Nonlinear Bilevel Problem Research Articles

Stochastic energy management of a multi-microgrid system with battery/ supercapacitor energy storages considering demand response and transactive energy

This paper proposes stochastic energy management for an electrical system consisting of a multi-microgrid and a distribution company (DISCO). The objective function aims at finding an optimal day-ahead energy schedule of dispatchable resources, energy storages, and demand response in the presence of intermittent renewable energy sources. Two different energy storage devices, i.e., battery and supercapacitor (SC), are modeled in the system to provide both high power and energy density. Wind speed, solar irradiation, and electrical demand uncertainties have been modeled using probability density functions of uncertain parameters, and the number of scenarios has been decreased using mixed-integer linear programming (MILP) based scenario reduction with a novel characteristic. In the proposed model, the DISCO, as a leader, attempts to maximize its profit while the microgrids (MGs), as followers, tend to minimize their operational costs. Karush-Kuhn-Tucker, strong duality, and Fortuny-Amat transformation are used to convert the non-linear bi-level problem into a linear single-level problem. The proposed model was tested on a hypothetical distribution system and the results showed the efficiency of the model in reducing operational costs of MGs by 7.5%, increasing the DISCO profit by 6.9%, and improving the reliability of the system by reducing the energy not supplied.

Bilevel Programming for Manufacturers Operating in an Omnichannel Retailing Environment

In this article, we present a joint production, pricing, and inventory control problem that can help manufacturers manage revenue in an omnichannel environment. This article puts forward a multiperiod Stackelberg game between a manufacturer and a retailer, assuming the former to be the leader. The leader–follower game is formulated as a mixed-integer nonlinear bilevel optimization problem wherein both players seek to maximize their respective profits. The article envisions an omnichannel retailing environment where online, offline, direct, and drop-shipping channels coexist. It then investigates how enabling showrooming and webrooming on the drop-shipping channel account can affect the supply-chain profitabilities. The analyses suggest that when the customers are aided with showrooming and webrooming services and are allowed to place drop-shipping orders from retailers’ stores, additional profit can be generated in the supply chain. To solve the proposed bilevel optimization problem, we investigate single-level reduction using Karush–Kuhn–Tucker conditions and hierarchical optimization technique based on the simulated annealing and randomized decomposition solver. To assess the efficacy of the solution techniques, a comparative analysis is carried out. Thereafter, with the aid of numerical experiments and sensitivity analyses, this article draws the key managerial insights for manufacturers.

Optimal operation of the green port system considering the multiport power electronic transformer in day-ahead markets

Increasing operation costs and carbon emissions have brought great challenges to the development of port systems, which are regarded as prosumers that comprise various renewable energy sources and diversiform electric loads. In this paper, an energy transaction framework for the green port with a multiport power electronic transformer (PET) is established to optimize the operation of the port system in day-ahead energy and reserve markets. The market behavior of the port system is formulated as a bi-level stochastic optimization model. The energy source schedule, berth allocation, and quay crane (QC) assignment are optimized to minimize the total operation cost at the upper level, whereas energy and reserve market clearing problems are settled at the lower level. The aforementioned nonlinear bi-level optimization problem is solved as a mathematical program with equilibrium constraints (MPECs). Karush–Kuhn–Tucker conditions and duality theory convert the proposed nonlinear bi-level problem into a linear single-level problem. Numerical simulations show that the proposed strategy can achieve the lowest total operation cost for the green port.

Bi-level Energy Trading Model Incorporating Large-scale Biogas Plant and Demand Response Aggregator

