**1. Introduction**

In order to cope with energy shortage and environmental pollution, countries all over the world are intensively promoting renewable energy development [1]. Wind energy is considered as one of the most promising types of renewable energy. In wind power generation systems, doubly fed induction generators (DFIG) play a leading role due to their distinct advantages [2]. However, DFIG-based wind farms are always far away from the load center and need long-distance transmission, which can weaken power capacity and stability margin [3]. Series-compensated capacitors are generally applied in the DFIG-based wind farms transmission line to enhance the capacity and stability [4].

Series-compensated capacitors method can induce subsynchronous control interaction (SSCI), due to the interaction between DFIG's converter control and series-compensated transmission line [5,6]. In the SSCI, the frequency and attenuation rate are mainly codetermined by parameters of wind turbines and power transmission systems, irrelevant to natural modal frequency of shafting [7]. With no mechanical part involved, SSCI has a small damping effect, and also its divergence speed is faster than that of conventional subsynchronous resonance [8]. Thus, SSCI can cause more severe damage. From public reports, related accidents have been observed in America and China [5,9], causing equipment damage and loss of power generation.

In recent years, many efforts have been made on SSCI issues, e.g., frequency scan, eigenvalue analysis, complex torque coefficient method, and time domain simulation [10]. Based

**Citation:** Ma, R.; Han, Y.; Pan, W. Variable-Gain Super-Twisting Sliding Mode Damping Control of Series-Compensated DFIG-Based Wind Power System for SSCI Mitigation. *Energies* **2021**, *14*, 382. https://doi.org/10.3390/en14020382

Received: 3 December 2020 Accepted: 8 January 2021 Published: 12 January 2021

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on these SSCI analysis methods, scholars tried to study SSCI damping strategies [11–17]. Papers [5,18–21] carried out many pioneering studies on modeling and supplementary damping control for subsynchronous resonance analysis. Paper [22] discussed multi-input multioutput supplementary damping control for both the rotor-side converter (RSC) and grid-side converter (GSC). In order to reduce the influence of PI control parameters on SSCI, paper [23] presented an optimization algorithm of PI parameters based on the t-distributed stochastic neighbor embedding for enhancing the damping. Paper [24] proposed a SSCI damping control for both the two control channels in the inner current loop of RSC with particle swarm optimized control coefficients. In short, the above studies [18–21,23,24] are all based on conventional double closed-loop PI control, and the design processes for SSCI damping controllers are relatively simple. Yet, these linear control methods can be inoperative when system operating points are changed, since a series-compensated DFIG-based wind power system is a complex and highly nonlinear system, with strong coupling features in both the aerodynamic and electrical parts [25,26]. These nonlinear factors can be dealt with by feedback linearization control. Paper [27] adopted a partial feedback linearization method to design damping controllers for GSC. Considering that RSC control is actually the dominant factor for SSCI mitigation, paper [28] continued the study of [27] to design SSCI damping controllers for RSC, achieving a good damping effect. Paper [29] proposed nonlinear controllers for both GSC and RSC based on the state feedback linearization method and verified its superior performance compared with conventional PI control.

Although feedback linearization control is an effective method for solving nonlinear problems in SSCI mitigation, it is rather sensitive to uncertainties in series-compensated DFIG-based wind power systems. These uncertainties exist in generator parameters, transmission line parameters, series compensation level, wind speed, and multiple series capacitor compensated lines, which can deteriorate subsynchronous oscillation of the system [30]. Hence, robustness is the desired characteristic for the series-compensated DFIG-based wind power control system. There are several attempts in SSCI robust control, such as *H*∞ and active disturbance rejection methods [30–32].

The widely adopted sliding mode control [33,34], which possesses invariance property for system disturbances and parameter perturbation, is another good choice for robust control of SSCI. Papers [35,36] discussed SSCI mitigation strategies by combining feedback linearization control with sliding mode method. Paper [9] proposed first-order sliding mode controllers to track the reference rotor currents for damping SSCI. Control chattering of rotor voltage, which can damage electronic components and increase SSCI, is a big obstacle for these conventional sliding mode methods. Furthermore, the upper bounds of system uncertainties derivatives, which are actually hard to calculate beforehand, have to be known in advance for all the above robust control methods.

