Sensor fault reconstruction for wind turbine benchmark model using a modified sliding mode observer

ABSTRACT


INTRODUCTION
In the last decade, the most rapidly expanding renewable energy sources are wind turbines, supplied by an entirely arbitrary wind speed, operating in uncertain environments, nonlinear dynamics, and exposure to considerable disturbances are key properties of these systems [1], [2]. Several recent studies have been conducted to improve the power coefficient of wind turbines as well as electricity production, blade profile and augmentation strategies using optimization approaches based on artificial intelligence (AI) have been addressed [3], [4]. Notwithstanding the implementation of advanced technology in modern wind turbines, the maintenance of these turbines is still long and costly, which influences their electricity production [5].
Odgaard et al. presented two wind turbine benchmark models (WTBMs) in 2009 [6] and 2013 [7], the second model is more realistic. Several research papers have been published in fault detection and isolation (FDI), and fault tolerant control (FTC) based on these WTBMs [8]- [11]. The main objective of an FDI system is to raise an alarm when an abnormal operation occurs in the monitored process and to locate its  Taouil) 5067 source. A widely studied methodology is the observer-based approach, which analyzes the residuals that represent the difference between the actual and observer outputs of the monitored system [12], [13]. In this paper, an observer class known as the sliding mode observer [14] is adopted. This observer class aims to reconstruct the fault instead of examining the residual. An unknown input proportional integral observer for decoupling the unknown input was established in Sun [15], and an optimization of it is introduced to lessen the effects of sensor noise, the actuator faults of the WTBM were also estimated using the suggested observer. A Kalman-like observer and support vector machines-based FDI system were proposed in [16], [17]. Shi and Patton [18] proposed an observer based active fault tolerant control (AFTC) approach. By modeling the wind turbine as a linear parameter varying (LPV) model using linear matrix inequality linear matrix inequality (LMI), to evaluate the system states and faults, an extended state observer was established. To examine some faults in wind turbines, a deep learning fault detection and classification method based on the time series analysis method and convolutional neural networks (CNN) is provided [19]. Changes in wind turbine blade vibration responses WTB can be used to detect the presence of damage. Xu et al. [20] introduced a probabilistic analysis approach for wind turbine damage detection.
Sliding mode observers sliding mode observers (SMOs) are characterized by robustness to disturbances and modeling uncertainties as well as their ability to estimate unknown inputs. SMOs have been widely used for FDI [14], [21]- [24]. Using a Takagi-Sugeno SMO, the actuator parameter faults in the WTBM were only partially identified and reconstructed [25], the reconstruction's accuracy needs to be improved because the method lacks robustness regarding model uncertainty. SMO is used by Rahnavard et al. [26], [27] to address the fault detection (FD) of sensors and actuators in the WTBM.
In contrast to WTBM-based approaches that merely detect and isolate the fault without providing any information on its magnitude, this paper proposes an FDI scheme that, in addition to faithfully reconstructing the fault, provides the exact magnitude, making it exploitable in FTC schemes that require knowledge of the fault magnitude. A modified fault estimation scheme based on the SMO is presented in this paper, particularly to detect, isolate, and estimate the sensor faults of the WTBM. The proposed modification is related to the discontinuous switching term of the observer, which allows an accurate reconstruction, especially in the case of simultaneous faults. The aerodynamic torque is considered as an input, and the MATLAB/Simulink environment is used to implement the simulations. The paper is structured as: section 2 briefly describes the WTBM, section 3 presents the fault estimation scheme along with modifications to the SMO and numerical values of its parameters, section 4 addresses fault scenarios and simulation results, and section 5 concludes the paper.

WIND TURBINE MODEL
The model considered is similar to the one studied in [7], it is a horizontal axis wind turbine with three blades, Figure 1 depicts a system overview of this system. This benchmark model contains the following subsystems: blade and pitch system, drive train, converter, and generator, the wind turbine's aerodynamic characteristics are strongly dependent on the blade pitch angles, the rotor speed, and the wind speed, which is the driving force of the wind power system. The resulting aerodynamic torque is transmitted from the rotor to the generator via the drive train, and at the output, the electrical energy is obtained from the converter. Depending on the different operating requirements, a controller is set up to control the blade pitch angles and the generator torque [28].

