Generalized Benders Decomposition Algorithm 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.

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.

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.

Flexibility transformation strategy of thermal power units for typical scenario considering wind and solar consumption based on generalized Benders decomposition algorithm

Flexibility transformation strategy of thermal power units for typical scenario considering wind and solar consumption based on generalized Benders decomposition algorithm

Full-time scale resilience enhancement strategies for an integrated electricity-gas-transportation coupled system

The frequent occurrence of extreme disaster events with low probability and high risk poses a serious threat to the safe and stable operation of power grids. With the development of energy internet technology, the deep integration of power distribution, natural gas and transportation networks has opened a promising way for resilient distribution networks. This paper proposes a full-time scale resilience enhancement framework for an integrated electricity-gas-transportation coupled system oriented to energy internet. In this framework, a minimax regret robust optimization model considering the uncertainty of extreme disaster attacks is developed to make the line hardening strategy in the pre-disaster stage, which is solved by the column-and-constraint generation algorithm. Furthermore, the synthetic restoration model considering the co-optimization of dynamic dispatch of gas-fired units, network reconfiguration and damaged component repair is established, in which the coupling influence of the gas and transportation networks on the restoration strategy during the disaster and post disaster stages is taken into account. And the sequential cone programming method and generalized Benders decomposition algorithm is implemented to solve the above model. Finally, the proposed model and solution method are validated via a modified 33-bus power distribution network coupled with a 12-node transportation network and a 7-node gas network. The simulation results demonstrate that the proposed minmax regret robust optimization model can achieve a pre-disaster defensive strategy with less conservativeness in contrast to the conventional robust optimization model. Specifically, the cost under the worst-case attacks by the former has reduced by 11.52–22.29% compared to the latter. Moreover, the influence of the interdependent critical infrastructure networks on the synthetic restoration strategy in the disaster and post-disaster stages has been investigated. In particular, the power output by gas-fired units has reduced by 2699 kW when considering the operation constraints of the gas network. Additionally, due to the time-varying characteristics of traffic flow and commuting time in the transportation network, the repair time for the damaged components is shortened by 1.09 h while the preparation time of crews changes from 2.5 to 1.5 h.

Admissible HVDC infeed capacity evaluation of receiving-end power grids

To fulfil the cross-regional bulk power delivery of renewable energy and alleviate the power shortage pressure of load centers, a number of line-commutated converter-based high voltage direct current (LCC-HVDC) transmission projects have been deployed in worldwide. Due to the system-wide operating risk sharply escalated with the intensively infeed of multiple large-capacity HVDC links, a vital knowledge gap exists in understanding the maximal admissible HVDC infeed capacity (AHIC) of receiving-end power grids. To tackle this issue, this paper proposes a novel AHIC evaluation method, which determines the maximal allowed total number and capacity of HVDC links that the receiving-end grid can accommodate while taking into account multi-preconceived fault security requirements. An improved dynamic system strength criterion is developed to assessing the immunity of post-contingency HVDC commutation failures (CF), which creatively considers the support effect from all AC buses based on the voltage sensitivity. The original two-stage model is decomposed into a construction master problem and an operation optimization subproblem, and then being iteratively solved using the generalized Benders decomposition algorithm. The effectiveness of the proposed approach is demonstrated on a modified IEEE RTS-79 system and a practical large-scale system.

Two-Stage emergency material scheduling based on benders decomposition considering traffic congestion after a disaster

This study establishes a stochastic mixed integer nonlinear programming model for a two-stage emergency multiclass material scheduling problem considering traffic congestion with a Bureau of Public Roads (BPR) function. The generalized Benders decomposition (GBD) algorithm is used to solve the model on the basis of the characteristics of the decomposable structure and nonlinear convex function. To prove the effectiveness of the GBD algorithm, we set up a 30-node medium-scale case and a 50-node large-scale case for experimental demonstration and compared the GBD algorithm with the simple branch and bound (SBB) and discrete and continuous optimization solvers. Results show that for the medium-scale experiment, the gap between the solution results of the GBD algorithm and the ones of SBB or other solvers is less than 0.2%. The solution time of the GBD algorithm is only half of the time of SBB and other solvers. The large-scale case of 50 nodes cannot be solved by the SBB or other solvers, whereas the GBD algorithm can solve it normally, and the results are reasonable. In large-scale cases with more than 50 nodes, the GBD algorithm is more effective and efficient compared with SBB and other solvers. The sensitivity of the correlation coefficient α in the BPR function was analyzed. The actual expected total cost increases with the increase in the value of α. Its solving results are accurate and reasonable, proving that the results are more practical and applicable after adding the BPR function.

