Adaptive Sliding Mode Control for PMSG Wind Turbine Systems

Abstract

Generally, wind power turbine rotors have very large diameters, and wind speeds vary according to the location on the turbine from which wind speeds are measured, so wind speed measurements may not be a suitable value for maximum energy production, depending on the measurement location. [...]wind speed measurement is more suitable for systems with severe noise due to the environmental uncertainty caused by measurement location and by noise inherent in their sensors. Because wind power turbine systems have significant uncertainty in their mechanical elements, control techniques are required to overcome this problem. When β is maintained as a constant, we can see the power coefficient Cp has only one maximum value Cpmax that corresponds to the optimal value of Cp [34]. [...]Cpmax can be expressed as Cp−opt(λopt,β)=Cpmax,λopt=ωmopt Rbv, where λopt and ωmopt are the optimal values of tip speed ratio and rotor speed, respectively. Take the time derivative and by Assumption 1, we have V˙1=S⋅S˙=z1⋅z˙1,=z1(1Juτ∗(t)−FJωm+1J(Tm+ΔTm)−ω˙m*),=z1(1J(Fωm+Jω˙m*−Tm−γsgn(z1)−c1Jz1)−FJωm+1J(Tm+ΔTm)−ω˙m*),=z1(−c1 z1+1J(ΔTm−γsgn(z1))),≤−c1 z12−η2|z1|. Since V˙≤0 , the rotation speed error z1 converges to 0 exponentially as time goes to infinity.  □ As in the conventional SMC, if we choose uτ∗(t)=Fωm+Jω˙m*−Tm−(δ+η2)sgn(z1)−c1Jz1 for some positive constant α , then we can show that V˙1≤−ηV1 12 and the control law u∗(t) drives the speed error z1 to zero in finite time tr≤2α|z1(0)| .

1. Introduction

Wind turbines in wind power generation systems can be classified as inductive or synchronous depending on the rotor of the generator [1]. Initially, the wind power generation industry used doubly-fed induction generators, for which the operating speed of the generator can be controlled using a slip. However, permanent magnet synchronous generators are increasingly being adopted for scenarios requiring high generation capacity, easy maintenance management, and cost efficiency, such as offshore wind power generation complexes. Permanent magnet synchronous generators (PMSG) have the advantages of reduced production costs due to developments in processing technology, reduced operating noise due to the use of permanent magnets in the rotors, reduced generator weight, and reduced maintenance costs. They also have the advantage of being able to create high torque output compared to doubly-fed induction generators due to high magnetic flux density and efficiency [2,3,4,5,6,7].

Wind turbines can be designed to operate at fixed speed or at variable speed [2,8]. Fixed-speed operating systems transform wind energy into electrical energy using induction generators directly connected to a three-phase power grid. The rotor of the wind turbine is coupled to the generator shaft with a fixed ratio gearbox and operates at constant speed. By contrast, variable speed operating systems are systems which vary the rotation speed of the generator’s rotors, and they are mainly used in synchronous generators. Variable operating wind power generators do not connect directly with the grid, instead connecting with it through a converter, and the generator is operated through aerodynamic control to adjust the torque, speed, and power. This reduces the mechanical stress and aerodynamic noise, and the generator can be controlled such that the wind power turbine operates at the maximum output coefficient over a wider range of wind speeds [2,9].

Here, the control method for energy production at maximum efficiency is called maximum power point tracking (MPPT) [10]. An MPPT controller calculates the optimal rotor rotation speed for varying wind speeds. Therefore, the location and accuracy of the sensors which measure wind speed can be considered as important factors. To extract maximum power, linear controllers are designed based on an approximated linear model, such as conventional vector control with proportional-integral loops [3,9]. However, these control strategies may not provide satisfactory performances due to the system nonlinearity of the PMSG and wide-range operation points. To improve the performance, a feedback linearizing control based MPPT is proposed [11], where the mechanical rotation speed controller and current controllers are designed via linear control methods. However, this results in a complex control law and has weak robustness against parameter uncertainties and external disturbances. Generally, wind power turbine rotors have very large diameters, and wind speeds vary according to the location on the turbine from which wind speeds are measured, so wind speed measurements may not be a suitable value for maximum energy production, depending on the measurement location. Therefore, wind speed measurement is more suitable for systems with severe noise due to the environmental uncertainty caused by measurement location and by noise inherent in their sensors. Because wind power turbine systems have significant uncertainty in their mechanical elements, control techniques are required to overcome this problem.

Sliding mode control (SMC) has been frequently used as a robust control for disturbance and uncertainty among the various robust control methods. Some papers on the application of SMC to wind energy conversion system have been presented in recent decades [11,12,13,14,15,16,17,18,19,20,21,22]. In the early studies of [11,12], SMC is applied for MPPT in the wind energy conversion system with uncertainties. In [12], an optimal torque SMC strategy for a variable speed wind turbine system is proposed to implement MPPT tasks. In other words, SMC strategy was applied for controlling electromagnetic torque in MPPT for PMSG. In reference [11], an input-output linearization technique and SMC are applied to a wind energy conversion system with unstructured uncertainties. In [13], an SMC based on Enhanced Exponential Reaching Law was proposed and investigated on a grid-connected PMSG wind turbine system. In [14,15], SMC was used to improve robustness under various operating conditions such as parameter changes or load variation, and to reduce errors between desired command values and actual current for d- and q-axis currents. In [16], an induction generator (IG) speed drive was designed with the application of a sliding mode controller and a proposed artificial neural network controller. In [17], optimum torque and rotor speed were studied to be obtained by defining the appropriate sliding surface. In [18], an adaptive SMC was studied, first by designing an SMC for reducing the rotor speed error in PMSG, followed by a design for controller gain and mechanical torque estimation.

