*3.2. Control Design*

As analyzed above, SSCI can be well suppressed once the rotor current dynamic is constrained by following the prescribed values. The reference values of rotor current can be deduced from (2), where active power, *P*∗ *s* , is acquired by maximum power point tracking (MPPT) and reactive power, *Q*∗ *s*, is calculated according to grid demand.

 .

$$\begin{cases} \begin{array}{c} i\_{rq}^{\*} = -\frac{2L'\, \_sP\_s^\*}{3L\_mIL\_s} \\\ i\_{rd}^{\*} = \frac{\underline{U}\_s}{\omega\_1L\_m} - \frac{2L'\, \_sQ\_s^\*}{3L\_mIL\_s} \end{array} \tag{12}$$

SSCI is mainly caused by the interaction between the RSC control and series-compensated transmission line. Thus, RSC control signals are chosen as control variables. According to (3) and (4), the equation of series-compensated wind power systems can be represented as:

)*x* = *f*(*x*) + *g*(*x*)*u y* = *h*(*x*) (13) *f*(*x*) = ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ *<sup>ω</sup>*1*irq* − *R r L r ird* − *<sup>ω</sup>*1*ird* − *R r L r irq* 1 *C idc <sup>ω</sup>*1*isq* + *R s L s isd* + *usd L s* − *<sup>ω</sup>*1*isd* + *R s L s isq* + *usq L s* − *<sup>ω</sup>*1*igq* − *RGSC LGSC igd* − *ugd LGSC <sup>ω</sup>*1*igd* − *RGSC LGSC igq* − *ugq LGSC <sup>ω</sup>*1*iLq* + *RL LL iLd* + 1 *LL* (*uld* − *uscd* − *Ed*) − *<sup>ω</sup>*1*iLd* + *RL LL iLq* + 1 *LL ulq* − *uscq* − *Eq <sup>ω</sup>*1*uscq* + 1 *CSC iLd* − *ω*1*uscd* + 1 *CSC iLq* 1 2*HT* (*TT* − *Ksθs*) 1 2*HG* (*Ksθs* − *Te*) 2*π f*1 *ωT* − *ωr Ng* ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ (14) *g*(*x*) = ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ *udc* 2*L r* 0 0 *udc* 2*L r* − 1 *Cdc ird* − 1 *Cdc irq* 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ (15)

where state vector is *x* = [*ird irq udc isd isq igd igq iLd iLq uscd uscq ωT ωr <sup>θ</sup>s*]*<sup>T</sup>*, control variables are *u* = [*Srd Srq*]*<sup>T</sup>*, and output equations are *y* = [*ird* − *i*∗*rd irq* − *<sup>i</sup>*<sup>∗</sup>*rq*]*<sup>T</sup>*.

To choose sliding mode function:

$$
\sigma\_{rd} = (i\_{rd} - i\_{rd}^\*) + c\_1 \int (i\_{rd} - i\_{rd}^\*) dt \tag{16}
$$

$$
\sigma\_{rq} = \left( i\_{rq} - i\_{rq}^\* \right) + c\_2 \int \left( i\_{rq} - i\_{rq}^\* \right) dt \tag{17}
$$

where positive constants, *c*1 and *c*2, are weight coefficients of integral sliding mode items. This can help to remove steady state error. To calculate first-order derivatives of *σrd* and *<sup>σ</sup>rq*:

$$\dot{\sigma}\_{rd} = \omega\_1 \dot{i}\_{rq} - \frac{R\_{r}^{\prime}}{L\_{r}^{\prime}} \dot{i}\_{rd} - \dot{i}\_{rd}^{\*} + c\_1 (\dot{i}\_{rd} - \dot{i}\_{rd}^{\*}) + \frac{u\_{dc}}{2L\_{r}^{\prime}} S\_{rd} \tag{18}$$

$$\dot{\sigma}\_{rq} = -\omega\_1 \dot{i}\_{rd} - \frac{\mathcal{R}\_r'}{L\_r'} \dot{i}\_{rq} - \dot{\bar{i}}\_{rq}^\* + c\_2 \left( \dot{i}\_{rq} - \dot{i}\_{rq}^\* \right) + \frac{\mathcal{U}\_{dc}}{2L\_r'} S\_{rq} \tag{19}$$

