- Email: [email protected]
Finite time estimation of actuator faults, states, and aerodynamic load of a realistic wind turbine
Accepted Manuscript Finite time estimation of actuator faults, states, and aerodynamic load of a realistic wind turbine Mostafa Rahnavard, Moosa Ayati, Mohammad Reza Hairi Yazdi, Mohammad Mousavi PII:
S0960-1481(18)30698-0
DOI:
10.1016/j.renene.2018.06.053
Reference:
RENE 10209
To appear in:
Renewable Energy
Received Date: 7 November 2017 Revised Date:
12 June 2018
Accepted Date: 13 June 2018
Please cite this article as: Rahnavard M, Ayati M, Hairi Yazdi MR, Mousavi M, Finite time estimation of actuator faults, states, and aerodynamic load of a realistic wind turbine, Renewable Energy (2018), doi: 10.1016/j.renene.2018.06.053. This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
ACCEPTED MANUSCRIPT
Finite Time Estimation of Actuator Faults, States, and Aerodynamic Load of a Realistic
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Wind Turbine
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Mostafa Rahnavard, Moosa Ayati*, Mohammad Reza Hairi Yazdi, Mohammad Mousavi
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School of Mechanical Engineering, Collage of Engineering, University of Tehran, North Kargar Street, Tehran, Iran
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*Corresponding author: [email protected], Tel.: +982161119920, P.O.B.: 11155-4563
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Abstract:
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This paper provides finite time estimation of wind turbine actuator faults and unknown aerodynamic load. Furthermore, finite-time
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state estimation of drivetrain, generator, and pitch subsystems are addressed in the contrary of asymptotic state/fault estimation in
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previous works. A realistic wind turbine model, incorporating the aero-elastic FAST simulator, is considered as the simulation
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example. Generally, aerodynamic load is not measurable in real applications due to instrument limitations, then, it is considered as an
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unknown input in this study. A novel terminal sliding mode observer is introduced for finite-time estimation of generator/convertor
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states, faults, and unknown aerodynamic load. Pitch actuator hydraulic pressure drop is modelled as an additive fault, by introducing a
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fault indicator. Then, two cascaded sliding mode observers are exploited for each pitch subsystem, to provide finite time state and
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fault reconstructions. Sufficient number of design parameters helps to achieve desired accuracy and convergence time. Finally,
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simulation results authenticate finite time estimation of wind turbine states and simultaneous actuator faults.
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Keywords: Fault detection and isolation, Fault reconstruction, Terminal sliding mode observer, Lyapunov stability theorem, Wind
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turbine.
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1- Introduction
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Nowadays, wind turbines (WTs) are the most growing renewable energy generators which contribute to the world power production
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in large scale as shown in Figure 1. Meanwhile, there is a strong demand on enhancing the reliability and efficiency as well as
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reducing the operation and maintenance costs [1].
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Figure 1: Total installed capacity of wind turbines during 1997-2020 [GW] [1].
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Recently, two wind turbine benchmark models (WTBMs) have been presented by Odgaard et al. in 2009 [2] and Odgaard and
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Johnson in 2013 [3]. Benchmark-2013 incorporates the aero-elastic simulator FAST and provides a higher fidelity and more realistic
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wind turbine model. Thus, it requires more sophisticated fault detection schemes. Many researches have been published with the scope
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of fault detection and isolation (FDI), fault reconstruction and estimation (FRE), and fault tolerant control (FTC) based on these
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WTBMs.
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Shi and Patton [4] have proposed an observer based active FTC approach. By modelling the wind turbine as a linear parameter
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varying (LPV) model, an extended state observer was designed to estimate system states and faults. Simani et al. [5] presented a FDI
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scheme based on identified fuzzy models. Also, fuzzy theory is exploited to provide an uncertain model of wind turbine and to
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manipulate noisy data and Takagi-Sugeno prototypes represent the residual generators for FDI purposes. In [6] a FTC scheme has
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been proposed for regulation of generator torque when generator/convertor actuator faults occur. The scheme includes two
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approaches. The first approach relies on fuzzy model reference adaptive control, while the second approach exploits identification
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method and fuzzy modeling. An adaptive actuator fault estimation scheme based on nonlinear geometric approach was proposed by
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Simani and Castaldi [7]. Wu et al. [8] presented a FTC method using virtual actuators for the problem of pitch actuator failures.
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Nonlinear wind turbine model is converted to a linear parameter varying model and FTC structure of virtual actuators is obtained by
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applying a state feedback law.
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Observers and filters have been extensively investigated during recent decades to address state/fault estimation and control of
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dynamical systems [9–15]. Sliding mode observers (SMOs) have been widely used for FRE purposes due to their robustness to
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disturbances and modelling uncertainties and their capability of estimating unknown inputs [16–19]. Some researchers have exploited
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SMOs for FRE and FTC of WTBMs [12,13,20–22]. Lan et. al [12] proposed a FTC scheme to compensate the pitch actuator faults.
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The proposed scheme incorporates a traditional Proportional-Integral (PI) controller to achieve nominal pitch performance, along with
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a fault compensator to eliminate the actuator fault effects. The pitch subsystem states and fault index are estimated using an adaptive
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SMO. In [20] a Takagi-Sugeno sliding mode observer with weighted switching action was implemented to partially reconstruct the
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parameters’ faults in hydraulic pitch actuator and generator subsystems. The method lacks robustness to model uncertainties and
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significant fault estimation errors are resulted. In [21], sensor and actuator faults were reconstructed using the classic SMO.
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Estimation of faults in pitch angle sensors and actuators is obtained using physical redundancy of pitch sensors. However, the method
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is not robust to model uncertainties and disturbances. Rahnavard et al. [13] proposed a robust FRE scheme based on modified SMO, to
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address reconstruction of actuator and sensor faults of WTBM presented in [2]. Pitch actuator low pressure faults and several
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simultaneous sensor faults, with different orders of magnitudes, are investigated.
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Classic SMOs guarantee finite time convergence of output estimation errors, while non-output errors converge to zero
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asymptotically. This implies that the states which do not directly participate in the output converge to actual values as time goes to
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infinity. Terminal sliding mode (TSM) concepts enable finite time convergence of both measured and unmeasured state estimation
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errors [22–25]. This is realized by injecting a non-smooth discontinuous switching term, including fractional powers, to the classic
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observer. A terminal sliding mode observer (TSMO) for a certain class of nonlinear systems was proposed in [23]. The system under
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consideration has n-degree-of-freedom, where position and velocity are measured and unmeasured states, respectively. This leads to a
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special state-space representation. A discontinuous term by use of fractional powers is injected into the classic observer, which
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guarantees finite time convergence of non-output estimation errors. 2
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In aforementioned SMO-based FRE and FTC papers of wind turbines, asymptotic estimation of states and faults are addressed. This
2
paper aims at finite time estimation of states and actuator faults for a realistic WT model. The proposed fault diagnosis scheme is
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evaluated on 2013 wind turbine benchmark [3] which is a more realistic WT model rather than 2009 benchmark. The aerodynamic torque is the main excitation source of WTs and is not usually measured in reality. Since, there is not any sensor
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to report aerodynamic torque accurately; it is regarded as an unknown input in 2013 benchmark model. Generator and convertor faults
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cause serious problems and lead to non-optimal power production of WT. These faults are commonly dynamical changes or offset in
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generator torque [2,3]. The proposed TSMO reconstructs generator common faults and aerodynamic torque in finite time as long as
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the states of drivetrain and generator/convertor subsystems.
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The pitch actuator faults are due to pressure drop in hydraulic system or increase of air content in oil. These faults change pitch
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actuators’ dynamics and reduce the speed of control actions. Furthermore, in the case of abrupt pressure drop, a safe and fast shut
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down of wind turbine is required [3,12]. Estimation of hydraulic pitch actuator faults are addressed in some researches, specially
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[4,8,12,26]. In [26], pitch actuator fault is partially reconstructed, however, the estimation accuracy requires improvement. In [4], the
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considered fault is high air content in oil and the pitch fault signal is reconstructed with considerable error. Comparing the error
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amplitude, [12] and [8] provide a better fault estimation rather than [26] and [4]. In this paper, two cascade SMOs are designed for
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each pitch subsystem to reconstruct actuator low pressure and abrupt pressure drop faults. The cascade observers provide simultaneous
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finite time estimation of states and actuator faults of three pitch subsystems by enhanced accuracy.
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The proposed fault diagnosis schemes are evaluated on benchmark-2013 WT and can be implemented in FTC methods, where fast
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and exact information about the fault magnitude is required. These schemes can also be utilized in benchmark-2009 WT model or any
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wind turbine whose subsystems are modelled similar to those of benchmark-2013.
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This paper is structured as follows. Section 2 briefly describes the enhanced WTBM. Theory of finite time fault reconstruction is
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presented in Section 3. First, a TSMO is proposed to reconstruct the generator and convertor faults as well as unknown aerodynamic
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torque. Then, two SMOs are exploited in a cascade form to reconstruct the pitch actuator low-pressure faults. Simulation results are
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vector and |. | refers to the absolute value of a scalar.
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2- Wind Turbine Description
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reported in Section 4. Finally, Section 5 concludes the paper. Throughout the paper, the symbol ‖. ‖ denotes the Euclidean norm of a
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Main components of a utility-scale wind turbine are depicted in Figure 2. The wind turbine model of this paper is a 5 MW offshore
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wind turbine benchmark proposed by Odgaard and Johnson [3]. The model is a variable speed three-bladed horizontal axis wind
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turbine with full span blade pitch control and generator torque regulation. FAST (Fatigue, Aerodynamics, Structures, and Turbulence)
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is an aero-elastic wind turbine simulator designed by NREL National Wind Technology Center, which is widely used for studying
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wind turbine control systems [3]. The benchmark incorporates FAST software [27] to model the turbine with 24 degrees of freedom.
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Figure 3 describes how the proposed fault estimation scheme is implemented in FAST & MATLAB Simulink environment. The
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benchmark’s main subsystems are described as follows.
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Figure 2: Main components of a utility-scale wind turbine [28].
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Figure 3: Fault diagnosis scheme block diagram.
The hydraulic pitch actuator is a piston servo system modelled as (1), which is a second-order closed loop transfer function from
(the pitch reference angle) to pitch angle .
