Lower-level Subproblem Research Articles

Optimal location and operation of energy storage and transmission switching for minimizing wind power spillage

This paper proposes a bi-level multi-objective optimization model to improve the integration of wind power generators in electrical networks based on the optimal location and operation of the energy storage system and transmission switching strategies. The minimization of wind power spillage, load shedding, power losses, and the improvement of the voltage stability margin are the objectives of the upper-level subproblem. While the lower-level subproblem aims to maximize social welfare under different scenarios and overall system constraints. Under the uncertainty of the wind power and load demand, a collection of lower-level subproblems that represent the market clearing conditions is used to constrain the upper-level subproblem. The effectiveness of the proposed algorithm is examined on a modified IEEE 24-bus test system. The results show that the applicability of the proposed algorithm in aiding power system improvement planning for minimizing wind power spillage to integrate wind energy with maximizing the social welfare and improving the loadability and the voltage stability.

A Bilevel EV Charging Station and DC Fast Charger Planning Model for Highway Network Considering Dynamic Traffic Demand and User Equilibrium

The rapid electric vehicle (EV) adoption and relatively short EV driving range urge city planners to expand charging infrastructure (i.e., EV charging stations (EVCSs) and direct current fast chargers (DCFCs)) in the highway network for supporting long-distance intercity EV travels. To properly address distinct roles as well as interactions of the EVCS planner, traffic network, power distribution networks (PDNs), and drivers, this paper explores a bilevel planning model in which the upper level determines EVCS and DCFC construction plans and the two lower-level subproblems describe the user equilibriumbased traffic assignment model of the traffic network and the operation model of PDNs. Moreover, the statistical charging time is modeled in the traffic network subproblem as a function of statistical charging demand and the number of DCFCs, leveraging the macro traffic pattern and micro charging behavior of individual drivers with balanced accuracy and computational complexity. The proposed model also captures the impacts of social factors, charging infrastructure, and driver behaviors on dynamic EV traffic demands. Reformulation and linearization techniques are applied to convert the proposed bilevel problem into a tractable single-level mixed-integer linear programming model for effective computation. The effectiveness of the proposed approach is tested on an intercity highway network with multiple PDNs. The findings from numerical case studies highlight the important interplay among the construction plans of EVCSs and DCFCs, the PDN constraints, the user equilibrium traffic network, and the EV drivers. Case studies illustrate the feasibility and necessity of considering the number of DCFCs to calculate the macro-perspective charging time in the user equilibrium traffic network model. The dynamic traffic demand, induced by social factors and traffic demand elasticity, was also found to be a crucial factor in reshaping EV traffic. Thus, properly considering these important factors would lead to a more accurate quantification of traffic demands and ultimately result in a more reasonable charging facility construction plan.

Value-Function-Based Sequential Minimization for Bi-Level Optimization.

Gradient-based Bi-Level Optimization (BLO) methods have been widely applied to handle modern learning tasks. However, most existing strategies are theoretically designed based on restrictive assumptions (e.g., convexity of the lower-level sub-problem), and computationally not applicable for high-dimensional tasks. Moreover, there are almost no gradient-based methods able to solve BLO in those challenging scenarios, such as BLO with functional constraints and pessimistic BLO. In this work, by reformulating BLO into approximated single-level problems, we provide a new algorithm, named Bi-level Value-Function-based Sequential Minimization (BVFSM), to address the above issues. Specifically, BVFSM constructs a series of value-function-based approximations, and thus avoids repeated calculations of recurrent gradient and Hessian inverse required by existing approaches, time-consuming especially for high-dimensional tasks. We also extend BVFSM to address BLO with additional functional constraints. More importantly, BVFSM can be used for the challenging pessimistic BLO, which has never been properly solved before. In theory, we prove the convergence of BVFSM on these types of BLO, in which the restrictive lower-level convexity assumption is discarded. To our best knowledge, this is the first gradient-based algorithm that can solve different kinds of BLO (e.g., optimistic, pessimistic, and with constraints) with solid convergence guarantees. Extensive experiments verify the theoretical investigations and demonstrate our superiority on various real-world applications.

