Large-scale Mixed-integer Linear Programming Research Articles

Multistage Stochastic optimization for mid-term integrated generation and maintenance scheduling of cascaded hydroelectric system with renewable energy uncertainty

The uncertainties resulting from the escalating penetration of renewable energy resources pose severe challenges to the efficient operation of modern power systems. Hydroelectricity is characterized by its flexibility, controllability, and reliability, and thus becomes one of the most ideal energy resources to hedge against such uncertainties. This paper studies the mid-term integrated generation and maintenance scheduling of a cascaded hydroelectric system (CHS) consisting of multiple cascaded reservoirs and hydroelectric units. To precisely describe the mid-term water regulation policies, the hydraulic coupling relationship and water-energy nexus of CHS are incorporated into the proposed optimization model. The uncertainties of natural water inflow and the power outputs of wind/solar energy generation are taken into consideration and captured via a stochastic process modeled by a scenario tree. A multistage stochastic optimization (MSO) approach is developed to coordinate the complementary operations of multiple energy resources, by optimizing the mid-term water resource management, generation scheduling, and maintenance scheduling of CHS. The proposed MSO model is formulated as a large-scale mixed-integer linear program that presents significant computational intractability. To address this issue, a tailored Benders decomposition algorithm is developed. Two real-world case studies are conducted to demonstrate the capability and characteristics of the proposed model and algorithm. The computational results show that the proposed MSO model can exploit the flexibility of hydroelectricity to efficiently respond to variable wind and solar power, and reserve water resources for the generation in peak months to reduce the consumption of fossil fuel. The proposed solution approach also exhibits promising computational efficiency when handling large-scale models.

Optimal design of aggregated energy systems with (N-1) reliability: MILP models and decomposition algorithms

This work investigates the design optimization of aggregated energy systems (multi-energy systems, microgrids, energy districts, etc.) with (N-1)-reliability requirements. The problem is formulated as a two-stage stochastic Mixed Integer Linear Program which optimizes design (first stage variables) and operation variables (second stage variables) simultaneously considering a set of typical and extreme days. The analysis proposes and compares different approaches to include the (N-1) reliability requirement in the optimization problem. Moreover, the paper proposes two effective decomposition algorithms to solve the large-scale Mixed Integer Linear Program suitable for design problems with and without (N-1) reliability requirements. Depending on the instance, such decomposition algorithms allow reducing the computational time by one or more orders of magnitude (from days to a few hours, in the worst cases tested in this work). The proposed methodology is tested to design the aggregated energy system for a real case study considering both a grid-connected and off-grid installation. Results indicate that the actual reliability of the design solutions depends by the profiles of energy demand and renewable production considered in the failure scenarios included in the design problem. Including N-1 reliability requirements causes an increase in the total annual cost in the range 15–20%, due to the increase in capital costs.

Multi-period stochastic programming for relief delivery considering evolving transportation network and temporary facility relocation/closure

In this study, we address a dynamic network design problem pertaining to the distribution of essential supplies during the post-disaster response phase. We present a two-stage, stochastic multi-period model that minimizes both expected unmet demand and costs associated with operating emergency response facilities and distributing relief supplies, taking into account evolving road conditions as well as the closure and relocation of temporary relief facilities. To address the computational intractability inherent in large-scale mixed-integer linear programming (MILP), we implement an accelerated branch-and-Benders-cut algorithm, facilitating efficient problem-solving processes. We validate this approach through extensive numerical analysis and its application to a real-world scenario, demonstrating its superiority over deterministic methods, especially when considering the trade-off between nominal operational costs and out-of-sample reliability. Results show that dynamically adjusting the number of operating facilities and their locations may enhance cost efficiency while decreasing unmet demand during the relief delivery process as the transportation network changes with time. Additionally, we show that utilizing stochastic models for decision-making can substantially hedge against demand uncertainty, both in meeting demand and reducing operational costs. However, the benefits of stochastic solutions diminish as the disruption rate increases.

