Bi-level Optimization Method Research Articles

Toward a Dynamic Reversible Lane Management Strategy by Empowering Learning-Based Predictive Assignment Scheme

Traffic congestion is a long-lasting worldwide problem and even becoming more severe in well-developed regions. Reversible lanes have been used worldwide on various road types to mitigate the effects of congestion and optimize mobility since the 1930s. However, with the limitation of traditional control and management methods, existing solutions can not meet the increasing travel demands. Therefore, the paper introduces a Predictive Empowered Assignment scheme for Reversible Lane (PEARL). By integrating the advanced traffic flow prediction module and bi-level optimization model, PEARL can be a more flexible dynamic lane assignment strategy with foresight compared to traditional lane management methods. In the prediction module, taking advantage of the development of machine learning technologies, the input of PEARL covers not only historical sequence and real-time data but also the environmental conditions in the region. The advanced Bi-directional Long Short-term Memory (Abi-LSTM) model is employed for short-term traffic flow prediction. Then, the study introduces a bi-level optimization method to maximize the total throughput in both directions and minimize the total user costs which determine the lane deployment. The iterations on the prediction module and optimization module can help PEARL coordinate the future lane control plan and make the best decision. Finally, the paper builds up an experiment platform to simulate PEARL in a real-world scenario with heavy input flow for its performance evaluation.

Optimal sizing of residential battery energy storage systems for long-term operational planning

Appropriate battery storage capacity plays an important role in the performance and cost of residential energy systems. However, the load demand and renewable energy generation vary seasonally. To address the long-term operational planning problem of battery energy storage, two battery sizing methods are developed based on the consensus alternating direction method of multipliers (C-ADMM). The residential system layout and convex battery model considering cycling aging are first established. Then, for the one-year horizon planning, C-ADMM decomposes the original large-scale problem into 365 sub-problems. The global optimal battery size and operation strategy are obtained by coordinating these sub-problems in parallel until convergence. In addition, to obtain the optimal battery size based on receding horizon control (RHC) strategies in a year, a new bi-level optimization method is developed, which uses the particle swarm optimization (PSO) algorithm in the outer loop and C-ADMM in the inner loop. Thus the inner loop can be accelerated through parallel computing. The results show that the long-term storage planning problem of residential systems can be solved quickly through C-ADMM due to the parallel computing capability. Furthermore, the optimal battery size calculated under the RHC strategy in a year can contribute to more economic benefits than the global battery size when applied online.

Bi-level Optimizations of Multi-Components Virtual Power Plant for Economic Dispatch and Renewable Utilization

The significant injections of renewable power have been leading to risks of safety and stability of power system by curtailing system inertia. Based on the requirement of ‘carbon neutralization’ policy, independent participants including energy storage (ES), load demand response (LDR) and virtual power plant (VPP) are introduced into system as innovative ancillary service methods. This paper proposes a bi-level optimization method to achieve the economic dispatch of multi-components virtual power plant consisting with gas generator, PV, wind turbine, energy storage and electricity vehicle in the day-ahead optimization, while achieving the maximum renewable power utilization in the real-time optimization by adjusting dispatch plan. This method considers constraints including solar/wind power forecasting, load level forecasting, ES charging/discharging constraints, and DC power flow limits. A case study based on practical project is present, showing that the bi-level optimization is capable of achieving diversified resources aggregation in extended region, as well as the economic optimization and green energy consumption target.

A New Robotic Knee Impedance Control Parameter Optimization Method Facilitated by Inverse Reinforcement Learning

Recent efforts in the design of intelligent controllers for configuring robotic prostheses have demonstrated new possibilities in improving mobility and restoring locomotion for individuals with lower-limb disabilities. In these efforts, personalizing the controller of the robotic device is a crucial step in order to meet individual user's needs and physical conditions. Reinforcement learning (RL) based control designs are among some of the most promising approaches to achieving real-time, optimal adaptive tuning capability. However, such designs to date rely on subjectively determining human-robot walking performance measures, commonly in a quadratic form. To further automate the RL design for robotic knee control parameter tuning and potentially improve human-robot locomotion performance, this study introduces a new bilevel optimization method to objectively specify such control design performance measures via inverse reinforcement learning (IRL), which in turn, will be used in low level (forward) RL design of the impedance control parameters. We demonstrate the effectiveness of the bilevel optimization approach with improved human-robot walking performance using systematic OpenSim simulation studies.