Robust location and sizing of electric vehicle battery swapping stations considering users’ choice behaviors

Battery swapping electric vehicle (EV) is an effective way to achieve vehicle electrification due to its advantage in reducing the waiting time and the upfront purchase cost, and alleviating users’ range anxiety. This paper proposes a robust battery swapping stations location and sizing model for the EV battery swapping service network design problem considering the users’ choice behaviors. Specifically, this paper develops a multinomial logit model to characterize the interaction between the service provider’s decisions and the EV users’ choice, and derives closed-form results for the sizing decisions to satisfy the pre-specified service level criteria by utilizing the distributionally robust chance constraint method and GI∖G∖m queueing model. This paper develops an equivalent mixed integer second-order cone programming (SOCP) reformulation and a generalized Benders decomposition algorithm with speedup strategies to solve the non-convex robust model efficiently. We extend the proposed model to consider the power network constraints, and propose a data-driven algorithm to estimate EV users’ swapping demands from traffic network data. Experimental results show that the proposed algorithm outperforms the state-of-the-art Gurobi solver in terms of run time and solution quality. This paper conducts a case study using real-world data to examine the influence of EV users’ choice behavior, service levels, charging rate, and battery swapping service rate on the optimal decisions and the expected profit, and presents useful managerial implications for practitioners.

Collaborative EV Routing and Charging Scheduling With Power Distribution and Traffic Networks Interaction

The increasing of electric vehicles (EVs) alleviates the faced environmental problems but brings challenges to the optimal operation of transportation network (TN) and distribution network (DN). However, the most of existing research works consider EV charging station assignment and navigation services in the TN separately from charging station power scheduling services in the DN. To overcome this research gap, this paper proposes a collaborative optimal routing and scheduling (CORS) method, providing optimal route to charging stations and designing optimized charging scheduling schemes for each EV. In the order of reporting, whenever an EV reports its charging demand, a CORS optimization model is built and solved so that a specific charging scheme is designed for that EV. Then, the TN and DN status is updated to guide the subsequent EVs operating. The proposed CORS integrates the real-time state of the TN and DN, and effects positive benefits in helping EVs to avoid traffic congestion, improving the utilization level of charging facilities and enhancing charging economy. The combined distributed biased min consensus algorithm and generalized benders decomposition algorithm are adopted to solve the complex nonlinear optimization problem. Through comparing with the existing methods, better effectiveness is verified by simulation results.

Optimization of a multiperiod refinery planning problem under uncertainty

AbstractUncertainty in refinery planning presents a significant challenge in determining the day‐to‐day operations of an oil refinery. Deterministic modeling techniques often fail to account for this uncertainty, potentially resulting in reduced profit. The stochastic programming framework explicitly incorporates parameter uncertainty in the problem formulation, thus giving preference to robust solutions. In this work, a nonlinear, multiperiod, industrial refinery problem is extended to a two‐stage stochastic problem, formulated as a mixed‐integer nonlinear program. A crude‐oil sequencing case study is developed with binary scheduling decisions in both stages of the stochastic programming problem. Solution via a decomposition strategy based on the generalized Benders decomposition (GBD) algorithm is proposed. The binary decisions are designated as complicating variables that, when fixed, reduce the full‐space problem to a series of independent scenario subproblems. Through the application of the GBD algorithm, a feasible mixed‐integer solution is obtained that is more robust to uncertainty than its deterministic counterpart.

Enhancing Resilience of Distribution Networks by Coordinating Microgrids and Demand Response Programs in Service Restoration

In case of high-impact low-probability events, in order to restore the critical loads of the distribution network as much as possible, it is necessary to employ all available resources such as microgrids and distributed generations. This article presents a two-stage method for critical load restoration after an extreme event utilizing the coordination of all available distributed generations, microgrids, and demand response programs. In the first stage, the postdisaster reconfiguration of the network is determined, and in the second stage, the demand response outputs of the responsive loads and the restoration status of all critical loads are ascertained. In the first stage, an algorithm is proposed to determine electrical islands, and in the second stage, a new model based on the generalized Benders decomposition algorithm is proposed, which considers demand response programs and all operational constraints and aims to maximize the restored critical loads. The effectiveness of the proposed method is verified by simulating the 33- and 69-bus test systems.

