Performance Comparison of PI, Fuzzy Logic, and Sliding Mode Controls for Wind Turbine Power Management

2.1 Doubly-fed asynchronous generator (DFIG)

The double-fed asynchronous machine is significant in industrial applications owing to its numerous advantages [23]. Among its benefits [24] is easy access to both the rotor and the stator, allowing for the measurement and control of currents. This provides excellent flexibility and precision for managing the flux and electromagnetic torque [25]. It offers several reconfiguration possibilities, making it applicable in various fields. It can operate at constant torque beyond the nominal speed with a slip (±30%) [26].

The double-fed induction machine (DFIM) exhibits diverse operating regimes, characterized by the slip, which denotes the speed difference between the rotor and the stator magnetic field. These regimes include stationary (g=1), sub-synchronous (0<g<1), synchronous (g=0), super-synchronous (g<0) [27]. Figure 1 depicts the mathematical model of the DFIG.

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Figure 1. Used model of the DFIG

2.2 Methods presentation

2.2.1 Active and reactive power controls by the classic regulator (PI)

The temporal description of the traditional PI regulator consists of directly linking the control signal u(t) to the error signal e(t) [28]:

$u(t)=k_p\left(e(t)+\frac{1}{T_i} \int_0^n e(t) d t\right)$      (1)

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Figure 2. Structure of a PI regulator in block diagram

The classic PI controller is characterized by its proportional and integral parameters (k_p and ki), which determine its performance (Figure 2). An increase in k_p speeds up the process response and reduces the static error (e) but can result in excessive oscillations.

Finding the optimal k_p value is necessary for a fast and well-damped response. Integral action eliminates steady-state error. However, excessive ki (decrease in T_i ) increase leads to system instability.

These two parameters must be adjusted to optimize the system's response without introducing instability [29].

We can formulate the gains of the correctors using the machine parameters and the response time as follows [30]:

$k_p=\frac{1}{\tau_x} \frac{L_{s t}\left(L_{r t}-\frac{M^2}{L_{s t}}\right)}{M V_s} ; k_i=\frac{1}{\tau_r} \frac{R_{r t} L_{s t}}{M V_s}$      (2)

- The direct method [31] is as shown in Figure 3.

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Figure 3. Block diagram of the direct control

- The indirect method: Control without a power loop [32], which is as illustrated in Figure 4.

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Figure 4. Block diagram of indirect control without power loop

- The indirect method: Control with a power loop [31, 32]. which is as depicted in Figure 5.

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Figure 5. Block diagram of indirect control with power loop

2.2.2 Active and reactive power controls by the fuzzy regulator

Originating from the work of Lotfi Zadeh in the 1960s at the University of Berkeley, fuzzy logic stands as a vital branch of artificial intelligence that manages the representation and processing of uncertain and imprecise knowledge through linguistic variables and fuzzy sets, making it adept at handling nonlinear and ambiguous processes [33]. As outlined in Figure 6 and 7, the design of a fuzzy regulator is intricate, demanding the delineation of regulatory strategies based on objectives and input-output observations while navigating challenges in selecting linguistic formalism and mathematical operators for undefined variables. Despite these complexities, fuzzy logic, by harnessing expert knowledge, can achieve remarkable results without requiring a detailed mathematical model of the system [34].

To further elucidate the application of fuzzy logic in DFIG control, it's crucial to delve into the specifics of its implementation. The selection of linguistic variables, rule bases, and defuzzification methods is based on the unique characteristics and requirements of DFIG systems. Linguistic variables are chosen to represent critical parameters of the DFIG accurately, while the rule base is tailored to capture the system's dynamic behavior under varying conditions. The defuzzification method is selected to balance response accuracy and computational efficiency. This detailed approach underlines the adaptability and precision of fuzzy logic in managing the complexities inherent in DFIG control.

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Figure 6. Synoptic diagram of a fuzzy regulator

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Figure 7. Block diagram for fuzzy control

Courtesy of Figures 8, 9 and 10, we will integrate fuzzy controllers into the DFIG vector control block for direct and indirect methods, ensuring independent active and reactive power control.

- The direct method

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Figure 8. Block diagram of the fuzzy regulator used in direct control

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Figure 9. Description of fuzzy logic control in direct control of the DFIG

- The indirect command

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Figure 10. Description of the fuzzy logic control in the indirect control of the DFIG

Table 1 shows the notations that characterize the linguistic values used for the "fuzzification" of inputs in direct control. Table 2 shows the inference matrix of the regulator used in direct control.

Table 1. The inference matrix

2.2.3 Control of the wind system using the sliding mode approach

According to a study by El-Alami et al. [17], sliding mode control is a technique that guides the state trajectory of a system towards a sliding surface, rendering the system robust against uncertainties and disturbances by creating a surface that facilitates tracking, management, and stability, and defining a control rule that attracts and sustains state trajectories on this surface, with the design involving the formulation of the sliding surface as a scalar function, typically based on the errors between the variables to be controlled [35].

