Multi-cut Benders Research Articles

Machine Learning-Enhanced Benders Decomposition Approach for the Multi-Stage Stochastic Transmission Expansion Planning Problem

The necessary decarbonization efforts in energy sectors entail integrating flexible assets and increased levels of uncertainty for the planning and operation of power systems. To cope with this in a cost-effective manner, transmission expansion planning (TEP) models need to incorporate progressively more details to represent potential long-term system developments and the operation of power grids with intermittent renewable generation. However, the increased modeling complexities of TEP exercises can easily lead to computationally intractable optimization problems. Currently, most techniques that address computational intractability alter the original problem, thus neglecting critical modeling aspects or affecting the structure of the optimal solution. In this paper, we propose an alternative approach to significantly alleviate the computational burden of large-scale TEP problems. Our approach integrates machine learning (ML) with the well-established Benders decomposition to manage the problem size while preserving solution quality. The proposed ML-enhanced Multicut Benders Decomposition algorithm improves computational efficiency by identifying effective and ineffective optimality cuts via supervised learning techniques. We illustrate the benefits of the proposed methodology by solving multi-stage TEP problems of different sizes based on the IEEE24 and IEEE118 test systems, while also considering energy storage investment options..

Optimal sizing of hybrid PV–diesel–biomass gasification plants for electrification of off-grid communities: An efficient approach based on Benders’ decomposition

Nowadays, millions of people in remote areas do not enjoy an uninterrupted power supply due to the lack of connectivity to the main power grid. Under such circumstances, the only feasible way to access electricity is typically local power generation, often relying on diesel engine–generator sets or photovoltaic arrays. However, on many occasions, this configuration does not fully exploit all available local resources, including biomass. Indeed, most isolated areas have access to local biomass production from agricultural activities, which can be used for local electricity generation through gasification. This paper addresses this challenge by developing an innovative optimal sizing tool for hybrid power plants integrating biomass gasifiers, specifically designed for isolated areas with access to local biomass production. The novel approach models the particular features of biomass gasification technologies, including long on/off times or restrictive ramping limits. To this end, an efficient methodology based on representative weeks is proposed, which is combined with a solution strategy based on the multi-cut Benders’ decomposition, thus resulting in a tractable framework that can deal with a huge amount of data efficiently. One of the most salient features of the new proposal is the consideration of local biomass production, which is included in the methodology through an original algorithm. Accordingly, a certain amount of biomass is sourced locally, leading to more accurate and reliable results. The new methodology is applied to a benchmark off-grid community in Ghana. The results demonstrate that the use of gasifiers reduces the project cost notably (by 90%) driven by the reduced biomass cost, which can be supplemented by locally generated biomass from agricultural activities. In addition, this technology constitutes a clean source of energy, reducing the total CO2 emissions by 83% compared to a baseline case in which only diesel generators are used. Moreover, it is demonstrated that biomass gasification can effectively act as base load power generation technology to reliably cover most of the local demand, thereby enabling a clean and inexpensive dispatchable local power generation. Finally, a sensitivity analysis reveals that the economic feasibility of the plant is more sensitive to the biomass cost than the selling price of biochar, resulting in a 33% increment in the total project cost when the price of biomass increases from 0 to 0.4 $/kg. Nevertheless, gasification remains as the predominant power generation technology even under unfavorable prices.

Establishing a hierarchical local market structure using multi-cut Benders decomposition

Local electricity markets (LEMs) such as peer-to-peer (P2P) and community-based markets allow prosumers and consumers to exchange electricity products and services locally. In order to coordinate electricity trading and flexibility services, this paper proposes a hierarchical prosumer-centric market framework with a hybrid LEM and a local flexibility market (LFM). Multi-cut Benders decomposition (MCBD) is employed to decompose the integrated hybrid LEM into a centralized P2P market and multiple community-based markets. The aggregators coordinate energy sources and demands of households in low voltage (LV) distribution networks (DN) as virtual power plants (VPPs) and engage in trading through a P2P market over the medium voltage (MV) DN. In addition, a modified MCBD (M-MCBD) approach is proposed to accelerate the convergence process. The LFM is operated by the distribution system operator (DSO) and is formulated as a mixed-integer nonlinear programming (MINLP) problem which is further relaxed to a mixed-integer second-order cone programming (MI-SOCP) problem. The case study demonstrates that aggregators were able to collaborate on trading within the hybrid LEM to minimize the costs incurred by prosumers within the network. Furthermore, the proposed M-MCBD method improves the scalability of the MCBD by enhancing its convergence speed and accuracy, as demonstrated by testing on problems of varying scales.

