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\section{POG modeling of an automotive hybrid system}

The planetary gear is the key element for modeling the hybrid
powertrain structure shown in Fig.~\ref{schema_epicicloidale}.
\begin{figure}[tbp]
 \centering
 \psfrag{r}[c]{R}
 \psfrag{c}[c]{C}
 \psfrag{s}[c]{S}
 \psfrag{i}[l]{ICE}
 \psfrag{p}[c]{PMSM}
 \psfrag{k}[b]{$K_{2}$}
 \psfrag{h}[b]{$K_{1}$}
 \psfrag{a}[b]{\nrsec{7}}
 \psfrag{g}[b]{\nrsec{1}}
 \psfrag{d}[b]{\nrsec{2}}
 \psfrag{b}[b]{\nrsec{3}}
 \psfrag{e}[b]{\nrsec{4}}
 \psfrag{f}[b]{\nrsec{5}}
 \includegraphics[width=7.5cm, angle=0]{epicicloidale.eps}
 \caption{Scheme of the considered power structure of the vehicle}
 \label{schema_epicicloidale}
\end{figure}
The considered structure includes an internal combustion engine
(ICE) and a multi-phase permanent magnet synchronous machine (PMSM).
The planetary gear is introduced as the key element connecting the
two motors and the driving wheels. The ICE is rigidly connected to
the Carrier (C), the PMSM is connected to the Sun (S) by means of an
elastic element $K_{1}$ and the driving axle is connected to the
Ring (R) by means of an elastic element $K_{2}$. It is possible to
represent this physical scheme with an ``high level'' POG
representation shown in Fig.~\ref{POG_hybrid}. The first block
represents the control input to the PMSM. The synchronous machine
present between sections \nrsec{1} and \nrsec{2} corresponds to the
POG scheme given in Fig.~\ref{POG_PMSM}. The elaboration block
between sections \nrsec{2} and \nrsec{3} represents the elasticity
of the shaft connecting the electric motor to the Planetary Gear
Sun. The POG model of the Planetary Gear is between sections
\nrsec{3} and \nrsec{4} and it is connected to the driving shaft of
the vehicle through the elasticity represented by the elaboration
block between sections \nrsec{4} and \nrsec{5}. The dynamics of the
vehicle are considered in the block between sections \nrsec{5} and
\nrsec{6}: the corresponding POG is presented in
\cite{2009_ELECTROMOTION}. Finally the air resistance is represented
in the last block.
%
 \begin{figure*}[t]
        \begin{center}
 \schemaPOG{(0,-4)(58,14)}{3mm}{
 \blosuretor{\begin{minipage}[c]{1.1cm} Control \end{minipage}}{\V}{\I}
 \nrsezxgiu{0.5}{$\scr 1$}{-1.75}
  \scrivigiu{8}{2}{\scriptsize Synchronous machine}
 \bloxboxgiu{5}{\begin{minipage}[c]{1cm} PMSM \end{minipage}}{}{\omega_{m}}
 \nrsezxgiu{0.5}{$\scr 2$}{-1.75}
  \scrivigiu{8}{2}{\scriptsize Elasticity}
 \blosuattdue{}{K_{1}}{b_{1}}{}{\tau_{s}}{\dst}
 \nrsezxgiu{0.5}{$\scr 3$}{-1.75}
  \scrivigiu{8}{2}{\scriptsize Planetary Gear}
\put(2,10.5){\vector(0,1){1.5}} \put(6,12){\vector(0,-1){1.5}}
\psline[linestyle=dashed, linewidth=0.2pt](1,11.25)(7,11.25)
\put(2,12.5){\makebox(0,0)[c]{$\omega_{sc}$}}
\put(6,12.5){\makebox(0,0)[c]{$\tau_{sc}$}}
%\put(7.6,11){\makebox(0,0)[c]{$\nrsec{7}$}}
 \bloxboxgiu{5}{\begin{minipage}[l]{0.7cm} Plan. \\ Gear \end{minipage}}{\omega_{c}}{\omega_{r}}
 \nrsezxgiu{0.5}{$\scr 4$}{-1.75}
 \put(-0.5,0.25){\nrsezxsu{0.5}{$\scr 7$}{-0.2}}
  \scrivigiu{8}{2}{\scriptsize Elasticity}
 \blosuattdue{}{K_{2}}{b_{2}}{}{\tau_{r}}{\dst}
 \nrsezxgiu{0.5}{$\scr 5$}{-1.75}
  \scrivigiu{10}{2}{\scriptsize Vehicle}
 \bloxboxgiu{7}{ \begin{picture}(5,2) \includegraphics[width=1.5cm]{icon_car.eps} \end{picture}}{\omega_{1}}{}
 \nrsezxgiu{0.5}{$\scr 6$}{-1.75}
  \scrivigiu{6}{2}{\scriptsize Air Resistance}
 \blosuretao{\begin{minipage}[l]{0.7cm} Air \\ Res. \end{minipage}}{\F_{res}}{\dot{\X}}
 }
  \end{center}
  %\vspace{4.6cm}
 \caption{\label{POG_hybrid} Model of the hybrid vehicle.}
 \end{figure*}
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
\begin{figure*}[t]
 \centering
 \includegraphics[width=5cm, angle=-90]{architetture_epi.eps}
 \caption{Scheme of the considered vehicle}
 \label{architetture_epi}
\end{figure*}
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
\begin{figure}[htbp]
   \begin{center}
 \schemaPOG{(0,-4)(25,12)}{3.0mm}{
 \thicklines
 \bloin{\V}{\I}
 \nrsezgiu{$\scr 1$}{-1.75}
 \scrivigiu{10}{1.8}{ \footnotesize Electrical part}
 \blovigiumxm{\ds\L\muno}{}{\dst}{\frac{1}{s}}{ \tiny \boldsymbol{\Phi}}
 \bloivsuaor{\R}{}{}
 %\nrsezgiu{$\scr 2$}{-1.75}
 \scrivigiu{6}{1.8}{\footnotesize Energy \\ \footnotesize conversion}
 \blokxyaor{4}{3}{\ds \K_{\tau}(\theta)}
                 {\ds \K_{\tau}\tras(\theta)}{\E}{\tau_{m}}
 %\nrsezgiu{$\scr 3$}{-1.75}
 \scrivigiu{10}{1.8}{ \footnotesize Mechanical part}
 \blovisumxm{\ds\frac{1}{J_m}}{\omega_{m}}{\dst}{\frac{1}{s}}{}
 \bloivgiuor{b_m}{}{}
 \nrsezgiu{$\scr 2$}{-1.75}
 \bloout{}{\tau_{s}}
 \scrivigiu{}{1.8}{}
 }
  \end{center}
  \caption{POG scheme of the multi-phase synchronous motor (PMSM).}\label{POG_PMSM}
\end{figure}