A Power Grid Partitioning Method for Short-Circuit Current Considering Multi-Scenario Security with an Improved Direct Current Model

In order to ensure the security of power grids and control the level of short-circuit currents, a multi-objective optimization method for power grid partitioning is proposed. This method takes into consideration both short-circuit currents and multi-scenario safety constraints. A power grid partitioning optimization model is established to achieve objectives such as minimizing disconnected lines, maximizing safety margins, and ensuring load balance in the main transformers. The model aims to satisfy constraints related to short-circuit current levels, base-case power flow, and N-1 security. To address the significant deviation in the static security constraint model caused by large amounts of active power losses in large-scale power grids, an improved direct current model is proposed to reduce these errors and meet the accuracy requirements for grid partitioning optimization. Additionally, to adapt to the variability of renewable energy output, an optimization method is proposed, combining three scenarios of renewable energy generation while satisfying short-circuit current and static security constraints. The power grid partitioning model is mathematically formulated as a large-scale mixed-integer linear programming problem, which presents challenges in terms of hardware requirements and computational complexity when solved directly. To mitigate these challenges, equivalent WARD values are assigned to the short-circuit current constraints, base-case constraints, and anticipated fault-induced power flow constraints. Anticipated faults and bottleneck branches are accurately incorporated, and the problem is decomposed into smaller-scale mixed-integer linear programming problems, solved in a stepwise iterative manner. This approach significantly improves computational efficiency and meets the requirements of practical large-scale power grid applications. To validate the proposed model and algorithm, a simulation program is developed using C++, and a simulation analysis of a regional transmission network is conducted. The program ensures the correctness of the proposed model and demonstrates the effectiveness of the algorithm.

Retracted: A Hybrid Algorithm of Ant Colony and Benders Decomposition for Large-Scale Mixed Integer Linear Programming.

[This retracts the article DOI: 10.1155/2022/1634995.].

A Hybrid Algorithm of Ant Colony and Benders Decomposition for Large-Scale Mixed-Integer Linear Programming

We propose a hybrid approach for solving problems of mixed-integer linear programming with two types of variables (MILPTTV). The hybrid approach includes three algorithms: (1) Benders decomposition; (2) the ant colony algorithm; (3) the feasible heuristic method. We decompose the programming problem into a master problem and a subproblem using Benders decomposition. The ant colony algorithm generates an initial solution for the master problem. A feasible heuristic is used to obtain feasible solutions as input to the subproblem, whereas the subproblem is solved to optimality by using a linear programming solver that is named Linprog function in MATLAB R2016a. Over successive iterations, the master problem is refined by adding cuts from the subproblem and the master problem is solved by a linear programming solver Cplexmilp function. We compare the performance of the proposed hybrid approach against a general Benders decomposition approach as well as against a mixed-integer programming solver CPLEX (version 12.6) that is invoked by MATLAB on MIPLIB test problem mas76, mas74 and some large-scale numerical examples.

Modeling and optimization of carbon-negative NGCC plant enabled by modular direct air capture

Natural gas combined cycle (NGCC) plants are the most prevalent source of electricity in the United States and are expected to continue playing a key role in the future energy market. However, they are also one of the main sources of CO2 emissions in the power market. Recently, the authors proposed a retrofit design for an existing NGCC to allow power generation with negative CO2 emissions by utilizing both post-combustion carbon capture (PCC) and direct air capture (DAC). The DAC unit captures CO2 directly from the atmosphere using low-grade heat from the NGCC via a novel heat integration scheme. The heat integration scheme and modular DAC design introduce significant flexibility, allowing the system to focus on either power generation or CO2 capture in response to electricity price changes. However, this creates a complex trade-off between the cost of DAC capacity and the enhanced operational flexibility it provides. In this work, we model the proposed retrofit systematically and co-optimize the design and operation of the system for an entire year under different electricity price signals and CO2 prices. The DAC operations are modeled with a novel formulation to account for the sorbent dynamics in its temperature swing cycles. The solutions for the resulting large-scale mixed-integer linear program show that a large DAC unit is preferred in most situations. The extra flexibility leads to longer dispatch time and higher profits compared with the base NGCC.

