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\DOI{10.5802/crphys.188}
\datereceived{2023-07-18}
\daterevised{2024-02-19}
\datererevised{2024-05-14}
\dateaccepted{2024-05-15}
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\dateposted{2024-08-22}
\begin{document}

\begin{noXML}

%\makeatletter
%\def\TITREspecial{\relax}
%\def\cdr@specialtitle@english{Energy in the heart of EM waves: modelling, measurements and management}
%\def\cdr@specialtitle@french{L'\'energie au c{\oe}ur des ondes \'electromagn\'etiques : mod\'elisation, mesures et gestion}
%\makeatother

\CDRsetmeta{articletype}{research-article}

\title{Analysis of inductive power transfer systems by metamodeling
techniques{\vspace*{-3pt}}}

\alttitle{Analyse de syst\`{e}mes de transfert de puissance inductifs
par des techniques de m\'{e}tamod\'{e}lisation{\vspace*{-3pt}}}

\author{\firstname{Yao} \lastname{Pei}\CDRorcid{0000-0003-0099-4001}\IsCorresp} 
\address{Universit\'{e} Paris-Saclay, CentraleSup\'{e}lec, CNRS,
Laboratoire de G\'{e}nie Electrique et Electronique de Paris, 91192,
Gif-sur-Yvette, France}
\address{Sorbonne Universit\'{e}, CNRS, Laboratoire de G\'{e}nie
Electrique et Electronique de Paris, 75252, Paris, France}
\email[Y. Pei]{yao.pei@centralesupelec.fr}

\author{\firstname{Lionel} \lastname{Pichon}\CDRorcid{0000-0002-3402-5498}} 
\addressSameAs{1}{Universit\'{e} Paris-Saclay, CentraleSup\'{e}lec, CNRS,
Laboratoire de G\'{e}nie Electrique et Electronique de Paris, 91192,
Gif-sur-Yvette, France}
\addressSameAs{2}{Sorbonne Universit\'{e}, CNRS, Laboratoire de G\'{e}nie
Electrique et Electronique de Paris, 75252, Paris, France}

\author{\firstname{Mohamed} \lastname{Bensetti}\CDRorcid{0000-0002-4755-5113}} 
\addressSameAs{1}{Universit\'{e} Paris-Saclay, CentraleSup\'{e}lec, CNRS,
Laboratoire de G\'{e}nie Electrique et Electronique de Paris, 91192,
Gif-sur-Yvette, France}
\addressSameAs{2}{Sorbonne Universit\'{e}, CNRS, Laboratoire de G\'{e}nie
Electrique et Electronique de Paris, 75252, Paris, France}

\author{\firstname{Yann} \lastname{Le Bihan}\CDRorcid{0000-0001-5563-9192}}
\addressSameAs{1}{Universit\'{e} Paris-Saclay, CentraleSup\'{e}lec, CNRS,
Laboratoire de G\'{e}nie Electrique et Electronique de Paris, 91192,
Gif-sur-Yvette, France}
\addressSameAs{2}{Sorbonne Universit\'{e}, CNRS, Laboratoire de G\'{e}nie
Electrique et Electronique de Paris, 75252, Paris, France}

\begin{abstract}
{This paper presents some metamodeling techniques to analyze the
variability of the performances of an inductive power transfer (IPT)
system, considering the sources of uncertainty (misalignment between
the coils, the variation in air gap, and the rotation on the receiver).
For IPT systems, one of the key issues is transmission efficiency,
which is greatly influenced by many sources of uncertainty. So, it is
meaningful to find a metamodeling technique to quickly evaluate the
system's performances. According to the comparison of Support Vector
Regression, Multigene Genetic Programming Algorithm, and sparse
Polynomial Chaos Expansions (PCE), sparse PCE is recommended for the
analysis due to the tradeoff between the computational time and the
accuracy of the metamodel.}
{\vspace*{-2pt}}
\end{abstract}

\begin{altabstract} 
Ce papier pr\'{e}sente diff\'{e}rentes techniques de
m\'{e}tamod\'{e}lisation afin d'analyser la variabilit\'{e} des
performances d'un syst\`{e}me de transfert de puissance par induction
(IPT), en tenant compte des sources d'incertitude (d\'{e}centrage des
bobines, la variation de l'entrefer et la rotation du r\'{e}cepteur).
Pour les syst\`{e}mes IPT, l'une des questions cl\'{e}s est
l'efficacit\'{e} de la transmission, qui est fortement influenc\'{e}e
par les nombreuses sources d'incertitude. Il est donc important de
d\'{e}terminer une technique de m\'{e}tamod\'{e}lisation susceptible
d'\'{e}valuer rapidement les performances du syst\`{e}me. Trois
techniques de m\'{e}tamod\'{e}lisation sont compar\'{e}es~: la
r\'{e}gression \`{a} vecteurs de support, l'algorithme de
programmation g\'{e}n\'{e}tique multig\'{e}nique et les
d\'{e}veloppements du chaos polynomial (PCE), il ressort que la
technique PCE est recommand\'{e}e pour une telle analyse en raison du
compromis entre le temps de calcul et la pr\'{e}cision du
m\'{e}tamod\`{e}le.
{\vspace*{-3pt}}
\end{altabstract} 

\shortrunauthors

\keywords{\kwd{Wireless power transfer}\kwd{Metamodels}\kwd{Polynomial
chaos expansions}\kwd{Support vector regression}\kwd{Multigene genetic
programming algorithm}}