For uncertain nonlinear systems with known bounds, classical sliding mode control provides robustness. Recently, the adaptive SMC designs adapt the switching gain online without any predefined knowledge of the bound of uncertainty [23,24]. To avoid over-estimation of gain, super twisting controllers [25,26] and Time-Delayed controllers [27,28,29,30] are proposed. [31] tracked the control of a class of uncertain nonlinear systems where the upper bound of the system uncertainty has explicit dependency on the system states.

In this paper, an adaptive robust sliding mode controller is proposed for a wind turbine system with parametric uncertainties and external disturbance. In the real system, some parameters, such as, mechanical inertia, viscous friction, stator resistance, inductance, are affected by operating conditions and manufacturing tolerance managed within predefined bounds. To improve the robustness and remove the effect of the noise and uncertainty which exist in wind speed measurements, an adaptive sliding mode controller is proposed. The wind speed sensor error is translated to an external torque disturbance with amplitude bound. To reject the torque disturbance, an SMC is designed for the nominal plant using the q-axis current as a virtual input. Next, the real physical control inputs, q-axis voltage and d-axis voltage, are designed to track the designed q-axis current and to regulate the d-axis current, respectively. An adaptation law is designed based on back-stepping technique while accounting for unknown parameters of the wind turbine system. The proposed controller is a dynamic controller which is composed of an SMC controller and parameter adaptation. The rotation speed error of rotor exponentially converges to the reference speed in spite of the external disturbance and unknown parameters.

The remainder of this paper is organized as follows. Chapter 2 briefly describes the wind energy conversion system in wind power generation systems. Chapter 3 defines the problem of noise and uncertainty in controller design, and describes the design of a controller to resolve this problem. Chapter 4 verifies the controller through simulations using MATLAB/Simulink, and Chapter 5 concludes the paper.

2. Modeling of a Wind Energy Conversion System

The wind power generation system model is shown in Figure 1. It is a system that converts wind energy into mechanical energy, and creates electrical energy through a generator. This system can be broadly divided into two parts: the generator side and the electrical grid side. This study focuses on controlling the generator side.

2.1. Wind Turbine Modeling

The kinetic energy of wind and the power of the wind can be expressed by Equations (1) and (2),

E=12m^v2=12(ρAvt)^v2=12ρA^v3t,

_Pw=Et=12ρA^v3=12ρ(π_Rb2)^v3=12ρπ_Rb2 ^v3,

where E is the wind’s kinetic energy, ρ is the air density, A is the area that the wind passes through, v is the velocity of the wind, t is the time, _Pw is the power of the wind which can potentially be used, and _Rb is the radius of the wind power turbine through which the wind passes. The power of the wind expressed in Equation (2) only shows the maximum potential power that the wind has. In actuality, only part of this potential power can be used as electricity via the wind power turbine. The ratio of the wind’s power and the mechanical power which can be generated by the turbine is the power coefficient _Cp . The maximum value of the output coefficient is 59.26%, which is referred to as Betz's limit; however, it actually exists in the range of approximately 25 to 45%, which can be expressed as follows [32,33]:

_Cp=_PmPw,

_Cp(λ,β)=_c1(_c21_λi−_c3β−_c4)^e−_c5λi,1_λi=1λ+0.08β−0.035^β3+1.

where _Pm is the output mechanical power, λ is the tip speed ratio, β is the blade pitch angle, _c1=0.5 , _c2=116 , _c3=0.4 , _c4=5 and _c5=21 . The wind power turbine mechanical energy that can be extracted from the wind can be expressed via Equations (2) and (3) as shown below:

_Pm=12ρπ_{Rb2 Cp}(λ,β)^v3,

Output power according to the rotor rotation speed for each wind speed is shown in Figure 2.

It can be seen that the maximum output energy in Figure 2 varies according to wind speed. The generator’s rotation speed at the maximum output energy is referred to as the optimal rotor speed ( _ωopt=_ωref ), and the technique for maintaining this speed at each wind speed, obtaining maximum output energy, is MPPT (Maximum Power Point Tracking) [10,34]. According to the variation of β , typical characteristics of aerodynamic power coefficient correspond to tip speed ratio are illustrated in Figure 3. When β is maintained as a constant, we can see the power coefficient _Cp has only one maximum value _Cpmax that corresponds to the optimal value of _Cp [34]. Therefore, _Cpmax can be expressed as

_Cp−opt(_λopt,β)=_Cpmax,_λopt=_{ωmopt Rb}v,

where _λopt and _ωmopt are the optimal values of tip speed ratio and rotor speed, respectively. By choosing the optimal values of tip speed ratio for the maximum power coefficient, maximum power can be extracted from the optimal rotor speed.