Increasing intermittent renewable energy sources (RESs) intensifies the imbalance between demand and generation, entailing the diversification of the deployment of electrical energy storage systems (ESSs). A large-scale biogas plant (LBP) installed with heating devices and biogas energy storage (BES) usually exhibits a storage-like characteristic of accommodating an increasing penetration level of RES in rural areas, which is addressed in this paper. By utilizing the temperature-sensitive characteristic of anaerobic digestion that enables the LBP to exhibit a storage-like characteristic, this paper proposes a bi-level energy trading model incorporating LBP and demand response aggregator (DRA) simultaneously. In this model, social welfare is maximized at the upper level while the profit of DRA is maximized at the lower level. Compared with cases only with DRA, the results show that the proposed model with the LBP improves the on-site accommodation capacity of photovoltaic (PV) generation up to 6.3%, 18.1%, and 18.9% at 30%, 40%, and 50% PV penetration levels, respectively, with a better economic performance. This nonlinear bi-level problem is finally recast by a single-level mathematical program with equilibrium constraints (MPEC) using Karush-Kuhn-Tucker (KKT) conditions and solved by the Cplex solver. The effectiveness of the proposed model is validated using a 33-bus test system and a sensitivity analysis is provided for analyzing what parameter influences the accommodation capacity most.

Energy management of hydrogen refueling stations in a distribution system: A bilevel chance-constrained approach

The increasing demand for hydrogen vehicles has made the planning and operation of hydrogen refueling stations (HRS) an essential task. In this paper, a comprehensive model of HRSs including on-site electrolyzers, storages, compressors, vaporizers, chillers, and dispensers has been model in a hydrogen distribution system (HDS). Optimizing the operation of HRSs and the layout of the HDS in the retail hydrogen market is a hierarchical decision-making problem. Thus, a bilevel model is proposed where the upper-level problem is maximizing the profit of the HDS, and the lower-level problem is minimizing the operation costs of the HRSs. A chance-constrained approach is implemented to cope with the uncertainties of hydrogen demand and renewable power generation. The proposed model is transformed from a nonlinear bilevel problem to a linear single-level model by applying the Karush–Kuhn–Tucker conditions and dual theory to the lower-level problem. Simulation results demonstrated the practicality of the proposed bilevel model compared with centralized models. In addition, it is shown that increasing the confidence level increases both the HDS profit and HRS operating costs.

Second order optimality conditions for a bilevel optimization problem in terms of approximate Hessians

In this paper, we are concerned with a nonlinear bilevel optimization problem (P). Using the Karush-Kuhn-Tucker (KKT) conditions associated to the lower-level problem, we reformulate the initial problem (P) into a one-level problem with equality and inequality constraints (P⁎). Applying approximate Hessians, introduced by Jeyakumar and Luc, we establish second order necessary and sufficient optimality conditions for (P⁎). To illustrate our findings, examples are also given.

Formulating false data injection cyberattacks on pumps’ flow rate resulting in cascading failures in smart water systems

As digital interconnections of smart cities greatly enhance the quality of life for residents, they also introduce challenges related to cybersecurity that may adversely impact critical infrastructure. Smart cities must secure their critical infrastructures against cyberattacks. Water distribution is one of these infrastructures. In this paper, a bi-level optimization model is developed for cyberattacks that can result in physical consequence of multiple node failures in water networks. Unlike the existing research that considered random cyberattacks, the proposed cyber–physical attack model injects targeted false data into pumps’ measurements causing pressure head buildup in targeted nodes. In addition, the injected false data will ensure bypassing the existing water system’s state-estimation and bad data detection methods, a condition that was not considered in previously developed attacks. The findings indicate the minimum number of targeted pumps required to create the most severe cascading failures across the water network. The model will identify strategic pumps that are likely to be targeted by cyberattacks, and therefore should be given extra layers of protection by water authorities. • A cybersecurity attack model in water distribution systems is proposed. • Minimum false data is injected on pump’s measurements to increase nodal heads. • Attack model bypasses state estimation causing cascading failures in water systems. • A nonlinear bi-level optimization problem is developed as the cyberattack model.

Semismooth Newton-type method for bilevel optimization: global convergence and extensive numerical experiments

We consider the standard optimistic bilevel optimization problem, in particular upper- and lower-level constraints can be coupled. By means of the lower-level value function, the problem is transformed into a single-level optimization problem with a penalization of the value function constraint. For treating the latter problem, we develop a framework that does not rely on the direct computation of the lower-level value function or its derivatives. For each penalty parameter, the framework leads to a semismooth system of equations. This allows us to extend the semismooth Newton method to bilevel optimization. Besides global convergence properties of the method, we focus on achieving local superlinear convergence to a solution of the semismooth system. To this end, we formulate an appropriate CD-regularity assumption and derive sufficient conditions so that it is fulfilled. Moreover, we develop conditions to guarantee that a solution of the semismooth system is a local solution of the bilevel optimization problem. Extensive numerical experiments on 124 examples of nonlinear bilevel optimization problems from the literature show that this approach exhibits a remarkable performance, where only a few penalty parameters need to be considered.