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( ) = ( ) + 2. . +
(1)
and are the damping factor and natural frequency, respectively. A transfer function is associated to each of pitch subsystems,
which are same in fault-free conditions. In the fault-free condition, the parameters are = 0.6 and = 11.11 / . The pitch
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actuator faults are modelled by changing the parameters and in the transfer function of (1) during the faults interval. The
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On the system level of wind turbine, the generator and convertor dynamics are modelled as a first-order transfer function of (2).
9
parameters associated with the hydraulic abrupt pressure drop are = 0.45 and = 5.73 / .
4
ACCEPTED MANUSCRIPT ! ( )
2
"! ,
=
,
and
(2) !
are the generator & converter constant, generator command torque, and generator torque, respectively. It is
concluded from [29] that, FAST models the drivetrain subsystem by a two-mass spring damper model as follows. # $ (%) =
&' (%) −
)*+ ,(%) − -*+ . (%) −
! (%) )*+ -*+ ,(%) + . (%) − 0− / / / 1 ,$(%) = (%) − ! (%) / #! $ ! (%) =
3
"! + "!
&'
! (%) 0 /
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( )
! (%)
(3) (4) (5)
is the aerodynamic torque acting on blades, is rotor angular speed, ! is generator angular speed, , is twisting (torsion)
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TurbSim software [30]. Wind speed inputs with mean value of 11, 14, 17 m/s are generated by TurbSim using IEC von Karman
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turbulence model and are provided for use in the benchmark model.
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3- Sliding Mode Observers for Wind Turbine Finite Time Fault Reconstruction
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5
angle of drive train, # is rotor moment of inertia, #! is generator moment of inertia, / is gearbox ratio, )*+ is torsion stiffness, and -*+
is torsion damping of drive train. The wind speed utilized in this benchmark is a full-field turbulent wind speed profile generated by
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3-1 Generator/convertor Fault Reconstruction and Aerodynamic Torque Estimation
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torque of 2013 benchmark model in a finite time. Augmenting the generator model of (2) and the drivetrain model of (3)-(5), leads to
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a system of the form:
13
5(%) = 61(%)
where, 1(%) = 7,, ! , ,
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:
, 3(%) =
, (%)
=
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1$ (%) = 21(%) + -3(%) + 4(%)
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In this section, an observer is designed using terminal sliding mode concepts to reconstruct the generator faults and aerodynamic
&' /# ,
(6)
and the system matrices are:
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disturbance, affecting system of (6). The disturbance signal is estimated using a terminal observer proposed in the following.
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E 0 D GHI : - = ;0, 0, 0, "! < , 6 = =0>×@ , A> B, 4 = =0, 0, 1, 0B: , 2 = D FJK D− GHI D JN C 0
The aerodynamic torque
&'
−
−
@
F LHI
1
LHI
FM JK LHI
FJK LHI
−
FJN
0
JN
0
0 Q @ − P JK P 0 P P −"! O
is unknown in the benchmark model. Thus, the term (%) =
&' /#
in (6) is considered as a
The generator and convertor system may experience two kinds of faults: the change of dynamics due to a change in generator
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parameter "! or an offset on the convertor torque. These faults are originated from internal problems, such as faults in convertor
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change of dynamics results in a slow torque control. This is expressed as a fault with high severity and fast development time in the
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electronics or an offset on convertor torque estimation [3]. Generator and convertor faults cause serious problems. Particularly, the
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benchmark model. Offset fault results in non-optimal power production of WT and is denoted as a fault with medium severity and fast
2
time of development. Without loss of generality, the aforementioned faults affecting the generator actuation are modelled as:
4
= −"!
!
+ "!
!,
+ RSK
(7)
where, RSK represents a general fault signal. It can be converted to generator faults, including change of actuation dynamics and torque :
offset. Defining an augmented fault vector as: R& = T, RSK U , system of (6) is rearranged as:
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$ !
1$ (%) = 21(%) + -3(%) + V& R& (%)
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R& (%) = T
0 0 0 0
&' /# ,
1 0
:
0: U is fault distribution matrix. In the following subsection, a TSMO is proposed to estimate 1(%) and 1
RSK U in finite time.
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6
where, V& = T
Remark 1: Consider the offset fault on generator torque, denoted as ∆ ! . It can be related to the general fault RSK as: ∆ This means that if RSK is reconstructed, ∆ This implies that if
!
!
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matrix c and system matrices in the new coordinate have the structure:
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XK
SK,N\] ^SK
. .
TSMO design, a change of coordinate c1(%) ⟶ e(t) is defined, where e = =e@ e: B: , e@ ∈ h@×@ , e ∈ h>×@ . The transformation c 1 0 0
0 0 1 0
0 k 0 j,k = l @ k> 0 1
k k 0 0 n , k> = l >@ n , ℬ = - = l @ n , ℱ& = V& = l n , V = l n , q = 6 km V A k> -
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c@ 0 c=i 0 0
c@ and c are design scalars and should be adjusted to stabilize k@ . k@ ∈ h@×@ , k ∈ h@×>, k> ∈ h>×@ , km ∈ h >×> , k>@ ∈ h@×@ ,
and k> ∈ h ×@ . It should be noted that certain structure of T results in: e = 1 . The transformation must satisfy the conditions: ) k@ is stable, r) k>@ is full-rank (nonzero).
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XK
YZ[K
The matrices C and V& are full rank and _`(6V& ) = _`(V& ) = 2. Assume R& is bounded as ‖R& ‖ ≤ b. To make a platform for
3-1-1 Terminal Sliding Mode Observer
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=−
and RSK are known, then ∆"! is determined. Therefore, RSK represents both offset and change of dynamics faults.
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!
is also estimated. Furthermore, let ∆"! denote the change of "! , it holds: ∆"! =
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(8)
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Remark 2: The transformation matrix c is easily obtained to fulfill the conditions ) and r). In the new system: k = c2c ^@ . With
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The system (8) is represented in the new coordinates as:
respect to the certain structure of c and 2, it stands: k@ =
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:M GHI
:s FJK
, k>@ =
@ GHI
:s FJK
, and k@ is stable if
:M :s
< 0.
e$@ (%) = k@ e@ (%) + k e (%) + -@ 3(%)
e$ (%) = k> e@ (%) + km e (%) + - 3(%) + V R& (e, %)
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5(%) = e (%)
Consider an observer with the structure:
(9) [
u$@ (%) = k@ e u@ (%) + k e u (%) + -@ 3(%) + v@ w (%) + v x y e
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ACCEPTED MANUSCRIPT u$ (%) = k> e u@ (%) + km e u (%) + - 3(%) + v> w (%) + vm x(%) e u (%) 5z(%) = e
2 3
4 5
:
In above equations, w (%) = 5z(%) − 5(%) = (%) = {|K , |N , SK } is output estimation error, where |K = ~! − ! , |N = ~ −
, and SK = !̂ −
with the structure of: x(%) = T _ {|K } ,
!.
" and (" < ) are odd integers. x(%) ∈ h >×@ is the discontinuous switching signal injected to the observer
_7|N 9,
_ {SK }U
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(10)
:
where, _( ) = /| | for ≠ 0 and _( ) = 0 for = 0. v , = 1,2,3,4, are observer gains, which are defined later. The error system is obtained as:
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[
$@ (%) = k@ @ (%) + k (%) + v@ (%) + v x y
$ (%) = k> @ (%) + km (%) + v> (%) + vm x(%) − V R& (e, %)
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Theorem 1. For the error system of (12), (%) = =@: , : B is bounded in finite time if: v@ = −k , v> = −km + km , where km ∈
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h >×> is an arbitrary stable matrix. For positive design scalars @ , , and > , = (@ , , > ) and vm = −.
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Proof 1. The error system is rewritten as: [ 0 v $ = + l n x + T U x y − V& R& vm 0
where, = lk@
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calculated as:
>
@ k + v@ 0 n is stable, and V& = l n. Define a Lyapunov function candidate = : . The time-derivative of is V km [
y $ = : $ = : + : v x − : V& R& vm x
(14)
Define a positive scalar = −̅(), where ̅(.) and (.) denote the maximum and minimum eigenvalues. Applying the Rayleigh’s
inequality on the first term of (14) and the Cauchy-Schwarz inequality on the other terms, results in:
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(13)
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k
10
12
$ < −‖‖ {‖‖ − 7√3() + ‖V& ‖b + √3‖v ‖9}
15 16
(15)
√>7‖M ‖Z()9Z‖ ‖ It is clear that @$ < 0 if: ‖‖ > . This means that ‖‖ is bounded by the ball of radius R as: ‖‖ ≤ h =
√>7‖M ‖Z()9Z‖ ‖
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. It should be noted that the error ‖‖ converges to the ball of R asymptotically. Consequently, the error converges
to the ball of h = h + in finite time, where is a positive scalar. If ‖‖ > h , then $ (%) < −2 . Denote % as the time of
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convergence to h and (0) as the initial value of Lyapunov function, then it holds: % <
18
complete.
19
( ()^ ) ¡
. Finally, the proof of Theorem 1 is
Theorem 2. converges to zero in finite-time and sliding motion takes place on = 0 if () > ¢@ +
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(12)
u@ − e@ , (%) = e u − e are estimation errors for unmeasured and measured states, respectively. where, @ (%) = e
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(11)
√3‖k> ‖‖v ‖ + ‖V& ‖b(1 + ‖k> ‖)
(16)
− √3‖k> ‖
where, ¢@ is a positive scalar.
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1
@
Proof 2. Consider the Lyapunov function candidate = : for subsystem of (8). Then, $ =
2
: $
=
: kmY
+ : k> @
− : Φ; _7,@ 9
:
_7, 9 _7,> 9< − : V R&
(17)
Considering that kmY is stable and Cauchy-Schwarz inequality, it yields $ < ‖ ‖{‖k> @ ‖ − () + ‖V ‖b}
→ () > ¢@ +
5
8
− √3‖k> ‖
@
7
√3‖k> ‖‖v ‖ + ‖V& ‖b(1 + ‖k> ‖)
Noting that = ‖ ‖ , it yields: 2
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Using the sub-multiplicative property of induced norms we have ‖k> @ ‖ ≤ ‖k> ‖‖@ ‖, replacing h from Theorem 1 results in:
<−
¢@
√2
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For positive scalar ¢@ , $ < −¢@ ‖ ‖ if () > ‖k> @ ‖ + ‖V ‖b + ¢@ . From Theorem 1 it stands: ‖@ ‖ ≤ h , since ‖@ ‖ ≤ ‖‖.
