Lower-level Problem Research Articles

Robust Design Optimization of Electric Machines with Isogeometric Analysis

In electric machine design, efficient methods for the optimization of the geometry and associated parameters are essential. Nowadays, it is necessary to address the uncertainty caused by manufacturing or material tolerances. This work presents a robust optimization strategy to address uncertainty in the design of a three-phase, six-pole permanent magnet synchronous motor (PMSM). The geometry is constructed in a two-dimensional framework within MATLAB®, employing isogeometric analysis (IGA) to enable flexible shape optimization. The main contributions of this research are twofold. First, we integrate shape optimization with parameter optimization to enhance the performance of PMSM designs. Second, we use robust optimization, which creates a min–max problem, to ensure that the motor maintains its performance when facing uncertainties. To solve this bilevel problem, we work with the maximal value functions of the lower-level maximization problems and apply a version of Danskin’s theorem for the computation of generalized derivatives. Additionally, the adjoint method is employed to efficiently solve the lower-level problems with gradient-based optimization. The paper concludes by presenting numerical results showcasing the efficacy of the proposed robust optimization framework. The results indicate that the optimized PMSM designs not only perform competitively compared to their non-robust counterparts but also show resilience to operational and manufacturing uncertainties, making them attractive for industrial applications.

A driving method for gray scale multiplication of electrowetting display based on hybrid modulation

In order to solve the problem of low gray level due to the few driving chips developed based on the photoelectric characteristics of electrowetting display, a driving method based on modulation is proposed to double the gray level of electrowetting display. In this method, the driving waveform based on pulse amplitude modulation and pulse width modulation hybrid modulation is designed, and the gray level–luminance curve of the electrowetting display is measured and analyzed. On this basis, the luminance nonlinear correction is carried out, and the improvement of 64 Gy levels to 127 Gy levels is realized by the principle of human visual persistence phenomenon. The experimental results show that the proposed driving scheme can break through the limitation of the driving chip and realize the multiplication of gray levels, in which 96% gray levels increase steadily with an average luminance difference of 0.07, and at the same time enhance the contrast and improve the display effect.

Interactive demand response and dynamic thermal line rating for minimizing the wind power spillage and carbon emissions

Spilling has already occurred as a result of rising the penetration of intermittent renewable generation, and it is anticipated that the level of renewable energy curtailment will continue to soar. This leads to an increase in operating costs, CO2 emissions, and not good utilization of renewable energy resources. A bi-level multi-objective optimization model is proposed in this paper to reduce wind power spillage (WPS) based on demand response (DR) and dynamic thermal line rating (DTLR). In the upper level, multiple objectives will be satisfied based on the optimal allocation and time of DR programs considering DTLR obtained in the lower level. The minimization of WPS, load shedding, power losses, and CO2 emissions are the objectives of this level. While the lower-level aims to maximize social welfare under different scenarios and overall system constraints. Under the uncertainty of the wind power and load demand, a collection of lower-level problems that represent the market clearing conditions is used to constrain the upper-level. The effectiveness of the proposed algorithm is examined on a modified two-area IEEE 24-bus test system. Results depict that the suggested bi-level model enables considerable reductions in the WPS by up to 32.7 %. Also, there is an enhancement in load shedding, power losses, and CO2 emissions by 28.93 %, 23.07 %, and 13.9 % respectively. Finally, the social welfare increased by up to 36.6 %.

Stackelberg Game-based Coordinative Dispatch of EV Charging Stations and Urban Distribution Systems Considering Temporal-spatial Dynamic Traffic Flow

This paper proposes a Stackelberg Game-based coordinative dispatch model of electric vehicle charging stations (EVCSs) and urban distribution systems (UDS) considering temporal-spatial dynamic traffic flow. The model is formulated hierarchically and consists of an upper-level problem and two lower-level problems. Firstly, a detailed EVCS dispatching model considering both electricity prices and charging service prices is developed in the upper-level. Then the electricity pricing model of UDS and travel decision model of EV users are proposed in the lower model. Particularly, a temporal-spatial dynamic model is developed to capture the traffic flow evolution in urban transport network preciously. EVCS, UDS operator (DSO) and EV users can make decisions independently as different stakeholders. Finally, a multi-layer iterative framework is designed to capture the tripartite benefit equilibrium of distribution grid, EV charging stations, and EV users, and to realize the effective solution of the model. Numerical studies based on a modified IEEE 33-bus system demonstrate the effectiveness and accuracy of our proposed approach.

