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Strong Stationarity Conditions for the Optimal Control of a Cahn-Hilliard-Navier-Stokes System

APPLIED MATHEMATICS AND OPTIMIZATION 2024年第1期89卷 12-12页

作者： Hintermueller, Michael Keil, Tobias Weierstrass Inst Appl Anal & Stochast Mohrenstr 39 D-10117 Berlin Germany Humboldt Univ Inst Math Unter Linden 6 D-10099 Berlin Germany

This paper is concerned with the distributed optimal control of a time-discrete Cahn-Hilliard-Navier-Stokes system with variable densities. It focuses on the double-obstacle potential which yields an optimal control problem for a variational inequality of fourth order and the Navier-Stokes equation. The existence of solutions to the primal system and of optimal controls is established. The Lipschitz continuity of the constraint mapping is derived and used to characterize the directional derivative of the constraint mapping via a system of variational inequalities and partial differential equations. Finally, strong stationarity conditions are presented following an approach from Mignot and Puel.

关键词： Cahn-Hilliard Strong stationarity mathematical programming with equilibrium constraints Navier-Stokes Non-matched densities Non-smooth potentials Optimal control Semidiscretization in time Directional differentiability

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Optimal bidding of plug-in electric vehicle aggregator in day-ahead and regulation markets

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INTERNATIONAL JOURNAL OF ELECTRIC AND HYBRID VEHICLES 2015年第3期7卷 209-232页

作者： Vaya, Marina Gonzalez Andersson, Goeran Swiss Fed Inst Technol Power Syst Lab CH-8092 Zurich Switzerland

We address the optimal bidding problem of an aggregator that manages the charging of a group of plug-in electric vehicles. The aggregator purchases energy in the day-ahead market, and offers capacity in the frequency regulation market. The charging flexibility of the fleet is expressed as a set of time-varying power and energy constraints. The impact of the aggregator's demand on dayahead market prices is endogenously modelled by formulating the problem as a bilevel problem: The upper level represents the aggregator's cost minimisation, and the lower level represents the market clearing. We explore several settings of the regulation market, analysing the impact of restrictions that may apply on the capacity offers (e.g., symmetric bids). Moreover, we compare the options of providing regulation with unidirectional charging, and with vehicle-to-grid (V2G). We test the proposed method in a case study for Switzerland, with historical day-ahead and regulation market data, and realistic driving patterns.

关键词： electric vehicles aggregator smart-charging demand side management ancillary services frequency regulation strategic bidding bilevel optimisation MPEC mathematical programming with equilibrium constraints

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Implementing Maximum Likelihood Estimation of Empirical Matching Models MPEC versus NFXP

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COMPUTATIONAL ECONOMICS 2022年第1期59卷 71-102页

作者： Dong, Baiyu Hsieh, Yu-Wei Zhang, Xing Univ Southern Calif Dept Econ Los Angeles CA 90007 USA Microsoft Corp Redmond WA 98052 USA

We propose two mathematical programming with equilibrium constraints (MPEC) formulations: the MPEC-Sparse and the MPEC-Dense to estimate a class of separable matching models. We compare MPEC with the Nested Fixed-Point (NFXP) algorithm-a well-received method in the literature of structural estimation. Using both simulated and actual data, we find that MPEC is more robust than NFXP in terms of convergence and solution quality. In terms of computing time, MPEC-Dense is 9 to 20 times faster than NFXP in simulations. For practitioners, MPEC is considerably simpler to program.

关键词： Aggregate matching Two-sided matching Separable matching models mathematical programming with equilibrium constraints Nested fixed-point algorithm Constrained optimization

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Generalized stationary points and an interior-point method for mathematical programs with equilibrium constraints

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mathematical programming 2004年第1期101卷 231-261页

作者： Liu, XW Sun, J Natl Univ Singapore Sch Business Singapore 117548 Singapore Hebei Univ Technol Dept Appl Math Tianjin Peoples R China Natl Univ Singapore Singapore MIT Alliance Singapore 117548 Singapore

Generalized stationary points of the mathematical program with equilibrium constraints (MPEC) are studied to better describe the limit points produced by interior point methods for MPEC. A primal-dual interior-point method is then proposed, which solves a sequence of relaxed barrier problems derived from MPEC. Global convergence results are deduced under fairly general conditions other than strict complementarity or the linear independence constraint qualification for MPEC (MPEC-LICQ). It is shown that every limit point of the generated sequence is a strong stationary point of MPEC if the penalty parameter of the merit function is bounded. Otherwise, a point with certain stationarity can be obtained. Preliminary numerical results are reported. which include a case analyzed by Leyffer for which the penalty interior-point algorithm failed to find a stationary point.