The wind power turbine’s mechanical energy can be expressed as a product of the torque and rotation speed:

_Pm=_{Tm ωm},

where _Tm is the wind power turbine’s mechanical torque, and _ωm is the wind power turbine rotor’s rotation speed. From Equations (5) and (7), the turbine’s torque is

_Tm=ρπ^R2 _Cp(λ,β)^v32_ωm.

The wind power generator’s basic dynamic equation is

_Tm=_Te+F_ωm+Jd_ωmdt,

where F is the viscous friction coefficient, J is the total inertia, and _Te is the electromagnetic torque.

2.2. PMSG Modeling

Normally, the currents and the voltages in 3-phase systems are represented in dq frame using dq transformation. In dq transformation, the d axis is set as the generator rotor’s N pole, and the q axis is perpendicular to the electrical angle, and according to the stator’s 3-phase current and the electrical elements in a synchronously rotating coordinate axis, are transformed as shown in Figure 4. The PMSG kinetic equation, having been dq transformed, is shown below [1,4,7,9]:

d_iddt=−RL_id+P_{ωm iq}+1L_ud,

d_iqdt=−RL_iq−P_{ωm id}+1LP_{ψm ωm}+1L_uq,

where _id is the d-axis current flowing to the stator, _iq is the q-axis current flowing to the stator, _ud is the input voltage for the stator’s d axis, _uq is the input voltage for the stator’s q axis, _ωm is the generator’s rotor speed, R is the resistance, L is the inductance, P is the number of pole pairs, and _ψm is the magnetic flux of the PMSG.

Then, the electromagnetic torque is

_Te=32P(_{ψm iq}+(_Ld−_Lq)_{iq id}),

where _Te is the electromagnetic torque of the generator. If the inductance of each axis has the same value ( L=_Ld=_Lq ), then Equation (12) can be simplified as

_Te=1.5P_{ψm iq}.

Substitute Equation (13) into Equation (9), then the dynamics of the wind turbine are represented as

d_ωmdt=1J(_Tm−1.5P_{ψm iq}−F_ωm).

From Equations (10), (11), and (14), the whole model is

d_ωmdt=−1J1.5P_{ψm iq}−FJ_ωm+1J(_Tm+Δ_Tm),d_iqdt=−RL_iq−P_{ωm id}+1LP_{ψm ωm}+1L_uq,d_iddt=−RL_id+P_{ωm iq}+1L_ud.

In this model, the viscous friction coefficient F and the total inertia J are known a priori. However, in practical cases, the true values are not the same as the designed ones, because of production variation and changes according to time. Furthermore, the mechanical torque _Tm includes disturbances such as wind speed changes. In the steady state, the mechanical torque _Tm is obtained using wind speed v and rotating speed _ωm from Equation (8). Thus, Δ_Tm represents the time varying signal with bounded amplitude, which is induced by the measurement error of v and the rotation speed error. For the stable and robust operation, we should consider the unknown parameters F and J in the controller design stage.

3. Controller Design

For the nominal turbine’s mechanical torque _Tm , we can obtain from the measured wind speed v and rotor rotation speed _ωm given by MPPT algorithm. However, the measured wind speed has some errors due to sensor noise and may not represent the effective wind speed according to the measurement sensor’s location. When the torque input error is Δ_Tm , the actual mechanical torque input is expressed as the sum of the known nominal value _Tm and the disturbance Δ_Tm . Under these conditions, the controller should track the reference rotation speed _ωm∗ to generate the maximum power.

To have a robustly stabilizing controller, we find a reference current input to stabilize the speed error based on the sliding mode control technique. The obtained control is the reference signal of q-axis current _iq to control the rotation speed. Since the q-axis current _iq is not the control input, the q-axis voltage _uq is designed to make _iq track the reference signal. The other control input _uq , the d-axis voltage, regulates the d-axis current _id . For the unknown parameters, a parameter estimator is designed based on backstepping.

3.1. Sliding Mode Controller Design for the Rotation Speed Error Regulation

A robust torque controller is designed based on the sliding mode control technique to overcome torque disturbances. In general, sliding mode controllers achieve excellent tracking performances and robustness against modeling uncertainty and disturbances. If we define the sliding surface S as a rotating speed error and the system satisfies stable sliding mode condition, then the rotating error slides on the sliding surface, i.e., rotating speed error remains 0. To make a system satisfy the condition of achieving a stable sliding mode, S⋅S˙<0 , where S is the sliding mode plane, and S˙ is the time derivative of S [35,36,37,38].

If the uncertainties in the mechanical torque are taken into account, the wind turbine PMSG system is represented as

d_ωmdt=−1J1.5P_{ψm iq}−FJ_ωm+1J(_Tm+Δ_Tm).

Define the rotor’s rotation speed error (speed tracking error)

_z1=_ωm−_ωm*,

where _ωm* is the reference rotation speed which is obtained by MPPT algorithm based on the measured wind speed, which may include sensor noises and wind speed variation. To have a torque controller, take the time derivative of Equation (17)

_z˙1=_ω˙m−_ω˙m*,=−1J1.5P_{ψm iq}−FJ_ωm+1J(_Tm+Δ_Tm)−_ω˙m*,=1Ju(t)−FJ_ωm+1J(_Tm+Δ_Tm)−_ω˙m*.

