Bilevel Formulation Research Articles

Distribution and transmission coordinated dispatch under joint electricity and carbon day-ahead markets

Worldwide efforts to attain a more renewable-based energy matrix have led to the creation of carbon emission-related markets. Simultaneously, the increasing penetration, mostly of renewable-based sources, in distribution networks may allow distribution system operators (DSOs) to change their traditional participation in wholesale markets as passive buyers. Over the last few years, many authors have investigated and proposed optimization models for the strategic participation of DSOs in electricity markets. Nonetheless, proposals regarding the DSO as a strategic agent in both electricity and carbon-related markets are rare. In this paper, we propose a stochastic bilevel formulation to address this gap. The economic and operational constraints of each stakeholder are considered in the model. Additionally, the problem is written in a convex formulation for which finite convergence to optimality is guaranteed. The formulation was tested for a 14-node transmission system and a 34-node distribution system in which dispatchable and non-dispatchable energy sources, energy storage systems, and demand response units are installed. The results show the DSO’s capacity to place strategic bids in both market environments to maximize its profit, which features intentionally taking financial losses in the carbon allowance market to maximize the overall profit due to the income maximization in the electricity market. Finally, an additional case study with a 123-node distribution system shows the proposed method’s scalability.

A logistics provider’s profit maximization facility location problem with random utility maximizing followers

We introduce a strategic decision-making problem faced by logistics providers (LPs) seeking facility location decisions that lead to profitable operations. The profitability depends on the revenue generated through agreements with shippers, and the costs arising when satisfying these agreements. The latter depends, in turn, on the service levels and characteristics of the shippers’ customers. However, at a strategic level, the LP has imperfect information thereof.We propose a stochastic bilevel formulation where a given LP (leader) anticipates the decisions of shippers (followers) arising from a random utility maximization model. Using a sample average approximation and properties of the associated optimal solutions, we propose a heuristic that can compute high-quality solutions. It is based on a reduced single-level mixed integer linear programming formulation that can be solved by a general-purpose solver after a preprocessing phase. We can quickly identify situations that lead to zero expected profit for the LP. Experimental results show that expected profit highly depends on shippers’ price sensitivities. Underestimating price sensitivities can lead to an overestimation of the expected profit.

How many clues to give? A bilevel formulation for the minimum Sudoku clue problem

It has been shown that any 9 by 9 Sudoku puzzle must contain at least 17 clues to have a unique solution. This paper investigates the more specific question: given a particular completed Sudoku grid, what is the minimum number of clues in any puzzle whose unique solution is the given grid? We call this problem the Minimum Sudoku Clue Problem (MSCP). We formulate MSCP as a binary bilevel linear program, present a class of globally valid inequalities, and provide a computational study on 50 MSCP instances of 9 by 9 Sudoku grids. Using a general bilevel solver, we solve 95% of instances to optimality, and show that the solution process benefits from the addition of a moderate amount of inequalities. Finally, we extend the proposed model to other combinatorial problems in which uniqueness of the solution is of interest.

Locating network trees by a bilevel scheme

AbstractIn this paper we investigate how to choose an optimal position of a specific facility that is constrained to a network tree connecting some given demand points in a given area. A bilevel formulation is provided and existence results are given together with some properties when a density describes the construction cost of the networks in the area. This includes the presence of an obstacle or of free regions. To prove existence of a solution of the bilevel problem, that is framed in Euclidean spaces, a lower semicontinuity property is required. This is obtained proving an extension of Goła̧b’s theorem in the general setting of metric spaces, which allows for considering a density function.