A Survey on Mixed-Integer Programming Techniques in Bilevel Optimization

Bilevel optimization is a field of mathematical programming in which some variables are constrained to be the solution of another optimization problem. As a consequence, bilevel optimization is able to model hierarchical decision processes. This is appealing for modeling real-world problems, but it also makes the resulting optimization models hard to solve in theory and practice. The scientific interest in computational bilevel optimization increased a lot over the last decade and is still growing. Independent of whether the bilevel problem itself contains integer variables or not, many state-of-the-art solution approaches for bilevel optimization make use of techniques that originate from mixed-integer programming. These techniques include branch-and-bound methods, cutting planes and, thus, branch-and-cut approaches, or problem-specific decomposition methods. In this survey article, we review bilevel-tailored approaches that exploit these mixed-integer programming techniques to solve bilevel optimization problems. To this end, we first consider bilevel problems with convex or, in particular, linear lower-level problems. The discussed solution methods in this field stem from original works from the 1980’s but, on the other hand, are still actively researched today. Second, we review modern algorithmic approaches to solve mixed-integer bilevel problems that contain integrality constraints in the lower level. Moreover, we also briefly discuss the area of mixed-integer nonlinear bilevel problems. Third, we devote some attention to more specific fields such as pricing or interdiction models that genuinely contain bilinear and thus nonconvex aspects. Finally, we sketch a list of open questions from the areas of algorithmic and computational bilevel optimization, which may lead to interesting future research that will further propel this fascinating and active field of research.

A Solving Algorithm for Nonlinear Bilevel Programing Problems Based on Human Evolutionary Model

An algorithm based on the human evolutionary model is proposed for solving nonlinear bilevel programing problems. In view of the hierarchical structure of this problem, the algorithm is designed through feeding back the optimal solution of the lower-level problem to the upper-level. Based on the quality of individuals at each iteration, this proposed algorithm can independently change the population size to achieve the balance between global and local searching ability during the progress of evolution, which can perform an exhaustive search in the whole landscape through creating an individual by using the tabu search method. Finally, we test four typical bilevel programing problems by using the proposed algorithm to verify its feasibility. The experimental results indicate the proposed algorithm can not only solve bilevel programing problems but also get the global optimal solution.

Bi-level optimization model for the coordination between transmission and distribution systems interacting with local energy markets

The coordination between distribution system and transmission system operation in the presence of distributed energy resources (DERs) is a new framework that needs appropriate modeling. Moreover, local energy market models are emerging, and there is the need to describe the decision-making occurring in active distribution systems including the distribution company (Disco) and the DER aggregators. This paper investigates the coordination between transmission, distribution, and DER aggregators that interact in a local market model. The individual objectives of the decision-makers are conflicting with each other. For this purpose, a bi-level optimization approach is proposed, in which the operation problem of the Disco and the day-ahead market clearing managed by the wholesale market operator are considered as the upper- and lower-levels problems, respectively. Moreover, to model the uncertainties of output power of renewable energy sources in the Disco’s problem, the information gap decision theory is used. The resulting model is a non-linear bi-level problem, which is transformed into a linear single-level one through the exploitation of the Karush-Kuhn-Tucker conditions and the duality theory. To investigate the effectiveness of the model, two case studies are defined in which the IEEE 33-bus and a real 43-bus distribution systems are connected to the RTS 24-bus power system.

