Nested Benders Decomposition Research Articles

Joint optimization of UPF placement and traffic routing for 5G core network user plane

This paper investigates the dynamic deployment of the 5G core network user plane functions (UPFs) in the edge network under time-varying traffic conditions. The objective is to minimize the total cost, including energy consumption, user plane latency, and UPF deployment costs by jointly optimizing UPF placement and traffic routing over the entire planning horizon. We formulate the optimization problem as a mixed-integer linear programming (MILP) problem, which is proven to be NP-hard. To obtain the optimal solution for the problem, we propose a Benders decomposition-based algorithm. However, the high computational complexity of this algorithm limits its practical applicability for large-scale networks. Therefore, we propose a low-complexity algorithm based on nested Benders decomposition that can solve the problem suboptimally. The simulation results demonstrate the convergence and computational efficiency of the proposed algorithm and show that it outperforms two existing algorithms. Moreover, we demonstrate that cost-saving can be achieved by using more power-efficient servers or deploying more servers. Additionally, we show that adjusting the cost parameters can achieve a good tradeoff between energy consumption and user plane latency.

Three-Stage Stochastic Unit Commitment for Microgrids Toward Frequency Security via Renewable Energy Deloading

Renewable energy is boosting the deployment of microgrids (MGs) with stochastic and low inertia nature. To improve the operational efficiency of MGs while guaranteeing frequency security, a three-stage stochastic unit commitment problem is proposed, where renewable energy can be deloaded. In the first stage, the diesel generators (DGs) are scheduled, responding to uncertainties of loads and photovoltaic generator output. In the second stage, the outputs of DGs are optimized to reduce the operational cost under uncertain disturbances. In the third stage, a novel PV deloading strategy is proposed to test the frequency security of MGs. The three-stage optimization problem is formulated as a multi-stage stochastic optimization problem with recourse. This problem is then solved using a novel nested Benders decomposition algorithm with both dual cuts and partial primal cuts. Simulations are performed on an AC MG under different PV penetration levels and disturbances. The results verify the effectiveness of the proposed model in balancing operational efficiency, renewable energy utilization, and frequency security.

A nested Benders decomposition-based algorithm to solve the three-stage stochastic optimisation problem modeling population-based breast cancer screening

Population-based cancer screening programmes invite high-risk population groups to screenings in order to increase the probability of an early diagnosis. In this paper, we study the organisation of preventive breast cancer screening, accounting for both scheduling of patients and planning of resources. Mammography screening comprises a two-stage healthcare process, encompassing a patient scan in a mammography unit and a scan examination by a central coordination center. Objectives are to minimise patient flow times and maximise resource efficiency and number of patients treated. Performance of population-based screening programmes is hampered due to large rates of patient no-shows, which is partially remedied by giving patients the option to cancel or reschedule their appointment. We model the scheduling problem as a three-stage stochastic optimisation problem and propose a diving heuristic relying on Sample Average Approximation and nested Benders decomposition to find high-quality integer solutions. Computational experimentation is performed on real-life instances to benchmark the proposed method to alternative methodologies. Results demonstrate that the proposed heuristic yields a stable performance for instances of different sizes. However, integrating the three decision stages increases significantly the complexity, such that larger-sized instances are preferably solved via two separate solution stages to improve resource efficiency. In addition, insights are provided in the value of stochastic optimisation and strategies to cancel or reschedule appointments mitigating the impact of no-show uncertainty.

Robust Scheduling with Temporal Decomposition of Integrated Electrical-Heating System Based on Dynamic Programming Formulation

To realize the potential of the integrated electrical-heating systems (IEHS) for coping with the uncertain renewable energy, a reserve scheduling model based on the dynamic programming transformed multi-stage adaptive robust optimization (DMRO) is proposed in this paper. The DMRO model enforces the non-anticipativity of scheduling process via temporal decomposed framework, and enhances the formulation flexibility for time-coupled equipment, such as energy storage systems (ESS). The whole scheduling model is split into two problems under the decomposition constructure, which consists of a reserve interval generation (RG) problem for making the unit commitment and systems' reserve plan, and a multi-stage economic operation (EO) problem aiming with simulating the worst-case scheduling process. To obtain the global optimal, a nested Benders decomposition algorithm is employed to solve the DMRO model. Case studies on the 6-Bus-5-node IEHS and the 118-Bus-24-node IEHS demonstrate the effectiveness of the proposed model and the solution methodology.