Blade and pitch subsystem
This block contains the aerodynamic model, blades, and pitch system. The aerodynamic torque is given by the relation: ( , ) is the torque coefficient, the profile used for ( , ) is shown in Figure 2, is a nonlinear function as a function of the pitch angle and on tip speed ratio , therefore is a nonlinear function of , and the wind speed .
The pitch system is a hydraulic system consisting of three identical actuators, each with an internal controller, the actuator i adjusts the pitch angle βi(i=1, 2, 3) of the blades by rotating them. This subsystem can be represented by the following second order transfer function: where denote the pitch angle, , denote the reference to the pitch angle, denote the natural frequency of the pitch actuator model [rad/s], and denote the damping ratio of the pitch actuator model. All , all and all are equal in free fault, otherwise are different. In the following only one pitch actuator is considered.

Drive train subsystem
The drive train allows to transfer the aerodynamic torque to the generator to ensure a high speed of rotation required by the generator, the model is built of a slow shaft and a fast shaft linked by a multiplier (the gearbox). This subsystem is modeled by (4)- (6): θ̇= ω r − 1 N g ω g . Table 1 gathers the meaning of the parameters evoked in the (4)-(6).

Generator and converter subsystem
In this subsystem we have a mechanical to electrical conversion, the converter and the generator are modeled by a transfer function of 1 st order: where is the generator torque, is the reference generator torque, and 1 α gc ⁄ is the first order system's time constant. The power available at the generator output is given by (8): denotes the generator's efficiency. By integrating the subsystems described above, the wind system is modeled in the state space as (9), (10): where x = [ ω g ω r θ β̇β τ g ] T is the state vector, u = [ τ gref τ r β r] T denote the control input vector,  3. FAULT ESTIMATION SCHEME 3.1. Sliding mode observer design Consider the system of (11) and (12) which describes a nominal linear system vulnerable to sensor and actuator faults: where, A ∈ ℝ n×n , B ∈ ℝ n×m , C ∈ ℝ p×n , D ∈ ℝ n×q , with p≥q and the matrices B, C and D are full rank. The functions f act and f sen present respectively an actuator fault and a sensor fault, f act and f sen are bounded. A priori, only the u(t) and y(t) signals are provided, and it is assumed that the system's state is unknown. The objective is to synthesize an observer that allows for an estimated state vector x and an estimated output vector ŷ, such that the output error: ε y (t) = ŷ(t) − y(t) tends to zero in a finite time when the sliding mode is attained, even in the presence of faults. It is shown in [14] that a change in coordinates exists x → T. x such that in the new coordinate system, the previous system is written as (13)-(15): y = x 2 .
where 11 is stable The coordinate system above will be used as a platform for the design of a SMO. The system of (11) and (12) in f act undergoes two transformations, the first one with the matrix T and the second one with T * so T = T * . T, T and T * can be calculated by (16) and (17), more details can be found in [14], [21].
Finally, the resulting structure is: where the gains G I and G n are calculated as (19): where is a positive scalar greater than the norm of the function that represents the fault. Ultimately, the sensorand actuator faults that have been reconstructed can be roughly estimated [16], [29] by (21) The matrix D is chosen such that: D=B.

Modifications for the observer
The outputs of the wind turbine have very different orders of magnitude 10 2 / , 1 / , 1 , and 10 4 . respectively for , , , the reconstructed sensor and actuator faults are given by (21) and (22). These two relations shows that the reconstructed faults are highly dependent on the scalar gain, whose value is roughly equal to the fault magnitude's maximum value. The choice of κ changes according to the considered output and its value must be chosen with precision by the designer. Therefore, in the classical SMO structure, the parameter is taken as fixed, which introduces a limitation for the fault reconstruction.
In f act , choosing a fixed κ does not allow for precise reconstruction of the faults of all the outputs; thus, it is necessary to redefine κ to adapt to each output, which is impractical and a priori inaccessible in the case where multiple faults affect multiple outputs at the same time. To remedy this problem, a modification to the parameter κ is proposed to be replaced by: ‖ 2 ‖ ‖ 2 ε y ‖ and the switching term become:  (24) where α is a scalar is taken equal to 1 450 and δ is a small scalar.

Observer design
The main source of the wind system's energy is the aerodynamic torque, which is obtained from relation (1) and represents the 2 nd input for (9), the 1 st and 3 rd inputs are provided by the wind turbine controller. The technical specifications and parameter numerical values of the wind turbine simulated in this paper are given in Odgaard et al. [7]. The proposed observer has the structure (18), considering the modification of the switching term (24) and putting the system of (11) and (12) in the canonical forms (13), (14), and (15). Using an algorithm similar to the one described in [21], the obtained state space matrices: .
The design parameters of the observer are taken as: , and = 1 −6 The gains obtained:

Wind input
The wind speed profile adopted in the simulation shown in Figure 3 is highly random and issued from a wind park's real wind speed measurement [7]. The wind speed considered covers a range of 3-18 m/s, which represents a good coverage of the normal operation of a wind turbine. Depending on the wind speed, in the interval (3, 12.5 m/s) the power generated by the wind turbine will be optimized, the wind turbine will be controlled to maintain a constant energy production in (12.5, 25 m/s), and if the speed exceeds 25 m/s the wind turbine will be parked in order to avoid any damage.

