# Application of Effective Wind Speed Estimation and New Sliding Mode Observer in Wind Energy Conversion System

Authors: Fengshuo Bian; Keqilao Meng; Yan Jia; Jianlong Ma; Rihan Hai
Journal: Mathematical Problems in Engineering
 (2022)

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## Abstract

The wind speed information measured by the wind speed sensor in the wind turbine generator may differ from sufficient wind speed. Therefore, this paper proposes a new effective wind speed calculation method and applies it to the optimal maximum power point tracking (MPPT) of wind energy. After estimating the turbine torque and its rotor speed, the process reverses the turbine’s aerodynamic model. The extended state observer (ESO) based on sliding mode control is used to estimate the aerodynamic torque, which solves complex and challenging tuning of the traditional ESO parameters, and replaces the conventional PI controller, thereby improving the anti-interference and robustness of the system. The sliding mode observer (SMO) is used to estimate the rotor speed. While satisfying the conditions of Lyapunov’s inequality, the design of the SMO is discussed in detail. The wind speed estimation method proposed for evaluating permanent magnet semidirect drive wind turbine and its application performance in maximum power tracking. The simulation was carried out in MATLAB/Simulink; the simulation confirmed that the estimated wind speed under different wind conditions is accurate. Compared with the traditional PI control, the utilization rate of wind energy is increased by 2%, which can be used for the MPPT control of the wind energy conversion system.


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## Body

## 1. Introduction

With the aggravation of resource crisis and environmental pollution, energy has been paid more and more attention worldwide. As the most promising renewable energy source, wind energy has been widely studied. In recent years, wind power generation technology has developed rapidly [1–4]. Variable-speed wind turbines (VSWTs) have a market share in the current wind power industry due to their potential to extract maximum energy from wind power [5–7]. Scholars have attached great importance to improving wind efficiency in varying wind speeds or maximum power tracking [8–10].The optimal tip speed ratio (TSR) is an impractical but straightforward method that maintains the optimal TSR by adjusting the generator speed [11, 12]. This control method is easy to understand, but it needs accurate wind speed and rotor information. The rotor speed and position information of wind turbines need to be measured by mechanical position sensors. These automated sensors need regular maintenance and inspection to ensure the accurate execution of algorithms. However, wind turbines are generally installed in remote mountainous or hilly areas, bringing great difficulties to the regular maintenance and inspection notion of sensors [13, 14]. The wind speed sensor and the mechanical sensor for measuring the speed increase the unit’s design cost, and the wind speed measured by the wind speed sensor is the point wind speed, not the sufficient wind speed. The wind speed sensor is sensitive to the axial component’s wind speed due to the wind wheel’s blocking effect and the other problems; the variable wind speed measurement makes a big problem in the actual situation. In recent years, the MPPT controller’s research based on sensorless wind speed estimation has become a hot spot.One effective wind speed estimation method is based on machine learning algorithms, such as artificial neural networks and support vector regression. In reference [15], a radial basis function neural network (RBFNN) based on a hybrid intelligent controller based on improved particle swarm optimization (PSO) is proposed to estimate the excellent v and ωref. In reference [16], a virtual anemometer based on a growing neural network is presented. Using it to replace the traditional wind speed sensor can reduce the volume/weight and reduce the cost. This method aims to establish the nonlinear relationship between wind turbines’ output data and sufficient wind speed. However, most of these methods only use the rotor speed and mechanical power as the