## 2 Single Motor Hybrid Powertrain Configuration with Dual Planetary Gears
### Topology Design Method
#### 2.2.1 Automatic Modeling of Operating Mode  
Because of the large number of operating modes in the basic configuration scheme, to explain the proposed topology design method more conveniently, this paper selects the operating mode shown in Figure 5 as an example to illustrate. According to the proposed topology design method, the dynamics relationship of this mode is shown in Equation (2). The fundamental matrix \(A\) is composed of rotational inertia matrix \(J\), matrix \(D\)[27; 28; 29], torque constraint matrix \(T_{con}\) and speed constraint matrix \(S_{con}\). where \(I\), \(\omega\), \(T\) are the rotational inertia, the speed, and the torque, respectively; the subscripts represent corresponding components and nodes. It should be noted that because the \(R_{1}\) node is grounded, \(T_{R1}\) represents the braking torque that locks the R1 node. F\({}_{1}\) and F\({}_{2}\) represent the internal force of PG1 and PG2, respectively, notice that they are intermediate variables for solving equation; \(T_{pg}\), \(\omega_{pg}\) are output torque and speed of the PG, respectively; \(-R_{i}\), \(R_{i}+S_{i}\), \(-S_{i}\) are the note coefficients of the ring gear, carrier and sun gear of PGi, respectively. The value of note coefficients is related to the radius of the nodes [27]. \(J\) is a diagonal matrix of size \(3n\times 3n\). In this paper, the first three diagonal elements in matrix \(J\) from top left to bottom right are the sum of the inertia of engine, output shaft, MG, and the inertia of the notes they connect, respectively. Therefore, the first three diagonal elements of matrix \(J\) in Equation (2) are \(l_{e}+I_{C1}\), \(I_{out}+I_{C2}\), and \(I_{mg}+I_{S1}\), respectively. The latter three diagonal elements are the inertia of the nodes which are not connected to the components. They are arranged in matrix \(J\) in the sequence of the ring gear, carrier, and sun gear from PG1 to PG2. In the initial state, the torque constraint matrix \(T_{con}\) is a negative unit matrix of size \(3n\times 3n\). The diagonal elements of the matrix \(T_{con}\) correspond to the diagonal elements of the matrix \(J\) one by one. When any two nodes of PG are connected, the extended intersection \(T_{con\ i,\ j}\) of corresponding diagonal elements \(T_{con\ i,\ i}\), \(T_{con\ j,\ j}\) (\(i<j\)) for the two nodes in the matrix \(T_{con}\) is transformed from zero to one. The speed constraint matrix \(S_{con}\) is composed of connection constraint and locking constraint, because the carrier and sun gear of PG1 are connected with the ring gear and sun gear of PG2, respectively, and the ring gear of PG1 is grounded in this mode. Therefore, the first two rows of the matrixare expressed as connection constraint of nodes. The third row of the matrix \(S_{con}\) is expressed as the locking constraints of the PG1 ring gear. The size of matrix \(D\) is \(3n\times n\). When the components are connected to the ring gear, carrier, and sun gear of PG\(i\), respectively, the node coefficients \(-R_{i}\), \(R_{i}+S_{i}\), \(-S_{i}\) are filled at the corresponding positions in column \(i\) of matrix \(D\), respectively. The nodes which are not connected to the components fill in their own node coefficients at the corresponding positions of matrix \(D\). Matrix \(a\) is composed of matrix \(\Omega\) and matrix \(T\). The size of matrix \(\Omega\) is \(3n\times 1\), and the matrix element is angular acceleration. The size of matrix \(T\) is \(4n\times 1\), and matrix elements are composed of torque and internal force \(F_{1}\) and \(F_{2}\). The dynamic equation of the system can be rewritten to Equation (3). \[\left\{ \begin{array}{l}Aa=H\\ A=\left[\begin{array}{ll}J&D_{1}\\ D_{2}&0\end{array}\right],D_{1}=\left[\begin{array}{ll}T_{con}&D\end{array} \right],D_{2}=\left[\begin{array}{ll}S_{con}&D^{T}\end{array}\right]^{T}\\ =\left[\begin{array}{ll}J&T_{con}&D\\ S_{con}&0&0\\ D^{T}&0&0\end{array}\right]\\ a=\left[\begin{array}{ll}\Omega\\ T\end{array}\right]\\ H=\left[\begin{array}{ll}H_{T}\\ H_{S}\end{array}\right]\end{array}