Grid-aware pricing scheme in future distribution systems based on real-time power tracing and bi-level optimization

In this paper, a market-clearing framework based on power tracing and loss allocation has been introduced. Growth in the participation of multiple prosumers in the distribution level with diverse sets of distributed energy resources (DERs) needs to be addressed by considering technical and economic factors in the proposed framework. The grid information of the distribution network is used in the pricing mechanism, which is empowered by power tracing and loss allocation. The calculated prices at each bus reflect the contribution of load/generation at the corresponding buy-in of the total operating costs of the distribution network. The distribution system operator (DSO) sends the selling/buying prices to the individual prosumers. Then prosumers, regarding their objective function, economically dispatches their resources through an iterative manner. A bi-level optimization method is proposed to satisfy the benefits of both DSO and prosumers. In bi-level programming, the upper level represents DSO’s problems, and prosumers problems are presented in the lower level of programming. Furthermore, this market-clearing method is also tested under real-time conditions to evaluate its performance in such situations. The proposed methods are tested on the IEEE 33-bus system.

Community Battery Storage Systems Planning for Voltage Regulation in Low Voltage Distribution Systems

The regulation of the grid voltage within operational limits becomes increasingly challenging as residential photovoltaic (PV) adoption rises. Therefore, this study proposes a method for the efficient planning of multiple community battery energy storage systems (BESS) in low voltage distribution systems embedded with high residential rooftop PV units. A bi-level optimization method based on a Neural Network Optimization Algorithm is developed to regulate the voltage in grid-connected solar PV. Since BESS characteristics are crucial for the reliable operation of the distribution networks, the objective of the bi-level optimization problem is to optimally place and operate BESS collectively at a distributed system level. The charging/discharging protocol of the batteries management system is obtained utilizing linear programming that minimizes the daily voltage signal. Simulations were carried out on a modified IEEE low voltage test feeder to examine the impact of PV integration and BESS installation on the voltage profile. Experimental results show the efficacy of the proposed method in enabling the utility to determine the optimal location, capacity, and number of BESS in the distribution system to keep the network voltage magnitude within acceptable bounds. In addition, results demonstrate that the network topology, load profiles, and amount of PV power highly influence the BESS characteristics.

Game-theoretic multi-energy trading framework for strategic biogas-solar renewable energy provider with heterogeneous consumers

This paper proposes a game-theoretic multi-energy trading framework for a biogas-solar renewable energy provider with heterogeneous consumers to promote the efficient utilization of local renewable energy resources. Within the proposed framework, the multi-energy provider utilizes biogas-solar complementarities to enhance the operational flexibility for electricity, biogas, and heat energy supplies, and consumers are enabled to actively participate in demand response under different multi-energy prices from the provider and UCs. A multi-energy pricing model is then proposed based on the Stackelberg game to interactively and dynamically determine the internal trading prices for optimal multi-energy trading between the provider and consumers. Furthermore, a bi-level optimization method is formulated to solve the Stackelberg game-based multi-energy trading problem to maximize the provider's profit at the upper level and the welfare of each consumer at the lower level. Case studies show that the provider's profit is improved by 33.7% using the proposed scheme compared with the benchmark scheme, and meanwhile the average welfare of consumers for consuming biogas and electricity are approximately 3.0 and 1.4 times greater than those using the benchmark scheme.

An optimal control approach for determining the source term in fractional diffusion equation by different cost functionals

This work is devoted to the mathematical analysis of an inverse source problem governed by a time-fractional diffusion equation. The aims of this paper are to identify the source function from additional data based on a regularized optimal control approach, and to determine the regularization parameters using bi-level optimization. To do this, firstly we formulate our inverse problem to an optimal control one with two types of fidelity terms, which are the Least-Squares fitting and L1 norm. Secondly, we establish the existence of the minimizer for the optimal control problem corresponding to the L1 cost function, since it is the general case. Thirdly, we present some numerical results for the different cases of the cost function. These results are based on solving the optimal control model by gradient method for the L2 case and the primal-dual method for non-smooth L1 norm. Finally, we propose an adaptive selection strategy for the regularization parameter based on the bi-level optimization method, and we give some numerical results to show the effectiveness of the proposed approach.

