\chapter{Full Electric Car Modeling}
\section{General Model Description}
\subsection{Bicycle Model}
In the first time the discussion of vehicle fundamentals will be restricted to one-dimensional movement (bicycle model). The movement behaviour of a vehicle along its moving direction is completely determined by all the forces acting on it in this direction.

According to the Newton's second law, vehicle acceleration can be written as

\begin{equation}
\frac{\displaystyle dV}{\displaystyle dt}=\frac{\sum F_t - \sum F_r}{\displaystyle \delta M}
\end{equation}

where V is the speed of the vehicle, $\sum F_t$ is the total traction effort of the vehicle, $\sum F_r$ is the total resistance, M is the total mass of the vehicle, and $\delta M$ is the mass factor that equivalently converts the rotational inertias of rotating components into translational mass.

\begin{figure}[ht]
    \center
    \includegraphics[width=12cm]{images/forces_car.png}
    \caption{Forces acting on a vehicle moving uphill}
    \label{fig:forces}
\end{figure}

As shown in figure~\ref{fig:forces}, the total resistance is composed mainly from rolling resistance $F_r$ (which combines the torques $Tr_f$ and $Tr_r$), wind resistance $F_w$, and climbing resistance $F_c$ (the term $Mg\sin\alpha$). In the opposite direction acts total traction effort $F_t$ which is composed from the front and rear tires efforts $F_{tf}$ and $F_{tr}$ generated by the prime movers\footnote{This force can be also negative for example in the case of braking.} and minus all friction losses in the powertrain. By developing the equation above, we obtain the dynamic equation it the form:

\begin{equation}
M\frac{\displaystyle dV}{\displaystyle dt}= F_{t} - (F_{r} + F_{w} + F_{c})
\end{equation}

Using the dynamic equation, we can express the maximal traction effort for both front and rear wheels. Then we can also examine the tire-ground adhesion which is connected with the maximal traction effort. The detailed description of the characteristics summarized in this chapter is presented in \cite{ehsani10}.

\subsection{Longitudinal Model}
\label{subsec:long_model}
The vehicular model of longitudinal dynamics is constructed based on the theory of vehicle multi-body dynamics as seen before.

\subsubsection{Wind Resistance}
The wind resistance is caused generally by two facts: the viscous friction of the surrounding air on the vehicle surface and the losses by the pressure difference between the front and the rear of the vehicle, generated by a separation of the air flow. 

The resistance expression can be usually simplified by considering a vehicle as a prismatic body with a frontal area $A_f$. The pressure difference force can by multiplied by an aerodynamic drag coefficient $C_d$ also known as Reynold's coefficient (estimated by simulation or experiment in wind tunnel) so the wind resistance can be written as

\begin{equation}
F_w(v)=\frac{1}{2} \cdot \rho \cdot A_f \cdot C_d \cdot v^2
\label{eq:Fa}
\end{equation}

where $v$ is the vehicle speed and $\rho$ is the density of the ambient air. 

\subsubsection{Tire Rolling Resistance}
The tire rolling resistance can be expressed as

\begin{equation}
F_r(v,p,...)=\mu_r(v,p,...) \cdot M \cdot g \cdot \cos (\Theta),\; v > 0
\label{eq:Fr}
\end{equation}

where M is the vehicle mass, g the acceleration due to gravity, term $\cos (\Theta)$ is road slope influence and $\mu_r$ is the rolling friction coefficient. 

The rolling friction coefficient depends on many variables. Especially vehicle speed $v$, tire pressure $p$, and road
surface conditions. 