Observed from (18) and (19), the relative degrees with respect to *σrd* and *<sup>σ</sup>rq* are both 1. They are less than the system order, which is 14. System dynamics can be divided into external dynamics and internal dynamics, according to zero dynamics stability theory. External dynamics are normally demanded to be stable and have good dynamic quality, while internal dynamics can only satisfy asymptotic stability. This paper will not go into details about asymptotic stability of internal dynamics for series-compensated DFIG wind power systems, which has been stated in papers [28,29]. Next, the design procedure for VGSTSM control law will be presented in detail. Here, *irq* control design is taken as an example because the design procedure is similar for *ird*.

Considering parameter perturbation, measuring error, and external disturbance, the lumped uncertainty is represented by <sup>Δ</sup>*dq*. Then, formula (19) was rewritten as:

$$\dot{\sigma}\_{rq} = -\omega\_1 \dot{i}\_{rd} - \frac{R'\_r}{L'\_r} \dot{i}\_{rq} - \dot{\dot{i}}^\*\_{rq} + c\_2 \left( \dot{i}\_{rq} - \dot{i}^\*\_{rq} \right) + \frac{\mu\_{dc}}{2L'\_r} S\_{rq} + \Delta d\_q \tag{20}$$

Taking state feedback control into account, this gave:

$$S\_{r\eta} = \frac{2L\_r'}{\imath\_{dc}} \left( \omega\_1 i\_{rd} + \frac{R\_r'}{L\_r'} i\_{r\eta} + \dot{i}\_{r\eta}^\* - c\_2 \left( i\_{r\eta} - i\_{r\eta}^\* \right) + \upsilon\_{r\eta} \right) \tag{21}$$

Then:

$$
\dot{\sigma}\_{r\eta} = \upsilon\_{r\eta} + \Delta d\_{\eta} \tag{22}
$$

The next step was to design auxiliary control law, *vrq*, for (22). RSC control chattering can be rather serious if conventional first-order sliding mode method is adopted. Thus, super-twisting algorithm with continuous control effect and small chattering was employed to construct *vrq*:

$$\begin{cases} \boldsymbol{\upsilon}\_{rq} = -\boldsymbol{\alpha}\_{q} \boldsymbol{\gamma}\_{q} \left| \boldsymbol{\sigma}\_{rq} \right|^{1/2} \text{sign}(\boldsymbol{\sigma}\_{rq}) + \boldsymbol{\upsilon}\_{rq2} \\ \boldsymbol{\dot{\upsilon}}\_{rq2} = -\boldsymbol{\beta}\_{q} \boldsymbol{\gamma}\_{q}^{2} \text{sign}(\boldsymbol{\sigma}\_{rq}) \end{cases} \tag{23}$$

The upper bound *Dqup* of Δ *dq* was demanded to be known in this control law. If control parameters, *<sup>α</sup>q* and *β<sup>q</sup>*, were chosen as 1.5 and 1.1, and *γq* was set as *Dqup*, then finite time stability and second-order sliding mode with respect to *<sup>σ</sup>rq* can be established [37]. However, this upper bound *Dqup* is hard to acquire in series-compensated DFIG-based wind power systems. In case the value for *Dqup* is conservative, RSC will produce excessive control effect, increase unnecessary chattering, and damage electromechanical devices. Therefore, the super-twisting control gains should be constructed as adaptive ones. Control gains can increase or decrease according to upper bound of uncertainty derivatives. This

.

adaptive strategy does not only satisfy control requirement, but also restrains chattering, and the superiority of super-twisting algorithm is fully developed.