For the simple notations, the quantity −1.5P_{ψm iq} is defined as _uτ(t) , which can be interpreted as torque control input to track the reference rotation speed _ωm∗ . To have a robustly stabilizing sliding mode controller, we define the sliding surface as the speed tracking error

S(t)=_z1=ω−_ωm*.

Assumption 1.

For the mechanical torque disturbance Δ_Tm(t) , there exists a positive real number δ that satisfies Equation (20).

|Δ_Tm(t)|≤δ.

The Assumption 1 implies that the influence on the torque by wind speed variation, sensor noises, and model uncertainties is limited by a bounded magnitude. Given the practical environment, this assumption is reasonable and acceptable. Now, we propose a torque control input _uτ∗(t) for a given δ ,

_uτ∗(t)=F_ωm+J_ω˙m*−_Tm−γsgn(_z1)−_c1J_z1,

where _c1>0 , η>0 , and γ=δ+η2 .

Proposition 1.

Suppose that a system (18) satisfies assumption 1 for a positive real number δ . Then, the error system (18) is robustly exponentially stable by the feedback input _uτ∗(t) given as (21).

Proof of Proposition 1.

Define a Lyapunov function candidate as

_V1=12^S2.

Take the time derivative and by Assumption 1, we have

_V˙1=S⋅S˙=_z1⋅_z˙1,=_z1(1J_uτ∗(t)−FJ_ωm+1J(_Tm+Δ_Tm)−_ω˙m*),=_z1(1J(F_ωm+J_ω˙m*−_Tm−γsgn(_z1)−_c1J_z1)−FJ_ωm+1J(_Tm+Δ_Tm)−_ω˙m*),=_z1(−_{c1 z1}+1J(Δ_Tm−γsgn(_z1))),≤−_{c1 z12}−η2|_z1|.

Since V˙≤0 , the rotation speed error _z1 converges to 0 exponentially as time goes to infinity.  □

As in the conventional SMC, if we choose _uτ∗(t)=F_ωm+J_ω˙m*−_Tm−(δ+η2)sgn(_z1)−_c1J_z1 for some positive constant α , then we can show that _V˙1≤−η_V1 ¹² and the control law ^u∗(t) drives the speed error _z1 to zero in finite time _tr≤2α|_z1(0)| .

Remark 1.

From Equation (21), we have the reference q-axis current _iq* to control the torque of PMSG as in the following Equation.

_iq*=−11.5P_ψm(F_ωm+J_ω˙m*−_Tm−γsgn(_z1)−_c1J_z1).

3.2. Voltage Controller Design and Adaptative Estimation for Unknown Parameters

In practical cases, many parameters are different from the designed values or changes over time. In this subsection, we assume that the viscous friction coefficient F and the total inertia J in Equation (16) have unknown values due to manufacturing tolerance. If the coefficients F and J are substituted with estimated values, the torque control input Equation (20) is

_u^τ ^∗(t)=F^_ωm+J^_ω˙m*−_Tm−γsgn(_z1)−_c1J^_z1,

where F^ is the estimated viscous friction F , and J^ is the estimated total inertia J .

Define the input error _z2 as

_z2(t)=_uτ(t)−_u^τ*(t).

Substitute Equation (26) into Equation (18); we have

_z˙1=_ω˙m−_ω˙m*,=1J_uτ(t)−FJ_ωm+1J(_Tm+Δ_Tm)−_ω˙m*,=1J(_z2+_u^τ∗(t))−FJ_ωm+1J(_Tm+Δ_Tm)−_ω˙m*,=1J_z2+1J(F^_ωm+J^_ω˙m*−_Tm−γsgn(_z1)−_c1J^_z1)−FJ_ωm+1J(_Tm+Δ_Tm)−_ω˙m*,=−_{c1 z1}+1J(_z2+(Δ_Tm−γsgn(_z1))+J˜(_{c1 z1}−_ω˙m*)−F˜_ωm),

where J˜=J−J^ , and F˜=F−F^ . Take derivative Equation (25),

ddt(_u^τ∗(t))=F^˙_ωm+F^_ω˙m+J^˙_ω˙m*+J^_ω¨m*−2γδ(_z1)−_c1J^˙_z1−_c1J^_z˙1,=F^˙_ωm+J^˙_ω˙m*+J^_ω¨m*+F^_ω˙m*+(−F^−J^˙+_c1J^)_{c1 z1}+(FJ−_c1)_z2+1J(F^−J^_c1)(Δ_Tm−γsgn(_z1))−2γδ(_z1)+J˜J((_{c1 z2}−_c1J^(_{c1 z1}−_ω˙m*))+F^(_{c1 z1}−_ω˙m*))+F˜J(_c1J^_ωm−(_z2+F^_ωm)).

From Equations (27) and (28), the time derivative of _z2 is

_z˙2=_u˙τ(t)−_u^˙τ∗(t),=−1.5P_ψm(−P_{ωm id}−RL_iq−P_ψmL_ωm+1L_uq)−(F^˙_ωm+J^˙_ω˙m*+J^_ω¨m*+F^_ω˙m*−2γδ(_z1))−((−F^−J^˙+_c1J^)_{c1 z1}+(FJ−_c1)_z2+1J(F^−J^_c1)(Δ_Tm−γsgn(_z1)))−J˜J((_{c1 z2}−_c1J^(_{c1 z1}−_ω˙m*))+F^(_{c1 z1}−_ω˙m*))−F˜J(_c1J^_ωm−(_z2+F^_ωm)).