training data, ignoring other useful output data and affecting the accuracy of the wind speed estimation. An effective wind speed estimation algorithm is based on aerodynamic torque estimation [17]. The whole wind turbine is regarded as an anemometer. The aerodynamic torque obtained by the state observer can be used to solve the sufficient wind speed through the mathematical expressions of wind speed and aerodynamic moment. In reference [18], the aerodynamic torque is estimated by an unscented Kalman filter. The Newton iteration method estimates the TSR and calculates the wind speed through aerodynamic torque and TSR. In literature [19], the nonlinear aerodynamic model is combined with the transmission system’s linear model, and the extended Kalman filter is used for wind speed estimation. Most control strategies use Kalman filters to estimate aerodynamic torque and solve for sufficient wind speed. However, the extended Kalman filter (EKF) method’s main disadvantage is linearizing each iteration system model to approximate the nonlinear system. Therefore, the effective wind speed estimation algorithm based on traditional EKF is not the best in highly nonlinear systems such as wind energy conversion system (WECS).In complex nonlinear systems, traditional PI control cannot meet the system’s requirements, so many controllers based on modern control theories are proposed to improve the system’s performance [20, 21]. In the underactuated system with cascaded external disturbances, a novel sliding mode control (SMC) method based on first-order differential equations is proposed in the literature [22]. Simulation results show that the proposed controller can effectively deal with system disturbances and nonlinearity. Literature [23] puts forward a novel nonsingular fast terminal SMC method. It can overcome the singularity problem of fast terminal SMC technology and eliminate the chattering problem without knowing the upper limit of interference. Simulation n results verify the effectiveness of this method. Compared with the PI controller, the active disturbance suppression control strategy does not depend on the wind turbine’s specific mathematical model. The extended state observer estimates the total disturbance of the system. The feedback controller is used to compensate for the trouble, thereby improving the speed tracking performance.The active disturbance rejection controller has fast convergence and robustness to the system’s parameter changes and has a particular application value in wind energy maximum power tracking [24, 25]. In literature, the wind speed estimation based on ESO in the active interference suppression controller has not been explored. In the position sensorless strategy, SMO is not sensitive to parameter changes and has strong robustness. They are widely used in permanent magnet synchronous motors without position sensors. However, the traditional sliding mode observer adopts discrete switch control, and chattering becomes an inherent characteristic of the sliding mode variable structure system. It cannot be eliminated. The SMO is insensitive to parameter changes and has strong robustness [26, 27]. They are widely used in permanent magnet synchronous motors without position sensors. Therefore, in this article, the author proposes a sensorless position algorithm based on active disturbance suppression control of sliding mode observer and extended state observer to estimate the sufficient wind speed of aerodynamic torque and apply it in the MPPT [28, 29].The structure of this article is as follows. In the second part, the principle and mathematical model of permanent magnet direct-drive wind turbines are explained. The third part discusses creating the active disturbance rejection controller for permanent magnet semidirect drive wind turbines and proposes aerodynamic torque estimation based on ESO. The calculation process of wind speed is explained at the end of this section. The fourth section discusses designing a new type of SMO for permanent magnet semidirect drive wind turbines and proves the observer’s stability. The fifth part introduces simulation research and experimental procedures to evaluate the proposed algorithm and analyze its results. Section six summarizes the article.