The capacitated [formula omitted]-hub interdiction problem with congestion: Models and solution approaches

We study the r-hub interdiction problem under the case of possible congestion. Hub interdiction problems are modeled as attacker-defender problems to identify a set of r critical hubs from a set of p hubs, which when attacked, causes maximum damage to network restoration activities of the defender. In this work we consider that in addition to the routing cost, the defender also aims to minimize the congestion cost. Incorporating the congestion cost in the problem introduces non-linearity in the objective function of the interdiction problem, which makes the problem challenging to solve. To address this, we propose two alternate exact solution approaches. The first approach is an inner-approximation-based approach (IBA), which overestimates the convex non-linear objective function and provides an upper bound. A lower bound is obtained from solving the lower-level problem exactly corresponding to the upper bound solution. The upper bound is tightened using improved approximation with new points generated in successive iterations. In the second approach (referred to as SBA), the problem is reformulated as a second-order conic program, which can be solved using an off-the-shelf solver. From our computational experiments on benchmark datasets (CAB and AP), we demonstrate the efficacy of both the proposed methods. However, IBA consistently outperforms SBA by a significant margin.

Micro-grid tri-level EPEC based model for finding equilibria in the day-ahead and balancing markets

This paper presents a model that explores the strategic bidding equilibrium within a microgrid and its interactions with other strategic and non-strategic rivals in a joint energy and balancing market. The model utilizes a tri-level mathematical program with equilibrium constraints (MPEC) to represent the behavior of each strategic producer. Upper level maximizes the profit of each strategic producer, including microgrid and another strategic rival. The first lower-level problem involves maximizing social welfare through the day-ahead market clearing process, while the second lower-level problem deals with the balancing market clearing process. These objectives form the core of the proposed model. To simplify the tri-level problem, duality theory and the Karush-Kuhn-Tucker (KKT) optimally conditions are employed to transform it into a mixed integer linear programming (MILP) problem. By simultaneously solving all MPECs, an equilibrium problem with equilibrium constraints (EPEC) is formulated. The resulting EPEC is then addressed using a diagonalization algorithm and game theory to obtain a market Nash equilibrium, which constitutes the secondary objective of this paper. The effectiveness of the model is evaluated using the 6-bus test system as a case study. The results indicate that, at the equilibrium point, both the microgrid and rivals experience reduced profits compared to the initial state.

Inexact direct-search methods for bilevel optimization problems

In this work, we introduce new direct-search schemes for the solution of bilevel optimization (BO) problems. Our methods rely on a fixed accuracy blackbox oracle for the lower-level problem, and deal both with smooth and potentially nonsmooth true objectives. We thus analyze for the first time in the literature direct-search schemes in these settings, giving convergence guarantees to approximate stationary points, as well as complexity bounds in the smooth case. We also propose the first adaptation of mesh adaptive direct-search schemes for BO. Some preliminary numerical results on a standard set of bilevel optimization problems show the effectiveness of our new approaches.

