## 2 Mathematical Models
### Dynamic Characteristics Analysis of Open Loop System  
The dynamic behavior of the compensation power is analyzed by means of the open loop power state equation of the compensation system. This provides a clear understanding of the power flow of the compensation system while designing power compensation control algorithms that adapt to load changes. First, (12) is rewritten in the compact state space form as:  
\[\left\{\begin{array}{ll}\frac{dp_{1}}{dt}=-\frac{R}{L}p_{1}-\omega q_{1}- \frac{9}{2L}U_{m}^{2}+\frac{3}{2L}\sqrt{\frac{3}{2}}U_{m}u_{d}+\frac{3U_{m0}U _{\phi 0}\cos\delta_{0}}{X}(\omega-\omega_{0})\\ \frac{dq_{1}}{dt}=\omega p_{1}-\frac{R}{L}q_{1}+\frac{3}{2L}\sqrt{\frac{3}{2} }U_{m}u_{q}+\frac{3U_{m0}U_{\phi 0}\sin\delta_{0}}{X}(\omega-\omega_{0})\end{array}\right. \tag{12}\]  
\[\dot{x}_{0}=A_{0}x_{0}+b_{0} \tag{13}\]  
where  
\[x_{0}=\left[p_{1}\quad q_{1}\right]^{T} \tag{13a}\] \[A_{o}=\left[\begin{array}{cc}-\frac{R}{L}&-\omega\\ \omega&-\frac{R}{L}\end{array}\right] \tag{13b}\]\[b_{0}=\begin{bmatrix}\frac{3\pi}{2L}\sqrt{\frac{3}{2}}U_{m}u_{d}+\frac{3U_{m}U_{d} \cos\delta_{0}}{X}(\omega-\omega_{0})-\frac{9}{2L}U_{m}^{2}\\ \frac{3}{2L}\sqrt{\frac{3}{2}}U_{m}u_{q}+\frac{3U_{m}U_{d}\sin\delta_{0}}{X}( \omega-\omega_{0})\end{bmatrix} \tag{13c}\]  
Second, the constant solution (fixed point) of the compensated system is calculated as (14), and the eigenvalues of the open-loop state matrix of the compensation system are calculated as (15).  
Equation (14) indicates that there is an equilibrium point in the system in case the denominator of the equation is not zero. However, there is the \((R/L-\omega)\) term in (15), so the system is not necessarily convergent. In this section, two cases of eigenvalues are considered as follows:  
Case 1: \(R/L>\omega\). The voltage frequency of the generator is 50 Hz, so it is assumed that the actual value of \(\omega\) is also approximately equal to 100\(\pi\) rad\(/\)s. Therefore, the case holds when the line resistance and filter inductance satisfy a 314-fold relationship. It can be inferred from (15) that the eigenvalues (\(\lambda_{A_{01}}\), \(\lambda_{A_{02}}\)) of the state matrix are all less than zero. Overall, the output power (\(p_{1}\), \(q_{1}\)) of the compensation system converges to the equilibrium point (\(p_{1}^{*}\), \(q_{1}^{*}\)) as time goes to infinity.  
Case 2: \(R/L<\omega\). In this case, the range of eigenvalues (\(\lambda_{A_{01}}\), \(\lambda_{A_{02}}\)) is difficult to determine and leads to the system no longer converges. Therefore, the compensation system requires an additional controller. The added controller not only changes the eigenvalues of the state matrix but also makes the output power of the system converge to the new equilibrium point (\(p_{1}^{*}\), \(q_{1}^{*}\)). According to the above two categories of analysis, line impedance and filter have a crucial role in the stability of the system. To complicate matters further, it is also coupled to the frequency of the generator. Note that, the relationship between impedance and frequency is not continued to be considered in this paper.  
\[\begin{bmatrix}p_{1}^{*}\\ q_{1}^{*}\end{bmatrix}=\begin{bmatrix}\frac{\frac{3\pi}{2L}\left(\sqrt{2}U_{m}u_ {d}+\frac{3U_{m}U_{d}\cos\delta_{0}}{X}(\omega-\omega_{0})-3U_{m}^{2}\right)}{ \left(\frac{\pi}{2}+\omega^{2}\right.}-\frac{\omega\left(\frac{3}{2}\sqrt{2}U_ {m}u_{q}+\frac{3U_{m}U_{d}\sin\delta_{0}}{X}(\omega-\omega_{0})\right)}{\left( \frac{\pi}{2}\right)^{2}+\omega^{2}}\\ \frac{\mathbb{E}\left(\frac{3}{2L}\sqrt{2}U_{m}u_{q}+\frac{3U_{m}U_{d}\cos \delta_{0}}{X}(\omega-\omega_{0})\right)}{\left(\frac{\pi}{2}\right)^{2}+ \omega^{2}}+\frac{\frac{3\pi}{2L}\left(\sqrt{2}U_{m}u_{d}+\frac{3U_{m}U_{d} \cos\delta_{0}}{X}(\omega-\omega_{0})-3U_{m}^{2}\right)}{\left(\frac{\pi}{2} \right)^{2}+\omega^{2}}\end{bmatrix} \tag{14}\]  
\[\begin{bmatrix}tr(A_{o})\\ \det(A_{o})\end{bmatrix}=\begin{bmatrix}-\frac{2R}{L}\\ \left(\frac{R}{L}+\omega\right)\left(\frac{R}{L}-\omega\right)\end{bmatrix} \tag{15}\]