\altkeywords{\kwd{Transfert d'\'{e}nergie sans
contact}\kwd{M\'{e}tamod\`{e}les}\kwd{D\'{e}veloppements du chaos
polynomial}\kwd{R\'{e}gression \`{a} vecteurs de
support}\kwd{L'algorithme de programmation g\'{e}n\'{e}tique
multig\'{e}nique}}

\def\thanksname{Note}
\thanks{This article follows the URSI-France workshop held on 21 and 22
March 2023 at Paris-Saclay.}

\maketitle

\end{noXML}

\section{Introduction}\label{sec1}

Around 800 million vehicles with internal combustion engines (ICEs) are
used worldwide~\cite{1}. These vehicles are a major source of
greenhouse gases, especially CO$_2$. Thus, a practical way of dealing with
the global warming problem is to replace ICE-powered vehicles with
electric vehicles (EVs)~\cite{1,2,3}. The use of electric cars also
improves the quality of air around major cities. To replace ICEs, many
vehicle companies are developing ``plug-in'' EVs, which use lithium-ion
(or polymer) batteries that can be recharged at home or at charging
stations~\cite{2,3}. But the promotion and adoption of plug-in EVs
raise many questions. First, the cost of lithium batteries is high.
Second, the batteries are heavy. Third, to decrease the charging time
for the battery, it requires an expensive infrastructure for charging
stations~\cite{2,3,4,5,6,7}. Finally, mishandling the high-power cables will
lead to security problems~\cite{3,4, 8,9}. So, due to these problems,
inductive power transfer (IPT) has been introduced as an alternative
technology, allowing power flow in a contactless manner. Using a
resonant IPT system seems to be an effective technology for the growth
of the EVs market~\cite{10}. Moreover, its application for the charge
during the vehicle's motion is promising to overcome the barriers of
heavy onboard battery storage and the long recharging time~\cite{3,4,
8,9}. IPT is essentially based on the resonance of two magnetically
coupled inductors (constituting the coupler): the transmitter, placed
on the ground, and the receiver, placed under the vehicle floor.

In a real IPT system scenario, various receiver positions may happen
during the park or driving~\cite{9}. A careful design process of IPT
systems requires considering multiple parameters (misalignment,
relative rotation of the receiver, air gap, etc.). So, when using
simulation tools, multiple 3D numerical computations are needed to
assess the performance of the IPT system when these situations happen.
Nevertheless, using complex simulation tools (such as the finite
element method) leads to high computational costs for wide parametric
analysis. In this case, a standard Monte Carlo (MC) analysis becomes
challenging regarding computer resources and simulation
time~\cite{11,12}. 

To solve this problem, ``metamodeling techniques'' for the parametric
and statistical analysis in complex nonlinear problems have been
developed. These approaches can reduce the computational cost by
substituting an expensive computational model with a so-called
metamodel, an analytical approximation of the original model that is
much faster to evaluate~\cite{11,12,13,14,15}. 
The metamodels are constructed by
learning the varying trend from input parameters and their
corresponding model responses, for example, generated from running 3D
FEM computations. Because it is faster to evaluate, a metamodel allows
more sophisticated analyses, such as sensitivity analysis~\cite{16,17}.
In recent literature, several metamodeling techniques have been applied
to generate some metamodels trained with a limited set of simulation
results, such as Support Vector Regression (SVR)~\cite{18}, Multigene
Genetic Programming Algorithm (MGPA)~\cite{19}, Polynomial Chaos
Expansions (PCE)~\cite{20}, and so on. Reference~\cite{14} focuses on
applying the SVR with polynomial kernels to the uncertainty
quantification and the parametric modeling of structures. Then,
references~\cite{14,15} compare the accuracy and robustness to noise
among the SVR, the least squares SVR~\cite{21}, and the sparse PCE. In
the domain of inductive power transfer, a MGPA metamodel is
investigated to express the self-inductance and the mutual inductance
of the IPT system versus geometrical parameters of the ferrite and
coils, so new equations are proposed for evaluating these values of the
inductances~\cite{22,23,24}. In~\cite{25}, sources of uncertainties have
been analyzed with different shapes of small-scale couplers by the
sparse PCE. Reference~\cite{26} has compared the MGPA and sparse PCE
metamodeling techniques for the design of IPT systems. Since the design
of realistic IPT systems involves complex configurations including many
parameters, choosing a fast metamodel is a key point. Comparing
different metamodeling techniques in the real-scale couplers for IPT
systems is required, and it is important to prove the efficiency in a
realistic experiment. 

Therefore, the paper aims to compare several metamodeling techniques
for analyzing the mutual inductance of IPT systems, considering the
sources of uncertainty (misalignment along the $X/Y$-axis, variation in
air gap, and receiver rotation). First, in Section~\ref{sec2}, a 3D
model of a practical IPT system is built and numerical predictions
related to the electrical parameters and stray magnetic field are
validated against experimental measurements. Then, different
metamodeling techniques about SVR with the Gaussian radial base
function (RBF) kernel, MGPA, and PCE are introduced in
Section~\ref{sec3}. After, in Section~\ref{sec4}, a comparison is
presented among these different metamodels for the analysis of square
couplers, considering the sources of uncertainty. Finally, the
conclusion is given in Section~\ref{sec5}. 


