Original articles
Behavioral modeling of Wireless Power Transfer System coils

https://doi.org/10.1016/j.matcom.2020.01.004Get rights and content

Highlights

  • Wireless power transfer system coils were modeled.

  • An analytical model was identified to describe the mutual inductance of the coils.

  • A multi-objective genetic programming algorithm was used to identify the model.

  • The model is able to easily analyze the effects of the coils misalignment on the mutual inductance.

Abstract

This paper proposes a technique to derive behavioral models for describing the mutual inductance between the coupled coils used in Wireless Power Transfer Systems for the electrical recharging of vehicles. These models describe analytically the dependence of the mutual inductance with respect to geometrical parameters related to the coils misalignments, to take into account the real operating conditions of such recharging systems. A Multi-Objective Genetic Programming (MOGP) algorithm has been adopted to discover behavioral models offering optimal trade-off between accuracy and complexity. The behavioral models are identified from a set of data evaluated by using literature analytical models and are then validated by using another set of such data and also by comparing the results with full 3D Finite Element numerical simulations.

Introduction

Wireless Power Transfer (WPT) is an emerging technology for battery recharging that nowadays is undergoing intense investigations, given its potential breakthrough impact on the electrical mobility [4], [7]. Being such a technology based on the inductive coupling, the derivation of suitable models of the coupled coils is essential in view of a reliable design and an effective performance analysis of the overall WPT System (WPTS) [18], including the sensitivity analysis with respect to variations of main operating parameters and coils mutual coupling [11]. Accurate and efficient calculations of the coupled coils are in particular needed to develop reliable techniques for measuring the power transferred on board the vehicle during the inductive charging and evaluate the compliance to human exposure standards, which are the primary goals of the H2020-EMPIR project “Metrology for Inductive Charging of Electric Vehicles” (MICEV) [17], [24], [25].

The mutual inductance, M, is a key parameter influencing the performance of the WPTSs, in terms of efficiency, power transfer, harmonic distortion and so on [7], [18]. Therefore the design and the optimization of such systems require the mapping of M in the range of variability of the geometrical and physical parameters. In static recharging systems, for instance, M can be expressed as function of the misalignment of the electrical vehicle with respect to the nominal position, and as function of the ferromagnetic and conducting properties of the materials. In dynamic systems, M should be also mapped in the different points of the vehicle trajectory. The above requests may be easily addressed if an analytical solution can be found for M, expressed in terms of the above geometrical and physical parameters. Unfortunately, analytical models (e.g., based on the classical Biot–Savart law [19]) or semi-analytical ones (e.g., based on Bessel and Struve functions [9] or Heuman’s lambda function [3]) are only available when simple coupling pair structures are analyzed. Instead, when studying real world WPTS, in all the cases of practical interest the coil pair systems are characterized by the presence of complicated 3D geometries including metallic parts (e.g., the vehicle chassis) and ferrite beds for controlling the magnetic flux lines. As a consequence, analytical or semi-analytical expressions of M are not available, and its evaluation requires experimental characterizations and/or numerical full 3D solutions of a magneto-quasi-static (MQS) problem [20]. Finite element solutions of the MQS problem are the most common approach used nowadays for studying these coil systems. For example, this approach is adopted in [23] to analyze different values of coils lateral misalignment and tilt angle, in [5] to optimize the ferrite arrangements and determine the coil turn number for multi-turn spiral coils, and in [15], where a parametric performance evaluation of different resonant coil designs have been performed for different values of axial, longitudinal and lateral misalignments of the coils. In all these cases, the mapping is provided by look-up tables of points obtained via full numerical simulation, possibly refined by means of a numerical interpolation of the look-up table values.

In this paper, we propose a way to combine the accuracy of the full numerical models to the features of the analytical solutions, by deriving suitable behavioral models, widely adopted in computational electromagnetics. Specifically we propose a technique to identify analytical behavioral models for WPTS coils mutual inductance, as a function of the coil mismatch, expressed in terms of vertical and horizontal shifts and of reciprocal rotation. The models are obtained by means of a Multi-Objective Genetic Programming (MOGP) algorithm [14], by imposing trade-off conditions on the metrics associated to all the possible solutions provided by such an algorithm. This evolutionary algorithm was previously used for discovering power loss behavioral models of IGBTs, power inductors and inverter power modules [12], [21], [22]. An alternative way could be the use of Artificial Neural Networks (ANNs) or Support Vector Machines (SVMs) [13], but both do not provide compact behavioral model expressions and have not been investigated in this study.

The paper is organized as follows. In Section 2, a WPTS coils pair studied in the MICEV project is briefly presented, and the problem is formulated. In Section 3, two alternative behavioral models are derived expressing the dependence of the mutual inductance on the axial and lateral misalignments of the two coils and on their reciprocal rotation angle. Finally, the results are discussed in Section 4.

Section snippets

WPTS coils and problem formulation

The coils analyzed here are taken from one of the WPTSs studied in the MICEV project [6]. The geometry is shown in Fig. 1, highlighting the surfaces associated to the transmitting (TX) and receiving (RX) coils. The model also includes an aluminum structure (which simulates the vehicle chassis) and some ferrite parts. Their main parameters are summarized in Table 1, while Table 2 lists the measured values of the inductances [6] along with the simulated ones, obtained by means of a full numerical

Data set generation

In order to build a reliable behavioral model, large data sets are needed for the identification and validation phases, covering the range of interest of the displacement variables Δx, Δz and α with adequate resolution. Let us note that no longitudinal misalignment (Δy) has been considered for the analysis, as only the static WPT process has been simulated (no movement of the vehicle along the charging lane). As pointed out, the use of 3D solvers like CARIDDI would require long computation

Approach 1

The coils misalignment parameters chosen to assemble the training data set T of the GP algorithm are listed in the left-hand side of Table 3: all the combinations of such values have been simulated, resulting in 300 test conditions. The right-hand side of Table 3 lists the misalignment parameters describing the validation data set V adopted in Approach 1, consisting of 120 test conditions. It should be noted that the values of Δx and Δz, chosen for the validation data set, are different from

Conclusions and perspectives

In this paper, two novel behavioral models of the mutual inductance for rectangular-shaped coils used in Wireless Power Transfer Systems (WPTSs) are proposed, valid for different types of coils misalignment. Such models have been discovered by means of a Multi-Objective Genetic Programming (MOGP) algorithm. One of the two behavioral models expresses the mutual inductance as a function of the axial and lateral misalignments of the coils, parameterized with respect to their rotation angle. The

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgments

The results here presented are developed in the framework of the 16ENG08 MICEV Project. The latter received funding from the EMPIR programme cofinanced by the Participating States and from the European Union’s Horizon 2020 research and innovation programme. The University of Salerno partially supported this work through the project funds “Sistemi di Carica Induttiva di Veicoli Elettrici” (300638FRB17DICAPUA, 300638FRB18DICAPUA). The University of Naples Federico II partially supported this

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