%
= 0, yields % <
M () ¥s
is tuned adjusting the design parameter ¢@ . When the sliding motion is achieved on = 0, the error system is reduced to: [ y
0 = k> @ + vm x¦M − V R&
where, x¦M is the continuous equivalent value of the switching term during = 0. Theorem 3. Partition v as: v = =v@
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positive scalar defined in the proof.
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Proof 3. Partition k> as k> = l
13
as (22) and the error system becomes as (23).
0BΦ
^
[ y
[
(22)
[
^ ̅@ [ y y l ̅ n y = k@ @ + @ v@ ̅@ R&
(23)
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$@ = k@ @ + =v@
Multiplying (23) by k>@ gives: ^
[
[
y y ̅@$ = k@ ̅@ + @ k>@ v@ ̅@
15 16
(24)
Since k@ is stable, there exists a pair of positive definite scalars ¨@ and ¬@ such that ¨@ k@ + k@: ¨@ = 2k@ ¨@ ≤ −¬@ . Assume the
Lyapunov function candidate as @ = ¨@ ̅@ for system (24). Then, the time-derivative of @ is derived as: ^
[
y @$ = 2 ¨@ k@ ̅@ + 2@ ¨@ k>@ v@ ̅@
17
©ª«
k>@ n and define two new signals as ̅@ = k>@ @ and R&̅ = k> @ − R& , R&̅ ∈ h . Then, x¦M is obtained k>
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̅@ x¦M = Φ^@ l ̅ n R&
(21)
0@× B where v@ ∈ R@×@ . @ converges to zero in finite time if k>@ v@ = −¨@M« , where ¨@ is a
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14
(20)
. This implies that converges to zero in the finite-time ‘% ’. It is obvious that the convergence time of
$@ = k@ @ + v x¦M
9
(19)
Define % as the time of zeroing and taking place the sliding motion on = 0. Integrating both sides of (20) and replacing
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(18)
[Zy y
(25)
Satisfying the condition of this Theorem, it holds: ^
[ [Zy [Zy
y y y @$ ≤ −¬@ ̅@ − 2@ ¨@ ̅@
(26)
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1
^
%@ <
2 3
©
^
©
Then, @$ < −2@ « (¨@ ̅@ ) M« and @$ < −2@ « @ M« . By integration it holds: [
©¯«
©¯«
y^[
^ y y (0) @ @ −"
(27)
where, @ (0) is the initial value of @ after convergence of , and %@ is the time that @ converges to zero. This implies that ̅@ and @
converges to zero in finite time which proves the Theorem. As seen in (27), %@ depends on the design parameters ", , and @ . Therefore, the desired convergence time of @ is achieved by tuning these parameters.
5
Remark 3: Note that the condition k>@ v@ = −¨@M« is easily satisfied. The pair of positive scalars ¨@ and ¬@ are obtained from the
6
algebraic inequality 2k@ ¨@ ≤ −¬@ . Then, v@ is determined from the algebraic equation k>@ v@ = −¨@M« .
9 10
Once the sliding motion is achieved on @ = = 0, the actuator fault R³& (%) is reconstructed as: R³& (%) = −VZ Φx´
approximated to any degree of accuracy by replacing _( ) =
15 16 17 18
µ
|µ |Z·µ
. ¸ is a small positive constant to avoid
(29) (30) :
Remark 4: The observer term vmx(%) in the second part of (12) attempts to achieve the sliding motion on = T|K , |N , SK U = 0.
Note that the outputs and consequently output estimation errors |K , |N , and SK have different orders of magnitude. Assume vm is
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with _( ) ≈
defined as: vm = −, where is a positive scalar. A small cannot force SK to converge to zero, while a large results in large chattering of |N . Hence, an individual and appropriate gain should be dedicated for each error with respect to the error magnitude. It
leads to convergence of all estimations without considerable chattering problem. As a consequence, vm is defined as a diagonal matrix of the form: vm = −(@ , , > ).
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13
µ
|µ |
c 0 zero denominator and is chosen with respect to the order of magnitude of . Since V = l n and R³& = T³ , R³SK U , then from (28): A
R³SK = −> x´,>
12
(28)
where, VZ = (V: V )^@ V: and x´ is the equivalent injection switching term required to hold the sliding motion on = 0. x´ can be
³ = − x´, → ̂&' = −# x´,
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Theorems 1-3 proved that the sliding motion is achieved on (%) = 0 in finite time, which implies that all the states and fault are
estimated accurately in finite time. Representing the TSMO in the original coordinate: [
5z(%) = 61z(%)
19
20 21
AC C
1z$(%) = 21z(%) + -3(%) + ¹@ w (%) + ¹ x(%) + ¹> x y
(31)
The observer gains ¹@ , ¹ , ¹> are defined as: v 0 v ¹@ = c ^@ l @ n, ¹ = c ^@ l n, ¹> = c ^@ T U vm v> 0
(32)
where, v@ and v> are determined from Theorem 1, v is determined from Theorem 3 and vm is obtained from Theorems 1 and 2. The TSMO design process is summarized in left flowchart of Figure 6.
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1 2
3-2 Pitch Actuator Fault Reconstruction Scheme
3
pressure drop in hydraulic systems or increase of air content in the oil. These faults change the dynamics of pitch actuators, cause
4
fluctuations in the generator speed, and affect capturing wind power [3,12]. In this paper low pressure fault, especially abrupt pressure
5
drop is considered which is faster and more severe than high air content in the oil. Pressure drop slows actuation of control commands.
6
It is too severe and an urgent shutdown is required [3].
7 8
3-2-1 Pitch Actuator Low Pressure fault formulation For each pitch actuator, a state space model of the following form is represented
5(%) = 61(%) =
10 11
: Where, 1(%) = ;,$ < , 3(%) = , 2 = l−2. − n, - = l n, and 6 = =0 1B.
1
0
0
The actuator malfunction caused by low pressure fault, changes the performance of pitch actuators and is modelled by changing the pitch parameters and . Consider and as the pitch parameters in low pressure condition. Let = 0.6 and =
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(33)
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1$ (%) = 21(%) + -3(%)
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The benchmark model contains three individual hydraulic pitch actuators, each is modelled as (1). The pitch actuator faults are
12
11.11 / denote the parameters in fault-free conditions, while º = 0.45 _ º = 5.73 / denote the parameters
13
during abrupt pressure drop [3]. Note that abrupt pressure drop is defined as the extreme level of low pressure faults, i.e. the worst
14
condition. Low pressure parameters are modelled as a function of parameters in fault-free and abrupt pressure drop conditions. This is
15
realized by introducing a fault index , as follows: = + , ( º − )
16
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2. = 2. + , (2. º º − 2. )
(34)
where, , ∈ =0,1B implies that ∈ =0.45, 0.6B, ∈ =5.73, 11.11B. , = 0 corresponds to the normal pressure, i.e. fault-free case and , = 1 refers to the abrupt pressure drop fault. In fact, using this formulation, low pressure fault is dependent to the fault index ,
18
and is identified by estimating the value of this fault index [13]. Figure 4 shows unit step responses of the pitch actuator model for
19
different low pressure conditions for several fault indices , . It is clear that the pressure drop slows down the dynamics of pitch
20
actuator and increases the overshoot. Thus, the performance of pitch subsystem degrades and the pitch angle reference is not
21
accurately followed.
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Figure 4: Unit step responses of pitch actuator for different faulty conditions. Define » = º − , 2. º º ( 2. . Then, replace the faulty parameters expressed as (34) in system matrices 2 and
-. Consider 2 2 Δ2, - - Δ-, where 2 and - contain healthy parameters and . Then, by denoting Δ21%
6
Δ-3% VR% [31], the pitch parameter fault is transformed into an additive fault R with the following description.
7
(, Δ2 l 0
9
The fault R is unknown but bounded according to |R| a ", " is the upper bound. Finally the state space model in the faulty
condition is expressed as:
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(», », (1 n , Δ- l n , V T U , R% , 1@ »1 (
0 0 0
1$ % 2 1% - 3% VR%
(35)
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Where, 2 l(2. ( n and - l n. The state 1% and fault R% are going to be estimated in finite time by the proposed
11
cascade SMOs.
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3-2-2 Finite time Fault Reconstruction by Cascade Sliding Mode Observers
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In this section, two SMOs are introduced in cascade to reconstruct the fault R% in (35) for each pitch subsystem. In the system of
(35), _`6V 0, _`V 1, thus _`6V _`V . This means that the fault is an unmatched signal. To reconstruct
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such a fault, two SMOs are exploited in cascade [32]. Defining 2@ (2 , 2 ( , -@ , 1@ $, and 1 , the
16
system of (35) is rewritten as:
1$@ % 2@ 1@ 2 1 -@ 3 ( R%
1$ % 1@
17
(36)
Primary Observer: The primary observer is structured as: 1z$@ % 2@ 1z@ 2 1z -@ 3 1z$ % 1z@ ( @ x
(37)
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The discontinuous switching term x is defined as: x = _(w ). w (%) = (%) = 1z (%) − 1 (%) is the output estimation error.
Defining @ (%) = 1z@ (%) − 1@ (%), the error system is derived as: $@ (%) = 2@ @ + 2 + R $ (%) = @ − @ x
Theorem 4: The output estimation error w (%) converges to zero in finite time and the sliding motion occurs on w (%) = (%) = 0 if
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(38)
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@ > |@| ½, where ½ is an arbitrary positive scalar.
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Proof 4: Consider the Lyapunov function as @ % w % . It is concluded that @$ % w $w w @ ( @ x and @$ < ¾w ¾|@ | ( @ .
