Open AccessArticle

Fault-Tolerant Control Implemented for Sustainable Active and Reactive Regulation of a Wind Energy Generation System

Adolfo R. Lopez

Jesse Y. Rumbo-Morales

Gerardo Ortiz-Torres

Jesus E. Valdez-Resendiz

^3,*

Gerardo Vazquez

and

Julio C. Rosas-Caro

⁴

Laboratory of Electrical and Power Electronics, Tecnologico Nacional de Mexico/ITS de Irapuato, Irapuato 36821, Mexico

Centro Universitario de los Valles, Universidad de Guadalajara, Carretera Guadalajara-Ameca Km 45.5, Ameca 46600, Mexico

School of Engineering and Sciences, Tecnologico de Monterrey, Monterrey 64700, Mexico

⁴

Facultad de Ingenieria, Universidad Panamericana, Zapopan 45010, Mexico

Author to whom correspondence should be addressed.

Sustainability 2024, 16(24), 10875; https://doi.org/10.3390/su162410875

Submission received: 30 October 2024 / Revised: 21 November 2024 / Accepted: 26 November 2024 / Published: 12 December 2024

(This article belongs to the Special Issue Power Electronics on Recent Sustainable Energy Conversion Systems)

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Browse Figures

Figure 1
Stages of a wind energy conversion system. "> Figure 2
WECS based on DFIG and back-to-back converter. "> Figure 3
Vector diagram of the DFIG variables oriented to the stator flux. "> Figure 4
RSC PI control scheme. "> Figure 5
RSC sliding-mode control scheme. "> Figure 6
RSC state-feedback control scheme. "> Figure 7
FTC scheme applied to the RSC system, with (1) the RSC system, (2) the nominal controller, and (3) the active fault-tolerant control system. "> Figure 8
Simulation model of the FTC scheme with the state-feedback controller, developed in Matlab/Simulink 2023B, with (1) the RSC system, (2) the nominal controller, and (3) the active fault-tolerant control system. "> Figure 9
Scenario 1—comparison between PI, SMC and nominal state-feedback controller. "> Figure 10
Scenario 1—input signal comparison between PI, SMC and nominal state-feedback controller. "> Figure 11
Scenario 2—additive fault estimation. "> Figure 12
Scenario 2—comparison between PI, SMC and state-feedback controller with FTC. "> Figure 13
Scenario 2—input signal comparison between PI, SMC and state-feedback with FTC. "> Figure 14
Scenario 2—comparison of absolute tracking error between the SMC and the state-feedback with FTC. ">

Versions Notes

Abstract

This paper presents the design of a fault-tolerant control system based on fault estimation, aimed at enhancing the sustainability and efficiency of a wind energy conversion system using a doubly-fed induction generator. The control architecture comprises a rotor-side converter (RSC) and a grid-side converter (GSC). The RSC is responsible for regulating both active and reactive power, and its model incorporates two linear subsystem representations. A fault-tolerant control (FTC) scheme is developed using a state-feedback controller; this controller is applied to regulate stator and rotor currents. Additionally, for comparison purposes, Proportional–Integral (PI) and Sliding-Mode Controllers (SMCs) are designed to analyze the performance of each controller. Furthermore, a proportional integral observer is employed in the proposed fault-tolerant scheme for actuator fault estimation. Fault detection is achieved by comparing the fault estimation signal with a predefined threshold. The main contribution of this work is the design and validation of a comprehensive active FTC scheme that enhances system reliability and sustainability. It also includes a performance analysis comparing three controllers (PI, SMC, and state-feedback) applied to the RSC. These controllers are evaluated for their ability to regulate active and reactive power in a wind energy conversion system under conditions of non-constant actuator faults.

Keywords:

wind energy generation; sustainable energy; resilient renewable systems; power control; reactive power control; fault-tolerant control

1. Introduction

Nowadays, the use of renewable energies has made a significant advance in recent years due to the high cost and negative impact of typical fossil fuels. The total electricity generation in the world increased by 2.3% in 2022, reaching 29,165 TWh; renewable energy sources made up 92% of this electricity increase [1].

Wind Energy Conversion Systems (WECS) involve converting the kinetic energy of the wind into electrical energy with the help of a generator. In recent years, technological advances have helped to develop more economical wind generation systems of great power and higher efficiency. These WECS are divided into three stages: the aerodynamic stage, the mechanical stage, and finally, the electrical stage (see Figure 1).

The generators commonly employed in Wind Energy Conversion Systems (WECS) can be broadly classified into two main categories: synchronous generators and asynchronous generators. The asynchronous generators extensively utilized in these systems include the Squirrel Cage Induction Generators (SCIG), the Wound Rotor Induction Generators (WRIG), and the Doubly-Fed Induction Generators (DFIG) [2].

1.1. Case Study

The case study of this paper focuses on a WECS based on a DFIG; this generator is connected to a multiplier gearbox attached to the rotation propeller shaft. The advantage of the DFIG is that both the rotor and stator windings can be accessed; in this case, the stator is fed directly to the network while the rotor is connected to a back-to-back converter. This converter consists of a Rotor-Side Converter (RSC), a Grid-Side Converter (GSC), and a DC bus. With this configuration, the converter only needs to manipulate the slip power, which oscillates around 30% of the nominal power of the machine [3,4]. Figure 2 shows the described WECS as well as the location of the RSC and the GSC.

Regarding control, the RSC is pivotal in overseeing and managing the active and reactive power transfer between the stator of the DFIG and the grid. This control is effectively executed by manipulating the rotor currents. Simultaneously, the GSC regulates the DC bus. Furthermore, the GSC controls the reactive power exchanged between the converter and the grid. This dual functionality underscores the importance of the GSC in maintaining stability and efficient power transfer within the system [5,6].