Optimal planning of collective photovoltaic arrays in energy communities through a multi-cut benders’ decomposition strategy

With the deregulation and the urgent necessity of decarbonizing the electricity sector, end users have been empowered to actively partake in the operation of the system. In this context, energy communities arise as a regulatory framework that allows aggregating a group of domestic consumers (or prosumers) in order to share own resources such as storage or photovoltaics. However, such collective assets should be properly designed in order to exploit them in an optimal way as well as avoid oversizing components, which may result in higher expenditures economically harmful. This new paradigm calls for developing proper computational tools for optimally planning and operating energy communities. In this paper, a novel tool for optimally planning collective photovoltaic arrays in cooperative energy communities is developed. This planning problem encompasses continuous and binary variables, which are evaluated over long time horizons and characterized through representative sets. These features make the problem computationally costly or even unaffordable for average machines. To overcome this issue, a novel solution strategy based on the multi-cut Benders’ decomposition is developed. The new proposal exploits the particular structure of the multi-year problem to produce dense cuts strategies that rely on multiple computational benefits, while the optimization framework allows including long-time parameters (such as panel degradation) in an intuitive way. The new approach is tested on a number of scenarios, comparing it with the traditional solution method without applying decomposition. Results reveals that the original intractable problem becomes manageable by average machines when applying the proposed decomposition technique, thus demonstrating its practicability and importance. In addition, the results obtained serve to highlight the importance of increasing local generation in energy communities, in order to reduce the dependency of the grid as well as reducing the total energy cost for users. In particular, the total project cost reduces by 19 % in our particular case study. Other aspects have been also highlighted, as the importance of photovoltaics to increase the exportable capacity of the community observing an increment of 96 % in the exportable energy in our case study. In this sense, the developed tool results valuable for community planners, allowing them to take decisions on pursuing to improve the economy and efficiency of the entire community.

Modeling the Reserve Capacity of Wind Power and the Inherent Decision-Dependent Uncertainty in the Power System Economic Dispatch

The ever-increasing penetration of renewable generation is demanding additional flexibility for the operation of power systems. In this sense, it has been proved that wind power generators (WPGs) can provide part of the required operating reserve through pitch angle control (PAC) under a deloaded strategy. Unlike conventional operating reserve sources, however, the operating reserve provided by WPGs can't always be considered reliable, which renders the power system dispatch more challenging. In this research, the reserve capacity of the WPGs is modeled considering the decision-dependent uncertainty (DDU). The random wind speed introduces uncertainty to the available reserve capacity of the WPGs while the decisions on the deloading degrees of the WPGs directly affect their coefficients between wind speed and usable wind power, making the uncertainty decision-dependent. Moreover, the operational technical limits of the WPGs and the multiple possible wind speed scenarios are also considered in the wind power reserve capacity modeling. Then, a two-stage reliability constrained stochastic economic dispatch model with WPGs' reserve provision (SED-WRP) is developed to jointly dispatch the generation and reserve. A group-based multi-cut Benders decomposition method is applied to improve the convergence performance of the solution algorithm. Case studies are solved to quantify the impact of considering the reserve provision of WPGs and the inherent DDU in power system dispatch decisions.

Hybrid quantum-classical multi-cut Benders approach with a power system application

Leveraging the current generation of quantum devices to solve optimization problems of practical interest necessitates the development of hybrid quantum-classical (HQC) solution approaches. In this paper, a multi-cut Benders decomposition (BD) approach that exploits multiple feasible solutions of the master problem (MP) to generate multiple valid cuts is adapted, so as to be used as an HQC solver for general mixed-integer linear programming (MILP) problems. The use of different cut selection criteria and strategies to manage the size of the MP by eliciting a subset of cuts to be added in each iteration of the BD scheme using quantum computing is discussed. The HQC optimization algorithm is applied to the Unit Commitment (UC) problem. UC is a prototypical use case of optimization applied to electrical power systems, a critical sector that may benefit from advances in quantum computing. The proposed approach is demonstrated using the D-Wave Advantage 4.1 quantum annealer.

The Benders by batch algorithm: Design and stabilization of an enhanced algorithm to solve multicut Benders reformulation of two-stage stochastic programs

This paper introduces a new exact algorithm to solve two-stage stochastic linear programs. Based on the multicut Benders reformulation of such problems, with one subproblem for each scenario, this method relies on a partition of the subproblems into batches. The key idea is to solve at most iterations only a small proportion of the subproblems by detecting as soon as possible that a first-stage candidate solution cannot be proven optimal. We also propose a general framework to stabilize our algorithm, and show its finite convergence and exact behavior. We report an extensive computational study on large-scale instances of stochastic optimization literature that shows the efficiency of the proposed algorithm compared to nine alternative algorithms from the literature. We also obtain significant additional computational time savings using the primal stabilization schemes.