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If @ > |@| ½, then @$ < (½¾w ¾. Since ¾w ¾ 2@ , it holds: @$ < −½2@ . By integration both sides of the last inequality, the convergence time of @ is %@ < in finite time of %@ .
s () ¿
, where @ (0) is the initial value of Lyapunov function. This implies that w (%) converges to zero
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Once the sliding motion is achieved on w (%) = 0, it holds from (38) that: @ = @ x´ , where x´ is the equivalent value of
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@
switching term required to maintain the sliding motion on w (%) = 0. x´ can be approximated to any degree of accuracy by replacing x = _7w 9 with
À
|À |Zs
, where @ is a small positive scalar. The error system related to @ is reduced to:
$@ % 2@ @ % R%
(39)
Now, define 5´ x´ @^@ @ as an output for the error system of (39).
13
Supplementary Observer: For the error system of (39) with the output 5´ @^@ @ , the supplementary observer is introduced as: ̂@$ % 2@ ̂@ ( Á 5z´ @^@ ̂@
(40)
Defining  ̂@ ( @ , Âw 5z´ ( 5´ @^@ Â, and Á _Âw , the error system becomes as: Â$ 2@  ( Á R
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(41)
Theorem 5: For an arbitrary positive scalar Ã, the estimation error  converges to zero in finite time if > " Ã.
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@ Proof 5: Assume the Lyapunov function as: Â . Then, $ ÂÂ$ Â2@ Â ( Â Á ÂR. Since 2@ is stable, it holds: $ <
17
|Â|" ( . If > " Ã, then $ < (Ã|Â|. This implies that at the finite time % <
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proof is complete.
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Ä
, the error  converges to zero and the
Convergence of  to zero means that ̂@ % → @ % . Hence, the error @ is accurately estimated by the supplementary observer.
Since @ % 1z@ % ( 1@ % , it stands: 1@ % 1z@ % ( @ % , while 1z@ % is estimated by the primary observer. Define 1̅@ % 1z@ % ( ̂@ % , where ̂@ % is estimated by the supplementary observer. Since ̂@ % → @ % , then it is concluded that 1̅@ % → 1@ %
and 1̅@ % is an accurate estimation of 1@ % in finite time. Note that 1z % provided by the primary observer is an exact estimation of 1 % , while 1z@ % is not an accurate estimation of 1@ % . The exact estimation of 1@ % (denoted as 1̅@ % ) is provided using the 12
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supplementary observer. The implementation of cascade observers is shown in Figure 5. The design process for pitch actuators fault
2
diagnosis is summarized in right flowchart of Figure 6. By setting  = 0 in (41), the fault R(%) is reconstructed in finite time as: R³(%) = Á´
Á´ is the continuous form of the switching term in the supplementary observer, which is approximated by replacing Á = , where is a small positive scalar. Finally the fault index ,Æ and the faulty pitch system parameters ~ , ³
_7Âw 9 with
5
are estimated from (43) and (44). R³(%) (1z@ − ̂@ ) + »(1z − )
~ = Ç + ,Æ »,
³ =
2 + ,Æ 2 ~
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,Æ =
|ÅÀ |ZM
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Hydraulic pitch system
βref Pitch fault reconstruction
(43)
(44)
βˆ
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βm
Primary observer
βˆ&
veq
weq
Supplementary observer
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eˆ1
β&
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Figure 5: Implementation of cascade observers for pitch fault reconstruction.
9
Remark 5: Note that each pitch subsystem is generally modelled as (33). For each individual pitch system, two observers are employed in cascade in the prescribed structure.
11
Remark 6: Note that the proposed scheme is capable to be applied to benchmark-2009 [3]. Generally, the proposed TSMO and
12
cascade SMOs can be used for any wind turbine system, where generator, drive-train, and pitch subsystems are modelled with the
13
structure similar to (1)-(5). Moreover, the method considers low pressure condition as a typical pitch fault. Whilst pump wear, high air
14
content in oil and etc. are modelled in the prescribed form of (34) [13].
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Start
Start
Find T from Section 3.1 (Remark 2)
Find ρ1 from Theorem 4 and design the primary observer in the form of (37)
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Determine l1, l3 from Theorem 1, l2 from Theorem 2 and l4 from Theorems 1&2
Choose an appropriate λ1 to convert the discontinuous function v(t) into a continuous function veq(t)
Choose an appropriate δ to convert the discontinuous function v(t) into a continuous function veq(t)
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Find ρ2 from Theorem 5 and design the supplementary observer in the form of (40)
Use (31) & (32) to revert the observer into the original coordinates
Use (30) for generator fault reconstruction
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Use (29) for aerodynamic load estimation
Choose an appropriate λ2 to convert the discontinuous function w(t) into a continuous function weq(t)
Use (43) & (44) for pitch fault reconstruction
Figure 6: Design algorithm flowcharts for fault reconstruction: Left: drivetrain and generator subsystems faults; right: pitch subsystems faults.
4- Simulation Results
4 5
4-1 Generator fault reconstruction/aerodynamic load estimation
6
[3]. Simulations are carried out with a stochastic full-field wind speed profile, generated by TurbSim [30] with mean value of 14
m/sec as in Figure 7. The observer parameters are designed as: @ = 13, = 13, > = 1.24, v@ = 1, km = −A> , " = 1, = 3, c@ = −10, c = 6.8 − 4, and ¸ = 1 − 2.
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The performance of the proposed TSMO described in section 3-1 is now evaluated on the offshore wind turbine benchmark model
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Figure 7: The stochastic wind speed profile exploited in simulations.
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A fault scenario is simulated in the generator system of the benchmark model. Figure 8 presents the states’ estimations provided by
2
the proposed TSMO. As is shown in the figure, the estimation errors converge to zero in finite time and the sliding motion is achieved.
3
The estimations of twisting angle, generator speed, rotor speed, and generator torque converge in 0.45, 0.45, 0.01, and 0.16 s,
4
respectively. Note that the twisting angle is accurately estimated, while it is not measured in wind turbines. The generator fault RSK and its estimation R³SK are illustrated in Figure 9. The estimation signal converges within 0.16 s. As pointed in
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Remark 1, the generator fault R³SK can be converted to an offset fault ∆
7
formulation. As stated before, the aerodynamic load (torque)
8
an output in the benchmark model, although FAST simulator has the capability of reporting it. In this study,
11
&'
of aerodynamic torque ̂&' is illustrated in Figure 10, which converges in 0.45 s.
As seen from Figure 8-Figure 10, the proposed TSMO provides accurate and finite time estimations of states, generator fault, and aerodynamic torque.
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is modelled as an
unknown input and augmented into the fault vector R& , as (8). Then, it is estimated from (29) by the proposed TSMO. The estimation
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is not measurable in the reality. Hence, it is not normally defined as
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or change of dynamics fault ∆"! by the proposed
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Figure 8: Twisting angle (É), generator speed (ÊË ), rotor speed (ÊÌ ), and generator torque (ÍË ) estimations.
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Figure 9: Actual and estimated generator fault ÎÍË .
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Figure 10: Actual and estimated aerodynamic torque ÍÏÐÌÑ .
5 6
4-2 Pitch actuators fault reconstruction
7
of the benchmark model [3]. The faults are simulated according to the actual fault indicators depicted in Figure 11.
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Three low pressure fault scenarios (including abrupt pressure drop fault) are simulated simultaneously for the three pitch actuators
1.4 1.2 1
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Actual pitch fault indicator 1 Estimated pitch fault indicator 1 Actual pitch fault indicator 2 Estimated pitch fault indicator 2 Actual pitch fault indicator 3 Estimated pitch fault indicator 3
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0.4 0.2 0
0
8 9 10 11
0.5
1
1.5
2
2.5
3
3.5
4
Time(s)
Figure 11: Actual and estimated fault indicators for pitch actuators.
The cascade observers proposed in section 3-2 are exploited to estimate the faults of each pitch subsystem. The parameters of observers for each pitch subsystem are designed as follows.
12
•
13
•
Pitch actuator 1: @ = 300, = 3000, @ = 1 − 4, = 1 − 3. Pitch actuator 2: @ = 250, = 2500, @ = 1 − 4, = 1 − 3. 16
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•
Pitch actuator 3: @ = 250, = 2500, @ = 1 − 4, = 1 − 3.
Estimations of the pitch angles and pitch rates together with estimation errors are shown in Figure 12 and Figure 13. The estimation
3
errors are almost zero, which clarifies that the cascade observers estimate the states accurately. These errors converge to zero within a
4
small finite time at initial samples of simulation. However, the errors increase when faults occur. For example, the fault of pitch
5
system 2 is present in interval [0.5-3.5] s. Hence, the estimation error of pitch angle 2 deviates from zero within this interval.
6 7
In addition to states and $ , the additive fault R(%) (in (35)) is reconstructed by the cascade observers for each pitch subsystem.
Then, the fault indicators ,Æ are estimated from (43), which are shown in Figure 11. The estimated indicators follow the actual ones precisely. Once the indicators are determined, the pitch parameters ³ and ~ are computed from (44).
Figure 12: Pitch angles’ estimations.
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Figure 13: Pitch rates’ estimations.
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5- Conclusion
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Wind turbine actuator faults in generator/convertor and pitch subsystems emerge slow actuation and non-optimal power production.
3
This paper aims at finite time reconstruction of these faults for a realistic wind turbine model. A terminal sliding mode observer is
4
proposed which faithfully observes the generator and drivetrain subsystems’ states, generator faults, and unknown aerodynamic
5
torque. In addition, two sliding mode observers are suggested in a cascade form to estimate the blade pitch angle, pitch rate, and fault
6
indicator of each hydraulic pitch actuator. Simulation results confirm that all the reconstructed signals, including the states and
7
simultaneous faults are estimated accurately in finite time. Compared to previous studies, this paper addresses the observation and
8
actuator fault reconstruction of wind turbine in finite-time, which makes prompt and appropriate actions possible. The proposed
9
schemes are also applicable in fault tolerant control methods, which require fast and accurate fault diagnosis.
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References
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[1]
S. Gsanger, J.-D. Pitteloud, World Wind Energy Report 2011, Bonn, Germany, 2012.
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P.F. Odgaard, K.E. Johnson, Wind turbine fault detection and fault tolerant control - an enhanced benchmark challenge, 2013 Am. Control Conf. (2013) 4447–4452.
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[30]
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• •
This paper proposes a fault detection algorithm for generator and pitch actuator faults. A novel fault tolerant observer for fault detection is proposed. Generator faults are detected and reconstructed via terminal sliding mode observer. Hydraulic pressure drop is modelled as an additive fault by introducing a fault indicator for each of the pitch systems. For every pitch actuator, two cascaded sliding mode observers are exploited to achieve finite time fault reconstruction. The novel terminal sliding mode observer is used to reconstruct the states, aerodynamic torque, and faults. Numerical simulations are performed in FAST & MATLAB Simulink.