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.

Robust Stochastic Facility Location: Sensitivity Analysis and Exact Solution

This work focuses on a broad class of facility location problems in the context of adaptive robust stochastic optimization under the state-dependent demand uncertainty. The demand is assumed to be significantly affected by related state information, such as the seasonal or socio-economic information. In particular, a state-wise ambiguity set is adopted for modeling the distributional uncertainty associated with the demand in different states. The conditional distributional characteristics in each state are described by a support, as well as by mean and dispersion measures, which are assumed to be conic representable. A robust sensitivity analysis is performed, in which, on the one hand, we analyze the impact of the change in ambiguity-set parameters (e.g., state probabilities, mean value abounds, and dispersion bounds in different states) onto the optimal worst-case expected total cost using the ambiguity dual variables. On the other hand, we analyze the impact of the change in location design onto the worst-case expected second-stage cost and show that the sensitivity bounds are fully described as the worst-case expected shadow-capacity cost. As for the solution approach, we propose a nested Benders decomposition algorithm for solving the model exactly, which leverages the subgradients of the worst-case expected second-stage cost at the location decisions formed insightfully by the associated worst-case distributions. The nested Benders decomposition approach ensures a finite-step convergence, which can also be regarded as an extension of the classic L-shaped algorithm for two-stage stochastic programming to our state-wise, robust stochastic facility location problem with conic representable ambiguity. Finally, the results of a series of numerical experiments are presented that justify the value of the state-wise distributional information incorporated in our robust stochastic facility location model, the robustness of the model, and the performance of the exact solution approach.

Non-convex nested Benders decomposition

We propose a new decomposition method to solve multistage non-convex mixed-integer (stochastic) nonlinear programming problems (MINLPs). We call this algorithm non-convex nested Benders decomposition (NC-NBD). NC-NBD is based on solving dynamically improved mixed-integer linear outer approximations of the MINLP, obtained by piecewise linear relaxations of nonlinear functions. Those MILPs are solved to global optimality using an enhancement of nested Benders decomposition, in which regularization, dynamically refined binary approximations of the state variables and Lagrangian cut techniques are combined to generate Lipschitz continuous non-convex approximations of the value functions. Those approximations are then used to decide whether the approximating MILP has to be dynamically refined and in order to compute feasible solutions for the original MINLP. We prove that NC-NBD converges to an varepsilon -optimal solution in a finite number of steps. We provide promising computational results for some unit commitment problems of moderate size.

Long-Term Expansion Planning of the Transmission Network in India under Multi-Dimensional Uncertainty

Considerable investment in India’s electricity system may be required in the coming decades in order to help accommodate the expected increase of renewables capacity as part of the country’s commitment to decarbonize its energy sector. In addition, electricity demand is geared to significantly increase due to the ongoing electrification of the transport sector, the growing population, and the improving economy. However, the multi-dimensional uncertainty surrounding these aspects gives rise to the prospect of stranded investments and underutilized network assets, rendering investment decision making challenging for network planners. In this work, a stochastic optimization model is applied to the transmission network in India to identify the optimal expansion strategy in the period from 2020 until 2060, considering conventional network reinforcements as well as energy storage investments. An advanced Nested Benders decomposition algorithm was used to overcome the complexity of the multistage stochastic optimization problem. The model additionally considers the uncertainty around the future investment cost of energy storage. The case study shows that deployment of energy storage is expected on a wide scale across India as it provides a range of benefits, including strategic investment flexibility and increased output from renewables, thereby reducing total expected system costs; this economic benefit of planning with energy storage under uncertainty is quantified as Option Value and is found to be in excess of GBP 12.9 bn. The key message of this work is that under potential high integration of wind and solar in India, there is significant economic benefit associated with the wide-scale deployment of storage in the system.