\subsubsection{Climbing Resistance}
The climbing resistance is conservative force induced by gravity and considerably influences the vehicle behaviour.

\begin{equation}
F_c(\Theta)= M \cdot g \cdot \ \sin (\Theta)
\label{eq:Fc}
\end{equation}


\subsubsection{Acceleration Resistance}
The acceleration resistance due to the inertia of the vehicle and of all rotating parts inside the vehicle causes
frictious (d'Alembert) forces. The rotating parts are here represented by 

\begin{equation}
F_a(\dot{v})= M(1+\delta_{eqm}) \cdot \frac{dv}{dt}
\label{eq:Fc2}
\end{equation}

where $\delta_{eqm}$ is the vehicle equivalent moment inertia representing the rotating parts such as wheels and powertrain inertia. 
\\

According to an analysis of the summation of performing forces acting on the vehicle body in the longitudinal direction, the power balance for the controlled vehicle is governed by:

\begin{equation}
\begin{tabular}{ r l }
\(P_D\left(t\right)\) & \(=f_{P_D}\left(v,\dot{v},M,\Theta\right)=\left(F_w+F_r+F_c+F_a\right)v\) \\
 & \(=\left[\frac{\displaystyle \rho}{\displaystyle 2}A_fC_dv^3+Mg\left(\mu_r+\cos\Theta+\sin\Theta\right)v\right]+M\left(1+\delta_{eqm}\right)\dot{v}v\)
\end{tabular}
\label{eq:Pd}
\end{equation}

And the following relationships are satisfied (for the topology in figure~\ref{fig:topology3}):

\begin{align}
P_D & = P_{CV}+P_B, P_m = \frac{P_D}{\varepsilon_m} \notag \\
T_m & = \frac{(F_w+F_r+F_c+F_a)r}{i_0i_s} \\
\omega_m & = \frac{vi_0i_s}{r_t}\notag
\end{align}

Where $P_D$ is the demanded power, $P_{CV}$ is the power delivered by DC/DC converter, $P_B$ is the power delivered by battery, $P_m$ is the motor demanded power, $\varepsilon_m$ is the motor efficiency, $T_m$ is the motor traction force, $\omega_m$ is the  motor's rotational speed, $F_w$ is the wind resistance, $F_r$ is the tire rolling resistance, $F_c$ is the climbing resistance, $F_a$ is the acceleration  resistance, $\rho$ is the air density, $A_f$ is the windward area for the vehicle, $C_d$ is the Reynolds coefficient, $g=9.8 m/s^2$, $\mu_r$ is the rolling resistance coefficient, $v$ is the velocity, $\dot{v}$ is the acceleration, $M$ is the mass, $\Theta$ is the road grade, $\delta_{eqm}$ is the vehicle equivalent moment inertia, $i_0$ is the final reduction ratio, $r_t$ is the effective tire radius, is $(s=1,...,6)$ is the gear position ratio.

\subsubsection{Position-based Power Calculation} 
As an important parameter in equation~\eqref{eq:Pd}, the road grade may change frequently at the actual road environment, especially in the mountain terrain, and has become a major impact on the energy consumption. In general, the proper measurement to record the road grade by GPS and 3D-map is dependent on position rather than time. Therefore, the time-based equation~\eqref{eq:Pd} is not suitable for the purpose of control algorithm design, and a power calculation depended on position is preferable. Using the transformation

\begin{equation}
\label{eq:trans1}
v = \frac{ds}{dt}
\end{equation}

\begin{equation}
\label{eq:trans2}
\frac{dv}{ds} = \frac{dt}{ds}\frac{dv}{dt} = \frac{1}{v}\frac{dv}{dt},v \ne 0
\end{equation}

equation~\eqref{eq:Pd} can be transferred into a position-based form

\begin{equation}
\label{eq:Pdtrans}
\begin{array}{l}
 P_D \left( s \right) = \left[ {\frac{\rho }{2}A_f C_d v^2 + Mg\left( {\mu
_r \cos \theta + \sin \theta } \right)} \right] \\
 {\begin{array}{*{20}c}
 \hfill & \hfill \\
\end{array} } + M\left( {1 + \delta _{eqm} } \right)v\frac{dv}{ds} \\
 \end{array}
\end{equation}


