# Maximum Wind Power Tracking of Doubly Fed Wind Turbine System Based on Adaptive Gain Second-Order Sliding Mode

Authors: Hongchang Sun; Yaozhen Han; Lvyuan Zhang
Journal: Journal of Control Science and Engineering
 (2018)

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## Abstract

This paper proposes an adaptive gain second-order sliding mode control strategy to track optimal electromagnetic torque and regulate reactive power of doubly fed wind turbine system. Firstly, wind turbine aerodynamic characteristics and doubly fed induction generator (DFIG) modeling are presented. Then, electromagnetic torque error and reactive power error are chosen as sliding variables, and fixed gain super-twisting sliding mode control scheme is designed. Considering that uncertainty upper bound is unknown and is hard to be estimated in actual doubly fed wind turbine system, a gain scheduled law is proposed to compel control parameters variation according to uncertainty upper bound real-time. Adaptive gain second-order sliding mode rotor voltage control method is constructed in detail and finite time stability of doubly fed wind turbine control system is strictly proved. The superiority and robustness of the proposed control scheme are finally evaluated on a 1.5 MW DFIG wind turbine system.


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## Body

## 1. Introduction

In the 21st century, as human population grows and nonregeneration energy is rapidly consumed, climate warming and environmental degradation problems become increasingly serious. Renewable energy such as wind energy, hydroenergy, and nuclear energy is highly favored. Particularly, wind energy, as one of the clean renewable energy sources, is attracting more and more attention [1]. The newly installed capacity of wind power reaches 54.6 GW in 2016 [2].Among the many adopted wind power technologies, doubly fed induction generator (DFIG) has been widely used in the wind power industry and dominated the market due to its excellent operating performances such as light weight, small size, highly cost effective, and small capacity of converter. In order to enhance generating efficiency, wind energy conversion optimization is one of the key issues in the DFIG wind power generation control system [3, 4].Many scholars study various control methods for optimizing wind energy conversion [5]. However, due to the randomness of wind, nonlinearity and uncertainty of generator set, it is hard to achieve wind energy capture rapidly and accurately via linear control theory. Nonlinear control techniques such as fuzzy logic [6], neural networks [7], feedback linearization [8], and backstepping control [9] are widely employed to achieve optimal conversion of wind power system. Despite the recognized advantage of using nonlinear controllers to cope with nonlinear systems, many of these techniques produce control laws with a rather high computational burden. These calculi commonly depend on the system states and also on several model parameters having the secondary effect of reducing control robustness. Other controllers, like fuzzy and neural networks methods, present the drawback of neither being systematic in the design stage nor providing a rigorous stability proof.Sliding mode control is a nonlinear robust control method and has already successfully applied to doubly fed wind turbine system [10–12]. However, as an on-off control, control effect of conventional first-order sliding mode is discontinuous and chattering phenomenon is serious. These drawbacks inevitably increase mechanical and electrical wear and shorten service life.Higher-order sliding mode can cancel the requirement of compelling relative degree to be one, enhance sliding accuracy, attenuate chattering phenomenon, and make control effective continuous [13]. Super-twisting algorithm, called higher-order sliding mode of the second order, has been adopted to achieve maximum wind power tracking and reactive power regulation [14–19]. However, control gains for these schemes cannot be real-time adjusted along with variation of system uncertainty. The rotor current or rotor voltage is conservatively chosen, and this will aggravate control chattering. Paper [20] proposes an adaptive second-order sliding mode control strategy to maximize the energy production simultaneously reducing the mechanical stress on the shaft. Yet, preliminary results of this proposal are applied to a simple single-input single-output topology and based on a unidirectional DFIG. Paper [21] presents a novel adaptive higher-order sliding mode control strategy for DFIG energy conversion system. Adaptive natures of the gains formulate a zero steady state transient performance of the system gifted by a reduced settling time. However, the system cannot converge in finite time. Evangelista et al. [22] achieve adaptive second-order sliding mode control for doubly fed wind turbine in finite time based on receding horizon adaptation time windows algorithm. Yet, the selection of adaptation period parameter should be kept in mind while tuning the whole control system.In this context, to face the stringent specifications of maximum wind power tracking and reactive power control problems of doubly fed wind turbine system where the uncertainties are subjected to fast variations, this paper proposes a new adaptive control gain second-order sliding mode scheme. It is not necessary to know the upper bound of uncertainty in advance by choosing suitable sliding function and designing adaptive gain second-order sliding mode controller. Control parameters can be adjusted according to variation of uncertainty upper bound. The control objectives of maximum wind power tracking and reactive power regulation are achieved by controlling rotor voltage, and the output chattering is greatly restrained.This paper is organized as follows. Section2 presents system modeling of doubly fed wind turbine. The detailed fixed gain and adaptive gain control strategies are explained in Section 3. The proposed control strategy is validated in MATLAB environment which is illustrated in Section 4.