Statistical inference of travelers’ route choice preferences with system-level data

Traditional network models encapsulate travel behavior among all origin–destination pairs based on a simplified and generic travelers’ utility function. Typically, the utility function consists of travel time solely, and its coefficients are equated to estimates obtained from discrete choice models and stated preference data. While this modeling strategy is reasonable, the inherent sampling bias in individual-level experimental data may be further amplified over network flow aggregation, leading to inaccurate flow estimates. In addition, individual-level data must be collected from surveys or travel diaries, which may be labor-intensive, costly, and limited to a small time horizon. To address these limitations, this study extends classical bi-level formulations to estimate travelers’ utility functions with multiple attributes using system-level data. This data tends to be less subject to sampling bias than individual-level data, it is cheaper to collect and it has become increasingly diverse and available. To leverage system-level data, we formulate a methodology grounded on non-linear least squares to statistically infer travelers’ utility function in the network context using traffic counts, traffic speeds, the number of traffic incidents, and sociodemographic information obtained from the US Census, among other attributes. The analysis of the mathematical properties of the optimization problem and its pseudo-convexity motivates the use of normalized gradient descent, an algorithm developed in the machine learning community that is suitable for pseudo-convex programs. More importantly, we develop a hypothesis test framework to examine the statistical properties of coefficients attached to utility terms and to perform attribute selection. Experiments on synthetic data show that the travelers’ utility function coefficients can be consistently recovered and that hypothesis tests are reliable statistics to identify which attributes are determinants of travelers’ route choices. Besides, a series of Monte-Carlo experiments showed that statistical inference is robust to various levels of sensor coverage and to noises in the Origin-Destination matrix and the traffic count measurements. The methodology is also deployed at a large scale using real-world multi-source data in Fresno, CA, collected before and during the COVID-19 outbreak.

Co-design of a multirotor UAV with robust control considering handling qualities and motor failure

Over the last few decades, multirotor UAVs have become significantly popular because of their various applications. However, multirotors suffer from poor endurance performance as compared to other aircraft configurations with lifting surfaces. This limitation can be addressed by improving the technology of components or by developing design optimization approaches at the vehicle level. This paper presents a co-design approach for the multirotor structure and propulsive system to improve the vehicle's energetic efficiency while respecting dynamic performance requirements. For this purpose, the approach uses a bi-level formulation to optimize the design and control concurrently. The methodology offers the possibility to activate or not a passive fault tolerant strategy in the controller synthesis to be tolerant to single rotor failure. In contrast to conventional design approaches that sequentially design each part, the co-design approach leads to a better optimal solution by considering the interactions and couplings between design and control. With the proposed co-design approach, the results show the soundness of the proposed design methodology and a significant reduction in the consumed energy compared to a reference non-optimal design while maintaining the same requirements in terms of dynamic performance.

A novel framework for the day-ahead market clearing process featuring the participation of distribution system operators and a hybrid pricing mechanism

The electricity market liberalization propelled by the increased operational flexibility observed in distribution systems urges the development of mathematical models for competitive electricity markets accounting for the participation of distribution systems operators. Most existing approaches model the interaction between the agents and the market as a bilevel programming problem, wherein the market is characterized at the lower level. Hence, the upper-level agent’s objective function is favored over minimizing the costs of supplying the system’s demand. In this paper, a new bilevel formulation is proposed. In the upper level, the market operator determines the optimal energy price that ensures the demand supply and maximizes the overall social welfare. In the lower level, the generation companies and distribution systems operators determine the amount of energy injected/extracted into/from the transmission system to maximize profits/minimize operational costs. A new pricing method is proposed in this paper as an alternative to the transmission system’s dual variables, which are inaccessible under the proposed framework. The resulting nonlinear mixed-integer bilevel programming formulation is transformed into an equivalent single-level mixed-integer linear program. Numerical results illustrate the proposed approach’s effectiveness in maintaining the minimum operational costs while distributing the payoff amongst the agents.

