Upper-level Problem Research Articles

Bilevel hyperparameter optimization for support vector classification: theoretical analysis and a solution method

Support vector classification (SVC) is a classical and well-performed learning method for classification problems. A regularization parameter, which significantly affects the classification performance, has to be chosen and this is usually done by the cross-validation procedure. In this paper, we reformulate the hyperparameter selection problem for support vector classification as a bilevel optimization problem in which the upper-level problem minimizes the average number of misclassified data points over all the cross-validation folds, and the lower-level problems are the l_1-loss SVC problems, with each one for each fold in T-fold cross-validation. The resulting bilevel optimization model is then converted to a mathematical program with equilibrium constraints (MPEC). To solve this MPEC, we propose a global relaxation cross-validation algorithm (GR–CV) based on the well-know Sholtes-type global relaxation method (GRM). It is proven to converge to a C-stationary point. Moreover, we prove that the MPEC-tailored version of the Mangasarian–Fromovitz constraint qualification (MFCQ), which is a key property to guarantee the convergence of the GRM, automatically holds at each feasible point of this MPEC. Extensive numerical results verify the efficiency of the proposed approach. In particular, compared with other methods, our algorithm enjoys superior generalization performance over almost all the data sets used in this paper.

Market-based hosting capacity maximization of renewable generation in power grids with energy storage integration

In an attempt to achieve net zero, the operation and planning of the energy system face techno-economic challenges brought by integrating large-scale distributed energy resources (DERs) with low carbon footprints. Previous work has analyzed the technical challenges including hosting capacity (HC) for DERs. In light of the deregulation of the power industry and the transition to power system with renewables at its center, this article takes the lead to maximizing renewable integration in power grids from a market viewpoint. It solves a significant problem brought forth by the fall in electricity prices, resulting from increasing renewable penetration that jeopardizes investment cost recovery and prevents sustainable grid integration of renewables. To this end, a novel bi-level optimization model is formulated, where the upper-level problem aims to maximize the HC of renewables ensuring the recovery of investment, and the lower-level problem describes the market clearing process considering network constraints. The optimal solution of devised bi-level problem can be found after reformulating it to a single-level mixed-integer linear problem (MILP) using the strong duality theorem and a special ordered set-type 1 (SOS1) founded linearization approach. Case studies confirm the significance of the devised model and quantitatively analyze the impact of different network capacities, renewable subsidies, and energy storage, respectively, on the market-based HC obeying its profitability constraint.

Time-Optimal Coordination for Connected and Automated Vehicles at Adjacent Intersections

In this paper, we provide a hierarchical coordination framework for connected and automated vehicles (CAVs) at two adjacent intersections. This framework consists of an upper-level scheduling problem and a low-level optimal control problem. By partitioning the area around two adjacent intersections into different zones, we formulate a scheduling problem for each individual CAV aimed at minimizing its total travel time. For each CAV, the solution of the upper-level problem designates the arrival times at each zones on its path which becomes the inputs of the low-level problem. The solution of the low-level problem yields the optimal control input (acceleration/deceleration) of each CAV to exit the intersections at the time specified in the upper-level scheduling problem. We validate the performance of our proposed hierarchical framework through extensive numerical simulations and comparison with signalized intersections, centralized scheduling, and FIFO queuing policy.

Bi-level scheduling in high-end equipment R&D: when more algorithm strategies may not be better

Motivated by the practical research and development (R&D) process in high-end equipment manufacturing, this study investigates a bi-level scheduling problem in a complex R&D project network, where each project contains multiple modules with a complete task network. In the bi-level scheduling problem, the upper-level problem is that the R&D project leader makes the decision on allocating all R&D project modules to limited R&D researchers and the objective is to minimise the total penalty cost of all projects, and the lower-level problem is that the researchers schedule and sort the assigned tasks to minimise their minimum makespan. The different capacity of researchers is considered, and some structural properties are derived based on the capacity analytics. To tackle this complex scheduling problem, an effective Variable Neighborhood Search algorithm based on the ‘less is more' concept is proposed, where a Multi-Greedy Heuristic is incorporated. Interestingly, we observe that simpler algorithmic strategies may lead to better algorithmic performance. Computational experiments are carried out to demonstrate that the performance of the proposed algorithm is efficient and stable.