%\section{Mechanical Energy Demand}
%As it is introduced in~\cite{guzzela07}, the mechanical energy is in general used to determine the first estimations of fuel consumption required to make a vehicle follow the MVEG-95 cycle\footnote{The combined cycle proposed by the motor vehicle expert group in 1995 (MVEG-95) repeats the ECE (the European urban driving cycle) four times (the first with a cold start)
%and adds an extra-urban portion referred to as the EUDC.}. The energy is expressed by the mean tractive force $F_{trac}$ 
%
%\begin{equation}
%\bar{F}_{trac} = \frac{1}{x_{tot}}\cdot \int_{t\in trac} F(t)\cdot v(t)dt
%\label{eq:F_trac}
%\end{equation}
%
%where $x_{tot} = \int_{0}^{t_{max}}v(t)dt$ is the total distance of the cycle and trac is the set of all time intervals where $F(t) > 0$, i.e., those parts of the cycle in which the vehicle drives in traction mode.
%
%The velocity profile of a test cycle is defined for the given time instants 
%$v(t_i) = v_i, t_i = i \cdot h, i = 0, ..., n$. Accordingly, the speed used to compute the mean tractive force is the average speed
%
%\begin{equation}
%v(t)=\bar{v}_{i} = \frac{v_i + v_{i-1}}{2}, \; \forall \; t \in \left[t_{i-1}, t_i\right)
%\label{eq:v_i}
%\end{equation}
%
%Similarly, the acceleration is approximated by
%
%\begin{equation}
%a(t)=\bar{a}_{i} = \frac{v_i - v_{i-1}}{h}, \; \forall \; t \in \left[t_{i-1}, t_i\right)
%\label{eq:a_i}
%\end{equation}
%
%
%An approximation of the tractive force~\eqref{eq:F_trac} can be found using the expression
%
%\begin{equation}
%\bar{F}_{trac} \approx \frac{1}{x_{tot}}\cdot \sum_{t\in trac} \bar{F}_{trac,i}\cdot \bar{v}_i \cdot h_i
%\label{eq:F_trac_approx}
%\end{equation}
%
%where the partial summation $i \in trac$ only covers the time intervals during
%which the vehicle is in traction mode.
%
%\subsection{No Recuperation}
%The total tractive force $\bar{F}_{trac}$ includes contributions from three different effects
%
%\begin{equation}
%\bar{F}_{trac} = F_{trac,a} + F_{trac,r} + F_{trac,m}
%\label{eq:F_trac_no_recup}
%\end{equation}
%
%with the three forces caused by aerodynamic and rolling friction and acceleration
%resistance defined by
%
%\begin{align}
%	F_{trac,a} & \approx \frac{1}{x_{tot}}\cdot \frac{1}{2} \cdot \rho_a \cdot A_f \cdot c_d \cdot \sum_{i\in trac} \bar{v}_i^3 \cdot h, \notag \\
%	F_{trac,r} & \approx \frac{1}{x_{tot}}\cdot m_v \cdot g \cdot \mu_r \cdot \sum_{i\in trac} \bar{v}_i \cdot h, \\
%	F_{trac,m} & \approx \frac{1}{x_{tot}}\cdot m_v \cdot \sum_{i\in trac} \bar{a}_i \cdot \bar{v}_i \cdot h. \notag
%	\label{eq:F_resis1}
%\end{align}
%
%The sums and the distance $x_0$ in these equations depend only on the driving
%cycle and not on the vehicle parameters. For the MVEG-95, the ECE cycle,
%and the EUDC, the following numerical values can easily be found
%
%\begin{align}
%	\frac{1}{x_{tot}}\cdot \sum_{i\in trac} \bar{v}_i^3 \cdot h & \approx \{319, 82.9, 455\}, \notag \\
%	\frac{1}{x_{tot}}\cdot \sum_{i\in trac} \bar{v}_i \cdot h & \approx \{0.856, 0.81, 0.88\}, \\
%	\frac{1}{x_{tot}}\cdot \sum_{i\in trac} \bar{a}_i \cdot \bar{v}_i \cdot h & \approx \{0.101, 0.126, 0.086\}. \notag
%\end{align}
%
%Once the driving cycle is chosen, \eqref{eq:F_trac_approx} allows for a simple estimation of the mean tractive force as a function of the vehicle parameters $m_v$, $A_f$ , $c_d$, and $\mu_r$. The mean tractive force is equal to $\bar{E}$, the average energy consumed per distance travelled. When the latter is expressed using the units $kJ/100~km$, the relationship between the two quantities is $\bar{E} = 100 \cdot \bar{F}_{trac}$. In these units, the energy consumed in the MVEG-95 cycle is given by
%
%\begin{equation}
%\bar{E}_{MVEG-95} \approx A_f \cdot c_d \cdot 1.9 \cdot 10^4 + m_v \cdot \mu_r \cdot 8.4 \cdot 10^2 + m_v \cdot 10 \left(kJ/100 km\right)
%\label{eq:E_no_recup}
%\end{equation}
%
%where, to simplify matters, the physical parameters air density $\rho_a$ (at sea level) and acceleration $g$ have been integrated in the constants. Considering typical values for the vehicle parameters, the three contributions in this sum are of the same order of magnitude.
%
%\subsection{Perfect Recuperation}
