## 2 The Composition and Mathematical Model of Doubly-Fed Variable-Speed Pumped Storage Units
### Mathematical Model of the Reversible Pump-Turbine  
When the output power of DFVSPS unit is continuously adjusted, the speed of the unit will fluctuate constantly. Therefore, the regulating system is required to make the pump turbine adapt to the fluctuations of output power and adjust the speed to stabilize near the speed reference. The pump turbine regulating system is mainly comprised of a governor, electro-hydraulic servo system, DFIM, pump turbine and water diversion system, and the connection diagram of each part is shown in Figure 2.  
\(n\) is the speed of DFVSPS unit; \(n_{ref}\) is the speed reference of DFVSPS unit; \(T_{m}\) is mechanical torque; \(T_{e}\) is electromagnetic torque. The mainstream proportion integration differentiation (PID) governor is adopted in the governor part. The PID governor has the advantages of being easy to use, strong stability and good robustness, and can be adjusted according to the deviation between the given reference value and the actual value. The function of the electro-hydraulic servo system is to convert the electrical signal of the governor into a mechanical signal for output, thereby controlling the guide vane opening. When an ac servo motor/electro-hydraulic actuator is used in the electro-hydraulic servo system, shown in Figure 3, in order to simplify the model for the convenience of research, the nonlinear links such as saturation limiting of a servomotor, dead zone and gap nonlinearity are ignored, and the linear transfer function can be simplified into the form shown in Formula (1). Where: \(y_{pid}\) is the output guide vane opening of governor; \(y\) is the output guide vane opening of servo system; \(y_{1}\) is the output guide vane opening of AC servo mechanism. There is a dead zone of governor output and a saturation limiting of main servomotor. \(T_{y1}\) is the AC servo mechanism response time constant; \(T_{y}\) is the main servomotor time constant:  
\[\frac{y(s)}{y_{pid}(s)}=\frac{1}{T_{y}s(T_{y1}s+1)+1}. \tag{1}\]  
Figure 3: ac servo motor/ electro-hydraulic actuator type servo system. Figure 2: Schematic diagram of pump turbine regulating system. The rigid water hammer model is adopted for a water traced pipeline, and the dynamic equation of the water traced pipeline can be expressed by Formula (2):  
\[\frac{H(s)}{Q(s)}=-T_{w}s, \tag{2}\]  
where: \(H\) is the head of pump turbine; \(Q\) is the water flow of pump turbine; \(T_{w}\) is water flow inertia time constant. The structure of the pump turbine is complex, the operating conditions change frequently during the operation, the internal flow field is complex, and it is affected by the effects of water hammer, water guiding mechanism and cavitation erosion, etc. For the internal characteristics of the pump turbine, many internal fluid factors need to be considered, so it is difficult to get a mathematical model consistent with the actual situation. The pump turbine model is generally expressed quantitatively through a model test. The model test is to carry out the simulation test on the experimental table according to the actual possible various conditions (including the switching process), and record the relevant data to form the full characteristic curve. The full characteristic curve contains all features of the pump turbine during the operation process. Using the full characteristic curve to study the switching process of pump turbine is now a common method. The full-characteristic curve of pump-turbine obtained by the model test method is shown in Figure 4.  
In Figure 4, the red line indicates the path of the pump turbine switching process, and numbers in Figure 4 indicate the various stages of the switching process. The details will be described later. The mechanical torque \(T_{m}\) of the pump turbine and the flow \(Q\) are related to the guide vane opening \(y\), head \(H\), and the rotational speed \(n\), and the relationship between them can be expressed by Formula (3):  
\[\left\{ \begin{array}{l}T_{m}=T_{m}(y,H,n)\\ Q=Q(y,H,n)\end{array}