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DOI:
10.1016/j.renene.2018.06.053
Reference:
RENE 10209
To appear in:
Renewable Energy
Received Date: 7 November 2017 Revised Date:
12 June 2018
Accepted Date: 13 June 2018
Please cite this article as: Rahnavard M, Ayati M, Hairi Yazdi MR, Mousavi M, Finite time estimation of actuator faults, states, and aerodynamic load of a realistic wind turbine, Renewable Energy (2018), doi: 10.1016/j.renene.2018.06.053. This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
ACCEPTED MANUSCRIPT
Finite Time Estimation of Actuator Faults, States, and Aerodynamic Load of a Realistic
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Wind Turbine
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Mostafa Rahnavard, Moosa Ayati*, Mohammad Reza Hairi Yazdi, Mohammad Mousavi
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School of Mechanical Engineering, Collage of Engineering, University of Tehran, North Kargar Street, Tehran, Iran
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*Corresponding author: [email protected], Tel.: +982161119920, P.O.B.: 11155-4563
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Abstract:
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This paper provides finite time estimation of wind turbine actuator faults and unknown aerodynamic load. Furthermore, finite-time
8
state estimation of drivetrain, generator, and pitch subsystems are addressed in the contrary of asymptotic state/fault estimation in
9
previous works. A realistic wind turbine model, incorporating the aero-elastic FAST simulator, is considered as the simulation
10
example. Generally, aerodynamic load is not measurable in real applications due to instrument limitations, then, it is considered as an
11
unknown input in this study. A novel terminal sliding mode observer is introduced for finite-time estimation of generator/convertor
12
states, faults, and unknown aerodynamic load. Pitch actuator hydraulic pressure drop is modelled as an additive fault, by introducing a
13
fault indicator. Then, two cascaded sliding mode observers are exploited for each pitch subsystem, to provide finite time state and
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fault reconstructions. Sufficient number of design parameters helps to achieve desired accuracy and convergence time. Finally,
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simulation results authenticate finite time estimation of wind turbine states and simultaneous actuator faults.
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Keywords: Fault detection and isolation, Fault reconstruction, Terminal sliding mode observer, Lyapunov stability theorem, Wind
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turbine.
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1- Introduction
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Nowadays, wind turbines (WTs) are the most growing renewable energy generators which contribute to the world power production
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in large scale as shown in Figure 1. Meanwhile, there is a strong demand on enhancing the reliability and efficiency as well as
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reducing the operation and maintenance costs [1].
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Figure 1: Total installed capacity of wind turbines during 1997-2020 [GW] [1].
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Recently, two wind turbine benchmark models (WTBMs) have been presented by Odgaard et al. in 2009 [2] and Odgaard and
2
Johnson in 2013 [3]. Benchmark-2013 incorporates the aero-elastic simulator FAST and provides a higher fidelity and more realistic
3
wind turbine model. Thus, it requires more sophisticated fault detection schemes. Many researches have been published with the scope
4
of fault detection and isolation (FDI), fault reconstruction and estimation (FRE), and fault tolerant control (FTC) based on these
5
WTBMs.
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Shi and Patton [4] have proposed an observer based active FTC approach. By modelling the wind turbine as a linear parameter
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varying (LPV) model, an extended state observer was designed to estimate system states and faults. Simani et al. [5] presented a FDI
8
scheme based on identified fuzzy models. Also, fuzzy theory is exploited to provide an uncertain model of wind turbine and to
9
manipulate noisy data and Takagi-Sugeno prototypes represent the residual generators for FDI purposes. In [6] a FTC scheme has
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been proposed for regulation of generator torque when generator/convertor actuator faults occur. The scheme includes two
11
approaches. The first approach relies on fuzzy model reference adaptive control, while the second approach exploits identification
12
method and fuzzy modeling. An adaptive actuator fault estimation scheme based on nonlinear geometric approach was proposed by
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Simani and Castaldi [7]. Wu et al. [8] presented a FTC method using virtual actuators for the problem of pitch actuator failures.
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Nonlinear wind turbine model is converted to a linear parameter varying model and FTC structure of virtual actuators is obtained by
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applying a state feedback law.
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Observers and filters have been extensively investigated during recent decades to address state/fault estimation and control of
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dynamical systems [9–15]. Sliding mode observers (SMOs) have been widely used for FRE purposes due to their robustness to
18
disturbances and modelling uncertainties and their capability of estimating unknown inputs [16–19]. Some researchers have exploited
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SMOs for FRE and FTC of WTBMs [12,13,20–22]. Lan et. al [12] proposed a FTC scheme to compensate the pitch actuator faults.
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The proposed scheme incorporates a traditional Proportional-Integral (PI) controller to achieve nominal pitch performance, along with
21
a fault compensator to eliminate the actuator fault effects. The pitch subsystem states and fault index are estimated using an adaptive
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SMO. In [20] a Takagi-Sugeno sliding mode observer with weighted switching action was implemented to partially reconstruct the
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parameters’ faults in hydraulic pitch actuator and generator subsystems. The method lacks robustness to model uncertainties and
24
significant fault estimation errors are resulted. In [21], sensor and actuator faults were reconstructed using the classic SMO.
25
Estimation of faults in pitch angle sensors and actuators is obtained using physical redundancy of pitch sensors. However, the method
26
is not robust to model uncertainties and disturbances. Rahnavard et al. [13] proposed a robust FRE scheme based on modified SMO, to
27
address reconstruction of actuator and sensor faults of WTBM presented in [2]. Pitch actuator low pressure faults and several
28
simultaneous sensor faults, with different orders of magnitudes, are investigated.
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Classic SMOs guarantee finite time convergence of output estimation errors, while non-output errors converge to zero
30
asymptotically. This implies that the states which do not directly participate in the output converge to actual values as time goes to
31
infinity. Terminal sliding mode (TSM) concepts enable finite time convergence of both measured and unmeasured state estimation
32
errors [22–25]. This is realized by injecting a non-smooth discontinuous switching term, including fractional powers, to the classic
33
observer. A terminal sliding mode observer (TSMO) for a certain class of nonlinear systems was proposed in [23]. The system under
34
consideration has n-degree-of-freedom, where position and velocity are measured and unmeasured states, respectively. This leads to a
35
special state-space representation. A discontinuous term by use of fractional powers is injected into the classic observer, which
36
guarantees finite time convergence of non-output estimation errors. 2
ACCEPTED MANUSCRIPT
1
In aforementioned SMO-based FRE and FTC papers of wind turbines, asymptotic estimation of states and faults are addressed. This
2
paper aims at finite time estimation of states and actuator faults for a realistic WT model. The proposed fault diagnosis scheme is
3
evaluated on 2013 wind turbine benchmark [3] which is a more realistic WT model rather than 2009 benchmark. The aerodynamic torque is the main excitation source of WTs and is not usually measured in reality. Since, there is not any sensor
5
to report aerodynamic torque accurately; it is regarded as an unknown input in 2013 benchmark model. Generator and convertor faults
6
cause serious problems and lead to non-optimal power production of WT. These faults are commonly dynamical changes or offset in
7
generator torque [2,3]. The proposed TSMO reconstructs generator common faults and aerodynamic torque in finite time as long as
8
the states of drivetrain and generator/convertor subsystems.
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4
The pitch actuator faults are due to pressure drop in hydraulic system or increase of air content in oil. These faults change pitch
10
actuators’ dynamics and reduce the speed of control actions. Furthermore, in the case of abrupt pressure drop, a safe and fast shut
11
down of wind turbine is required [3,12]. Estimation of hydraulic pitch actuator faults are addressed in some researches, specially
12
[4,8,12,26]. In [26], pitch actuator fault is partially reconstructed, however, the estimation accuracy requires improvement. In [4], the
13
considered fault is high air content in oil and the pitch fault signal is reconstructed with considerable error. Comparing the error
14
amplitude, [12] and [8] provide a better fault estimation rather than [26] and [4]. In this paper, two cascade SMOs are designed for
15
each pitch subsystem to reconstruct actuator low pressure and abrupt pressure drop faults. The cascade observers provide simultaneous
16
finite time estimation of states and actuator faults of three pitch subsystems by enhanced accuracy.
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The proposed fault diagnosis schemes are evaluated on benchmark-2013 WT and can be implemented in FTC methods, where fast
18
and exact information about the fault magnitude is required. These schemes can also be utilized in benchmark-2009 WT model or any
19
wind turbine whose subsystems are modelled similar to those of benchmark-2013.
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This paper is structured as follows. Section 2 briefly describes the enhanced WTBM. Theory of finite time fault reconstruction is
21
presented in Section 3. First, a TSMO is proposed to reconstruct the generator and convertor faults as well as unknown aerodynamic
22
torque. Then, two SMOs are exploited in a cascade form to reconstruct the pitch actuator low-pressure faults. Simulation results are
23
vector and |. | refers to the absolute value of a scalar.
25
2- Wind Turbine Description
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reported in Section 4. Finally, Section 5 concludes the paper. Throughout the paper, the symbol ‖. ‖ denotes the Euclidean norm of a
26
Main components of a utility-scale wind turbine are depicted in Figure 2. The wind turbine model of this paper is a 5 MW offshore
27
wind turbine benchmark proposed by Odgaard and Johnson [3]. The model is a variable speed three-bladed horizontal axis wind
28
turbine with full span blade pitch control and generator torque regulation. FAST (Fatigue, Aerodynamics, Structures, and Turbulence)
29
is an aero-elastic wind turbine simulator designed by NREL National Wind Technology Center, which is widely used for studying
30
wind turbine control systems [3]. The benchmark incorporates FAST software [27] to model the turbine with 24 degrees of freedom.
31
Figure 3 describes how the proposed fault estimation scheme is implemented in FAST & MATLAB Simulink environment. The
32
benchmark’s main subsystems are described as follows.
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Figure 2: Main components of a utility-scale wind turbine [28].
6
7 8
Figure 3: Fault diagnosis scheme block diagram.
The hydraulic pitch actuator is a piston servo system modelled as (1), which is a second-order closed loop transfer function from
(the pitch reference angle) to pitch angle .