Ranking constraint relaxations for mixed integer programs using a machine learning approach

Solving large-scale Mixed Integer Linear Programs (MIP) can be difficult without advanced algorithms such as decomposition based techniques. Even if a decomposition technique might be appropriate, there are still many possible decompositions for any large MIP and it may not be obvious which will be the most effective. The quality of a decomposition depends on both the tightness of the dual bound, in our case generated via Lagrangian Relaxation, and the computational time required to produce that bound. Both of these factors are difficult to predict, motivating the use of a Machine Learning function to predict decomposition quality based a score that combines both bound quality and computational time. This paper presents a comprehensive analysis of the predictive capabilities of a ML function for predicting the quality of MIP decompositions created via constraint relaxation. In this analysis, the role of instance similarity and ML prediction quality is explored, as well as the benchmarking of a ML ranking function against existing heuristic functions. For this analysis, a new dataset consisting of over 40000 unique decompositions sampled from across 24 instances from the MIPLIB2017 library has been established. These decompostions have been created by both a greedy relaxation algorithm as well as a population based multi-objective algorithm, which has previously been shown to produce high quality decompositions. In this paper, we demonstrate that a ML ranking function is able to provide state-of-the-art predictions when benchmarked against existing heuristic ranking functions. Additionally, we demonstrate that by only considering a small set of features related to the relaxed constraints in each decomposition, a ML ranking function is still able to be competitive with heuristic techniques. Such a finding is promising for future constraint relaxation approaches, as these features can be used to guide decomposition creation. Finally, we highlight where a ML ranking function would be beneficial in a decomposition creation framework.

Learning-Assisted Variables Reduction Method for Large-Scale MILP Unit Commitment

The security-constrained unit commitment (SCUC) challenge is solved repeatedly several times every day, for operations in a limited time. Typical mixed-integer linear programming (MILP) formulations are intertemporal in nature and have complex and discrete solution spaces that exponentially increase with system size. Improvements in the SCUC formulation and/or solution method that yield a faster solution hold immense economic value, as less time can be spent finding the best-known solution. Most machine learning (ML) methods in the literature either provide a warm start or convert the MILP-SCUC formulation to a continuous formulation, possibly leading to sub-optimality and/or infeasibility. In this paper, we propose a novel ML-based variables reduction method that accurately determines the optimal schedule for a subset of trusted generators, shrinking the MILP-SCUC formulation and dramatically reducing the search space. ML indicators sets are created to shrink the MILP-SCUC model, leading to improvement in the solution quality. Test results on IEEE systems with 14, 118, and 300 busses, the Ontario system, and Polish systems with 2383 and 3012 busses report significant reductions in solution times in the range of 48% to 98%. This is a promising tool for system operators to solve the MILP-SCUC with a lower optimality gap in a limited-time operation, leading to economic benefits.

Accelerated dual dynamic integer programming applied to short-term power generation scheduling

The short-term generation scheduling (STGS) problem defines which units must operate and how much power they must deliver to satisfy the system demand over a planning horizon of up to two weeks. The problem is typically formulated as a large-scale mixed-integer linear programming problem, where off-the-shelf commercial solvers generally struggle to efficiently solve realistic instances of the STGS, mainly due to the large-scale of these models. Thus, decomposition approaches that break the model into smaller instances that are more easily handled are attractive alternatives to directly employing these solvers. This paper proposes a dual dynamic integer programming (DDiP) framework for solving the STGS problem efficiently. As in the standard DDiP approach, we use a nested Benders decomposition over the time horizon but introduce multiperiod stages and overlap strategies to accelerate the method. Simulations performed on the IEEE-118 system show that the proposed approach is significantly faster than standard DDiP and can deliver near-optimal solutions.

A Hybrid Algorithm of Ant Colony and Benders Decomposition for Large-Scale Mixed Integer Linear Programming.