Remark 2.

In practical control systems, it is impossible to implement the signum function sgn(·) and a generalized function δ(·) . By replacing the signum function with sigmoid function tanh(·) , we obtain similar results. If we use a sigmoid function tanh(·) , the Dirac delta function δ(·) will not appear in _u^˙τ∗(t) . However, because of this approximation, the states are absolutely bounded instead of displaying exponential convergence. Please refer to Appendix A.

Assumption 2.

The variation ranges for total inertia J and viscous friction coefficient F are known, i.e. J∈[_Jmin,_Jmax] and F∈[_Fmin,_Fmax] .

As observed in Equation (27) and (29), the error system contains control input voltage _uq and time derivatives J^˙ and F^˙ . For some Lyapunov function, if we can make the time derivative of Lyapunov function be negative by choosing appropriate functions for _uq , J^˙ , and F^˙ , then the error system is stable in the sense of Lyapunov. Additionally, since PMSG uses a permanent magnet, the d-axis current may be zero. Therefore, the d­-axis current is regulated by control input voltage _ud . In other words, if d-axis current tracking error is _z3 such that _z3=_id−_id* then _z3 goes to zero as time goes to infinity.

Now we propose the d,q-axis control input voltage _ud,_uq , and the estimator of J^,F^ as follows:

_uq=L1.5P_ψm(−F^˙_ωm−J^˙_ω˙m*−J^_ω¨m*+_c1J^˙_z1−_c12J^_z1+2γδ(_z1))+L1.5P_ψm(_{c1 z2}−F^(_ω˙m*−_{c1 z1})+2γ_Jmin|F^−J^_c1|sgn(_z2)+_{c2 z2})+PL_{ωm id}+R_iq+P_{ψm ωm},

_ud=R_id−PL_{ωm iq}−_{c3 z3},

J^˙=−_{c1 z22}+(_c1J^_z2−F^_z2+_z1)(_{c1 z1}−_ω˙m*),

F^˙=−_z2(_c1J^_ωm−_z2−F^_ωm)−_{z1 ωm}.

Proposition 2.

Suppose that Assumption 1 and Assumption 2 are satisfied. By the control inputs (30) and (31) and the parameter estimator (32) and (33), the tracking error states, _z1 , _z2 , _z3 exponentially decay to the origin and the estimated parameters J^,F^ converge to some bounded values.

Proof of Proposition 2. Define a Lyapunov function candidate as

_V2=12_z12+12_z22+12_z32+12J^J˜2+12J^F˜2.

Equation (34) is differentiated as follows:

_V˙2=_{z1 z˙1}+_{z2 z˙2}+_{z3 z˙3}−J˜JJ^˙−F˜JF^˙,=_z1(−_{c1 z1}+1J(_z2+(Δ_Tm−γsgn(_z1))+J˜(_{c1 z1}−_ω˙m*)−F˜_ωm))+_z2(−1.5P_ψm(−P_{ωm id}−RL_iq−P_ψmL_ωm+1L_uq)−(F^˙_ωm+J^˙_ω˙m*+J^_ω¨m*+F^_ω˙m*−2γδ(_z1)))+_z2(−(−F^−J^˙+_c1J^)_{c1 z1}−(FJ−_c1)_z2−1J(F^−J^_c1)(Δ_Tm−γsgn(_z1)))−J˜J_z2((_{c1 z2}−_c1J^(_{c1 z1}−_ω˙m*))+F^(_{c1 z1}−_ω˙m*))−F˜J_z2(_c1J^_ωm−(_z2+F^_ωm))+_z3(−RL_id+P_{ωm iq}+1L_ud)−J˜JJ^˙−F˜JF^˙.

By Assumption 1, we have

|Δ_Tm−γsgn(_z1)|≤2γ.

Hence, _V˙2 can be written as:

_V˙2≤_z1(−_{c1 z1}+1J(_z2+(Δ_Tm−γsgn(_z1))+J˜(_{c1 z1}−_ω˙m*)−F˜_ωm))+_z2(−1.5P_ψm(−P_{ωm id}−RL_iq−P_ψmL_ωm+1L_uq)−(F^˙_ωm+J^˙_ω˙m*+J^_ω¨m*+F^_ω˙m*−2γδ(_z1)))+_z2(−(−F^−J^˙+_c1J^)_{c1 z1}−(FJ−_c1)_z2+2γJ|F^−J^_c1|sgn(_z2))+_z3(−RL_id+P_{ωm iq}+1L_ud)−J˜J((_{c1 z2}−_c1J^(_{c1 z1}−_ω˙m*))_z2+F^(_{c1 z1}−_ω˙m*)_z2+J^˙)−F˜J(_c1J^_{ωm z2}−(_z2+F^_ωm)_z2+F^˙)

From Equations (30)–(33), Equation (37) is rewritten as

_V˙2≤−_{c1 z12}+1J_{z1 z2}+1J_z1(Δ_Tm−γsgn(_z1))−FJ_z22+2γ|F^−J^_c1|(1J−1_Jmin)sgn(_z2)_z2−_{c3 z32},≤−1J(γ−|Δ_Tm|)|_z1|−2γ|F^−J^_c1|(1_Jmin−1J)|_z2|+1J_{z1 z2}−_{c1 z12}−FJ_z22−_{c2 z22}−_{c3 z32}.