Mixed-integer linear programming models and algorithms for generation and transmission expansion planning of power systems

With the increasing penetration of renewable generating units, especially in remote areas not well connected with load demand, there are growing interests to co-optimize generation and transmission expansion planning (GTEP) in power systems. Due to the volatility in renewable generation, a planner needs to include the operating decisions into the planning model to guarantee feasibility. However, solving the GTEP problem with hourly operating decisions throughout the planning horizon is computationally intractable. Therefore, we propose several spatial and temporal simplifications to the problem. Built on the generation expansion planning (GEP) formulation of Lara et al. (2018), we propose a mixed-integer linear programming formulation for the GTEP problem. Three different formulations, i.e., a big-M formulation, a hull formulation, and an alternative big-M formulation, are reported for transmission expansion. We theoretically compare the tightness of the LP relaxations of the three formulations. The proposed MILP GTEP model typically involves millions or tens of millions of variables, which makes the model not directly solvable by the commercial solvers. To address this computational challenge, we propose a nested Benders decomposition algorithm and a tailored Benders decomposition algorithm that exploit the structure of the GTEP problem. Using a case study from Electric Reliability Council of Texas (ERCOT), we are able to show that the proposed tailored Benders decomposition outperforms the nested Benders decomposition. The coordination in the optimal generation and transmission expansion decisions from the ERCOT study implies that there is an additional value in solving GEP and TEP simultaneously.

A time-consistent Benders decomposition method for multistage distributionally robust stochastic optimization with a scenario tree structure

A computational method is developed for solving time consistent distributionally robust multistage stochastic linear programs with discrete distribution. The stochastic structure of the uncertain parameters is described by a scenario tree. At each node of this tree, an ambiguity set is defined by conditional moment constraints to guarantee time consistency. This method employs the idea of nested Benders decomposition that incorporates forward and backward steps. The backward steps solve some conic programming problems to approximate the cost-to-go function at each node, while the forward steps are used to generate additional trial points. A new framework of convergence analysis is developed to establish the global convergence of the approximation procedure, which does not depend on the assumption of polyhedral structure of the original problem. Numerical results of a practical inventory model are reported to demonstrate the effectiveness of the proposed method.

Deterministic electric power infrastructure planning: Mixed-integer programming model and nested decomposition algorithm

This paper addresses the long-term planning of electric power infrastructures considering high renewable penetration. To capture the intermittency of these sources, we propose a deterministic multi-scale Mixed-Integer Linear Programming (MILP) formulation that simultaneously considers annual generation investment decisions and hourly operational decisions. We adopt judicious approximations and aggregations to improve its tractability. Moreover, to overcome the computational challenges of treating hourly operational decisions within a monolithic multi-year planning horizon, we propose a decomposition algorithm based on Nested Benders Decomposition for multi-period MILP problems to allow the solution of larger instances. Our decomposition adapts previous nested Benders methods by handling integer and continuous state variables, although at the expense of losing its finite convergence property due to potential duality gap. We apply the proposed modeling framework to a case study in the Electric Reliability Council of Texas (ERCOT) region, and demonstrate massive computational savings from our decomposition.