Considering the characteristic of barrier function [37], adaptive gain super-twisting sliding mode control law is designed as:

$$\begin{cases} \left. \upsilon\_{rq} = -1.5 \sqrt{\gamma\_q} \right| \upsilon\_{rq} \Big|^{1/2} \text{sign}(\upsilon\_{rq}) + \upsilon\_{rq2} \\ \left. \dot{\upsilon}\_{rq2} = -1.1 \gamma\_q \text{sign}(\upsilon\_{rq}) \end{cases} \tag{24}$$

Adaptive control gain is constructed as:

$$\begin{cases} \dot{\gamma}\_q = \gamma\_{q0\_{\prime}} & \text{if } 0 < t \le t\_{rs} \\\ \gamma\_q = \frac{b\_q \varepsilon\_q}{\varepsilon\_q - \left| \sigma\_{rq} \right|} & \text{if } t\_{rs} < t \end{cases} \tag{25}$$

where *trs* is the time that *<sup>σ</sup>rq* reaches *<sup>ε</sup>q*/2. *γq*<sup>0</sup> and *bq*,*ε<sup>q</sup>* are positive constants. Then, for any *εq* > 0, *trs* > 0 for any initial status, *<sup>σ</sup>rq*(0). When *t* ≥ *trs*, then *<sup>σ</sup>rq*<sup>&</sup>lt; *εq* is satisfied. It was indicated that *irq* can converge to the error range of its reference value in finite time and achieve actual tracking for *i*∗*rq*.

The proof for the above conclusion is followed below. Firstly, let us prove that *<sup>σ</sup>rq* can reach *<sup>ε</sup>q*/2 in finite time, *trs*. It is supposed that *<sup>σ</sup>rq*(0)<sup>&</sup>gt; *εq*2 is satisfied, then adaptive control gain is determined by .*γq* = *γq*0, according to (25).

> ⎧⎨⎩

Consider the following variable transformation:

$$\begin{aligned} z\_{q1} &= \frac{\sigma\_q}{\gamma\_q^2} \\ z\_{q2} &= \frac{\dot{\sigma}\_q}{\gamma\_q^2} \end{aligned} \tag{26}$$

The derivatives for *zq*1 and *zq*2 can be denoted as:

$$\begin{cases} \dot{z}\_{q1} = -\alpha\_q \Big| z\_{q1} \Big|\_{1/2} \text{sign}(z\_{q1}) + z\_{q2} - \frac{2\dot{\gamma}\_q}{\gamma\_q} z\_{q1} \\\ \dot{z}\_{q2} = -\beta\_q \text{sign}(z\_{q1}) - \frac{\Delta \dot{d}\_q}{\gamma\_q^2} - \frac{2\dot{\gamma}\_q}{\gamma\_q} z\_{q2} \end{cases} \tag{27}$$

Choose Lyapunov function:

$$V\_{q1} = \chi\_q^T(t) P\_q \chi\_q(t) \tag{28}$$

.

where *Pq* is constant symmetric positive definite matrix, *χTq* (*t*) = \$ *zq*11/2*sign*(*zq*1) *zq*2 %. Then, time derivative of *<sup>χ</sup>q*(*t*) can be deduced as:

$$\dot{\chi}\_q(t) = \frac{1}{2|z\_{q1}|^{1/2}} K\_{\emptyset} \chi\_{\emptyset} + \frac{\dot{\chi}\_q}{\gamma\_q} \Lambda\_{\emptyset} \chi\_{\emptyset} - \frac{M\_{\emptyset}}{\gamma\_q^2} \tag{29}$$

where *Kq* = <sup>−</sup>*αq*/2 1/2 <sup>−</sup>*βq* 0 , <sup>Λ</sup>*q* = −1/2 0 0 −1 , and *Mq* = 0Δ .*dq* . Time derivative of *Vq*1 is:

$$\dot{V}\_{q1} = -\frac{1}{2|z\_{q1}|^{1/2}} \chi\_q^T Q\_q \chi\_q - \frac{\dot{\gamma}\_q}{\gamma\_q} \chi\_q^T R\_q \chi\_q - \frac{2\Delta d\_q}{\gamma\_q^2} P\_q \chi\_q \tag{30}$$