By the Cauchy-Schwarz inequality,

_V˙2≤−1J(γ−|Δ_Tm|)|_z1|−2γ|F^−J^_c1|(1_Jmin−1J)|_z2|−(_c1−1J)_z12−(_c2+F−1J)_z22−_{c3 z32},≤−(_c1−1J)_z12−(_c2+F−1J)_z22−_{c3 z32}.

For _c1 , _c2 and _c3 such that _c1>1_Jmin,_c2>max{1−_FmaxJmin,1−_FminJmin},_c3>0 , _V˙2≤0 .

Thus, the error states _z1,_z2 and _z3 decay to zero exponentially. Furthermore, if the error state z=^(_z1z2z3)′ is zero, then J^˙ and F^˙ are also zero. By LaSalle’s Invariance Principle, we can say the estimator output J^,F^ converges to some bounded point. □

Hence, it can be concluded that the proposed controller (30)–(33) of the model of the wind energy conversion system with uncertainties (10), (11) and (16) guarantees that the rotating speed converges to the reference rotating speed _ωm* exponentially. However, the parameter estimation converges to some values which may not the true ones.

4. Simulation Results

In order to verify the proposed adaptive sliding mode controller, a PMSG wind turbine system is simulated using MATLAB/Simulink, where the system parameters and the designed controller gains are presented in Table 1 and Table 2, respectively. The torque input used in the simulation is depicted in Figure 5, which starts at 1000[N⋅m] and drops to 900[N⋅m] at time t=1[sec] . The reference rotor speeds are 75[rad/s] and 70[rad/s] for each torque value. The sinusoidal input 5sin(44t)+5sin(20t)+5sin(52t) is also applied as an external disturbance Δ_Tm which is bounded by δ=17 . The initial values of the controller states F^ and J^ are assumed to be zero.

For the performance analysis, we use an SMC controller designed based on nominal values _Jnom=90 and _Fnom=9 without parameter adaption. As the simulation results, each of the system states and the estimated values are shown in Figure 6. In Figure 6a, the controller wind turbine rotor speeds _ωm and the reference rotor speed _ωm* are shown as a solid line and a dotted line, respectively. The reference rotor speed is initially 75[rad/s], and at time instant t=1[sec] , it changes to 70[rad/s]. In the presence of a torque disturbance and unknown parameters, the rotor rotating speed _ωm exponentially converges to the reference speed _ωm* in a short time using adaptive SMC. However, in the case of generic SMC, the rotor speed converged to 74.8[rad/s] and 69.8[rad/s] with remaining steady state error. The q- and d-axis currents _iq,_id and reference values _iq*,_idref are plotted in Figure 6b,c. The actual currents are solid lines, and the reference values are dotted lines. The q-, d-axis currents converge to the reference values in exponential way. The conversion rate is determined by choosing proper control parameters _c1,_c2,_c3 , as shown in the proof of Proposition 2. The control inputs q-axis voltage _uq and d-axis voltage _ud are depicted in Figure 6d,e. Due to the _z2 , chattering is observed during a reaching phase. The result of parameter estimator is shown in Figure 6d,e. As mentioned before, the estimated parameters converge to some bounded values which may different from the true ones.

5. Conclusions

An adaptive robust sliding mode controller is proposed for a PMSG wind turbine system which has parametric uncertainties and external torque disturbance. The torque disturbance includes the wind speed measurement error and mechanical vibrations, which influence the mechanical torque. The control purpose is to track the reference rotor speed which is given by the MPPT algorithm and to regulate the d-axis current. To obtain a robust controller, a q-axis current is used as virtual control input and q-axis voltage is designed for the q-axis current to track the designed virtual input using an adaptive SMC. Regarding the d-axis current, d-axis voltage is designed for d-axis current regulation. For the unknown parameters J and F, an adaptation law is designed to make the closed loop system be Lyapunov stable. The performance of the controller is verified through generator rotor speed tracking via a simulation using MATLAB/Simulink (R2017b, MathWorks). The error states converge to the origin exponentially, and estimated parameters converged to some bounded values. In addition, for a practical implementation sigmoid function, ultimate boundedness of error system is shown in Appendix A.

In the future, it will be necessary to extend to a complete wind power generation system including a grid side converter connected to a grid.

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Parameter	Value	Unit
The resistance of the PMSG(R)	0.15	Ω
The inductance of the PMSG(L)	5.3	mH
The magnetic flux of the PMSG(ψm)	1.314	wb
The number of pole pairs(P)	4	-
The total inertia(J)	100	kg⋅m2
The minimum of inertia(Jmin)	80	kg⋅m2
The viscous friction coefficient(F)	10	N⋅m⋅s/rad
The upper bound of disturbance (δ)	17	-

Parameter	Value
γ	20
c1	20
c2	20
c3	10

Author Contributions

Conceptualization, K-H.C.; Formal analysis, K-H.C.; Software, Sung-W.L.; Visualization, Sung-Won Lee; Writing - review & editing, K-H.C.

Funding

This research was funded by the Korea Electrical Power Corporation through the Korea Electrical Engineering and Science Research Institute, grant number R18XA06-56.

