## 2 The Composition and Mathematical Model of Doubly-Fed Variable-Speed Pumped Storage Units
### Mathematical Model of the Reversible Pump-Turbine
\right.. \tag{3}\]  
Different pump turbine models can be obtained by analyzing Equation (3) with different methods. In order to simplify the calculation, the pump turbine model is divided into two parts: the turbine operating condition model and the pump operating condition model. The characteristic curve of pump turbine during its turbine operating condition is similar to the conventional turbine in the low speed region [17], as shown in Figure 5. Since the transition process in this paper is the transition process during the normal operation of the unit, all in the low speed region, so the model of the conventional turbine can be used to replace the model of pump turbine under turbine operating conditions. Figure 4: Full-characteristic curve of pump-turbine and main transition process path. The conventional turbine mathematical model commonly used at present is a simplified nonlinear model, which has a simple structure and can better reflect the fluid characteristics inside the turbine [18]. This model is proposed by IEEE [19] and the nonlinear turbine model can be expressed as Formula (4) in case of the loss of water head is ignored:  
\[\left\{ \begin{array}{l}q=A_{t}y\sqrt{h}\\ P_{m}=(q-q_{nl})h\end{array} \right., \tag{4}\]  
where: \(q\) is the flow; \(y\) is the ideal guide vane opening; \(h\) is the head; \(A_{t}=1/(y_{FL}-y_{NL})\), \(y_{FL}\) is an ideal guide vane opening; \(y_{NL}\) is the no-load guide vane opening; \(P_{m}\) is the mechanical power; \(q_{nl}\) is the no-load flow. Each variable in the model of pump turbine is represented by the per unit value (taking the rated value of the unit as the base value). When connected with DFIM, the output of pump turbine needs to be converted to an actual value. The block diagram of this model using the rigid water hammer in Formula (2) is shown in Figure 6.  
Where \(G\) is the actual guide vane opening and \(h_{s}\) is the static head of water. The mathematical model of the pump turbine under pump condition is: according to the full characteristic curve of the pump turbine in the positive pump area, it can be found that, when the guide vane opening is small, the guide vane opening can regulate the flow of a pump turbine. However, after the guide vane opening becomes larger, the flow curves corresponding to different guide vane opening coincide with each other. Therefore, in the area of large guide vane opening, the guide vane opening has no obvious adjustment effect on the flow. At this time, there is an obvious regulating relationship between the speed and the flow of the unit. Therefore, the model of the pump turbine under pump mode can be divided into two parts: the pump mode model in small guide vane opening and the pump mode model in large guide vane opening. According to the relevant literature [20; 21], different pump turbine, different guide vane opening of the characteristics curves start to coincide seriously. In general, the DFVSPS unit enters the stable operation stage when the guide vane opening is about 50% under start-up stage in pump mode. Pump mode model under small guide vane opening is consistent with turbine operation mode, except that the direction of water flow is opposite to the turbine operation mode. When the pump turbine operates at small guide vane opening in pump mode, and the head loss increases sharply, which will have a great impact on the torque and flow of the pump turbine. Therefore, the head loss  
Figure 5: Comparison of operating characteristic curves between conventional turbine and pump turbine. Figure 6: Simplified nonlinear turbine model. caused by the change of guide vane opening needs to be taken into account at this time. The pump mode model under small guide vane opening is shown in Figure 7.  
Where the other quantities are consistent with generating mode, \(f_{g}\) is the head loss coefficient, \(f_{g}=(G_{max}-G)/G\), and \(G_{max}\) is the maximum guide vane opening.