## 3 Mathematical Modeling
### Mathematical Modeling of Synchronous Generator  
In hydraulic wind turbine systems, the integration of the variable-displacement motor and the excitation synchronous generator plays a critical role in energy conversion. The rotational shaft of the variable-displacement motor is coupled directly or via a linkage to the rotor shaft of the excitation synchronous generator, ensuring synchronous rotation between the two components. This synchronous rotation is critical for efficient energy transfer. When driven by hydraulic fluid, the variable-displacement motor transfers its rotational speed and torque directly to the rotor of the excitation synchronous generator. The rotor's rotation induces a magnetic field via the excitation winding, generating an alternating voltage in the stator. This process enables the sequential conversion of energy from hydraulic to mechanical and finally to electrical forms. The working principle of the excitation synchronous generator is illustrated in Figure 3b. The coaxial connection design enhances energy transfer efficiency between the variable-displacement motor and the excitation synchronous generator, significantly reducing energy losses, such as mechanical friction and magnetic hysteresis. This reduction in energy losses further enhances the system's overall efficiency. Moreover, the precise synchronization of rotational speeds ensures that the output frequency of the excitation synchronous gen  
Figure 3: The working principle of the excitation synchronous generator. (**a**) The control schematic diagram of the excitation synchronous generator. (**b**) The schematic diagram of synchronous generator structure: 1—stator; 2—rotor; 3—air gap; 4—armature winding. erator matches the grid frequency, thereby ensuring stable and reliable grid integration. The electrical energy is then conditioned by transformers and power electronics for grid integration and end-use. The voltage equation of a three-phase synchronous generator is as follows:  
\[ \begin{bmatrix}u_{a}\\ u_{b}\\ u_{c}\\ u_{f}\\ 0\\ 0\end{bmatrix} = \begin{bmatrix}r_{a}&0&0&0&0&0\\ 0&r_{b}&0&0&0&0\\ 0&0&r_{c}&0&0&0\\ 0&0&0&r_{f}&0&0\\ 0&0&0&0&r_{D}&0\\ 0&0&0&0&0&r_{Q}\end{bmatrix} \cdot \begin{bmatrix}-i_{a}\\ -i_{b}\\ -i_{c}\\ i_{f}\\ i_{D}\\ i_{Q}\end{bmatrix} + \begin{bmatrix}\psi_{a}\\ \psi_{b}\\ \psi_{c}\\ \psi_{f}\\ \psi_{D}\\ \psi_{Q}\end{bmatrix}  \tag{16}\]  
The flux linkage equation is as follows:  
\[ \begin{bmatrix}\psi_{a}\\ \psi_{b}\\ \psi_{c}\\ \psi_{f}\\ \psi_{D}\\ \psi_{Q}\end{bmatrix} = \begin{bmatrix}L_{aa}&M_{ab}&M_{ac}&M_{af}&M_{aD}&M_{aQ }\\ M_{ba}&L_{bb}&M_{bc}&M_{bf}&M_{bb}&M_{bD}&M_{bQ}\\ M_{ca}&M_{cb}&L_{cc}&M_{cf}&M_{cD}&M_{cQ}\\ M_{fa}&M_{fb}&M_{fc}&L_{ff}&M_{fD}&M_{fQ}\\ M_{Da}&M_{Db}&M_{Dc}&M_{Df}&L_{DD}&M_{DQ}\\ M_{Qa}&M_{Qb}&M_{Qc}&M_{Qf}&M_{QD}&L_{QQ}\end{bmatrix} \cdot \begin{bmatrix}i_{ a}\\ i_{b}\\ i_{c}\\ i_{D}\\ i_{Q}\end{bmatrix}  \tag{17}\]  
where \(u\) is voltage, \(i\) is current, \(\psi\) is flux linkage, \(r\) is resistance, \(L\) is self-inductance, \(M\) is mutual inductance, \(a\) is \(a\)-axis, \(b\) is \(b\)-axis, \(c\) is \(c\)-axis, \(f\) is field winding, \(D\) is longitudinal damping factor, and \(Q\) is transverse damping factor. To perform the Park reduction-order transformation on Equations (16) and (17), transforming the three-phase \(a\), \(b\), and \(c\) coordinate system into the \(d\), \(q\) and \(0\) coordinate system, the transformation formula is as follows:  
\[ \begin{bmatrix}i_{d}\\ i_{q}\end{bmatrix} =\sqrt{\frac{2}{3}} \begin{bmatrix}\cos\theta&\cos\left( \theta-\frac{2}{3}\pi\right)&\cos\left(\theta+\frac{2}{3}\pi\right)\\ -\sin\theta&-\sin\left(\theta-\frac{2}{3}\pi\right)&-\sin\left(\theta+\frac{ 2}{3}\pi\right)\end{bmatrix} \cdot \begin{bmatrix}i_{a}\\ i_{b}\\ i_{c}\\ i_{c}\end{bmatrix}  \tag{18}\]  
where \(\theta\) is the angle between the d-axis and the a-axis. After the transformation, the voltage equation of the synchronous generator is as follows:  
\[ \begin{bmatrix}u_{d}\\ u_{q}\\ u_{0}\\ u_{f}\\ 0\\ 0\end{bmatrix} =