A General Descent Aggregation Framework for Gradient-Based Bi-Level Optimization.

In recent years, a variety of gradient-based methods have been developed to solve Bi-Level Optimization (BLO) problems in machine learning and computer vision areas. However, the theoretical correctness and practical effectiveness of these existing approaches always rely on some restrictive conditions (e.g., Lower-Level Singleton, LLS), which could hardly be satisfied in real-world applications. Moreover, previous literature only proves theoretical results based on their specific iteration strategies, thus lack a general recipe to uniformly analyze the convergence behaviors of different gradient-based BLOs. In this work, we formulate BLOs from an optimistic bi-level viewpoint and establish a new gradient-based algorithmic framework, named Bi-level Descent Aggregation (BDA), to partially address the above issues. Specifically, BDA provides a modularized structure to hierarchically aggregate both the upper- and lower-level subproblems to generate our bi-level iterative dynamics. Theoretically, we establish a general convergence analysis template and derive a new proof recipe to investigate the essential theoretical properties of gradient-based BLO methods. Furthermore, this work systematically explores the convergence behavior of BDA in different optimization scenarios, i.e., considering various solution qualities (i.e., global/local/stationary solution) returned from solving approximation subproblems. Extensive experiments justify our theoretical results and demonstrate the superiority of the proposed algorithm for hyper-parameter optimization and meta-learning tasks. Source code is available at https://github.com/vis-opt-group/BDA.

A hierarchy model to use local resources by DSO and TSO in the balancing market

Due to the increasing penetration level of local generating resources at the distribution network, the transmission system operator (TSO) and distribution system operators (DSOs) need to coordinate their operations to use these resources and provide the required energy and regulation for the power system. This work proposes a hierarchy model for scheduling the local flexible resources at the distribution and transmission levels. In the first stage, the optimal bidding strategy of balance service providers (BSPs) and the distribution balancing market-clearing problem are formulated by a bi-level optimization problem, respectively. At this stage, the lower-level sub-problem is replaced by the Karush–Kuhn–Tucker (KKT) conditions, and the bi-level problem is formulated as a mathematical programming with equilibrium constraints (MPEC) problem. Results of the first stage are considered as fixed parameters for the second stage problem, which specifies the feasible bids of local resources that the TSO can dispatch. At the second stage, DSO checks the feasibility of remaining bids, which are not dispatched in the distribution balancing market, and determines the maximum regulation capability of the local resources that the TSO can use. Simulation results indicate that in the IEEE 33-bus distribution test system with the total imbalance energy of 4.672 MWh, 32.36% and 67.64% of the required power are compensated by transmission balancing market and local resources, which results in a reduction of 24.32% in balancing costs.

Confidence-Based Ant Colony Optimization for Capacitated Electric Vehicle Routing Problem With Comparison of Different Encoding Schemes

The blossoming of electric vehicles gives rise to a new vehicle routing problem (VRP) called capacitated electric VRP. Since charging is not as convenient as refueling, both the service of customers and the recharging of vehicles should be considered. In this article, we propose a confidence-based bilevel ant colony optimization (ACO) algorithm to solve the problem. It divides the whole problem into the upper level subproblem capacitated VRP and the lower level subproblem fixed routing vehicle charging problem. For the upper level subproblem, an ACO algorithm is used to generate customer service sequence. Both the direct encoding scheme and the order-first split-second encoding scheme are implemented to make a guideline of their applicable scenes. For the lower level subproblem, a new heuristic called simple enumeration is proposed to generate recharging schedules for vehicles. Between the two subproblems, a confidence-based selection method is proposed to select promising customer service sequence to conduct local search and lower level optimization. By setting adaptive confidence thresholds, the inferior service sequences that have little chance to become the iteration best are eliminated during the execution. The experiments show that the proposed algorithm has reached the state-of-the-art level and updated eight best known solutions of the benchmark.