Variable heat pricing to steer the flexibility of heat demand response in district heating systems

Flexibility from low-temperature district heating (LTDH) systems favors the integration of renewable energy in power systems. Flexibility from LTDH systems may be revealed and steered through dynamic pricing for heat. For the purpose of optimal heat pricing, the heat market with variable heat pricing is modeled as a Stackelberg game (i.e., with a bi-level structure). The upper-level problem of the heat provider is to maximize net profit by issuing hourly heat prices day-ahead, while the lower-level problem of heat consumers is to minimize costs by adjusting heat demand based on the dynamic price. The proposed model is tested on a groundwater-based LTDH system with one provider and three consumers. It is evaluated based on the operational cost of the heat provider and the average heat unit price for the consumers, compared to flat heat pricing.

Expansion planning of a price-maker virtual power plant in energy and reserve markets

This paper addresses the expansion planning problem of a virtual power plant considering the possibility of building new assets such as conventional, renewable, and energy storage units. The VPP is modeled as a price-maker player that strategically participates in energy and reserve markets by altering the prices of these markets to its own benefit. The VPP includes flexible demands, conventional and renewable generating units, and energy storage units. Variabilities in renewable production levels are modeled using a modified clustering K-medoids method, while the uncertain reserve request coefficients are represented using a set of scenarios. The present problem is formulated as a two-stage stochastic bi-level model, where the upper-level problem represents the expected profit of the VPP to be maximized by building new assets, while the lower-level problems model the clearing of the energy and reserve electricity markets for different market conditions, which aim to maximize the social welfare and reduce the cost of the ancillary systems, respectively, and determine the market prices of both markets. The stochastic bi-level model is converted to a single-level mixed-integer linear programming problem by replacing the lower-level problems by their Karush–Kuhn–Tucker optimality conditions and by applying exact linearization techniques. Numerical results show that considering the VPP as a price-maker player influences its expansion decisions and allows the VPP to increase its profit up to 108.8% in comparison with a price-taker behavior. Also, the numerical results indicate that the participation in the reserve market increases a 26.76% the expected profit achieved by the VPP.

Routing and Charging Facility Location for EVs Under Nodal Pricing of Electricity: A Bilevel Model Solved Using Special Ordered Set

We consider the problem of identifying optimal location of electric vehicle (EV) charging stations, while accounting for (i) route optimization and (ii) charging cost optimization by the EV fleets, where the electricity price is obtained endogenously by an optimal power flow (OPF) model. We solve the problem using a bi-objective bilevel programming framework with the objectives being one of minimising travel time and the other of minimising EV charging cost. The upper level problem consists of the facility location and the transportation model and the lower level problem consists of the OPF model. After reformulating this computational hard problem as a mathematical program with equilibrium constraints (MPEC), we solve the problem using a special ordered sets-type 1 (SOS1)-based approach. We record the significant improvement in speed by our method, as opposed to the standard Big-M approach. Finally, we apply the technique to the Sioux Falls transportation network with the IEEE 14-bus electricity network embedded on it. We observe that solutions through our models results in as much 37% lower operating costs for the EVs.

Designing Customised Bus Routes for Urban Commuters with the Existence of Multimodal Network – A Bi-Level Programming Approach

Customised bus (CB) is a cutting-edge mean of transportation and has been implemented worldwide. To support the spread of the CB system, methodologies for CB network design have been conducted. However, a majority of them cannot be adopted directly for multi-modal transportation environment. In this paper, we proposed a bi-level programming model to fill this gap. The upper-level problem is to maximise the usage of the CB system with the limitation of operation constraints. Meanwhile, the lower-level problem is to capture the traveller’s choice by minimising traveller’s generalised cost during travel. A solving procedure via genetic algorithm is further proposed and validated via the metro data at Shanghai. The results indicated that the proposed CB route network would attract nearly 5,000 users during morning peak period under the given metro transaction data. We further studied the features of the selected routes and found that the CB network mainly served residence to commercial or industrial parks travellers and would provide travel service with fewer stops, and higher travel efficiency by travelling through expressway.