Conflicts of Interest

The authors declare no conflict of interest.

Appendix A

For the practical environments, all propositions are modified using sigmoid function tanh(⋅) . The torque control input in Equation (21) is

_uτ∗(t)=F_ωm+J_ω˙m*-_Tm-γtanh(_z1ϕ)-_c1J_z1,

where ϕ is the positive scalar to determine the boundary layer thickness.

Proposition A1.

Suppose that a system (18) satisfies assumption 1 for a positive real number δ . Then, the rotating speed error _z1 is ultimately bounded by the feedback input ^u∗(t) given as (A1).

Proof of Proposition A1.

Define a Lyapunov function candidate as

_V3=12^S2.

Take the time derivative; we have

_V˙3=S⋅S˙=_z1⋅_z˙1,=_z1(1J^u*(t)-FJ_ωm+1J(_Tm+Δ_Tm)-_ω˙m*),=_z1(-_{c1 z1}+1J(Δ_Tm-γtanh(_z1ϕ))),≤-_{c1 z12}+2γJ|_z1|.

Since _V˙3≤0 for all _z1,∍,|_z1|≤2γJ_c1 , the solution _z1 is ultimately bounded. □

The control input Equation (25) is rewritten as the following equation:

_u^τ ^∗(t)=F^_ωm+J^_ω˙m*-_Tm-γtanh(_z1)-_c1J^_z1

Substitute Equation (A4) into Equation (27), we have

_z˙1=_ω˙m-_ω˙m*,=1J_z2+1J(F^_ωm+J^_ω˙m*-_Tm-γtanh(_z1ϕ)-_c1J^_z1)-FJ_ωm+1J(_Tm+Δ_Tm)-_ω˙m*,=-_{c1 z1}+1J(_z2+(Δ_Tm-γtanh(_z1ϕ))+J˜(_{c1 z1}-_ω˙m*)-F˜_ωm).

Take derivative Equation (A4),

ddt(_u^τ∗(t))=F^˙_ωm+F^_ω˙m+J^˙_ω˙m*+J^_ω¨m*-γϕ(1-^tanh2(_z1ϕ))_z˙1-_c1J^˙_z1-_c1J^_z˙1,=F^˙_ωm+J^˙(_ω˙m*-_{c1 z1})+J^_ω¨m*+_c12J^_z1-_{c1 z2}+FJ_z2+F^_ω˙m*-F^_{c1 z1}-γϕ(1-^tanh2(_z1ϕ))(-_{c1 z1}+1J_z2+1J(Δ_Tm-γtanh(_z1ϕ)))-(Δ_Tm-γtanh(_z1ϕ))(J^J_c1-F^J)+F˜J(_c1J^_ωm-_z2-F^_ωm+_ωmγϕ(1-^tanh2(_z1ϕ)))+J˜J(_{c1 z2}-_c1J^(_{c1 z1}-_ω˙m*)+F^(_{c1 z1}-_ω˙m*)-γϕ(1-^tanh2(_z1ϕ))(_{c1 z1}-_ω˙m*))

From Equations (A5) and (A6), the time derivative of _z2 is

_z˙2=u˙(t)-_u^˙τ∗(t),=-1.5P_ψm(-P_{ωm id}-RL_iq-P_ψmL_ωm+1L_uq)-F^˙_ωm-J^˙(_ω˙m*-_{c1 z1})-J^_ω¨m*-F^_ω˙m*+F^_{c1 z1}-_c12J^_z1+_{c1 z2}-FJ_z2+γϕ(1-^tanh2(_z1ϕ))(-_{c1 z1}+1J_z2+1J(Δ_Tm-γtanh(_z1ϕ)))+1J(Δ_Tm-γtanh(_z1ϕ))(J^_c1-F^)-F˜J(_c1J^_ωm-_z2-F^_ωm+_ωmγϕ(1-^tanh2(_z1ϕ)))-J˜J(_{c1 z2}-_c1J^(_{c1 z1}-_ω˙m*)+F^(_{c1 z1}-_ω˙m*)-γϕ(1-^tanh2(_z1ϕ))(_{c1 z1}-_ω˙m*)).

Now we propose the d,q-axis control input voltage _ud,_uq , and the estimator of J^,F^ as follows:

_uq=L1.5P_ψm(-F^˙_ωm-J^˙(_ω˙m*-_{c1 z1})-J^_ω¨m*-_c12J^_z1+_{c1 z2}-F^_ω˙m*+F^_{c1 z1}-γϕ(1-^tanh2(_z1ϕ))_{c1 z1})+L1.5P_ψm(2γ_Jmin|J^_c1-F^+γϕ(1-^tanh2(_z1ϕ))|tanh(_z2θ)+_{c2 z2})+PL_{ωm id}+R_iq+P_{ψm ωm},

_ud=R_id-PL_{ωm iq}-_{c3 z3},

J^˙=-_{c1 z22}+_c1J^_z2(_{c1 z1}-_ω˙m*)-F^_z2(_{c1 z1}-_ω˙m*)+_z2γϕ(1-^tanh2(_z1ϕ))(_{c1 z1}-_ω˙m*)+_{c1 z12}-_{ω˙m* z1},

F^˙=_z22-(_c1J^-F^+γϕ(1-^tanh2(_z1ϕ)))_{ωm z2}-_{z1 ωm},

where ϕ and θ are positive scalars to determine each boundary layer thickness.