A Bilevel Ant Colony Optimization Algorithm for Capacitated Electric Vehicle Routing Problem.

The development of electric vehicle (EV) techniques has led to a new vehicle routing problem (VRP) called the capacitated EV routing problem (CEVRP). Because of the limited number of charging stations and the limited cruising range of EVs, not only the service order of customers but also the recharging schedules of EVs should be considered. However, solving these two aspects of the problem together is very difficult. To address the above issue, we treat CEVRP as a bilevel optimization problem and propose a novel bilevel ant colony optimization algorithm in this article, which divides CEVRP into two levels of subproblem: 1) capacitated VRP and 2) fixed route vehicle charging problem. For the upper level subproblem, the electricity constraint is ignored and an order-first split-second max-min ant system algorithm is designed to generate routes that fulfill the demands of customers. For the lower level subproblem, a new effective heuristic is designed to decide the charging schedule in the generated routes to satisfy the electricity constraint. The objective values of the resultant solutions are used to update the pheromone information for the ant system algorithm in the upper level. Through good orchestration of the two components, the proposed algorithm can significantly outperform state-of-the-art algorithms on a wide range of benchmark instances.

Dealing With Link Blockage in mmWave Networks: A Combination of D2D Relaying, Multi-Beam Reflection, and Handover

In this paper, we consider adaptive user equipments (UE) link selection and user association in millimeter-wave (mmWave) networks. We formulate a joint optimization of link selection, resource allocation, and user association, aiming to maximize the sum logarithmic rate of all UEs. The formulated problem is solved by decomposing it into two levels of subproblems. The lower-level subproblem is link selection and resource allocation with a given user association, which is solved by a three-stage process. In the first stage, we establish the D2D relaying architecture by assuming that all UEs are served via D2D relaying. Based on the relaying architecture, we derive the optimal resource allocation in the second stage. Finally, an adaptive link selection algorithm is proposed in the third stage to determine the set of UEs that switch from D2D relaying to multi-beam reflection. The high-level subproblem is user association, for which we solve it with a dual decomposition-based approach. Simulation results indicate that compared to benchmark schemes, the average data rate achieved by the proposed scheme is significantly higher than the benchmark schemes and is close to an upper bound. Besides, the proposed scheme achieves a good tradeoff between system performance and fairness.

Bi-level convex optimization of eco-driving for connected Fuel Cell Hybrid Electric Vehicles through signalized intersections

Eco-driving for connected Fuel Cell Hybrid Electric Vehicles (FCHEVs) is a coupled problem of speed planning and energy management. To reduce the computational burden, bi-level optimization decouples and hierarchically solves the upper-level and lower-level subproblems. This paper proposes a bi-level convex approach for eco-driving of a connected FCHEV proceeding through multiple signalized intersections. On the upper level, the non-linear traffic light constraints are transformed into time-varying linear state constraints and the cost function becomes quadratic after using the average speed. On the lower level, model convexification is carried out for the fuel cell system and battery. Then the upper-level speed planning and lower-level energy management are sequentially solved by the MOSEK solver and the Alternating Direction Method of Multipliers (ADMM) algorithm. The results show that the proposed bi-level convex approach greatly reduces the computational cost while maintaining high energy efficiency, with only 6.59% computational time and almost the same fuel economy compared to the bi-level Dynamic Programming (DP) method.