The mixed integer linear programming (MILP) has been widely applied in many fields such as supply chain management and robot control, while how to develop a more efficient algorithm to solve large-scale MILP is still in discussion. This study addresses a hybrid algorithm of the ant colony and Benders decomposition to improve the efficiency. We firstly introduce the design of our algorithm, in which the Benders algorithm decomposes the MILP into a master problem and a slack problem, the ant colony algorithm generates initial solutions for the master problem, and heuristic rules obtain feasible solutions for the slack problem. Then, the computational experiments are carried out to verify efficiency, with a benchmark test and some medium-large scale examples. Compared with other algorithms like CPLEX, GUROBI, and traditional ACA, our algorithm shows a better performance with a 0.3%–4.0% optimality gap, as well as a significant decrease of 54.3% and 33.6% on average in the CPU time and iterations, respectively. Our contribution is to provide a low-workload, time-saving, and high-accuracy hybrid algorithm to solve MILP problems with a large amount of variables, which can be widely used in more commercial solvers and promote the utilization of the artificial intelligence.

Integrated Parametric Graph Closure and Branch-and-Cut Algorithm for Open Pit Mine Scheduling under Uncertainty

Open pit mine production scheduling is a computationally expensive large-scale mixed-integer linear programming problem. This research develops a computationally efficient algorithm to solve open pit production scheduling problems under uncertain geological parameters. The proposed solution approach for production scheduling is a two-stage process. The stochastic production scheduling problem is iteratively solved in the first stage after relaxing resource constraints using a parametric graph closure algorithm. Finally, the branch-and-cut algorithm is applied to respect the resource constraints, which might be violated during the first stage of the algorithm. Six small-scale production scheduling problems from iron and copper mines were used to validate the proposed stochastic production scheduling model. The results demonstrated that the proposed method could significantly improve the computational time with a reasonable optimality gap (the maximum gap is 4%). In addition, the proposed stochastic method is tested using industrial-scale copper data and compared with its deterministic model. The results show that the net present value for the stochastic model improved by 6% compared to the deterministic model.

Hyper-Graph Partitioning for a Multi-Agent Reformulation of Large-Scale MILPs

This letter addresses the challenge of solving large-scale Mixed Integer Linear Programs (MILPs). A resolution scheme is proposed for the class of MILPs with a hidden constraint-coupled multi-agent structure. In particular, we focus on the problem of disclosing such a structure to then apply a computationally efficient decentralized optimization algorithm recently proposed in the literature. The multi-agent reformulation problem consists in manipulating the matrix defining the linear constraints of the MILP so as to put it in a singly-bordered block-angular form, where the blocks define local constraints and decision variables of the agents, whereas the border defines the coupling constraints. We translate the matrix reformulation problem into a hyper-graph partitioning problem and introduce a novel algorithm which accounts for the specific requirements on the singly-bordered block-angular form to best take advantage of the decentralized optimization approach. Numerical results show the effectiveness of the proposed hyper-graph partitioning algorithm.

Optimal Balancing of Wind Parks with Virtual Power Plants

In this paper, we explore the optimization of virtual power plants (VPP), consisting of a portfolio of biogas power plants and a battery whose goal is to balance a wind park while maximizing their revenues. We operate under price and wind production uncertainty and in order to handle it, methods of machine learning are employed. For price modeling, we take into account the latest trends in the field and the most up-to-date events affecting the day-ahead and intra-day prices. The performance of our price models is demonstrated by both statistical methods and improvements in the profits of the virtual power plant. Optimization methods will take price and imbalance forecasts as input and conduct parallelization, decomposition, and splitting methods in order to handle sufficiently large numbers of assets in a VPP. The main focus is on the speed of computing optimal solutions of large-scale mixed-integer linear programming problems, and the best speed-up is in two orders of magnitude enabled by our method which we called Gradual Increase.

