Super-Twisting Sliding Mode Control for a DSIG-Based Dual-Rotor Wind Turbine System

Driven by the growing demand for clean and sustainable electricity, wind energy has experienced remarkable global expansion outpacing other forms of renewable energy. Power capacities have swiftly evolved from a few kilowatts to several hundred megawatts within a short period [1]. Currently, conventional single-rotor wind turbines (WTs) with three blades and a horizontal axis dominate the global WT market. These turbines achieve an energy efficiency of approximately 40%, constrained by Betz's limit, which has a theoretical maximum of 59%; nevertheless, this limitation can be effectively addressed [2]. Recent research has focused on designing more advanced WTs with improved power characteristics [3]. One promising innovation is the dual-rotor wind turbine (DRWT), which, still under development, has the potential to surpass Betz's limit (0.59) in terms of energy efficiency. This work specifically explores the use of this advanced turbine technology [3, 4]. Given the continuous fluctuations in wind speed, variable-speed WT systems are well suited to maintaining efficient energy extraction over a wide operating range [3, 4]. Numerous designs employing diverse generator types exist in the wind energy sector. Among these, the doubly-fed induction generator (DFIG) is the most prevalently implemented [5], especially for high-capacity applications, due to several benefits, including a 30% reduction in the costs associated with the machine-side inverter. In recent times, alternative generator technologies have significantly evolved and now rival the DFIG. These include permanent magnet synchronous generators (PMSG) and other specialized machines, such as multiphase systems and doubly-excited machines, among others [6]. Owing to their remarkable benefits over standard three-phase generators, such as dividing power delivery across phases, lowering per-phase current without increasing voltage, enhanced dependability, reduced rotor current harmonics, smaller torque ripple amplitudes, higher pulsation frequencies, and reduced direct current distortions, multiphase machines have attracted significant interest for integration into WT systems. Of these, the dual-star induction generator (DSIG) has emerged as the most frequently adopted in such implementations [7]. This study emphasizes a WTS powered by a DSIG. This generator features dual stator windings displaced electrically by a 30-degree angle and linked to the electrical network via a bidirectional converter. Such an arrangement facilitates optimal power harnessing from the wind while dynamically regulating voltage magnitude and frequency. The most commonly utilized control strategy in wind power applications is vector control (VC) augmented with proportional-integral (PI) controllers. However, despite its extensive deployment in industrial contexts, this methodology exhibits significant robustness challenges owing to PI controller limitations [8]. Several advanced solutions have been suggested recently to overcome this issue. Among these, SMC, introduced by Utkin in 1977, has seen substantial adoption. SMC has showcased impressive reliability and adaptability in managing rotating machinery. Nevertheless, its applicability has been constrained by the adverse effects of the "chattering" phenomenon [9]. To address this drawback caused by the discontinuity in the control mechanism, researchers have proposed various approaches, such as smoothing the sign function approximation, employing fuzzy systems, leveraging neural networks, and utilizing second-order sliding mode strategies [10]. More recently, groundbreaking methods have emerged; these developments demonstrate promising progress, including adaptive terminal and exponential variable discrete-time SMC laws, fractional-order SMC, and adaptive higher-order sliding modes. Additionally, hybrid sliding mode techniques have gained traction in modern studies, including combinations like backstepping sliding mode, passivity-oriented sliding mode strategies, H∞ sliding mode approaches, and predictive sliding mode frameworks [11]. Among the advanced sliding mode control techniques, the super-twisting continuous sliding mode controller (STCSMC) has emerged as an effective solution for significantly reducing the chattering effect. Initially introduced by Levant [12], this method has demonstrated excellent performance, particularly in the management of rotating electrical machines. Consequently, it has been adopted for the purposes of this study.