\section{IPT system}\label{sec2}

\subsection{IPT system introduction}\label{sec2.1}

Figure~\ref{fig1} shows the block diagram of an IPT system for EVs. The
system consists of a transmitter, a receiver, converters, and
compensation networks for the transmitter and the receiver. The
electrical network provides a voltage through the AC/DC converter. The
magnetic field produced by the transmitter induces an alternating
magnetic field in the receiver. The converters then rectify the AC
power to charge the battery. Due to the dimensions of the coils, the
parasitic capacitance is insufficient to ensure the resonance in the
operational frequency range. Consequently, compensation networks (such
as capacitors) are connected to the transmitter and receiver (the
self-inductance of the transmitter and the receiver) to adjust the
operational frequency and create the resonant state~\cite{3,4, 8,9,
27,28,29,30}. It minimizes the reactive power supply and improves both the
transmission efficiency and the power transfer capability.


\begin{figure}
\includegraphics{fig01}
\caption{\label{fig1}Block diagram of an IPT system.}
\end{figure}


{In the system, the magnetic coupler (indicated in the red line frame)
is the key part, which normally includes a pair of coils, ferrite
plates, and shielding~\cite{3,4, 8,9, 27,28,29,30}. The geometry and
configuration of the coils are crucial for determining the magnetic
field of the IPT system and its efficiency. The ferrite plates
influence the efficiency and prevent magnetic flux leakage. The
shielding is used to prevent magnetic flux leakage, which is usually
placed above the receiver to minimize the flux leakage around the
system. Some of the IPT systems take the vehicle chassis as conductive
material for shielding~\cite{31}}. {For the IPT system transmission
efficiency, circuit models with lumped parameters are often used, and
the compensation networks are designed to minimize the reactive
component of the power supply. Following~\cite{3,4, 8,9, 27,
29,30,31,32,33,34}, the series--series (SS) compensation network is
taken into account to analyze the power transmission efficiency of the
system, as shown in Figure~\ref{fig2}. Indeed, according 
to~\cite{3,4, 8,9, 27,
29,30,31,32,33,34}, it is suitable for the IPT systems, and the
condition of resonance in the SS compensation remains constant,
independently of the variations of the mutual inductance and the load}
${{R}}_{{L}}$. ${{L}}_{1}$ {and} ${{L}}_{2}$
{represent the self-inductances of the transmitter and the receiver,
respectively;} ${{R}}_{1}$ {and} ${{R}}_{2}$ {represent the
resistances of the transmitter and the receiver, respectively;}
${{C}}_{1}$ {and} ${{C}}_{2}$ {are the resonant capacitors; $M$
is the mutual inductance between the transmitter and the receiver.}


\begin{figure}
\includegraphics{fig02}
\caption{\label{fig2}Equivalent electrical circuit in the SS
compensation topology~\cite{3,4, 8,9, 27, 29,30,31,32,33,34}.}
\end{figure}



So, the equation to calculate the maximum transmission efficiency
$\eta _{\max }$ of the system in Figure~\ref{fig2} can be achieved as below
when the transmitter and the receiver are identical~\cite{32,33,34}: 
{\begin{equation}
\label{eq1}
\eta _{\max }\approx 1-\frac{2}{{kQ}}=1-\frac{2\sqrt{R_{1}R_{2}}}
{w_{0}M}=1-\frac{{{R}}_{1}}{\rmpi {{f}}_{0}{M}}
\end{equation}}\unskip
where the coupling coefficient
${k}={{M}}/{\sqrt{{{L}}_{1}{{L}}_{2}}}$;
the system quality factor ${Q}=2\rmpi
{{f}}_{0}\sqrt{{{{L}}_{1}{{L}}_{2}}/
{{{R}}_{1}{{R}}_{2}}}$~\cite{35,36,37}; ${{f}}_{0}$ is
the resonant frequency. 

\subsection{IPT system experimental validation}\label{sec2.2}

In this work, a square coupler was built in the GeePs laboratory, and
shown in Figure~\ref{fig3}. The square shape is proven in~\cite{33,34}
to be well-suited for IPT systems. The parameters of the coupler are
summarized in Table~\ref{tab1}~\cite{33,34}. The square coupler has six turns
arranged in two layers. These turns are made with litz wires composed
of 1250 strands, and the strand's diameter is 0.1~mm, so it is smaller
than the skin depth at the operating frequency~\cite{34}. The operating
frequency of the IPT system is around 85~kHz~\cite{33,34,35,36, 38}. 


\begin{figure}
\includegraphics{fig03}
\caption{\label{fig3}Simulation model and experimental model of the
square magnetic coupler.}
\end{figure}


\begin{table}
\caption{\label{tab1} Parameters of the magnetic coupler}
\begin{tabular}{cc}
\thead
{Parameter} & {Value {(}Unit{)}} \\
\endthead
Exterior length of the coils $l_{\mathrm{ex}}$ & 468 ({mm}) \\ 
Interior length of the coils $l_{\mathrm{in}}$ & 442 ({mm}) \\ 
Coil thickness $d_{c}$ & 13 ({mm}) \\ 
Ferrite length $l_{f}$ & 600 ({mm}) \\ 
Ferrite width $w_{f}$ & 500 ({mm}) \\ 
Ferrite thickness $t_{f}$ & 2 ({mm}) \\ 
Distance between the coils and the ferrite  & 8 {(}mm{)} \\ 
Ferrite relative permeability $\mu _{r}$ & 2000 \\ 
Cross-sectional area of the wires $S$ & 9.82 $(\mathrm{mm}^{2})$ \\ 
Air gap & 150 ({mm})
\botline
\end{tabular}
\end{table}

Then, the magnetic parameters of the square coupler are simulated in
COMSOL~6.1~\cite{39} and measured by a RLC meter (Wayne Kerr~4300), and
the results are given in Table~\ref{tab2}. The relative errors between the
measurement and the simulation are around 10\%, which may be caused by
the real coil shape that is not exactly the same as defined in the
simulation. Also, some uncertainty exists about the winding arrangement
in the real coil, which may not be well positioned in two layers. 