The control of the RSC involves meticulously computing the rotor references and generating the stator currents necessary to achieve the targeted levels of active and reactive power. The derivation of these references is facilitated through algebraic relations that intricately incorporate power variables, along with the rotor and stator currents of the DFIG. This methodical approach ensures a precise and coordinated control mechanism, where the algebraic relationships form the basis for determining the requisite parameters to optimize the performance of the DFIG [5,7]. The active and reactive power control of the RSC in WECS has been widely used in the literature [6,8,9,10], and good performance of such a controller is needed to obtain the needed value.

Given these characteristics and the increase in performance requirements at different operation conditions, when connected to the grid, WECS has become progressively vulnerable to faults. These faults are usually voltage harmonics, which in turn can generate voltage ripple, power or torque pulsations, overheating, and increased losses in the stator windings. This inevitably influences process stability, reliability, and safety dynamics [11].

1.2. Literature Review

When discussing the Fault Ride-Through (FRT) mechanism implemented in WECS based on DFIG, in [12], the authors implement a series converter on the stator side during a grid fault, while other papers demonstrate the use of linear control techniques to achieve optimal performance [13,14].

Fault-Tolerant Control (FTC) methods can be considered to identify malfunctions at any time and improve reliability and safety in a process. The FTC techniques are classified into two types [15]: passive and active. On the one hand, in the passive techniques, a fixed controller is designed to tolerate changes in the plant dynamics, and then, the stabilized system could satisfy its goals under all faulty conditions. This approach needs neither Fault Diagnosis (FD) schemes nor controller reconfiguration, but it has limited capabilities because it uses fixed parameters. On the other hand, active techniques adapt or reconfigure the controller parameters based on information from the FD system. This ensures the system’s stability and maintains acceptable performance despite the fault. According to [10], different names can be used to distinguish diagnostic steps: fault detection, fault isolation, and fault identification (also known as fault estimation).

In the context of fault detection and isolation applied to WECS, ref. [16] proposed a diagnostic approach for multiple open-circuit faults in the back-to-back converters of turbines. This diagnostic was implemented by using Permanent Magnet Synchronous Generators (PMSGs). The proposed fault detection method uses the information contained in the instantaneous amplitude (IA) in both the PMSG- and grid-side converters. This method enables accurate distinction between different states of any switch without being affected by speed variations. Simulation and experimental results showed that the proposed strategy detects and locates multiple faults.

In [17], a fault detection methodology of the power converter within a wind turbine chain equipped with a DFIG is presented. In this case, a short-circuit failure in one of the power semiconductor switches is considered and the fault detection method is obtained by the identification of the

α

β

signals of the RSC. Simulation results have been used to analyze the fault detection and identification method. Additionally, a passive FTC system has been proposed to ensure continuity of service. Fault diagnosis method-based detection for sensors, actuators, and system faults in a wind turbine machine is presented in [18]. Physical and analytical redundancy of sensors and actuators is used to achieve the proposed detection method. The simulation results show the efficiency of the diagnosis scheme.

While a state-of-the-art review on the fault diagnosis of wind power converters is presented in [19], this paper studies the typical fault modes of wind power converters, including short-circuit and open-circuit faults in power switches. Reliability and robustness performance are discussed.

Recent works on FTC design for WECS subjected to faults can be found in [20,21,22,23]. In [20], a nonlinear fault-tolerant approach is presented using a fractional-order non-singular terminal sliding mode controller. This method is designed to maximize power extraction in wind energy conversion systems operating below their rated wind speeds, even in the presence of actuator faults. The sliding surface is proposed to ensure fast finite-time convergence, and the fractional calculus enhances the convergence speed of system states without chattering. Numerical simulation is carried out in the WECS system in both fault-free and faulty scenarios.

A passive FTC system using Fuzzy-PID (Fuzzy-Proportional–Integral Derivative) controller for controlling the DFIG rotor-side converter in fault conditions is presented in [21]. The controller is added only in fault conditions and it reduces the current of the rotor circuit with a proper control of the RSC. The proposed fault control system controls the amplitude of the rotor current when it enters the circuit and prevents it from getting too high. The simulation results guarantee the effectiveness of the proposed method.

In [22] an optimal fault-tolerant pitch control strategy is addressed to adjust the pitch angle of the wind turbine system. The proposed scheme uses a fractional-calculus extended memory of pitch angles with a fractional-order PID controller to enhance the performance of the wind turbine. Fault-tolerant load frequency control design of interconnected wind power systems is investigated in [23]. The FTC is designed with the presence of known and unknown actuator faults to improve the frequency stability of the DFIG-based interconnected wind power systems, including the load disturbances and wind speed fluctuations.

The importance of the FTC in WECS is due to its efficiency. This is because the failure of the components of the WECS leads to low performance. In this case, the FTC keeps an optimal performance by dynamically adjusting the control strategy.

Another topic of importance of the FTC in WECS is security; this is because a failure in WECS could be catastrophic. In this case, the FTC improves security by detecting and mitigating the failures before it becomes a critical problem [24].

One gap in the current research of FTC in WECS is the lack of complex failure scenarios. The research of FTC in WECS focuses on simple or individual failures. More research needs to be carried out under complex circumstances [24,25].

Previous studies on active and reactive power control in wind energy conversion systems, incorporating fault detection, fault estimation, and fault-tolerant control, have seldom addressed non-constant faults in the actuator. In this case, this paper will exclusively focus on the RSC as the actuator. During a fault in the RSC, the DFIG cannot capture the maximum wind power, resulting in inaccurate measurements of active and reactive power. In general, an adequate function of the controller ensures the stability of the overall system [26,27]. Therefore, the motivation of this paper is to consider the RSC actuator partial faults with variable dynamics.

Additionally, this paper presents the design of a feedback controller and an FTC strategy using a proportional–integral observer for detection, estimation, accommodation, and reconfiguration of the fault. The main contribution of this work is the design and validation of a complete active FTC scheme and the performance analysis between three controllers (PI, SMC, and state-feedback) applied to the rotor-side converter for regulating the active and reactive power of the wind energy conversion system in the presence of non-constant actuator faults. The proposed scheme is easy to implement and also offers relevant characteristics compared to the robust controllers presented in this work (PI and SMC); our proposed FTC system provides information about the time of the fault occurrence and its dynamics, and it is possible to accommodate the fault by adding a signal to the nominal control law. The structure of the article is as follows. Section 2 presents the control model of the rotor-side converter, as well as the PI, the sliding-mode controller and the nominal pole placement controller. Section 3 validates the FTC scheme in simulation and the main graphs illustrating the performance of the system. Finally, the discussion and the conclusions are presented in Section 4 and Section 5, respectively.