Notes on the value function approach to multiobjective bilevel optimization

This paper is concerned with the value function approach to multiobjective bilevel optimization which exploits a lower-level frontier-type mapping in order to replace the hierarchical model of two interdependent multiobjective optimization problems by a single-level multiobjective optimization problem. As a starting point, different value-function-type reformulations are suggested and their relations are discussed. Here, we focus on the situations where the lower-level problem is solved up to efficiency or weak efficiency, and an intermediate solution concept is suggested as well. We study the graph-closedness of the associated efficiency-type and frontier-type mappings. These findings are then used for two purposes. First, we investigate existence results in multiobjective bilevel optimization. Second, for the derivation of necessary optimality conditions via the value function approach, it is inherent to differentiate frontier-type mappings in a generalized way. Here, we are concerned with the computation of upper coderivative estimates for the frontier-type mapping associated with the setting where the lower-level problem is solved up to weak efficiency. We proceed in two ways, relying, on the one hand, on a weak domination property and, on the other hand, on a scalarization approach. Illustrative examples visualize our findings and some flaws in the related literature.

Reinforcement learning based bilevel real-time pricing strategy for a smart grid with distributed energy resources

The integration of flexible loads, distributed energy resources, and other technologies is becoming common in advance power and energy systems. However, the integration also presents significant challenges due to the increasing complexity and uncertainty. To effectively manage these resources, an adaptive pricing mechanism is needed that can account for their variable availability. Based on this, we propose a new bilevel real-time pricing model that considers different distributed energy resources, uncertainty of renewable energy generation, carbon trading mechanisms, and grid fluctuations. Specifically, the upper-level optimization problem aims to maximize the total profit of the supplier that applies Q-learning algorithm. The lower-level optimization problem addresses the need for each user to make optimal power consumption decisions by constructing an individual Markov Decision Process framework for each user. The bilevel model achieves an effective balance of interests between the supplier and users by simultaneously considering both the upper-level and lower-level optimization problems. Additionally, our model can be efficiently solved using the distributed algorithm without the need to acquire transition probabilities. Simulation results show that the method is highly effective in balancing power supply and demand between the supplier and users, reducing carbon emissions, and mitigating power fluctuations.

Market design for ancillary service provisions of inertia and frequency response via virtual power plants: A non-convex bi-level optimisation approach

The Great Britain (GB) government is paving the path to decarbonisation by actively promoting the integration of wind power into its generation mix. This sharp transition to renewable energy, however, introduces specific challenges. The characteristics of non-synchronous wind turbines have the potential to impact grid stability and frequency security due to the reduction in system inertia. In response to these challenges, the deployment of virtual power plants (VPPs) is envisioned in deregulated power systems, which allow for the aggregation and coordinated control of diverse distributed energy resources, enhancing grid flexibility, reliability, and efficiency. Moreover, VPPs offer the provision of system inertia and frequency response services at the national level. This study drops this assumption and leverages the localised flexibility inherent in VPPs to meet the ancillary service requirements of the low-carbon power system. The formulated problem is constructed as a non-convex bi-level optimisation problem, where the upper-level problem represents the operation of a frequency-constrained unit commitment model, and the lower-level problem embodies the energy dispatches and frequency responses of a group of VPPs, guided by the dual price signals of cleared energy and ancillary services. To address the non-convex nature of the unit commitment problem, a two-fold approach is employed. First, binary commitment status decision variables are relaxed to continuous versions. Subsequently, the duality gap between the original non-convex unit commitment problem and its relaxed dual form is minimised, yielding a solution closely approximating the optimal solution of the original problem. Second, the relaxed bi-level optimisation problem is transformed into a single-level mathematical programs with equilibrium constraints by replacing the lower-level VPP problems with their equivalent Karush-Kuhn–Tucker optimality conditions. The case studies conducted in this work encompass a comprehensive scope. Initially, the study evaluates the effectiveness of VPPs in delivering ancillary services within the projected context of the GB power system for 2030. The findings reveal a substantial 28.28% decrease in system operation costs when compared to the benchmark case lacking VPP flexibility. Additionally, the integration of VPPs leads to significantly lower price signals for inertia, primary frequency response, and enhanced frequency response, reduced by 81.21%, 79.13%, and 86.27%, respectively, in contrast to the benchmark case. The analysis then delves into a sensitivity exploration, investigating the inherent flexibility of VPPs. The profit of local VPPs is expected to increase with VPP flexibility. Finally, the study illustrates how VPPs can adapt to and derive benefits from an increasing level of wind power penetration by providing a higher amount of frequency response.