Joint Charging Infrastructure and Autonomous Fleet Planning Considering Distributed Renewable Resources

Considering its unique operational characteristics, the autonomous mobility-on-demand (AMoD) system is playing a more and more critical role in the future transportation system. In this article, we consider a joint planning problem of autonomous fleet (i.e., vehicle type and fleet size) and charging infrastructure (i.e., the size of charging piles with various charging rates, distributed renewable resources) while accounting for the spatial–temporal demand-side flexibility of the AMoD system. We adopt an augmented network flow model to capture the autonomous fleet's charging, order serving, and rebalancing decisions. The proposed planning problem is modeled as a two-stage stochastic program and solved by multicut Benders decomposition techniques. Simulation experiments based on real-world data are performed to demonstrate the effectiveness of the proposed planning scheme.

Block-Based Multicut Benders Decomposition Algorithm for Transmission and Energy Storage Co-Planning

This study proposes a block-based multicut Benders decomposition algorithm to solve the co-planning of transmission expansion and energy storage problem in a bi-level approach. The proposal breaks the chronological representative period into multiple subperiods blocks. This division makes it possible to use parallel computation methods to solve each block simultaneously, reducing the simulation time, which allows the use of a more extensive time window to model the variability of random variables of the system, such as wind and load. In the proposed algorithm, the master problem defines the State of Charge (SoC) of the energy storage devices between the blocks and the investment in transmission and energy storage devices. To demonstrate the effectiveness of the proposed method, different sizes of representative periods are evaluated in three test systems: Garver 6-bus, IEEE-RTS 24-bus, and IEEE-118 188-bus. The tests compare the performance of the proposed block-based multicut Benders decomposition algorithm with the usual approach applied in the literature considering Benders decomposition and the complete problem formulated as a Mixed-Integer Linear Programming (MILP) problem.

A complementarity model for electric power transmission-distribution coordination under uncertainty

The growing penetration of stochastic renewable energy sources increases the need for operational flexibility to cope with imbalances. Existing proposals for flexibility procurement are envisioning markets where the transmission system operator (TSO) can access flexible resources located at the distribution system operator (DSO)-level and vice versa, but the coordination between these two entities is a matter of active research. We consider two trading floors, i.e., day-ahead and real-time markets, and propose a method for day-ahead coordination on how to share flexible resources, described as a complementarity model. The proposed coordination approach is to optimize prices and capacity limits at the physical interface of TSO and DSO, the so-called “coordination variables”. For given values of these variables, the DSO pre-qualifies the participation of DSO-level resources in the day-ahead market by capping their quantity bids. This way, the DSO ensures that the constraints of its system, modeled by a conic program, will be respected. Pursuing computational tractability, we decompose the model using a multi-cut Benders’ decomposition approach. It separates the conic modeling of real-time power flows under each scenario from the mixed-integer linear formulation of the day-ahead market-clearing problem. We quantify the potential benefit of the proposed coordination method in terms of improved social welfare. Using an ex-post out-of-sample simulation, the performance of the proposed coordination method is assessed against two benchmarks: (i) a fully uncoordinated scheme which obtains a lower bound for the expected social welfare, and (ii) an ideal benchmark which co-optimizes the TSO and DSO problems, providing an upper bound for the expected social welfare.

A two-stage stochastic MILP model for generation and transmission expansion planning with high shares of renewables

AbstractThis paper is concerned with the generation and transmission expansion planning of large-scale energy systems with high penetration of renewable energy sources. Since expansion plans are usually provided for a long-term planning horizon, the system conditions are generally uncertain at the time the expansion plans are decided. In this work, we focus on the uncertainty of thermal power plants production costs, because of the important role they play in the generation and transmission expansion planning by affecting the merit order of thermal plants and the economic viability of renewable generation. To deal with this long-term uncertainty, we consider different scenarios and we define capacity expansion decisions using a two-stage stochastic programming model that aims at minimizing the sum of investment, decommissioning and fixed costs and the expected value of operational costs. To be computationally tractable most of the existing expansion planning models employ a low level of temporal and technical detail. However, this approach is no more an appropriate approximation for power systems analysis, since it does not allow to accurately study all the challenges related to integrating high shares of intermittent energy sources, underestimating the need for flexible resources and the expected costs. To provide more accurate expansion plans for power systems with large penetration of renewables, in our analysis, we consider a high level of temporal detail and we include unit commitment constraints on a plant-by-plant level into the expansion planning framework. To maintain the problem computationally tractable, we use representative days and we implement a multi-cut Benders decomposition algorithm, decomposing the original problem both by year and by scenario. Results obtained with our methodology in the Italian energy system under a 21-year planning horizon show how the proposed model can offer professional guidance and support in strategic decisions to the different actors involved in electricity transmission and generation.

