## 3 Modeling and Control Strategy Design
### ECMS Problem Formulation  
#### 3.3.1 Equivalent Consumption Minimization Strategy  
For SHEVs with the powertrain structure shown in Figure 1, the power balance equation can be written as:  
\[p_{d}(t)=p_{g}(t)+p_{b}(t) \tag{8}\]  
where \(p_{g}\) is the output power of the engine-generator set. Combing Equations (5), (7), and (8), the dynamics of the powertrain energy system can be written as a differential equation for SOC, as shown in Equation (9):  
\[\dot{SOC}=\frac{U_{OC}^{2}(t)-U_{OC}(t)\sqrt{U_{OC}^{2}(t)-4R(t)\big{(}p_{d}(t) -p_{g}(t)\big{)}}}{2R(t)} \tag{9}\]  
and it can be further represented in a state space form:  
\[\dot{x}(t)=f(x(t),u(t)) \tag{10}\]  
where the state, \(x\), is the battery SOC; the control input, \(u\), is the output power of the engine-generator set, which is \(p_{g}\) in Equation (9). The battery open-circuit voltage and resistance are SOC-related; therefore, the right-hand side of Equation (9) can be represented by a function of state and input. In general, the constraints for the state and the control variables include:  
\[\left\{ \begin{array}{c}SOC_{\min}\leq SOC\leq SOC_{\max}\\ 0\leq p_{g}(t)\leq p_{g}\max\\ p_{\min}(t)\leq p_{b}(t)\leq p_{b\max}(t)\end{array} \right. \tag{11}\]  
where the state-variable SOC is constrained within a narrow window in the charge-sustaining mode and the width of \([SOC_{\min},SOC_{\max}]\) is chosen to be 5% in this study; the engine-generator set's output power is limited by \(p_{g\max}\); this is 120 kW in this study according to \(p_{\max}\), as defined in Table 1 along with the generator efficiency; the battery output power is constrained by the instantaneous SOP, which was discussed in Section 3.2. Then, the optimal control of fuel economy can have a cost function in the general form of:  
\[J=\int_{t_{0}}^{t_{f}}\dot{m}_{fuel}(u(t),t)dt \tag{12}\] \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Component** & **Parameter** & **Notation** & **Value** \\ \hline \multirow{8}{*}{Vehicle} & Total mass & \(M\) & 2040 (kg) \\  & Final drive/gear ratio & \(i_{0}\) & 4.2 \\  & Radius of the tire & \(r\) & 0.371 (m) \\  & Frontal area & \(A\) & 2.1 (m\({}^{2}\)) \\  & Rolling resistance & \(f\) & 0.0098 \\  & coefficient & \(C_{d}\) & 0.28 \\ \hline \multirow{2}{*}{Engine} & Peak power & \(p_{\max}\) & 125 (kW) \\  & Maximum speed & \(\omega_{\max}\) & 5000 (rpm) \\  & Maximum torque & \(T_{\max}\) & 256 (Nm) \\ \hline \multirow{2}{*}{Motor} & Peak power & \(p_{\rm immax}\) & 300 (kW) \\  & Maximum speed & \(\omega_{\rm immax}\) & 6000 (rpm) \\  & Maximum torque & \(T_{\rm immax}\) & 1340 (Nm) \\ \hline \multirow{2}{*}{Battery pack} & Total capacity & \(E_{b}\) & 19.7 (kWh) \\  & Nominal voltage & \(U_{b}\) & 346.5 (V) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Main parameters of the target SHEV. where the control input is the same as \(p_{g}\) in the state equation. When operating in the charge-sustaining mode, the total consumed energy from the battery is negligible; hence, the battery is acting as an auxiliary reversible fuel tank: if the engine's output power is higher than the required power, with extra power to charge the battery, the charged energy will be released in the future and fuel will be \"saved\"; if the engine's output power is lower than the required power, the battery discharges at this moment and needs to be recharged in the future to \"consume\" the fuel. Based on this mechanism, the ECMS introduces an equivalent factor (EF) to transform electric energy into an equivalent amount of fuel, and the global optimization problem is converted to a local optimization problem with a cost function in the form of:  
\[\dot{m}_{eqv}(t)=\dot{m}_{fuel}(t)+s(t)\dot{m}_{elc}=\dot{m}_{fuel}(t)+s(t) \frac{p_{b}(t)}{Q_{LHV}} \tag{13}\]  
where \(m_{eqv}\) is the equivalent fuel consumption rate; \(m_{elc}\) is the converted equivalent fuel consumption from electric energy, which is equal to the battery power divided by the fuel low heating value; and \(s\) is the equivalent factor.