Sensor fault reconstruction
The objective is to reconstruct sensor faults, f act (t) is considered null in (11), four sensors are implemented to measure generator speed ( ), rotor speed ( ), blade pitch angle ( ), and generator torque ( ), for each sensor, a fault is proposed, in the last case, the faults are considered simultaneously, the magnitude of the fault is chosen in a logical way according to the amplitude of the variable considered, and it is noted that the fault profile is a priori unknown by the system. The sensor faults simulated in this paper are the following: Case 1: Figure 4 (fault case 1) shows the real and estimated faults of the generator torque sensor, which starts at 100 s and ends at 255 s; this fault realizes a constant amplitude bias, the maximum real fault amplitude is 400 N.m, this value represents 10% of the maximum value of the generator torque in the considered interval, the obtained result shows that the reconstructed fault follows faithfully and accurately the real fault.
Case 2: The generator speed sensor fault is reconstructed. It runs from 400 to 600 s as shown in Figure 4 (fault case 2). It is an intermittent fault that realizes a constant amplitude bias with maximum amplitude of 25 rad/s, which represents 32% of the maximum value of the generator speed in the considered interval. The results of the simulation show that the fault is reconstructed accurately with a relative gap that does not exceed 2.8%. However, At the moment of a sudden change in the real signal, an overflow is observed for the reconstructed signal. It is also noted that when a fault occurs for the rotor speed sensor, a perturbation appears in the signal reconstructed for the generator speed sensor fault, and vice versa, due to the fact that the two quantities are coupled by (5) and (6).
Case 3: The rotor speed sensor fault is also simulated. It starts at 1,400 s and ends at 1,600 s as shown in Figure 4 (fault case 3), and the fault amplitude is 0.2 rad/s, which is 20% of the maximum measured rotor speed value. A part from the overshoot at the time of state change, and the disturbance in the interval (400, 600 s) due to the fault in the rotor speed sensor which occurs in this interval, the rotor speed sensor fault is reconstructed with good accuracy, the relative gap is: 2.5%.
Case 4: The fault profile of the pitch position sensor is shown in Figure 4 (fault case 4). Starting at 1,100 s and ending at 1,200 s, with a maximum amplitude of 0.8°. The real signal and their reconstructed values are perfectly confused.
Case 5: In this case, the sensor faults are considered simultaneously. Figure 5 illustrates this situation: the generator speed sensor and rotor speed sensor faults are considered between 400 and 600 s, the generator torque sensor fault starts at 510 s and ends at 680 s, and the pitch position sensor fault is considered between 520 and 650 s. The simulation results show that the generator torque sensor and pitch position sensor faults are reconstructed accurately. The overshoot is still observed for the generator speed sensor and rotor speed sensor faults; however, the relative gaps for these two faults are respectively evaluated at 0.08% and 0.07%, which shows that the reconstruction accuracy is higher compared to the case of the faults considered separately. This allows us to conclude that the effect of the modification brought to the observer, already evoked in paragraph 3.2, is more significant in the case of simultaneous faults.

CONCLUSION
The wind turbine faults affecting the generator torque sensor, generator speed sensor, rotor speed sensor, and pitch angle sensor result in non-optimal operation of the wind turbine system. In this paper, a sliding mode observer with a modified switching term to fit the different magnitudes of the sensor faults is implemented in order to reconstruct in real time the aforementioned faults. To validate the proposed modification, five sensor fault scenarios are proposed. These scenarios illustrate two situations: in the first one, the faults are considered individually, and in the second one, the faults are considered simultaneously. Throughout the simulation, the switching term is taken to be of the same value even though the sensor faults have quite different magnitudes. The results of the simulation show that the faults are detected, isolated, and reconstructed with precision in the situation of the faults considered individually, except for the overshoot observed for the generator speed and rotor speed sensor faults, which disappears in a very short time. In the case of the simultaneous faults (case 5), the reconstruction is more precise and done without changing the parameters of the SMO, which justifies the modification of the switching term proposed for this SMO.