Worst-case traffic assignment model for mixed traffic flow of human-driven vehicles and connected and autonomous vehicles by factoring in the uncertain link capacity

Connected and autonomous vehicles (CAVs) can form platoons to reduce the time headway and improve the link capacity. However, in a mixed traffic flow environment where both human-driven vehicles (HDVs) and CAVs exist, the platoon intensity is significantly impacted by the stochastic order of the HDVs and CAVs (i.e., the fleet sequence). Therefore, the link capacity involves a large uncertainty even under the same HDV and CAV flow. This uncertain link capacity can cause a large variation in network flow. In the literature, traffic assignment models for mixed traffic flows of HDVs and CAVs are developed based on expected link capacity models, in which the computed link capacity is deterministic for given HDV and CAV flows. These models ignore the impacts of uncertain link capacity on the network performance and network flow distribution, which can dramatically reduce the effectiveness of the corresponding planning strategies. To address this problem, this study proposes a worst-case mixed traffic assignment model. It aims to compute the worst network performance and corresponding equilibrium flow that may occur due to uncertain link capacity. The worst-case mixed traffic assignment is formulated as a bilevel programming problem, where the low-level problem is a variational inequality problem presented to compute the equilibrium results based on a fixed link capacity while the upper-level problem is to find the optimal input for all link capacities within their ranges to minimize the network performance. The partition-based norm relaxed method of the feasible direction solution algorithm is proposed to solve the bilevel worst-case mixed traffic assignment problem. A numerical application shows that the uncertain link capacity has drastic effects on the network flows and network performance, and the proposed algorithm can effectively and efficiently solve the bilevel worst-case mixed traffic assignment problem to compute the worst-case network equilibrium flows and network performance. These results can help traffic managers design robust planning strategies to ensure a minimum level of network performance under the impacts of uncertain link capacity.

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.

Increasing Flexibility of Combined Heat and Power Systems Through Optimal Dispatch With Variable Mass Flow

In combined heat and power systems, varying mass flow can better make use of the heating system’s inertia to increase the flexibility of electric power systems. This is challenging, however, due to integer variables and bilinear constraints in existing optimal dispatch models. In this paper, we incorporate an improved heat pipeline model to eliminate integer variables without compromising on accuracy. Subsequently, a novel decomposition method is proposed to solve the resulting optimization model with bilinear constraints. Namely, the original optimization model is decomposed into a lower-level convex sub-problem with the fixed mass flow and a simple upper-level problem to search for the optimal mass flow. Numerical tests show that the proposed method can achieve lower generation costs than existing variable mass flow dispatch methods with outstanding computational efficiency.

Bidding Strategy of a Wind-Thermal GENCO considering Piecewise Linear AC Power Flow and Correlated Uncertainties

In deregulated electricity markets, generation companies (GENCO) try to maximize their economic benefits considering the electricity demand, transmission network condition, and other participants’ behaviors. The increasing penetration of renewable sources such as wind power generation with intermittent nature poses several challenges to the participation of GENCOs in the electricity market. Thus, this paper presents a stochastic bilevel optimization model to determine the coordinated bidding strategy of a wind-thermal GENCO with the aim of maximizing its profit in the day-ahead and real-time balancing market. Herein, the model aims to maximize the profit of GENCO in the day-ahead and the balancing market in the upper-level problem while minimizing the operation cost of the system in the lower-level problem. The uncertainties of wind power generation and electricity demand are modeled by defining a set of scenarios considering their mutual correlation using the copula technique. Additionally, incorporating AC power flow constraints in the proposed optimization model offers a better solution to the coordinated bidding strategy of the wind-thermal GENCO. Further, the nonlinear AC power flow equations are linearized using the piecewise approximation technique to reduce the computational complexity and enhance the accuracy of the optimal solution. In the end, the developed algorithm is implemented on the IEEE 24-bus RTS, and the simulation results are provided to validate the efficiency and applicability of the proposed coordinated bidding strategy model. The results advocate that the participation of the thermal unit along with the wind farm might mitigate the risk of uncertainties, but it causes an intense increase in the locational marginal price of the system. Importantly, the simulation results indicate the computational efficiency of the model by developing an exact AC power flow model without compromising the results. Notably, it has been found that the profit of the wind-thermal GENCO would be increased by 35.2% employing the copula technique to model the mutual correlation of uncertain parameters.

Dynamic pricing based EV load management in distribution network

This paper presents an optimization model to optimally manage the electric vehicle (EV) charging load in distribution networks. We propose a bi-level programming model where the upper level is the load management problem, and the lower level is the charging station selection problem for EVs. The optimal solution of lower-level problem is a function in the charging price that is determined in the upper-level problem. Simulations on a power-transportation system are carried out to validate the proposed model, and analysis on load management is explored.