where *K<sup>T</sup> q Pq* + *PqKp* = − *Qp* and Λ*<sup>T</sup> q Pq* + *Pq*<sup>Λ</sup>*q* = − *Rq*. Then, thanks to the studies in [12], symmetric positive definite matrix, *Pq*, exits, and then *Qp* are *Rq* are positive definite. Then:

$$\dot{V}\_{q1} \le -k\_{q1} V\_{q1}^{\frac{1}{2}} + 2k\_{q3} \frac{D\_{qup}}{\gamma\_q^2} V\_{q1}^{\frac{1}{2}} - \frac{\dot{\gamma}\_q}{\gamma\_q} k\_{q2} V\_{q1} \tag{31}$$

where *kq*1 = *<sup>λ</sup>*min(*Qq*) 2 √*p*11*λ*max(*Pq*), *kq*2 = *<sup>λ</sup>*min(*Rq*) *<sup>λ</sup>*max(*Pq*), and *kq*3 = *<sup>λ</sup>*max(*Rq*) *<sup>λ</sup>*min(*Pq*) 1 2 . *λ*min and *λ*max are minimum eigenvalue and maximum eigenvalue of the relative matrix, respectively. *p*11 is the first elementofmatrix

*Pq*.The first item of the right side in (31) is negative, while the second item is positive. Here, the second item will decrease following increasement of adaptive control gain. Adaptive control gain becomes big enough to conquer uncertainties. Thus, the second item becomes very small. The third item will be negative and further reduced when . *γq* is negative.

As discussed, the first item will be bigger than the second one, and the third item will become smaller. Then, the right side of (31) will be negative and . *Vq*1 ≤ −*aqV*1/2 *q*1 satisfied, which means finite time stability is achieved. *Vq*1 will continue to decrease and then *σrq* can reach *<sup>ε</sup>q*/2.

It was proved above that *σrq* can reach *<sup>ε</sup>q*/2 when the time is *t* = *trs*. The second step is to prove that *σrq* ≤ *εq* can be satisfied after *t* ≥ *trs*.

Choose Lyapunov function:

$$V\_{q2} = \frac{1}{2} \sigma\_{r\eta}^2 \tag{32}$$

Then:

$$\dot{V}\_{q2} = \sigma\_{rq}\dot{\sigma}\_{rq} = \sigma\_{rq}\left(-a\_{q}\gamma\_{q}\left|\sigma\_{rq}\right|^{1/2}\text{sign}(\sigma\_{rq}) + \sigma\_{rq2}\right) \tag{33}$$

where *<sup>σ</sup>rq*<sup>2</sup> = *vrq*2 + <sup>Δ</sup>*dq*. Then:

$$\dot{V}\_{q2} \le \left| \sigma\_{rq} \left| \left( -a\_{\overline{q}} \gamma\_{\overline{q}} \left| \sigma\_{rq} \right|^{1/2} \text{sign}(\sigma\_{rq}) + \left| \sigma\_{rq2} \right| \right) \right. \tag{34} \right|$$

According to the barrier function, *γq* = *bq<sup>ε</sup>q <sup>ε</sup>q*−|*<sup>σ</sup>rq* |, of (25):

$$\begin{array}{ll} \dot{V}\_{q2} & \leq \frac{|\sigma\_{rq}|}{\varepsilon\_{q} - |\sigma\_{rq}|} \left( \alpha\_{q} \varepsilon\_{q} b\_{q} \left| \sigma\_{rq} \right|^{1/2} - \left| \sigma\_{rq2} \right| \varepsilon\_{q} + \left| \sigma\_{rq} \right| \left| \sigma\_{rq2} \right| \right) \\ & = -\frac{|\sigma\_{rq}| \left| \sigma\_{rq2} \right|}{\varepsilon\_{q} - |\sigma\_{rq}|} \left( \frac{a\_{q} \varepsilon\_{q} b\_{q} \left| \sigma\_{rq} \right|^{1/2}}{\left| \sigma\_{rq2} \right|} - \varepsilon\_{q} + \left| \sigma\_{rq} \right| \right) \end{array} \tag{35}$$