Adopting a Bi-level Optimization Method for the Freight Transportation Problem: A Multi-objective Programming Approach

Nowadays, various companies find it a crucial task to elevate their distribution system efficiency to survive in what is an intensely competitive environment. Freight transportation via road systems may not always be a practical option for the shipper because of financial, social, and environmental aspects. So, in addition to the road system, the shipper can benefit greatly from the rail system. Thus, taking advantage of different transportation modes can help shippers to tackle, to some extent, arising issues. In this paper, we aim at investigating a freight transportation network including rail and road transportation modes in which the shipper’s choice affects the decisions deeply. Given that this problem as a bi-level programming model, the rail operator acts as a leader and the shipper is treated as a follower. Once the rail operator reveals the rail transportation price, an appropriate response is received from the shipper and this pattern is repeated until an equilibrium is achieved. From the sustainability view, a multi-objective mixed-integer programming model is developed to simulate their interactions considering a sustainable triple bottom line. Employing the Karush–Kuhn–Tucker approach, we converted the formulated bi-level programming model into a single one. Several numerical examples are investigated to evaluate the validity of the developed model and approaches. Different in-depth sensitivity analyses are carried out and the impacts of several factors, such as line and train capacity, on the rail operator and shipper’s decisions are investigated. The results reveal that the rail operator could increase their market share by a wise long-term investment in the rail infrastructure.

Blockchain Based Peer-To-Peer Energy Trading Between Wind Power Producer and Prosumers in Short-Term Market

With the increase of wind power penetration, the deviation caused by its volatility and intermittency poses a growing threat to the grid. Energy trading in short-term markets for wind power and backup storage helps compensate for the deviations. Furthermore, the introduction of peer-to-peer (P2P) energy trading can effectively reduce the risk of centralized market management. However, the higher cost of energy storage are not conducive to wind power producers (WPPs). And P2P trading also suffers from trust and efficiency problems. This paper provides a blockchain based short-term energy trading market, which resolves generation deviations through efficient and trusted real-time transactions between WPP and prosumers. The blockchain-based energy trading market is a trustless P2P structure, and the trading is triggered by smart contracts to ensure efficiency. Furthermore, a reputation mechanism is designed to incentivize WPP’s generation forecasts to be accurate and prosumers to participate in the market. A bilevel optimization method is designed to increase the revenue of WPP and reduce the costs of prosumers. The market can effectively balance the deviation of wind power generation, increase the revenue of WPP by 9.55%, and reduce the costs for consumers by 5.6%.

Surrogate-based bilevel shape optimization for blended-wing–body underwater gliders

For blended-wing–body underwater gliders (BWBUGs), a high-performance shape design necessitates not only a higher lift-to-drag ratio (LDR), but also additional carrying space. However, regardless of how the parameters alter with the fixed layout, some potential optimal solutions may be lost. It is feasible to achieve the genuine global optimum when the parameter space of the layout is liberated. On the other hand, the shape and layout parameters have different physical properties. The complexity of optimization can be reduced if these can be organized according to their properties. Therefore, a surrogate-based bilevel shape optimization (SBSO) method is presented, in which surrogate models are updated iteratively until the optimum is found. The upper level aims at maximizing the LDR, while the lower level aims to maximize the volume. SBSO is tested on 12 benchmark cases, several existing algorithms and shape optimization of a BWBUG, and all show excellent performance.

A Resource Allocation Scheme for Real-Time Energy-Aware Offloading in Vehicular Networks with MEC

With the emergence of new vehicular applications, computation offloading based on mobile edge computing (MEC) has become a promising paradigm in resource-constrained vehicular networks. However, an unreasonable offloading strategy in offloading can cause serious energy consumption and latency. A real-time energy-aware offloading scheme for vehicle networks, based on MEC, is proposed to optimize communication and computation resource to decrease energy consumption and latency. Because the problem of computation offloading and resource allocation is the mixed-integer nonlinear problem (MINLP), this article uses a bi-level optimization method to transform the original MINLP into two subproblems. Furthermore, considering the mobility of vehicle users (V-UEs) and the availability of cloud resources, an offloading scheme based on deep reinforcement learning (DRL) is adopted to help users make the optimal offloading decisions. The simulation results show that the proposed bi-level optimization algorithm reduces the total overhead by nearly 40% to the compared algorithm.