\begin{table}
\caption{\label{tab2} Comparison of the magnetic parameters in the
simulation and the measurement}
\begin{tabular}{cccc}
\thead
Frequency \smath{=} 85 {(}kHz{)} & 
\parbox[t]{2.6cm}{\centering Self-inductance $L$ {(}${\rmmu}$H{)}} & 
\parbox[t]{2.8cm}{\centering Mutual inductance $M$ {(}${\rmmu}$H{)}} & 
\parbox[t]{1.9cm}{\centering Coupling coefficient $k$}\vspace*{2pt} \\
\endthead 
Simulation & 63.7 & 13.4 & 0.21 \\ 
Measurement & 58.5 & 12.1 & 0.20 \\ 
Relative error of the experiment (\%)
& 8.1\% & 9.7\% & 4.8\% 
\botline
\end{tabular}
\end{table}

The magnetic flux density distribution is measured at 150~mm above the
receiver as shown in Figure~\ref{fig4}. The method to do the
measurement in the near-field test bench is presented in~\cite{34}. The
magnetic probe measures the magnetic field and the robot moves the
magnetic probe automatically for different measurement positions. 


\begin{figure}
\includegraphics{fig04}
\caption{\label{fig4}Measurement line of the magnetic flux density
above the receiver.}
\end{figure}


{Figure~\ref{fig5} compares the measured and simulated}
$B_{\mathrm{norm}}$ {(the norm of the magnetic flux density) on the
measurement line as depicted in Figure~\ref{fig4}. The relative error
between the measured and simulated} $B_{\mathrm{norm}}$ {is around
10\%. There exist differences in the position and the amplitude of the
magnetic flux density, probably because the coils of the coupler are
made by hand. Hence, they are not exactly the same as those defined in
the simulation. Moreover, the ferrite plate in the experiment is not as
flat as it is in the simulation. Consequently, the magnetic flux
density values of the measurement line are not the same as those in the
simulation.} 


\begin{figure}
\includegraphics{fig05}
\caption{\label{fig5}Comparison of the magnetic flux density
$B_{\mathrm{norm}}$ obtained by the simulation and measurement.}
\end{figure}


From the results presented above, the reliability of the 3D coupler
model has been confirmed through the comparison between the simulated
values and the measurement values both for the mutual inductance and
the magnetic flux density.

\subsection{Uncertain parameters in the IPT System}\label{sec2.3}


{To investigate the efficiency of the IPT system, it is mandatory to
consider the sources of uncertainty, such as variations in the
misalignment of the receiver due to imperfect parking alignment and
variations in the air gap due to loading or unloading the vehicle.
Figure~\ref{fig6} shows the rotation angle along the $Z$-axis} $\alpha
${, the misalignment along the $X$-axis} $\rmDelta x${, the misalignment
along the $Y$-axis} $\rmDelta y$, {and the air gap between two coils}
$\rmDelta z$ {for the couplers.} 


\begin{figure}
\includegraphics{fig06}
\caption{\label{fig6}Influencing factors for the square couplers.}
\end{figure}


Before performing the uncertainty analysis, it is necessary to assume a
probability distribution for the sources of uncertainty. Here, a
Gaussian distribution is chosen for these influencing factors, which
conforms to the probability that may happen in reality. The statistical
parameters of the influencing factors are displayed in Table~\ref{tab3}. The
range of the air gap and the rotation angle along the $Z$-axis are
referred to~\cite{38}. Meanwhile, the range for the misalignment along
the $X/Y$-axis is considered reasonable due to the size of the parking
space and the size of the EV chassis. 


\begin{table}
\caption{\label{tab3} Properties of the influencing factors}
\begin{tabular}{ccccc}
\thead
Parameters & {Symbol} & {Distribution} & {Mean value} & {{Standard deviation}}\\
\endthead
Misalignment along $X$-axis {(}mm{)} & ${\rmDelta }{x}$ & Gaussian & 0 & 150 \\
Misalignment along $Y$-axis {(}mm{)} & ${\rmDelta }{y}$ & Gaussian & 0 & 150 \\
Air gap between two coils {(}mm{)} & ${\rmDelta}{z}$ & Gaussian & 150 & 20 \\
Rotation angle along $Z$-axis {(}deg{)} & ${\alpha }$ & Gaussian & 0 & 3
\botline
\end{tabular}
\end{table}

A {parametric sweep for all these influencing factors is very
time-consuming. So, it is relevant to build a metamodel using the
COMSOl simulation results, which will help save computational time. The
following section presents a detailed study of the metamodels of the
mutual inductance $M$, considering sources of uncertainty.}

\section{Metamodeling techniques introduction}\label{sec3}

This section provides an overview of the mathematical framework behind
three metamodeling techniques:

\begin{itemize}
\item 
{Support Vector Regression (SVR) with RBF kernel;}
\item 
{Multigene Genetic Programming Algorithm (MGPA);}
\item 
{Sparse Polynomial Chaos Expansions (sparse PCE).}
\end{itemize}
They are considered promising techniques which allow building
metamodels for the nonlinear system responses with several
variables~\cite{14,15, 20,21,22,23,24,25,26}. 

\subsection{Support vector regression metamodeling}\label{sec3.1}

Support vector regression (SVR) is a metamodeling technique
approximating an unknown or expensive-to-evaluate model. It represents
a class of learning techniques for regression tasks developed by
Vapnik~\cite{40}. This method provides significant generalization
capabilities, thus making it less likely to overfit data. 


