Sliding Mode Controller and Fuzzy PID for Doubly Fed Induction Generator

Due to the increasing economic and environmental costs of fossil fuels, along with the global push to replace them, the energy landscape has undergone significant changes in recent decades. Renewable energy sources have emerged as a key solution to these challenges, helping to meet growing energy demands while also mitigating the looming energy crisis [1]. Among renewable technologies, wind energy has become a leading option due to its abundance, environmental benefits, and rapid global expansion [2, 3].

At the same time, a significant portion of electrical energy is transformed into mechanical energy using motors. Among these, doubly fed induction generator (DFIG) is especially popular in industrial applications due to its suitability for variable-speed operations, namely in electric vehicles, wind turbine systems, and marine propulsion. The doubly-fed induction machine (DFIM) offers several advantages, including the following [4-7]: The DFIG can be supplied and controlled through the stator, rotor, or a combination of both; it is capable of operating over a wide range of speed variations around the synchronous speed (up to ± 30%), enabling maximum power generation at varying wind speeds in turbine systems; it permits decoupled regulation of reactive and active power; and it enables independent control of flux, torque, and power factor.

In the field of control systems, the challenges associated with nonlinear and uncertain processes remain a key priority. Classical methods, such as PID control, have been widely used because of their efficiency and simplicity, especially for linear systems. However, when dealing with highly nonlinear, time-varying, or uncertain systems, such as those found in magnetohydrodynamic flows [8], more advanced control strategies are required. Lotfi et al. [9, 10] optimized the PID controller using metaheuristic algorithms applied to the speed control of a WECS and DFIM. Larabi et al. [11] applied an LQG-based LMI technique to a doubly-fed induction generator (BDFIG). Arabi et al. [12] proposed a nonlinear PI regulator enhanced by meta-heuristic methods for wind turbine speed regulation.

Two such strategies are sliding mode control (SMC) and the fuzzy PID controller.

SMC is a widely used control method due to its efficiency and robustness. The SMC control law is designed using the system model and consists of two distinct components. The first is the equivalent control, which forces the system dynamics to reach the selected sliding surface, ensuring convergence. The second is called the switching term; it keeps the system states on the sliding surface once it is reached. This control scheme has been applied in various studies to manage a wide range of systems, including robotic systems [13], magnetic levitation (MAGLEV) systems [14], and permanent magnet synchronous motors (PMSMs) [15].

In recent years, numerous studies have proposed and implemented advanced control strategies for DFIGs. Karakasis et al. [16] developed a comprehensive control scheme based on maximum power point tracking (MPPT) and loss minimization (LM) for wind systems equipped with DFIGs, aiming to maximize electrical energy production. Bouderbala et al. [17] introduced both direct and indirect field-oriented control (FOC) methods to enhance the reactive and active power outputs of DFIGs connected to variable-speed wind turbines. Jeon et al. [18] proposed a proportional–integral (PI) controller for frequency smoothing in DFIGs. Sadeghi et al. [19] applied a super-twisting sliding mode control technique combined with direct power control (DPC) to brushless doubly-fed induction generators (BDFIGs).

Their approach addressed the drawbacks of conventional DPC, such as power ripples and current distortion, while improving robustness compared to vector control. El Ouanjli et al. [20] introduced fuzzy direct torque control (FDTC) for doubly-fed induction machines (DFIMs) equipped with two voltage source inverters (VSIs), focusing on reducing electromagnetic torque ripples and improving total harmonic distortion (THD). Benbouhenni and Bizon [21] applied direct vector control to extract optimal active and reactive power from dual-rotor wind power (DRWP) systems using DFIGs. Their approach involved rotor current control through four-level fuzzy pulse-width modulation (PWM) and replaced the conventional PI controller with a neural network. To address the drawbacks of field-oriented control, namely significant torque and flux oscillations, Ayrir et al. [22] proposed a fuzzy direct torque regulation method for DFIG-based wind turbines. Tamalouzt et al. [23] implemented a direct reactive power regulation approach using a three-level inverter topology for DFIG-based wind turbines. Xiahou and Wu [24] introduced a fault-tolerant control scheme based on a Kalman filter to handle voltage and current sensor faults in DFIGs. Herizi et al. [25] employed a backstepping control technique enhanced with fuzzy logic to improve the efficiency of DFIG-based systems. Motivated by the above discussion, the present study proposes the application of SMC for controlling the DFIG system. The stability of the applied controller is established with the help of a Lyapunov function. The performance of SMC is compared with that of a fuzzy PID controller, focusing on key parameters, namely overshoot, rise time, and steady-state error. The models are simulated using Simscape in MATLAB/Simulink. The simulation results demonstrate that SMC outperforms the fuzzy PID controller in terms of rise time and steady-state error.