$S(x)=\left(\lambda+\frac{d}{d t}\right)^{r-1} e(x)$      (3)

e(x) the difference between the reference and measured value, r the relative degree of the system, λ positive constant, its value is linked to the speed of convergence of the state trajectories.

Convergence conditions are pivotal in ensuring the system's dynamics converge towards the sliding surfaces and remain unaffected by external disturbances, a process fundamentally grounded in driving the sliding surface towards zero through a convergence dynamic, typically expressed in a specific form.

$\left\{\begin{array}{l}\dot{\mathrm{S}}(x)>0 \text { if } S(x)<0 \\ \dot{\mathrm{S}}(x)<0 \text { if } S(x)>0\end{array}\right.$      (4)

It can also be used to demonstrate the existence and stability of the system [17, 36]. The concept consists of selecting a positive scalar function V(x)>0 to ensure the attraction of the variable to be controlled towards its reference value.

Then, a U command is developed to reduce this function $\dot{V}$(x)＜0. The Lyapunov function V(x) is used, defined as follows:

$V(x)=\frac{1}{2} \mathrm{~S}(\mathrm{x})^2$      (5)

It is possible to express the derivative in the following form:

$\dot{V}(x)=S(x) S(x)$      (6)

The SMC control law U(t) is composed of two main parts, $U_c(t)$ and $U_d(t)$. These two parts are determined as follows:

$U(t)=U_c(t)+U_d(t)$      (7)

$U_c(t)$: The continuous part of the controller. This part maintains the system's output on the sliding surface. $U_d(t)$: The discontinuous part of the SMC control law includes the nonlinear switching element of the control law and is characterized by its erratic nature on the sliding surface.

In this part, we will control the active and reactive power of the DFIG by replacing the PI regulator with a nonlinear regulator SMC of order 1 (r=1).

And reactive $Q_S$ power sliding surfaces $P_S$ are determined using the rotor current tracking errors $\left(I_{d r}, I_{q r}\right)$ respectively by El Alami et al. [17]:

$\left\{\begin{array}{l}S\left(I_{q r}\right)=e_{I_{q r}}=I_{q r}{ }^*-I_{q r} \\ S\left(I_{d r}\right)=e_{I_{d r}}=I_{d r}{ }^*-I_{d r}\end{array}\right.$      (8)

The derivative of our sliding surface:

$\left\{\begin{array}{l}\dot{S}\left(I_{q r}\right)=\dot{e}_{I_{q r}}={I^·}_{q r}{ }^*-{I^·}_{q r} \\ \dot{S}\left(I_{d r}\right)=\dot{e}_{I_{d r}}={I^·}_{d r}{ }^*-{I^·}_{d r}\end{array}\right.$      (9)

Lyapunov function will then be:

$\left\{\begin{array}{l}V\left(S_{I_{q r}}\right)=\frac{1}{2} S_{I_{q r}}{ }^2 \\ V\left(S_{I_{d r}}\right)=\frac{1}{2} S_{I_{d r}}{ }^2\end{array}\right.$      (10)

The derivative of the Lyapunov function:

$\left\{\begin{array}{l}\dot{V}\left(S_{I_{q r}}\right)=S_{I_{q r}} \dot{S}_{I_{q r}} \\ \dot{V}\left(S_{I_{d r}}\right)=S_{I_{d r}} \dot{S}_{I_{d r}}\end{array}\right.$      (11)

By substituting the expression for the derivative of the currents $\left(I_{q r}, I_{d r}\right)$ of the rotor voltage equations $\left(V_{q r}, V_{d r}\right)$, we obtain the following formulation [36]:

$\left\{\begin{array}{c}\dot{S}_{I_{q r}}=\dot{I}_{q r}{ }^*-\frac{1}{s i g L_{r t}}\left(V_{q r}-R_{r t} I_{q r}-sig L_{r t} \omega_r I_{d r}-\omega_r \frac{M V_s}{\omega_s L_{s t}}\right) \\ \dot{S}_{I_{d r}}=\dot{I}_{d r}{ }^*-\frac{1}{sigL_{rt}}\left(V_{d r}-R_{r t} I_{d r}-sigL_{rt} \omega_r I_{d r}\right)\end{array}\right.$      (12)

The control law consists of the combination of the equivalent switching control and the discontinuous regulation:

$\left\{\begin{array}{c}\dot{S}_{I q r}={I^·}_{q r}{ }^*-\frac{1}{s i g L_{r t}}\left(\begin{array}{c}\left(V_{q r}^{e q}+V_{q r}^d\right)-R_{r t} I_{q r}-sigL_{rt} \omega_r I_{d r}-\omega_r \frac{M V_s}{\omega_s L_{s t}}\end{array}\right) \\ \dot{S}_{I_{d r}}={I^·}_{d r}{ }^*-\frac{1}{s i g L_{r t}}\left(\begin{array}{c}\left(V_{d r}^{e q}+V_{d r}^d\right)-R_{r t} I_{d r} -sigL_{rt} \omega_r I_{d r}\end{array}\right)\end{array}\right.$        (13)