2. Materials and Methods

2.1. Rotor-Side Converter Control Model

This section shows the control model of the rotor-side converter, which is essential to understanding the design of the different controllers shown in Section 2.2.3.

The model of the DFIG is the same as that of an SCIG type; however, in the case of the DFIG, the rotor is made up of a triphasic wound, which can be accessed through slip rings. The three-phase model of the DFIG is given in [28], which for the sake of brevity is not presented in this paper. This three-phase model can be rewritten in the reference frame

d q 0

, which is the product of the transformation

K_{T} (Θ_{s}) : f_{a b c} \to f_{d q 0}

of the three-phase model.

The

d q 0

reference frame is a mathematical transformation used to represent the three-phase quantities (e.g., voltages or currents) in a rotating reference frame. This transformation simplifies analysis and control by converting three-phase AC signals (

a b c

) into two orthogonal components (

d q

), and a zero-sequence component. This approach is particularly useful in control systems, as it reduces the complexity of the calculations required for the DFIG.

This reference frame is orthogonal and rotating, with angular displacement

Θ_{s}

. Assuming a balanced three-phase system, the zero-sequence component vanishes, obtaining the following

d q

model:

\begin{matrix} v_{d s} & = R_{s} i_{d s} - ω λ_{q s} + {\dot{λ}}_{d s}, \end{matrix}

(1)

\begin{matrix} v_{q s} & = R_{s} i_{q s} + ω λ_{d s +} {\dot{λ}}_{q s}, \end{matrix}

(2)

\begin{matrix} v_{d r} & = R_{r} i_{d r} - (ω - ω_{r}) λ_{q r} + {\dot{λ}}_{d r}, \end{matrix}

(3)

\begin{matrix} v_{q r} & = R_{r} i_{q r} + (ω - ω_{r}) λ_{d r} + {\dot{λ}}_{q r}, \end{matrix}

(4)

\begin{matrix} λ_{d s} & = (L l_{s} + \frac{3}{2} L m_{s}) i_{d s} + \frac{3}{2} L m_{s} i_{d r}, \end{matrix}

(5)

\begin{matrix} λ_{q s} & = (L l_{s} + \frac{3}{2} L m_{s}) i_{q s} + \frac{3}{2} L m_{s} i_{q r}, \end{matrix}

(6)

\begin{matrix} λ_{d r} & = (L l_{r} + \frac{3}{2} L m_{s}) i_{d r} + \frac{3}{2} L m_{s} i_{d s}, \end{matrix}

(7)

\begin{matrix} λ_{q r} & = (L l_{r} + \frac{3}{2} L m_{s}) i_{q r} + \frac{3}{2} L m_{s} i_{q s}, \end{matrix}

(8)

where

i_{d, q s}

and

i_{d, q r}

are the stator and rotor currents, while

V_{d, q s}

and

V_{d, q r}

are the stator and rotor voltages.

λ_{d, q s}

and

λ_{d, q r}

represent the rotor flux links;

R_{r}

and

R_{s}

are the resistances of the rotor and the stator.

L l_{r}

is the rotor inductance,

L l_{s}

is the stator inductance,

L m_{s}

is the mutual inductance. In addition,

ω

is the angular velocity of the arbitrary reference frame, and

ω_{r}

is the angular velocity of the rotor.

As mentioned before, the control of the RSC involves controlling the current that flows through the stator of the DFIG. Because this converter is connected to the rotor terminals, the stator current is controlled indirectly through the rotor currents.

This RSC control involves calculating the current references in the rotor; those references are obtained from algebraic relationships that involve the power variables, as well as the stator and rotor currents [5]. In this case, it was considered to orient the DFIG variables to the stator flow (Figure 3).

Due to this orientation, the equations of

λ_{d s}

and

λ_{q s}

are modified, obtaining the following:

\begin{matrix} λ_{d s} & = L_{s} i_{d s} + L_{M} i_{d r} = λ_{s} = L_{M} i_{m s}, \end{matrix}

(9)

\begin{matrix} λ_{q s} & = L_{s} i_{q s} + L_{M} i_{q r} = 0, \end{matrix}

(10)

where

L_{s} = L l_{s} + \frac{3}{2} L m_{s}, L_{M} = \frac{3}{2} L m_{s}

. Clearing

i_{d s}

and

i_{q s}

from (9) and (10), and substituting in the equations of

λ_{d r}

and

λ_{q r}

, described in (7) and (8), respectively, the following is obtained:

\begin{matrix} V_{d r} & = R_{r} i_{d r} + σ L_{r} \frac{d i_{d r}}{d t} - c_{1}, \end{matrix}

(11)

\begin{matrix} V_{q r} & = R_{r} i_{q r} + σ L_{r} \frac{d i_{q r}}{d t} + c_{2}, \end{matrix}

(12)

where

\begin{matrix} σ = (\frac{1 - L_{M}^{2}}{L_{s} L_{r}}), L_{r} = L_{r} + \frac{3}{2} L m_{s}, \end{matrix}

(13)

and

c_{1}

and

c_{2}

are compensation terms, defined as:

\begin{matrix} c_{1} & = (ω - ω_{r}) (L_{M} i_{m s} + σ L_{r} i_{d r}), \end{matrix}

(14)

\begin{matrix} c_{2} & = (ω - ω_{r}) (σ L_{r} i_{q r}) . \end{matrix}

(15)