Research on a Bi-Level Collaborative Optimization Method for Planning and Operation of Multi-Energy Complementary Systems

Multi-energy complementary systems (MCSs) are complex multilevel systems. In the process of system planning, many aspects—such as power planning, investment cost, and environmental impact—should be considered. However, different decision makers tend to have different levels of control objectives, and the multilevel problems of the system need to be solved effectively with comprehensive judgment. Therefore, based on the terminal MCS energy structure model, the optimization method of MCS planning and operation coordination, considering the influence of planning and operation in the system’s life cycle, is studied in this paper. Consequently, the research on the collaborative optimization strategy of MCS construction and operation was carried out based on the bi-level multi-objective optimization theory in this paper. Considering the mutual restraint and correlation between system construction and operation in practical engineering, a bi-level optimization model for collaborative optimization of MCS construction and operation was constructed. To solve the model effectively, the existing non-dominated sorting genetic algorithm III (NSGA-III) was improved by the authors on the basis of previous research, which could enhance the global search ability and convergence speed of the algorithm. To effectively improve and strengthen the reliability of energy supply, and increase the comprehensive energy utilization of the system, the effects of carbon transaction cost and renewable energy penetration were considered in the optimization process. Based on an engineering example, the bi-level model was solved and analyzed. It should be noted that the optimization results of the model were verified to be applicable and effective by comparison with the single-level multi-objective programming optimization. The findings of this paper could provide theoretical reference and practical guidance for the planning and operation of MCSs, making them significant for social application.

Learning regularization parameters of inverse problems via deep neural networks

In this work, we describe a new approach that uses deep neural networks (DNN) to obtain regularization parameters for solving inverse problems. We consider a supervised learning approach, where a network is trained to approximate the mapping from observation data to regularization parameters. Once the network is trained, regularization parameters for newly obtained data are computed by efficient forward propagation of the DNN. We show that a wide variety of regularization functionals, forward models, and noise models may be considered. The network-obtained regularization parameters can be computed more efficiently and may even lead to more accurate solutions compared to existing regularization parameter selection methods. We emphasize that the key advantage of using DNNs for learning regularization parameters, compared to previous works on learning via bilevel optimization or empirical Bayes risk minimization, is greater generalizability. That is, rather than computing one set of parameters that is optimal with respect to one particular design objective, DNN-computed regularization parameters are tailored to the specific features or properties of the newly observed data. Thus, our approach may better handle cases where the observation is not a close representation of the training set. Furthermore, we avoid the need for expensive and challenging bilevel optimization methods as utilized in other existing training approaches. Numerical results demonstrate that trained DNNs can predict regularization parameters faster and better than existing methods, hence resulting in more accurate solutions to inverse problems.

Improved Penalty Method via Doubly Stochastic Gradients for Bilevel Hyperparameter Optimization

Hyperparameter optimization (HO) is an important problem in machine learning which is normally formulated as a bilevel optimization problem. Gradient-based methods are dominant in bilevel optimization due to their high scalability to the number of hyperparameters, especially in a deep learning problem. However, traditional gradient-based bilevel optimization methods need intermediate steps to obtain the exact or approximate gradient of hyperparameters, namely hypergradient, for the upper-level objective, whose complexity is high especially for high dimensional datasets. Recently, a penalty method has been proposed to avoid the computation of the hypergradient, which speeds up the gradient-based BHO methods. However, the penalty method may result in a very large number of constraints, which greatly limits the efficiency of this method, especially for high dimensional data problems. To address this limitation, in this paper, we propose a doubly stochastic gradient descent algorithm (DSGPHO) to improve the efficiency of the penalty method. Importantly, we not only prove the proposed method can converge to the KKT condition of the original problem in a convex setting, but also provide the convergence rate of DSGPHO which is the first result in the references of gradient-based bilevel optimization as far as we know. We compare our method with three state-of-the-art gradient-based methods in three tasks, i.e., data denoising, few-shot learning, and training data poisoning, using several large-scale benchmark datasets. All the results demonstrate that our method outperforms or is comparable to the existing methods in terms of accuracy and efficiency.

Theoretical framework about optimal model of operation after intentional attacks considering transmission system switching

This research focuses on optimal post-intentional attack operation considering switching transmission systems. The applicable models for this process focus on the application of bi-level optimization methods that are capable of analyzing two possible scenarios in order to reduce the time of loss or departure from the demand of the electricity system. The main objective of this work is related to maintaining the minimum requirements that allow the operation of the Electric Power System, for this the approach of equations will be carried out that allows establishing the mathematical models in the event of intentional attacks, it must maintain the operation of the Electric System and react in the event of contingencies through the Optimal Switching of Transmission Lines.