Proposition A2.

Suppose that Assumption 1 and Assumption 2 are satisfied. By the control inputs (A8) and (A9) and the parameter estimator (A10) and (A11), the tracking error states, _z1 , _z2 , _z3 are ultimately bounded.

Proof of Proposition A2.

Define a Lyapunov function candidate as

_V4=12_z12+12_z22+12_z32+12J^J˜2+12J^F˜2

Equation (A12) is differentiated as follows:

_V˙4=_{z1 z˙1}+_{z2 z˙2}+_{z3 z˙3}-J˜JJ^˙-F˜JF^˙,=_z1(-_{c1 z1}+1J(_z2+(Δ_Tm-γtanh(_z1ϕ))+J˜(_{c1 z1}-_ω˙m*)-F˜_ωm))+_z2(-1.5P_ψm(-P_{ωm id}-RL_iq-P_ψmL_ωm+1L_uq)-F^˙_ωm-J^˙(_ω˙m*-_{c1 z1}))+_z2(-J^_ω¨m*-F^_ω˙m*+F^_{c1 z1}-_c12J^_z1+_{c1 z2}-FJ_z2+γJϕ(1-^tanh2(_z1ϕ))_z2)+_z2(γϕ(1-^tanh2(_z1ϕ))(-_{c1 z1}+1J(Δ_Tm-γtanh(_z1ϕ))))+_z21J(Δ_Tm-γtanh(_z1ϕ))(J^_c1-F^)-F˜J_z2(_c1J^_ωm-_z2-F^_ωm+_ωmγϕ(1-^tanh2(_z1ϕ)))-J˜J_z2(_{c1 z2}-_c1J^(_{c1 z1}-_ω˙m*)+F^(_{c1 z1}-_ω˙m*)-γϕ(1-^tanh2(_z1ϕ))(_{c1 z1}-_ω˙m*))+_z3(-RL_id+P_{ωm iq}+1L_ud)-J˜JJ^˙-F˜JF^˙.

By using the Equation (36), _V˙4 can be written as:

_V˙4≤-_{c1 z12}+1J_{z1 z2}+1J(Δ_Tm-γtanh(_z1ϕ))_z1+(-FJ+γJϕ(1-^tanh2(_z1ϕ)))_z22+_z2(-1.5P_ψm(-P_{ωm id}-RL_iq-P_ψmL_ωm+1L_uq))+_z2(-F^˙_ωm-J^˙(_ω˙m*-_{c1 z1})-J^_ω¨m*-_c12J^_z1+_{c1 z2}-F^_ω˙m*+F^_{c1 z1}-γϕ(1-^tanh2(_z1ϕ))_{c1 z1})+|_z2|J2γ|γϕ(1-^tanh2(_z1ϕ))+J^_c1-F^|+_z3(-RL_id+P_{ωm iq}+1L_ud)-F˜J(_c1J^_{ωm z2}-_z22-F^_{ωm z2}+_{ωm z2}γϕ(1-^tanh2(_z1ϕ))+_{z1 ωm}+F^˙)-J˜J(_{c1 z22}-_c1J^_z2(_{c1 z1}-_ω˙m*)+F^_z2(_{c1 z1}-_ω˙m*)-_z2γϕ(1-^tanh2(_z1ϕ))(_{c1 z1}-_ω˙m*)-_{c1 z12}+_{ω˙m* z1}+J^˙)

From Equations (A8)-(A11), Equation (A14) is rewritten as:

_V˙4≤1J(Δ_Tm-γtanh(_z1ϕ))_z1-_{c1 z12}+1J_{z1 z2}+(γJϕ(1-^tanh2(_z1ϕ))-FJ-_c2)_z22-_{c3 z32}+2γ|γϕ(1-^tanh2(_z1ϕ))+J^_c1-F^|(1Jsgn(_z2)-1_Jmintanh(_z2θ))_z2,≤1J(Δ_Tm-γtanh(_z1ϕ))_z1-_{c1 z12}+1J_{z1 z2}+(γJϕ-FJ-_c2)_z22-_{c3 z32}+2γ|γϕ+J^_c1-F^|(1J+1_Jmin)|_z2|,≤2γJ|_z1|+2γ|γϕ+J^_c1-F^|(1J+1_Jmin)|_z2|-_{c1 z12}+1J_{z1 z2}+(γJϕ-FJ-_c2)_z22-_{c3 z32}.

By the Cauchy-Schwarz inequality,

_V˙4≤-(_c1-1J)_z12+2γJ|_z1|-(_c2-1J(γϕ+F-1))_z22+2γ|γϕ+J^_c1-F^|(1J+1_Jmin)|_z2|-_{c3 z32}.

Let _α1=_c1-1J , _β1=2γJ , _α2=_c2-1J(γϕ+F-1) , _β2=2γ|γϕ+J^_c1-F^|(1J+1_Jmin) . Then we can rewrite Equation (A16) as

_V˙4≤-_{α1 z12}+_β1|_z1|-_{α2 z22}+_β2|_z2|-_{c3 z32}.

Therefore, the states are ultimately bounded. □

© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

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