A Maximum Principle for a Time-Optimal Bilevel Sweeping Control Problem

In this article, we investigate a time-optimal state-constrained bilevel optimal control problem whose lower-level dynamics feature a sweeping control process involving a truncated normal cone. By bilevel, it is meant that the optimization of the upper level problem is carried out over the solution set of the lower level problem.This problem instance arises in structured crowd motion control problems in a confined space. We establish the corresponding necessary optimality conditions in the Gamkrelidze’s form. The analysis relies on the smooth approximation of the lower level sweeping control system, thereby dealing with the resulting lack of Lipschitzianity with respect to the state variable inherent to the sweeping process, and on the flattening of the bilevel structure via an exact penalization technique. Necessary conditions of optimality in the Gamkrelidze’s form are applied to the resulting standard approximating penalized state-constrained single-level problem, and the main result of this article is obtained by passing to the limit.

ALSO-X and ALSO-X+: Better Convex Approximations for Chance Constrained Programs

In a chance constrained program (CCP), decision makers seek the best decision whose probability of violating the uncertainty constraints is within the prespecified risk level. As a CCP is often nonconvex and is difficult to solve to optimality, much effort has been devoted to developing convex inner approximations for a CCP, among which the conditional value-at-risk ([Formula: see text]) has been known to be the best for more than a decade. This paper studies and generalizes the [Formula: see text], originally proposed by Ahmed, Luedtke, SOng, and Xie in 2017 , for solving a CCP. We first show that the [Formula: see text] resembles a bilevel optimization, where the upper-level problem is to find the best objective function value and enforce the feasibility of a CCP for a given decision from the lower-level problem, and the lower-level problem is to minimize the expectation of constraint violations subject to the upper bound of the objective function value provided by the upper-level problem. This interpretation motivates us to prove that when uncertain constraints are convex in the decision variables, [Formula: see text] always outperforms the [Formula: see text] approximation. We further show (i) sufficient conditions under which [Formula: see text] can recover an optimal solution to a CCP; (ii) an equivalent bilinear programming formulation of a CCP, inspiring us to enhance [Formula: see text] with a convergent alternating minimization method ([Formula: see text]); and (iii) an extension of [Formula: see text] and [Formula: see text] to distributionally robust chance constrained programs (DRCCPs) under the [Formula: see text]Wasserstein ambiguity set. Our numerical study demonstrates the effectiveness of the proposed methods. Funding: This work was supported by the Division of Civil, Mechanical and Manufacturing Innovation [Grant 2046426]. Supplemental Material: The e-companion is available at https://doi.org/10.1287/opre.2021.2225 .

Benders Subproblem Decomposition for Bilevel Problems with Convex Follower

Bilevel optimization formulates hierarchical decision-making processes that arise in many real-world applications, such as pricing, network design, and infrastructure defense planning. In this paper, we consider a class of bilevel optimization problems in which the upper level problem features some integer variables and the lower level problem enjoys strong duality. We propose a dedicated Benders decomposition method for solving this class of bilevel problems, which decomposes the Benders subproblem into two more tractable, sequentially solvable problems that can be interpreted as the upper and lower level problems. We show that the Benders subproblem decomposition carries over to an interesting extension of bilevel problems, which connects the upper level solution with the lower level dual solution, and discuss some special cases of bilevel problems that allow sequence-independent subproblem decomposition. Several novel schemes for generating numerically stable cuts, finding a good incumbent solution, and accelerating the search tree are discussed. A computational study demonstrates the computational benefits of the proposed method over a state-of-the-art, bilevel-tailored, branch-and-cut method; a commercial solver; and the standard Benders method on standard test cases and the motivating applications in sequential energy markets.

Optimal pricing for bidirectional wireless charging lanes in coupled transportation and power networks

As electric vehicles (EVs) become more prevalent, there is growing concern regarding their impact on transportation and power networks. Using vehicle-to-grid (V2G) technology, EVs can operate as additional energy storage sources that make the power grid more efficient and reliable. With their onboard battery, EVs can relocate energy supply spatially and temporally. In this study, we envision that the future deployment of bidirectional charging lanes is capable of transferring power wirelessly and bidirectionally in regional road networks. Additionally, in this study, these futuristic bidirectional charging lanes are controlled by a government agency that manages both transportation and power systems. The government agency attempts to minimize total system costs by using an electricity pricing scheme to buy energy from and sell energy to EVs, which can be modeled as a bi-level optimization problem. In this study, we developed a new user equilibrium (UE) model to depict the equilibrium conditions in a road network in which bidirectional charging lanes are deployed. When traveling between their origin and destination, EV drivers are assumed to select routes and decide on battery charging/discharging plans to minimize their trip costs while ensuring the completion of their trips without running out of range. Trip costs consist of trip travel time and the purchase price of electricity minus the benefits of discharging energy to the power grid. Having established UE conditions, we next formulated the equilibrium conditions of a transportation network coupled with an electricity network as a non-linear complementarity system, which formed our lower-level problem. In such a network, the agency tries to provide energy to both networks with the objective of minimizing total system costs. To formulate the agency’s decision-making process, an additional layer of optimization is added on top of the coupled networks, which is our upper-level problem. This layer minimizes the total travel time and the price of electricity paid to EVs and generators. Next, to efficiently solve the model, a meta-heuristic solution algorithm based on gray wolf optimizer and manifold sub-optimization was developed. The model and the solution algorithm were evaluated based on three different numerical examples. Numerical results demonstrate that bidirectional charging lanes have significant potential to reduce peak load and charging costs associated with EVs.