Consequently, this paper proposes a novel variable-gain super-twisting damping control strategy for SSCI mitigation. It can greatly reduce sliding mode chattering and does not need the unknown upper bounds of uncertainty derivatives. The SSCI mechanism was firstly analyzed with the aid of the presented series-compensated DFIG-based wind farm model. Rotor current dynamics constraint was identified as the dominant factor for SSCI mitigation. Super-twisting control laws were then constructed to track the prescribed rotor currents under *dq* direction. Adaptive control laws were subsequently conceived via barrier function. Then, the control gains can be self-adjusted following the upper bounds of uncertainty derivatives. SSCI mitigation was achieved without conservative RSC control signals. The performance of the newly designed variable-gain super-twisting sliding mode (VGSTSM) damping control scheme was evaluated under different wind speed, series compensation level, and short circuit fault. Comparative studies with conventional PI method, feedback linearization control, and first-order sliding mode control were also completed to verify the superiority.

This paper is organized as follows. Modeling for series-compensated DFIG-based wind farm is stated in Section 2. Section 3 details SSCI mechanism, design procedure of the proposed control strategy, and stability proof of the closed-loop power system. The demonstration of effectiveness and superiority of the proposed VGSTSM damping control strategy for SSCI mitigation is shown in Section 4. Some conclusions are finally drawn in Section 5.

#### **2. Series-Compensated DFIG-Based Wind Power System Modeling**

The model of a series-compensated DFIG-based wind farm is shown in Figure 1 [18,36]. It mainly includes wind turbine, shafting, induction generator, RSC, DC bus link, GSC, and series-compensated transmission line. DFIG represents the 100 MW equivalent lumped model of 50 generators (2 MW for each unit). *Rs* and *Ls* are stator resistance and inductance, respectively. *RRSC* and *LRSC* are RSC link resistance and inductance, respectively. *RGSC* and *LGSC* are GSC link resistance and inductance, respectively. *RL* and *LL* are equivalent resistance and inductance of series-compensated transmission line, respectively. *Cdc* is DC bus capacitor, and *CSC* is the series-compensated capacitor. *e* is the grid voltage. System equations are all analyzed under the synchronous rotating reference frame.

Power fluctuation and subsynchronous rotor current will be induced when subsynchronous disturbance current occurs in the series-compensated transmission line. The affected RSC generates corresponding output voltage, and then injects subsynchronous current into the rotor, and finally induces the superposition of the stator side and the original disturbance. It will increase the original disturbance and form a divergent subsynchronous oscillation if the amplitude of the superimposed current is larger than that of the original disturbance current.

Electrical dynamic of series-compensated DFIG-based wind farms can be deduced via Kirchhoff's laws under a synchronous rotating reference frame.

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩*dird dt* = *<sup>ω</sup>*1*irq* − *Rr Lr ird* − *urd Lr* + *udc Lr Sd dirq dt* = −*ω*1*ird* − *Rr Lr irq* − *urq Lr* + *udc Lr Sq dudc dt* = 1*Cdc idc* − 1*Cdc irdSd* − 1*Cdc irqSq disd dt* = *<sup>ω</sup>*1*isq* + *Rs Ls isd* + *usd Ls disq dt* = −*ω*1*isd* + *Rs Ls isq* + *usq Ls digd dt* = −*ω*1*igq* − *RGSC LGSC igd* − *ugd LGSC digq dt* = *<sup>ω</sup>*1*igd* − *RGSC LGSC igq* − *ugq LGSC diLd dt* = *<sup>ω</sup>*1*iLq* + *RL LL iLd* + 1*LL* (*uld* − *uscd* − *Ed*) *diLq dt* = −*ω*1*iLd* + *RL LL iLq* + 1*LL ulq* − *uscq* − *Eq duscd dt* = *<sup>ω</sup>*1*uscq* + 1*CSC iLd duscq dt* = −*ω*1*uscd* + 1*CSC iLq* (1)

where *ug*, *ig*, *us*, *is*, *ur*, *ir*, *udc*, and *idc* are the voltages and currents of GRC, stator, rotor, and DC bus capacitor, respectively. *uSC* is defined as series-compensated voltage. *Rr* = *Rr* + *RRSC*, *Rs* = *Rs* + *RL*, *Ls* = *Ls* + *LL* − 1/*ω*21*CSC*, and *Lr* = *Lr* + *LRSC* − *<sup>L</sup>*2*m*/*<sup>L</sup><sup>s</sup>*.

