# Battery Sizing for Mild P2 HEVs Considering the Battery Pack Thermal Limitations
## 3 Development of Supervisory Control Strategy
### ECMS Control Strategy with Thermal Limitations Using Temperature Threshold  
To avoid the thermal runaway of the battery, limitations can be applied by setting the battery temperature \(\theta_{bat}\) threshold to 55 \({}^{\circ}\)C. Over this temperature, the usage of electric power is limited. Hence, the required power is provided mainly by the ICE. This is implemented by setting the penalty factor \(PF_{SO}\) to a huge number when the battery temperature is over the given threshold as shown in Equation (23). As can be seen, if the battery temperature is \(\theta_{bat}>55\)\({}^{\circ}\)C the penalty factor is set to a very high number i.e., to 1000, regardless of the battery SOC.  
\[PF_{soc}^{\theta}=\left\{\begin{array}{ccc}1-0.15\;\overline{SOC}^{3}+0.05 \;\overline{SOC}^{4},&if\;\theta_{bat}\leq 55\;^{\circ}\mathrm{C}&and\; SOC<0.6\\ 1,&if\;\theta_{bat}\leq 55\;^{\circ}\mathrm{C}&and\; SOC\geq 0.6\\ 1000,&\theta_{bat}>55\;^{\circ}\mathrm{C}&\end{array}\right. \tag{23}\]  
Then, the penalty factor \(PF_{soc}\) in Equation (22) is replaced with the penalty factor \(PF_{soc}^{\theta}\) which considers the thermal condition of the battery as well. By implementing such a limitation, the penalty factor works as an on/off switch triggered by the temperature threshold and battery operation outside of the designed operating temperature range can be avoided. However, this limitation does not include the rate of temperature change, which might result in a temperature overshoot due to a previous power drain from the battery. In the next step, the temperature change rate is also limited.  
ECMS Control Strategy with Thermal Limitations Using Penalty Function on Temperature and Temperature Change Rate  
To avoid thermal shocks in the battery, the penalty factor is represented as a cumulative product of penalty factors on the temperature \(PF_{\theta}\), on the temperature change rate \(PF_{\hat{\theta}}\) and on the SOC \(PF_{soc}\) are applied to equivalent fuel consumption (\(\dot{m}_{eqv}\)) of the electric motor:  
\[J_{ecms}=\dot{m}_{ice}\;(P_{ice})+PF_{\theta}\cdot PF_{\hat{\theta}}\cdot PF_{ soc}\cdot\dot{m}_{eqv}(P_{bat}) \tag{24}\]  
The penalty function on temperature \(PF_{\theta}\) is defined as:  
\[PF_{\theta}=1+1.75\cdot\overline{\theta}^{3} \tag{25}\]  
Figure 11: Penalty function on SOC.  
where \(\overline{\theta}\) is normalized temperature calculated as a function of upper \(\theta_{high}\) and lower \(\theta_{low}\) temperature limit as [10]:  
\[\overline{\theta}=\frac{2\cdot\theta_{bat}-(\theta_{high}+\theta_{low})}{\theta_{ high}-\theta_{low}} \tag{26}\]  
where: \(\theta_{low}=10\)\({}^{\circ}\)C and \(\theta_{high}=60\)\({}^{\circ}\)C.  
Function of \(PF_{\theta}\) is plotted in Figure 12a, it has a value of about 1 when the battery temperature is lower than 40 \({}^{\circ}\)C. As the battery temperature increases beyond 40 \({}^{\circ}\)C, a correction factor \(PF_{\theta}\) gets higher value, so usage of EM power will be limited.  
The penalty on the temperature change rate \(PF_{\dot{\theta}}\) is used to avoid a high-temperature change rate which could induce the thermal shock on the battery is:  
\[PF_{\dot{\theta}}=1+1\cdot\overline{\dot{\theta}}^{3} \tag{27}\]  
where \(\overline{\dot{\theta}}\) is normalized temperature change rate, a function of upper \(\dot{\theta}_{high}\) and lower \(\dot{\theta}_{low}\) temperature change limits [10]:  
\[\overline{\dot{\theta}}=\frac{2\cdot\dot{\theta}-(\dot{\theta}_{high}+\dot{ \theta}_{low})}{\dot{\theta}_{high}-\dot{\theta}_{low}} \tag{28}\]  
where for upper and lower limits for rate of temperature change are:  
\[\dot{\theta}_{low}=0\)\({}^{\circ}\)C/s _and_\(\dot{\theta}_{high}=6\)\({}^{\circ}\)C/s  
The graphical representation of \(PF_{\dot{\theta}}\) is given in Figure 12b.