关键词： global convergence interior-point methods mathematical programming with equilibrium constraints stationary point

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An MPEC approach to second-best toll pricing

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mathematical programming 2004年第1期101卷 33-55页

作者： Lawphongpanich, S Hearn, DW Univ Florida Dept Ind & Syst Engn Gainesville FL 32611 USA

This paper addresses two second-best toll pricing problems, one with fixed and the other with elastic travel demands, as mathematical programs with equilibrium constraints. Several equivalent nonlinear programming formulations for the two problems are discussed. One formulation leads to properties that are of interest to transportation economists. Another produces an algorithm that is capable of solving large problems and easy to implement with existing software for linear and nonlinear programming problems. Numerical results using transportation networks from the literature are also presented.

关键词： congestion pricing traffic equilibrium mathematical programming with equilibrium constraints

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Adaptive grids for the estimation of dynamic models

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QME-QUANTITATIVE MARKETING AND ECONOMICS 2022年第2期20卷 179-238页

作者： Lanz, Andreas Reich, Gregor Wilms, Ole HEC Paris Jouy En Josas France Tsumcor Res AG Schwerzenbach Switzerland Univ Hamburg Hamburg Germany Tilburg Univ Tilburg Netherlands

This paper develops a method to flexibly adapt interpolation grids of value function approximations in the estimation of dynamic models using either NFXP (Rust, Econometrica: Journal of the Econometric Society, 55, 999-1033, 1987) or MPEC (Su & Judd, Econometrica: Journal of the Econometric Society, 80, 2213-2230, 2012). Since MPEC requires the grid structure for the value function approximation to be hard-coded into the constraints, one cannot apply iterative node insertion for grid refinement;for NFXP, grid adaption by (iteratively) inserting new grid nodes will generally lead to discontinuous likelihood functions. Therefore, we show how to continuously adapt the grid by moving the nodes, a technique referred to as r-adaption. We demonstrate how to obtain optimal grids based on the balanced error principle, and implement this approach by including additional constraints to the likelihood maximization problem. The method is applied to two models: (i) the bus engine replacement model (Rust, 1987), modified to feature a continuous mileage state, and (ii) to a dynamic model of content consumption using original data from one of the world's leading user-generated content networks in the domain of music.

关键词： Numerical dynamic programming mathematical programming with equilibrium constraints r-adaptive grid refinement Equi-oscillation

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An MPEC Model for the Optimal Operation of Unbalanced Three-phase Distribution Systems

An MPEC Model for the Optimal Operation of Unbalanced Three-...

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14th IEEE Madrid PowerTech Conference (IEEE POWERTECH)

作者： Lopez, Juan Camilo Arias, Nataly Banol Rider, Marcos J. State Univ Campinas UNICAMP Sch Elect & Comp Engn FEEC Campinas SP Brazil

ISBN: (纸本)9781665435970

In this paper, a mathematical programming with equilibrium constraints (MPEC) model for the optimal operation of unbalanced three-phase electrical distributed systems is presented. First, the problem is formulated as a non-linear programming (NLP) problem, considering photo-voltaic (PV) generation, dispatchable distributed generation (DG) units, and day-ahead minimization of total active power losses. Then, through a set of efficient linearization techniques, the unbalanced three-phase AC power flow is transformed into a linear programming (LP) model. Finally, through Karush-Kuhn-Tucker (KKT) conditions, the proposed MPEC model is formulated as the combination of the LP primal constraints, the LP dual constraints and complementary constraints. MPEC models are able to optimize without an explicit objective function. Thus, they could be used as inner constraints in bi-level problems, for formulating robust optimization models and in electricity market design for distribution systems. Results show that the proposed MPEC model obtains the same optimal solution of the primal LP problem, without an explicit objective function.