Dynamic participation in local energy communities with peer-to-peer trading.

Energy communities and local electricity markets (e.g., as peer-to-peer trading) are on the rise due to increasingly decentralized electricity generation and favorable adjustment of the legal framework in many European countries. This work applies a bi-level optimization model for dynamic participation in peer-to-peer electricity trading to determine the optimal parameters of new participants who want to join an energy community, based on the preferences of the members of the original community (e.g., environmental, economic, or mixed preference). The upper-level problem chooses optimal parameters by minimizing an objective function that includes the prosumers' cost-saving and emission-saving preferences, while the lower level problem maximizes community welfare by optimally allocating locally generated photovoltaic (PV) electricity between members according to their willingness-to-pay. The bi-level problem is solved by transforming the lower level problem by its corresponding Karush-Kuhn-Tucker (KKT) conditions. The results demonstrate that environment-oriented prosumers opt for a new prosumer with high PV capacities installed and low electricity demand, whereas profit-oriented prosumers prefer a new member with high demand but no PV system capacity, presenting a new source of income. Sensitivity analyses indicate that new prosumers' willingness-to-pay has an important influence when the community must decide between two new members. The added value of this work is that the proposed method can be seen as a basis for a selection process between a large number of potential new community members. Most important future work will include optimization of energy communities over the horizon several years.

A Novel Multi-Hierarchical Bidding Strategy for Peer-to-Peer Energy Trading Among Communities

Recently, several market types and regulations have been developed in an attempt to handle the increased carbon effect. End-users can also actively participate in the existing distribution system thanks to Peer-to-Peer (P2P) energy trading, which is one of the new emerging market types. In this paper, a novel dual bidding strategy for multi-hierarchical P2P energy trading that includes both intra community and inter communities is proposed considering uncertainties in solar irradiance and temperature. While the lower-level problem consists of both optimal bids of the households to the own Local Market Operators (LMO) for intra community trades and optimal bids of the LMOs to the Central Market Operator (CMO) for inter community trades, profit of both the LMOs and CMO is maximized by clearing the market prices at the upper-level problem. To prove the validity of the devised model, a set of case studies are created. Moreover, the results suggest that the proposed bi-level model is robust, and a remarkable amount of cost savings could be provided by integrating the model.

Optimal Price-maker Trading Strategy of Wind Power Producer Using Virtual Bidding

This paper proposes a stochastic optimization model for generating the optimal price-maker trading strategy for a wind power producer using virtual bidding, which is a kind of financial tool available in most electricity markets of the United States. In the proposed model, virtual bidding is used to improve the wind power producer's market power in the day-ahead (DA) market by trading at multiple buses, which are not limited to the locations of the wind units. The optimal joint wind power and virtual trading strategy is generated by solving a bi-level nonlinear stochastic optimization model. The upper-level problem maximizes the total expected profit of the wind power and virtual bidding while using the conditional value at risk (CVaR) for risk management. The lower-level problem represents the clearing process of the DA market. By using the Ka-rush-Kuhn-Tucker (KKT) conditions, duality theory, and big- <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$M$</tex> method, the bi-level nonlinear stochastic model is firstly transferred into an equivalent single-level stochastic mathematical program with the equilibrium constraints (MPEC) model and then a mixed-integer linear programming (MILP) model, which can be solved by existing commercial solvers. To reduce the computational cost of solving the proposed stochastic optimization model for large systems, a method of reducing the number of buses considered for virtual bidding is proposed to simplify the stochastic MPEC model by reducing its decision variables and constraints related to virtual bidding. Case studies are performed to show the effectiveness of the proposed model and the method of reducing the number of buses considered for virtual bidding. The impacts of the transmission limits, wind unit location, risk aversion parameters, wind power volatility, and wind and virtual capacities on the price-maker trading strategy are also studied through case studies.