Budgeting in International Humanitarian Organizations

Problem definition: International humanitarian organizations (IHOs) prepare a detailed annual allocation plan for operations that are conducted in the countries they serve. The annual plan is strongly affected by the available financial budget. The budget of IHOs is derived from donations, which are typically limited, uncertain, and to a large extent earmarked for specific countries or programs. These factors, together with the specific utility function of IHOs, render budgeting for IHOs a challenging managerial problem. In this paper, we develop an approach to optimize budget allocation plans for each country of operations. Academic/practical relevance: The current research provides a better understanding of the budgeting problem in IHOs given the increasing interest of the operations management community for nonprofit operations. Methodology: We model the problem as a two-stage stochastic optimization model with a concave utility function and identify a number of analytical properties for the problem. We develop an efficient generalized Benders decomposition algorithm as well as a fast heuristic. Results: Using data from the International Committee of the Red Cross, our results indicate 21.3% improvement in the IHO’s utility by adopting stochastic programming instead of the expected value solution. Moreover, our solution approach is computationally more efficient than other approaches. Managerial implications: Our analysis highlights the importance of nonearmarked donations for the overall performance of IHOs. We also find that putting pressure on IHOs to fulfill the targeted missions (e.g., by donors or media) results in lower beneficiaries’ welfare. Moreover, the IHOs benefit from negative correlation among donations. Finally, our findings indicate that, if donors allow the IHO to allocate unused earmarked donations to other delegations, the performance of the IHO improves significantly.

A Distributionally Robust Chance-Constrained Unit Commitment with N-1 Security and Renewable Generation

With the increasing penetration of renewable energy generation, one of the major challenges is the problem of how to express the stochastic process of wind power and photovoltaic output as the exact probability density and distribution, in order to improve the security and accuracy of unit commitment results, a distributed robust security-constrained optimization model based on moment uncertainty is proposed, in which the uncertainty of wind and photovoltaic power is captured by two uncertain sets of first- and second-order moments, respectively. The two sets contain the probability distribution of the forecast error of the wind and photovoltaic power, and in the model, the energy storage is considered. In order to solve the model effectively, firstly, based on the traditional chance-constrained second-order cone transformation, according to the first- and second-order moments polyhedron expression of the distribution set, a cutting plane method is proposed to solve the distributed robust chance constraints. Secondly, the modified IEEE-RTS 24 bus system is selected to establish a simulation example, an improved generalized Benders decomposition algorithm is developed to solve the model to optimality. The results show that the unit commitment results with different emphasis on economy and security can be obtained by setting different conservative coefficients and confidence levels and, then, provide a reasonable decision-making basis for dispatching operation.

A stochastic programming model for emergency supply planning considering transportation network mitigation and traffic congestion

How to conduct effective and efficient emergency supply planning is a challenging task. In this paper, we tackle a general emergency supply planning problem. The problem not only integrates the decisions of transportation network mitigation and emergency supply pre-positioning before disasters, but also considers post-disaster dynamic transportation planning with traffic congestion effects incorporated. We formulate this problem as a two-stage stochastic programming model, which aims to minimize the expected total cost related to various disaster mitigation, preparedness, and response decisions. A variant of the model is optimally solved by applying a generalized Benders decomposition algorithm, which significantly outperforms state-of-the-art global optimization solvers. Finally, a case study for a hurricane threat in the southeastern U.S. is conducted to demonstrate the advantages of our model and to illustrate insights on the optimal network mitigation and pre-positioning plan as well as the transportation plan. It is shown that considering traffic congestion effects and dynamic transportation plans brings about spatial and temporal flexibility for achieving better emergency supply plans.

Generalized Benders decomposition for competitive facility location with concave demand and zone-specialized variable attractiveness

We study a competitive facility location problem, in which a company enters a market where competitor’s facilities exist. Customers with elastic buying powers make choices following the gravity rule. Each facility, once open, has an intrinsic fixed attraction to customers. Besides, zone-specialized variable attractiveness can be provided to increase the total attraction of a facility to a specific customer zone. The objective of the company is to maximize profit by determining the locations of the facilities and the facility-customer pairwise attractiveness level, accounting for the expected revenue and the cost. The problem is formulated as a mixed-integer nonlinear program and subsequently solved by a tailored generalized Benders decomposition algorithm with tunable parameters. We then conduct extensive computational studies to demonstrate the efficiency of the algorithm. Finally, we analyze the solution structures under different scenarios and provide managerial implications for real-world applications.

A Thinned Irregular Array Synthesis Approach Based on Benders Decomposition

A thinned irregular tiled array synthesis approach is proposed for 5G base station array design. To fit the needs of multiusers, different sizes of vertical tiles with thinned elements ae considered. The tiling configuration and excitations are optimized to achieve high gain, low sidelobes, and wide-angle scanning simultaneously. A mixed integer second-order cone programming (MISOCP) model is obtained from convexity analysis and transformation. Besides, a generalized Benders decomposition algorithm is built to decouple the complex original problem into mixed integer linear programming (MILP) master problems and convex second-order cone programming (SOCP) subproblems, which is used to solve the large-scale problems efficiently. Irregular partition optimization is based on full-wave simulations of regular finite arrays. The joint effects of active VSWR and radiation patterns are considered to reduce the mismatch between element simulations and final performances. Numerical results show that the 5G array synthesized by the proposed approach is superior to those reported by the state-of-the-art methods in terms of both E- and H-plane scans and the number of T/R modules.