{SVR attempts to approximate the relationship between the input
variables} $x=[x_{1},\ldots ,x_{d}]\in {\mathbb{R}}^{d}$ {and the output}
$y\in \mathbb{R}$ {given a training data set of} $N$ {samples}
${\{({\mathbi{x}}_{i},y_{i})\}}_{i=1}^{N}$
($y=M(\mathbi{x})$  {is the model response of the system supposed
to be a scalar quantity with a finite variance, where} $M$ {is a
numerical model presenting the observed phenomenon). It achieves this
through the equation~\cite{14, 18, 21}:}
{\begin{equation}
\label{eq2}
\mathcal{M}^{\mathrm{SVR}}(\mathbi{x})={\mathbi{w}}^{T}\rmPhi (\mathbi{x})+b
\end{equation}}\unskip
{where} $\rmPhi (\mathbi{x})=[\phi _{1}(\mathbi{x}),\ldots ,\phi
_{D}(\mathbi{x})]$ {is a nonlinear mapping function} $\rmPhi (\cdot )$:
${\mathbb{R}}^{d}\rightarrow {\mathbb{R}}^{D}$ {which maps the parameter
space of dimension} $d$ {into the corresponding} {\textit{feature
space}} {of dimension} $D$; $\mathbi{w}\in {\mathbb{R}}^{D}$ {is a
vector collecting the unknown coefficients of the nonlinear
regression;} $b\in \mathbb{R}$ {is a bias term that is retrieved as a
by-product of the solution in~\cite{14,15, 18};}
${\mathbi{w}}^{T}\rmPhi (\mathbi{x})$ {is defined as the
inner product in} ${\mathbb{R}}^{D}$~\cite{18}. {The dimensionality of
the} {\textit{feature space}} $D$ {is defined by the nonlinear map}
$\rmPhi (\mathbi{x})$.


{Assuming that we can tolerate a deviation of at most} $\varepsilon $
{between} $\mathcal{M}^{\mathrm{SVR}}(\mathbi{x})$ {and} $y${, so only
when the absolute value of the difference between}
$\mathcal{M}^{\mathrm{SVR}}(\mathbi{x})$ {and} $y$ {is greater than}
$\varepsilon ${, it needs to reduce this deviation. Then, the SVR model
expression can be formalized to find} $\mathbi{w}$ {following the
minimization equation~\cite{18}:}
{\begin{equation}
\label{eq3}
\min_{\mathbi{w}}
\frac{1}{2}\|\mathbi{w}\|^{2}+C\sum _{i=1}^{N}{\mathcal{L}}^{\varepsilon }
(\mathcal{M}^{\mathrm{SVR}}({\mathbi{x}}_{i}),y_{i})
\end{equation}}\unskip
{where} $C\in {\mathbb{R}}^{+}$ {is a regularization parameter, chosen by
cross-validation, which provides a trade-off between the accuracy of
the model on the training data set and its flatness to avoid
overfitting leading to an oscillating behavior~\cite{18};}
${\mathcal{L}}^{\varepsilon }$ {is the} $\varepsilon ${-insensitive
loss function, which is most widely used as follows (called:}
$\mathcal{L}_{1}${-penalization)~\cite{14,15, 18}:}
{\begin{equation}
\label{eq4}
{\mathcal{L}}_{1}^{\varepsilon }(\mathbi{x};y)=\begin{cases} 0&
\mbox{if}\ | \mathcal{M}^{\mathrm{SVR}}(\mathbi{x})-y| < \varepsilon,\\
(| \mathcal{M}^{\mathrm{SVR}}(\mathbi{x})-y| -\varepsilon ) & 
\mbox{otherwise}.
\end{cases}
\end{equation}}\unskip

{A nonlinear regressor considering this loss function is illustrated in
Figure~\ref{fig7}(a). Any point that is outside the} $\varepsilon
${-insensitive tube needs to be penalized, illustrated in
Figure~\ref{fig7}(b). The best combination of the parameters}
$(\mathbi{w},b)$ {minimizes the deviation of the model predictions
from the training samples outside the ${\varepsilon}$-intensive
zone.} 


\begin{figure}
\includegraphics{fig07}
\caption{\label{fig7}(a) Only the vectors outside the $\varepsilon $
-insensitive tube (dotted line area) are penalized; (b) Penalization of
deviations larger than $\varepsilon $ for ${\mathcal{L}}^{\varepsilon
}$ loss function~\cite{14,15, 18}.}
\end{figure}


The parameters for building an SVR metamodel in this chapter are shown
below, implemented within UQLAB version~2.0~\cite{18, 41}, which is
fully compatible with the MATLAB environment. UQLAB is a general
purpose Uncertainty Quantification framework developed at ETH Zurich
(Switzerland), which is made of open-source scientific modules. 
\begin{itemize}
\item 
{Loss function:} $L_{1}$\ $\varepsilon ${-insensitive.} 
\item 
{Kernel function: Gaussian radial basis function (RBF)}
{\begin{equation}
\label{eq5}
k_{\mathrm{Gaussian}}({\mathbi{x}}_{i},{\mathbi{x}}_{j})=
\exp \left(-\frac{\|{\mathbi{x}}_{i}-{\mathbi{x}}_{j}\|^{2}}{2\sigma
^{2}}\right)
\end{equation}}\unskip
{where} $\|{\mathbi{x}}_{i}-{\mathbi{x}}_{j}\|$ {is the
Euclidean distance between} ${\mathbi{x}}_{i}$ {and}
${\mathbi{x}}_{j}${. The larger this distance, the smaller the
value of RBF.} $\sigma > 0$ {is the width of the RBF. The smoothness of
the Gaussian RBF is controlled by the magnitude of} $\sigma $ {(the
higher} $\sigma ${, the smoother the Gaussian RBF).}
\end{itemize}