## 2. Permanent Magnet Semi-direct Drive Wind Turbine

### 2.1. Mathematical Model of Wind Turbine

The mechanical powerPm is as follows:(1)Pm=12ρπR2v3CPλ,β.The expression of TSRλ is as follows:(2)λ=ωRv.The aerodynamic torqueTa is as follows:(3)Ta=Pmω=12pπR2v3CPλ,βω.There is a gearbox between the generator and the wind wheel of the permanent magnet semidirect drive wind turbine, and the gearbox transformation ratio isk.β is the pitch angle. λ is the TSR. Pm is the mechanical power absorbed by the wind turbine. CP is the wind energy utilization coefficient. R is the wind wheel’s radius. Ta is the aerodynamic torque; ρ is the air density. v is the wind speed.According to a large number of statistical data, theCP can be approximately expressed as(4)Cpβ,λ=0.5176.116.1n−0.4β−5.e−211/n,1n=1λ+0.08β−0.035β3+1.Figure1 is a graph of the wind energy utilization coefficient.Figure 1
Wind energy utilization coefficient.The equation of the transmission system can be approximately expressed as(5)Ta=Te+Bω+Jdωdt,where J is the moment of inertia.


### 2.2. Mathematical Model of Permanent Magnet Synchronous Motor

To make our control easier, we assume three-phase symmetrical distribution of stator winding, electromagnetic symmetry, ignoring the influence of temperature and other factors on the permanent magnet flux linkage. Under this ideal condition, the stator electric equation inα β coordinate system is shown as follows:(6)Ldiαdt=−Rsiα−eα+uαLdiβdt=−Rsiβ−eβ+uβeα=−ψfωesinθe,eβ=ψfωecosθe..Because the stator current response speed of the permanent magnet synchronous motor is much faster than the rotor response speed, the derivative ofωe can be assumed to be 0, and the back EMF can be converted to(7)e˙α=−ωeeβe˙β=ωeeα.Te expression of electromagnetic torque is as follows:(8)Te=32piqidLd−Lq+ψf.In the hidden pole PMSM, the inductance of the d-axis and q-axis is equal to(9)Te=32pψfiq=kMiq,where iα, iβ is α-axis and β-axis current; id, iq is d-axis and q-axis current; RS is stator resistance; uα and uβ are stator voltage in the static two-phase coordinate system; L is stator winding inductance; Ld and Lq are dq axis inductance; ψf is permanent magnet flux chain; θe is the electrical angle of the rotor; ωe is the angular speed of the motor rotor; Te is the electromagnetic torque; p is the number of poles of the motor.


### 2.3. Wind Speed Estimation Based on EKF

## 2.1. Mathematical Model of Wind Turbine

The mechanical powerPm is as follows:(1)Pm=12ρπR2v3CPλ,β.The expression of TSRλ is as follows:(2)λ=ωRv.The aerodynamic torqueTa is as follows:(3)Ta=Pmω=12pπR2v3CPλ,βω.There is a gearbox between the generator and the wind wheel of the permanent magnet semidirect drive wind turbine, and the gearbox transformation ratio isk.β is the pitch angle. λ is the TSR. Pm is the mechanical power absorbed by the wind turbine. CP is the wind energy utilization coefficient. R is the wind wheel’s radius. Ta is the aerodynamic torque; ρ is the air density. v is the wind speed.According to a large number of statistical data, theCP can be approximately expressed as(4)Cpβ,λ=0.5176.116.1n−0.4β−5.e−211/n,1n=1λ+0.08β−0.035β3+1.Figure1 is a graph of the wind energy utilization coefficient.Figure 1
Wind energy utilization coefficient.The equation of the transmission system can be approximately expressed as(5)Ta=Te+Bω+Jdωdt,where J is the moment of inertia.


## 2.2. Mathematical Model of Permanent Magnet Synchronous Motor

To make our control easier, we assume three-phase symmetrical distribution of stator winding, electromagnetic symmetry, ignoring the influence of temperature and other factors on the permanent magnet flux linkage. Under this ideal condition, the stator electric equation inα β coordinate system is shown as follows:(6)Ldiαdt=−Rsiα−eα+uαLdiβdt=−Rsiβ−eβ+uβeα=−ψfωesinθe,eβ=ψfωecosθe..Because the stator current response speed of the permanent magnet synchronous motor is much faster than the rotor response speed, the derivative ofωe can be assumed to be 0, and the back EMF can be converted to(7)e˙α=−ωeeβe˙β=ωeeα.Te expression of electromagnetic torque is as follows:(8)Te=32piqidLd−Lq+ψf.In the hidden pole PMSM, the inductance of the d-axis and q-axis is equal to(9)Te=32pψfiq=kMiq,where iα, iβ is α-axis and β-axis current; id, iq is d-axis and q-axis current; RS is stator resistance; uα and uβ are stator voltage in the static two-phase coordinate system; L is stator winding inductance; Ld and Lq are dq axis inductance; ψf is permanent magnet flux chain; θe is the electrical angle of the rotor; ωe is the angular speed of the motor rotor; Te is the electromagnetic torque; p is the number of poles of the motor.


## 2.3. Wind Speed Estimation Based on EKF

## 3. Design of Active Disturbance Rejection Controller for Wind Turbine

When the system is not dependent on the plant’s input and output, the system can still be estimated accurately under the condition of uncertain input and output. ADRC is mainly divided into three parts: tracking differentiator (TD), nonlinear state error feedback (NLSEF), and extended state observer (ESO).The id = 0 control method is adopted in the optimal TSR control method for MPPT of permanent magnet direct-drive wind turbines. Due to the nonlinearity, strong coupling, and uncertainty of wind turbines, the ADRC controller is used to replace the traditional PI speed controller. The design is as follows:(10)ω˙=1JTa−BJω−kMJiq,ω˙=at+biq,(11)at=1JTa−BJωb=−kMJ,where a(t) is the total disturbance and b is the control gain.The principle of ADRC is to use ESO to estimate the disturbance and then compensate for it. The standard form is constructed as follows:(12)ω˙=biq.From the above equation, it can be seen that the disturbance a(t) contains aerodynamic torque, a moment of inertia, friction coefficient, and other information. The transition process is reasonably arranged to realize the real-time tracking of the input signal. The disturbance is estimated and compensated by ESO to improve the antijamming ability of the system.


