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A column generation algorithm for the estimation of origin-destination matrices in congested traffic networks

EUROPEAN JOURNAL OF OPERATIONAL RESEARCH 2008年第3期184卷 860-878页

作者： Garcia-Rodenas, Ricardo Verastegui-Rayo, Doroteo Univ Castilla La Mancha Escuela Super Informat E-13071 Ciudad Real Spain Univ Castilla La Mancha Escuela Univ Politecn E-13071 Ciudad Real Spain

The general problem of estimating origin-destination (O-D) matrices in congested traffic networks is formulated as a mathematical programme with equilibrium constraints, referred to as the demand adjustment problem (DAP). This approach integrates the O-D matrix estimation and the network equilibrium assignment into one process. In this paper, a column generation algorithm for the DAP is presented. This algorithm iteratively solves a deterministic user equilibrium model for a given O-D matrix and a DAP restricted to the previously generated paths, whose solution generates a new O-D trip matrix estimation. The restricted DAP is formulated via a single level optimization problem. The convergence on local minimum of the proposed algorithm requires only the continuity of the link travel cost functions and the gauges used in the definition of the DAP. In order to analyze the convergence and performance of the proposed algorithm, various numerical tests were carried out on small scale problems. (C) 2007 Elsevier B.V. All rights reserved.

关键词： traffic adjustment of origin-destination trip matrix mathematical programming with equilibrium constraints large scale optimization robustness and sensitivity analysis

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DUAL-WEIGHTED GOAL-ORIENTED ADAPTIVE FINITE ELEMENTS FOR OPTIMAL CONTROL OF ELLIPTIC VARIATIONAL INEQUALITIES*

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ESAIM-CONTROL OPTIMISATION AND CALCULUS OF VARIATIONS 2014年第2期20卷 524-546页

作者： Hintermueller, M. Hoppe, R. H. W. Loebhard, C. Humboldt Univ Dept Math D-10099 Berlin Germany Univ Houston Dept Math Houston TX 77204 USA Univ Augsburg Inst Math D-86152 Augsburg Germany

A dual-weighted residual approach for goal-oriented adaptive finite elements for a class of optimal control problems for elliptic variational inequalities is studied. The development is based on the concept of C-stationarity. The overall error representation depends on primal residuals weighted by approximate dual quantities and vice versa as well as various complementarity mismatch errors. Also, a priori bounds for C-stationary points and associated multipliers are derived. Details on the numerical realization of the adaptive concept are provided and a report on numerical tests including the critical cases of biactivity are presented.

关键词： Adaptive finite element method C-stationarity goal-oriented error estimation mathematical programming with equilibrium constraints optimal control of variational inequalities.

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How much is enough? Optimal support payments in a renewable-rich power system

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ENERGY 2016年第Part1期117卷 300-313页

作者： Rintamaki, Tuomas Siddiqui, Afzal S. Salo, Ahti Aalto Univ Sch Sci Dept Math & Syst Anal Syst Anal Lab Helsinki Finland UCL Dept Stat Sci London England Stockholm Univ Dept Comp & Syst Sci S-10691 Stockholm Sweden

The large-scale deployment of intermittent renewable energy sources may cause substantial power imbalance. Together with the transmission grid congestion caused by the remoteness of these sources from load centers, this creates a need for fast-adjusting conventional capacity such as gas-fired plants. However, these plants have become unprofitable because of lower power prices due to the zero marginal costs of renewables. Consequently, policymakers are proposing new measures for mitigating balancing costs and securing supply. In this paper, we take the perspective of the regulator to assess the effectiveness of support payments to flexible generators. Using data on the German power system, we implement a bi-level programming model, which shows that such payments for gas-fired plants in southern Germany reduce balancing costs and can be used as part of policy to integrate renewable energy. (C) 2016 Elsevier Ltd. All rights reserved.

关键词： Renewable energy Support payments Day-ahead market Balancing market Congestion management mathematical programming with equilibrium constraints

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Distribution LMP-Based Demand Management in Industrial Park via a Bi-Level programming Approach

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IEEE TRANSACTIONS ON SUSTAINABLE ENERGY 2021年第3期12卷 1695-1706页

作者： Wei, Jingdong Zhang, Yao Wang, Jianxue Wu, Lei Xi An Jiao Tong Univ Sch Elect Engn Xian 710049 Peoples R China Stevens Inst Technol Elect & Comp Engn Dept Hoboken NJ 07030 USA

To enhance industrial park's economic gains and effectively allocate its electricity bill among industrial users with combined heat and power (CHP) units and photovoltaic (PV) panels, this paper proposes a distribution locational marginal price (DLMP)-based bi-level demand management approach. The upper level optimizes dispatching decisions of industrial users with the objective of minimizing their energy bills, and the lower level is a DLMP-based market clearing problem to minimize the two-part tariff cost of the industrial park operator. In order to solve the proposed bi-level model efficiently, it is first equivalently converted into a single-level mathematical programming with equilibrium constraints (MPEC), and then reformulated as a mixed-integer second-order conic programming (MISOCP) model by linearizing bilinear terms. Numerical results demonstrate the effectiveness of our proposed bi-level method in lowering industrial park's electricity bill and achieving effective allocation among users.