\subsection{Multigene genetic programming algorithm metamodeling}\label{sec3.2}


{In MGPA, each prediction} $M^{\mathrm{MGPA}}$ {of the model output is formed by
the weighted output of the genes plus a bias term. Each gene is a
function of the} $d$ {input variables} $\mathbi{x}=\{x_{1},\ldots
,x_{d}\}$ {of the system. Given a training data set of} $N$ {samples}
${\{({\mathbi{x}}_{i},y_{i})\}}_{i=1}^{N}$
($y=M(\mathbi{x})$  {is the model response of the system supposed
to be a scalar quantity with a finite variance, where} $M$ {is a
numerical model presenting the observed phenomenon), the MGPA metamodel
can be expressed as~\cite{19}:}
{\begin{fxequation}
\begin{array}{rcl}
M^{\mathrm{MGPA}}(\mathbi{x})&=&
d_{0}+d_{1}\times \mathrm{gene}~1+\cdots +d_{Q}\times \mathrm{gene}~Q\\ 
&=& d_{0}+d_{1}\times  {\inlinefig{fx01}} 
+ d_{2}\times {\inlinefig{fx02}} 
+\cdots +d_{Q}\times {\inlinefig{fx03}} 
\end{array}
\label{eq6}
\end{fxequation}}\unskip
{where} $d_{0}$ {is the bias term,} $d_{1},\ldots ,d_{Q}$ {are the gene
weights and} ${Q}$ {is the number of genes. The weights}
$\mathbi{d}(\mathbi{d}=[d_{0}d_{1}\cdots d_{Q}])$ {for the
genes are automatically determined by using an ordinary least square
method to regress the genes against a training data set~\cite{19}. Each
gene combines a set of elementary functions with the input variables
(such as sum, multiplication, division, logarithm, arctangent,
hyperbolic tangent, sine, exponential, power function, etc.), and the
gene depth is the number of levels in the gene structure. The
expression of the MGPA metamodel is evolved automatically by using the
training data set~\cite{19, 22,23}.}

The process of building a metamodel with the MGPA method is~\cite{19}: 
\begin{enumerate}[{(1)}]
\item 
{Load the training data set (a set of existing input values and
corresponding model response values);} 
\item 
{The genetic algorithm works on a population} {of metamodels, each one
representing a potential} {solution for expressing the relationship}
{between the input variables and the model response. The initial
population of the metamodels is evolved automatically by using the
training data set. During its evolution, this algorithm transforms the
current population of metamodels into a new population by applying the
classical genetic operations (selections, cross over, mutation,
etc.)~\cite{42}. When it achieves the maximum generation, the MGPA
metamodel will be picked out in terms of high coefficient of
determination (}$R^{2}${) and low model complexity~\cite{19}. The model
complexity is computed as the simple sum of the number of nodes (the
number of elementary functions plus the number of occurrences of the
input variables) inside its constituent genes~\cite{43}, and} $R^{2}$
{is calculated as below~\cite{19}:}
{\begin{equation}
\label{eq7}
R^{2}=1-\frac{{\sum }_{i=1}^{N}(M({\mathbi{x}}_{i})-{M^{\mathrm{MGPA}}}
({\mathbi{x}}_{i}))^{2}}{{\sum }_{i=1}^{N}\left(M({\mathbi{x}}_{i})
-\frac{1}{N}{\sum }_{i=1}^{N}M({\mathbi{x}}_{i})\right)^{2}}
\end{equation}}\unskip
{where} $M({\mathbi{x}}_{i})$ {is the} $i${th} {value
from the studied system,} $M^{\mathrm{MGPA}}({\mathbi{x}}_{i})$
{is the predicted value on the MGPA metamodel, and} $N$ {is the number
of samples in the training data set. This value ranges from 0 to 1.} 
\end{enumerate}
The MGPA toolbox is provided by GPTIPS, which is a free, open-source
MATLAB-based software platform~\cite{19}. It can automatically evolve
both the structure and the parameters of the mathematical model from
the training data set. However, how to appropriately define the maximum
number of genes and the maximum gene depth for an accurate MGPA
metamodel needs to be carefully considered. 

\subsection{Sparse polynomial chaos expansions metamodeling}\label{sec3.3}

Polynomial Chaos Expansions (PCE) is a metamodeling technique that
provides a functional approximation for the relationship between the
input variables and model output in a non-intrusive way~\cite{20, 44}.
It means that it focuses only on the one-to-one mapping relationship
between input and output. Furthermore, the post-process of the PCE
metamodel can also help to find the most influential input variable to
the model output.