%Equation \eqref{eq:F_trac_approx} is valid for the case that none of the vehicle - kinetic energy
%is recuperated when braking. In the opposite case, with \textit{perfect recuperation}
%(recuperation device of zero mass and 100\% efficiency), the energy spent to
%accelerate the vehicle is completely recuperated during the braking phases. As
%a consequence, the mean force $\bar{F}$ does not have any contributions $\bar{F}_m$ caused
%by acceleration losses, i.e.,
%
%\begin{equation}
%\bar{F} = \bar{F}_a + \bar{F}_r
%\label{eq:F_trac_recup}
%\end{equation}
%
%However, in the full recuperation case, the mean force must include the losses caused by aerodynamic and rolling resistances also during the braking phases. In this case, \eqref{eq:F_resis1} must be replaced by
%
%\begin{align}
%	\bar{F}_{a} & \approx \frac{1}{x_{tot}}\cdot \frac{1}{2} \cdot \rho_a \cdot A_f \cdot c_d \cdot \sum_{i=1}^n \bar{v}_i^3 \cdot h, \notag \\
%	\bar{F}_{r} & \approx \frac{1}{x_{tot}}\cdot m_v \cdot g \cdot \mu_r \cdot \sum_{i=1}^n \bar{v}_i \cdot h.
%\end{align}
%
%with the following numerical values valid for the MVEG-5, the ECE, and the EUDC cycles
%
%\begin{align}
%	\frac{1}{x_{tot}}\cdot \sum_{i=1}^n \bar{v}_i^3 \cdot h & \approx \{363,100,515\}, \notag \\
%	\frac{1}{x_{tot}}\cdot \sum_{i=1}^n \bar{v}_i \cdot h & \approx \{1,1,1\}.
%\end{align}
%
%With these results, an approximation can be derived that is similar to
%\eqref{eq:E_no_recup} of the mechanical energy $\bar{E}_{rec,MVEG-95}$ needed to drive $100~km$ in the
%MVEG-95 cycle. In the case of full recuperation this quantity is given by
%
%\begin{equation}
%\bar{E}_{rec,MVEG-95} \approx A_f \cdot c_d \cdot 2.2 \cdot 10^4 + m_v \cdot \mu_r \cdot 9.81\cdot 10^2 \left(kJ/100 km\right)
%\label{eq:E_recup}
%\end{equation}
%
%\subsection{Mean Tractive Force Sensitivity}
%The mean tractive force necessary to drive a given cycle depends on the vehicle parameters $A_f \cdot c_d$, $\mu_r$, and $m_v$. It is possible to examine their influence by sensitivity analysis defined as
%
%\begin{equation}
%S_p = \lim_{\delta p \rightarrow 0} \frac {\left[\bar{E}_{MVEG-95}(p+\delta p)-\bar{E}_{MVEG-95}(p)\right]/\bar{E}_{MVEG-95}(p)}{\delta p/p} 
%\label{eq:S_p}
%\end{equation}
%
%where $\bar{E}_{MVEG-95}$ is the cycle energy as defined by \eqref{eq:E_no_recup}. The variable $p$
%stands for any of the three parameters $A_f \cdot c_d$, $\mu_r$, or $m_v$. Its variation is
%denoted by $\delta p$.
%
%According to the studies published in \cite{guzzela07}, we can say that
%
%\begin{itemize}
%	\item In the case of a standard vehicle, the sensitivity $S_{m_v}$ is by far the most important. Accordingly, the most promising approach to reduce the mechanical energy consumed in a cycle is to reduce the vehicle's mass. This approach implies of course also the velocity and acceleration reduction which significantly influence especially the aerodynamic resistance and the rolling and acceleration resistances.
%	\item The relative dominance of the vehicle's mass on the energy consumption is not reduced when an advanced vehicle concept is analyzed. For this reason, the idea of recuperating the vehicle - kinetic energy will remain interesting for this vehicle class as well.
%\end{itemize}
%
%\subsection{Energy Recuperation}
%In the previous sections, it was supposed, that the recuperation device can recuperate all the energy. This is of course not possible in real applications, where the recuperation efficiency is around 60~\%. In addition, the recuperation device will increase the vehicle mass and thus the energy losses. 
%
%The maximal theoretic recuperated energy is given by
%
%\begin{equation}
%\Delta \bar{E}_{rec,max} = \bar{E}_{MVEG-95}-\bar{E}_{rec,MVEG-95} \; (m_{rec}=0)
%\label{eq:E_rec}
%\end{equation}
%
%The actual value of the energy $\bar{E} (\eta_{rec},m_{rec})$ necessary to drive the MVEG-95 cycle when a recuperation device with efficiency $\eta_{rec}$ and mass $m_{rec}$ is installed is obtained by linear interpolation between the two extreme cases \eqref{eq:E_no_recup} and \eqref{eq:E_recup} and by considering the additional mass of the recuperation device.