The layout of the paper is as follows: Section 2 presents the dynamics of DFIG. Section 3 discusses the design and system stability of the implemented approach. Section 4 provides the application of fuzzy PID on DFIG, and Section 5 interprets the simulation results along with a detailed discussion. Finally, Section 6 concludes the paper with the main findings.

SMC is a popular control method designed using two control terms. The first is the equivalent control, which drives the system dynamics to converge to the selected tracking errors (sliding surface). The second is the switching control, which maintains the system dynamics along the chosen sliding surface [26, 27]. The need for such robust stability guarantees is a recurring theme in engineering systems, as highlighted by recent reviews leveraging computational intelligence for stability analysis in other complex domains [28].

Table 1 summarizes the control objectives of this study.

Figures 1 and 2 illustrate the control architecture of the proposed approach.

Table 1. The control objective

1.png

Figure 1. Sliding mode control diagram

2.png

Figure 2. Sliding mode control (SMC) based doubly fed induction generator (DFIG) control diagram

3.1 Speed controller design

The sliding surface is expressed by:

$\mathrm{s}_1=\mathrm{e}_1=\Omega_{\mathrm{ref}}-\Omega$          (8)

Differentiating the above tracking error yields:

$\dot{\mathrm{s}}_1=\frac{\mathrm{d} \Omega_{\mathrm{ref}}}{\mathrm{dt}}-\frac{\mathrm{d} \Omega}{\mathrm{dt}}$          (9)

After substituting Eq. (3) into Eq. (9), we get:

$\dot{\mathrm{s}}_1=\frac{\mathrm{d} \Omega_{\mathrm{ref}}}{\mathrm{dt}}-\left(-\frac{\mathrm{PM}}{\mathrm{JL}_{\mathrm{s}}} \mathrm{i}_{\mathrm{qr}} \varphi_{\mathrm{ds}}-\frac{\mathrm{f}}{\mathrm{j}} \Omega\right)$          (10)

Considering the definition of SMC, the equivalent and switching controls can be expressed as:

$\mathrm{i}_{\mathrm{qre}}^{\mathrm{eq}}=-\frac{\mathrm{JL}_{\mathrm{s}}}{\mathrm{PM} \varphi_{\mathrm{ds}}}\left(\dot{\Omega}_{\mathrm{ref}}+\frac{\mathrm{f}}{\mathrm{j}} \Omega\right)$          (11)

$\mathrm{i}_{\mathrm{qre}}^{\mathrm{sw}}=\rho_1 \operatorname{sign}\left(\mathrm{~s}_1\right)$          (12)

3.2 Current controller design

To establish the current control law, the following sliding surfaces are selected:

$\mathrm{s}_2=\mathrm{e}_2=\mathrm{i}_{\mathrm{drref}}-\mathrm{i}_{\mathrm{dr}}$          (13)

$\mathrm{s}_3=\mathrm{e}_3=\mathrm{i}_{\mathrm{qrref}}-\mathrm{i}_{\mathrm{qr}}$          (14)

The time derivatives of Eqs. (13) and (14) are calculated as follows:

$\dot{\mathrm{s}}_2=\frac{\mathrm{di}_{\mathrm{drref}}}{\mathrm{dt}}-\frac{\mathrm{di}_{\mathrm{dr}}}{\mathrm{dt}}$          (15)

$\dot{\mathrm{s}}_3=\frac{\mathrm{di}_{\mathrm{qrref}}}{\mathrm{dt}}-\frac{\mathrm{di}_{\mathrm{qr}}}{\mathrm{dt}}$          (16)

By substituting Eq. (7) into Eqs. (15) and (16), we obtain:

$\dot{\mathrm{s}}_2=\frac{\mathrm{di}_{\mathrm{drref}}}{\mathrm{dt}}-\frac{1}{\mathrm{~L}_{\mathrm{r}} \sigma}\left(\mathrm{v}_{\mathrm{dr}}-\mathrm{R}_{\mathrm{r}} \mathrm{i}_{\mathrm{dr}}+\mathrm{L}_{\mathrm{r}}\left(\mathrm{w}_{\mathrm{s}}-\mathrm{w}\right) \sigma \mathrm{i}_{\mathrm{qr}}\right)$          (17)

