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Discretizing the magnetic vector potential formulation of eddy current problems in space results in an infinitely stiff differential algebraic equation system that is integrated in time using implicit time integration methods. Applying a generalized Schur complement to the differential algebraic equation system yields an ordinary differential equation (ODE) system. This ODE system can be integrated in time using explicit time integration schemes by which the solution of high-dimensional nonlinear algebraic systems of equations is avoided. The purpose of this paper is to further investigate the explicit time integration of eddy current problems.

The resulting magnetoquasistatic Schur complement ODE system is integrated in time using the explicit Euler method taking into account the Courant–Friedrich–Levy (CFL) stability criterion. The maximum stable CFL time step can be rather small for magnetoquasistatic field problems owing to its proportionality to the smallest edge length in the mesh. Ferromagnetic materials require updating the reluctivity matrix in nonlinear material in every time step. Because of the small time-step size, it is proposed to only selectively update the reluctivity matrix, keeping it constant for as many time steps as possible.

Numerical simulations of the TEAM 10 benchmark problem show that the proposed selective update strategy decreases computation time while maintaining good accuracy for different dynamics of the source current excitation.

The explicit time integration of the Schur complement vector potential formulation of the eddy current problem is accelerated by updating the reluctivity matrix selectively. A strategy for this is proposed and investigated.

The maximum entropy snapshot sampling (MESS) method aims to reduce the computational cost required for obtaining the reduced basis for the purpose of model reduction. Hence, it can significantly reduce the original system dimension whilst maintaining an adequate level of accuracy. The purpose of this paper is to show how these beneficial results are obtained.

The so-called MESS method is used for reducing two nonlinear circuit models. The MESS directly reduces the number of snapshots by recursively identifying and selecting the snapshots that strictly increase an estimate of the correlation entropy of the considered systems. Reduced bases are then obtained with the orthogonal-triangular decomposition.

Two case studies have been used for validating the reduction performance of the MESS. These numerical experiments verify the performance of the advocated approach, in terms of computational costs and accuracy, relative to gappy proper orthogonal decomposition.

The novel MESS has been successfully used for reducing two nonlinear circuits: in particular, a diode chain model and a thermal-electric coupled system. In both cases, the MESS removed unnecessary data, and hence, it reduced the snapshot matrix, before calling the QR basis generation routine. As a result, the QR-decomposition has been called on a reduced snapshot matrix, and the offline stage has been significantly scaled down, in terms of central processing unit time.

A transient magneto-quasistatic vector potential formulation involving nonlinear material is spatially discretized using the finite element method of first and second polynomial order. By applying a generalized Schur complement the resulting system of differential algebraic equations is reformulated into a system of ordinary differential equations (ODE). The ODE system is integrated in time by using explicit time integration schemes. The purpose of this paper is to investigate explicit time integration for eddy current problems with respect to the performance of the first-order explicit Euler scheme and the Runge-Kutta-Chebyshev (RKC) method of higher order.

The ODE system is integrated in time using the explicit Euler scheme, which is conditionally stable by a maximum time step size. To overcome this limit, an explicit multistage RKC time integration method of higher order is used to enlarge the maximum stable time step size. Both time integration methods are compared regarding the overall computational effort.

The numerical simulations show that a finer spatial discretization forces smaller time step sizes. In comparison to the explicit Euler time integration scheme, the multistage RKC method provides larger stable time step sizes to diminish the overall computation time.

The explicit time integration of the Schur complement vector potential formulation of eddy current problems is accelerated by a multistage RKC method.

In high voltage direct current cable systems, cable joints are known as the least reliable components due to the use of multiple dielectrics. Resulting from the electric field and temperature depending conductivity of the different dielectrics, field enhancement at critical areas, e.g. triple points, may result in accelerated aging and the failure of the component. To reduce the stress, different field grading techniques are applied. The purpose of this study is to investigate different grading techniques for cable joints. Different shapes of the electrode and a varying nonlinear conductivity of field grading materials (FGM) are used for the simulation of the electric field.

Coupled electro-thermal field simulations are applied for different joint geometries, to obtain the stationary electric field. Electric field simulations in cable joint using geometric and nonlinear field grading techniques are shown.