\section{Components}
HEV uses an electric motor for traction, and chemical batteries, fuel cells, ultra capacitors, and/or flywheels for their corresponding energy sources. In figure~\ref{fig:hevconfig} is shown the illustration of a general HEV divided into three parts (electric propulsion, energy source and auxiliary subsystem). The electronic power converter regulates the power flow between the electric motor and energy source on the base of two control inputs - brake and accelerator pedals.

The backward power flow is due to the regenerative braking of the HEV and this regenerated energy can be restored into the energy source, provided the energy source is receptive. Most HEV batteries as well as ultra capacitors are supported to accept regenerative energy.

\subsection{Topology of HEV Energy System}
The basic topology of HEV energy system in combining a SC system with a battery system as a power supplement device is to simply connect both systems in parallel as shows figure~\ref{fig:topology}.

\begin{figure}[ht]
    \center
    \includegraphics[width=12cm]{images/hevconfig.png}
    \caption{Conceptual illustration of a general HEV configuration~\cite{guzzela07}}
    \label{fig:hevconfig}
\end{figure}

\begin{figure}[ht]
    \center
    \includegraphics[width=8cm]{images/topology.png}
    \caption{Topology of battery and SC in parallel}
    \label{fig:topology}
\end{figure}

After embedding the DC/DC converter, in this way, two kinds of topologies are proposed. The first type has the SC connected directly across the DC bus and the battery connected through a bi-directional DC/DC converter. The second configuration has the battery on the main DC bus instead. Two configurations are shown in figure~\ref{fig:topology2}.

\begin{figure}[ht]
    \center
    \includegraphics[width=12cm]{images/topology2.png}
    \caption{Two kinds of topology with different connections of DC/DC converter}
    \label{fig:topology2}
\end{figure}

Previous research has verified that having the SC connected to the bus cable through DC/DC converter as shown in figure~\ref{fig:topology2} (b) is likely to yield high efficiency. Notable, in the configuration of figure~\ref{fig:topology2} (a), the entire battery power has to be transferred through the DC/DC converter and hence lowering the total efficiency of the battery pack.

