## 2 The Harmonic Circulating Current between Stator Branches-Based Protection Method  
When the variable-speed pumped storage unit is in normal operation, the magnetomotive force of a single coil is a rectangular wave that can be decomposed into the superposition of multiple fractional and integer harmonic (including the fundamental wave) magnetomotive forces. However, for the whole branch of stator and rotor windings, the fractional and low-order harmonic (excluding fundamental wave) magnetomotive force components of each coil in the branch offset each other, and the total magnetomotive force of the branch exhibits the fundamental wave characteristic. Therefore, the current flowing through the single-turn rotor coil only contains the fundamental frequency component during normal operation, which can be denoted as \(i_{r}=I_{r}\cos\omega_{r}t\). \(\omega_{r}\) is the electric angular frequency of the rotor. After a short-circuit fault occurs in rotor windings, a large short-circuit current flows through the short-circuit coils, which can be recorded as \(i_{f}=I_{f}\cos(\omega_{r}t)\). According to Fourier series decomposition, the magnetomotive force of the short-circuit coils can be decomposed into:  
\[ \begin{array}{ll}F(\theta,t)&=\sum\limits_{k}\frac{2 Nf_{f}}{k\pi P}k_{fk}\cos (k\theta)\cos(\omega_{r}t)\\ &=\sum\limits_{k}\frac{Nf_{f}}{k\pi P}k_{fk}\cos(k\theta\pm\omega_{r}t)\\ &k=\frac{1}{P},\frac{2}{P},\frac{3}{P},\cdots\end{array}  \tag{1}\]  
where \(N\) is the turn number of short-circuit winding, \(k_{fk}\) is the harmonic winding factor of the short-circuit winding, \(\theta\) is the space electrical angle of the rotor, \(P\) is the number of pole pairs, and \(k\) is the harmonic order. Assuming that \(\varphi\) represents the stator space electrical angle, then:  
\[\varphi=\theta-\frac{1-s}{s}\omega_{r}t \tag{2}\]  
where \(s\) is the slip ratio. Since there is a relationship between the electric angular frequency of stator and rotor: \(\omega_{r}=s\omega_{s}\), where \(\omega_{s}\) is the stator electric angular frequency. Then, Equation (2) can be expressed as:  
\[\varphi=\theta-(1-s)\omega_{s}t \tag{3}\]  
Substituting Equation (3) into Equation (1), the magnetomotive force of the short-circuit coils can be expressed in the stator space electric angle as:  
\[F(\varphi,t)=\sum\limits_{k}\frac{Nl_{f}}{k\pi P}k_{f\kappa}\cos(k\varphi+k( 1-s)\omega_{s}t\ \pm\ s\omega_{s}t) \tag{4}\]  
According to Equation (4), the characteristic frequency of the induced potential corresponding to the harmonic magnetic field with order \(k\) is:  
\[f_{k}=[k(1-s)\pm s]f_{s}\quad k=\frac{1}{P},\frac{2}{P},\cdots \tag{5}\]  
where \(f_{s}\) is the fundamental wave frequency of the stator side, equal to 50 Hz. The fractional and integer harmonic magnetic fields generated by the short-circuit current are known. For the integer harmonic magnetic field, \(k\) = 1, 2..., since the amplitude and phase of the overall induced potential of each stator in-phase branch are equal, there will be no circulating current between the stator branches. For the fractional harmonic magnetic fields, \(k\) = 1/\(P\)..., \((P\ -\ 1)/P\), \((P\ +\ 1)/P\),... since the different phases of the overall induced potential of each stator in-phase branch, the circulating current will begenerated between the stator branches. The theoretical characteristic frequency of the circulating current is:  
\[f_{k}=[k(1-s)\pm s]f_{s}\quad k=\frac{1}{p},\frac{2}{p},\ldots\ k\
eq 1,2,\ldots \tag{7}\]  
The circulating current can be measured directly by the split-phase transverse differential protection without additional installation of a measurement device. When an internal short-circuit fault occurs in rotor windings, the circulating current between stator branches has the characteristic harmonic frequency as shown in Equation (7). Therefore, the protection criterion for rotor windings' internal short-circuit faults can be designed as follows:  
\[