Anticipated Motives for Gambling Treatment in Adults from the U.S.

Prior works note that identifying problematic play is a leading barrier to treatment seeking, contributing to low treatment rates in those with gambling problems (e.g., Bijker et al., 2022; Suurvali et al., 2012a). While research has looked at treatment seekers’ motives for treatment (Gainsbury et al., 2014; Suurvali et al., 2012b), the situations or signs (anticipated motives) individuals look for that suggest they would need treatment in the future remains unknown. Participants were gamblers (N = 1,791) from a census-matched U.S. sample of adults who reported no concerns about their gambling. Participants completed questions assessing how much money they would have to lose while gambling to think they might have a problem and what factors might motivate them to pursue treatment for gambling problems in the future. Participants reported a wide range of financial loss that would suggest they had a problem, and higher income men who gambled more frequently reported higher necessary losses. There was little variation in endorsement (40–60%) of 14 situations that may lead them to seek treatment in the future (e.g., felt guilty). However, income, gender identity, and problem gambling behavior were linked to the endorsement of some of these anticipated motives, with some differences in endorsement between those engaging in high- and low-level problem gambling. Collectively, results are consistent with the inference that many individuals may not be aware of what problematic gambling would look like for them, though income, gender identity, and problem gambling behavior may impact their consideration of anticipated motives.

Bilevel optimization with a multi-objective lower-level problem: risk-neutral and risk-averse formulations

In this work, we propose different formulations and gradient-based algorithms for deterministic and stochastic bilevel problems with conflicting objectives in the lower level. Such problems have received little attention in the deterministic case and have never been studied from a stochastic approximation viewpoint despite the recent advances in stochastic methods for single-level, bilevel, and multi-objective optimization. To solve bilevel problems with a multi-objective lower level, different approaches can be considered depending on the interpretation of the lower-level optimality. An optimistic formulation that was previously introduced for the deterministic case consists of minimizing the upper-level function over all non-dominated lower-level solutions. In this paper, we develop new risk-neutral and risk-averse formulations, address their main computational challenges, and develop the corresponding deterministic and stochastic gradient-based algorithms.

Penyuluhan Kesehatan Reproduksi Pada Wanita Menopause

Menopause is the period when menstruation stops which occurs in women aged 48 to 55 years. Menopause is a phase where women do not experience menstruation. It is hoped that the benefits and objectives of this counseling will be useful for improving and increasing knowledge about Reproductive Health in Menopausal Women. Menopause is a phase where women do not experience menstruation. Often women face menopause with anxiety and fear because they are entering old age and are no longer able to bear children. The method used in this program is to provide health education through counseling about Reproductive Health in Menopausal Women. Based on the results of the Self-Awareness Survey, there are several problems, namely the large number of teenagers who consume unhealthy food, teenagers who do not know about reproductive health, the lack of knowledge of pregnant women and mothers with toddlers regarding the contents of MCH books and the lack of knowledge of menopausal women in caring for reproductive health. The priority problem is the lack of knowledge of menopausal women in caring for reproductive health. With 110 people interviewed, there were 101 menopausal women who said they did not know about caring for reproductive health. From the SMD results, we found the problem of low levels of knowledge of reproductive health in menopausal women. To overcome this, we made one effort to provide education regarding reproductive health in menopausal women. After the counseling was carried out, participants looked enthusiastic about the counseling material presented and were able to practice how to care for reproductive health.