Take note of the right side of (35), define:

$$F\_{\eta} = \frac{\alpha\_{\eta} \varepsilon\_{\eta} b\_{\eta} \left| \sigma\_{rq} \right|^{1/2}}{\left| \sigma\_{rq2} \right|} - \varepsilon\_{\eta} + \left| \sigma\_{rq} \right| \tag{36}$$

*Fq* = 0 is a quadratic equation, and the two roots are: 

$$\left|\varepsilon\_{11}\right|^{1/2} = \frac{1}{2} \left( \frac{-\alpha\_q \varepsilon\_q b\_q}{\left|\sigma\_{rq2}\right|} + \left( \left(\frac{a\_q \varepsilon\_q b\_q}{\left|\sigma\_{rq2}\right|}\right)^2 + 4\varepsilon\_q \right)^{1/2} \right) \tag{37}$$

$$|\varepsilon\_{12}|^{1/2} = \frac{1}{2} \left( \frac{-a\_q \varepsilon\_q b\_q}{|\sigma\_{rq2}|} - \left( \left(\frac{a\_q \varepsilon\_q b\_q}{|\sigma\_{rq2}|}\right)^2 + 4\varepsilon\_q \right)^{1/2} \right) \tag{38}$$

It can be easily observed that the second root is negative, and then only the second root needed to be paid more attention. According to (37):

$$\varepsilon\_{11} = \pm \frac{1}{4} \left( \frac{-a\_{\theta} \varepsilon\_{q} b\_{q}}{|\sigma\_{rq2}|} + \left( \left( \frac{a\_{\theta} \varepsilon\_{q} b\_{q}}{|\sigma\_{rq2}|} \right)^{2} + 4 \varepsilon\_{q} \right)^{1/2} \right)^{2} \tag{39}$$

According to the well-known inequation, *a*2 + *b*2 ≤ (*a* + *b*)2:

$$
\left(\frac{a\_q \varepsilon\_q b\_q}{\left|\sigma\_{rq2}\right|}\right)^2 + \left(2\varepsilon\_q^{1/2}\right)^2 \le \left(\frac{a\_q \varepsilon\_q b\_q}{\left|\sigma\_{rq2}\right|} + 2\varepsilon\_q^{1/2}\right)^2\tag{40}
$$

With the aid of (40), the upper bound of |*<sup>e</sup>*11| can be written as:

$$\varepsilon\_{\|c\|1\|} \le \left(\frac{\frac{-s\_{\|c\|}s\_{\mathbb{Q}}}{\lceil \frac{\sigma\_{\|c\|}2}{\lceil \sigma\_{\|}2 \rceil} \rceil + 2\binom{1}{\lceil \frac{\sigma\_{\|c\|}s\_{\mathbb{Q}}}{\lceil \sigma\_{\|}2 \rceil}} \rceil^2}{2}}{2}\right)^2 = \left(\frac{\frac{-s\_{\mathbb{Q}}s\_{\mathbb{Q}}}{\lceil \sigma\_{\|}2 \rceil} + \left(\left(\frac{\lceil \frac{\sigma\_{\mathbb{Q}}s\_{\mathbb{Q}}}{\lceil \sigma\_{\|}2 \rceil} \rceil} + 2\iota\_q^{1/2}\right)\right)^2}{2}\right)^2 = \left(\frac{2\iota\_q^{1/2}}{2}\right)^2 = \varepsilon\_{\mathbb{Q}}\tag{41}$$

Finally, the inequation from (41) can be deduced as:

$$|\mathfrak{e}\_{11}| \le \left(\frac{2\mathfrak{e}\_q^{1/2}}{2}\right)^2 \le \mathfrak{e}\_q \tag{42}$$