## 2. Doubly Fed Wind Turbine System Modeling

Structure diagram of doubly fed wind turbine system is shown in Figure1, which mainly includes wind turbine, gearbox, DFIG, and converter. Firstly, wind energy is converted to mechanical energy via wind turbine, and then mechanical energy is converted to electric energy by means of DFIG. DFIG stator is directly connected to grid, and rotor is by way of converter. DFIG can feed into the grid via stator and rotor under hypersynchronous mode.Figure 1
Schematic diagram of doubly feed wind turbine system.According to Betz’ Law, the captured mechanical power of wind turbine is(1)Pa=12πρR2Cpλ,βv3,where ρ is air density (kg/m3), R represents blade radius (m), v is wind speed (m/s), and Cp is power conversion efficiency which is nonlinear function of tip speed ratio λ and pitch angle β. Tip speed ratio λ is defined as(2)λ=ωmrRv=ωrRkgbv,where ωmr is rotating speed (rad/s), ωr is DFIG rotor speed, and kgb is gear ratio.Power conversion efficiencyCp can be expressed as [23](3)Cpλ,β=c1c21λs-c3β-c4e-c5/λs+c6λ,where 1/λs=1/(λ+0.08β)-0.035/(1+β3), c1=0.5176, c2=116, c3=0.4, c4=5, c5=21.Wind turbine mechanical power can also be defined as(4)Pa=Taωmr.Then, wind turbine mechanical torque is(5)Ta=Paωmr=12kgbπρR3Ctλ,βv2,where Ct(λ,β)=Cp(λ,β)/λ is wind turbine torque coefficient.Dynamic characteristics of doubly fed wind turbine system can be accurately described via fifth-order differential equations [24], four of which are expressed by flux linkage under d-q synchronous rotating coordinate axis:(6)ψ˙ds=uds-Rsids+ωsψqs,ψ˙qs=uqs-Rsiqs-ωsψds,ψ˙dr=udr-Rridr+ωs-pωrψqr,ψ˙qr=uqr-Rriqr-ωs-pωrψdr,where(7)ψds=Lsids+Lmidr,ψqs=Lsiqs+Lmiqr,ψdr=Lridr+Lmids,ψqr=Lriqr+Lmiqs.In formulas (6) and (7), subscripts sandr denote stator and rotor physical quantities. d and q are physical quantities of d-axis and q-axis. u, i, r, ψ denote voltage, current, resistance, and flux linkage, respectively, ωs is power grid angular frequency, Ls and Lr are self-inductance of stator winding and rotor winding, and Lm is mutual inductance of stator and rotor.The fifth differential equation is kinematic equation:(8)ω˙r=1JTa-Te,where J represents inertia of the whole rotating part and Te is DFIG electromagnetic torque:(9)Te=32pLmiqsidr-idsiqr,where p is number of pole pairs.For achieving decoupling control,d-axis under synchronous rotating coordinate system is oriented to stator flux linkage vector and stator resistance is neglected. Reduced-order model of doubly fed wind turbine system can be deduced from formula (5) to formula (9):(10)i˙qr=-LmVsLeq+ωsidr1-pωsωr-RrLsLeqiqr+LsLequqr,i˙dr=ωsiqr1-pωsωr-RrLsLeqidr+LsLequdr,ω˙r=1JTa-3pLmVs2ωsLsiqr,where Vs is network line voltage, Leq=LsLr-Lm2. Then stator current is calculated as(11)iqs=-LmLsiqr,ids=VsωsLs-LmLsidr.Reactive power is expressed as(12)Qs=3Vs22ωsLs-3LmVs2Lsidr.