A New Nested Benders Decomposition Strategy for Parallel Processing Applied to the Hydrothermal Scheduling Problem

Several optimization strategies are suitable for parallel processing, such as mathematical-programming based decomposition strategies, like Lagrangian relaxation and two-stage Benders decomposition, and evolutionary programming-like algorithms. In such cases, most subproblems can be solved independently by different processors. However, parallel processing is not suitable for some problems solved by nested Benders decomposition, due to the hierarchy between subproblems related to different time steps. In particular, for the deterministic case, parallel programming has been restricted to thread processing in lower-level machine computation and optimization solver codes. This paper proposes a new nested Benders decomposition strategy that is suitable for parallel processing, where time-coupled stages are solved simultaneously and an alternative procedure is employed to share initial conditions for the next stages as well as Benders cuts for the previous stages. The methodology is applied to the deterministic short term hydrothermal scheduling problem for the real large-scale Brazilian system, where the advantages of the proposed approach become more evident. In order to show the applicability of the method in low-budget and small parallel environments, the case study was processed with up to 24 process on multi-core computers.

Dynamic convexification within nested Benders decomposition using Lagrangian relaxation: An application to the strategic bidding problem

Many decomposition algorithms like Benders decomposition and stochastic dual dynamic programming are limited to convex optimization problems. In this paper, we utilize a dynamic convexification method that makes use of Lagrangian relaxation to overcome this limitation and enables the modeling of non-convex multi-stage problems using decomposition algorithms. Though the algorithm is confined by the duality gap of the problem being studied, the computed upper bounds (for maximization problems) are at least as good as those found via a linear programming relaxation approach. We apply the method to the strategic bidding problem for a hydroelectric producer, in which we ask: What is the revenue-maximizing production schedule for a single price-maker hydroelectric producer in a deregulated, bid-based market? Because the price-maker’s future revenue function has a sawtooth shape, we model it using mixed-integer linear programming. To remedy the non-concavity issues associated with modeling the future revenue function as a mixed-integer linear program, we model the price-maker’s bidding decision utilizing both Benders decomposition and Lagrangian relaxation. We demonstrate the utility of our algorithm through an illustrative example and through three case studies in which we model electricity markets in El Salvador, Honduras, and Nicaragua.

Hierarchical Benders Decomposition for Open-Pit Mine Block Sequencing

The open-pit mine block sequencing problem (OPBS) models a deposit of ore and surrounding material near the Earth’s surface as a three-dimensional grid of blocks. A solution in discretized time identifies a profit-maximizing extraction (mining) schedule for the blocks. Our model variant, a mixed-integer program (MIP), presumes a predetermined destination for each extracted block, namely, processing plant or waste dump. The MIP incorporates standard constructs but also adds not-so-standard lower bounds on resource consumption in each time period and allows fractional block extraction in a novel fashion while still enforcing pit-wall slope restrictions. A new extension of nested Benders decomposition, “hierarchical” Benders decomposition (HBD), solves the MIP’s linear-programming relaxation. HBD exploits time-aggregated variables and can recursively decompose a model into a master problem and two subproblems rather than the usual single subproblem. A specialized branch-and-bound heuristic then produces high-quality, mixed-integer solutions. Medium-sized problems (e.g., 25,000 blocks and 20 time periods) solve to near optimality in minutes. To the best of our knowledge, these computational results are the best known for instances of OPBS that enforce lower bounds on resource consumption.

Solving multistage quantified linear optimization problems with the alpha–beta nested Benders decomposition

Quantified linear programs (QLPs) are linear programs with variables being either existentially or universally quantified. QLPs are convex multistage decision problems on the one side, and two-person zero-sum games between an existential and a universal player on the other side. Solutions of feasible QLPs are so-called winning strategies for the existential player that specify how to react on moves—fixations of universally quantified variables—of the universal player to certainly win the game. To find a certain best one among different winning strategies, we propose the extension of the QLP decision problem by an objective function. To solve the resulting QLP optimization problem, we exploit the problem’s hybrid nature and combine linear programming techniques with solution techniques from game-tree search. As a result, we present an extension of the nested Benders decomposition algorithm by the $$\alpha \beta $$ -heuristic and move-ordering, two techniques that are successfully used in game-tree search to solve minimax trees. We furthermore exploit solution information from QLP relaxations obtained by quantifier shifting. The applicability is examined in an experimental evaluation.