In steady state, we obtain:

$\left\{\begin{array}{l}S\left(I_{q r}\right)=0 ; \dot{S}\left(I_{q r}\right)=0 ; V_{q r}^d=0 \\ S\left(I_{d r}\right)=0 ; \dot{S}\left(I_{d r}\right)=0 ; V_{d r}^d=0\end{array}\right.$      (14)

To ensure the convergence of the Lyapunov function, an assumption is made on the form of the discontinuous law function, as defined in [37], with:

$\left\{\begin{array}{l}V_{q r}^d=K_{I_{q r}} Sign\left(S\left(I_{q r}\right)\right) \\ V_{d r}^d=K_{I_{d r}} Sign\left(S\left(I_{d r}\right)\right)\end{array}\right.$      (15)

Finally, the DFIG order formula is as follows:

$\left\{\begin{array}{l}\begin{gathered}V_{q r}=K_{I_{q r}} Sign\left(S\left(I_{q r}\right)\right)+\operatorname{sig} L_{r t}\left({I^·}_{q r}{ }^*+\frac{R_{r t}}{sig L_{r t}} I_{q r}\right. \left.+\omega_r I_{d r}+\omega_r \frac{M V_s}{\omega_s L_{s t}}\right)\end{gathered} \\ \begin{gathered}V_{d r}=K_{I_{d r}} Sign\left(S\left(I_{d r}\right)\right)+sig L_{r t}\left({I^·}_{d r}{ }^*+\frac{R_{r t}}{sigL_{r t}} I_{d r}\right. \left.-\omega_r I_{q r}\right)\end{gathered}\end{array}\right.$      (16)

Applying sliding mode control (SMC) in DFIG systems offers distinct advantages, particularly in robustness and stability. SMC's ability to counteract system uncertainties and disturbances makes it highly suitable for DFIGs, which often operate under variable and unpredictable conditions. The introduction of the Lyapunov function in this context is pivotal, as it provides a mathematical foundation to ensure system stability. By demonstrating that the system's energy decreases over time according to the Lyapunov function, we can assert the stability of DFIG systems under the influence of SMC. This connection underscores the relevance of SMC in maintaining consistent performance and reliability of DFIG systems in diverse operational scenarios.

Figure 11 shows a block diagram of the SMC command we used to control the DFIG.

Figure 12 shows the block diagram that we created using MATLAB Simulink.

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Figure 11. The DFIG control diagram with control (SMC):

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Figure 12. The DFIG control diagram with control (SMC): (a) Control of reactive power Q_s; (b) Controls the active power P_s

Control Mechanisms for DFIG: Harnessing the full potential of wind energy necessitates advanced control systems, especially when dealing with the complexities of a doubly-fed induction generator (DFIG). Our investigation evaluated three distinct regulators: Proportional-Integral (PI), fuzzy logic, and sliding mode.

PI Control: The PI control mechanism, a traditional approach, offers stability and simplicity. While it is adept at handling linear systems and disturbances, its performance can be compromised in nonlinear scenarios or during rapid wind fluctuations.

Fuzzy Logic Control: Fuzzy logic, a more modern approach, is designed to handle uncertainties and nonlinearities inherent in wind energy systems. Employing linguistic variables and rule-based systems offers a more adaptive control mechanism. However, its performance relies heavily on the rule base, and designing an optimal rule base can be challenging.

Sliding Mode Control: Sliding Mode is a nonlinear control strategy that offers robustness against system uncertainties and external disturbances. Its main advantage lies in its insensitivity to parameter variations and external disturbances. However, the challenge with sliding mode control is the chattering phenomenon, which can introduce high-frequency oscillations in the system.

Comparative Analysis: Our comparative analysis revealed that while each regulator has its strengths, their performance is context-dependent. The PI control offers reliable performance for steady wind conditions and linear scenarios. However, fuzzy logic and sliding mode controls exhibit superior adaptability and robustness in systems with rapid wind fluctuations or nonlinearities. This confirms the findings of the comparative studies [36, 37], which concluded that the sliding-mode control and fuzzy controller are capable of enhancing system robustness to parameter variations and wind speed fluctuations compared to classical vector control (PI).

It's worth noting that the optimal choice of regulator is also influenced by other factors, such as implementation complexity, computational requirements, and system specifications. For instance, while fuzzy logic offers adaptability, it might require more computational resources than PI control.

The choice of control mechanism for DFIG in wind turbines should be based on a holistic evaluation considering the operational conditions and system requirements. Our study provides a foundation for such assessments, shedding light on the intricacies of each control strategy and their applicability in real-world scenarios.