## 3. Adaptive Gain Higher-Order Sliding Mode Control Strategy

### 3.1. Control Objective Description

In the constant power zone, the objective is to adjust pitch angle to maintain wind turbine system operating at rated power, while, in the partial load zone, the objective is to track maximum wind energy. Control objectives of this paper are to achieve maximum wind power tracking and reactive power regulation according to grid demand and tolerance range of wind power system.The pitch angleβ is set as zero when wind turbine system operates at the partial load zone. Then wind turbine power coefficient Cp is only related to tip speed ratio λ. As is shown in Figure 2, there is always Cpmax related to optimal tip speed ratio λopt for certain wind turbine. In other words, wind turbine can only operate under specific revolving speed to achieve maximum wind energy conversion efficiency for specific wind speed.Figure 2
The relation between power coefficient and tip speed ratio.When wind turbine operates at maximum power point (Cp=Cpmax, λ=λopt), the corresponding torque is expressed as(13)Topt=πρR5Cpmax2kgb3λopt3ωr2=k0ωr2,where k0=πρR5Cpmax/2kgb3λopt3.The objective of maximum wind power tracking control is further presented as achievingTe=Topt for any wind speed at partial load zone. Considering power regulation ability enhancement and power system stability improvement, the other control objective is reactive power regulation. The choice of reactive power reference value mainly relies on certain evaluation function to attain the optimal one.


### 3.2. Controller Design

For doubly fed wind turbine system (10), this section firstly presents standard super-twisting second-order sliding mode design method. Then considering unknown uncertainty upper bound and system wear reduction, adaptive gain second-order sliding mode control scheme is detaily designed to achieve maximum wind power tracking and reactive power regulation. Finite time stability of the closed-loop system is also strictly analyzed and proved.


#### 3.2.1. Design Procedure under Fixed Control Gain

For tracking maximum wind power and regulating reactive power, the sliding variables are chosen as(14)s1=Topt-Te=k0ωr2-3pLmVs2ωsLsiqr,s2=Qref-Q=Qref-3pVs2LsVsωs-Lmidr.To calculate first-order time derivative ofs1 and s2(15)s˙1=2k0ωrω˙r-3pLmVs2ωsLsi˙qr=f1t,iqr,idr,ωr+g1t,iqr,idr,ωruqr,(16)s˙2=Q˙ref+3pLmVs2ωsLsi˙dr=f2t,iqr,idr,ωr+g2t,iqr,idr,ωrudr,where(17)f1t,iqr,idr,ωr=3pLmVs2ωsLsLmVsLeq+RrLsLeqiqr+ωs-pωr-pLmVsωsLeqωr+2k0ωrJTa-Te,g1t,iqr,idr,ωr=-3pLmVs2ωsLeq,f2t,iqr,idr,ωr=Q˙ref+3pLmVs2Lsωs-pωr-RrLsLeqidr,g2t,iqr,idr,ωr=3pLmVs2Leq.Then there exist positive constantsUMi, Ci, Γmi, ΓMi, qi∈(0,1), satisfying(18)f˙it,iqr,idr,ωr+g˙it,iqr,idr,ωrUMi≤Ci,0<Γmi≤git,iqr,idr,ωr≤ΓMi,fit,iqr,idr,ωrgit,iqr,idr,ωr<qiUMi,where i=1,2. To apply standard super-twisting second-order sliding mode control algorithm(19)uqr=-α1s11/2sign⁡s1+u1qr,u1qr=-β1sign⁡s1uqr≤UM1-uqruqr≤UM1,(20)udr=-α2s21/2sign⁡s2+u1dru1dr=-β2sign⁡s2udr≤UM2-udrudr≤UM2.According to sufficient condition for finite time stability of super-twisting algorithm [25, 26], when control parameters satisfy(21)Γmiβi>Ciαi>2Γmiβi-CiΓmiβi-CiΓMi1+qiΓmi21-qisecond-order sliding mode with respect to si can be established in finite time, and optimal torque tracking and reactive power regulation are achieved.It is observed that uncertainty upper bounds are needed when designing rotor voltage second-order sliding mode controller, yet the upper bounds are usually unknown in practical wind power system and hard to be estimated. Overestimation of these upper bounds may lead to conservative choice of controller parameters which will produce more control effective, aggravate control chattering, and shorten service cycle of wind turbines.