Joint Task Partitioning and User Association for Latency Minimization in Mobile Edge Computing Networks

Mobile edge computing (MEC) is a promising solution to support emerging delay-sensitive mobile applications, such as self-driving, augment/virtual reality, and various Internet of Things (IoT) applications. By deploying MEC servers at network edge, e.g., close to cellular base stations (BSs), the computational tasks generated by these applications can be offloaded to edge nodes (ENs) and be quickly executed there. At the same time, with the projected large number of IoT devices, the communication and computational resources allocated to each user can be quite limited, making it challenging to provide low-latency MEC services. In this paper, we investigate the problem of task partitioning and user association in an MEC system, aiming to minimize the average latency of all users. We assume that each task can be partitioned into multiple subtasks that can be executed on local devices (e.g., vehicles), MEC servers, and/or cloud servers; each user can be associated with one of the nearby ENs. The subtasks can be independent of or dependent on each other. For each case, we formulate the joint optimization of task partitioning ratios and user association as a mixed integer programming problem. Each problem is solved by decomposing it into two subproblems. The lower-level subproblem is task partitioning under a given user association, which can be solved optimally. The higher-level subproblem is user association, we propose a dual decomposition-based approach and a matching-based approach to derive near-optimal solutions. Simulation results show that compared to benchmark schemes, the proposed schemes reduce the average latency by about 50% and 40% for the cases of independent and dependent subtasks, respectively.

An optimal transmission line switching and bus splitting heuristic incorporating AC and N-1 contingency constraints

Optimal transmission line switching and/or bus splitting is shown to contribute in relieving congestion and reducing the operation cost by rerouting power flows throughout the network. Although bus splitting may be as powerful as line switching in congestion mitigation and is typically considered a smaller disturbance compared with line switching, it has received less attention in the literature in part due to the more complicated node-breaker modeling requirement. In this paper, an optimal transmission line switching and bus splitting heuristic is presented to minimize the operation cost while respecting AC and N-1 contingency constraints. We present a two-level solution method where switching decisions are made in the upper level problem formulated as a mixed integer second order cone programming master problem, while the resulting network topology is checked against AC and N-1 contingency constraints in lower level subproblems. Line switching and bus splitting are modeled as switching actions assuming double-bus double-breaker substation arrangements where all elements at a substation, including generators, loads, lines and shunt elements, are given switches to connect to either of the busbars if the respective substation is split. We also introduce additional constraints to model a breaker-and-a-half substation scheme. Furthermore, a pre-screening step is presented to limit the search space of the problem, thus accelerating the solution process. We demonstrate the application of the proposed method on IEEE standard test systems.

Detecting and solving aircraft conflicts using bilevel programming

We present two bilevel programming formulations for the aircraft deconfliction problem: one based on speed regulation in k dimensions, the other on heading angle changes in 2 dimensions. We propose three reformulations of each problem based on KKT conditions and on two different duals of the lower-level subproblems. We also propose a cut generation algorithm to solve the bilevel formulations. Finally, we present computational results on a variety of instances.

A residential energy management system with bi-level optimization-based bidding strategy for day-ahead bi-directional electricity trading

Bi-directional electricity trading of demand response (DR) and transactive energy (TE) frameworks allows the traditionally passive end-users of electricity to play an active role in the local power balance of the grid. Appropriate building energy management systems (BEMSs), coupled with an optimized bidding strategy, can provide significant cost savings for prosumers (consumers with on-site power generation and/or storage facility) when they participate in such bi-directional trading. This paper presents a BEMS with an optimization-based scheduling and bidding strategy for small-scale residential prosumers to determine optimal day-ahead energy-quantity bids considering the expected cost of real-time imbalance trading under uncertainty. The proposed scheduling and bidding strategy is formulated as a stochastic bi-level minimization problem that determines the day-ahead energy-quantity bids by minimizing the energy cost in the upper level considering expected cost of uncertainty, whereas a number of lower-level sub-problems ensure optimal operation of building loads and distributed energy resources (DERs) for comfort reservation, minimization of consumers’ inconveniences and degradation of residential storage units. A modified decomposition method is used to reformulate the nonlinear bi-level problem as a mixed-integer linear programming (MILP) problem and solved using ‘of the shelf’ commercial software. The effectiveness of the proposed BEMS model is verified via case studies for a residential prosumer in Sydney, Australia with real measurement data for building energy demand. The efficacy of the proposed method for overall financial savings is also validated by comparing its performance with state-of-the-art day-ahead scheduling strategies. Case studies indicate that the proposed method can provide up to 51% and 22% cost savings compared to inflexible non-optimal scheduling strategies and deterministic optimization-based methods respectively. Results also indicate that the proposed method offers better economic performance than standard cost minimization models and multi-objective methods for simultaneous minimization of energy cost and user inconveniences.