### 3.1. TD

In the MPPT of wind turbines, ADRC has a large control quantity and impact on the system due to the excessive error in the initial state. TD is used to arrange the transition process for the rated wind turbine speed.(13)e=ω1−ωrefω˙1=−afale,δ,h0.The expression of the nonlinear integrated control functionfal is as follows:(14)fale,δ,h0=eδsigne,>h0eh1−δ,e≤h0,where a is the velocity factor, ω1 is a tracking signal for ωref, h0 is the filtering factor, and δ is the nonlinear factor.


### 3.2. ESO

The extended state observer obtains the tracking signal and derivative signal of the system output signal and the system disturbance estimation signal to estimate the disturbance. The external disturbance and model error are extended to a new state variable. According to the ESO principle, the following equation can be obtained:(15)e1=z1−ωz˙1=z2−l1fale1,δ,h0+biqz˙2=−l2fale1,δ,h0,where l1 and l2 are the adjustable observer parameters; z1 is the estimated value of the speed ω; z2 is the estimated value of disturbance a(t).


### 3.3. SMC-ESO

In the unified order ADRC, the adjustable ESO and NLSEF are complex and difficult to tune. In this paper, SMC and ADRC are combined to reduce adjustable settings and improve system response speed. The sliding mode control method is introduced into ESO and NLSEF to improve the controller’s parameter tuning and control performance. Rewrite ESO above as(16)ex1=z1−ωz˙1=z2+biqz˙2=Qe3.Q(e) is used to replace the nonlinear comprehensive control functionfal to estimate the wind turbine’s disturbance. Let a˙t=st, and ensure s(t)≤θ. Because wind turbines’ disturbance value is limited, s(t) can be guaranteed to be a bounded function.Let the error equation be(17)ex1=z1−ωex2=z2−at.Combined with the above formula, we can get(18)e˙x1=ex2e˙x2=Qe3−st.The sliding surface is constructed as(19)s=cex1+ex2,where c is the sliding surface parameter and c > 0, and the optimal control function Q(e) is designed to ensure the sliding mode system to be stable quickly. Here, the expression Q(e) is(20)Qe=−cex2−εsigns−qs,where ε, q is the parameter, and ε, q > 0.The Lyapunov function isv = s2/2. When the following conditions are satisfied, the system is stable.(21)ss˙<0.According to the above formula, we can get that(22)ss˙=s−εsigns−at−qs=−εs−sat−qs2≤θ−εs.As long as we select the appropriateε > 0, we can guarantee that the Lyapunov stability condition is satisfied, and the system will gradually reach the equilibrium point in a finite time. The hyperbolic tangent function replaces the switch symbol-function sign (s) to suppress the chattering phenomenon.(23)tanhs=es−e−ses+e−s.


### 3.4. NLSEF

The TD and the ESO generate tracking signals and state variables, respectively. The nonlinear error feedback controller solves the control structure by using functions for the above two parameters’ errors.(24)e2=ω1−z1u0=qfale2,δ,h0u=u0−z2/b,q is the controller parameter; u is the controller output iq.In the MPPT of the wind turbine, the first-order ADRC controller design is shown in the figure.Figure2 is a block diagram of the first-order ADRC structure.Figure 2
First-order ADRC.


