders decomposition (GBD)-based joint passive and active precoding algorithm. The proposed algorithm offers near-optimal performance (≥ 99.9%) with significantly-reduced computational complexity. Simulation results show that the proposed algorithm achieves energy savings of up to 50% and 95%, compared to the benchmark by the maximum ratio transmission and that without IRSs, respectively. In addition, the energy savings increase with the number of reflecting elements packed on each IRS as well as that of codewords.

Optimization under uncertainty of a hybrid waste tire and natural gas feedstock flexible polygeneration system using a decomposition algorithm

Market uncertainties motivate the development of flexible polygeneration systems that are able to adjust operating conditions to favor production of the most profitable product portfolio. However, this operational flexibility comes at the cost of higher capital expenditure. A scenario-based two-stage stochastic nonconvex Mixed-Integer Nonlinear Programming (MINLP) approach lends itself naturally to optimizing these trade-offs. This work studies the optimal design and operation under uncertainty of a hybrid feedstock flexible polygeneration system producing electricity, methanol, dimethyl ether, olefins or liquefied (synthetic) natural gas. A recently developed C++ based software framework (named GOSSIP) is used for modeling the optimization problem as well as its efficient solution using the Nonconvex Generalized Benders Decomposition (NGBD) algorithm. Two different cases are studied: The first uses estimates of the means and variances of the uncertain parameters from historical data, whereas the second assesses the impact of increased uncertain parameter volatility. The value of implementing flexible designs characterized by the value of the stochastic solution (VSS) is in the range of 260–405 M$ for a scale of approximately 893 MW of thermal input. Increased price volatility around the same mean results in higher expected net present value and VSS as operational flexibility allows for asymmetric exploitation of price peaks.

Chance-Constrained Joint Dispatch of Generation and Wind Curtailment-Load Shedding Schemes With Large-Scale Wind Power Integration

Wind curtailment (WC) and load shedding (LS) are indispensable measures to mitigate the operational risk in high-renewable power systems. Moreover, WC and LS schemes should be pre-scheduled and confirmed by related entities to make them applicable. In this paper, we propose a novel chance-constrained economic dispatch (CCED) model which can generate optimal WC and LS schemes accounting for the reserve shortage and transmission congestion problems. In the proposed method, WC and LS power are formulated as random decision variables, with which the infeasibility issues of conventional CCED are fully addressed. To solve the proposed model, we first convert the complicated chance constraints into a set of deterministic inequalities equivalently by employing the conditional Value-at-Risk (CVaR) representation and duality theory. Then, a two-layer iterative algorithm is proposed to solve the equivalent problem efficiently, which is based on the generalized Benders decomposition (GBD) framework. Numerical tests demonstrate the effectiveness and efficiency of the proposed method.

Grid-friendly peer-to-peer transactive energy market for inhibiting players’ strategic behavior

With the development of distributed renewable energy resources (RERs) in smart cities, the peer-to-peer (P2P) transactive energy market has become an effective way for RER owners to consume the surplus produced power. Nevertheless, the influence of grid usage costs in P2P energy trading has not been considered in prosumers’ strategic behaviors. To solve this issue, we analyze the network usage of P2P energy transactions and model a Nash–Stackelberg game to simulate the trading process of market participants. Firstly, we assume the prosumers can join the P2P energy market and determine their roles freely. The market manager adjusts the grid usage prices to influence the prosumers’ exchanged energy amounts. We then establish a monolithic VCG mechanism to inhibit the prosumers’ strategic behaviors while reducing the computational difficulty of the VCG mechanism. Finally, we develop a best-response (BR)-based algorithm to solve one energy prosumer’s cost minimization subproblem at a time to seek market equilibrium. Since the energy prosumers and market manager interact on the trading amount, the prosumers’ responses are modeled as bi-level mixed-integer nonlinear programming (BLMINLP). We employ the generalized Benders decomposition method to reduce the solution difficulty. Case studies verify the theoretical properties and show the practicability of the proposed algorithms.

Open capacity measurement and evaluation of n-1 security criteria considering distribution network dispatching security

In order to ensure the safety of distribution network dispatching, the open capacity evaluation, one of the distribution network operation and maintenance tools, is analyzed. Aiming at the shortcomings of traditional reactive power capacity evaluation that is too simple and cannot effectively handle multiple interconnected feeders, a new method is proposed to consider N-1 to establish security and network reconfiguration. Considering the safety factor and the minimum network loss, the open capacity is evaluated, and the interconnection switch is regenerated, which provides the best scheme for the maintenance of the distribution network. This model not only considers the power flow constraints during normal operation but also considers the switching operation constraints under fault conditions. It has the advantages of high solution quality and feasible calculation results. In order to solve the model, a two-level hybrid algorithm based on generalized Benders decomposition, a dichotomous approximation is proposed. The algorithm adopts the idea of upper approximation, lower test, and two-layer interaction. It continuously reduces the solution space through alternate iterations until the optimal solution is obtained. The test results of 22-node and 70-node systems verify the correctness and effectiveness of the model and algorithm in this paper.