If *<sup>σ</sup>q*(*t*) ≥ |*<sup>e</sup>*11|, then *Fq* is positive definite. Consequently, . *Vq*2 < 0 is satisfied for |*<sup>e</sup>*11| ≤ *<sup>σ</sup>q*(*t*) < *<sup>ε</sup>q*. Hence, *<sup>σ</sup>q*(*t*) will always satisfy *<sup>σ</sup>q*(*t*) < |*<sup>e</sup>*11| the rest of the time, and |*<sup>e</sup>*11| is smaller than *εq* for any derivative of <sup>Δ</sup>*dq*.

Therefore, real sliding mode, with respect to *<sup>σ</sup>q*(*t*), is established in finite time. The *q*-axis rotor current, *irq*, allows us to track for the prescribed *i*∗*rq* with unknown upper bound of uncertainty derivative.

Adaptive gain control law for *σrd* can be designed in a similar way. State feedback control is: 

$$S\_{rd} = \frac{2L'\_r}{\mu\_{dc}} \left( -\omega\_1 i\_{rq} + \frac{R'\_r}{L'\_r} i\_{rd} + \dot{i}^\*\_{rd} - \varepsilon\_1 (\dot{i}\_{rd} - \dot{i}^\*\_{rd}) + \upsilon\_{rd} \right) \tag{43}$$

Sliding mode control law and adaptive control gain are:

$$\begin{cases} \left. v\_{rd} = -1.5 \sqrt{\gamma\_d} |\sigma\_{rd}|^{1/2} \operatorname{sign}(\sigma\_{rd}) + v\_{rd2} \\ \left. \dot{v}\_{rd2} = -1.1 \gamma\_d \operatorname{sign}(\sigma\_{rd}) \right. \end{cases} \tag{44}$$

$$\begin{cases} \dot{\gamma}\_d = \gamma\_{d0}, & \text{if } 0 < t \le t\_{rs1} \\\ \gamma\_d = \frac{b\_d \varepsilon\_d}{\varepsilon\_d - |\sigma\_{rd}|}, & \text{if } t\_{rs1} < t \end{cases} \tag{45}$$

*ird* can converge to the demanded neighborhood in finite time. As mentioned above, the internal dynamics of the system are asymptotically stable, and the external dynamics are finite time stable. Thus, the stability of the whole control system is guaranteed.

## **4. Time-Domain Simulation**

Time-domain simulation is one of the best measures for dynamic stability analysis of a power system. Nonlinear mathematical models can be employed in time-domain simulations, which is very suitable for the nonlinear and complex characteristics of the DFIG power system. The 100 MW aggregated model was adopted to verify effectiveness. Superiority of the proposed control strategy was also compared with PI [19], feedback linearization [27], and conventional sliding mode methods [9] under MATLAB/Simulink. Simulation parameters for series-compensated DFIG-based wind power systems is referred

**Table**

**1.**

to Table 1. The schematic diagram of the proposed VGSTSM damping control scheme is depicted as Figure 2. Firstly, the sliding mode functions were calculated, then the auxiliary control quantities were obtained according to the adaptive law and super-twisting sliding mode control laws, and finally the RSC control signals were obtained through the feedback control. Control parameters were chosen as *εq* = 0.001, *γq* = 2.2, *bq* = 2.0, *εd* = 0.001,*γd* = 2.5, and *bd* = 2.3.

 Series-compensated power system parameters.

wind

DFIG-based


**Figure 2.** Schematic diagram of the proposed damping control scheme.

The wind speed was set as 9 m/s. Series-compensated capacitator was injected into the DFIG-based wind power transmission line at 2.5 s, forming 40% series-compensated level. Transient responses of active power, reactive power, electromagnetic torque, rotor angular speed, DC bus voltage, and transmission line current are shown in Figures 3 and 4. As observed, all of these variables start oscillation when this switch capacitator is put into the system, they can then be rapidly stabilized under the proposed control strategy. When the series-compensated level is increased to 85%, the variables can still converge to steady state, as shown in Figures 5 and 6, though the oscillation time somewhat increased. Figures 3–6 indicate that the proposed strategy was effective for SSCI mitigation under different series-compensated level.