Joint Optimization for Coordinated Charging Control of Commercial Electric Vehicles Under Distributed Hydrogen Energy Supply

The transition to the zero-carbon power system is underway accelerating recently. Hydrogen energy and electric vehicles (EVs) are promising solutions on the supply and demand sides. This brief presents a novel architecture that includes hydrogen production stations, fast-charging stations, and commercial EVs. The proposed architecture jointly optimizes the hydrogen energy dispatch and the EV charging location selection and is formulated by a time-varying bilevel bipartite graph (T-BBG) model. We develop a bilevel iteration optimization method combining the linear programming (LP) and the Kuhn–Munkres (KM) algorithm to solve the joint problem whose optimality is proved theoretically. The effectiveness of the proposed architecture on reducing the operating cost is verified via case studies in Shanghai. The proposed method outperforms other strategies and improves the performance by at least 13%, which shows the potential economic benefits of the joint architecture. The convergence and impact of parameters are assessed.

Matching Uncertain Renewable Supply with Electric Vehicle Charging Demand—A Bi-Level Event-Based Optimization Method

The matching between dynamic supply of renewable power generation and flexible charging demand of the Electric Vehicles (EVs) can not only increase the penetration of renewables but also reduce the load to the state electric power grid. The challenges herein are the curse of dimensionality, the multiple decision making stages involved, and the uncertainty of both the supply and demand sides. Event-Based Optimization (EBO) provides a new way to solve large-scale Markov decision process. Considering different spatial scales, we develop a bi-level EBO model in this paper which can both catch the changes on the macro and micro levels. By proper definition, the size of event space stays fixed with the scale of the problem, which shows good scalability in online optimization. Then a bi-level Q-learning method is developed to solve the problem iteratively. We demonstrate the performance of the method by numerical examples. Our method outperforms other methods both in performance and scalability.

Selection and assessment of pedestrian areas in urban environments: a model - and expert-opinion-based approach

Many cities worldwide have implemented pedestrian zones in the past years, as part of an effort to ease the impacts of congestion. However, as pedestrianisation directly affects traffic circulation, operation and accessibility, the issue of optimally selecting pedestrianisation areas within an urban road network is pivotal. The present paper applies a bi-level optimization method for identifying road links and axes within an urban road network, which, if selected for pedestrianisation, would result in the minimum impacts on overall traffic and accessibility. The method is based on a unified network performance measure for calculating the criticality of network links, using the total network demand and the difference in travel time as a consequence of a link closure. Outcomes of the proposed method are presented for a case study in Thessaloniki, Greece. Using a detailed traffic model, the authors present parts of the road network where traffic is restricted for private vehicles and turned into pedestrian zones, causing less additional travel times to the entire network. Results are expressed in traffic-related performance measures (e.g. network travel time). Road links for pedestrianisation are ranked and validated through a survey, targeted to experts and representatives of the responsible local governmental authority.

Bi-Level Optimization Framework for Buildings to Heating Grid Integration in Integrated Community Energy Systems

This paper proposes a bi-level optimization framework for buildings to heating grid (B2HG) integration in integrated heating/electricity community energy systems. The proposed B2HG framework enables buildings to carry out heating demand response to the integrated community energy systems (ICES). In upper level, ICES operator with energy conversion equipment devotes to maximizing its profits by optimizing the energy generation/supply schedules and the sale heating price. In lower level, a detailed physical model of the building with adjustable indoor radiators is developed while considering the building thermal dynamics. Consumers minimize their heating costs by adjusting the flow rates of their radiators according to the heating price provided by the upper level. Then, the proposed bi-level optimization problem is converted into a mixed-integer linear programming using Karush-Kuhn-Tucker conditions and strong duality theorem. The piecewise linearization is used to eliminate the nonlinearity of heating network constraints. Numerical results show that consumers can provide heating demand response to ICES operator by considering the thermal dynamics. With the multiple energy conversion equipment, the ICES operator can unlock the flexibility from the supply side. Moreover, the proposed bi-level optimization method can combine the flexibility from both the supply and demand side, and benefits both the ICES operator and building consumers.