关键词： Cogeneration Reactive power Tariffs programming Energy consumption Resistance heating Natural gas Bi-level optimization combined heat and power distribution locational marginal price mathematical programming with equilibrium constraints mixed-integer second-order conic programming solar photovoltaics

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Solving multi-leader-common-follower games

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OPTIMIZATION METHODS & SOFTWARE 2010年第4期25卷 601-623页

作者： Leyffer, Sven Munson, Todd Argonne Natl Lab Math & Comp Sci Div Argonne IL 60439 USA

Multi-leader-common-follower games arise when modelling two or more competitive firms, the leaders, that commit to their decisions prior to another group of competitive firms, the followers, that react to the decisions made by the leaders. These problems lead in a natural way to equilibrium problems with equilibrium constraints (EPECs). We develop a characterization of the solution sets for these problems and examine a variety of nonlinear optimization and nonlinear complementarity formulations of EPECs. We distinguish two broad cases: problems where the leaders can cost-differentiate and problems with price-consistent followers. We demonstrate the practical viability of our approach by solving a range of medium-sized test problems.

关键词： Nash games Stackelberg games nonlinear programming nonlinear complementarity problems mathematical programming with equilibrium constraints equilibrium problems with equilibrium constraints

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Strategic Generation Investment in Energy Markets: A Multiparametric programming Approach

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IEEE TRANSACTIONS ON POWER SYSTEMS 2022年第4期37卷 2590-2600页

作者： Taheri, Sina Kekatos, Vassilis Veeramachaneni, Sriharsha Virginia Tech Bradley Dept ECE Blacksburg VA 24061 USA NextEra Analyt St Paul MN 55107 USA

An investor has to carefully select the location and size of new generation units it intends to build, since adding capacity in a market affects the profit from units this investor may already own. To capture this closed-loop characteristic, strategic investment (SI) of generation can be posed as a bilevel optimization. By analytically studying a small market, we first show that its objective function can be non-convex and discontinuous. Realizing that existing mixed-integer problem formulations become impractical for larger markets and number of instances, this work put forth two SI solvers: a grid search to handle setups where the candidate investment locations are few, and a stochastic gradient descent approach for otherwise. Both solvers leverage powerful results of multiparametric programming (MPP), each in a unique way. The grid search entails finding the primal/dual solutions for a large number of optimal power flow (OPF) problems, which nonetheless can be efficiently computed several at once thanks to the properties of MPP. The same properties facilitate the rapid calculation of gradients in a mini-batch fashion, thus accelerating the implementation of a stochastic (sub)-gradient descent search. Tests on the IEEE 30- and 118-bus systems using real-world data corroborate the advantages of the novel solvers.

关键词： Investment Task analysis Costs Schedules Generators programming Optimization mathematical programming with equilibrium constraints bilevel programming locational marginal prices stochastic gradient descent strategic investment multiparametric programming

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Optimal distance tolls under congestion pricing and continuously distributed value of time

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TRANSPORTATION RESEARCH PART E-LOGISTICS AND TRANSPORTATION REVIEW 2012年第5期48卷 937-957页

作者： Meng, Qiang Liu, Zhiyuan Wang, Shuaian Natl Univ Singapore Dept Civil & Environm Engn Singapore 117576 Singapore

This paper addresses the optimal distance-based toll design problem for cordon-based congestion pricing schemes. The optimal distance tolls are determined by a positive and non-decreasing toll-charge function with respect to the travel distance. Each feasible toll-charge function is evaluated by a probit-based SUE (Stochastic User equilibrium) problem with elastic demand, asymmetric link travel time functions, and continuously distributed VOT, solved by a convergent Cost Averaging (CA) method. The toll design problem is formulated as a mixed-integer mathematical programming with equilibrium constraints (MPEC) model, which is solved by a Hybrid GA (Genetic Algorithm) CA method. Finally, the proposed models and algorithms are assessed by two numerical examples. Crown Copyright (C) 2012 Published by Elsevier Ltd. All rights reserved.