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( ) = ( ) + 2. . +
(1)
and are the damping factor and natural frequency, respectively. A transfer function is associated to each of pitch subsystems,
which are same in fault-free conditions. In the fault-free condition, the parameters are = 0.6 and = 11.11 / . The pitch
10
actuator faults are modelled by changing the parameters and in the transfer function of (1) during the faults interval. The
11
On the system level of wind turbine, the generator and convertor dynamics are modelled as a first-order transfer function of (2).
9
parameters associated with the hydraulic abrupt pressure drop are = 0.45 and = 5.73 / .
4
ACCEPTED MANUSCRIPT ! ( )
2
"! ,
=
,
and
(2) !
are the generator & converter constant, generator command torque, and generator torque, respectively. It is
concluded from [29] that, FAST models the drivetrain subsystem by a two-mass spring damper model as follows. # $ (%) =
&' (%) −
)*+ ,(%) − -*+ . (%) −
! (%) )*+ -*+ ,(%) + . (%) − 0− / / / 1 ,$(%) = (%) − ! (%) / #! $ ! (%) =
3
"! + "!
&'
! (%) 0 /
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1
( )
! (%)
(3) (4) (5)
is the aerodynamic torque acting on blades, is rotor angular speed, ! is generator angular speed, , is twisting (torsion)
6
TurbSim software [30]. Wind speed inputs with mean value of 11, 14, 17 m/s are generated by TurbSim using IEC von Karman
7
turbulence model and are provided for use in the benchmark model.
8
3- Sliding Mode Observers for Wind Turbine Finite Time Fault Reconstruction
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5
angle of drive train, # is rotor moment of inertia, #! is generator moment of inertia, / is gearbox ratio, )*+ is torsion stiffness, and -*+
is torsion damping of drive train. The wind speed utilized in this benchmark is a full-field turbulent wind speed profile generated by
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3-1 Generator/convertor Fault Reconstruction and Aerodynamic Torque Estimation
11
torque of 2013 benchmark model in a finite time. Augmenting the generator model of (2) and the drivetrain model of (3)-(5), leads to
12
a system of the form:
13
5(%) = 61(%)
where, 1(%) = 7,, ! , ,
!9
:
, 3(%) =
, (%)
=
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1$ (%) = 21(%) + -3(%) + 4(%)
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In this section, an observer is designed using terminal sliding mode concepts to reconstruct the generator faults and aerodynamic
&' /# ,
(6)
and the system matrices are:
16
disturbance, affecting system of (6). The disturbance signal is estimated using a terminal observer proposed in the following.
14
17
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E 0 D GHI : - = ;0, 0, 0, "! < , 6 = =0>×@ , A> B, 4 = =0, 0, 1, 0B: , 2 = D FJK D− GHI D JN C 0
The aerodynamic torque
&'
−
−
@
F LHI
1
LHI
FM JK LHI
FJK LHI
−
FJN
0
JN
0
0 Q @ − P JK P 0 P P −"! O
is unknown in the benchmark model. Thus, the term (%) =
&' /#
in (6) is considered as a
The generator and convertor system may experience two kinds of faults: the change of dynamics due to a change in generator
19
parameter "! or an offset on the convertor torque. These faults are originated from internal problems, such as faults in convertor
20
change of dynamics results in a slow torque control. This is expressed as a fault with high severity and fast development time in the
18
electronics or an offset on convertor torque estimation [3]. Generator and convertor faults cause serious problems. Particularly, the
5
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1
benchmark model. Offset fault results in non-optimal power production of WT and is denoted as a fault with medium severity and fast
2
time of development. Without loss of generality, the aforementioned faults affecting the generator actuation are modelled as:
4
= −"!
!
+ "!
!,
+ RSK
(7)
where, RSK represents a general fault signal. It can be converted to generator faults, including change of actuation dynamics and torque :
offset. Defining an augmented fault vector as: R& = T, RSK U , system of (6) is rearranged as:
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$ !
1$ (%) = 21(%) + -3(%) + V& R& (%)
7 8 9
R& (%) = T
0 0 0 0
&' /# ,
1 0
:
0: U is fault distribution matrix. In the following subsection, a TSMO is proposed to estimate 1(%) and 1
RSK U in finite time.
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6
where, V& = T
Remark 1: Consider the offset fault on generator torque, denoted as ∆ ! . It can be related to the general fault RSK as: ∆ This means that if RSK is reconstructed, ∆ This implies that if
!
!
12
matrix c and system matrices in the new coordinate have the structure:
16
XK
SK,N\] ^SK
. .
TSMO design, a change of coordinate c1(%) ⟶ e(t) is defined, where e = =e@ e: B: , e@ ∈ h@×@ , e ∈ h>×@ . The transformation c 1 0 0
0 0 1 0
0 k 0 j,k = l @ k> 0 1
k k 0 0 n , k> = l >@ n , ℬ = - = l @ n , ℱ& = V& = l n , V = l n , q = 6 km V A k> -
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c@ 0 c=i 0 0
c@ and c are design scalars and should be adjusted to stabilize k@ . k@ ∈ h@×@ , k ∈ h@×>, k> ∈ h>×@ , km ∈ h >×> , k>@ ∈ h@×@ ,
and k> ∈ h ×@ . It should be noted that certain structure of T results in: e = 1 . The transformation must satisfy the conditions: ) k@ is stable, r) k>@ is full-rank (nonzero).
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XK
YZ[K
The matrices C and V& are full rank and _`(6V& ) = _`(V& ) = 2. Assume R& is bounded as ‖R& ‖ ≤ b. To make a platform for
3-1-1 Terminal Sliding Mode Observer
14
=−
and RSK are known, then ∆"! is determined. Therefore, RSK represents both offset and change of dynamics faults.
10 11 13
!
is also estimated. Furthermore, let ∆"! denote the change of "! , it holds: ∆"! =
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(8)
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Remark 2: The transformation matrix c is easily obtained to fulfill the conditions ) and r). In the new system: k = c2c ^@ . With
19
The system (8) is represented in the new coordinates as:
respect to the certain structure of c and 2, it stands: k@ =
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:M GHI
:s FJK
, k>@ =
@ GHI
:s FJK
, and k@ is stable if
:M :s
< 0.
e$@ (%) = k@ e@ (%) + k e (%) + -@ 3(%)
e$ (%) = k> e@ (%) + km e (%) + - 3(%) + V R& (e, %)
20
5(%) = e (%)
Consider an observer with the structure:
(9) [
u$@ (%) = k@ e u@ (%) + k e u (%) + -@ 3(%) + v@ w (%) + v x y e
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ACCEPTED MANUSCRIPT u$ (%) = k> e u@ (%) + km e u (%) + - 3(%) + v> w (%) + vm x(%) e u (%) 5z(%) = e
2 3
4 5
:
In above equations, w (%) = 5z(%) − 5(%) = (%) = {|K , |N , SK } is output estimation error, where |K = ~! − ! , |N = ~ −
, and SK = !̂ −
with the structure of: x(%) = T _ {|K } ,
!.
" and (" < ) are odd integers. x(%) ∈ h >×@ is the discontinuous switching signal injected to the observer
_7|N 9,
_ {SK }U
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(10)
:
where, _( ) = /| | for ≠ 0 and _( ) = 0 for = 0. v , = 1,2,3,4, are observer gains, which are defined later. The error system is obtained as:
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[
$@ (%) = k@ @ (%) + k (%) + v@ (%) + v x y
$ (%) = k> @ (%) + km (%) + v> (%) + vm x(%) − V R& (e, %)
7
Theorem 1. For the error system of (12), (%) = =@: , : B is bounded in finite time if: v@ = −k , v> = −km + km , where km ∈
8
h >×> is an arbitrary stable matrix. For positive design scalars @ , , and > , = (@ , , > ) and vm = −.
9
Proof 1. The error system is rewritten as: [ 0 v $ = + l n x + T U x y − V& R& vm 0
where, = lk@
11
calculated as:
>
@ k + v@ 0 n is stable, and V& = l n. Define a Lyapunov function candidate = : . The time-derivative of is V km [
y $ = : $ = : + : v x − : V& R& vm x
(14)
Define a positive scalar = −̅(), where ̅(.) and (.) denote the maximum and minimum eigenvalues. Applying the Rayleigh’s
inequality on the first term of (14) and the Cauchy-Schwarz inequality on the other terms, results in:
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(13)
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k
10
12
$ < −‖‖ {‖‖ − 7√3() + ‖V& ‖b + √3‖v ‖9}
15 16
(15)
√>7‖M ‖Z()9Z‖ ‖ It is clear that @$ < 0 if: ‖‖ > . This means that ‖‖ is bounded by the ball of radius R as: ‖‖ ≤ h =
√>7‖M ‖Z()9Z‖ ‖
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. It should be noted that the error ‖‖ converges to the ball of R asymptotically. Consequently, the error converges
to the ball of h = h + in finite time, where is a positive scalar. If ‖‖ > h , then $ (%) < −2 . Denote % as the time of
17
convergence to h and (0) as the initial value of Lyapunov function, then it holds: % <
18
complete.
19
( ()^ ) ¡
. Finally, the proof of Theorem 1 is
Theorem 2. converges to zero in finite-time and sliding motion takes place on = 0 if () > ¢@ +
20
(12)
u@ − e@ , (%) = e u − e are estimation errors for unmeasured and measured states, respectively. where, @ (%) = e
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(11)
√3‖k> ‖‖v ‖ + ‖V& ‖b(1 + ‖k> ‖)
(16)
− √3‖k> ‖
where, ¢@ is a positive scalar.
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1
@
Proof 2. Consider the Lyapunov function candidate = : for subsystem of (8). Then, $ =
2
: $
=
: kmY
+ : k> @
− : Φ; _7,@ 9
:
_7, 9 _7,> 9< − : V R&
(17)
Considering that kmY is stable and Cauchy-Schwarz inequality, it yields $ < ‖ ‖{‖k> @ ‖ − () + ‖V ‖b}
→ () > ¢@ +
5
8
− √3‖k> ‖
@
7
√3‖k> ‖‖v ‖ + ‖V& ‖b(1 + ‖k> ‖)
Noting that = ‖ ‖ , it yields: 2
6
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Using the sub-multiplicative property of induced norms we have ‖k> @ ‖ ≤ ‖k> ‖‖@ ‖, replacing h from Theorem 1 results in:
<−
¢@
√2
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4
For positive scalar ¢@ , $ < −¢@ ‖ ‖ if () > ‖k> @ ‖ + ‖V ‖b + ¢@ . From Theorem 1 it stands: ‖@ ‖ ≤ h , since ‖@ ‖ ≤ ‖‖.