#### 3.2.2. Adaptive Gain Second-Order Sliding Mode Control Design

Adaptive control gain can deal with unknown uncertainty upper bound problem and further restrain rotor voltage chattering. Hereuqr is redesigned as an example because the system has been decoupled, and design procedure of udr is similar to uqr.Due to the existence of parameter uncertainty and external disturbance,f1(t,iqr,idr,ωr) and g1(t,iqr,idr,ωr) may be expressed as(22)f1t,iqr,idr,ωr=f-1t,iqr,idr,ωr+Δf1,(23)g1t,iqr,idr,ωr=g-1t,iqr,idr,ωr+Δg1,where Δf1 and Δg1 mainly include fluctuation of Lm, Vs, ωs, Leq and satisfy(24)Δg1g-1t,iqr,idr,ωr=χqt,iqr,idr,ωr≤χq1<1,where χq1 is unknown upper bound.Carrying out feedback control for system (15)(25)uqr=g-1-1t,iqr,idr,ωr-f-1t,iqr,idr,ωr+vqr;then(26)σ˙1=Δf1-Δg1g-1t,iqr,idr,ωrf-1t,iqr,idr,ωr+1+Δg1g-1t,iqr,idr,ωrvqr=Δf1-χqt,iqr,idr,ωrf-1t,iqr,idr,ωr+1+χqt,iqr,idr,ωrvqr=a~q+b~qvqr,where a~q=Δf1-χq(t,iqr,idr,ωr)f-1(t,iqr,idr,ωr) and b~q=1+χq(t,iqr,idr,ωr).In practice, parametersLm, Vs, ωs, Ls, J vary slowly if they are influenced by uncertainties, and iqr, Ta, Te are also continuous, such that a~q satisfies(27)a~˙≤aq1,where aq1 exists and is unknown.Then aiming at formula (26), control law vqr is designed to establish second-order sliding mode with respect to s1(28)vqr=-αqrs1/2sign⁡s1+vsm1,v˙sm1=-βqr2sign⁡s1,where αqr and βqr are adaptive control gains.b
~
q can be regarded as an uncertain piecewise constant function. Then formulas (26) and (28) may be written as(29)s˙1=-αqrb~qs11/2sign⁡s1+ωq∗,ω˙q∗=-βqrb~q2sign⁡s1+a~˙q+b~˙qvsm1,ωq∗=a~q+b~qvsm1,ωq∗0=0.b
~
˙
q
v
s
m
1 is supposed to have uncertain upper bound χq2>0; meanwhile, control gain βqr is bounded, meaning βqr≤βqr∗, βqr∗>0; then in any finite time interval(30)b~˙qτsm1≤12b~˙q∫0tβqrds1≤12b~˙qβqr∗t≤χq2.Thus, uncertain functionγ˙q=a~˙q+b~˙qτsm1 possesses unknown upper bound; then(31)γ˙q≤aq1+χq2=χq3.Second-order sliding mode control problem of system (26) is converted to design adaptive gain super-twisting algorithm for system (29) to achieve s1→0, s˙1→0 in finite time under conditional formulas (24), (27), (30), and (31).Theorem1 provides adaptive law of control gain αqr, βqr, and guarantees establishment of second-order sliding mode with respect to s1.Theorem 1.
In order to achieve optimal torque tracking of doubly fed wind power system,q-axis rotor voltage uqr is designed as formula (25), auxiliary control law is formula (28), and adaptive control gain αqr, βqr are designed as(32)α˙qr=ωq1δq12signs1-μqαqr>αqrmηqαqr≤αqrm,βqr=2εqαqr;then actual second-order sliding mode with respect to s1 is established after finite time tf>0 and αqr>(2χ3+λq+4εq2)2/12εqλq(1-χq1)-εq(4χ3+1)/λq(1-χq1), where s1≤ηq1, s˙q1≤ηq2, αqrm, ωq1, εq are positive constant.Proof.