Dynamic optimization of fluid catalytic cracking unit using a nonconvex sensitivity-based generalized Benders decomposition

Fluid catalytic cracking unit, whose batch operations are operated in a multirate mode, is a typical continuous process with batch operations. The integrated optimization of this problem can be formulated as a hybrid parametric dynamic optimization. To obtain a high-quality solution, adaptive direct methods are usually required to solve the problem iteratively. By exploiting the decomposable structure, a novel framework is proposed in this paper, which can obtain an equivalent or better precision solution with relatively coarse discretization. In detail, by designating the batch operations as complicating variables, an optimal solution and sensitivity information about batch operations are obtained by a nonconvex sensitivity-based generalized Benders decomposition algorithm. Then the optimal continuous operations are implemented as extra closed-loop controllers by tracking the necessary conditions of optimality, while the optimal batch operations are improved by a line search method. The practical potential of the framework is demonstrated with several operation modes.

Towards Optimal Planning of EV Charging Stations under Grid Constraints

At present, the public charging network can not fully satisfy the charging demands of electric vehicles (EVs), which hinders the further development of EVs. In fact, as the key roles of charging service market, the operators need to plan charging stations properly to improve the profit while ensuring qualified charging service. Meanwhile, as the power supplier, the grid requires the charging stations to be deployed properly to lower the generation cost while ensuring safe and stable grid operation. This paper aims to plan the EV charging station (CS) network to improve the comprehensive profit by taking both the operator and the grid into consideration. Firstly, the Voronoi diagram method is used to divide the area to be planned based on the candidate set. Then, a mathematical profit maximization model with electric physical constraints is designed to distribute appropriate capacities for each candidate EV CS locations. The generalized Benders decomposition algorithm is applied to obtain the optimal solution. Finally, the simulation results demonstrate the effectiveness of the proposed algorithm based on a case study which consists of a 56-node distribution system and Xiamen traffic network system.

Harvest-or-Transmit Policy for Cognitive Radio Networks: A Learning Theoretic Approach

We consider an underlay cognitive radio network where the secondary user (SU) harvests energy from the environment. We consider a slotted-mode of operation where each slot of SU is used for either energy harvesting or data transmission. Considering block fading with memory, we model the energy arrival and fading processes as a stationary Markov process of first-order. We propose a harvest-or-transmit policy for the SU along with optimal transmit powers that maximize its expected throughput under three different settings. First, we consider a learning-theoretic approach where we do not assume any a priori knowledge about the underlying Markov processes. In this case, we obtain an online policy using Q-learning . Then, we assume that the full statistical knowledge of the governing Markov process is known a priori . Under this assumption, we obtain an optimal online policy using infinite horizon stochastic dynamic programming . Finally, we obtain an optimal offline policy using the generalized Benders decomposition algorithm. The offline policy assumes that for a given time deadline, the energy arrivals and channel states are known in advance at all the transmitters. Finally, we compare all policies and study the effects of various system parameters on the system performance.

Budgeting in International Humanitarian Organizations

Problem definition: International Humanitarian Organizations (IHOs) prepare a detailed annual allocation plan for the operations to be conducted in the countries they serve. The annual plan is strongly affected by the available financial budget. Budget of IHOs comes from donations, which are typically limited, uncertain, and to a large extent earmarked for specific countries or programs. These factors, together with the specific utility function of IHOs, render budgeting for IHOs a challenging managerial problem. In this paper, we develop an approach to optimize budget allocation plans for each country of operations. Academic relevance: The current research provides a better understanding of the budgeting problem in IHOs, given the increasing interest of the operations management community for nonprofit operations. Methodology: We model the problem as a two-stage decision process, and identify a number of analytical properties for the problem. We then develop an efficient generalized Benders decomposition algorithm as well as a fast heuristic to solve the problem. We also propose a simple framework to calculate the model parameters from field data, which we apply to data of the International Committee of the Red Cross. Results: Our results indicate 21.3% improvement in ICRC’s utility by adopting stochastic programming instead of expected value solution. Moreover, our solution approach is computationally more efficient than other alternatives. Managerial insights: Our analysis indicates the importance of non-earmarked donations for the overall performance of IHOs. We also find that putting a high pressure on IHOs to fulfill the targeted missions (by donors, media, etc.) results in a lower welfare for beneficiaries. Finally, our findings indicate that if donors would allow the IHO to allocate excessive earmarked donations to other delegations, the performance of IHO would improve significantly, and that the advantage of such a scheme is increasing in the size of the IHO.