Transmission expansion planning in a wind-dominated power system: A closed-loop approach

The current paradigm for the transmission expansion planning (TEP) problem follows an open-loop (OL) approach. This approach first executes an accuracy-oriented forecasting method to predict the available wind power, demand, and spinning and non-spinning reserve (SR and NSR) requirements. Afterward, the TEP problem is solved based on these predictions. However, this OL model does not necessarily obtain a better expansion plan in terms of the real system cost (RSC) against the actual realization of wind and demand. This article contributes to the existing literature by developing a closed-loop (CL) strategy for the TEP problem using a cost-oriented prediction method. In this novel cost-oriented methodology, the RSC assesses the forecasting data quality rather than the statistical indices (accuracy-oriented). In this regard, a bilevel programming model is established to train the predicted data to minimize the system RSC. The upper level trains the predictors for wind power and SR/NSR requirements, while the two lower-level problems formulate the unit commitment (UC)-based TEP and the economic redispatch (ERD) problem. The bilevel programming that contains integer variables at both levels is solved by the reformulation and decomposition (R&D) technique. The numerical results on the IEEE 118-bus grid illustrate the cost-effective advantages of the CL-TEP method.

Solving Bilevel Programs Based on Lower-Level Mond-Weir Duality

This paper focuses on developing effective algorithms for solving a bilevel program. The most popular approach is to replace the lower-level problem with its Karush-Kuhn-Tucker conditions to generate a mathematical program with complementarity constraints (MPCC). However, MPCC does not satisfy the Mangasarian-Fromovitz constraint qualification (MFCQ) at any feasible point. In this paper, inspired by a recent work using the lower-level Wolfe duality (WDP), we apply the lower-level Mond-Weir duality to present a new reformulation, called MDP, for bilevel program. It is shown that, under mild assumptions, they are equivalent in globally or locally optimal sense. An example is given to show that, different from MPCC, MDP may satisfy the MFCQ at its feasible points. Relations among MDP, WDP, and MPCC are investigated. On this basis, we extend the MDP reformulation to present another new reformulation (called eMDP), which has similar properties to MDP. Furthermore, to compare two new reformulations with the MPCC and WDP approaches, we design a procedure to generate 150 tested problems randomly and comprehensive numerical experiments show that MDP has quite evident advantages over MPCC and WDP in terms of feasibility to the original bilevel programs, success efficiency, and average CPU time, whereas eMDP is far superior to all other three reformulations. History: Accepted by Pascal Van Hentenryck, Area Editor for Computational Modeling: Methods & Analysis. Funding: This work was supported by the National Natural Science Foundation of China [Grants 12071280 and 11901380]. Supplemental Material: The software that supports the findings of this study is available within the paper and its Supplemental Information ( https://pubsonline.informs.org/doi/suppl/10.1287/ijoc.2023.0108 ) as well as from the IJOC GitHub software repository ( https://github.com/INFORMSJoC/2023.0108 ). The complete IJOC Software and Data Repository is available at https://informsjoc.github.io/ .

Optimal bidding strategy of a wind power producer in Chinese spot market considering green certificate trading.

In China, wind power producers will participate in the spot market as strategic producers. They should submit offering prices and forecasted production to the independent system operator. Intraprovincial and interprovincial green certificate trading, as a mechanism to promote the development of wind power, is advanced in parallel with the spot market. Studying the bidding strategies of wind power producers in the spot market, especially with the introduction of intraprovincial and interprovincial green certificate trading, has great practical significance for the stable operation of wind power producers and the construction of a renewable energy-friendly electricity market. This paper proposes a bi-level stochastic model to simulate the bidding strategy of a wind power producer in the spot market. The upper level problem of the model describes the decision-making process of the wind power producer, and the lower level problems include the clearing process of the spot market and the TGC market, in which the spot market includes the day-ahead market and the balance market, and the TGC market includes the intraprovincial and interprovincial TGC markets. Then, using Karush-Kuhn-Tucker (KKT) conditions, the strong duality theorem, and the big M method, the bi-level model is recast into a mixed integer linear (MIL) model convenient to be solved. Finally, an example is given to analyze the wind power producer's optimal bidding strategy under different electricity market environments and different wind power penetration levels.

Efficient Surrogate Modeling Method for Evolutionary Algorithm to Solve Bilevel Optimization Problems.