$\dot{\mathrm{s}}_3=\frac{\mathrm{di}_{\mathrm{qrref}}}{\mathrm{dt}}-\frac{1}{\mathrm{~L}_{\mathrm{r}} \sigma}\left(\mathrm{v}_{\mathrm{qr}}-\mathrm{R}_{\mathrm{r}} \mathrm{i}_{\mathrm{qr}}+\mathrm{L}_{\mathrm{r}}\left(\mathrm{w}_{\mathrm{s}}-\mathrm{w}\right) \sigma \mathrm{i}_{\mathrm{dr}}-\frac{\mathrm{M}}{\mathrm{L}_{\mathrm{s}}}\left(\mathrm{w}_{\mathrm{s}}-\mathrm{w}\right) \varphi_{\mathrm{ds}}\right)$          (18)

As mentioned previously, the equivalent control attempts to drive the system dynamics onto the considered sliding surface, thereby making the latter converge to zero [29-31]. Therefore, the equivalent control laws are given by:

$\mathrm{v}_{\mathrm{dr}}^{\mathrm{eq}}=\mathrm{L}_{\mathrm{r}} \sigma\left(\left(\mathrm{R}_{\mathrm{r}} \mathrm{i}_{\mathrm{qr}}-\mathrm{L}_{\mathrm{r}}\left(\mathrm{w}_{\mathrm{s}}-\mathrm{w}\right) \sigma \mathrm{i}_{\mathrm{dr}}\right)-\frac{\mathrm{di}_{\mathrm{drref}}}{\mathrm{dt}}\right)$          (19)

$\mathrm{v}_{\mathrm{qr}}^{\mathrm{eq}}=\mathrm{L}_{\mathrm{r}} \sigma\left(\left(\mathrm{R}_{\mathrm{r}} \mathrm{i}_{\mathrm{dr}}-\mathrm{L}_{\mathrm{r}}\left(\mathrm{w}_{\mathrm{s}}-\mathrm{w}\right) \sigma \mathrm{i}_{\mathrm{qr}}+\frac{\mathrm{M}}{\mathrm{L}_{\mathrm{s}}} \frac{\mathrm{d}}{\mathrm{dt}} \varphi_{\mathrm{ds}}\right)\right)-\frac{\mathrm{di}_{\mathrm{qrref}}}{\mathrm{dt}}$          (20)

In the reaching step, the switching term is used to maintain the system dynamics on the selected sliding surface [31], and it is expressed as:

$\mathrm{v}_{\mathrm{dr}}^{\mathrm{sw}}=\mathrm{L}_{\mathrm{r}} \sigma\left(\rho_1 \operatorname{sgn}\left(\mathrm{~s}_1\right)\right)$          (21)

$\mathrm{v}_{\mathrm{qr}}^{\mathrm{sw}}=\mathrm{L}_{\mathrm{r}} \sigma\left(\rho_2 \operatorname{sgn}\left(\mathrm{~s}_2\right)\right)$          (22)

The final control laws are obtained by adding the equivalent terms to the switching terms.

3.3 Stability analysis

In this paper, the stability of the proposed control laws is established using the following Lyapunov function [32]:

$\mathrm{V}=\sum_{\mathrm{i}=1}^3 \frac{1}{2} \mathrm{~s}_{\mathrm{i}}^2$          (23)

After computing the derivative of Eq. (23), we obtain:

$\dot{\mathrm{V}}=\sum_{\mathrm{i}=1}^3 \mathrm{~s}_{\mathrm{i}} \dot{\mathrm{s}}_{\mathrm{i}}=s_1 \dot{\mathrm{~s}}_1+s_2 \dot{\mathrm{~s}}_2+s_3 \dot{\mathrm{~s}}_3$          (24)

After substituting Eqs. (10), (17), and (18) into Eq. (24), we obtain:

$\begin{gathered}\dot{\mathrm{V}}=s_1\left(\frac{\mathrm{~d} \Omega_{\mathrm{ref}}}{\mathrm{dt}}-\left(-\frac{\mathrm{PM}}{\mathrm{JL}_{\mathrm{s}}} \mathrm{i}_{\mathrm{qr}} \varphi_{\mathrm{ds}}-\frac{\mathrm{f}}{\mathrm{j}} \Omega\right)+\delta_1\right)+s_2\left(\frac{\mathrm{di}_{\mathrm{drref}}}{\mathrm{dt}}-\frac{1}{\mathrm{~L}_{\mathrm{r}} \sigma}\left(\mathrm{v}_{\mathrm{dr}}-\right.\right. \\ \left.\left.\mathrm{R}_{\mathrm{r}} \mathrm{i}_{\mathrm{dr}}+\mathrm{L}_{\mathrm{r}}\left(\mathrm{w}_{\mathrm{s}}-\mathrm{w}\right) \sigma \mathrm{i}_{\mathrm{qr}}\right)+\delta_2\right)+s_3\left(\frac{\mathrm{di}_{\mathrm{qrref}}}{\mathrm{dt}}-\frac{1}{\mathrm{~L}_{\mathrm{r} \sigma}}\left(\mathrm{v}_{\mathrm{qr}}-\right.\right. \\ \left.\left.\mathrm{R}_{\mathrm{r}} \mathrm{i}_{\mathrm{qr}}+\mathrm{L}_{\mathrm{r}}\left(\mathrm{w}_{\mathrm{s}}-\mathrm{w}\right) \sigma \mathrm{i}_{\mathrm{dr}}-\frac{\mathrm{M}}{\mathrm{L}_{\mathrm{s}}}\left(\mathrm{w}_{\mathrm{s}}-\mathrm{w}\right) \varphi_{\mathrm{ds}}\right) \delta_3\right)\end{gathered}$          (25)