To design effective controllers for the linear subsystems (11) and (12), these are represented as transfer functions and as a continuous state-space model. Applying the Laplace transform on (11) and (12), without considering the compensation terms, the following are obtained [5]:

\begin{matrix} \frac{I_{d r} (s)}{V_{d r} (s)} = \frac{1}{σ L_{r} s + R_{r}}, \frac{I_{q r} (s)}{V_{q r} (s)} = \frac{1}{σ L_{r} s + R_{r}} . \end{matrix}

(16)

Also, the nominal state-space model of the previous subsystems (11) and (12) can be represented by:

\begin{matrix} \dot{x} & = A x + B u + P, \\ y & = C x, \end{matrix}

(17)

with

\begin{matrix} A & = [\begin{matrix} - \frac{R_{r}}{σ L_{r}} & 0 \\ 0 & - \frac{R_{r}}{σ L_{r}} \end{matrix}], B = [\begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix}], \\ P & = [\begin{matrix} - \frac{c_{1}}{σ L_{r}} \\ \frac{c_{2}}{σ L_{r}} \end{matrix}], C = [\begin{matrix} \frac{1}{σ L_{r}} & 0 \\ 0 & \frac{1}{σ L_{r}} \end{matrix}], \end{matrix}

where

x = {[i_{d r}, i_{q r}]}^{⊤}

is the state vector,

u = {[V_{d r}, V_{q r}]}^{⊤}

is the input vector of the system and y denotes the measurable output vector. In this representation, the compensation terms (14) and (15) are known perturbation vector P.

2.2. Rotor-Side Converter Controller Design

2.2.1. PI Controller

From (16), it can be observed that

V_{d r}

and

V_{q r}

depend on

i_{q r}

, and therefore, the design of the linear controller of the RSC is obtained as [5,7]. The RSC’s linear Proportional–Integral (PI) controllers are designed, obtaining the control loop with the following transfer function:

G_{c} (s) = \frac{K_{p} s + K_{i}}{s^{2} (σ L_{r}) + s (K_{p} + R_{r}) + K_{i}} .

(18)

Figure 4 shows the control scheme, where the conversion of the rotor current from the

a b c

scheme to the

d q

reference frame can be observed.

The reference values of

P_{s}^{*}

and

Q_{s}^{*}

come from the expressions in the

d q

reference frame that describe the active and reactive power at the DFIG stator terminal; these are:

\begin{matrix} P_{s} = \frac{3}{2} (V_{d s} i_{d s} + V_{q s} i_{q s}), Q_{s} = \frac{3}{2} (V_{q s} i_{d s} - V_{d s} i_{q s}) . \end{matrix}

(19)

Because the stator flux is aligned with the

d q

reference,

V_{d s} = 0

and

λ_{q s} = 0

R_{s}

is very small and therefore considered zero; so,

V_{d s}

and

V_{q s}

from (1) and (2) are simplified as follows:

\begin{matrix} V_{d s} = 0, V_{q s} = ω λ_{d s} = ω L_{M} i_{m s} . \end{matrix}

(20)

From (9) and (10), the expressions that describe

i_{d s}

and

i_{q s}

are obtained:

\begin{matrix} i_{d s} = \frac{L_{M}}{L_{S}} (i_{m s} - i_{d r}), i_{q s} = - \frac{L_{M}}{L_{S}} i_{q r} . \end{matrix}

(21)

Substituting (20) and (21) in (19), the equations are obtained with which it is possible to calculate reference currents

i_{d r}^{*}

and

i_{q r}^{*}

from the reference powers

P_{s}^{*}

and

Q_{s}^{*}

; these are shown below [29]:

\begin{matrix} i_{d r}^{*} = i_{m s} - \frac{2}{3} \frac{Q_{s}^{*} L_{s}}{v_{q r} L_{M}}, i_{q r}^{*} = - \frac{2}{3} \frac{P_{s}^{*} L_{s}}{v_{q r} L_{M}} . \end{matrix}

(22)

In this case, a speed of 4200 rpm with an active power reference of 100 W and reactive power of 50

V A R

was considered in the WECS. It is important to notice that in this paper, the GSC control, as well as the equations that define the value of the DC link capacitor, will not be aboard.

2.2.2. Sliding-Mode Controller

The basic idea behind SMC is to choose a sliding surface along which the system can be slid to its desired value (see Figure 5). For simplicity, this subsection will only explain the sliding-mode controller applied to the system (11), but the same analysis can be considered for the system (12). Consider the error expressed as

e_{s} = P_{s}^{*} - (1 / σ L_{r}) i_{d r}

. In order to make the error and derivative of error equal to zero, the following sliding-mode surface function is defined:

\begin{matrix} s = γ e_{s}, \end{matrix}

(23)

where

γ > 0

. Then, the time derivative of the sliding surface is

\begin{matrix} \dot{s} = γ {\dot{e}}_{s} = γ {\dot{P}}_{s}^{*} - \frac{γ}{σ L_{r}} {\dot{i}}_{d r}, \end{matrix}

(24)

and supposing that the reference value is constant, then by using (11), Equation (24) is rewritten as:

\begin{matrix} \dot{s} = - \frac{γ}{σ L_{r}} (- \frac{R_{r}}{σ L_{r}} i_{d r} + V_{d r} + \frac{c_{1}}{σ L_{r}}) . \end{matrix}

(25)

Now, the reaching law is given by [30]:

\begin{matrix} \dot{s} = - K_{s} sgn (s) - ϵ {| s |}^{b sgn (| s | - 1)} s, \end{matrix}

(26)

where

K_{s}

and

ϵ

are positive parameters, with

0 < b < 1

. The stability condition can be consulted in [30]. By using (25) and (26), and solving for the input

V_{d r}

, the control law is defined as:

\begin{matrix} V_{d r} = & (- \frac{σ L_{r}}{γ}) (- K_{s} sgn (s) - ϵ {| s |}^{b sgn (| s | - 1)} s - \frac{γ R_{r}}{σ^{2} L_{r}^{2}} i_{d r} + \frac{γ c_{1}}{σ^{2} L_{r}^{2}}) . \end{matrix}

(27)