The primary contributions of this research are outlined as follows:

• Design of a resilient and high-performance control framework for a DSIG-driven WT system utilizing STCSMC.
• Comparative analysis of the performance of PI, STSMC, and the proposed STCSMC applied to the DSIG-based DRWT under variable wind speed conditions and robustness tests involving a 50% increase in the system inertia.

The remainder of this paper is organized as follows. Section II presents the mathematical model of the DSIG-based wind energy conversion system. Section III provides an in-depth explanation of the VC method applied to the DSIG. Section IV describes the control approach implemented for the Grid-Side Converter (GSC). Section V introduces an advanced control strategy for the DSIG WT system utilizing Sliding Mode Control with Super Twisting (STCSMC). Section VI discusses the simulation results and their analysis. Finally, Section VII concludes the study with key findings.

The fundamental purpose of the VC method is to emulate the decoupled torque and flux control characteristics of a DC machine in the DSIG. This is achieved by decoupling the flux and torque control. A decoupled control structure is obtained by orienting the rotor flux along the direct axis. Under this condition, the direct-axis flux component is regulated to its reference value, whereas the quadrature-axis component is maintained at zero.

${{\phi }_{dr}}=\phi _{r}^{*}\text{  and  }{{\phi }_{qr}}=0$                       (13)

Substituting Eqs. (13) into (9), we can express:

${{i}_{dr}}=0$                               (14)

${{i}_{qr}}=-\frac{\omega _{sl}^{*}\phi _{r}^{*}}{{{R}_{r}}}$                   (15)

where, ${{\omega }^{*}}_{sl}={{\omega }^{*}}_{s}-{{\omega }_{r}}$, with ${{\omega }^{*}}_{sl}$ denoting the reference slip speed. By incorporating Eqs. (13) into (10), the rotor current components can be written as follows:

${{i}_{qr}}=-\frac{{{L}_{m}}}{({{L}_{m}}+{{L}_{r}})}\left( {{i}_{qs1}}+{{i}_{qs2}} \right)$                            (16)

${{i}_{qr}}=-\frac{{{L}_{m}}}{({{L}_{m}}+{{L}_{r}})}\left( {{i}_{qs1}}+{{i}_{qs2}} \right)$                       (17)

By substituting Eqs. (17) into (15), the following expression is obtained:

$\omega _{sl}^{*}=\frac{{{R}_{r}}{{L}_{m}}}{({{L}_{m}}+{{L}_{r}})}\frac{\left( {{i}_{qs1}}+{{i}_{qs2}} \right)}{\phi _{r}^{*}}$                           (18)

By incorporating Eqs. (13) into (11), the electromagnetic torque expression is reformulated as:

$T_{em}^{*}=p\frac{{{L}_{m}}}{{{L}_{m}}+{{L}_{r}}}\left( \left( {{i}_{qs1}}+{{i}_{qs2}} \right)\phi _{r}^{*} \right)$                      (19)

The application of rotor field orientation enables the derivation of the system's electrical and mechanical dynamic model, expressed by the following equations:

$\left\{ \begin{align}  & \frac{d{{i}_{ds1}}}{dt}=\frac{1}{{{L}_{1}}}\left( U_{ds1}^{*}-{{R}_{s1}}{{i}_{ds1}}+\omega _{s}^{*}\left( {{L}_{1}}{{i}_{qs1}}+{{T}_{r}}\phi _{r}^{*}\omega _{sl}^{*} \right) \right) \\ & \frac{d{{i}_{qs1}}}{dt}=\frac{1}{{{L}_{1}}}\left( U_{qs1}^{*}-{{R}_{s1}}{{i}_{qs1}}+\omega _{s}^{*}\left( {{L}_{1}}{{i}_{ds1}}+\phi _{r}^{*} \right) \right) \\ & \frac{d{{i}_{ds2}}}{dt}=\frac{1}{{{L}_{2}}}\left( U_{ds2}^{*}-{{R}_{s2}}{{i}_{ds2}}+\omega _{s}^{*}\left( {{L}_{2}}{{i}_{qs2}}+{{T}_{r}}\phi _{r}^{*}\omega _{sl}^{*} \right) \right) \\ & \frac{d{{i}_{qs2}}}{dt}=\frac{1}{{{L}_{2}}}\left( U_{qs2}^{*}-{{R}_{s2}}{{i}_{qs2}}-\omega _{s}^{*}\left( {{L}_{2}}{{i}_{ds2}}+\phi _{r}^{*} \right) \right) \\ & \frac{d{{\phi }_{r}}}{dt}=-\frac{{{R}_{r}}}{\left( {{L}_{r}}+{{L}_{m}} \right)}{{\phi }_{r}}+\frac{{{R}_{r}}{{L}_{m}}}{\left( {{L}_{r}}+{{L}_{m}} \right)}\left( {{i}_{ds1}}+{{i}_{ds2}} \right) \\ & \frac{d{{\Omega }_{g}}}{dt}=\frac{1}{J}\left( {{T}_{g}}-{{f}_{r}}{{\Omega }_{g}}-p\frac{{{L}_{m}}}{\left( {{L}_{r}}+{{L}_{m}} \right)}\left( {{i}_{ds1}}+{{i}_{ds2}} \right)\phi _{r}^{*} \right) \\ \end{align} \right.$                  (20)

where, T_r = L_r/R_r.

5.1 Principle of Super-twisting continuous sliding mode controller

As highlighted in Section 1, extensive research over recent years has focused on overcoming the Chattering problem associated with traditional SMC. One of the principal drawbacks of sliding mode control in electrical drive applications is the chattering phenomenon, which produces high-frequency switching, increases power losses, and generates electromagnetic interference [16, 17]. To address this issue, high-order sliding mode control (HOSMC) has emerged as an advanced technique. Unlike traditional SMC, which acts on the first derivative of the system's deviation from the desired state, HOSMC operates on higher-order derivatives. This refinement has gained significant attention for its potential to reduce chattering. Numerous studies have validated the efficacy of HOSMC, yielding promising theoretical and practical results [18, 19]. Despite these advances, certain challenges persist in implementing HOSMC. One key issue lies in the requirement for extensive information about the sliding surface. Specifically, synthesizing an n-order controller demands knowledge of n derivatives of the sliding surface. [18, 19] To overcome this limitation, the super-twisting algorithm (STA), combined with quasi-continuous functions, is employed in this study. This algorithm offers a robust solution by mitigating the dependency on higher-order derivative information while effectively reducing chattering. The proposed control law for this algorithm, as applied in this work, is described by Eq. (30) [12].

$u\left( t \right)=-{{l}_{1}}{{\left| S \right|}^{{{a}_{1}}}}\text{sgn(}S\text{)}\cdots \text{-}-{{l}_{n}}{{\left| {{S}^{(n-1)}} \right|}^{{{a}_{n}}}}\text{sgn(}{{S}^{(n-1)}}\text{)-}z$                     (30)

With z defined by Eq. (31) and (a₁, a₂,..., a_n) as constants satisfying the condition in Eq. (32). Furthermore, (l₁, l₂,..., l_n) represent the parameters of the Hurwitz polynomial: sⁿ + l_ns^n-1 + ...+ l₂s+l₁.

$\left\{ \begin{align}  & z\left( t \right)=k\cdot {{\left| S \right|}^{1/2}}\text{sgn(}S\text{)}+{{z}_{1}}\left( t \right) \\ & {{{\dot{z}}}_{1}}\left( t \right)=\alpha \cdot \text{sgn(}S\text{)} \\ \end{align} \right.$                       (31)

$\left\{ \begin{align}  & {{a}_{i-1}}=\frac{{{a}_{i}}{{a}_{i-1}}}{2{{a}_{i+1}}-{{a}_{i}}},\text{ }i=2,...,n \\ & {{a}_{n}}+1=1\text{   and   }{{a}_{n}}=a \\ \end{align} \right.$                           (32)

5.2 Application to the dual-star induction generator-based dual-rotor wind turbine

Figure 3 depicts the application of the STCSMC to the DSIG-based DRWT system. In this configuration, the proposed control architecture departs from the conventional PI-based design by employing STCSM controllers for both the mechanical speed and current regulation of the DSIG. These controllers are specifically designed to generate the reference values for the q-axis current and voltages, as defined in the corresponding Eqs. (33)-(37).