Detailed topology is shown in figure~\ref{fig:topology3}. The main source composed by a battery pack and an auxiliary source implemented with a bi-directional DC/DC converter and SC bank. The battery pack connected directly across DC bus, and the SC bank connected through DC/DC converter. The output power of SC bank can be regulated by the energy management system (EMS), which acts on DC/DC converter by the way of the current control signal. The energy management algorithm takes into account multiple route information, current, voltage and state of charge (SoC) of battery and SC. They suffer from different energy loss and efficiency issues while the power transmits among them.

\begin{figure}[ht]
    \center
    \includegraphics[width=12cm]{images/topology3.png}
    \caption{Topology of EV system}
    \label{fig:topology3}
\end{figure}

\subsection{Batteries}
Electrochemical batteries are a key component of electric vehicles. Batteries are devices that transform chemical energy into electrical energy and vice versa. They represent a reversible electrical energy storage system. For HEV applications, the first consideration is the specific energy of a traction battery, because it limits the vehicle range. Other desirable attributes are high specific power, long calendar and cycle life, low initial and replacement costs, high reliability, and high robustness. Among other current technical challenges, a key point is developing accurate techniques to determine the capacity or the state of charge (SoC) of batteries during their operation.

SoC is defined as the ratio of remaining capacity to fully charged capacity and can be expressed as:

\begin{equation}
SoC = SoC_0 - \int\frac{i}{Q(i)}dt
\end{equation}

where $Q(i)$ is the ampere-hour capacity of the battery or SC at current rate $i$ and $SoC_0$  is the initial value of the SoC. For discharging, $i$ is positive, and for charging, $i$ is negative.

The energy capacity is considered to be more important than the coulometric capacity (ampere-hours), because it is directly associated with vehicle operation and the coulometric capacity usually depends on different discharging current rates~\cite{ehsani10}. The energy delivered from the battery can be expressed as:

\begin{equation}
E_c = \int_{0}^{t}V(i,SoC)i(t)dt
\end{equation}

where $V(i,SoC)$ is the voltage at the battery terminals, which is a function of battery current and SoC.

The basic equivalent circuit model (Thevenin model) of a Li battery is shown in figure~\ref{fig:batt_eq_circuit}. It is represented by an ideal open circuit voltage source ($V_{oc}$) and a series internal resistance ($R_{int}$).

\begin{figure}[ht]
    \center
    \includegraphics[width=7cm]{images/batt_eq_circuit.png}
    \caption{Equivalent circuit model of Li battery}
    \label{fig:batt_eq_circuit}
\end{figure}

The equivalent circuit loop holds true to Kirchhoff's voltage law to produce a terminal voltage expressed as:

\begin{align}
V_B & = V_{oc} - I_BR_{int} \\
V_{oc} & = f_{voc}(SoC_{Qb},I_B)
\end{align}

where $SoC_{Qb}$ is state of charge, $I_B$ is battery current, $R_{int}$ is internal resistance, $V_{oc}$ is open circuit voltage and $V_B$ is terminal voltage.

The state variable of $SoC_{Qb}$ can be defined as:

\begin{equation}
SoC_{Qb}(t) = \frac{Q(t)}{Q_0}
\label{eq:SoC}
\end{equation}

where $Q$ is the remaining battery charge, $Q_0$ is the total charge for a fully charged battery.

Both $V_{oc}$ and $R_{int}$ are dependent upon the battery instantaneous $SoC_{Qb}$. The open circuit voltage $V_{oc}$ can be obtained from Nernst equations \cite{hamann07} or by empirical methods. The internal resistance $R_{int}$ can be expressed as a function of the battery $SoC_{Qb}$, operating temperature as well as the charge and discharge currents.

To proceed modelling and simulations, it is necessary to know precise parameters of the given battery, namely capacity, voltage, energy and internal resistance of the battery.

The key to minimize the energy consumption for the batteries manage system is the efficiency of the battery charge/discharge. If we let the Faradic efficiency be 1, we can express the battery efficiency as energy efficiency which is defined as a ratio of electrical energy delivered by a battery from a particular $SoC_{Qb}$ to the electrical energy required to return the battery to the same $SoC_{Qb}$. It can be written as:

\begin{equation}
\eta_{BE}=f_{\eta BE}(SoC,P_E)
\end{equation}

\begin{equation}
P_B=I_BV_B
\end{equation}

where $P_B$ is battery power.