### 3.5. Calculation of Sufficient Wind Speed Based on ESO Aerodynamic Torque Estimation

The ESO can estimate the actual disturbance a(t) in the above formula. The expression of aerodynamic torque is shown as follows:(25)T^a=Jat+BJω,Ta=Pmω=12pπR2v3CPλ,βω.In the MPPT control system of a permanent magnet semidirect drive wind turbine, the pitch angle is 0°. Therefore, the expression equation ofCP is as follows:(26)CPβ,λ=0.5176.1161λ−0.035−5.e−211/λ−0.035.The Newton–Raphson method is sufficient. Newton–Raphson method is a method of approximately solving nonlinear algebraic equations. This method expands nonlinear equations through the Taylor formula, discarding the high term and converting them into linear equations. Solving equations is transformed into a process of iterative solving of linear equations to find the equations’ roots.(27)fv=Ta−12pπR2Cpλ,βv3ω=0,fv=fv0+f′v0v−v0.(28)v1=v0−fv0f′v0,v2=v1−fv1f′v1.The real-time wind speedv is solved by successively iterating the Newton–Raphson method.


## 3.1. TD

In the MPPT of wind turbines, ADRC has a large control quantity and impact on the system due to the excessive error in the initial state. TD is used to arrange the transition process for the rated wind turbine speed.(13)e=ω1−ωrefω˙1=−afale,δ,h0.The expression of the nonlinear integrated control functionfal is as follows:(14)fale,δ,h0=eδsigne,>h0eh1−δ,e≤h0,where a is the velocity factor, ω1 is a tracking signal for ωref, h0 is the filtering factor, and δ is the nonlinear factor.


## 3.2. ESO

The extended state observer obtains the tracking signal and derivative signal of the system output signal and the system disturbance estimation signal to estimate the disturbance. The external disturbance and model error are extended to a new state variable. According to the ESO principle, the following equation can be obtained:(15)e1=z1−ωz˙1=z2−l1fale1,δ,h0+biqz˙2=−l2fale1,δ,h0,where l1 and l2 are the adjustable observer parameters; z1 is the estimated value of the speed ω; z2 is the estimated value of disturbance a(t).


## 3.3. SMC-ESO

In the unified order ADRC, the adjustable ESO and NLSEF are complex and difficult to tune. In this paper, SMC and ADRC are combined to reduce adjustable settings and improve system response speed. The sliding mode control method is introduced into ESO and NLSEF to improve the controller’s parameter tuning and control performance. Rewrite ESO above as(16)ex1=z1−ωz˙1=z2+biqz˙2=Qe3.Q(e) is used to replace the nonlinear comprehensive control functionfal to estimate the wind turbine’s disturbance. Let a˙t=st, and ensure s(t)≤θ. Because wind turbines’ disturbance value is limited, s(t) can be guaranteed to be a bounded function.Let the error equation be(17)ex1=z1−ωex2=z2−at.Combined with the above formula, we can get(18)e˙x1=ex2e˙x2=Qe3−st.The sliding surface is constructed as(19)s=cex1+ex2,where c is the sliding surface parameter and c > 0, and the optimal control function Q(e) is designed to ensure the sliding mode system to be stable quickly. Here, the expression Q(e) is(20)Qe=−cex2−εsigns−qs,where ε, q is the parameter, and ε, q > 0.The Lyapunov function isv = s2/2. When the following conditions are satisfied, the system is stable.(21)ss˙<0.According to the above formula, we can get that(22)ss˙=s−εsigns−at−qs=−εs−sat−qs2≤θ−εs.As long as we select the appropriateε > 0, we can guarantee that the Lyapunov stability condition is satisfied, and the system will gradually reach the equilibrium point in a finite time. The hyperbolic tangent function replaces the switch symbol-function sign (s) to suppress the chattering phenomenon.(23)tanhs=es−e−ses+e−s.


## 3.4. NLSEF

The TD and the ESO generate tracking signals and state variables, respectively. The nonlinear error feedback controller solves the control structure by using functions for the above two parameters’ errors.(24)e2=ω1−z1u0=qfale2,δ,h0u=u0−z2/b,q is the controller parameter; u is the controller output iq.In the MPPT of the wind turbine, the first-order ADRC controller design is shown in the figure.Figure2 is a block diagram of the first-order ADRC structure.Figure 2
First-order ADRC.