$\begin{align}  & {{i}_{q}}^{*}=-{{l}_{1}}{{\left| {{S}_{\omega }} \right|}^{{{a}_{1}}}}\text{sign}\left( {{S}_{\omega }} \right)- \\ & {{k}_{1}}\cdot \text{sign}{{\left| {{S}_{\omega }} \right|}^{1/2}}+\int{{{\alpha }_{1}}\cdot \text{sign}\left( {{S}_{\omega }} \right)} \\ \end{align}$                         (33)

With: i_q^* = i_qs1^* + i_qs2^*.

$\begin{align}  & {{U}_{ds1}}^{*}=-{{l}_{2}}{{\left| {{S}_{ids1}} \right|}^{{{a}_{1}}}}\text{sign}\left( {{S}_{ids1}} \right) \\ & -{{k}_{2}}\cdot \text{sign}{{\left| {{S}_{ids1}} \right|}^{1/2}}+\int{{{\alpha }_{2}}\cdot \text{sign}\left( {{S}_{ids1}} \right)} \\ \end{align}$                (34)

$\begin{align}  & {{U}_{qs1}}^{*}=-{{l}_{2}}{{\left| {{S}_{iqs1}} \right|}^{{{a}_{1}}}}\text{sign}\left( {{S}_{iqs1}} \right)- \\ & {{k}_{2}}\cdot \text{sign}{{\left| {{S}_{iqs1}} \right|}^{1/2}}+\int{{{\alpha }_{2}}\cdot \text{sign}\left( {{S}_{iqs1}} \right)} \\ \end{align}$                     (35)

$\begin{align}  & {{U}_{ds2}}^{*}=-{{l}_{2}}{{\left| {{S}_{ids2}} \right|}^{{{a}_{1}}}}\text{sign}\left( {{S}_{ids2}} \right) \\ & -{{k}_{2}}\cdot \text{sign}{{\left| {{S}_{ids2}} \right|}^{1/2}}+\int{{{\alpha }_{2}}\cdot \text{sign}\left( {{S}_{ids2}} \right)} \\ \end{align}$                  (36)

$\begin{align}  & {{U}_{qs2}}^{*}=-{{l}_{2}}{{\left| {{S}_{iqs2}} \right|}^{{{a}_{1}}}}\text{sign}\left( {{S}_{iqs2}} \right) \\ & -{{k}_{2}}\cdot \text{sign}{{\left| {{S}_{iqs2}} \right|}^{1/2}}+\int{{{\alpha }_{2}}\cdot \text{sign}\left( {{S}_{iqs2}} \right)} \\ \end{align}$                        (37)

where, S_ω, S_ids1, S_iqs1, S_ids2, S_iqs2 are the sliding mode surfaces corresponding to the mechanical speed and stator currents, respectively. The constants k₁ and k₂ must be properly selected to satisfy the stability criteria.

To ensure a consistent comparison of the investigated control strategies, the GSC control structure is maintained unchanged for all simulation cases. The PI regulators used in the GSC are dedicated to DC-link voltage and grid-side power regulation and are therefore not included in the controller comparison. The machine-side DSIG control is instead evaluated using three different strategies: conventional PI, STSMC, and the proposed STCSMC. Hence, the control cases differ only in the machine-side regulation law, while the GSC, system parameters, operating conditions, and reference signals remain identical. This configuration provides a consistent basis for assessing the contribution of the proposed STCSMC in terms of tracking accuracy, dynamic response, robustness, and chattering reduction.