2.2.3. Nominal State-Feedback Controller

In this subsection, a feedback controller is designed for the RSC system, such that the stator and rotor currents tend to the desired reference, as shown in Figure 6. The state-feedback controller is a pole placement technique known as tracking controller as it requires output to follow the input command signal. The integral action reduces the finite error to zero [31]. This controller will be the nominal controller for the active FTC. When a partial fault is detected by the observer, then an alarm is activated and the observer will estimate the magnitude of the fault. Subsequently, a new control signal will be added to the state-feedback control law in order to compensate the fault. The integrator comparator is used to reach the desired time-varying values by defining the following error dynamics:

\dot{e} = d_{c} - C x,

(28)

where the error dynamics vector is

\dot{e} = {[{\dot{e}}_{1}, {\dot{e}}_{2}]}^{⊤}

, and the desired reference vector is represented by

d_{c} = {[P_{s}^{*}, Q_{s}^{*}]}^{⊤}

. By using (17), an augmented system

x_{c} = {[x, e]}^{⊤}

can be proposed as [32]:

\begin{matrix} {\dot{x}}_{c} & = {\bar{A}}_{c} x_{c} + {\bar{B}}_{c} u + \bar{P} + \bar{R} d_{c}, \\ y_{c} & = {\bar{C}}_{c} x_{c}, \end{matrix}

(29)

with

\begin{matrix} {\bar{A}}_{c} & = [\begin{matrix} A & 0_{2 \times 2} \\ - C & 0_{2 \times 2} \end{matrix}], {\bar{B}}_{c} = [\begin{matrix} B \\ 0_{2 \times 2} \end{matrix}], {\bar{P}}_{c} = [\begin{matrix} P \\ 0_{2 \times 1} \end{matrix}], \\ {\bar{R}}_{c} & = [\begin{matrix} 0_{2 \times 1} \\ 1_{2 \times 1} \end{matrix}], {\bar{C}}_{c} = [\begin{matrix} C & 0_{2 \times 2} \end{matrix}] . \end{matrix}

If the pair

({\bar{A}}_{c}, {\bar{B}}_{c})

is controllable, then a nominal control law u can be obtained as follows [33]:

\begin{matrix} u & = - {\bar{K}}_{c} x_{c} = - [\begin{matrix} K_{1} & K_{2} \end{matrix}] [\begin{matrix} x \\ e \end{matrix}] - B_{c}^{+} P_{c}, \end{matrix}

(30)

where the State-Feedback (SF) gain matrices are defined as

K_{1}

and

K_{2}

, and

B_{c}^{+}

is the pseudoinverse of the matrix

B_{c}

. The term

B_{c}^{+} P_{c}

is proposed to eliminate the effect of the vector

P_{c}

on the controller design. Then, the problem is reduced to determine optimal values of the controller gains.

Then, synthesizing (30) in the linear system (29) and assuming that

d_{c} = 0

gives the closed-loop system represented by:

\begin{matrix} {\dot{x}}_{c} & = ({\bar{A}}_{c} + {\bar{B}}_{c} {\bar{K}}_{c}) x_{c} . \end{matrix}

(31)

Global asymptotic convergence of the tracking controller is guaranteed by the following Theorem 1.

Theorem 1.

Given the system (29) and the comparator integrator control law (30), the closed-loop system (31) is asymptotically stable if there exist matrices

X = X^{⊤} > 0

and M, such that

\begin{matrix} X {\bar{A}}_{c} + {\bar{A}}_{c}^{⊤} X - {\bar{B}}_{c} M - M^{⊤} {\bar{B}}_{c}^{⊤} < 0 . \end{matrix}

(32)

The gain matrix for the controller is computed by

{\bar{K}}_{c} = M X^{- 1}

The proof can be consulted in [34].

Usually, the Linear Matrix Inequality (LMI) (32) produces slow dynamics. This problem was solved using

α

-stability, as defined in [35], with the eigenvalues of

({\bar{A}}_{c} - {\bar{B}}_{c} {\bar{K}}_{c})

located in the desired region, expressed as:

\begin{matrix} λ ({\bar{A}}_{c} - {\bar{B}}_{c} {\bar{K}}_{c}) \in D, \end{matrix}

(33)

where

D

is the

α

-stability. Then, to avoid slow dynamics, (32) is replaced with (34). Therefore, the following corollary of Theorem 1 is obtained:

Corollary 1.

Given the system (29) and the comparator integrator control law (30), the closed-loop error system (31) is asymptotically stable if there exist a positive scalar α, matrices

X = X^{⊤} > 0

and M, such that the condition is met.

X {\bar{A}}_{c} + {\bar{A}}_{c}^{⊤} X - {\bar{B}}_{c} M - M^{⊤} {\bar{B}}_{c}^{⊤} + 2 α X < 0,

(34)

and condition (33) holds. The gain matrix of the controller is computed by

{\bar{K}}_{c} = M X^{- 1}

2.3. Active Fault-Tolerant Control Design

The linear state-space representation of the RSC system (17) is used to design and validate the feasibility of a fault-tolerant control (FTC) algorithm in simulation. The nominal pole placement SF controller is considered to validate the proposed FTC algorithm. A Fault Accommodation (FA) scheme is proposed to achieve this goal. In Figure 7, a general FA structure is depicted. Notice here that the scheme is composed of three main blocks: (1) the RSC system, (2) the nominal controller, and (3) the active fault-tolerant control system constructed by the fault detection (FD), estimation (FE), and accommodation subsystems (FA). This scheme is based on our previous work [36].

Only the nominal SF controller is applied when the RSC is in a fault-free case. Once the actuator fault occurs, the FE signal generated by a linear proportional–integral observer (PIO) detects and accommodates the actuator fault. The actuator fault is detected when the FE signal has a more precise value than a predefined threshold. Finally, the FTC law is computed using the FE signal when the fault is detected (faulty case).