关键词： Complementary constraints Karush-Kuhn-Tucker conditions mathematical programming with equilibrium constraints unbalanced three-phase electrical distributed systems

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An MPEC approach to second-best toll pricing

An MPEC approach to second-best toll pricing

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3rd International Conference on Complementarity Problems

作者： Lawphongpanich, S Hearn, DW Univ Florida Dept Ind & Syst Engn Gainesville FL 32611 USA

关键词： congestion pricing traffic equilibrium mathematical programming with equilibrium constraints

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Machine Learning for Optimal Inverter Operation in Distribution Grids 55

Machine Learning for Optimal Inverter Operation in Distribut...

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55th Annual Conference on Information Sciences and Systems (CISS)

作者： Singh, Manish K. Gupta, Sarthak Kekatos, Vassilis Virginia Tech Bradley Dept Elect & Comp Engn Blacksburg VA 24061 USA

ISBN: (纸本)9781665412681

Smart inverters have been advocated as a fast-responding voltage control solution in distribution grids with renewables. Nevertheless, finding the optimal inverter setpoints for reactive power in near real time can be computationally and communication-wise taxing. We put forth two deep neural network (DNN)-based approaches to facilitate inverter control. The first approach is a learning-to-optimize scheme, where a DNN is used by the electric utility to predict the optimal power flow (OPF) minimizers once presented with the new grid loading conditions. The novel feature here is that this DNN is trained to match not only the minimizers, but also their sensitivities with respect to the OPF problem parameters. This is a unique feature for the learning-to-optimize setup yielding a dramatic improvement on data efficiency. The second approach targets a decentralized control scheme, where inverter control curves are modeled as DNNs that are jointly trained through an OPF formulation via stochastic primal-dual updates. Thanks to a flexible DNN architecture, the inverters can be partially driven by a utility control signal depending on the available communication specifications. © 2021 IEEE.

关键词： mathematical programming with equilibrium constraints bilevel programming locational marginal prices

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Data Driven Solution to Market equilibrium via Deep Reinforcement Learning 2

Data Driven Solution to Market Equilibrium via Deep Reinforc...

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IEEE 2nd International Conference on Power Science and Technology (ICPST)

作者： Wen, Lin Wang, Jianxiao Lin, Li Zou, Yang Gao, Feng Hong, Qiteng North China Elect Power Univ Sch Elect & Elect Engn Beijing Peoples R China Peking Univ Natl Engn Lab Big Data Anal & Applicat Beijing Peoples R China Peking Univ Coll Engn Dept Ind Engn & Management Beijing Peoples R China Univ Strathclyde Dept Elect & Elect Engn Glasgow Lanark Scotland

ISBN: (纸本)9798350349047;9798350349030

equilibrium analysis is an effective tool to analyze the operation efficiency of the electricity market, but how to efficiently solve the power market equilibrium model with multiagent participation remains to be further studied. The traditional model transformation method has low computational efficiency and poor ability to deal with strong uncertainty. The optimality and convergence of the solution are not fully guaranteed. Therefore, this paper intends to use a multi-agent deep reinforcement learning algorithm to reach a fast solution to the market equilibrium model. Here we take the interaction between multiple virtual power plants (VPPs) and the upstream power grid as an example. A bi-level mathematical programming with equilibrium constraints (MPEC) is constructed. The upper level aims to minimize the cost of running the VPPs, the lower level aims to minimize the cost of running the power grid. A multi-agent reinforcement learning algorithm is employed to solve the MPEC based on deep deterministic policy gradient (DDPG), which is a data-driven, selflearning and model-free approach without the modeling and computational complexity caused by existing methods. Finally, an example of iteration between an IEEE-30 node power grid system and a VPP is given to verify the rationality and effectiveness of the proposed algorithm.

关键词： Market equilibrium mathematical programming with equilibrium constraints reinforcement learning virtual power plant

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