5.3 Stability analysis and gain selection

Consider the following dynamic system described in state-space form:

$\frac{dx}{dt}=a(x,t)+b(x,t)u,\text{    }y=c\text{(}x\text{,}t\text{)}$                     (38)

where, x, u and y represent the system’s state, input, and output, respectively.

The objective of the proposed controller is to determine a control input that drives the tracking error to zero within finite time. The sliding variable is defined as S = y-y^∗, where y∗ denotes the reference signal. The STA does not require the explicit computation or numerical differentiation of the sliding variable. Instead, it uses the sliding variable directly, while its integral term generates the required corrective action. For the proposed current-control loops, the relative-degree-one assumption is satisfied because the control voltages directly affect the current dynamics, while the lumped perturbations remain bounded over the considered operating range. Under these conditions, the STA ensures finite-time convergence of the sliding variable without explicit numerical differentiation. Consequently, the controller can be implemented directly from the measured current errors, thereby avoiding numerical differentiation and reducing sensitivity to measurement noise. According to the finite-time convergence theory established by Levant [12], the STA guarantees finite-time convergence of the sliding variable and its first derivative under the stated assumptions. Therefore, the necessary condition for stability and convergence can be expressed as follows:

${{k}_{1}}>\frac{{{A}_{M}}}{{{B}_{m}}},\text{  }{{k}_{2}}\ge \frac{4{{A}_{M}}}{{{B}_{m}}^{2}}\cdot \frac{{{B}_{M}}({{K}_{1}}+{{A}_{M}})}{{{B}_{m}}({{K}_{1}}-{{A}_{M}})}$                           (39)

where, A_M denotes the upper bound of $\mid A\mid $, while B_M and B_m represent the upper and lower bounds of B, respectively, in the second-order dynamics of the output y.

This paper presented an advanced vector-control strategy for a DRWT based on a DSIG, using a STCSMC. The proposed strategy was developed to improve rotor-speed tracking, dynamic performance, chattering attenuation, and robustness against system parameter variations. The performance of the proposed controller was evaluated through MATLAB/Simulink simulations and compared with conventional PI and STSMC strategies under identical operating conditions.

The quantitative results confirm the effectiveness of the proposed STCSMC. The rise time was reduced to 0.049 s, compared with 0.151 s for the PI controller, corresponding to a reduction of approximately 67.5%. The settling time was also reduced from 2.61 s to 1.5 s, which meant an improvement of around 42.5%. Furthermore, the speed overshoot was reduced from 25% for PI and 50% for STSMC to approximately 1% with the proposed STCSMC, corresponding to reductions of 96% and 98%, respectively. The RMS tracking error was reduced from 3.51% for PI and 1.125% for STSMC to only 0.4% with STCSMC, representing reductions of approximately 88.6% and 64.4%, respectively. According to the findings, the steady-state tracking error was practically eliminated with the proposed controller.

The harmonic analysis also demonstrated a modest reduction in the THD, from 74.29% with STSMC to 73.96% with STCSMC, which is equivalent to a fall of 0.33 percentage points or approximately 0.44% in relative terms. Moreover, the proposed controller effectively reduces chattering and provides smoother electromagnetic torque and stator-flux responses. Even with an increased inertia by 50%, the STCSMC-based wind energy system maintains stable and accurate speed tracking, whereas the PI controller shows a more pronounced degradation in dynamic performance.

Overall, the obtained outcome demonstrates that the proposed STCSMC provides a faster, more accurate, and more robust control solution for the DSIG-based DRWT. The combination of reduced transient error, lower overshoot, negligible steady-state error, reduced chattering, and improved robustness makes the proposed strategy a promising candidate for high-performance wind-energy conversion systems. Future work will focus on experimental validation of the proposed controller and its implementation on a real-time control platform under a wider range of operating conditions.