## 3.5. Calculation of Sufficient Wind Speed Based on ESO Aerodynamic Torque Estimation

The ESO can estimate the actual disturbance a(t) in the above formula. The expression of aerodynamic torque is shown as follows:(25)T^a=Jat+BJω,Ta=Pmω=12pπR2v3CPλ,βω.In the MPPT control system of a permanent magnet semidirect drive wind turbine, the pitch angle is 0°. Therefore, the expression equation ofCP is as follows:(26)CPβ,λ=0.5176.1161λ−0.035−5.e−211/λ−0.035.The Newton–Raphson method is sufficient. Newton–Raphson method is a method of approximately solving nonlinear algebraic equations. This method expands nonlinear equations through the Taylor formula, discarding the high term and converting them into linear equations. Solving equations is transformed into a process of iterative solving of linear equations to find the equations’ roots.(27)fv=Ta−12pπR2Cpλ,βv3ω=0,fv=fv0+f′v0v−v0.(28)v1=v0−fv0f′v0,v2=v1−fv1f′v1.The real-time wind speedv is solved by successively iterating the Newton–Raphson method.


## 4. Design of a New Sliding Mode Observer

### 4.1. Traditional SMO

The traditional sliding mode observer is designed based on the stationary two-phase coordinate system’s space vector equation.(29)Ldi^αdt=−Rsi^α−k1sgni^α−iα+uαLdi^βdt=−Rsi^β−k1sgni^β−iβ+uβi^α, i^β is the estimated value of current in a static two-phase coordinate system. The estimated voltage is a static two-phase coordinate system, and k1 is the sliding mode observer’s gain.By subtracting the above formula [27],(30)Ldi^αdt−diαdt=−Rsi^α−iα−k1sgni^α−iα+uαLdi^βdt−diβdt=−Rsi^β−iβ−k1sgni^β−iβ+uβ.When the estimated current is equal to the actual value of the current, the system reaches the sliding mode surface [28].(31)eα=k1sgni^α−iαeβ=k1sgni^β−iβ.The back EMF in the two-phase stationary coordinate system contains the position information of the motor rotor.(32)θe=−arctaneαeβωe=dθedt,θe is the rotor’s electrical angle, and ωe is the rotor’s electrical angular velocity.Figure3 is a block diagram of the traditional SMC structure.Figure 3
Traditional SMO.


### 4.2. A New SMO

The symbol-function sign used in traditional sliding mode observers will lead to high-frequency chattering in practical application. A new saturation function is needed to replace the traditional symbolic switching function to reduce the chattering.The new functionG (s) is(33)Gs=1,s≥bs3b3tanhs,0≤s<b−s3b3tanhs,−b<s<0−1,s≤−b.where b is the thickness of the boundary layer.Considering the influence of inverter switching, the estimation accuracy will be reduced if the arctangent method is directly used to extract the rotor position information. Therefore, the PLL system is designed to extract rotor position information.Figure4 is a block diagram of the PLL structure.Figure 4
PLL structure block diagram.


### 4.3. Proof of Stability

The Lyapunov stability criterion is applied to analyze the system stability.The sliding surfaces is defined as(34)s=iα−i^αiβ−i^β.From the above formula,(35)Ls˙=−Rs+es−k1Gs.The Lyapunov functionV is established as(36)V=12s2.According to the Lyapunov stability condition, it is known that if the observer is to be stable,(37)V˙=ss˙<0.By combining the above formula,(38)V˙=ss˙=−RLs2+1Lses−k1Gs,−RLs2<0.Therefore, it is only necessary to satisfy the following formula to satisfy the stability condition:(39)1Lses−k1Gs<0.That is, the selection of parameterk1 must meet the following conditions:(40)k1>maxeα,eβ.