Adaptive Robust Data-Driven Building Control via Bilevel Reformulation: An Experimental Result

Data-driven control approaches for the minimization of energy consumption of buildings have the potential to significantly reduce deployment costs and increase the uptake of advanced control in this sector. A number of recent approaches based on the application of Willems’ fundamental lemma for data-driven controller design from input/output measurements are very promising for deterministic linear time-invariant (LTI) systems. This article proposes a systematic way to handle unknown measurement noise and measurable process noise, and extends these data-driven control schemes to adaptive building control via a robust bilevel formulation, whose upper level ensures robustness and whose lower level guarantees prediction quality. Corresponding numerical improvements and an active excitation mechanism are proposed to enable a computationally efficient reliable operation. The efficacy of the proposed scheme is validated by a multizone building simulation and a real-world experiment on a single-zone conference building on the École Polytechnique Fédérale de Lausanne (EPFL) campus. The real-world experiment includes a 20-day nonstop test, where, without extra modeling effort, our proposed controller improves 18.4% energy efficiency against an industry-standard controller while also robustly ensuring occupant comfort.

Iterative Backpropagation Method for Efficient Gradient Estimation in Bilevel Network Equilibrium Optimization Problems

Network optimization or network design with an embedded traffic assignment (TA) to model user equilibrium principle, sometimes expressed as bilevel problems or mathematical programs with equilibrium constraints (MPEC), is at the heart of transportation planning and operations. For applications to large-scale multimodal networks with high dimensional decision variables, the problem is nontrivial, to say the least. General-purpose algorithms and problem-specific bilevel formulations have been proposed in the past to solve small problems for demonstration purposes. Research gap, however, exists in developing efficient solution methods for large-scale problems in both static and dynamic contexts. This paper proposes an efficient gradient estimation method called Iterative Backpropagation (IB) for network optimization problems with an embedded static TA model. IB exploits the iterative structure of the TA solution procedure and simultaneously calculates the gradients while the TA process converges. IB does not require any additional function evaluation and consequently scales very well with higher dimensions. We apply the proposed approach to origin-destination (OD) estimation, an MPEC problem, of the Hong Kong multimodal network with 49,806 decision variables, 8,797 nodes, 18,207 links, 2,684 transit routes, and 165,509 OD pairs. The calibrated model performs well in matching the link counts. Specifically, the IB-gradient based optimization technique reduces the link volume squared error by 98%, mean absolute percentage error (MAPE) from 95.29% to 21.23%, and the average GEH statistics from 24.18 to 6.09 compared with the noncalibrated case. The framework, even though applied to OD estimation in this paper, is applicable to a wide variety of optimization problems with an embedded TA model, opening up an efficient way to solve large-scale MPEC or bilevel problems. Funding: The study is supported by IVADO Postdoctoral Fellowship scheme 2021, HSBC 150th Anniversary Charity Programme HKBF17RG01, National Science Foundation of China (71890970, 71890974), General Research Fund (16212819, 16207920) of the HKSAR Government, and the Hong Kong PhD Fellowship.

Algorithmic schemes for non-smooth simple bilevel programming problem and simple MPEC problem

ABSTRACT This article continues our study on simple bilevel and simple MPEC problems. In this article, we focus on developing algorithms. We show how using the idea of a gap function one can represent a simple MPEC as a simple bilevel problem with non-smooth data. This motivates us first to develop an algorithm for a simple bilevel problem with non-smooth data and modify the scheme for the same to develop an algorithm for the simple MPEC problem. We also discuss how the simple bilevel formulation of a simple MPEC can help us in formulating a stopping criteria for the simple MPEC problem.

Smart camera placement for building surveillance using OpenBIM and an efficient Bi-level optimization approach

Cameras have been widely used in buildings for safety and efficiency surveillance. Especially with the integration of artificial intelligence and building information modeling (BIM), many intelligent applications have emerged. However, optimizing the camera plan in buildings with various geometric and semantic properties remains challenging. This study proposes a smart camera placement framework that can be generally applied in different building scenarios to solve this problem. Firstly, a camera-integrated BIM is constructed based on a systematic literature review (SLR) and Industry Foundation Classes (IFC) schema, followed by a sample-based voxelization method to convert BIM models to semantic voxels. Secondly, a realistic camera coverage analysis method is developed to consider the object's properties, based on which a bi-level formulation is proposed. Finally, an efficient bi-level genetic algorithm (EBGA) is proposed to ensure both optimization accuracy and efficiency. The performance of the proposed placement framework is validated in a study of a multi-purpose laboratory and office floor. The validation demonstrated that EBGA obtained full coverage of essential objects and 21.4% more coverage values over the experience-based method. Compared with two benchmark GAs, EBGA saved 65.2% and 49.7% of computation time correspondingly in the experiment.

