## Appendix A Description of VGT Flow Oriented Model  
### Mass Flow Model  
The procedure developed by the authors of [30, 43] for modeling the turbine mass flow parameter is mainly based on viewing the turbine as a single equivalent nozzle that covers from station 0 to 4 of the radial turbine, as shown in Figure 11. The primary assumption of this model is the behavior of the turbine is quasi-steady throughout the nozzle simulating the turbine; from both the thermodynamic and fluid dynamics perspective. The fundamental approach followed by [PERSON] et al. [30] is calculating for every turbine operating point an equivalent nozzle flow area (\(A_{\text{eff}}\)) that represents the whole turbine expansion. Accordingly, this equivalent nozzle flow area (\(A_{\text{eff}}\)) should allow calculating the mass flow parameter for the entire operative range of the turbine; thus, it is possible to plot the whole mass flow curve. To do so, first, the continuity equation is applied to stator, rotor and equivalent nozzle, as shown in Equation (A1). Then, using the velocity definition and solving the mass flow, an expression for the equivalent nozzle area shown in Equation (A2) is obtained. By assuming a certain hypothesis, as described in the work of [PERSON] et al. [30], and doing some simplifications, a new expression of the throat area of the equivalent nozzle is obtained, as shown in Equation (A3). It mainly depends on the information available in a standard turbocharger map and the measurable geometry of the turbine with four fitting constants (\(a\), \(b\), \(c\), and \(d\)) in Equation (A3). \[\dot{m}=A_{2^{\prime}}\rho_{2^{\prime}}v_{2^{\prime}}=A_{4}\rho_{4}v_{4}=A_{Neg }\rho_{4}v_{Neg}\] (A1)\[A_{\text{Neq}}=A_{4}\sqrt{\frac{1+\left(\frac{u_{4}}{\mathcal{P}_{\text{Neq}}} \right)^{2}-\left(\frac{u_{3}}{\mathcal{P}_{\text{Neq}}}\right)^{2}+\left( \frac{w_{2}}{\mathcal{P}_{\text{Neq}}}\right)^{2}}{\left(\frac{A_{4}}{ \mathcal{A}_{2^{\prime}}}\right)^{2}\left(\frac{\rho_{4}}{\rho_{2^{\prime}}} \right)^{2}+1}} \tag{10}\]  
\[A_{\text{eff}}=\frac{a\cdot A_{4}^{\text{geom}}\cdot\sqrt{1+\frac{\sigma^{2} \left[\left(\frac{D_{4}}{\mathcal{D}_{3}}\right)^{2}-1\right]+b}{\eta_{ls}}}}{ \sqrt{1+\left(c\cdot\frac{A_{4}^{\text{geom}}}{A_{2^{\prime}}^{\text{geom}}} \right)^{2}\cdot\frac{\left(\frac{1}{\Pi_{2^{\prime},4}^{2}}\right)^{2}}{\left( 1-\eta_{ls}\cdot\left(1-\left(\frac{1}{\Pi_{2^{\prime},4}^{2}}\right)^{\frac {-1}{T}}\right)\right)^{2}}}} \tag{11}\]  
where \(A_{2^{\prime}}^{\text{geom}}\) is the geometrical throat section of the stator vanes and can be calculated for any VGT as described in [30]. The rotor outlet geometrical area (\(A_{4}^{\text{geom}}\)) is calculated by taking the arithmetic diameter between the turbine rotor shroud (\(D_{4}\)) and rotor hub (\(D_{nut}\)) diameters, as shown in Equation (10). \[A_{4}^{\text{geom}}=\pi\cdot\left(\frac{D_{4}^{2}-D_{nut}^{2}}{4}\right) \tag{12}\]  
In Equation (11), \(D_{3}\) is turbine rotor diameter, \(D_{4m}\) is the mean diameter between turbine rotor shroud (\(D_{4}\)) and rotor hub (\(D_{nut}\)) diameters, \(\eta_{ls}\) represents the turbine total-static adiabatic efficiency, \(\sigma\) is the blade to jet speed ratio calculated using the Equation (12), and \(\Pi_{2^{\prime},4}\) represents pressure ratio in the VGT rotor. \(\Pi_{2^{\prime},4}\) is calculated from the total to static turbine pressure ratio by using a fitting constant \"\(d\)\" as shown in Equation (13). It is obtained by making the hypothesis that the turbine stator pressure drop to the whole turbine total-to-static pressure drop ratio is constant for a VGT position [30]. A constant average value of turbine efficiency (\(\eta_{ls}=\overline{\eta}_{ls}=0.8\)) is assumed relaying on \"\(b\)\" coefficient to avoid the possible inconsistency during the extrapolations.