**Figure 1.** Structure of series-compensated doubly fed induction generator (DFIG)-based wind farm connected to grid.

After the applied stator field-orient, the stator active and reactive power for the DFIG model described in the *dq* reference frame can be represented as:

$$\begin{cases} \begin{array}{c} P\_{\text{s}} = -\frac{3L\_{\text{m}}}{2L\_{\text{s}}^{\*}} \mathbf{L} I\_{\text{s}} i\_{rq} \\ Q\_{\text{s}} = \frac{3L\_{\text{s}}^{\*}}{2L\_{\text{s}}^{\*} \omega\_{1}} (\mathbf{L}\_{\text{s}} - \omega\_{1} L\_{\text{m}} i\_{rd}) \end{array} \tag{2}$$

According to Betz theory, mechanical power captured by wind turbine is denoted as:

$$P\_T = \frac{1}{2} \mathcal{C}\_p S\_w \rho\_T v\_T^3 \tag{3}$$

where *Cp* is power coefficient, *Sw* is the blade sweep area, *ρT* is air density, and *υT* is wind speed. As is shown in Equation (3), the mechanical power, *PT*, is determined by power coefficient, *Cp*, under the fixed wind speed. *Cp* is related to tip speed ratio, *λT*, and blade pitch angle, *β<sup>T</sup>*, with the typical functional relation [27,36]:

$$\mathcal{C}\_p = 0.5176 \left( \frac{116}{\lambda\_i} - 0.4\beta\_T - 5 \right)^{\frac{-21}{\lambda\_i}} + 0.0068\lambda\_T \tag{4}$$

Two related equations are 1*λi* = 1 *λT*+0.08*βT* − 0.035 *β*<sup>3</sup>*T*+<sup>1</sup> and *λT* = *<sup>ω</sup>TRT υT* , where *ωT* is mechanical angular speed and *RT* is the rotor radius of the wind turbine. With the change of *λT* and fixed *β<sup>T</sup>*, power coefficient, *Cp*, has a maximum value, *Cp*max, and the corresponding *λT* is optimum tip speed ratio, *<sup>λ</sup>Topt*. In other words, for a specific wind speed, the wind turbine can only run under specific mechanical angular speed,*<sup>ω</sup>T*, to achieve maximum power point tracking (MPPT). Thus, generator rotor speed must be regulated timely with the change of wind speed to capture maximum power.

The mechanical drive system of the wind turbine transmits the captured kinetic energy to the generator via the gear box, high speed shaft, and low speed shaft. It is rather complicated, and the mechanical shaft dynamic can be modeled as one mass, two mass, and three mass, according to different modeling methods [18]. The two mass model is sufficient and widely praised in SSCI studies, and its dynamic is represented as:

$$\begin{cases} \frac{d\boldsymbol{\omega}\_{T}}{dt} = \frac{1}{2H\_{T}} (T\_{T} - K\_{s}\boldsymbol{\theta}\_{s})\\ \frac{d\boldsymbol{\omega}\_{T}}{dt} = \frac{1}{2H\_{G}} (K\_{s}\boldsymbol{\theta}\_{s} - T\_{c})\\ \frac{d\boldsymbol{\theta}\_{s}}{dt} = 2\pi f\_{1} \left(\boldsymbol{\omega}\_{T} - \frac{\boldsymbol{\omega}\_{r}}{N\_{\mathcal{S}}}\right) \end{cases} \tag{5}$$

where *HT* and *HG* are inertia time constants of wind turbine and generator, respectively; *Ks* is stiffness coefficient of shafting; *θs* is relative angular displacement of the two mass block; *TT* and *Te* denote mechanical torque and electromagnetic torque of the wind turbine

and generator, respectively; *ωr* is rotor angular speed of the generator; *Ng* represents gear ratio; and *f*1 is power frequency.

#### **3. SSCI Analysis and Control Design**