Using the geometric field grading, the shape of the stress cone determines the field values in critical areas (triple points). High stress reduction is obtained for a certain curvature of the stress cone. For the nonlinear stress control, materials with a higher conductivity in comparison to the cable and the joint material are used. A field reduction is obtained by increasing the total conductivity. On the other hand, this is also increasing the insulation losses within the total FGM. More applicable is the decrease of the switching field or the increase of nonlinearity, which is only locally increase the conductivity and the insulation losses. Furthermore, simulations results show that an approximately constant field reduction is obtained, if the nonlinearity is above a certain threshold.

This study is restricted to a field dependency of FGM only. For impulse voltages, high temperature and electric conductivity values my result in a thermal runaway. Furthermore, only direct current field grading techniques are studied.

The field grading of cable joints, using geometric and nonlinear techniques, is analyzed. A comparison between the electric field, by varying the curvature of the ground stress cone or the FGM conductivity constants in a complex joint geometry is novel. With its effect on the electric fields, general requirements for the geometry (geometric field grading) or the values of the FGM constants (nonlinear field grading) are defined to obtain a sufficient field grading.

In high voltage direct current (HVDC), power cables heat is generated inside the conductor and the insulation during operation. A higher amount of the generated heat in comparison to the dissipated one, results in a possible thermal breakdown. The accumulation of space charges inside the insulation results in an electric field that contributes to the geometric electric field, which comes from the applied voltage. The total electric field decreases in the vicinity of the conductor, while it increases near the sheath, causing a possible change of the breakdown voltage.

Here, the thermal breakdown is studied, also incorporating the presence of space charges. For a developed electro-thermal HVDC cable model, at different temperatures, the breakdown voltage is computed through numerical simulations.

The simulation results show a dependence of the breakdown voltage on the temperature at the location of the sheath. The results also show only limited influence of the space charges on the breakdown voltage.

The study is restricted to one-dimensional problems, using radial symmetry of the cable, and does not include any aging or long-term effect of space charges. Such aging effect can locally increase the electric field, resulting in a reduced breakdown voltage.

A comparison of the breakdown voltage with and without space charges is novel. The chosen approach allows for the first time to assess the influence of space charges and field inversion on the thermal breakdown.

In high-voltage direct current (HVDC) cable systems, space charges accumulate because of the constant applied voltage and the nonlinear electric conductivity of the insulating material. The change in the charge distribution results in a slowly time-varying electric field. Space charges accumulate within the insulation bulk and at interfaces. With an operation time of several years of HVDC systems, typically the stationary electric field is of interest. The purpose of this study is to investigate the influence of interfaces on the stationary electric field stress and space charge density.

An analytic description of the stationary electric field inside cable insulation is developed and numerical simulations of a cable joint geometry are applied, considering spatial variations of the conductivity in the vicinity of the electrodes and interfaces.

With increasing conductivity values toward the electrodes, the resulting field stress decreases, whereas a decreasing conductivity results in an increasing electric field. The increased electric field may cause partial discharge, resulting in accelerated aging of the insulation material. Thus, interfaces and surfaces are characterized as critical areas for the reliability of HVDC cable systems.

This study is restricted to stationary electric field and temperature distributions. The electric field variations during a polarity reversal or a time-varying temperature may result in an increased electric conductivity and electric field at interfaces and surfaces.

An analytical description of the electric field, considering surface effects, is developed. The used conductivity model is applicable for cable and cable-joint insulations, where homo- and hetero-charge effects are simulated. These simulations compare well against measurements.

High resolution simulations of body-internal electric field strengths induced by magneto-quasistatic fields from wireless power transfer systems are computationally expensive. The exposure simulation can be split into two separate simulation steps allowing the calculation of the magnetic flux density distribution, which serves as input into the second simulation step to calculate the body-internal electric fields. In this work, the magnetic flux density is interpolated from in situ measurements in combination with the scalar-potential finite difference scheme to calculate the resulting body-internal field. These calculations are supposed to take less than 5 s to achieve a near real-time visualization of these fields on mobile devices. The purpose of this work is to present an implementation of the simulation on graphics processing units (GPUs), allowing for the calculation of the body-internal field strength in about 3 s.

This work uses the co-simulation scalar-potential finite difference scheme to determine the body-internal electric field strength of human models with a voxel resolution of 2 × 2 × 2 mm³. The scheme is implemented on GPUs. This simulation scheme requires the magnetic flux density distribution as input, determined from radial basis functions.