Single-Level Differentiable Contact Simulation

We present a differentiable formulation of rigid-body contact dynamics for objects and robots represented as compositions of convex primitives. Classical physics engines rely on non-differentiable collision detection modules. More recent optimization-based approaches simulating contact between convex primitives rely on a bilevel formulation that separates collision detection and contact simulation. These latest approaches are unreliable in realistic contact simulation scenarios because isolating the collision detection problem introduces contact location non-uniqueness. Our approach combines contact simulation and collision detection into a unified single-level optimization problem. This disambiguates the collision detection problem in a physics-informed manner. Our formulation features improved simulation robustness and a reduction in computational complexity when compared to a similar <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">differentiable</i> simulation baseline. We illustrate the contact and collision differentiability on a robotic manipulation task requiring optimization-through-contact. We provide a numerically efficient implementation of our formulation called <uri xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">https://github.com/simon-lc/Silico.jl</uri> Silico.jl.

Optimising Electric Vehicle Charging Station Placement Using Advanced Discrete Choice Models

We present a new model for finding the optimal placement of electric vehicle charging stations across a multiperiod time frame so as to maximise electric vehicle adoption. Via the use of stochastic discrete choice models and user classes, this work allows for a granular modelling of user attributes and their preferences in regard to charging station characteristics. We adopt a simulation approach and precompute error terms for each option available to users for a given number of scenarios. This results in a bilevel optimisation model that is, however, intractable for all but the simplest instances. Our major contribution is a reformulation into a maximum covering model, which uses the precomputed error terms to calculate the users covered by each charging station. This allows solutions to be found more efficiently than for the bilevel formulation. The maximum covering formulation remains intractable in some instances, so we propose rolling horizon, greedy, and greedy randomised adaptive search procedure heuristics to obtain good-quality solutions more efficiently. Extensive computational results are provided, and they compare the maximum covering formulation with the current state of the art for both exact solutions and the heuristic methods. History: Accepted by Andrea Lodi, Area Editor for Design &amp; Analysis of Algorithms–Discrete. Funding: This work was supported by Hydro-Québec and the Natural Sciences and Engineering Research Council of Canada [Discovery Grant 2017-06054; Collaborative Research and Development Grant CRDPJ 536757–19]. Supplemental Material: The online appendix is available at https://doi.org/10.1287/ijoc.2022.0185 .

Flexibility Pricing of Grid-Connected Energy Hubs in the Presence of Uncertain Energy Resources

The paper expresses the problem of flexibility pricing in energy hubs (EHs) that are in connection with electricity, heat, and gas networks considering of uncertain energy generation sources. Scheme includes a bilevel formulation. Its upper-level states for modeling of the flexibility services are provided by various resources within the EH. The problem considers maximization of the expected profit of these resources in the flexibility market. The problem constraints include the flexibility model of flexible resources such as storage devices, responsive loads, and controllable distributed generations (DGs). The flexibility model of resources relies on their active and heat power. The lower-level problem calculates energy and flexibility prices and formulates the flexible operation of energy resources considering EHs. Here, constraints include optimal power flow equations in the energy networks; operation model of EHs with power sources, storage devices, and different responsive loads; and flexibility limits of EHs. Also, a linear approximation model is adopted in the suggested design using conventional linearization techniques. Next, the Karush–Kuhn–Tucker (KKT) method is used to derive a single-level model for the problem. The scheme adopts scenario-based stochastic programming (SBSP) so that uncertainties of renewable power, energy price, load, and energy consumption of mobile storage devices are properly modeled. Finally, the results validate the suggested design’s potential in modifying and enhancing the operation, flexibility, and economic situation of energy networks and EHs.