## 4.1. Traditional SMO

The traditional sliding mode observer is designed based on the stationary two-phase coordinate system’s space vector equation.(29)Ldi^αdt=−Rsi^α−k1sgni^α−iα+uαLdi^βdt=−Rsi^β−k1sgni^β−iβ+uβi^α, i^β is the estimated value of current in a static two-phase coordinate system. The estimated voltage is a static two-phase coordinate system, and k1 is the sliding mode observer’s gain.By subtracting the above formula [27],(30)Ldi^αdt−diαdt=−Rsi^α−iα−k1sgni^α−iα+uαLdi^βdt−diβdt=−Rsi^β−iβ−k1sgni^β−iβ+uβ.When the estimated current is equal to the actual value of the current, the system reaches the sliding mode surface [28].(31)eα=k1sgni^α−iαeβ=k1sgni^β−iβ.The back EMF in the two-phase stationary coordinate system contains the position information of the motor rotor.(32)θe=−arctaneαeβωe=dθedt,θe is the rotor’s electrical angle, and ωe is the rotor’s electrical angular velocity.Figure3 is a block diagram of the traditional SMC structure.Figure 3
Traditional SMO.


## 4.2. A New SMO

The symbol-function sign used in traditional sliding mode observers will lead to high-frequency chattering in practical application. A new saturation function is needed to replace the traditional symbolic switching function to reduce the chattering.The new functionG (s) is(33)Gs=1,s≥bs3b3tanhs,0≤s<b−s3b3tanhs,−b<s<0−1,s≤−b.where b is the thickness of the boundary layer.Considering the influence of inverter switching, the estimation accuracy will be reduced if the arctangent method is directly used to extract the rotor position information. Therefore, the PLL system is designed to extract rotor position information.Figure4 is a block diagram of the PLL structure.Figure 4
PLL structure block diagram.


## 4.3. Proof of Stability

The Lyapunov stability criterion is applied to analyze the system stability.The sliding surfaces is defined as(34)s=iα−i^αiβ−i^β.From the above formula,(35)Ls˙=−Rs+es−k1Gs.The Lyapunov functionV is established as(36)V=12s2.According to the Lyapunov stability condition, it is known that if the observer is to be stable,(37)V˙=ss˙<0.By combining the above formula,(38)V˙=ss˙=−RLs2+1Lses−k1Gs,−RLs2<0.Therefore, it is only necessary to satisfy the following formula to satisfy the stability condition:(39)1Lses−k1Gs<0.That is, the selection of parameterk1 must meet the following conditions:(40)k1>maxeα,eβ.


## 5. The Simulation

Aiming at the MPPT control strategy of the optimal TSR, the control system of id = 0 is adopted, and the MPPT control in the wind turbine is simulated in Matlab/Simulink.Figure5 is a block diagram of the improved MPPT control structure. Table 1 shows the system parameters.Figure 5
Block diagram of the overall structure.Table 1
Simulation parameters.
ParameterSymbolValueFriction coefficientB0.001 N.mPolar logarithmp4Stator resistanceRs0.958 ΩInductanceL0.012HMoment of inertiaJ0.003 kg.m2Permanent magnet flux linkageψf0.1827 WbWindwheel radiusR5mThe transmission ratio of the inductive gearboxk40Simulation timet60sAir densityρ1.25 kg/m3


### 5.1. Comparative Analysis: Estimation Based on Traditional SMO and New SMO

It can be seen from Figures6 and 7 that in the control system of the sliding mode observer based on the sign function, the error between the estimated speed and the actual speed changes sharply in the range of -1-1 r/min, and the chattering is enormous; in the control system based on the new sliding mode observer, the speed error is close to 0, and the chattering is weakened. The error of the proposed method is reduced by 3% compared with the traditional method.Figure 6
Speed estimation comparison.Figure 7
Speed error comparison.


### 5.2. Aerodynamic Torque Estimation and Wind Speed Estimation Based on EKF and SMC-ESO

Compared with the Kalman filter method, the SMC-ESO-based estimation method has nonlinear advantages in estimating system state. In EKF, the dynamic system modeling equations need to be linearized around each equilibrium point. The calculation of Jacobian matrix elements is more intensive at each new equilibrium point. Another disadvantage of the EKF-based estimation algorithm is that it is sensitive to the initial covariance matrix of process noise and measurement noise.As can be seen from Figures8 and 9, under the natural wind speed simulation, the EKF cannot estimate the aerodynamic torque and wind speed very well due to the strong robustness of sliding mode control, and the estimation accuracy based on SMC-ESO is greatly improved compared with EKF.Figure 8
Torque estimation comparison.Figure 9
Comparison of wind speed estimates.