Multi-objective optimization of a power-to-hydrogen system for mobility via two-stage stochastic programming

A systematic way for the optimal design of renewable-based hydrogen refuelling stations in the presence of uncertainty in the hydrogen demand is presented. A two-stage stochastic programming approach is used to simultaneously minimize the total annual cost and the CO2 footprint due to the electricity generation sources. The first-stage (design) variables correspond to the sizing of the devices, while the second-stage (operation) variables correspond to the scheduling of the installed system that is affected by uncertainties. The demand of a fleet of fuel cell vehicles is synthesized by means of a Poisson distribution and different scenarios are generated by random sampling. We formulate our problem as a large-scale mixed-integer linear program and we rely on a two-level approximation scheme to keep the problem computationally tractable. A solely deterministic setting which does not take into account uncertainties leads to underestimated device sizes, resulting in a significant fraction of demand remaining unserved with a consequent loss in revenue. The multi-objective optimization produces a convex Pareto front, showing that a reduction in carbon footprint comes with increasing costs and thus diminishing profit.

A deep learning-based robust optimization approach for refinery planning under uncertainty

Refinery planning under uncertainty has gained tremendous attention, and this paper bridges deep learning and robust optimization to address this issue. First, we propose a large-scale mixed-integer linear programming model for refinery planning, where the fixed-yield models of the processing units are used. Prices of final products are considered uncertain parameters in the developed model to enhance the solution's applicability. Second, historical data of different products are collected to construct the uncertainty set characterizing all possible realizations of uncertainty. Third, a deep learning method is employed to capture the uncertainties of product prices, which has been proven to be powerful for high-dimensional price data. Based on the constructed uncertainty set, a data-driven robust optimization model is further developed. Finally, an iterative constraint generation algorithm is applied to solve the data-driven robust optimization problem. Case studies from an actual refinery are presented to showcase the effectiveness of the proposed method, which owes particularly to the representation capability of deep learning.

Risk-averse optimization for resilience enhancement of complex engineering systems under uncertainties

With the growth of complexity and extent, large scale interconnected network systems, e.g., transportation networks or infrastructure networks, become more vulnerable to external disturbances. Hence, managing potential disruptive events during the design, operating, and recovery phase of an engineered system and therefore improving the system’s resilience is an important yet challenging task. To ensure system resilience after the occurrence of failure events, this study proposes a mixed-integer linear programming (MILP) based restoration framework using heterogeneous dispatchable agents. The scenario-based stochastic optimization (SO) technique is adopted to deal with the inherent uncertainties imposed on the recovery process from nature. Moreover, different from conventional SO using deterministic equivalent formulations, the CVaR risk measure is implemented for this study because of the temporal sparsity of the decision making in applications such as the recovery from extreme events. The resulting restoration framework involves a large-scale MILP problem and thus an adequate decomposition technique i.e. modified Lagrangian dual decomposition, is also employed to achieve tractable computational complexity. Case study results based on the IEEE 37-bus test feeder demonstrate the benefits of using the proposed framework for resilience improvement as well as the advantages of adopting SO formulations.

Distributed Primal Decomposition for Large-Scale MILPs

This article deals with a distributed Mixed-Integer Linear Programming (MILP) setup arising in several control applications. Agents of a network aim to minimize the sum of local linear cost functions subject to both individual constraints and a linear coupling constraint involving all the decision variables. A key, challenging feature of the considered setup is that some components of the decision variables must assume integer values. The addressed MILPs are NP-hard, nonconvex, and large-scale. Moreover, several additional challenges arise in a distributed framework due to the coupling constraint, so that feasible solutions with guaranteed suboptimality bounds are of interest. We propose a fully distributed algorithm based on a primal decomposition approach and an appropriate tightening of the coupling constraint. The algorithm is guaranteed to provide feasible solutions in finite time. Moreover, asymptotic and finite-time suboptimality bounds are established for the computed solution. Monte Carlo simulations highlight the extremely low suboptimality bounds achieved by the algorithm.