Optimal bidding strategy of a gas-fired power plant in interdependent low-carbon electricity and natural gas markets

This work presents a bi-level optimization framework to determine the optimal bidding strategies for a strategic gas-fired power plant, exerting market power in interdependent pool-based electricity and natural gas markets, under a carbon emission trading scheme (CETS). The upper-level problem aims at maximizing the profits of the strategic player, while at the lower-level problem, the day-ahead electricity and natural gas markets are cleared sequentially, considering the provision of carbon emission allowances for conventional power producers and high penetration of wind power generation. The bi-level formulation is initially recast into a mathematical program with equilibrium constraints (MPEC), using the Karush-Kuhn-Tucker optimality conditions and duality theory, and is further reformulated into a mixed integer linear program. The proposed algorithm is applied to a Pennsylvania-New Jersey-Maryland (PJM) 5-bus power grid, incurred by transmission constraints and a single node natural gas network. Numerical simulations provide CETS-embedded electricity clearing prices and optimal bidding decisions for the strategic gas-fired power plant, under plausible power transmission congestions and natural gas prices increment scenarios.

Mathematical programming formulations for the Collapsed k-Core Problem

In social network analysis, the size of the k-core, i.e., the maximal induced subgraph of the network with minimum degree at least k, is frequently adopted as a typical metric to evaluate the cohesiveness of a community. We address the Collapsed k-Core Problem, which seeks to find a subset of b users, namely the most critical users of the network, the removal of which results in the smallest possible k-core. For the first time, both the problem of finding the k-core of a network and the Collapsed k-Core Problem are formulated using mathematical programming. On the one hand, we model the Collapsed k-Core Problem as a natural deletion-round-indexed Integer Linear formulation. On the other hand, we provide two bilevel programs for the problem, which differ in the way in which the k-core identification problem is formulated at the lower level. The first bilevel formulation is reformulated as a single-level sparse model, exploiting a Benders-like decomposition approach. To derive the second bilevel model, we provide a linear formulation for finding the k-core and use it to state the lower-level problem. We then dualize the lower level and obtain a compact Mixed-Integer Nonlinear single-level problem reformulation. We additionally derive a combinatorial lower bound on the value of the optimal solution and describe some pre-processing procedures, and valid inequalities for the three formulations. The performance of the proposed formulations is compared on a set of benchmarking instances with the existing state-of-the-art solver for mixed-integer bilevel problems proposed in (Fischetti, Ljubić, Monaci, and Sinnl, 2017).

Two-Stage Participation of Grid-Connected Energy Hubs in Day-Ahead and Real-Time Energy Markets Based on Robust Modeling of Uncertainties

Optimal management of sources, storage, and responsive loads in the energy network leads to improvement of economic and technical situation of the network. However, note that single-layer energy management requires to provide the network operator with more amounts of information. This results in the increase in the processing speed for the network operator. Also, coordination of sources, storage, and responsive loads in the form of an aggregator such as an energy hub can play an important role for them in promoting the economic situation and energy efficiency. So, this paper presents the participation of grid-connected energy hubs (EHs) in the day-ahead (DA) and real-time (RT) energy markets. Scheme has a two-stage optimization framework. In the first stage, participation of EHs in the hourly DA energy market is executed based on a two-layer energy management system that is modeled as bilevel problem. EH model in DA market is formulated in upper-level problem aimed at maximizing their profit subject to meeting the linearized constraints of the EH’s operation. Lower-level problem deals with the operation of energy networks to minimize cost of energy purchase from the upstream network constrained to linearized optimal power flow equations. The second stage of participation of EHs in a 5 min RT market has an objective function to minimize the unbalance between the EHs’ profits in the mentioned two markets. This stage is subject to the upper-level constraints and the lower-level formulation in the first stage. Robust optimization models the uncertainties of load, renewable power, energy price, and energy demand of mobile storage to achieve an optimal robust solution. Benders decomposition algorithm solves bilevel formulation in two-stage scheme. Numerical results confirm capabilities of the scheme in enhancing operation status of the networks and economic situation of EHs. Hence, the Benders decomposition algorithm is able to extract the optimal solution in shorter computation time. Two-layer energy management needs shorter computation time than the single-layer management. Simultaneous modeling of two real-time and day-ahead energy markets reduces the unbalance cost between these markets. The proposed scheme can enhance the operating and economic indicators of the network by up to around 18% and 26% compared to power flow studies of energy networks.