**Figure 3.** Transient responses of active power, reactive power, and electromagnetic torque after 40% series compensation is switched.

**Figure 4.** Transient responses of rotor angular speed, DC bus voltage, and transmission line current after 40% series compensation is switched.

Wind speed increased to 11 m/s under 85% compensation to evaluate controller performances for different wind speeds. The responses of these variables are demonstrated in Figure 7. By comparing Figure 7 with Figures 5 and 6, it was observed that SSCI mitigation was better when wind speed was higher.

**Figure 5.** Transient responses of active power, reactive power, and electromagnetic torque after 85% series compensation is switched.

**Figure 6.** Transient responses of rotor angular speed, DC bus voltage, and transmission line current after 85% series compensation is switched.

After the system entered steady state, three-phase short circuit fault occurred at the high voltage side of transformer at t = 5 s to verify capacity for fault ride-through of the proposed control method. The duration of the fault was 20 ms. As shown in Figure 8, dynamic responses of active power, reactive power, and electromagnetic torque can all return to normal after a short transient fluctuation. This indicates that SSCI can be quickly suppressed under three-phase short circuit fault and the capacity for fault ride-through was enhanced under the proposed control method.

The performance for SSCI mitigation was compared to that of other control means based on PI control, partial feedback linearization, and first-order sliding mode. Figure 9 is the control structure of the classical double closed-loop PI scheme. The symbol \* means reference value. Control parameters for PI controllers are *Kp* = 0.1, *KQ* = 0.83, *Kiq* = 1.2, *Kid* = 5, *Tp* = 0.05, *Tiq* = 0.005, *TQ* = 0.025, and *Tid* = 0.0025. Figure 10 shows active and reactive power responses under PI (Proportional Integral) controller [19] and the proposed method, when wind speed is 7 m/s and capacitance compensation is 60%. Growing

oscillations are observed under PI control, while the effect for SSCI mitigation is good under the proposed method.

**Figure 7.** Transient responses under wind speed of 11 m/s and series-compensated level of 85%.

**Figure 8.** Dynamic responses of active power, reactive power, and electromagnetic torque under three-phase short circuit fault.

**Figure 9.** Control structure of PI scheme.

**Figure 10.** Active and reactive responses under PI (blue) and the proposed method (red), with wind speed of 7 m/s and series-compensated level of 60%.

To compare the control performance under the proposed method and partial feedback linearization method [27], the implementation block diagram of the damping controller based on the partial feedback linearization method is shown in Figure 11. The relative control laws are represented as:

$$\begin{cases} S\_q = \frac{L\_{rf}}{\underline{u}\_{dc}} \left( v\_1 + \omega\_1 i\_{rd} + \frac{R\_{rf}}{\underline{L}\_{rf}} i\_{rq} + \frac{v\_{rq}}{\underline{L}\_{rf}} \right) \\\ S\_d = \frac{L\_{rf}}{\underline{u}\_{dc}} \left( v\_2 - \omega\_1 i\_{rq} + \frac{R\_{rf}}{\underline{L}\_{rf}} i\_{rd} + \frac{v\_{rd}}{\underline{L}\_{rf}} \right) \end{cases} \tag{46}$$

$$\begin{cases} \upsilon\_1 = k\_{1p} \left( i\_{rq\\_ref} - i\_{rq} \right) + k\_{1i} \int\_0^t (i\_{rq\\_ref} - i\_{rq}) dt \\ \upsilon\_2 = k\_{2p} \left( i\_{rd\\_ref} - i\_{rd} \right) + k\_{2i} \int\_0^t (i\_{rd\\_ref} - i\_{rd}) dt \end{cases} \tag{47}$$

**Figure 11.** Implementation block diagram of the damping controller using partial feedback linearization method.

Figure 12 shows active power responses at 6 m/s wind speed and 60% compensation, and Figure 13 show the responses when parameter perturbation is considered. The variation ranges of *Lm*, *Ls*, *LRSC*, and *RRSC* are ±50% of the nominal values with combination of sine and cosine functions. The curves barely changed under the proposed method while it seems to be greatly affected under the partial feedback linearization method. This verified robustness to parameter perturbation of the proposed method.

