# Battery Sizing for Mild P2 HEVs Considering the Battery Pack Thermal Limitations
## 2 Vehicle Performance Modelling
### Battery Performance and Electro-Thermal Model
#### 2.5.2 Dynamic Component  
The dynamic component of the cell model accounts for the voltage losses, including joule loss, diffusion loss due to mass transfer, activation loss due to charge transfer and loss due to hysteresis. The output of the dynamic model is the predicted closed-circuit voltage that accounts for these losses. The model is developed from an RC circuit with a single RC branch in series with a resistive branch [37] as shown in Figure 8.  
Figure 7: OCV and closed-circuit voltage (CCV) across SOC in charge and discharge phases. The OCV from the datasheet is between the measured terminal voltage in charge and discharge phases.  
The _hyst_ component in the circuit accounts for the hysteretic contribution of the losses. The estimated dynamic voltage \(\tilde{v}\) is computed as.  
\[\tilde{v}(k)=OCV(SOC(k))\ \ +v_{loss}(k) \tag{11}\]  
\[v_{loss}(k)=Mh(k)+M_{0}s(k)-\sum_{j}R_{j}i_{kj}\ (k)-R_{0}i(k) \tag{12}\]  
where \(v_{loss}\) is the voltage loss, \(M_{0}\) is the instantaneous hysteresis voltage; \(M\) is the dynamic hysteresis magnitude; \(R_{0}\) is the instantaneous series resistor; \(i_{kj}(k)\) is the resistor diffusion current; \(R_{j}\) is the parallel branch resistance; \(h(k)\) is the hysteresis voltage; \(s(k)\) is sign of current input. It is 1 for positive current input, \(-1\) for negative input and zero otherwise.  
The first two components at the right-hand side of Equation (12) model the instantaneous and the dynamic hysteresis voltage. The third component models the RC branch of the circuit. The subscript \(j\) represents the number of branches. The RC branch models the losses due to mass transport and charge transfer. The diffusion-resistor current \(i_{kj}\) that passes through the RC branch is computed from the time constant of the RC branch. The fourth component is the series resistive loss that models the joule losses.  
The parameters of the model appear linearly according to Equation (13). \(v_{loss}\) is computed for the \(N\) data points of the experimental data and the parameters, \(M,\ M_{0},\ R_{j},\ R_{0}\) are computed by least square approximation.  
\[\left[\begin{array}{c}v_{loss}(1)\\ v_{loss}(2)\\ \vdots\\ v_{loss}(N)\end{array}\right]=\left[\begin{array}{cccc}h(1)&s(1)&-i_{R_{j}} ^{\rm T}(1)&-i(1)\\ h(2)&s(2)&-i_{R_{0}}^{T}(2)&-i(2)\\ &\vdots\\ h(N)&s(N)&-i_{R_{j}}^{T}(N)&-i(N)\end{array}\right]\left[\begin{array}{c}M \\ M_{0}\\ R_{j}\\ R_{0}\end{array}\right] \tag{13}\]  
By least square approximation, \(X=A\setminus Y\).  
Table 2 reports the corresponding values of the electric and the thermal model parameters.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline  & **Variable** & **Units** & **Value** \\ \hline \multirow{4}{*}{Electric Model} & Dynamic hysteresis magnitude, \(M\) & - & 0.017 \\  & Instantaneous hysteresis height, \(M_{0}\) & - & negligible \\  & Instantaneous series resistor, \(R_{0}\) & Ohms & 0.024 \\  & Parallel branch resistance, \(R_{j}\) & Ohm & 0.018 \\ \hline \multirow{3}{*}{Thermal Model} & Specific heat capacity, \(c_{p}\) & J/kg. K & 1200 \\  & Thermal resistance, \(R_{conv}\) & K/W & 14.6 \\ \cline{1-1}  & Cell mass, \(m_{cell}\) & kg & \(48.5\times 10^{-3}\) \\ \hline \hline \end{tabular}
\end{table}
Table 2: List of electrical and thermal model parameters with the estimated values.  
Figure 8: Equivalent circuit that describes the dynamic model designed for terminal voltage and voltage loss prediction [37].  
The dynamic current profile used for data collection during the experiment is applied as input to the model to compute \(v_{loss}\) according to Equation (12). This profile consists of a sequence of random charge and discharge current values applied in the range of \(-4.5\) A and \(+4.5\) A. The current profile and \(v_{loss}\) are shown in Figure 9.