where, $\delta_i=\left(\delta_1, \delta_2, \delta_3\right)$ denotes the system uncertainties.

In sliding mode, the control input is given by the sum of the equivalent term and the switching term; therefore, the control law is expressed as:

$\left\{\begin{array}{c}\mathrm{i}_{\mathrm{qr}}=\mathrm{i}_{\mathrm{qr}}^{\mathrm{eq}}+\mathrm{i}_{\mathrm{qr}}^{\mathrm{sw}} \\ \mathrm{v}_{\mathrm{dr}}=\mathrm{v}_{\mathrm{dr}}^{\mathrm{eq}}+\mathrm{v}_{\mathrm{dr}}^{\mathrm{sw}} \\ \mathrm{v}_{\mathrm{qr}}=\mathrm{v}_{\mathrm{qr}}^{\mathrm{eq}}+\mathrm{v}_{\mathrm{qr}}^{\mathrm{sw}}\end{array}\right.$          (26)

where, $\mathrm{i}_{\mathrm{qr}}^{\mathrm{eq}}, \mathrm{i}_{\mathrm{qr}}^{\mathrm{sw}}, \mathrm{v}_{\mathrm{dr}}^{\mathrm{eq}}, \mathrm{v}_{\mathrm{dr}}^{\mathrm{sw}}, \mathrm{v}_{\mathrm{qr}}^{\mathrm{eq}}$ and $\mathrm{v}_{\mathrm{qr}}^{\mathrm{sw}}$ are given in Eqs. (11) and (12) and Eqs. (19)-(22).

Substituting Eq. (25) into Eq. (24) yields:

$\begin{gathered}\dot{\mathrm{V}}=\mathrm{s}_1\left(-\rho_1 \operatorname{sgn}\left(\mathrm{~s}_1\right)+\delta_1\right)+\mathrm{s}_2\left(-\rho_2 \operatorname{sgn}\left(\mathrm{~s}_2\right)+\delta_2\right)  +\mathrm{s}_3\left(-\rho_3 \operatorname{sgn}\left(\mathrm{~s}_3\right)+\delta_3\right)\end{gathered}$          (27)

Eq. (27) can be rewritten as follows:

$\dot{\mathrm{V}}=\sum_{\mathrm{i}=1}^3\left(-\rho_{\mathrm{i}}\left|\mathrm{s}_{\mathrm{i}}\right|+\mathrm{s}_{\mathrm{i}} \delta_i\right)$          (28)

After selecting $\rho_{\mathrm{I}}>\max \left(\delta_{\mathrm{i}}\right)$, we have $\dot{V}<0$. Since $\mathrm{V}>0$ and $\dot{V}<0$, the system’s states are guaranteed to converge toward the selected sliding surface, thereby achieving the specified steady state in finite time.

To tune the PID gains using FLC, two inputs are selected: the error and the rate of error. The outputs of the FLC are K_p, K_i and K_d. Seven membership functions are used for both inputs and outputs. Seven linguistic variables are assigned for the inputs (BN: big negative, MN: medium negative, N: negative, ZE: zero, P: positive, MP: medium positive, and BP: big positive) and seven others for the outputs (EM: extremly marginal, VM: very marginal, M: marginal, A: avereage, H: hight, VH: very hight, and EH: extremly hight). Tables 2, 3 and 4 show the parameters of the inputs and outputs of the FLC and the fired fuzzy rules.

Table 2. The parameters of inputs and outputs of fuzzy logic controller (FLC)

Table 3. Fuzzy inferences for determining K_d [39]

Table 4. Fuzzy inferences for determining K_i and K_p [39]

Figures 4 and 5 depict the proposed control diagram for DFIG regulation.

4.png

Figure 4. The control loop of fuzzy PID

5.png

Figure 5. The control loop of fuzzy PID based DFIG