**Figure 12.** Active responses under the proposed method (blue) and partial feedback linearity (red) with wind speed of 6 m/s and compensation of 60%.

**Figure 13.** Active responses under the proposed method (blue) and partial feedback linearity (red) at 6 m/s wind speed and 60% compensation with parameter perturbation.

Conventional first-order sliding mode control (SMC) damping scheme [9] is shown as Figure 14. The control laws are:

$$\begin{array}{c} S\_{rq} = \frac{2}{L\_{nl}l\_{\rm c}}(-i\_{rd}\omega\_{1}(L\_{m}^{2}-L\_{r7}L\_{s})-L\_{m}(i\_{sq}\mathbf{R}\_{s}-\mathbf{u}\_{sq}+L\_{s}i\_{sd}\omega\_{r}) + L\_{s}i\_{l\eta}\mathbf{R}\_{r\ell}-L\_{r7}L\_{s}i\_{rd}\omega\_{r} \\ \qquad - i\_{rq}(L\_{m}^{2}-L\_{r7}L\_{s})-\rho\_{i\_{rq}}\text{sign}(v\_{i\_{rq}})(L\_{r7}L\_{s}-L\_{m}^{2}) \\\\ S\_{rd} = \frac{2}{L\_{nl}l\_{\rm c}}(-i\_{rq}\omega\_{1}(L\_{m}^{2}-L\_{r7}L\_{s})+L\_{m}(-i\_{sd}\mathbf{R}\_{s}+\mathbf{u}\_{sd}+L\_{s}i\_{sq}\omega\_{r}) + L\_{s}i\_{rd}\mathbf{R}\_{r\ell}+L\_{r7}L\_{s}i\_{rq}\omega\_{r} \\ \qquad - i\_{rd}(L\_{m}^{2}-L\_{r7}L\_{s})-\rho\_{i\_{rd}}\text{sign}(v\_{i\_{rd}})(L\_{r7}L\_{s}-L\_{m}^{2}) \end{array} \tag{48}$$

where *ρirq* = 8.5 × 10<sup>5</sup> and *ρird* = 2.3 × 106.

**Figure 14.** Conventional first-order SMC scheme.

The transient responses of active power, electromagnetic torque, and DC link voltage under both the proposed method and first-order sliding mode method [9] are shown as Figure 15 when wind speed is 10 m/s and compensation is 60%. The oscillation processes both quickly disappeared after about 0.5 s, and SSCI mitigation was achieved. However, under the proposed method, the control chattering of RSC control input was greatly restrained, and the upper bounds of uncertainty derivatives were not needed. Figure 16 shows the regulating process of adaptive gains of super-twisting control.

For evaluating the control performance, two indices have been defined as follows:

$$RMS\_{c\_i} = \sqrt{\frac{1}{n\_s} \sum\_{k=1}^{n\_s} c\_{i'}^2} \; RMS\_{u\_i} = \sqrt{\frac{1}{n\_s} \sum\_{k=1}^{n\_s} u\_i^2} \tag{50}$$

where *ns*, *ei*, and *ui* are the number of samples, root mean square (RMS) of tracking errors, and control quantities, respectively.

When the studied system works with series-compensated level of 60% and wind speed of 10 m/s, the RMS for tracking errors and control quantities are shown in Table 2, which exhibit superiority of the proposed method.

**Figure 15.** The transient responses of active power, electromagnetic torque, and DC link voltage under both the proposed method and first-order sliding mode method.

**Figure 16.** Adaptive gains.

**Table 2.** Root mean square (RMS) for tracking errors and control quantities.