### 5.3. Comparative Analysis: The Best Blade Tip Speed Ratio Control Strategy under Traditional PI Control and the Best Blade Tip Speed Ratio Control Strategy Based on the New Sliding Mode Observer and Wind Speed Estimation

As can be seen from Figures10 and 11, compared with the traditional PI speed controller, the SMC-ADRC speed controller has a better effect on speed tracking, no overshoot, faster response speed, and strong anti-interference ability under the rapidly changing natural wind. Under SMC-ADRC, the wind energy utilization coefficient CP and tip speed ratio can also reach the optimal value under the condition of a sudden change of wind speed, which improves the MPPT ability of wind turbines.Figure 10
Speed tracking performance comparison.Figure 11
Comparison of wind energy utilization rate.


## 5.1. Comparative Analysis: Estimation Based on Traditional SMO and New SMO

It can be seen from Figures6 and 7 that in the control system of the sliding mode observer based on the sign function, the error between the estimated speed and the actual speed changes sharply in the range of -1-1 r/min, and the chattering is enormous; in the control system based on the new sliding mode observer, the speed error is close to 0, and the chattering is weakened. The error of the proposed method is reduced by 3% compared with the traditional method.Figure 6
Speed estimation comparison.Figure 7
Speed error comparison.


## 5.2. Aerodynamic Torque Estimation and Wind Speed Estimation Based on EKF and SMC-ESO

Compared with the Kalman filter method, the SMC-ESO-based estimation method has nonlinear advantages in estimating system state. In EKF, the dynamic system modeling equations need to be linearized around each equilibrium point. The calculation of Jacobian matrix elements is more intensive at each new equilibrium point. Another disadvantage of the EKF-based estimation algorithm is that it is sensitive to the initial covariance matrix of process noise and measurement noise.As can be seen from Figures8 and 9, under the natural wind speed simulation, the EKF cannot estimate the aerodynamic torque and wind speed very well due to the strong robustness of sliding mode control, and the estimation accuracy based on SMC-ESO is greatly improved compared with EKF.Figure 8
Torque estimation comparison.Figure 9
Comparison of wind speed estimates.


## 5.3. Comparative Analysis: The Best Blade Tip Speed Ratio Control Strategy under Traditional PI Control and the Best Blade Tip Speed Ratio Control Strategy Based on the New Sliding Mode Observer and Wind Speed Estimation

As can be seen from Figures10 and 11, compared with the traditional PI speed controller, the SMC-ADRC speed controller has a better effect on speed tracking, no overshoot, faster response speed, and strong anti-interference ability under the rapidly changing natural wind. Under SMC-ADRC, the wind energy utilization coefficient CP and tip speed ratio can also reach the optimal value under the condition of a sudden change of wind speed, which improves the MPPT ability of wind turbines.Figure 10
Speed tracking performance comparison.Figure 11
Comparison of wind energy utilization rate.


## 6. Conclusion

In this paper, a new wind speed calculation method based on ESO in ADRC is proposed and applied to maximum power tracking of the wind energy conversion system. SMO, which is suitable for calculating the wind turbine’s stator speed, is designed and evaluated. The new transform function is used to replace the traditional switch function sign to improve rotor position estimation accuracy. The stability of the designed observer is analyzed by the Lyapunov stability analysis method. Based on SMC characteristics, the ADRC controller is intended to replace the traditional PI controller, SMC-ADRC is designed to estimate the wind speed, and the wind speed is fed back to the control system. Compared with the conventional strategy, the system’s anti-interference performance is effectively improved, and the cost of a wind speed sensor is reduced. Under the condition of wind speed, the estimation of wind speed and natural wind speed of the wind turbine is practical. Without mechanical sensors, the estimated wind speed and rotor speed signals can be used in the MPPT of the wind energy conversion system.In this paper, the sliding mode ADRC is only compared with the traditional PI control, but not with other advanced control methods such as fuzzy control, sliding mode control, and predictive control. Future work will focus on implementing the proposed controller based on the wind energy conversion system experimental platform’s useful wind speed estimation. We will also compare the proposed wind speed estimation strategy using other wind speed estimation strategies based on artificial intelligence technology in maximum power tracking.



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*Source: 8863163-2022-05-04.xml*