Benefits of Strategically Sizing Wind-Integrated Energy Storage and Transmission

We examine the behavior of a strategic firm that invests-in and operates wind, energy storage, and transmission. The capacity of the energy storage and transmission are co-optimized with the firm's wind-supply and energy-storage offers into a centrally dispatched electricity market. We employ a bi-level stochastic optimization model. The upper level determines the capacities and offering strategies to maximize the firm's expected profits. Multiple lower-level problems represent market clearing under different operating conditions, which capture uncertainties. The resulting large-scale optimization model is solved using multi-cut Benders's decomposition. The model is applied to a case study that is based on Alberta's electricity market.

Spacial coordination between data centers and power system considering uncertainties of both source and load sides

The management of the demand side in the power system, such as data centers (DCs), can improve the stability of the grid and bring economic benefits in planning and operation. Considering the increasing applications of big data and net-based technologies, the computational jobs in different region can be shifted as flexible load. In this paper, a new type of decomposition method is proposed to solve the model of the stochastic economic dispatch (ED) incorporating DCs, which considers the uncertainties of job arriving rates in DCs and wind generation in power systems. The original ED formulations are decomposed with Benders decomposition to improve computational performance, and a new type of group based multi-cut Benders decomposition is applied to improve the convergence. Case studies on a joint 18-bus system illustrate the benefits of spacial coordination between DCs and power system. The comparison between single-cut and multi-cut approach proved the effectiveness of the proposed method.

Do unit commitment constraints affect generation expansion planning? A scalable stochastic model

Due to increasing penetration of stochastic renewable energy sources in electric power systems, the need for flexible resources especially from fast-start conventional generation units (e.g., combined cycle gas turbine plants) is growing. The fast-start conventional units are being operated more frequently in order to respond to the variability and uncertainty of stochastic generation. This raises two important technical questions: as it is common in the literature, is it still an appropriate simplification to ignore the operational unit commitment (UC) constraints of conventional units within the generation expansion planning optimization? And if not, which UC constraint impacts most the expansion planning outcomes? To answer these questions, this paper aims at measuring the planning inefficiency (i.e., the underestimation of need for new generation capacity) caused by ignoring each UC constraint. To this purpose, we develop a centralized network-constrained generation expansion planning model incorporating UC constraints. In particular, we model start-up and shut-down costs, minimum production level and hourly ramping limits of conventional units. Wind power production is considered as the only source of uncertainty, and is modeled through a set of scenarios. A two-stage stochastic programming tool is used, whose first stage determines the long-term expansion and short-term UC decisions over different hours of representative days, while the second stage models the real-time operation for accommodating imbalances arising from wind deviation under different scenarios. Since this problem is potentially hard to solve especially with a large number of representative days and scenarios, a multi-cut Benders’ decomposition algorithm is implemented. The well-functioning of the proposed model and the impact of each UC constraint on planning outcomes are evaluated using an extensive numerical study. In our case studies, the exclusion of ramping constraints from planning optimization causes large error and is the most distorting simplification.

Benders decomposition for the Hazmat Transport Network Design Problem

We propose a new method for solving the Hazmat Transport Network Design Problem. In this problem, the government wants to reduce the risk of hazardous accidents for the population by restricting the shipment of hazardous goods on roads. When taking that decision, the government has to anticipate the reaction of the carriers who want to minimize the transportation costs by solving a shortest path problem. We use a bilevel formulation that guarantees stable solutions and transform this model into a mixed-integer linear program by applying the Karush–Kuhn–Tucker conditions. This model is solved to optimality with a multi-cut Benders decomposition. Valid inequalities and an acceleration approach for the master problem further reduce both the iteration number and the run time. Moreover, a partial decomposition technique for bilevel problems is introduced. The numerical study shows the computational benefits of the method and run time savings of more than 50% for large instances compared to a cutting plane method from the literature. The method is especially efficient for large number of shipments. Finally, we show that the bilevel model reduces the risk by 35% on average compared to a two-step decision process that does not anticipate the carriers’ reaction.