{It starts by considering the vector} $\mathbi{x}\in
{\mathbb{R}}^{d}$ {collecting} $d$ {independent input variables}
$\{x_{1},\ldots ,x_{d}\}$ {with a joint probability density function
(PDF)} $f_{\mathbi{X}}(x)${, representing the input
variables of the physic system. Given a training data set of} $N$
{samples} ${\{({\mathbi{x}}_{i},y_{i})\}}_{i=1}^{N}$
($y=M(\mathbi{x})$  {is the model response of the system supposed
to be a scalar quantity with a finite variance, where} $M$ {is a
numerical model presenting the observed phenomenon), the PCE metamodel
is established to simulate the varying trend of the model
response~\cite{20, 44}:}
{\begin{equation}
\label{eq8}
M^{\mathrm{PCE}}(\mathbi{x})=\sum _{\alpha \in \mathbb{N}^{d}}\hat{c}_{\alpha
}\rmPhi _{\alpha } (\mathbi{x})
\end{equation}}\unskip
{where} $\hat{c}_{\alpha }$ {are the unknown deterministic
coefficients, and} $\rmPhi _{\alpha }(\mathbi{x})$ {are multivariate
polynomials basis functions which are orthonormal with respect to the
joint PDF} $f_{\mathbi{x}}(x)$. $\alpha \in
\mathbb{N}^{d}$ {is a multi-index that identifies the components of the
multivariate polynomials} $\rmPhi _{\alpha }${. If the input variables
have a uniform distribution, the orthogonal polynomial is Legendre;
while if the input variables have a Gaussian distribution, the
orthogonal polynomial is Hermite~\cite{44}.} 


{The coefficients} $\hat{c}_{\alpha }$ {are obtained by post-processing
the experimental design}
${\{({\mathbi{x}}_{i},y_{i})\}}_{i=1}^{N}${, a training
data set consisting of} $N$ {samples of the input variables and the
corresponding model responses} $y${. From the set of model responses,
the coefficients can be estimated by the ordinary least square
regression method~\cite{16, 20, 44}. For this, the infinite series in
Equation (\ref{eq9}) has to be truncated. Choosing a maximum polynomial degree}
$p${, the usual truncation scheme preserves all polynomials associated
with the set~\cite{16, 20, 44}:}
{\begin{equation}
\label{eq9}
{\mathcal{A}}^{d,p}=\left\{\alpha \in \mathbb{N}^{d}\colon \|\alpha \|_{1}=
\sum _{i=1}^{d}\alpha _{i}\leq p\right\}.
\end{equation}}\unskip

{Thus, the cardinal of the set} ${\mathcal{A}}^{d,p}$ {denoted}
$L={(d+p)!}/{d!p!}$ {increases quickly with the number
of input variables} $d$ {and the degree} $p$ {of the
polynomials~\cite{16, 20}. This leads that the size of the PCE retained
in the set} ${\mathcal{A}}^{d,p}$ {will be too large when dealing with
high-dimensional problems.} 


{In order to overcome this limitation, a hyperbolic truncation
strategy} ${\mathcal{A}}_{q}^{d,p}$ {based on the total degree} $p$
{and a parameter} $q${, with} $0< q< 1${, allowing for reduction of the
size of the PCE basis is then defined as follows~\cite{16, 20, 44}:}
{\begin{equation}
\label{eq10}
{\mathcal{A}}_{q}^{d,p}=\left\{\alpha \in \mathbb{N}^{d}\colon \|\alpha \|_{q}=
\left(\sum _{i=1}^{d}{\alpha }_{i}^{q}\right)^{{1}/{q}}\leq p\right\}.
\end{equation}}\unskip

{This favors the most relevant effects and low-order interactions,
which are known to have the largest impact on the variability of the
model response according to the sparsity-of-effects
principle~\cite{44}. It is important to point out that lower values of}
$q$ {imply a larger number of neglected high-rank interactions. In
addition, when} $q=1${, this scheme is equivalent to the standard PCE.
When} $q< 1${, the retained terms of the polynomial basis can be
substantially reduced~\cite{20, 44}.} 

\subsection{Error estimates of a metamodel}\label{sec3.4}

After the metamodel is constructed, its accuracy can be quantified by
estimating the Root Mean Square Error (RMSE) obtained with the
metamodel on the training data set. It is defined as:
{\begin{equation}
\label{eq11}
\varepsilon _{\mathrm{RMSE}}=\sqrt{\frac{{\sum
}_{i=1}^{\mathrm{N}}({M^{\mathrm{metamodel}}}
({\mathbi{x}}_{i})-M({{\mathbi{x}}_{i}}))^{2}}{N}}
\end{equation}}\unskip
{where} $M({\mathbi{x}}_{i})$ {is the model response of the
training data set,}
$M^{\mathrm{metamodel}}({\mathbi{x}}_{i})$ {is the
prediction value} {from the metamodels above,} {and} $N$ {is the number
of samples in the training data set.}

Except for the training data set used to construct the metamodel, a
test data set different from the training samples, can be used to
validate the predictive performance. The test error between the test
data set and the predictive values on the metamodel can also be
calculated by RMSE.

\section{Metamodeling techniques for square couplers taking into
account sources of\newline uncertainty}\label{sec4}

Here, the SVR with RBF kernel, MGPA, and the sparse PCE metamodeling
techniques are implemented to build a metamodel for small-scale square
couplers and are compared below. The results given in this section were
done with a XEON~E5-1620, 8-core processor, working at 3.70~GHz. The 3D
model of the couplers is obtained by COMSOL~5.6, and the SVR and PCE
metamodels are calculated in MATLAB~2019b with the UQLAB Framework,
while the MGPA metamodel is calculated in MATLAB~2017b due to the
limitation of the GPTIPS toolbox functions. 