Co-optimization planning of integrated electricity and district heating systems based on improved quadratic convex relaxation

The rapid growth of combined heat and power (CHP) units has led to the development of integrated electricity and district heating systems (IEHS). To support the design of a highly efficient energy supply system, this paper proposes a long-term co-optimization planning model for an IEHS. Not only CHP units, non-CHP thermal generators, wind farms and electric boilers but also transmission lines and heat pipelines are considered as investment candidates to meet electricity and heat demands. Nonlinear hydraulic conditions and thermal conditions are adopted to precisely capture the characteristics of the heating system. To make the planning model tractable, the nonlinear hydraulic conditions are approximated through piecewise linearization. Based on the introduction of auxiliary variables, the nonconvex thermal conditions are reformulated into linear constraints through quadratic convex relaxation. Hence, the planning model is converted into a large-scale mixed integer linear programming (MILP) problem. Since the planning model is formulated based on independent load blocks, a parallel Benders decomposition algorithm combined with the sequential bound-tightening procedure is proposed to efficiently obtain high-quality solutions. Numerical cases are studied based on two IEHSs of different scales to validate the effectiveness of the proposed co-optimization planning model and the feasibility of the proposed solution methods for solving this complicated planning model for an IEHS.

Optimum Scheduling of Energy Production Using Hybrid Decomposition Methods - A Comparative Study.(Dept.M)

The production of energy exists in different shapes and forms, e.g. thermal, mechanical, electrical or other forms of energy. Each of these forms exists in different types and arts. The production and scheduling of energy types are decision processes which are basically concerned with the adoption or more precisely the allocation of the natural and industrial resources in order to best satisfy marketing and customer requircinents at minimum best economic conditions. There are many reasons that make energy production planning a challenging problem. The variable structure of both the demand and costs, the difficulty to precisely forecast the demand at the very detailed level, the less flexibility to modify the operating conditions, and that there are usually long and uncertain delays in obtaining the industrial resources (new machines, workers training, subcontracting capacities, raw materials item components, etc.) Researchers and seientists have paid a great part of attention for developing scheduling and planning systems to support such decision making processes. Two different approaches for production planning and scheduling are known. The first, called the monolithic approach, formulates the problems as a large scale mixed-integer linear programming problems and is usually solved approximately using Lagrangean relaxation to the mixed integer linear program. The second approach, called the hierarchical approach, partitions the planning and scheduling problem into a hierarchy of smaller subproblems. The upper hierarchy deals with strategic decisions for the planning horizon, while the lower hierarchy deals with the more detailed short-termed scheduling. The computational effort required for the monolithic approach is generally greater than that required for the hierarchical approach. The hierarchical approach may require less detailed demand input data. Also the uncertainty is better treated by the hierarchical approach. In this study, a comparison is presented between two different hybrid procedures for solving in hierarchical energy production planning and scheduling problem. A simplification in one of them is made. In the first procedure, hierarchical subproblems are included in an overall mixed integer linear p rogramming formulation. Accordingly, the problem is partitioned into two hierarchics; families in the lower hierarchy, which, in turn, are aggregated into types in the upper hierarchy. In this procedure, the set of inventory constraints are priced out using Lagrangeau multipliers in the main objective function, which is then partitioned into two subproblems. The first subproblem is an optimal control problem on the family level (lower hierarchy) while the other subproblem is a linear program on the type level (upper hierarchy). The two subproblems are linked together by an inventory aggregation constraint. By this approach, a feedback is included automatically in the solution procedure, but it does not pass directly from the lower level to the upper level since both subproblems are solved in parallel. In the second procedure, instead of using Lagrangean relaxation with respect to a group of constraints in the original model, the model is first partitioned, ill which cuts are priced out by means of a set of Lagrangean multipliers into the subproblem generated on the family level. Accordingly, one subproblem is trivial and the other subproblem is an incapacitated lot-sizing oplinal control problem. The size of this subproblem is exactly the same as that of the equivalent subproblem in the previous procedure. In both procedures, the optimal control subproblem is solved using dynamic programming and the set of Lagrangean multipliers are updated using subgradient optimization algorithm. Results show that both algorithms are quite acceptable and efficient. A comparison between the results of both algorithms shows considerable improvement compared to the result found in the literature. The enhancement of these results may be returned to the fact that step sizes are more suitably chosen and to the simplification done in the solution procedure.