*3.1. SSCI Mechanism*

For conventional double closed-loop PI control of RSC under stator-flux oriented method, the increments of rotor voltage and current under *dq* reference frame are [22,28]:

$$\begin{cases} \Delta u\_{rd} = R\_l \Delta i\_{rd} - k\_1 (\omega\_s - \omega\_r) \Delta i\_{rq} + k\_1 p \Delta i\_{rd} \\ \Delta u\_{rq} = R\_r \Delta i\_{rq} - k\_1 (\omega\_s - \omega\_r) \Delta i\_{rd} + k\_1 p \Delta i\_{rq} \\ \Delta i\_{rd} = \frac{1}{k\_2} \Delta i\_{sd} \\ \Delta i\_{rq} = \frac{1}{k\_2} \Delta i\_{sq} \end{cases} \tag{6}$$

where *k*1 = *Lr* − *<sup>L</sup>*2*m*/*Ls* and *k*2 = <sup>−</sup>*Lm*/*Ls*, and *p* is the differential operator.

The terminal voltage of DFIG is supposed to be a three-phase symmetric fundamental sinusoidal wave, and phase voltage is expressed as:

$$u\_{s1} = \sqrt{2}lI\_s \sin(\omega\_s t + \varphi\_{u0})\tag{7}$$

where *ϕu*0 is initial phase of fundamental voltage.

When current disturbance (with resonance angular frequency, *<sup>ω</sup>n*) appears in the fixed series-compensated transmission line, a phase current of DFIG can be expressed as:

$$i\_{sa} = \sqrt{2}I\_s \sin(\omega\_s t + \varphi\_{i0}) + \sqrt{2}I\_n \sin(\omega\_n t + \varphi\_{in}) = i\_{sa0} + i\_{sa\\_sub} \tag{8}$$

where *Is* and *ϕi*0 are effective value and initial phase of fundamental current, *isa*0, respectively. *In*, *ωn*, and *ϕin* are effective value, angular frequency, and initial phase of subsynchronous current, *isa*\_*sub*, respectively.

Under *dq* reference frame, subsynchronous voltage and current can be expressed as:

$$\begin{cases} \ u\_{sd} = 0\\ \ u\_{sq} = -\sqrt{3}I\_s \end{cases} \tag{9}$$

$$\begin{cases} \dot{i}\_{sd} = -\sqrt{3}I\_s \sin(\varphi\_{n0} - \varphi\_{i0}) - \sqrt{3}I\_n \sin[(\omega\_s - \omega\_n)t + \varphi\_i] = \dot{i}\_{sd0} + \dot{i}\_{sd\\_sub} \\\ i\_{sq} = -\sqrt{3}I\_s \cos(\varphi\_{n0} - \varphi\_{i0}) - \sqrt{3}I\_n \cos[(\omega\_s - \omega\_n)t + \varphi\_i] = \dot{i}\_{sq0} + \dot{i}\_{sq\\_sub} \end{cases} \tag{10}$$

where *ϕi* = *ϕu*0 − *ϕin*, *isd*0, and *isq*0 are direct current components of stator current under *dq* frame, and *isd*\_*sub* and *isq*\_*sub* are subsynchronous components with frequency *ωs* − *ωn*.

It is supposed that fundamental power can be accurately tracked, and variation of instantaneous active and reactive power only contains subsynchronous components.

$$\begin{cases} \Delta p\_s = 3lI\_sI\_n\cos[(\omega\_s - \omega\_n)t + \varphi\_i] = -\sqrt{3}lI\_si\_{sq\\_sub} \\ \Delta q\_s = 3lI\_sI\_n\sin[(\omega\_s - \omega\_n)t + \varphi\_i] = -\sqrt{3}lI\_si\_{sd\\_sub} \end{cases} \tag{11}$$

As is shown in Formula (11), subsynchronous current with angular frequency, *ωn*, can induce power fluctuation with angular frequency, *ωs* − *ωn*. Then, Δ*ps* and Δ*qs* can enter into the inner current control loop and turn into reference values of the rotor current. Meanwhile, a rotating magnetic field is formed via cutting rotor winding by subsynchronous current of the stator side, then three phase subsynchronous current with angular frequency,*<sup>ω</sup>r* − *ωn*, is induced in rotor winding, which can cause rotor voltage disturbances. These disturbances react upon rotor winding and impose subsynchronous current with angular frequency,*<sup>ω</sup>s* − *<sup>ω</sup>n*,which eventually cause a new subsynchronous current. Once this new subsynchronous current is added to original current disturbance, √2*In* sin(*<sup>ω</sup>n<sup>t</sup>* + *ϕin*), the current disturbance will be gradually increased. The DFIG controller and series-compensated transmission line interacts and stimulates each other, which causes diverging oscillation of active and reactive power.