2.3.1. Actuator Fault Estimation System

Following some ideas from [32], an observer has been designed for fault estimation. An actuator fault can be modeled as an additive external signal:

u_{f} = u + f

, where

f = {[f_{1}, f_{2}]}^{⊤} = - θ u

means the actuator fault vector that affects the RSC system and

θ = diag (θ_{1}, θ_{2})

is the actuator loss of effectiveness (LoE). The fault signal is assumed to be constant over time or at least slowly varying, such that

\dot{f} = 0

. The value of

θ_{i}

, with

i = 1

and 2, indicates (i) a total actuator fault of i-th actuator when

θ_{i} = 1

, (ii) the i-th actuator is healthy when

θ_{i} = 0

, and (iii) LoE of the actuator when

θ_{i} = [0, 1]

. In this paper, only the case of the actuator LoE presented in the RSC system is considered.

A linear proportional–integral observer will now be applied to the RSC linear system to estimate the actuator fault vector f. The linear system (17) is rewritten in the augmented faulty linear form

x_{a} = {[x, f]}^{⊤}

, as follows [32]:

\begin{matrix} {\dot{x}}_{a} & = {\overset{˘}{A}}_{a} x_{a} + {\overset{˘}{B}}_{a} u + {\overset{˘}{P}}_{a}, \\ y_{a} & = {\overset{˘}{C}}_{a} x_{a}, \end{matrix}

(35)

with

\begin{matrix} {\overset{˘}{A}}_{a} & = [\begin{matrix} A & B \\ 0_{2 \times 2} & 0_{2 \times 2} \end{matrix}], {\overset{˘}{B}}_{a} = [\begin{matrix} B \\ 0_{2 \times 2} \end{matrix}], \\ {\overset{˘}{P}}_{a} & = [\begin{matrix} P \\ 0_{2 \times 1} \end{matrix}], {\overset{˘}{C}}_{a} = [\begin{matrix} C & 0_{2 \times 2} \end{matrix}] . \end{matrix}

A linear PIO is written for the augmented system (35), as follows:

\begin{matrix} {\dot{\hat{x}}}_{a} & = {\overset{˘}{A}}_{a} {\hat{x}}_{a} + {\overset{˘}{B}}_{a} u + {\overset{˘}{L}}_{P I} e_{y} + {\overset{˘}{P}}_{a}, \\ {\hat{y}}_{a} & = {\overset{˘}{C}}_{a} {\hat{x}}_{a}, \end{matrix}

(36)

where

e_{y} = y_{a} - {\hat{y}}_{a}

is the estimation error vector, and

{\overset{˘}{L}}_{P I} = {[L_{P}, L_{I}]}^{⊤}

represents the observer gain to be computed for estimate f and

x_{a}

. The dynamics of the state estimation error

e_{a} = x_{a} - {\hat{x}}_{a}

between the augmented system (35) and the augmented PIO (36) are given by:

\begin{matrix} {\dot{e}}_{a} & = ({\overset{˘}{A}}_{a} - {\overset{˘}{L}}_{P I} {\overset{˘}{C}}_{a}) e_{a} . \end{matrix}

(37)

Sufficient conditions to guarantee asymptotic stability of the estimation error (37) are given through the following theorem:

Theorem 2.

Consider the linear system (35) and assume that the pair

({\overset{˘}{A}}_{a}, {\overset{˘}{C}}_{a})

is observable. The state and the fault estimation error dynamics (37) are asymptotically stable if there exists

S = S^{⊤}

and N, such that the following holds:

S {\overset{˘}{A}}_{a} + {\overset{˘}{A}}_{a}^{⊤} S - N {\overset{˘}{C}}_{a} - {\overset{˘}{C}}_{a}^{⊤} N^{⊤} < 0 .

(38)

The gain of the observer is computed by

{\overset{˘}{L}}_{P I} = S^{- 1} N

The proof can be seen in [32].

2.3.2. Actuator Fault Detection System

The actuator additive fault estimation signal vector is used to detect the actuator fault at any time, as follows:

\begin{matrix} | \hat{f_{i}} | & \geq μ_{i} \Rightarrow in faulty case (Alarm = 1), \\ | \hat{f_{i}} | & < μ_{i} \Rightarrow in fault-free case (Alarm = 0), \end{matrix}

(39)

where

μ_{i}

, with

i = 1

and 2, is a constant threshold, chosen according to simulation results. In a fault-free case, the estimated value

| \hat{f_{i}} |

is close to zero, while in a fault case, the estimated value has a greater value than the threshold to indicate the occurrence of a fault. If the fault estimation value is greater than the threshold, then it is considered a faulty case and the alarm indicator is one.

2.3.3. Fault Accommodation Control Law

When the partial fault is detected and estimated correctly, a new control law is added to the nominal controller to compensate for the effect of the fault on the system. The fault estimation signal is used with the nominal controller to ensure the tracking trajectory performance of the faulty system to the reference. Hence, the FA control law is expressed as

u_{f}^{★} = u_{f} - \hat{f}

, where the first part of the equation is the input with actuator fault and the second part is the additive fault estimation to be added in order to accommodate the fault; see Figure 7.

3. Results

In this section, simulation implementation of the proposed fault-tolerant strategy is presented and validated in Matlab/Simulink 2023B (see Figure 8). Table 1 shows the parameters of the DFIG considered in this paper. Recalling from Section 2, the velocity considered was 4200 rpm, with an active power of 100 W and a reactive power of 50

V A R

The PI controller gains are

K_{p} = 0.41

and

K_{i} = 68.54

. The sliding-mode parameters are established as

γ = 0.001, K_{s} = 0.008, b = 0.08

and

ϵ = 100

. The synthesis of the pole placement state-feedback controller and the observer gains have been solved with Yalmip Toolbox [37]. The controller gain is computed by solving Corollary 1 with

α = 3

to avoid slow dynamics. Thus, the following controller gain is obtained:

{\bar{K}}_{c} = [\begin{matrix} 1.11 & 0 & - 237.18 & 0 \\ 0 & 1.11 & 0 & - 237.18 \end{matrix}] .