关键词： Distance-based toll Cordon-based congestion pricing Stochastic user equilibrium Continuously distributed value-of-time mathematical programming with equilibrium constraints Genetic algorithm

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Adjustment of the link travel-time functions in traffic equilibrium assignment models

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TRANSPORTMETRICA A-TRANSPORT SCIENCE 2013年第9期9卷 798-824页

作者： Garcia-Rodenas, Ricardo Verastegui-Rayo, Doroteo Univ Castilla La Mancha Dept Math E-13071 Ciudad Real Spain

The network calibration problem (NCP) consists of adjusting (or estimating) the parameters of the link travel-time functions in a congested traffic network via the use of traffic link counts and equilibrium costs in a set of origin-destination demand pairs. In this article, a mathematical program with equilibrium constraints (MPEC) is proposed to model this problem. The proposed model is called NCP() and supposes that the available information has been obtained in several different periods (each period is defined by a single origin-destination demand matrix). The existence of the solutions to the stated NCP() under weak conditions in the link travel-time functions is established, and a sensitivity analysis of these solutions is carried out. A column generation algorithm (CGA) is proposed to solve the NCP() model, using sensitivity analysis to prove that this algorithm converges to a local optimum of the MPEC problem. Finally, the special case of estimating the link travel-time functions in additive models when information is only available for a single period is analysed in a numerical study. In this case the NCP() model is overspecified, that is, it has an infinite number of solutions. For this reason we propose, as an alternative, estimating the congestion costs in a subset of links, rather than estimating the parameterisation of the link travel-time functions . This special case has been named the NCP() model. The numerical study has been carried out in small and medium-sized networks, with the objective of studying both the proposed MPEC models and the computational aspects of the CGA.

关键词： calibration traffic assignment models mathematical programming with equilibrium constraints column generation procedures multicommodity network flow problems

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DISTRIBUTED OPTIMAL CONTROL OF THE CAHN-HILLIARD SYSTEM INCLUDING THE CASE OF A DOUBLE-OBSTACLE HOMOGENEOUS FREE ENERGY DENSITY

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SIAM JOURNAL ON CONTROL AND OPTIMIZATION 2012年第1期50卷 388-418页

作者： Hintermueller, M. Wegner, D. Humboldt Univ Dept Math D-10099 Berlin Germany Karl Franzens Univ Graz Dept Math & Sci Comp Graz Austria

In this paper we study the distributed optimal control for the Cahn-Hilliard system. A general class of free energy potentials is allowed which, in particular, includes the double-obstacle potential. The latter potential yields an optimal control problem of a parabolic variational inequality which is of fourth order in space. We show the existence of optimal controls to approximating problems where the potential is replaced by a mollified version of its Moreau-Yosida approximation. Corresponding first-order optimality conditions for the mollified problems are given. For this purpose a new result on the continuous Frechet differentiability of superposition operators with values in Sobolev spaces is established. Besides the convergence of optimal controls of the mollified problems to an optimal control of the original problem, we also derive first-order optimality conditions for the original problem by a limit process. The newly derived stationarity system corresponds to a function space version of C-stationarity.

关键词： Cahn-Hilliard system double-obstacle potential mathematical programming with equilibrium constraints distributed optimal control Yosida regularization C-stationarity

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Capacity Withholding Constrained by Operational Limits of Generation Under Financial Virtual Divestiture in a Day-Ahead Market

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IEEE TRANSACTIONS ON POWER SYSTEMS 2018年第1期33卷 771-780页

作者： Ameri, Massoud Rahimiyan, Morteza Latify, Mohammad Amin Shahrood Univ Technol Dept Elect Engn Shahrood *** Iran Isfahan Univ Technol Dept Elect & Comp Engn Esfahan *** Iran

This paper develops an equilibrium-based model for ex ante analysis of capacity withholding under financial virtual divestiture in a day-ahead market, including one dominant producer and a number of small producers considered as a price-taking competitive fringe. We model the offering problem of the dominant producer as a bilevel optimization problem. In the upper-level problem, the dominant producer derives optimal strategy of physical capacity withholding restricted by operational limits of its own generating units and conditions stated in the financial virtual divestiture. In the lower-level problem, market operator clears the day-ahead market on an hourly basis. We recast the bilevel optimization problem as a mathematical programming with equilibrium constraints problem and convert it to a mixed integer linear programming problem. We show how the application of the financial virtual divestiture may mitigate market power in the day-ahead market, through a case study. The numerical results demonstrate that the dominant producer can exploit technological diversification of its own generating units to exercise market power, even in the presence of this market power mitigation policy.

关键词： Financial virtual divestiture market power mathematical programming with equilibrium constraints

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