Combining sampling-based and scenario-based nested Benders decomposition methods: application to stochastic dual dynamic programming

Nested Benders decomposition is a widely used and accepted solution methodology for multi-stage stochastic linear programming problems. Motivated by large-scale applications in the context of hydro-thermal scheduling, in 1991, Pereira and Pinto introduced a sampling-based variant of the Benders decomposition method, known as stochastic dual dynamic programming (SDDP). In this paper, we embed the SDDP algorithm into the scenario tree framework, essentially combining the nested Benders decomposition method on trees with the sampling procedure of SDDP. This allows for the incorporation of different types of uncertainties in multi-stage stochastic optimization while still maintaining an efficient solution algorithm. We provide an illustration of the applicability of our method towards a least-cost hydro-thermal scheduling problem by examining an illustrative example combining both fuel cost with inflow uncertainty and by studying the Panama power system incorporating both electricity demand and inflow uncertainties.

A Computational Strategy to Solve Preventive Risk-Based Security-Constrained OPF

The benefit of risk-based (RB) security-constrained optimal power flow (SCOPF) model lies in its ability to improve the economic performance of a power system while enhancing the system's overall security level. However, the RB-SCOPF model is difficult to solve due to the following two characteristics: 1) the overload severity of a circuit changes with the loading condition on it, thus is hard to express with a deterministic function, and 2) the risk index is a function of the state variables in both normal and contingency states, which greatly increases the scale of optimization. To handle the first issue, a new expression of severity function is proposed so that it is possible to decompose the model into a SCOPF subproblem and a risk subproblem. To deal with the second issue, a nested Benders decomposition with multi-layer linear programming method is proposed. Illustrations use the ISO New England bulk system is provided to demonstrate the feasibility of the proposed method. Analysis is presented to demonstrate the merits of the RB-SCOPF over the traditional SCOPF model.

Production planning under uncertainty: the case of hydrothermal systems

This paper formulates the hydrothermal production planning problem as a multi-stage stochastic programming model. This problem is usually solved using algorithms that are based on dynamic programming and nested Benders decomposition (NBD). The hydrothermal planning problem is remodelled as a capacitated production planning model with convex costs; then a vanishing effect property of the first period decision is derived. Based on this property, a special implementation scheme of the NBD algorithm is proposed, providing faster convergence and less computational effort. The NBD algorithm is implemented for a large-scale hydrothermal power system of the Pacific Northwest in the USA with 20 thermal plants and 21 thermal plants.

A nested benders decomposition approach for telecommunication network planning

AbstractDespite its ability to result in more effective network plans, the telecommunication network planning problem with signal‐to‐interference ratio constraints gained less attention than the power‐based one because of its complexity. In this article, we provide an exact solution method for this class of problems that combines combinatorial Benders decomposition, classical Benders decomposition, and valid cuts in a nested way. Combinatorial Benders decomposition is first applied, leading to a binary master problem and a mixed integer subproblem. The subproblem is then decomposed using classical Benders decomposition. The algorithm is enhanced using valid cuts that are generated at the classical Benders subproblem and are added to the combinatorial Benders master problem. The valid cuts proved efficient in reducing the number of times the combinatorial Benders master problem is solved and in reducing the overall computational time. More than 120 instances of the W‐CDMA network planning problem ranging from 20 demand points and 10 base stations to 140 demand points and 30 base stations are solved to optimality. © 2010 Wiley Periodicals, Inc. Naval Research Logistics, 2010

Medium term scheduling of a hydro-thermal system using stochastic model predictive control

A multistage stochastic programming formulation is presented for monthly production planning of a hydro-thermal system. Stochasticity from variations in water reservoir inflows and fluctuations in demand of electric energy are considered explicitly. The problem can be solved efficiently via Nested Benders Decomposition. The solution is implemented in a model predictive control setup and performance of this control technique is demonstrated in simulations. Tuning parameters, such as prediction horizon and shape of the stochastic programming tree are identified and their effects are analyzed.