## 1 Introduction  
New power systems with renewable energy as the main body have the problem of large random fluctuations of power generation; thus, they have a huge demand for energy storage devices represented by pumped storage. The variable-speed pumped storage unit is different from the traditional constant-speed pumped storage unit, which can be used not only for planned peak regulation but also for dynamic peak regulation with system power fluctuation. It will be an important device for future power systems. Variable-speed pumped storage units adopt a doubly fed AC excitation structure, which has the characteristics of variable rotor speed, multi-branch rotor winding structure, and dense distribution of rotor windings [1]. Under high-speed operation conditions, the probability of internal short-circuit faults of the rotor windings is high; not only inter-turn short-circuit faults, but inter-phase short-circuit faults may occur. It will cause an unbalanced force on the rotor, partial overheating, and severe vibration of the unit [2, 3]. If left unchecked, it may evolve into serious faults such as large-scale short circuits and rotor core damage [4]. However, AC excitation system protection of variable-speed pumped-storage units is also used as the rotor windings' internal short-circuit fault protection in engineering, which lacks pertinence and has low sensitivity. Only a few short-circuit faultswith large phase currents can be covered, which is difficult to sensitively reflect the inter-turn short-circuit faults with weak fault characteristics. Therefore, special protection should be provided to detect the fault in the early stage and avoid causing significant economic losses [5]. At present, there have been several studies [6; 7; 8] on the internal short-circuit fault of synchronous generators. There are two kinds of methods based on non-electrical quantities and electrical quantities. The methods based on non-electrical quantities mainly include the vibration signal analysis method [9] and temperature measurement method [10]. Those based on the electrical quantities mainly include the detection coil-based method [11; 12; 13], recurrent surge oscillography method [14; 15], flux linkage difference vector-based method [16], machine learning method [17; 18; 19] and circulating current between stator branches-based method [20; 21]. The essence of the circulating current between stator branches-based method is to use the stator windings as the detection coil to monitor the distorted air-gap magnetic field caused by the rotor windings' internal short-circuit faults. This has the advantage of no additional equipment installation. However, the rotor winding of a variable speed pumped storage unit is in the form of three-phase AC excitation, while the excitation circuit of the conventional synchronous unit is single-phase DC excitation, which is completely different. When an internal short-circuit fault of a rotor winding occurs, the fault type and characteristics of variable-speed units are more complex and are completely different from those of a synchronous speed unit, which makes the existing inter-turn short-circuit protection of the excitation circuit no longer applicable. Variable-speed pumping storage units and doubly fed wind generators both use doubly fed AC excitation [22]. In [23], two measuring coils are used to obtain the magnetic field difference value. The short-circuited fault of the rotor windings is detected by analyzing the induced voltage difference of the measuring coils. In [24], the variation law of the flux linkage difference vector before and after the short-circuit fault of rotor windings is analyzed, then the inter-turn short-circuit fault can be easily identified. In [25; 26], the armature reaction mechanism of the air gap magnetic field is studied when the rotor windings of a doubly fed wind generator are short-circuited, and the characteristic frequencies of the electrical quantities on the stator side are summarized.