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\section{Simulation}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

The system shown in Fig.~\ref{schema_epicicloidale} has been
implemented in Matlab/Simulink as it is shown in
Fig.~\ref{architetture_epi}. All subsystems have been modeled with
POG and in particular the vehicle subsystem in the right part of the
scheme is a bicycle model of a car that includes the tire-road
elastic interaction, see \cite{2009_ELECTROMOTION}.






%\subsection{Notations}
%
%In this paper the following notations will be used:\\
%%- Row and column matrices:
%%%
%%\[
%% \matll{}{}{i}{1:n\hspace{-1mm}}{ R_i } = \mat{cccc}{R_{1}&R_{2}&\ldots &R_{n}},
%% \hspace{0.3cm}
%% \matll{i}{1:n\hspace{-1mm}}{}{}{ R_i } = \mat{c}{R_{1}\\R_{2}\\ \vdots \\R_{n}}
%%   \]
%%
%%- Diagonal matrices:
%%\[
%% \matll{i}{}{}{1:n\hspace{-1mm}}{ R_i } =
%% \mat{@{\,}c@{\,}c@{\,}c@{\,}c@{\,}}{R_{1}&&&\\&R_{2}&&\\ &&\ddots&
%% \\&&&R_{n}},
%% \hspace{0.3cm}
%% \matll{}{\hspace{-1mm}1:n}{i}{}{ R_i } = \mat{@{\,}c@{\,}c@{\,}c@{\,}c@{\,}}
%% {&&&R_{1}\\&&R_{2}&\\ &\cdots&& \\R_{n}&&&}
%% \]
%- Full matrices:
%%
%\[
% \matll{i}{1:n\hspace{-1mm}}{j}{1:m\hspace{-1mm}}{ R_{i,j} }
%  = \mat{cccc}{
%  R_{11}&R_{12}&\cdots &R_{1m}\\
%  R_{21}&R_{22}&\cdots &R_{2m}\\
%  \vdots&\vdots&\ddots &\vdots\\
%  R_{n1}&R_{n2}&\cdots &R_{nm}\\
%  }.
% \]
% %
% - The symbols $\j$ and $ e^{\j\theta}$ denote the following
%matrices:
%%
% \[
% \j =\left[\begin{array}{cc}
%  0 & -1 \\
%  1 &  0
% \end{array}\right],
% \hspace{0.6cm}
%  e^{\j\theta}=\left[\begin{array}{cc}
%  \cos{\theta} & -\sin{\theta} \\
%  \sin{\theta} & \cos{\theta}
% \end{array}\right].
%  \]
%%
%- The symbol $\I_{m}$ denotes an identity matrix of order $m$.