The observer gain is computed by solving Theorem 2, which relies on:

{\overset{˘}{L}}_{P I} = {[\begin{matrix} 1.87 & 0.01 & 4.78 & 0.02 \\ - 0.01 & 1.87 & 0.02 & 4.78 \end{matrix}]}^{⊤} .

For testing the effectiveness of the proposed scheme, two scenarios were considered.

3.1. Scenario 1—Fault-Free

Figure 9 and Figure 10 compare the PI, the sliding-mode and the nominal state-feedback controller applied to the RSC system in a fault-free case. It is easy to see that the three controllers fulfill their objective of regulating the stator and rotor currents. However, the nominal SF controller and the SMC provide fewer tracking errors between the output variables and the references (see Figure 9). The control law signal applied to the RSC system is presented in Figure 10. This figure shows that the sliding-mode control produces a chattering signal. This is a disadvantage compared to the feedback controller, as chattering can lead to wear on the actuators.

3.2. Scenario 2—Non-Constant Actuator Faults

In the second scenario non-constant additive faults are considered, applied as (40) (see Figure 11). In addition, as seen at the top Figure 11, a smaller error is achieved between the real fault and the estimate by using the PI observer. The bottom of Figure 11 shows the actuator fault detection with the alarm indicator.

\begin{matrix} f_{1} & = \{\begin{matrix} 0, & t < 0.032 s, \\ \sin (150 t), & 0.032 s \leq t \leq 0.1 s, \\ 78 t - 10.14, & 0.13 s \leq t \leq 0.18 s, \\ 0, & t \geq 0.032 s . \end{matrix} \\ f_{2} & = \{\begin{matrix} 0, & t < 0.032 s, \\ 60 t - 7.8, & 0.032 s \leq t \leq 0.1 s, \\ 1.6 \sin (150 t) + 1.6, & 0.13 s \leq t \leq 0.18 s, \\ 0, & t \geq 0.032 s . \end{matrix} \end{matrix}

(40)

These non-constant faults are applied to the RSC system in order to evaluate the performance of the nominal PI, the nominal SMC and the state-feedback controller adding the proposed FTC scheme. From Figure 12 and Figure 13, a comparison between the faulty RSC system with proportional–integral controller, with the sliding-mode controller, and with the state-feedback controller adding the FTC strategy shows that the proposed scheme is superior to regulating the stator and rotor currents, even with the presence of non-constant actuator faults.

As displayed in Figure 12, the proposed strategy significantly reduces the error between the references and the stator and rotor currents. The actuator fault detection is achieved after approximately 0.0016 s of its occurrence, and consequently, the fault accommodation is generated to compensate for the actuator faults. This reduction in error between the references allows for the achievement of the proposed active and reactive power in the RSC. In addition, Figure 12 shows that the nominal PI controller has bigger error between the desired and the real currents.

4. Discussion

Although the proposed controller and the sliding-mode controller are capable of regulating the faulty RSC system, the proposed FTC strategy can stabilize the system with less error than the SMC, as revealed in Figure 12 and Figure 14, where

| e_{d r} | = | P_{s}^{*} - i_{d r} |

, and

| e_{q r} | = | Q_{s}^{*} - i_{q r} |

. To compare the controller performances, the Integral of Absolute Error (IAE) and the Integral Time Absolute Error (ITAE) are obtained for the tracking error. Table 2 shows that the minimum values of the IAE and the ITAE are obtained for the state-feedback controller with the proposed fault-tolerant control strategy. These results show that the SF controller presents better performance for regulating the stator and rotor currents, even with the presence of non-constant actuator faults.

Figure 13 highlights the performance differences among the PI, SMC, and state-feedback with FTC controllers under non-constant actuator fault conditions. It can be observed that the state-feedback controller with FTC exhibits a rapid and stable response, maintaining control stability despite fault occurrences. In contrast, the SMC controller generates significant chattering, which may negatively impact the actuator’s lifespan and system stability over time. The PI controller, while stable, responds slower than the proposed state-feedback with FTC. Overall, the proposed fault-tolerant controller based on the state-feedback effectively mitigates the effects of faults, showcasing superior performance compared to the other controllers. The proposed controller can promptly command actuators to operate as expected, faster than the PI and the sliding mode controller, effectively reducing non-constant actuator faults.

The results show that the PI and sliding mode controllers are trying to compensate for the faults, but their performance is degraded due to the effect of the actuator fault. Even though the control regulation is robust against faults, these two controllers do not use the fault estimation system, so they cannot bring information about when the fault occurs or the dynamics of the fault. On the other hand, the proposed FTC scheme comprises the fault estimation system and the state-feedback controller. The fault estimation system is designed separately for the controller by considering the separation principle. The proposed scheme is robust against faults and can generate information about the time of the fault and the dynamics of the fault. With these data, fault compensation is achieved.

5. Conclusions

A fault-tolerant control strategy with a fault estimation and compensation for an RSC system of a WECS was proposed in this paper. Three controllers were designed to evaluate the system’s performance under actuator faults: PI, SMC and pole placement state-feedback controller with the proposed FTC system. Fault detection was performed by comparing the fault estimation signal with a predefined threshold. Upon occurrence of actuator faults, the FE signal generates a new control law. The actuator faults are accommodated, and then the performance of the system is improved. The effectiveness and performance of the proposed method and its feasibility in designing fault-tolerant control, fault detection, and identification have been demonstrated in simulation. The results indicate that the SF controller significantly reduces the error between the references and the stator and rotor currents. Additionally, the performance indexes IAE and ITAE applied to the PI, SMC and SF controller are obtained. These results highlight that the SF controller outperforms others in regulating the currents of the RSC system, even in the presence of non-constant actuator faults. Future work will consider the design of sensor fault diagnosis and sensor FTC for the RSC system.

Future work will focus on extending the proposed FTC scheme to address other fault types, such as sensor and electrical faults. It will also explore its application to different system configurations, including various wind turbine models and hybrid energy systems. Key challenges will involve managing computational complexity, ensuring scalability, and enhancing robustness to uncertainties. Additionally, the integration of machine learning and advanced sensor technologies will be explored to improve fault prediction and diagnosis, thereby contributing to more reliable and efficient wind energy systems.