Although the battery efficiency is not a straightforward parameter to quantify the performance of a battery, it is effective as a comparative measure of the various power and energy system strategies. That is why, for the purpose of energy consumption optimization, it is necessary to know the characteristic map of battery efficiency. In the figure~\ref{fig:batt_map} is shown an example of characteristic map of Li battery efficiency.

\begin{figure}[ht]
    \center
    \includegraphics[width=12cm]{images/batt_map.png}
    \caption{Characteristic map of a Li battery efficiency}
    \label{fig:batt_map}
\end{figure}

\subsection{Supercapacitors}
The energy source, mainly batteries and fuel cells, has high specific energy, whereas the power source has high specific power. The power source that has received wide attention is the SC. SC can be recharged from the energy source, which is in our case battery, during less demanding driving power or during regenerative braking. 

The SC is characterized by a much higher specific power but a much lower specific energy compared to batteries so it is difficult to use SC alone as energy storage. One promising application is the so-called battery and SC hybrid energy storage system for HEVs. Specific energy and specific power requirements can be decoupled, thus affording an opportunity to design a battery that is optimized for specific energy and cycle life with little attention being paid to specific power. Due to the load levelling effect of the SC, the high current discharging from the battery and the high current charging to the battery by regenerative braking are minimized so that available energy, endurance, and life of the battery can be significantly increased. Compared to the batteries, the SC can be charged and discharged in order of milliseconds and the number of charging cycles capabilities at 80~\% DoD (Depth of Discharge) can be more than $10^6$ cycles, while in the case of batteries it is only around $10^3$ cycles. Of course, by increasing the endurance of battery, the whole system cost and used battery replacement expenses decrease. 

The SC can be modelled by an ideal capacitance and an equivalent series resistance (ESR), as shown in figure~\ref{fig:sc_eq_circuit}.

\begin{figure}[ht]
    \center
    \includegraphics[width=7cm]{images/sc_eq_circuit.png}
    \caption{Equivalent circuit model of SC}
    \label{fig:sc_eq_circuit}
\end{figure}

The equivalent circuit loop holds true to Kirchhoff's voltage law to produce a terminal voltage expressed as:

\begin{align}
V_{SC} & = V_{oc} - I_{SC}R_{int} \\
V_{oc} & = f_{voc}(SoC_{Q_{SC}},I_{SC})
\end{align}

where $I_{SC}$ is SC current and $V_{SC}$ is terminal voltage.

The ESR accounts for resistive losses in the dielectric, plate material, and electrolytic solution. The actual capacitance and ESR are dependent on terminal voltage, voltage charge rate, current, and temperature. However within the capacitor's working region, fixed value capacitance and ESR can accurately model a real SC.

Again, for the purpose of energy consumption optimization, it is necessary to know the characteristic map of SC efficiency and SC parameters - capacity, voltage and internal resistance. In the figure~\ref{fig:sc_map} is shown an example of characteristic map of a SC efficiency.

\begin{figure}[ht]
    \center
    \includegraphics[width=12cm]{images/sc_map.png}
    \caption{Characteristic map of a SC efficiency}
    \label{fig:sc_map}
\end{figure}

From figure~\ref{fig:en_pow_char} is visible, that battery is high energy component (thus it is used to provide long duration electric energy) and SC is high power component (thus it is used to provide high frequent jerk power and absorb brake regeneration energy).

Compared to the Li battery, the terminal voltage of the SC has smaller variation during charge and discharge at a fixed $SOC_{Q_{SC}}$. However the SC has a much larger voltage variation when the $SOC_{Q_{SC}}$ varies. The SC has a higher efficiency than the Li battery, generally above 82\% if the $SOC_{Q_{SC}}$ is higher than 60\%. Also, the SC has a very rapid dynamic response and low charging time. 

\begin{figure}[ht]
    \center
    \includegraphics[width=12cm]{images/en_pow_char.png}
    \caption{Region diagram of energy and power characteristics of different components}
    \label{fig:en_pow_char}
\end{figure}