{The SVR with RBF kernel, MGPA, and the sparse PCE methods have been
adopted to quantify the impact of these uncertainty parameters on the
mutual inductance} $M$ {of small-scale square couplers. In addition,
the parameters on MGPA and sparse PCE methods are chosen considering
the metamodel accuracy and the computational time to build an accurate
metamodel. The SVR with RBF kernel builds a metamodel in light of the}
$L_{1}$\ $\varepsilon ${-insensitive loss function. The MGPA metamodel is
performed with the following settings: Population size~\smath{=}~300,
Number of generations~\smath{=}~100, Maximum number of
genes~\smath{=}~6, and Maximum gene depth~\smath{=}~4. The sparse PCE
metamodel is constructed by the adaptive degree method~\cite{20, 44},
in which the degree of PCE metamodel varies from 3 to 15 to select the
most accurate one. The hyperbolic scheme in Equation (\ref{eq10}) is
set to} $q=0.75$ {to reduce the size of the polynomial basis.} 


{1000 samples are chosen by the Latin hypercube sampling (LHS)
method~\cite{45} and formed as the database. All metamodels have been
trained from the same training dataset containing around one-third of
the samples of the database. This data set results from COMSOL
simulations with a computational cost of 6~h (one simulation with the
full solver takes about 50~s). To investigate the performance of the
obtained metamodels, their predictions are then compared with a test
data set containing the remaining database samples. 
Table~\ref{tab4} provides a
detailed comparison of the accuracy and the computational cost of the
proposed metamodeling techniques by collecting the RMSE on the training
data set and the test data set, along with the corresponding
computational time} $t_{\mathrm{training}}$ {and the predictive time}
$t_{\mathrm{predictive}}$ {(to predict one output) respectively. It
also shows that the sparse PCE metamodel turns out to use the least
computational time and to be the most accurate metamodel with a
training RMSE and a test RMSE, which is better than the accuracies
obtained by the SVR and MPGA metamodels.}


\begin{figure}
\includegraphics{fig08}
\caption{\label{fig8}Scatter plots of the mutual inductance providing a
comparison among the predictions of the SVR metamodel with RBF kernel
(red marker in (a)), the MGPA metamodel (black marker in (b)), the
sparse PCE metamodel (blue marker in (c)) and the results of COMSOL
computations.}
\end{figure}

\begin{table}
\caption{\label{tab4} Comparison of the accuracy and the computational
cost of the SVM, MPGA, and PCE metamodels computed for the square
couplers}
\begin{tabular}{ccccc}
\thead
{Method} & {Training RMSE} & {Test RMSE} & 
${{t}}_{{\mathrm{training}}}$ & ${{t}}_{{\mathrm{predictive}}}$ \\ 
\endthead
COMSOL computations & - & - & 6.45 h & 60~s \\ 
SVR (RBF) & 0.0266 & 0.0420 & 2.48~s & {\textless}1~s \\
MGPA & 0.0233 & 0.0475 & 313.54~s & {\textless}1~s \\
Sparse PCE & 0.0158 & 0.0270 & 0.357~s & ${<}$1~s 
\botline
\end{tabular}
\end{table}

Furthermore, another 1000 samples selected by the MC method are
computed in COMSOL to form a new data set. Figure~\ref{fig8} provides
the scatter plots for the metamodels of the mutual inductance on the
SVR, MGPA, and sparse PCE methods. These plots emphasize a good
agreement between these metamodels and this data set because the
samples are very close to the solid lines. 

Then, the impact of the influencing factors on the mutual inductance is
illustrated in Figure~\ref{fig9}, where the probability density
functions (PDFs) of the mutual inductance estimated via the SVR, MGPA,
and sparse PCE metamodels are compared with the PDF of the COMSOL
computations. It can be seen that the variability of the mutual
inductance is well captured by these metamodels, which confirms a good
estimation of the PDF of the mutual inductance with these metamodels
and highlights a similar level of accuracy. In terms of computational
cost, this data set, including 1000 samples, required about 14~h to
compute in COMSOL, while the metamodels on the SVR, MGPA, and sparse
PCE need less than 1 minute. It is worth noting that this computational
cost does not include the time to generate the training samples from
LHS in COMSOL, which were needed for constructing the metamodels.


\begin{figure}
\includegraphics{fig09}
\caption{\label{fig9}PDFs of the mutual inductance obtained from the
SVR (solid red curve), MGPA (solid black curve) and sparse PCE
metamodels (solid blue curve) compared with the PDF of COMSOL
computations (solid magenta curve).}
\end{figure}

{Compared to the other metamodeling techniques based on the same
dataset, the sparse PCE metamodeling technique uses less time to build
a metamodel and provides more accurate results. So, it is meaningful to
choose such an approach to analyse the performances of the coupler,
taking into account the sources of uncertainty.}

\section{Conclusion}\label{sec5}

This paper provides an overview of SVR with RBF kernel, MGPA, and PCE
metamodeling techniques and their application in case of analyzing an
IPT system. When the metamodels are built by these techniques, the ways
to evaluate the accuracy and build the PDF are also summarized. Some
metamodeling techniques (SVR with RBF kernel, MGPA, and sparse PCE) are
built and compared for analyzing the mutual inductance $M$ {of the
magnetic couplers} considering sources of uncertainty. Due to the
tradeoff between the computational time and the accuracy of the
metamodel, the sparse PCE metamodeling technique appears to be a very
useful tool in the analysis of IPT systems. Then, it will be used in
the next step to address the magnetic and thermal coupled field
analysis. Indeed, when a high-power IPT system works for a long time,
the heating of the magnetic coupler brings adverse effects on the
efficiency and stability of the system. An accurate prediction of such
a phenomenon will improve the assessment of IPT performances.

\section*{Declaration of interests}
The authors do not work for, advise, own shares in, or receive funds
from any organization that could benefit from this article, and have
declared no affiliations other than their research organizations.

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