Stochastic bi-level coordination of active distribution network and renewable-based microgrid considering eco-friendly Compressed Air Energy Storage system and Intelligent Parking Lot

The optimal operation of active distribution systems in the presence of private renewable-based entities is one of the primary challenges of future power networks. In this regard, developing a practical framework to deal with this kind of issue is essential. Hence, in this paper, a novel bi-level stochastic programming approach is presented for optimal energy and reserve scheduling of the active distribution system in the presence of different eco-efficient autonomous players. In the proposed model, the distribution system operator, as a leader, attempts to minimize its total operating costs. At the same time, the renewable-based microgrid owner, as an independent follower, tends to maximize its profit from exchanging energy and reserve with the distribution system operator. The suggested scheme is a non-linear bi-level problem which is transformed into a non-linear single-level problem through Karush–Kuhn–Tucker conditions. In order to find the global optima, the non-linear single-level problem is linearized by utilizing the Big-M method. Finally, to investigate the effectiveness of the provided model, it is tested on the modified IEEE 15-Bus active distribution system under different cases and scenarios. Obtained results indicate that the operation cost of the distribution system operator can be reduced up to 134.09$, from 10710.11$ to 10576.02$, and the profit of the microgrid owner can be increased significantly 906.93$, from 659.455$ to 1566.39$, by considering both environmentally friendly units, IPL and CAES.

DLMP Calculation and Congestion Minimization With EV Aggregator Loading in a Distribution Network Using Bilevel Program

This article develops a distributional locational marginal pricing (DLMP) based electric vehicle (EV) aggregator scheduling framework minimizing the congestion in the network. The article has two parts: 1) calculation of DLMPs by solving actual DistFlow equations; and 2) EV aggregator scheduling in a distribution network formulated as a bilevel problem considering EV aggregator as a leader and distribution system operator (DSO) as a follower. The upper level problem is the cost minimization of EV aggregator, whereas the lower level problem is the social welfare maximization of DSO while satisfying the network constraints. The nonlinear bilevel problem is reformulated into the single-level problem using Karush-Kuhn-Tucker conditions and duality theorem. The framework considers the uncertainty of the EV aggregator by using robust programming and tested with the 15-bus radial distribution network for congested and uncongested cases.

A bi-level programing for wildfire self-evacuation network design

A non-linear bi-level problem is suggested in this paper for wildfire self-evacuation planning, the upper problem of which includes binary variables and the lower problem includes continuous variables. In this model, the upper problem selects a number of links and adds them to the available evacuation network. It, moreover, predicts the traffic balance, and the time window of the links in the lower problem. A part of the objective function in the bi-level problem is non-linear which is linearized with a linear approximation method that does not require binary variables. Then the linear bi-level model is reformulated as a non- linear single level problem. This model is linearized and transferred into Mixed Integer Programing. The model is then used for the real case study of the Beechworth fire in 2009. The resulted outputs of the model are beneficial in planning design schemes for emergency evacuation to use the maximum potential of the available transportation network.

Stimulating sustainable energy at maritime ports by hybrid economic incentives: A bilevel optimization approach

Over the past few decades, maritime ports have faced ever-increasing critiques regarding their significant production of air emissions resulting from their energy activities. To promote sustainability at ports, regulatory authorities have introduced the concept of regulations and economic incentives. In this paper, we analyze the process in which a regulatory authority defines regulations, incentives, and tax policies to motivate one or more ports in the region to initiate energy sustainability and emission-reduction efforts. We model the behaviors of both the regulatory authority and the participating ports in the form of a multi-objective mixed-integer nonlinear bilevel optimization problem to capture the hierarchy of the policy-making process and the existing competitions among the ports. The proposed model finds the optimal incentive and tax policies for the policy-maker in the upper-level and provides the ports in the region with the optimal choice of sustainable energy solutions and service prices in the lower-level. Simulation results show that the proposed approach can effectively reduce the region-wide emission from the port activities while ensuring port entities’ welfare, competitiveness, and sustainable growth as regional energy hubs.

On Bilevel Optimization with Inexact Follower

Traditionally, in the bilevel optimization framework, a leader chooses her actions by solving an upper-level problem, assuming that a follower chooses an optimal reaction by solving a lower-level problem. However, in many settings, the lower-level problems might be nontrivial, thus requiring the use of tailored algorithms for their solution. More importantly, in practice, such problems might be inexactly solved by heuristics and approximation algorithms. Motivated by this consideration, we study a broad class of bilevel optimization problems where the follower might not optimally react to the leader’s actions. In particular, we present a modeling framework in which the leader considers that the follower might use one of a number of known algorithms to solve the lower-level problem, either approximately or heuristically. Thus, the leader can hedge against the follower’s use of suboptimal solutions. We provide algorithmic implementations of the framework for a class of nonlinear bilevel knapsack problem (BKP), and we illustrate the potential impact of incorporating this realistic feature through numerical experiments in the context of defender-attacker problems.