%
= 0, yields % <
M () ¥s
is tuned adjusting the design parameter ¢@ . When the sliding motion is achieved on = 0, the error system is reduced to: [ y
0 = k> @ + vm x¦M − V R&
where, x¦M is the continuous equivalent value of the switching term during = 0. Theorem 3. Partition v as: v = =v@
11
positive scalar defined in the proof.
12
Proof 3. Partition k> as k> = l
13
as (22) and the error system becomes as (23).
0BΦ
^
[ y
[
(22)
[
^ ̅@ [ y y l ̅ n y = k@ @ + @ v@ ̅@ R&
(23)
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$@ = k@ @ + =v@
Multiplying (23) by k>@ gives: ^
[
[
y y ̅@$ = k@ ̅@ + @ k>@ v@ ̅@
15 16
(24)
Since k@ is stable, there exists a pair of positive definite scalars ¨@ and ¬@ such that ¨@ k@ + k@: ¨@ = 2k@ ¨@ ≤ −¬@ . Assume the
Lyapunov function candidate as @ = ¨@ ̅@ for system (24). Then, the time-derivative of @ is derived as: ^
[
y @$ = 2 ¨@ k@ ̅@ + 2@ ¨@ k>@ v@ ̅@
17
©ª«
k>@ n and define two new signals as ̅@ = k>@ @ and R&̅ = k> @ − R& , R&̅ ∈ h . Then, x¦M is obtained k>
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̅@ x¦M = Φ^@ l ̅ n R&
(21)
0@× B where v@ ∈ R@×@ . @ converges to zero in finite time if k>@ v@ = −¨@M« , where ¨@ is a
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14
(20)
. This implies that converges to zero in the finite-time ‘% ’. It is obvious that the convergence time of
$@ = k@ @ + v x¦M
9
(19)
Define % as the time of zeroing and taking place the sliding motion on = 0. Integrating both sides of (20) and replacing
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(18)
[Zy y
(25)
Satisfying the condition of this Theorem, it holds: ^
[ [Zy [Zy
y y y @$ ≤ −¬@ ̅@ − 2@ ¨@ ̅@
(26)
8
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1
^
%@ <
2 3
©
^
©
Then, @$ < −2@ « (¨@ ̅@ ) M« and @$ < −2@ « @ M« . By integration it holds: [
©¯«
©¯«
y^[
^ y y (0) @ @ −"
(27)
where, @ (0) is the initial value of @ after convergence of , and %@ is the time that @ converges to zero. This implies that ̅@ and @
converges to zero in finite time which proves the Theorem. As seen in (27), %@ depends on the design parameters ", , and @ . Therefore, the desired convergence time of @ is achieved by tuning these parameters.
5
Remark 3: Note that the condition k>@ v@ = −¨@M« is easily satisfied. The pair of positive scalars ¨@ and ¬@ are obtained from the
6
algebraic inequality 2k@ ¨@ ≤ −¬@ . Then, v@ is determined from the algebraic equation k>@ v@ = −¨@M« .
9 10
Once the sliding motion is achieved on @ = = 0, the actuator fault R³& (%) is reconstructed as: R³& (%) = −VZ Φx´
approximated to any degree of accuracy by replacing _( ) =
15 16 17 18
µ
|µ |Z·µ
. ¸ is a small positive constant to avoid
(29) (30) :
Remark 4: The observer term vmx(%) in the second part of (12) attempts to achieve the sliding motion on = T|K , |N , SK U = 0.
Note that the outputs and consequently output estimation errors |K , |N , and SK have different orders of magnitude. Assume vm is
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with _( ) ≈
defined as: vm = −, where is a positive scalar. A small cannot force SK to converge to zero, while a large results in large chattering of |N . Hence, an individual and appropriate gain should be dedicated for each error with respect to the error magnitude. It
leads to convergence of all estimations without considerable chattering problem. As a consequence, vm is defined as a diagonal matrix of the form: vm = −(@ , , > ).
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13
µ
|µ |
c 0 zero denominator and is chosen with respect to the order of magnitude of . Since V = l n and R³& = T³ , R³SK U , then from (28): A
R³SK = −> x´,>
12
(28)
where, VZ = (V: V )^@ V: and x´ is the equivalent injection switching term required to hold the sliding motion on = 0. x´ can be
³ = − x´, → ̂&' = −# x´,
11
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Theorems 1-3 proved that the sliding motion is achieved on (%) = 0 in finite time, which implies that all the states and fault are
estimated accurately in finite time. Representing the TSMO in the original coordinate: [
5z(%) = 61z(%)
19
20 21
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1z$(%) = 21z(%) + -3(%) + ¹@ w (%) + ¹ x(%) + ¹> x y
(31)
The observer gains ¹@ , ¹ , ¹> are defined as: v 0 v ¹@ = c ^@ l @ n, ¹ = c ^@ l n, ¹> = c ^@ T U vm v> 0
(32)
where, v@ and v> are determined from Theorem 1, v is determined from Theorem 3 and vm is obtained from Theorems 1 and 2. The TSMO design process is summarized in left flowchart of Figure 6.
9
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1 2
3-2 Pitch Actuator Fault Reconstruction Scheme
3
pressure drop in hydraulic systems or increase of air content in the oil. These faults change the dynamics of pitch actuators, cause
4
fluctuations in the generator speed, and affect capturing wind power [3,12]. In this paper low pressure fault, especially abrupt pressure
5
drop is considered which is faster and more severe than high air content in the oil. Pressure drop slows actuation of control commands.
6
It is too severe and an urgent shutdown is required [3].
7 8
3-2-1 Pitch Actuator Low Pressure fault formulation For each pitch actuator, a state space model of the following form is represented
5(%) = 61(%) =
10 11
: Where, 1(%) = ;,$ < , 3(%) = , 2 = l−2. − n, - = l n, and 6 = =0 1B.
1
0
0
The actuator malfunction caused by low pressure fault, changes the performance of pitch actuators and is modelled by changing the pitch parameters and . Consider and as the pitch parameters in low pressure condition. Let = 0.6 and =
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1$ (%) = 21(%) + -3(%)
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The benchmark model contains three individual hydraulic pitch actuators, each is modelled as (1). The pitch actuator faults are
12
11.11 / denote the parameters in fault-free conditions, while º = 0.45 _ º = 5.73 / denote the parameters
13
during abrupt pressure drop [3]. Note that abrupt pressure drop is defined as the extreme level of low pressure faults, i.e. the worst
14
condition. Low pressure parameters are modelled as a function of parameters in fault-free and abrupt pressure drop conditions. This is
15
realized by introducing a fault index , as follows: = + , ( º − )
16
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2. = 2. + , (2. º º − 2. )
(34)
where, , ∈ =0,1B implies that ∈ =0.45, 0.6B, ∈ =5.73, 11.11B. , = 0 corresponds to the normal pressure, i.e. fault-free case and , = 1 refers to the abrupt pressure drop fault. In fact, using this formulation, low pressure fault is dependent to the fault index ,
18
and is identified by estimating the value of this fault index [13]. Figure 4 shows unit step responses of the pitch actuator model for
19
different low pressure conditions for several fault indices , . It is clear that the pressure drop slows down the dynamics of pitch
20
actuator and increases the overshoot. Thus, the performance of pitch subsystem degrades and the pitch angle reference is not
21
accurately followed.
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Figure 4: Unit step responses of pitch actuator for different faulty conditions. Define » = º − , 2. º º ( 2. . Then, replace the faulty parameters expressed as (34) in system matrices 2 and
-. Consider 2 2 Δ2, - - Δ-, where 2 and - contain healthy parameters and . Then, by denoting Δ21%
6
Δ-3% VR% [31], the pitch parameter fault is transformed into an additive fault R with the following description.
7
(, Δ2 l 0
9
The fault R is unknown but bounded according to |R| a ", " is the upper bound. Finally the state space model in the faulty
condition is expressed as:
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(», », (1 n , Δ- l n , V T U , R% , 1@ »1 (
0 0 0
1$ % 2 1% - 3% VR%
(35)
10
Where, 2 l(2. ( n and - l n. The state 1% and fault R% are going to be estimated in finite time by the proposed
11
cascade SMOs.
12 13
3-2-2 Finite time Fault Reconstruction by Cascade Sliding Mode Observers
0
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0
In this section, two SMOs are introduced in cascade to reconstruct the fault R% in (35) for each pitch subsystem. In the system of
(35), _`6V 0, _`V 1, thus _`6V _`V . This means that the fault is an unmatched signal. To reconstruct
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14
1
15
such a fault, two SMOs are exploited in cascade [32]. Defining 2@ (2 , 2 ( , -@ , 1@ $, and 1 , the
16
system of (35) is rewritten as:
1$@ % 2@ 1@ 2 1 -@ 3 ( R%
1$ % 1@
17
(36)
Primary Observer: The primary observer is structured as: 1z$@ % 2@ 1z@ 2 1z -@ 3 1z$ % 1z@ ( @ x
(37)
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ACCEPTED MANUSCRIPT
1 2
The discontinuous switching term x is defined as: x = _(w ). w (%) = (%) = 1z (%) − 1 (%) is the output estimation error.
Defining @ (%) = 1z@ (%) − 1@ (%), the error system is derived as: $@ (%) = 2@ @ + 2 + R $ (%) = @ − @ x
Theorem 4: The output estimation error w (%) converges to zero in finite time and the sliding motion occurs on w (%) = (%) = 0 if
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3
(38)
4
@ > |@| ½, where ½ is an arbitrary positive scalar.
5
Proof 4: Consider the Lyapunov function as @ % w % . It is concluded that @$ % w $w w @ ( @ x and @$ < ¾w ¾|@ | ( @ .