Scenario-based modeling for multiple allocation hub location problem under disruption risk: multiple cuts Benders decomposition approach

The hub location problem arises in a variety of domains such as transportation and telecommunication systems. In many real-world situations, hub facilities are subject to disruption. This paper deals with the multiple allocation hub location problem in the presence of facilities failure. To model the problem, a two-stage stochastic formulation is developed. In the proposed model, the number of scenarios grows exponentially with the number of facilities. To alleviate this issue, two approaches are applied simultaneously. The first approach is to apply sample average approximation to approximate the two stochastic problem via sampling. Then, by applying the multiple cuts Benders decomposition approach, computational performance is enhanced. Numerical studies show the effective performance of the SAA in terms of optimality gap for small problem instances with numerous scenarios. Moreover, performance of multi-cut Benders decomposition is assessed through comparison with the classic version and the computational results reveal the superiority of the multi-cut approach regarding the computational time and number of iterations.

A cross-decomposition scheme with integrated primal–dual multi-cuts for two-stage stochastic programming investment planning problems

We describe a cross-decomposition algorithm that combines Benders and scenario-based Lagrangean decomposition for two-stage stochastic programming investment planning problems with complete recourse, where the first-stage variables are mixed-integer and the second-stage variables are continuous. The algorithm is a novel cross-decomposition scheme and fully integrates primal and dual information in terms of primal---dual multi-cuts added to the Benders and the Lagrangean master problems for each scenario. The potential benefits of the cross-decomposition scheme are demonstrated with numerical experiments on a number of instances of a facility location problem under disruptions. In the original formulation, where the underlying LP relaxation is weak, the cross-decomposition method outperforms multi-cut Benders decomposition. If the formulation is improved with the addition of tightening constraints, the performance of both decomposition methods improves but cross-decomposition clearly remains the best method for large-scale problems.

Supplier selection in the processed food industry under uncertainty

This paper addresses an integrated framework for deciding about the supplier selection in the processed food industry under uncertainty. The relevance of including tactical production and distribution planning in this procurement decision is assessed. The contribution of this paper is three-fold. Firstly, we propose a new two-stage stochastic mixed-integer programming model for the supplier selection in the process food industry that maximizes profit and minimizes risk of low customer service. Secondly, we reiterate the importance of considering main complexities of food supply chain management such as: perishability of both raw materials and final products; uncertainty at both downstream and upstream parameters; and age dependent demand. Thirdly, we develop a solution method based on a multi-cut Benders decomposition and generalized disjunctive programming. Results indicate that sourcing and branding actions vary significantly between using an integrated and a decoupled approach. The proposed multi-cut Benders decomposition algorithm improved the solutions of the larger instances of this problem when compared with a classical Benders decomposition algorithm and with the solution of the monolithic model.

Risk-based integrated production scheduling and electricity procurement for continuous power-intensive processes

For optimal operation of power-intensive plants, production scheduling and electricity procurement have to be considered simultaneously. In addition, uncertainty needs to be taken into account. For this purpose, an integrated stochastic mixed-integer linear programming model is developed that considers the two most critical sources of uncertainty: spot electricity price, and product demand. Conditional value-at-risk is incorporated into the model as a measure of risk. Furthermore, scenario reduction and multicut Benders decomposition are implemented to solve large-scale real-world problems. The proposed model is applied to an illustrative example as well as an industrial air separation case. The results show the benefit from stochastic optimization and the effect of taking a risk-averse rather than a risk-neutral approach. An interesting insight from the analysis is that in risk-neutral optimization, accounting for electricity price uncertainty does not yield significant added value; however, in risk-averse optimization, modeling price uncertainty is crucial for obtaining good solutions.

Financial Hedging and Sustainability Modeling Considering Uncertainties: A Case Study of Ethanol Supply Chain

Incorporating financial hedging and sustainability in a supply chain is crucial for profit maximization or cost minimization. Uncertainties in supply chain develop into risks that affect the profit maximization or cost minimization expectations. In order to deliver end-products to destination markets in an efficient and effective manner, a supply chain management model that incorporates risk management measures is crucial. This paper develops a mathematical model that integrates hedging strategies in a biofuel supply chain with a corn and cellulosic raw material production setting. The paper is structured by first developing an optimization model considering maximization of the supply chain profit with risk without hedging for both corn and cellulosic biorefinery plants. Secondly, we incorporate sustainability concepts including environmental and social aspects. Finally, a heuristic method is developed for the hedging and a two-stage stochastic linear programming with Multi-cut Benders Decomposition Algorithm (MBD) is used to solve the problem. A case study in North Dakota is adopted for this study. The results for hedging and non-hedging are compared and sensitivity analyses conducted.