An Exact Approach for Solving Pickup-and-Delivery Traveling Salesman Problems with Neighborhoods

This paper studies a variant of the traveling salesman problem, called the pickup-and-delivery traveling salesman problem with neighborhoods, that combines traditional pickup and delivery requirements with the flexibility of visiting the customers at locations within compact neighborhoods of arbitrary shape. We derive two optimality conditions for the problem, a local condition that verifies whether a given tour is locally optimal at the visiting points and a global condition that can be used to cut off suboptimal regions of the neighborhoods. We model the problem as a mixed-integer nonlinear program and propose a generalized Benders decomposition to solve instances of the problem with convex and nonconvex neighborhoods. Finally, we conduct extensive computational experiments to demonstrate the efficacy of our solution framework. Supplemental Material: The e-companion is available at https://doi.org/10.1287/trsc.2022.0138 .

A categorized and decomposed algorithm for thermal system simulation based on generalized benders decomposition

Accurate and efficient analysis of thermal systems is increasingly difficult with the increasing system scale and complexity. Many studies have to sacrifice the accuracy for calculation efficiency by simplifications. This work proposes a categorized and decomposed (C&D) algorithm to simulate thermal systems efficiently and robustly without introducing any simplification. It converts the system simulation into an equivalent optimization problem, which categorizes system constraints according to whether they are linear, and decomposes the optimization problem into a subproblem and a master problem to handle nonlinearity. Besides, variables’ gradient information is used to accelerate the convergence and enhance the stability. The proposed algorithm is applied to a supercritical carbon dioxide (sCO2) recompression Brayton cycle, and results show that it consumes about 8.84% calculation time compared with sequential modular method and requires much fewer initial values (about two orders of magnitude) compared with simultaneous equations method. Compared with the hierarchical and categorized algorithm, the proposed algorithm has a 48% larger convergence range regarding the deviation of initial values, and is more efficient when the initial value deviation is small. The proposed C&D algorithm owns a much higher efficiency and robustness and is a promising tool for complex thermal system simulation.

Coordinated Dispatch Optimization between the Main Grid and Virtual Power Plants Based on Multi-Parametric Quadratic Programming

Virtual power plants (VPPs) are a critical technology for distribution systems that can integrate various renewable energy resourcescontrollable loads and energy storage systems into one specific power plant through a distributed energy management system. This paper proposes a coordinated dispatch optimization model between the main grid and VPPs aiming to minimize both the power generation cost and total system active loss. When the time of the equivalent dispatching model is not divisible due to the existence of a time coupling constraint inside the VPPs, this model can obtain the global optimal solution through iteration between the main grid and the VPPs. By employing multi-parametric quadratic programming to obtain accurate critical domains and optimal cost functions, the convergence speed and stability are significantly improved. Additionally, a reactive power and voltage optimization technique leveraging the generalized Benders decomposition is presented for the coordination of the main grid and the VPPs. Moreover, the impact of distributed energy resource (DER) clusters on the main grid was studied, from which we proved that the proposed approach can expeditiously abate energy production expenditure and system active dissipation whilst enhancing the system equilibrium.

Resilience-oriented operation of microgrids in the presence of power-to-hydrogen systems

This study presents a novel framework for improving the resilience of microgrids based on the power-to-hydrogen concept and the ability of microgrids to operate independently (i.e., islanded mode). For this purpose, a model is being developed for the resilient operation of microgrids in which the compressed hydrogen produced by power-to-hydrogen systems can either be used to generate electricity through fuel cells or sold to other industries. The model is a bi-objective optimization problem, which minimizes the cost of operation and resilience by (i) reducing the active power exchange with the main grid, (ii) reducing the ohmic power losses, and (iii) increasing the amount of hydrogen stored in the tanks. A solution approach is also developed to deal with the complexity of the bi-objective model, combining a goal programming approach and Generalized Benders Decomposition, due to the mixed-integer nonlinear nature of the optimization problem. The results indicate that the resilience approach, although increasing the operation cost, does not lead to load shedding in the event of main grid failures. The study concludes that integrating distributed power-to-hydrogen systems results in significant benefits, including emission reductions of up to 20 % and cost savings of up to 30 %. Additionally, the integration of the decomposition method improves computational performance by 54 % compared to using commercial solvers within the GAMS software.