## 2 Computational Model  
The present numerical study was performed on a 1D gas dynamic environment using the GT-Power software tool (v7.5.0, Gamma Technologies, LLC., Westmont, IL, USA, 2014). The model used for this study, shown in Figure 2, represents a 2.0 L spark-ignition turbocharged engine. The model, previously used in [17], was provided by the engine supplier, and it has been validated against experimental data in both steady-state and transients by POWERTECH Engineering (see Model Validation).  
Figure 1: Energy regeneration areas over the load/speed map of an engine: (a) current approach; (b) desired operation.  
Figure 2: 1D model of the baseline 2.0 L turbocharged engine used in the present study.  
The baseline 2.0 L engine model was used only for the Phase 1 study of the steady-state simulations, which included a wastegate enthalpy loss study for quantifying the energy availability across the speed/load map of the engine.  
For the Phase 2 and Phase 3 of the steady-state simulations, as well as the transient simulations included in this paper, the model was modified by implementing a motor/generator (M/G) component model. The motor/generator was connected directly to the turbocharger's shaft, as shown in Figure 3.  
The original model was fitted with a wastegate/boost pressure and a throttle/brake mean effective pressure proportional-integral-derivative (BMEP PID) controllers for achieving the targeted boost pressure and BMEP. The target values for different engine speeds and loads were provided to the model using look-up tables. For the electrically-assisted model, two new PID controllers were introduced or replaced the original controllers as necessary, depending on the type of study. These are:  
* A motor/generator (M/G)/boost pressure controller for achieving the desired intake pressure
* A WG/pre-turbine pressure controller for achieving the desired pre-turbine pressure  
The compressor and turbine mass flow multipliers were modified and spanned in the range of 0.7 to 1.3 for the purpose of undertaking parametric studies with different component sizing, as shown in Equation (1). By restricting or enhancing the mass flow rate within a component, the size of the component can be simulated.  
\[\text{Average mass flow rate }=\text{Multiplier }\times\frac{\int\dot{m}\, \text{d}t}{\int\text{d}t} \tag{1}\]  
where \(\dot{m}\) is the mass flow rate through the part (in the case of the turbine, the wastegate mass flow is excluded).  
Although this approach would not promise a high level of accuracy, it is deemed to be reliable to predict the trend of the overall system's behaviour when the turbine or compressor sizing is increased or decreased compared to the original components based on a highly-calibrated engine. Furthermore, the wastegate area was modified from completely closed to diameters larger than the baseline engine. However, due to the change of the size of critical components, the operation of the engine and its main parameters needs to be closely monitored to ensure realistic performance. For this reason, the parameters limits shown in Table 1 were set and established, which were not violated during the parametric studies.  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Parameter** & **Unit** & **Value** \\ \hline Engine torque & (Nm) & 304 \\ Turbine inlet temperature & (\({}^{\circ}\)C) & 1000 \\ Turbocharger speed & (krpm) & 200 \\ Pre-turbine pressure & (bar) & 3 \\ Post-compressor temperature & (\({}^{\circ}\)C) & 200 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Maximum limits set for the parametric studies.  
Figure 3: Motor/generator component model used for converting the model to an e-turbo engine.