8 9 10 11
If @ > |@| ½, then @$ < (½¾w ¾. Since ¾w ¾ 2@ , it holds: @$ < −½2@ . By integration both sides of the last inequality, the convergence time of @ is %@ < in finite time of %@ .
s () ¿
, where @ (0) is the initial value of Lyapunov function. This implies that w (%) converges to zero
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7
Once the sliding motion is achieved on w (%) = 0, it holds from (38) that: @ = @ x´ , where x´ is the equivalent value of
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@
switching term required to maintain the sliding motion on w (%) = 0. x´ can be approximated to any degree of accuracy by replacing x = _7w 9 with
À
|À |Zs
, where @ is a small positive scalar. The error system related to @ is reduced to:
$@ % 2@ @ % R%
(39)
Now, define 5´ x´ @^@ @ as an output for the error system of (39).
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Supplementary Observer: For the error system of (39) with the output 5´ @^@ @ , the supplementary observer is introduced as: ̂@$ % 2@ ̂@ ( Á 5z´ @^@ ̂@
(40)
Defining  ̂@ ( @ , Âw 5z´ ( 5´ @^@ Â, and Á _Âw , the error system becomes as: Â$ 2@  ( Á R
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Theorem 5: For an arbitrary positive scalar Ã, the estimation error  converges to zero in finite time if > " Ã.
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@ Proof 5: Assume the Lyapunov function as: Â . Then, $ ÂÂ$ Â2@ Â ( Â Á ÂR. Since 2@ is stable, it holds: $ <
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|Â|" ( . If > " Ã, then $ < (Ã|Â|. This implies that at the finite time % <
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proof is complete.
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, the error  converges to zero and the
Convergence of  to zero means that ̂@ % → @ % . Hence, the error @ is accurately estimated by the supplementary observer.
Since @ % 1z@ % ( 1@ % , it stands: 1@ % 1z@ % ( @ % , while 1z@ % is estimated by the primary observer. Define 1̅@ % 1z@ % ( ̂@ % , where ̂@ % is estimated by the supplementary observer. Since ̂@ % → @ % , then it is concluded that 1̅@ % → 1@ %
and 1̅@ % is an accurate estimation of 1@ % in finite time. Note that 1z % provided by the primary observer is an exact estimation of 1 % , while 1z@ % is not an accurate estimation of 1@ % . The exact estimation of 1@ % (denoted as 1̅@ % ) is provided using the 12
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supplementary observer. The implementation of cascade observers is shown in Figure 5. The design process for pitch actuators fault
2
diagnosis is summarized in right flowchart of Figure 6. By setting  = 0 in (41), the fault R(%) is reconstructed in finite time as: R³(%) = Á´
Á´ is the continuous form of the switching term in the supplementary observer, which is approximated by replacing Á = , where is a small positive scalar. Finally the fault index ,Æ and the faulty pitch system parameters ~ , ³
_7Âw 9 with
5
are estimated from (43) and (44). R³(%) (1z@ − ̂@ ) + »(1z − )
~ = Ç + ,Æ »,
³ =
2 + ,Æ 2 ~
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Hydraulic pitch system
βref Pitch fault reconstruction
(43)
(44)
βˆ
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βˆ&
veq
weq
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β&
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Figure 5: Implementation of cascade observers for pitch fault reconstruction.
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Remark 5: Note that each pitch subsystem is generally modelled as (33). For each individual pitch system, two observers are employed in cascade in the prescribed structure.
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Remark 6: Note that the proposed scheme is capable to be applied to benchmark-2009 [3]. Generally, the proposed TSMO and
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cascade SMOs can be used for any wind turbine system, where generator, drive-train, and pitch subsystems are modelled with the
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structure similar to (1)-(5). Moreover, the method considers low pressure condition as a typical pitch fault. Whilst pump wear, high air
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content in oil and etc. are modelled in the prescribed form of (34) [13].
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Start
Start
Find T from Section 3.1 (Remark 2)
Find ρ1 from Theorem 4 and design the primary observer in the form of (37)
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Determine l1, l3 from Theorem 1, l2 from Theorem 2 and l4 from Theorems 1&2
Choose an appropriate λ1 to convert the discontinuous function v(t) into a continuous function veq(t)
Choose an appropriate δ to convert the discontinuous function v(t) into a continuous function veq(t)
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Find ρ2 from Theorem 5 and design the supplementary observer in the form of (40)
Use (31) & (32) to revert the observer into the original coordinates
Use (30) for generator fault reconstruction
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Use (29) for aerodynamic load estimation
Choose an appropriate λ2 to convert the discontinuous function w(t) into a continuous function weq(t)
Use (43) & (44) for pitch fault reconstruction
Figure 6: Design algorithm flowcharts for fault reconstruction: Left: drivetrain and generator subsystems faults; right: pitch subsystems faults.
4- Simulation Results
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4-1 Generator fault reconstruction/aerodynamic load estimation
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[3]. Simulations are carried out with a stochastic full-field wind speed profile, generated by TurbSim [30] with mean value of 14
m/sec as in Figure 7. The observer parameters are designed as: @ = 13, = 13, > = 1.24, v@ = 1, km = −A> , " = 1, = 3, c@ = −10, c = 6.8 − 4, and ¸ = 1 − 2.
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The performance of the proposed TSMO described in section 3-1 is now evaluated on the offshore wind turbine benchmark model
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Figure 7: The stochastic wind speed profile exploited in simulations.
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A fault scenario is simulated in the generator system of the benchmark model. Figure 8 presents the states’ estimations provided by
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the proposed TSMO. As is shown in the figure, the estimation errors converge to zero in finite time and the sliding motion is achieved.
3
The estimations of twisting angle, generator speed, rotor speed, and generator torque converge in 0.45, 0.45, 0.01, and 0.16 s,
4
respectively. Note that the twisting angle is accurately estimated, while it is not measured in wind turbines. The generator fault RSK and its estimation R³SK are illustrated in Figure 9. The estimation signal converges within 0.16 s. As pointed in
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Remark 1, the generator fault R³SK can be converted to an offset fault ∆
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formulation. As stated before, the aerodynamic load (torque)
8
an output in the benchmark model, although FAST simulator has the capability of reporting it. In this study,
11
&'
of aerodynamic torque ̂&' is illustrated in Figure 10, which converges in 0.45 s.
As seen from Figure 8-Figure 10, the proposed TSMO provides accurate and finite time estimations of states, generator fault, and aerodynamic torque.
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is modelled as an
unknown input and augmented into the fault vector R& , as (8). Then, it is estimated from (29) by the proposed TSMO. The estimation
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is not measurable in the reality. Hence, it is not normally defined as
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or change of dynamics fault ∆"! by the proposed
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Figure 8: Twisting angle (É), generator speed (ÊË ), rotor speed (ÊÌ ), and generator torque (ÍË ) estimations.
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Figure 9: Actual and estimated generator fault ÎÍË .
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Figure 10: Actual and estimated aerodynamic torque ÍÏÐÌÑ .
5 6
4-2 Pitch actuators fault reconstruction
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of the benchmark model [3]. The faults are simulated according to the actual fault indicators depicted in Figure 11.
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Three low pressure fault scenarios (including abrupt pressure drop fault) are simulated simultaneously for the three pitch actuators
1.4 1.2 1
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Actual pitch fault indicator 1 Estimated pitch fault indicator 1 Actual pitch fault indicator 2 Estimated pitch fault indicator 2 Actual pitch fault indicator 3 Estimated pitch fault indicator 3
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0.4 0.2 0
0
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0.5
1
1.5
2
2.5
3
3.5
4
Time(s)
Figure 11: Actual and estimated fault indicators for pitch actuators.
The cascade observers proposed in section 3-2 are exploited to estimate the faults of each pitch subsystem. The parameters of observers for each pitch subsystem are designed as follows.
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Pitch actuator 1: @ = 300, = 3000, @ = 1 − 4, = 1 − 3. Pitch actuator 2: @ = 250, = 2500, @ = 1 − 4, = 1 − 3. 16
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Pitch actuator 3: @ = 250, = 2500, @ = 1 − 4, = 1 − 3.
Estimations of the pitch angles and pitch rates together with estimation errors are shown in Figure 12 and Figure 13. The estimation
3
errors are almost zero, which clarifies that the cascade observers estimate the states accurately. These errors converge to zero within a
4
small finite time at initial samples of simulation. However, the errors increase when faults occur. For example, the fault of pitch
5
system 2 is present in interval [0.5-3.5] s. Hence, the estimation error of pitch angle 2 deviates from zero within this interval.
6 7
In addition to states and $ , the additive fault R(%) (in (35)) is reconstructed by the cascade observers for each pitch subsystem.
Then, the fault indicators ,Æ are estimated from (43), which are shown in Figure 11. The estimated indicators follow the actual ones precisely. Once the indicators are determined, the pitch parameters ³ and ~ are computed from (44).
Figure 12: Pitch angles’ estimations.
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Figure 13: Pitch rates’ estimations.
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5- Conclusion
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Wind turbine actuator faults in generator/convertor and pitch subsystems emerge slow actuation and non-optimal power production.
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This paper aims at finite time reconstruction of these faults for a realistic wind turbine model. A terminal sliding mode observer is
4
proposed which faithfully observes the generator and drivetrain subsystems’ states, generator faults, and unknown aerodynamic
5
torque. In addition, two sliding mode observers are suggested in a cascade form to estimate the blade pitch angle, pitch rate, and fault
6
indicator of each hydraulic pitch actuator. Simulation results confirm that all the reconstructed signals, including the states and
7
simultaneous faults are estimated accurately in finite time. Compared to previous studies, this paper addresses the observation and
8
actuator fault reconstruction of wind turbine in finite-time, which makes prompt and appropriate actions possible. The proposed
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schemes are also applicable in fault tolerant control methods, which require fast and accurate fault diagnosis.
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This paper proposes a fault detection algorithm for generator and pitch actuator faults. A novel fault tolerant observer for fault detection is proposed. Generator faults are detected and reconstructed via terminal sliding mode observer. Hydraulic pressure drop is modelled as an additive fault by introducing a fault indicator for each of the pitch systems. For every pitch actuator, two cascaded sliding mode observers are exploited to achieve finite time fault reconstruction. The novel terminal sliding mode observer is used to reconstruct the states, aerodynamic torque, and faults. Numerical simulations are performed in FAST & MATLAB Simulink.
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