## 2 SISO Impedance Measurement and Oscillation Analysis of FSC-VSPSU
### Structure of the FSC-VSPSU  
Figure 1 presents the detailed structure of the FSC-VSPSU, which mainly consists of upper/lower reservoirs, a pump turbine, a motor, and a power electronic converter. In addition, the control systems are mainly composed of a turbine governor, an excitation system, and a converter controller. The upper and lower reservoirs store water resources, and the pump turbine is a mechanical transmission part. These two components can collaborate to facilitate the reciprocal transformation between potential energy and mechanical kinetic energy. Furthermore, the pump turbine drives the motor to rotate to generate the electric energy, and the power electronic converter can achieve the controllable energy exchange with the power grid. By coordinating the turbine governor, the excitation system, and the converter controller, the FSC-VSPSU can operate efficiently [13].  
(1) For the excitation system, it regulates the voltage at the motor terminal, with the corresponding structure depicted in Figure 2. In detail, \(H_{1}\)(s) represents the voltage sampling link; \(H_{2}\)(s) is the series compensation link; \(H_{3}\)(s) represents the power amplification link; and \(H_{4}\)(s) is the parallel correction link [11]. Among them, \(H_{2}\)(s) is crucial for the voltage control performance. Additionally, the limiter is designed to restrict occurrences of over-excitation and under-excitation.  
Figure 1: Structure diagram of the FSC-VSPSU.  
The expressions of \(H_{1}\)(s), \(H_{2}\)(s), \(H_{3}\)(s), and \(H_{4}\)(s) are listed as  
\[\left\{\begin{array}{l}H_{1}(s)=\frac{1}{1+sT_{\rm c}}\\ H_{2}(s)=K\frac{1+sT_{\rm a}}{K_{\rm c}+sT_{2}}\cdot\frac{1+sT_{\rm a}}{1+sT_{ \rm a}}\\ H_{3}(s)=\frac{K_{\rm c}}{1+sT_{\rm a}}\\ H_{4}(s)=\frac{sK_{\rm c}}{1+sT_{\rm f}}\end{array}\right. \tag{1}\]  
where \(T_{\rm c}\) is the time constant of the sampling link; \(T_{1}\), \(T_{2}\), \(T_{3}\), and \(T_{4}\) are the time constants of the series compensation link; \(T_{\rm a}\) is the time constant of the power amplification link; \(T_{\rm f}\) is the time constant of the parallel correction link; \(K\) is the DC gain of the series correction link; \(K_{\rm V}\) is the integration correction selection factor; \(K_{\rm a}\) is the gain of the power amplification link; and \(K_{\rm f}\) is the gain of the parallel correction link.  
(2) For the turbine governor, it realizes the speed/power regulation of the pump turbine, and the specific structure is shown in Figure 3. In detail, \(H_{5}\)(s) represents the speed/power sampling link; \(H_{6}\)(s) is the delay of the hydraulic system; \(H_{7}\)(s) is the transfer function to describe the water hammer effect; and \(T_{\rm c}/T_{\rm o}\) are the hydraulic servo motor's closing/opening times [11]. Furthermore, the PI control in the turbine governor is crucial to the control performance of the pump turbine.  
The expressions of \(H_{5}\)(s), \(H_{6}\)(s), and \(H_{7}\)(s) are listed as  
\[\left\{\begin{array}{l}H_{5}(s)=\frac{1}{1+sT_{\rm R}}\\ H_{6}(s)=\frac{1}{1+sT_{\rm P}}\\ H_{7}(s)=\frac{1-sT_{\rm w}}{1+0.5 sT_{\rm w}}\end{array}\right. \tag{2}\]  
where \(T_{\rm R}\) is the time constant of the collection link; \(T_{\rm P}\) is the time constant of the power delay of the hydraulic system; and \(T_{\rm w}\) is the time constant of the water hammer effect of the pump turbine.  
(3) For the converter controller, it can control the grid-connected power or the motor speed quickly. Taking the rapid control of the motor speed as an example, its structure diagram is presented in Figure 4. In detail, the grid-side converter facilitates the control of DC voltage through a dual-loop control strategy, wherein the inner loop governs the current, while the outer loop regulates the DC voltage. The machine-side converter controls the motor speed, which also adopts the double closed loop control, but with the rotation speed loop as the outer control loop [13].  
Figure 3: Structure diagram of the turbine governor.  
Figure 2: Structure diagram of the excitation system.