Computational Optimization of Systems Governed by Partial Differential Equations

Borzì, Alfio; Schulz, Volker

doi:10.1137/1.9781611972054

Cited by 236 publications

(358 citation statements)

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“…For the static wake model, gradients are obtained using finite differencing, which can be implemented efficiently because there are only N control variables per minimization. For the dynamic wake model, gradients are obtained using one backward simulation of the adjoint equations of the wake model using the formal Lagrangian method (Goit and Meyers, 2015;Borzì and Schulz, 2011). The full procedure is detailed in Shapiro et al (2017a).…”

Section: Controller Designsmentioning

confidence: 99%

Wind farms providing secondary frequency regulation: evaluating the performance of model-based receding horizon control

et al. 2018

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Abstract. This paper is an extended version of our paper presented at the 2016 TORQUE conference . We investigate the use of wind farms to provide secondary frequency regulation for a power grid using a model-based receding horizon control framework. In order to enable real-time implementation, the control actions are computed based on a time-varying one-dimensional wake model. This model describes wake advection and wake interactions, both of which play an important role in wind farm power production. In order to test the control strategy, it is implemented in a large-eddy simulation (LES) model of an 84-turbine wind farm using the actuator disk turbine representation. Rotor-averaged velocity measurements at each turbine are used to provide feedback for error correction. The importance of including the dynamics of wake advection in the underlying wake model is tested by comparing the performance of this dynamic-model control approach to a comparable static-model control approach that relies on a modified Jensen model. We compare the performance of both control approaches using two types of regulation signals, "RegA" and "RegD", which are used by PJM, an independent system operator in the eastern United States. The poor performance of the static-model control relative to the dynamic-model control demonstrates that modeling the dynamics of wake advection is key to providing the proposed type of model-based coordinated control of large wind farms. We further explore the performance of the dynamic-model control via composite performance scores used by PJM to qualify plants for regulation services or markets. Our results demonstrate that the dynamic-model-controlled wind farm consistently performs well, passing the qualification threshold for all fast-acting RegD signals. For the RegA signal, which changes over slower timescales, the dynamic-model control leads to average performance that surpasses the qualification threshold, but further work is needed to enable this controlled wind farm to achieve qualifying performance for all regulation signals.

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Section: Controller Designsmentioning

confidence: 99%

Wind farms providing secondary frequency regulation: evaluating the performance of model-based receding horizon control

et al. 2018

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“…There is the added complexity that the gradient here is an infinite-dimensional object, which raises some subtle functional analysis questions from a purely mathematical point of view; we refer the reader to the references [14,4] for rigorous justification of our manipulations.…”

Section: Maximizing the Mutual Informationmentioning

confidence: 99%

“…The literature on gradient-based optimization [20] and the subset on PDE-based gradient optimization [4] offers more sophisticated schemes, but for simplicity we start with this basic one. An example of a more sophisticated scheme is the nonlinear conjugate-gradient ascent, as recommended in the literature on Fokker-Planck control [2].…”

Section: Maximizing the Mutual Informationmentioning

confidence: 99%

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Optimal Design for Estimation in Diffusion Processes from First Hitting Times

Iolov¹,

Ditlevsen²,

Longtin³

2017

SIAM/ASA J. Uncertainty Quantification

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Abstract. We consider the optimal design problem for the Ornstein-Uhlenbeck process with fixed threshold, commonly used to describe a leaky, noisy integrate-and-fire neuron. We present a solution to the problem of devising the best external time-dependent perturbation to the process in order to facilitate the estimation of the characteristic time parameter for this process. The optimal design problem is constrained here by the fact that only the times between threshold crossings from below, known as hitting times, are observable. The optimal control is based on a maximization of the mutual information between the posterior of the unknown parameter given these observations and the distribution of the hitting times. Our approach is based on the adjoint method for computing the gradient of a functional of a solution to a Fokker-Planck partial differential equation with respect to an input function (i.e., to the control). Our method also enables the estimation of other parameters, in the case when more than one parameter is unknown.Key words. design of experiments, mutual information, leaky integrate-and-fire neuronal models, FokkerPlanck equation, probability density function control method, Ornstein-Uhlenbeck process AMS subject classifications. 62K05, 62L05, 93E20, 92C20, 92D25, 49L20, 49M25

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“…Books considering PDE-constrained optimization are, for instance, Hinze et al [13], Tröltzsch [38], Borzì and Schulz [4] and Leugering et al [22,21]. Besides the PDE-constraints, the optimization problems often include control constraints given through a non-empty, convex and closed subset of a Hilbert space.…”

Section: Introductionmentioning

confidence: 99%

A note on functional a posteriori estimates for elliptic optimal control problems

Wolfmayr

2016

Numerical Methods Partial

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Abstract. In this work, new theoretical results on functional type a posteriori estimates for elliptic optimal control problems with control constraints are presented. More precisely, we derive new, sharp, guaranteed and fully computable lower bounds for the cost functional in addition to the already existing upper bounds. Using both, the lower and the upper bounds, we arrive at two-sided estimates for the cost functional. We prove that these bounds finally lead to sharp, guaranteed and fully computable upper estimates for the discretization error in the state and the control of the optimal control problem. First numerical tests are presented confirming the efficiency of the a posteriori estimates derived.

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Computational Optimization of Systems Governed by Partial Differential Equations

Cited by 236 publications

References 0 publications

Wind farms providing secondary frequency regulation: evaluating the performance of model-based receding horizon control

Wind farms providing secondary frequency regulation: evaluating the performance of model-based receding horizon control

Optimal Design for Estimation in Diffusion Processes from First Hitting Times

A note on functional a posteriori estimates for elliptic optimal control problems

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