Robust capacitated Steiner trees and networks with uniform demands

AbstractWe are interested in the design of robust (or resilient) capacitated rooted Steiner networks in the case of terminals with uniform demands. Formally, we are given a graph, capacity, and cost functions on the edges, a root, a subset of vertices called terminals, and a bound on the number of possible edge failures. We first study the problem where and the network that we want to design must be a tree covering the root and the terminals: we give complexity results and propose models to optimize both the cost of the tree and the number of terminals disconnected from the root in the worst case of an edge failure, while respecting the capacity constraints on the edges. Secondly, we consider the problem of computing a minimum‐cost survivable network, that is, a network that covers the root and terminals even after the removal of any edges, while still respecting the capacity constraints on the edges. We also consider the possibility of protecting a given number of edges. We propose three different formulations: a bilevel formulation (with an attacker and a defender), a cutset‐based formulation and a flow‐based one. We compare the formulations from a theoretical point of view, and we propose algorithms to solve them and compare their efficiency in practice.

Charging Network Design and Service Pricing for Electric Vehicles With User-Equilibrium Decisions

This paper aims to investigate the electric vehicle (EV) charging network design and utilization management considering user-centric decisions. A hierarchical formulation is developed with the EV charging network design and demand-driven pricing scheme in the upper level and users’ charging decisions to minimize their own travel costs and charging expenses in the lower level. The model aims to minimize the facility deployment cost and maximize the charging income of the network operator while minimizing the user-centric costs. We have converted the proposed bi-level formulation into an equivalent single-level model using the lower-level objective function as complementary equations. Then, we have developed an iterative active-set based solution technique to determine the strategic decisions on charging network design. To partially overcome the computational burden, the arc travel times are estimated using a macroscopic fundamental diagram concept. The proposed integrated methodology is applied to a hypothetical and an empirical case study to evaluate its performance and solution quality. The numerical results indicate that the proposed algorithm can solve the problem efficiently and outperform a system-level bi-level optimization benchmark. Our experiments show a CPU time of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$2.3~hr$ </tex-math></inline-formula> for the proposed approach compared to <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$173.1~hr$ </tex-math></inline-formula> of the benchmark. Finally, a series of sensitivity analyses has been conducted to study the impact of input parameters on the solutions and draw managerial insights.

Optimal scheduling of multi-energy type virtual energy storage system in reconfigurable distribution networks for congestion management

The virtual energy storage system (VESS) is one of the emerging novel concepts among current energy storage systems (ESSs) due to the high effectiveness and reliability. In fact, VESS could store surplus energy and inject the energy during the shortages, at high power with larger capacities, compared to the conventional ESSs in smart grids. This study investigates the optimal operation of a multi-carrier VESS, including batteries, thermal energy storage (TES) systems, power to hydrogen (P2H) and hydrogen to power (H2P) technologies in hydrogen storage systems (HSS), and electric vehicles (EVs) in dynamic ESS. Further, demand response program (DRP) for electrical and thermal loads has been considered as a tool of VESS due to the similar behavior of physical ESS. In the market, three participants have considered such as electrical, thermal and hydrogen markets. In addition, the price uncertainties were calculated by means of scenarios as in stochastic programming, while the optimization process and the operational constraints were considered to calculate the operational costs in different ESSs. However, congestion in the power systems is often occurred due to the extreme load increments. Hence, this study proposes a bi-level formulation system, where independent system operators (ISO) manage the congestion in the upper level, while VESS operators deal with the financial goals in the lower level. Moreover, four case studies have considered to observe the effectiveness of each storage system and the simulation was modeled in the IEEE 33-bus system with CPLEX in GAMS.