Author Contributions

Conceptualization, A.R.L. and G.O.-T.; methodology, J.Y.R.-M.; software, G.O.-T.; validation, J.Y.R.-M. and J.C.R.-C.; formal analysis, A.R.L. and G.V.; investigation, A.R.L. and G.O.-T.; resources, J.E.V.-R.; writing—original draft preparation, A.R.L.; writing—review and editing, J.C.R.-C. and G.V.; visualization, G.O.-T.; supervision, J.E.V.-R.; project administration, A.R.L. and G.O.-T.; funding acquisition, J.E.V.-R. All authors have read and agreed to the published version of the manuscript.

Funding

This research was funded by Consejo Nacional de Humanidades, Ciencias y Tecnologías, under grant CF-2023-G-1344. Desarrollo de nuevos enfoques para análisis, modelado y control de convertidores electrónicos.

Institutional Review Board Statement

Not applicable.

Informed Consent Statement

Not applicable.

Data Availability Statement

The data presented in this study are available on request from the corresponding author.

Conflicts of Interest

The authors declare no conflicts of interest.

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Figure 1. Stages of a wind energy conversion system.

Figure 2. WECS based on DFIG and back-to-back converter.

Figure 3. Vector diagram of the DFIG variables oriented to the stator flux.

Figure 4. RSC PI control scheme.

Figure 5. RSC sliding-mode control scheme.

Figure 6. RSC state-feedback control scheme.

Figure 7. FTC scheme applied to the RSC system, with (1) the RSC system, (2) the nominal controller, and (3) the active fault-tolerant control system.

Figure 8. Simulation model of the FTC scheme with the state-feedback controller, developed in Matlab/Simulink 2023B, with (1) the RSC system, (2) the nominal controller, and (3) the active fault-tolerant control system.

Figure 9. Scenario 1—comparison between PI, SMC and nominal state-feedback controller.

Figure 10. Scenario 1—input signal comparison between PI, SMC and nominal state-feedback controller.

Figure 11. Scenario 2—additive fault estimation.

Figure 12. Scenario 2—comparison between PI, SMC and state-feedback controller with FTC.

Figure 13. Scenario 2—input signal comparison between PI, SMC and state-feedback with FTC.

Figure 14. Scenario 2—comparison of absolute tracking error between the SMC and the state-feedback with FTC.

Table 1. DFIG parameters.

Parameter	Symbol	Value	Unit
Rated DFIG Power	P	372	W
Number of poles	p	2	-
Stator voltage (60 Hz)	-	$34.29$	V
Rotor resistance	$R_{r}$	$0.312$	$Ω$
Stator resistance	$R_{s}$	$0.343$	$Ω$
Rotor inductance	$L l_{r}$	$1.198$	mH
Stator inductance	$L l_{s}$	$1.198$	mH
Mutual inductance	$L m_{s}$	$38.62$	mH

Table 2. Performance indexes of the PI, SMC and state-feedback controller with FTC applied to the faulty RSC system.

Error	IAE			ITAE
Error	PI	SMC	SF with FTC	PI	SMC	SF with FTC
$e_{d r} = P_{s}^{*} - i_{d r}$	0.2964	0.004743	0.000890	0.02281	0.000642	8.411 × $10^{- 6}$
$e_{q r} = Q_{s}^{*} - i_{q r}$	0.1514	0.004394	0.000573	0.02045	0.000557	7.147 × $10^{- 6}$

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Lopez, A.R.; Rumbo-Morales, J.Y.; Ortiz-Torres, G.; Valdez-Resendiz, J.E.; Vazquez, G.; Rosas-Caro, J.C. Fault-Tolerant Control Implemented for Sustainable Active and Reactive Regulation of a Wind Energy Generation System. Sustainability 2024, 16, 10875. https://doi.org/10.3390/su162410875

AMA Style

Lopez AR, Rumbo-Morales JY, Ortiz-Torres G, Valdez-Resendiz JE, Vazquez G, Rosas-Caro JC. Fault-Tolerant Control Implemented for Sustainable Active and Reactive Regulation of a Wind Energy Generation System. Sustainability. 2024; 16(24):10875. https://doi.org/10.3390/su162410875

Chicago/Turabian Style

Lopez, Adolfo R., Jesse Y. Rumbo-Morales, Gerardo Ortiz-Torres, Jesus E. Valdez-Resendiz, Gerardo Vazquez, and Julio C. Rosas-Caro. 2024. "Fault-Tolerant Control Implemented for Sustainable Active and Reactive Regulation of a Wind Energy Generation System" Sustainability 16, no. 24: 10875. https://doi.org/10.3390/su162410875

APA Style

Lopez, A. R., Rumbo-Morales, J. Y., Ortiz-Torres, G., Valdez-Resendiz, J. E., Vazquez, G., & Rosas-Caro, J. C. (2024). Fault-Tolerant Control Implemented for Sustainable Active and Reactive Regulation of a Wind Energy Generation System. Sustainability, 16(24), 10875. https://doi.org/10.3390/su162410875

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Fault-Tolerant Control Implemented for Sustainable Active and Reactive Regulation of a Wind Energy Generation System

Abstract

1. Introduction

1.1. Case Study

1.2. Literature Review

2. Materials and Methods

2.1. Rotor-Side Converter Control Model

2.2. Rotor-Side Converter Controller Design

2.2.1. PI Controller

2.2.2. Sliding-Mode Controller

2.2.3. Nominal State-Feedback Controller

2.3. Active Fault-Tolerant Control Design

2.3.1. Actuator Fault Estimation System

2.3.2. Actuator Fault Detection System

2.3.3. Fault Accommodation Control Law

3. Results

3.1. Scenario 1—Fault-Free

3.2. Scenario 2—Non-Constant Actuator Faults

4. Discussion

5. Conclusions

Author Contributions

Funding

Institutional Review Board Statement

Informed Consent Statement

Data Availability Statement

Conflicts of Interest

References

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