Using NVIDIA A100 GPUs, the body-internal electric field strength with high-resolution models and 8.9 million degrees of freedom can be determined in about 2.3 s.

This paper describes in detail the used scheme and its implementation to make use of the computational performance of modern GPUs.

This paper aims to present a fast and modular framework implementation for the thermal analyses of foreign metal objects in the context of wireless power transfer (WPT) to evaluate whether they pose a hazard to the system. This framework serves as a decision-making tool for determining the necessity of foreign object detection in certain applications and at certain transmitted power levels.

To assess the necessity of implementing foreign object detection, the considered WPT system is modeled, and Arnoldi-Krylov-based model order reduction is applied to generate separate reduced models of the ground and vehicle modules of the WPT system. This enables interoperable evaluations to be conducted. Further discussion on the implementation details of the system-level simulations used to evaluate the electrical and thermal characteristics is provided. The resulting modular implementation allows for efficient evaluation of the thermal behavior of the wireless charging system at various transferred power levels and under various boundary conditions.

Based on the transferred power level, the WPT model, the relative positioning between the vehicle and the charging pad and the charging time, it may be necessary to divide the area of the charging pad into multiple regions for the purpose of implementing foreign object detection.

While the tools and fundamentals of thermal analysis are widely known and used, their application to high-power WPT systems for electric vehicles has not yet been thoroughly discussed in this form in the literature. The approach presented in this paper is not limited to the specific WPT model discussed but rather is directly applicable to other WPT models as well.

Magneto-quasi-static fields emanated by inductive charging systems can be potentially harmful to the human body. Recent projects, such as TALAKO and MILAS, use the technique of wireless power transfer (WPT) to charge batteries of electrically powered vehicles. To ensure the safety of passengers, the exposing magnetic flux density needs to be measured in situ and compared to reference limit values. However, in the design phase of these systems, numerical simulations of the emanated magnetic flux density are inevitable. This study aims to present a tool along with a workflow, based on the Scaled-Frequency Finite Difference Time-Domain and Co-Simulation Scalar Potential Finite Difference schemes, to determine body-internal magnetic flux densities, electric field strengths and induced voltages into cardiac pacemakers. The simulations should be time efficient, with lower computational costs and minimal human workload.

The numerical assessment of the human exposure to magneto-quasi-static fields is computationally expensive, especially when considering high-resolution discretization models of vehicles and WPT systems. Incorporating human body models into the simulation further enhances the number of mesh cells by multiple millions. Hence, the number of simulations including all components and human models needs to be limited while efficient numerical schemes need to be applied.

This work presents and compares four exposure scenarios using the presented numerical methods. By efficiently combining numerical methods, the simulation time can be reduced by a factor of 3.5 and the required storage space by almost a factor of 4.

This work presents and discusses an efficient way to determine the exposure of human beings in the vicinity of wireless power transfer systems that saves computer simulation resources and human workload.

Inductive charging systems for electrically powered cars produce a magneto-quasistatic field and organism in the vicinity might be exposed to that field. Magneto-quasistatic fields induce electric fields in the human body that should not exceed limits given by the International Commission of Non-Ionizing Radiation protection (ICNIRP) to ensure that no harm is done to the human body. As these electric fields cannot be measured directly, they need to be derived from the measured magnetic flux densities. To get an almost real-time estimation of the harmfulness of the magnetic flux density to the human body, the electric field needs to be calculated within a minimal computing time. The purpose of this study is to identify fast linear equations solver for the discrete Poisson system of the Co-Simulation Scalar Potential Finite Difference scheme on different graphics processing unit systems.

The determination of the exposure requires a fast linear equations solver for the discrete Poisson system of the Co-Simulation Scalar Potential Finite Difference (Co-Sim. SPFD) scheme. Here, the use of the AmgX library on NVIDIA GPUs is presented for this task.

Using the AmgX library enables solving the equation system resulting from an ICNIRP recommended human voxel model resolution of 2 mm in less than 0.5 s on a single NVIDIA Tesla V100 GPU.

This work is one essential advancement to determine the exposure of humans from wireless charging system in near real-time from in situ magnetic flux density measurements.