The purpose of this study was to develop an evolutionary algorithm (EA) with bilevel surrogate modeling, called BL-SAEA, for tackling bilevel optimization problems (BLOPs), in which an upper level problem is to be solved subject to the optimality of a corresponding lower level problem. The motivation of this article is that the extensive lower level optimization required by each upper level solution consumes too many function evaluations, leading to poor optimization performance of EAs. To this end, during the upper level optimization, the BL-SAEA builds an upper level surrogate model to select several promising upper level solutions for the lower level optimization. Because only a small number of upper level solutions require the lower level optimization, the number of function evaluations can be considerably reduced. During the lower level optimization, the BL-SAEA constructs multiple lower level surrogate models to initialize the population of the lower level optimization, thus further decreasing the number of function evaluations. Experimental results on two widely used benchmarks and two real-world BLOPs demonstrate the superiority of our proposed algorithm over six state-of-the-art algorithms in terms of effectiveness and efficiency.

Risk and resilience-based restoration optimization of transportation infrastructures under uncertainty.

Disruptive events cause decreased functionality of transportation infrastructures and enormous financial losses. An effective way to reduce the effects of negative consequences is to establish an optimal restoration plan, which is recognized as a method for resilience enhancement and risk reduction in the transportation system. This study takes the total travel time as the resilience measure to formulate a bilevel optimization model for a given scenario. However, the uncertainties involved in restoration activities cannot be overlooked. In this context, the inherent uncertainty is represented with a set of scenarios generated via the Latin hypercube technique. To assess the risk under uncertainty, a conditional value at risk with regret (CVaR-R) measure is introduced when considering the existence of worst-case scenarios. Then, the bilevel programming model is transformed from the deterministic case to the stochastic case, where the upper-level problem determines the restoration sequence to minimize CVaR-R and the lower-level problem is a traffic assignment problem. An integrated framework based on a novel genetic algorithm and the Frank-Wolfe algorithm is designed to solve the stochastic model. Numerical experiments are conducted to demonstrate the properties of the proposed bilevel programming model and the performance of the solution algorithm. The proposed methodology provides new insights into the restoration optimization problem, which provides a reference for emergency decision-making.

Robust PID controller tuning using bilevel optimization

Tuning PID controllers for performance while satisfying robustness constraints is a difficult optimization problem. Previous authors have resorted to fine frequency discretization or global optimization algorithms to ensure robustness constraint satisfaction. In this work, we propose to formulate the tuning problem as a bilevel optimization problem, where the upper level cost function quantifies control performance, and the lower level problem defines frequencies at which robustness must be ensured. Here, the first-order optimality conditions for the peaks of the sensitivity functions (the lower level) are embedded as constraints to the performance optimization problem. Automatic differentiation is used to construct the problem and for efficient numerical solving, with convergence typically achieved in the order of 0.1 seconds. The methodology is applied to two case studies, where it is found to be suitable to improve existing controller designs both in terms of performance and robustness.

Capturing Electricity Market Dynamics in Strategic Market Participation Using Neural Network Constrained Optimization

In competitive electricity markets, the optimal bid or offer problem of a strategic agent is commonly formulated as a bi-level program and solved as a mathematical program with equilibrium constraints (MPEC). If the lower-level part of the problem can be well approximated as a convex problem, this approach leads to a global optimum. However, electricity markets are governed by non-convex (partially known) constraints and reward functions of the participating agents. In this paper, an alternative data-driven paradigm, labeled as a mathematical program with neural network constraint (MPNNC), is developed. The method uses a neural network to represent the mapping between the upper-level (agent) decisions and the lower-level (market) outcomes, i.e., it replaces the lower-level problem with a surrogate model. In the presented case studies, the proposed model is used to find the optimal load shedding strategy of a strategic load-serving entity. First, the MPNNC performance is compared to the MPEC approach, both in convex and non-convex environments, showing that the proposed MPNNC achieves similar performance to an ideal MPEC that has perfect knowledge of the simulated market environment. Then, aggregated supply curves from the Belgian spot exchange are used to assess the potential gains of using the developed model in real-life applications.