Optimal operation of smart distribution networks in the presence of demand response aggregators and microgrid owners: A multi follower Bi-Level approach

With utilizing the advanced automation services and communication technologies, distribution systems have been transferred from passive to smart networks. In this regard, owing to widespread and complicated interactions between varied entities, distribution system operators (DSOs) encounter various challenges in terms of energy management. Hence, in this paper, a novel Multi Follower Bi-Level framework is presented for operational scheduling of smart distribution networks (SDNs) in the presence of two players, the Demand Response Aggregator (DRA) and Microgrid Owner (MGO). In the proposed method, the upper-level problem minimizes the DSO's operating costs, while, the lower-levels maximize the MGO and DRA's profits from exchanging power. In this procedure, a non-profit agent, the Distribution-Independent System Operator is introduced to coordinate operational conflicts and interests of the network. The considered model is a non-linear Bi-Level problem which is converted into a linear Single-Level problem through KKT conditions and the Big-M method. The presented scheme is implemented on two modified SDNs. The results demonstrate that the Bi-level technique is appropriate for each entity to promote its benefit without any negative influence on the other players. Finally, to evaluate effectiveness of the proposed model, sensitivity analysis is conducted by transferring the DRA's interest to the upper-level.

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.

Optimal offer-bid strategy of an energy storage portfolio: A linear quasi-relaxation approach

This paper proposes a model of the behavior of an expected profit-maximizing merchant storage owner with the ability to exercise unilateral market power. The resulting non-linear bilevel optimization problem is transformed into a single-level stochastic bilinear program using the Karush-Kuhn-Tucker conditions of the lower-level Independent System Operator dispatch problem. By discretizing the offers and bids of the merchant storage owner, the problem is formulated as a stochastic disjunctive program. Using the disjunctive nature of the derived program, a specialized branch-and-bound algorithm that applies a linear quasi-relaxation of the merchant storage problem is proposed. Our solution algorithm is able to solve the problem in an efficient manner; returning the charge and discharge strategies for the merchant storage owner that yield the highest expected profits. Simulations of test systems reveal the various abilities of the merchant storage owner to exercise unilateral market power. Those include demand withholding, generation withholding and under-use which result in an increased congestion in both space and time when compared to the welfare-maximizing use of storage. Factors such as uncertain bids by other players, final state-of-charge requirements and arbitrage by other storage players are investigated. Moreover, numerical results demonstrate the superior computational performance of the proposed solution algorithm when benchmarked against current practices in the literature.

Risk management model for simultaneous participation of a distribution company in Day-ahead and Real-time markets

Abstract In Active Distribution Networks (ADNs), Distribution Company (Disco) follows two main strategies of dispatching of Distributed Energy Resources (DERs) and trading energy with wholesale energy markets, including Day-Ahead (DA) and Real-Time (RT) markets, to meet the demand. An attempt is made in this paper to model the strategic behavior of the Disco, in the wholesale DA and RT energy markets, through a bi-level optimization approach While the objective of the upper-level problem is to minimize the expected cost of the Disco, the lower-level problem (with two optimization problems) formulates to simultaneously maximize the social-welfare of the DA market and minimize the cost of the RT market. Furthermore, uncertain behavior of renewable energy sources as well as demand is tackled into the problem formulation. To this end, Disco decision-making represents as a risk-based two-stage stochastic problem where the Disco’s risk aversion is modeled using conditional value at risk (CVaR) method. Generally stated, the proposed model is a non-linear bi-level problem which may be transformed into a non-linear but single-level problem through Karush–Kuhn–Tucker (KKT) conditions and dual theory. Detailed numerical results on a 6-bus and RTS 24-bus power systems are used to demonstrate efficiency of the proposed model. Moreover, sensitivity analysis is carried out to investigate the effect of risk-aversion parameter on the decision making of the Disco and the offers/bids in both the DA and RT markets.