\subsection{DC/DC Converter}
To connect a SC to an external power system, it is necessary to boost the SC voltage or to increase the number of SC. The role of the DC/DC converter is to regulate the voltage to control the SC power. The transmission efficiency of the DC/DC converter can be influenced from multiple factors, like flow power, current and so on. For simplifying the algorithm, the power efficiency of DC/DC converter is defined as a function of the load power. It can be written as:

\begin{equation}
\varepsilon_{CV}=\eta_{CV}(P_{CV})
\end{equation}

where $\varepsilon_{CV}$ and $P_{CV}$ are the efficiency and the load power of DC/DC converter. We can notice that

\begin{equation}
P_{CV}=P_{SC}\varepsilon_{CV}
\end{equation}

For the purpose of energy consumption optimization, it is necessary to know the DC/DC converter power efficiency map. Example is shown in figure~\ref{fig:dcdc_map}.

\begin{figure}[ht]
    \center
    \includegraphics[width=12cm]{images/dcdc_map.png}
    \caption{DC/DC converter power efficiency map}
    \label{fig:dcdc_map}
\end{figure}

\subsection{Electric Motor}
The electric motor characteristics are based on the efficiency data as shown in figure~\ref{fig:motor_map}, which is a non-linear function of the electric motor torque and the rotational speed

\begin{equation}
\varepsilon_{m}=\eta_m(T_m,\omega_m)
\end{equation}

Then, the electric motor demanded power is

\begin{equation}
P_m=\frac{T_m,\omega_m}{\varepsilon_{m}}
\end{equation}

where, $T_m\in [T_{m min}, T_{m max}], \omega_m\in [ \omega_{m min}, \omega_{m max}]$, $\varepsilon_{m}$ and $P_m$ are the output torque (Unit: $N.m$), output rotational speed (Unit: $rad/s$), efficiency and the demanded power (Unit: $kW$) of the electric motor. $T_{m min}$, $T_{m max}$, $\omega_{m min}$ and $\omega_{m max}$ are the minimal and the maximum values of torque and rotational speed of the electric motor respectively.

The efficiency map of electric motor is given by manufacturer, and shown as the solid lines in figure~\ref{fig:motor_map}. The dashed line in figure~\ref{fig:motor_map} is the distribution of the optimal efficiency operating points corresponding to the given output powers. They can be obtained via interpolation method, and used to determine the optimal transmission ratio.

\begin{figure}[ht]
    \center
    \includegraphics[width=7cm]{images/motor_map.png}
    \caption{Electric motor efficiency map~\cite{bin10}}
    \label{fig:motor_map}
\end{figure}


\section{Simulation Methods and Tools}
While performing simulations, more approaches can be utilized. One of them are quasistatic simulations, where the input variables are the speed $v$ and the acceleration $a$ of the vehicle as well as the grade angle $\alpha$ of the road. With this information, the force $F_t$ is computed that has to be acting on the wheels to drive the chosen profile for a vehicle described by its main parameters $\{A_f \cdot c_d, c_r, m_v\}$. In this step the vehicle is assumed to run at constant speed $v$, acceleration $a$, and grade $\alpha$ for a (short) time period $h$. Once $F_t$ is known, the losses of the powertrain are considered by a power balance. The quasistatic method is well suited to the minimization of the fuel consumption of complex powertrain structures. With this approach it is possible to design supervisory control systems that optimize the power flows in the propulsion system. The influence of the driving pattern can be included in these calculations. The main drawback of this method is that the physical causality is not respected and the driving profile that has to be followed has to be known a priori. Therefore, this method is not able to handle feedback control problems or to correctly deal with state events.

Another possible approach is the dynamic approach, which is based on a ``correct'' mathematical description of
the system which is usually formulated by sets of ordinary differential equations in the state space form. That is the way how it can be described many dynamic effects in powertrains, which may or may not be relevant for fuel consumption. The description is very versatile. For example the design of feedback control systems or the detection and correct handling of state events in optimization problems can be solved using this method. Usual drawback of this method is high computational burden. This method is used only if other simpler methods are not available or sufficient.

For the reasons of simulations, efficient and powerful simulation software must be available. The software should fulfil basic requirements such a possibility of interconnection of different powertrain components, their scaling and parametrizing and visualization and numerical optimization integration. Such available often used software is for example ADVISOR or QSS Toolbox for Matlab/Simulink. 