Bi-Level Participatory Forest Management Planning Supported by Pareto Frontier Visualization

Abstract This research addresses the problem of forested landscape management planning in contexts characterized by multiple ecosystem services and multiple stakeholders. A new methodology for participatory landscape-level forest management is proposed. Specifically, a bilevel representation is used, whereas models of subsystems are used for constructing an integrated model of the master problem. Participatory workshops and interactive visualization of the Pareto frontier are used to support the solution of the multi-objective optimization upper- and lower-level problems. The visualization is implemented by a technique—Interactive Decision Maps—that displays interactively the Pareto frontier in the form of decision maps, that is, collections of the objectives’ tradeoff curves. Since the upper-level problem may be characterized by a large number of decision variables, we compare the Pareto frontier generated by the Interactive Decision Maps technique with the Pareto frontier generated by a decomposition approach that builds from the Pareto frontiers of the lower-level subproblems. The approach supports further the negotiation between upper- and lower-level goals. Results are discussed for a large-scale application in a forested landscape in northwest Portugal.

IGDT‐based economic dispatch considering the uncertainty of wind and demand response

Integration of wind generation and demand response (DR) poses challenges for the power system operation due to their uncertain characteristics. In this study, an economic dispatching method considering the uncertainties of wind and DR is proposed. The method based on the information gap decision theory (IGDT) can be used to evaluate risks associated with uncertainties from risk averse (RA) and risk-seeking (RS) perspectives. The RA IGDT-based model can provide maximum tolerable robustness region for the required cost target. The RS IGDT-based model can help achieve the lowest operation costs with desired uncertainties. The proposed model is bi-level. The upper level subproblem aims to maximise (minimise) the allowable uncertainty level to satisfy the pre-determined cost target, while the lower level subproblem is to maximise (minimise) possible cost considering the uncertainties. The bi-level model is then transformed into a single level mixed integer linear programming problem that can be solved through commercial solves. Finally, the authors evaluate the performance of the IGDT-based approach by simulations on the modified 6-bus and IEEE 118-bus systems. The results show that the proposed approach can provide suggestions for system operators to make appropriate scheduling plan based on the expected cost targets and risk preferences.

Optimal bidding strategy of wind power producers in pay-as-bid power markets

Abstract This paper presents a method to determine the optimal bidding strategy of the wind power producers with market power for a strategic presence in the day-ahead market with pay as bid method. Since the wind power producer is not capable of exact prediction of his power production, he has to trade the difference between the amount won in the day-ahead market and the actual production value in the balancing market. Uncertainties related to power generation is modeled by likely scenarios. However in order to model the punitive effect of trade in balancing market, the balancing market price is considered as a factor of the day-ahead market's clearing price. In the proposed model, optimal bidding strategy is formulated via a bi-level problem including the upper-level and lower-level sub-problems. The purpose of the upper-level sub-problem is to maximize the wind power producer's earning while the purpose of the lower-level sub-problem is to clear the day-ahead market. To solve both upper-level and lower-level problems, particle swarm optimization algorithm is applied. The results of three-bus test system and IEEE 24-bus RTS shows the efficiency of the proposed method.

Optimal Allocation Model for EV Charging Stations Coordinating Investor and User Benefits

Deploying charging stations (CSs) catering to the demand from electric vehicles plays an important role in modernizing energy infrastructures. In this paper, a bi-level optimal allocation model for allocating fast CSs is proposed aiming to maximize the CS investor benefit (upper layer sub-problem) by optimally allocating CSs with the coordinated determination of the expected efficiency of charging service supply (lower layer sub-problem). The efficiency is formulated as a charging performance index in terms of user satisfaction degree, which mathematically couples the upper level and lower level sub-problems. The proposed nonlinear bi-level model is reduced to a single-layer optimization model under the Karush–Kuhn–Tucker optimality conditions. An improved dynamic differential evolution algorithm with an adaptive update strategy is proposed to solve this single-layer optimization model. The proposed method is validated using a realistic case. The test results show that the proposed methodology is able to co-ordinate both objectives of interest, i.e., allocating CSs network with the maximized benefits and optimizing the fast charging service efficiency at the same time.

