Multiobjective Optimization of the Design of Nonlinear Electromagnetic Systems Using Parametric Reduced Order Models

This dissertation delivers a contribution to the field of order reduction of large-scale nonlinear models of electromagnetic devices. In particular, it enables applying model order reduction techniques to an important class of electromagnetic devices that contain moving components and materials with nonlinear magnetic properties. Such devices include among others rotating electrical machines, electromagnetic valves, electromagnetic solenoids, and electromechanical relays.The presented methods exploits the trajectory piecewise linear (TPWL) approach in approximating the nonlinear dependency of materials properties on the applied magnetic field. Additionally, the model nonlinearity that is caused by the movement of the device components is handled using a novel approach that updates the electromagnetic (EM) field model permanently according to the new components positions.The order of the large-scale electromagnetic field model is reduced by approximating the original electromagnetic field distribution by a linear combination of few virtual field distributions that are found using the proper orthogonal decomposition (POD) approach.The challenge of selecting the number and the position of the linearization points in the TPWL model is tackled using a new approach that considers the change in the magnetic properties of the device materials among all the simulated state-vectors.The new presented methods are extended to enable generating parametric reduced order models of moving nonlinear EM devices. Such models enable a fast and accurate prediction of the behavior of the EM device and its variations that result from changing the values of its design parameters. Additionally, several algorithms for generating an optimal reduction subspace of the parametric model are presented and compared.Finally, an approach for overcoming the challenge of generating reduced order models of EM devices while considering the strong influence of their power electronics driving circuits is introduced and applied to the example of a rotating electrical machine coupled to a power electronics driving circuit.The new methods presented in this work are validated by applying them on the examples of three industrial devices. An electrical transformer, an electromagnetic valve, and a rotating electrical machine. DEDICATIONTo my parents, my wife, and my daughter. ACKNOWLEDGMENTS

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“…It is worth mentioning that the presented results in this chapter have been partially published in our work [3] …”

Section: Voltage and Currentmentioning

confidence: 93%

Model-Order Reduction of Moving Nonlinear Electromagnetic Devices

et al. 2008

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“…Consequently, optimal control of these systems is all the more challenging, and considering multiple objectives further increases the cost. Due to this reason, only few problems have been addressed directly, see, e.g., [50][51][52][53]. A method exploiting special structure in the system dynamics has been proposed in [54], and a special case of Pareto optimal solutions -namely Nash equilibria -have been computed in [55,56] 1 .…”

Section: Solution Methodsmentioning

confidence: 99%

A Survey of Recent Trends in Multiobjective Optimal Control -- Surrogate Models, Feedback Control and Objective Reduction

Peitz¹,

Dellnitz²

2018

Preprint

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Abstract:Multiobjective optimization plays an increasingly important role in modern applications, where several criteria are often of equal importance. The task in multiobjective optimization and multiobjective optimal control is therefore to compute the set of optimal compromises (the Pareto set) between the conflicting objectives. The advances in algorithms and the increasing interest in Pareto optimal solutions have led to a wide range of new applications related to optimal and feedback control which results in new challenges such as expensive models or real-time applicability. Since the Pareto set generally consists of an infinite number of solutions, the computational effort can quickly become challenging which is particularly problematic when the objectives are costly to evaluate or when a solution has to be presented very quickly. This article gives an overview over recent developments in accelerating multiobjective optimal control for complex problems where either PDE constraints are present or where a feedback behavior has to be achieved. In the first case, surrogate models yield significant speed-ups. Besides classical meta-modeling techniques for multiobjective optimization, a promising alternative for control problems is to introduce a surrogate model for the system dynamics. In the case of real-time requirements, various promising model predictive control approaches have been proposed, using either fast online solvers or offline-online decomposition. We also briefly comment on dimension reduction in many-objective optimization problems as another technique for reducing the numerical effort.

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“…Inserting Equation (20) in Equation (19) results in m(e k (μ), e k (μ); μ) + t · a(e k (μ), e k (μ); μ)…”

Section: Energy Norm Estimatormentioning

confidence: 99%

Model order reduction and error estimation with an application to the parameter-dependent eddy current equation

Jung

Patera

Haasdonk

et al. 2011

Mathematical and Computer Modelling of Dynamical Systems

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In product development, engineers simulate the underlying partial differential equation many times with commercial tools for different geometries. Since the available computation time is limited, we look for reduced models with an error estimator that guarantees the accuracy of the reduced model. Using commercial tools the theoretical methods proposed by G. Rozza, D.B.P. Huynh and A.T. Patera [Reduced basis approximation and a posteriori error estimation for affinely parameterized elliptic coercive partial differential equations, Arch. Comput. Methods Eng. 15 (2008), pp. 229-275] lead to technical difficulties. We present how to overcome these challenges and validate the error estimator by applying it to a simple model of a solenoid actuator that is a part of a valve.Keywords: model order reduction; Krylov subspace method; reduced basis method; a posteriori error estimator; eddy current equation IntroductionModel order reduction methods are effective methods that reduce the computational complexity of partial differential equations (PDEs) and dynamical systems, respectively. In literature many different reduction methods have been presented. Here, we focus on the reduced basis method (RBM) and Krylov subspace methods. The RBM is discussed in detail in [1,2]. The basics of the local and global Krylov subspace methods are introduced in [3-6].In this article, we combine the ingredients of the RBM with the Krylov subspace methods and compare them. Additionally, we adapt the error estimator from [2, Section 4.4] and derive an 'optimized' L 2 -error estimator that reduces the increase of the effectivity with the time compared to a simpler L 2 -error estimator.In general, the theory is applicable to linear PDE or linear dynamical systems with geometrical parameters. As an example, we treat the eddy current equation in the time domain. All error estimators are applied to the eddy current equation in connection with the RBM, and the global and local Krylov subspace method. Afterwards, the three reduction methods are compared. To the best of our knowledge, we provide with this the first comparison of

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Multiobjective Optimization of the Design of Nonlinear Electromagnetic Systems Using Parametric Reduced Order Models

Cited by 19 publications

References 5 publications

Model-Order Reduction of Moving Nonlinear Electromagnetic Devices

Model-Order Reduction of Moving Nonlinear Electromagnetic Devices

A Survey of Recent Trends in Multiobjective Optimal Control -- Surrogate Models, Feedback Control and Objective Reduction

Model order reduction and error estimation with an application to the parameter-dependent eddy current equation

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