## 2 Multimode Hybrid Powertrain Model
### Mathematical Model  
The mathematical approach adopted to model the two-mode hybrid system is the same described by [PERSON] et al. (2018), where an automated routine is described to evaluate the dynamic equilibrium equations for a generic multimode powertrain architecture. The dynamic equilibrium of the two-mode hybrid system shown in Figure 1, with no clutches engaged, is expressed by (see also (2018)):  
\[\mathbf{A_{0}x_{0}=u_{0}} \tag{1}\]  
where the vector \(\mathbf{x_{0}}=\left[ \begin{array}{cccccc}\dot{\omega}_{c,2}&\dot{\omega}_{r,1}& \dot{\omega}_{s,1}&\dot{\omega}_{s,2}&\dot{\omega}_{c,1}&\dot{\omega}_{r,2}&F_{ 1}&F_{2}\end{array} \right]^{T}\) includes the angular acceleration of the two PGS carriers (\(\dot{\omega}_{c,1}\), \(\dot{\omega}_{c,2}\)), suns (\(\dot{\omega}_{s,1}\), \(\dot{\omega}_{s,2}\)), and rings (\(\dot{\omega}_{r,1}\), \(\dot{\omega}_{r,2}\)) and the internal forces, \(F_{1}\) and \(F_{2}\), exchanged between the ring and the pinions gears teeth of the first and the second PGS, respectively; \(\mathbf{u_{0}}=\left[ \begin{array}{cccccc}T_{load}&T_{ICE}&T_{MG1}&T_{MG2}&0&0&0 &0\end{array} \right]^{T}\) is the generalized torque vector, where \(T_{load}\) represents the equivalent vehicle motion resistance torque (see Equation (9)), \(T_{ICE}\) is the torque applied by the Internal Combustion Engine (ICE) and \(T_{MG1}\) and \(T_{MG2}\) are the torques applied on the output shaft of the two Motors/Generators (MGs), respectively. The \(8\times 8\) matrix \(\mathbf{A_{0}}\) includes geometric and inertial parameters of the powertrain components (see also (2018)):  
\[\mathbf{A_{0}}=\left[ \begin{array}{cccccc}J_{out}+J_{c,2}&0&0&0&0&0&r_{r,2}+r_{ s,2}\\ 0&J_{ICE}+J_{r,1}&0&0&0&0&-r_{r,1}&0\\ 0&0&J_{MG1}+J_{s,1}&0&0&0&-r_{s,1}&0\\ 0&0&0&J_{MG2}+J_{s,2}&0&0&0&-r_{s,2}\\ 0&0&0&0&J_{c,1}&0&r_{r,1}+r_{s,1}&0\\ 0&0&0&0&0&J_{r,2}&0&-r_{r,2}\\ 0&-r_{r,1}&-r_{s,1}&0&r_{r,1}+r_{s,1}&0&0&0\\ r_{r,2}+r_{s,2}&0&0&-r_{s,2}&0&-r_{r,2}&0&0\end{array} \right] \tag{2}\]  
where \(r_{r,1}\) and \(r_{r,2}\) are the ring gears radii and \(r_{s,1}\) and \(r_{s,2}\) the solar gears radii of the two PGSs, respectively; \(J_{r,1}\), \(J_{c,1}\), and \(J_{s,1}\) are the torsional moments of inertia of the ring, the carrier and the solar gears of the first PGS, meanwhile \(J_{r,2}\), \(J_{c,2}\), and \(J_{s,2}\) are the corresponding torsional moments of inertia of the second PGS; \(J_{ICE}\), \(J_{MG1}\), and \(J_{MG2}\) represent the torsional moments of inertia of the ICE, the MG1 and MG2, respectively; \(J_{out}=M\frac{R_{\rm{IN}}^{2}}{\gamma_{r}^{2}}\) is  
Figure 2: Scheme of the two powertrain modes: Input Split (**left**) and Compound Split (**right**). the equivalent vehicle torsional moment of inertia where \(M\) is the vehicle mass, \(\tau_{F}\) is the transmission final drive ratio and \(R_{W}\) the tire rolling radius. The presence of rigid connections or clutch engagements between the component \(i\) and \(j\) of the two PGSs, modifies Equation (2) by summing up the \(j_{th}\) row to the \(i_{th}\) row and removing the \(j_{th}\) row, thus reducing the number of system equation. Due to the presence of a grounding clutch that links the \(k_{th}\) component of the two PGSs to the chassis, i.e., clutch 2 in Figure 1, the number of system equations in Equation (2) is further reduced by removing the \(k_{th}\) row. For both Input Split and Compound Split modes, the system in Equation (2) is reduced to (see also [36]):  
\[\mathbf{Ax}=\mathbf{u} \tag{3}\]  
where \(\mathbf{x}=\left[ \begin{array}{cc}\dot{\omega}_{c,2}=\dot{\omega}_{c,1}=\dot{ \omega}_{out}&\dot{\omega}_{r,1}=\dot{\omega}_{ICE}&\dot{\omega}_{s,1}=\dot{ \omega}_{MG1}&\dot{\omega}_{s,2}=\dot{\omega}_{MG1}&F_{1}&F_{2}\end{array}  \right]^{T}\), \(\mathbf{u}=\left[ \begin{array}{cc}T_{load}&T_{ICE}&T_{MG1}&T_{MG2}&0&0\end{array}  \right]^{T}\) and \(\mathbf{A}\) is reduced to a \(6\times 6\) matrix.