## 1 Introduction  
A wide range of flexible power regulations under both pumping mode and generation mode can be achieved by a variable-speed pumped-storage generator-motor (VPSGSM) [1], which is more efficient than the traditional constant-speed pumped-storage generator-motor [2]. It is a key component in building a future power system with newer energy [3]. In the 'pumped storage medium and long-term development plan 2021-2035' issued by the State Energy Administration of China, it is pointed out that 'it is necessary to explore and innovate the development mode of pumped storage and master variable speed pumped storage technology'. At present, there are few studies on the relay protection of VPSGM. The rotor winding of VPSGSM adopts an AC excitation structure [4]; the excitation voltage is high. In the process of high-speed rotation, the rotor winding is prone to insulation degradation, and internal short-circuit faults are easy to be generated [5]. The short-circuit faults will cause serious problems, such as partial overheating of the windings [6], abnormal vibration of the generator-motor [7], and even rotor damage [8]. Due to the large capacity of VPSGSM, the cost of the rotor equipment is very high. Thus, targeted protection schemes should be provided to avoid the economic loss caused by rotor damage. The stator and rotor windings of VPSGSM are both multibranch structures, but the rotor winding cannot be configured with transverse differential protection and longitudinal differential protection, which are used for the stator winding. The main reason is that the internal space of the rotor is compact, and the condition of installing the current transformer at the neutral point side is not available [9]. Thus, the short-circuit fault of the rotor winding is in the 'weak protection' state. The fault is addressed by the instantaneous over-current protection of the power electronic excitation device. It can only act when the instantaneous value of the rotor phase current reaches three times the rated current, which has low sensitivity and has difficulty meeting the safety requirements. Since the working principle of the VPSGSM is similar to that of the doubly fed induction generator (DFIG), the related research on DFIG has certain reference values. In [10], a differential protection method considering the rotor current frequency is proposed for the DFIG with a flux-oriented vector control strategy. This method relies on the accurate measurement of the motor speed, which may not be suitable for the units using a voltage-oriented vector control strategy. In [11], a rotor winding interturn short-circuit-fault diagnosis method based on the change in flux observation difference before and after the fault is proposed, and the faults can be detected rapidly. In [12], the armature reaction mechanism of the air gap magnetic field in the DFIG rotor winding short circuit is analyzed. In [13], the characteristic frequencies of the stator side current are summarized for the DFIG rotor winding short-circuit fault. The above studies focus on the rotor fault diagnosis of DFIG. However, the capacity of the VPSGSM is much larger than that of the DFIG. The short-circuit current of VPSGSM is huge and the fault is more serious. Therefore, a protection method with a clear criterion is needed more for the VPSGSM. In [14], the internal fault simulation calculation model of the VPSGSM is established based on the multiloop theory. In [15], the low-frequency component of the stator branch current is used to judge whether a rotor winding short-circuit fault occurs. In [16], a specific protection criterion is proposed for the VPSGSM based on the harmonic characteristics of the circulating current between stator branches. This protection method has high reliability, but poor rapidity. The existing studies only use the electrical quantity of the stator side or the rotor side for fault identification. However, the combined protection of the stator and rotor side is a new solution.