A bi-level programming for multistage co-expansion planning of the integrated gas and electricity system

This paper focuses on the coordinated expansion planning of the integrated natural gas and electrical power systems with bi-directional energy conversion. Both the Gas-fired Power Generations (GPGs) and Power-to-Gas stations (P2Gs) are considered as the linkages between the natural gas and electric power systems. The system operation is optimized and embedded in the planning horizon. A bi-level multi-stage programming problem is formulated to minimize the investment cost plus the operational cost. The upper-level optimizes the expansion plan and determines the network topology as well as the generation capacities, while the lower-level is formulated as an optimal economic dispatch under the operational constraints given by the upper-level decision. To solve the bi-level multi-stage programming problem, a hybrid algorithm is proposed combining the modified binary particle swarm optimization (BPSO) and the interior point method (IPM). The BPSO is used for the upper-level sub-problem, and the IPM is adopted for the lower-level sub-problem. Numerical case studies have been carried out on the practical gas and electricity transmission network in western Denmark. Simulation results demonstrate the effectiveness of the proposed approach.

The hierarchical decomposition method and its application in recognizing Marcellus Shale lithofacies through combining with neural network

Abstract The three dimensional (3-D) distribution of shale lithofacies has been approved to be helpful for recognizing shale gas productive areas at basin and regional scales. To build the 3-D model, the successful prediction of shale lithofacies by conventional logs is critical. Thus, a hierarchical decomposition (HD) method for multi-class classification was proposed and developed through decomposing the original multi-class problem into several simpler sub-problems on the basis of the class hierarchy. This method can effectively combine the mathematical methods and professional knowledge, and also overcome several drawbacks existing in the common decomposition methods, such as one-against-all, one-against-one and error-correcting output code. Due to the feature that the misclassification in upper-level sub-problems will be handed down into the lower-level sub-problems, the error control and recovery algorithms were generated in this HD method, including setting higher threshold in upper levels, deciding classes through voting method based on the K -fold cross validation, and re-classifying the meta-class when its child class cannot be decided. By using the HD, the cross validation correct ratio of core data sample and pulsed neutron spectroscopy data sample was improved from 85.7% to 89.0% and from 77.8% to 79.5%, respectively, which strongly supports the HD as a promising method for solving multi-class classification problems. In this paper, the comprehensive methodology of the HD for multi-class classification was explained, and the enhanced recognition of Marcellus Shale lithofacies in the Appalachian Basin was also demonstrated.

Two-level decomposition algorithm for crew rostering problems with fair working condition

A typical railway crew scheduling problem consists of two phases: a crew pairing problem to determine a set of crew duties and a crew rostering problem. The crew rostering problem aims to find a set of rosters that forms workforce assignment of crew duties and rest periods satisfying several working regulations. In this paper, we present a two-level decomposition approach to solve railway crew rostering problem with the objective of fair working condition. To reduce computational efforts, the original problem is decomposed into the upper-level master problem and the lower-level subproblem. The subproblem can be further decomposed into several subproblems for each roster. These problems are iteratively solved by incorporating cuts into the master problem. We show that the relaxed problem of the master problem can be formulated as a uniform parallel machine scheduling problem to minimize makespan, which is NP-hard. An efficient branch-and-bound algorithm is applied to solve the master problem. Effective valid cuts are developed to reduce feasible search space to tighten the duality gap. Using data provided by the railway company, we demonstrate the effectiveness of the proposed method compared with that of constraint programming techniques for large-scale problems through computational experiments.