To bring in new state vectorz(33)z=z1z2T=s11/2sign⁡s1ωq∗T.
Then formula (29) can be expressed as(34)z˙1=12z1-αqrb~qz1+z2,z˙2=-βqrb~q2z1z1+γ˙q.
Formula (34) is rewritten as matrix form(35)z˙1z˙2=12z1-αqrb~q1-βqrb~q0z1z2+12z11002z10γ˙q.
Considering|a~˙q|≤aq1 and |γ˙q|≤aq1+χq2=χq3, then γ˙q is expressed as(36)γ˙q=ρq22sign⁡s1=ρq22z1z1,where 0<ρq2≤2χq3; then formula (35) is further denoted as(37)z˙1z˙2=A-z1z1z2,where A-(z1)=12z1-αqrb~q1-βqrb~q+ρq20.
It is observed that ifz1,z2→0 in finite time, then s1,s˙1→0 is set up in finite time, and |z1|=|s1|1/2, sign(z1)=sign(s1) are both established.
To choose Lyapunov function(38)Vz1,z2,αqr,βqr=V0+12δ1αqr-αqr∗2+12δ2βqr-βqr∗2,where V0=(λq+4εq2)z12+z22-4εqz1z2=zTPz, P=λq+4εq2-2εq-2εq1, and αqr∗, βqr∗ are positive constant. It is noted that matrix P is always positive if λq>0 and εq is arbitrary real number.
To calculate time derivative of Lyapunov function (38)(39)V˙z1,z2,αqr,βqr=zTA-Tz1P+PA-z1z+1δ1αqr-αqr∗α˙qr+1δ2βqr-βqr∗β˙qr.
Considering formulas (35) and (37)(40)V˙0=zTA-Tz1P+PA-z1z≤-12z1zTΘz.
Considering|a~˙q|≤aq1 and formula (36), Symmetric matrix Θ is calculated as(41)Θ=Θ11Θ12Θ214εq,where Θ11=2λqαqrb~q+4εqb~q(2εqαqr-βqr)+4εqρq2, Θ12=Θ21=(βqrb~q-2εqαqrb~q-λq-4εq2)-ρq2.
To guarantee positive definiteness of matrixΘ, βqr is designed as(42)βqr=2εqαqr.
Then, ifαqr satisfies(43)αqr>2χ3+λq+4εq2212εqλq1-χq1-εq4χ3+1λq1-χq1matrix Θ is positive definite, and minimum eigenvalue λmin(Θ)≥2εq is satisfied.
In view of formulas (40), (42), and (43) being established, then(44)V˙0≤-12z1zTΘz≤-2εq2z1zTz=-εqz1z2,λminPz2≤zTPz≤λmaxPz2,where z2=z12+z22=s1+z22, and(45)z1=s11/2≤z≤V01/2zλmin1/2P.
Then(46)V˙0≤-rqV01/2,where rq=εqλmin1/2(P)/λmax(P).
To combine formulas (39) and (46), then(47)V˙z,α,β=z˙TPz+zTPz˙+1δ1εαα˙qr+1δ2εββ˙qr≤-1z1zTΘz+1δ1εαα˙qr+1δ2εββ˙qr≤-rqV01/2+1δ1εαα˙qr+1δ2εββ˙qr=-rqV01/2-ωq12δ1εα-ωq22δ2εβ+1δ1εαα˙qr+1δ2εββ˙qr+ωq12δ1εα+ωq22δ2εβ,where εα=αqr-αqr∗εβ=βqr-βqr∗.
Considering inequation(a2+b2+c2)1/2≤a+b+c and formula (38), then(48)-rqV01/2-ωq12δ1εα-ωq22δ2εβ≤-ηq0Vz,αqr,βqr,where ηq0=min(rq,ωq1,ωq2). Then(49)V˙z1,z2,αqr,βqr≤-ηq0Vz1,z2,αqr,βqr+1δ1εαα˙qr+1δ2εββ˙qr+ωq12δ1εα+ωq22δ2εβ.
One solution of formula (32) is(50)αqr=αqr0+ωq1δ12·t0≤t≤tf.
Hence,αqr, βqr are both bounded; namely, αqr-αqr∗<0, βqr-βqr∗<0. Then formula (49) is expressed as(51)V˙z1,z2,αqr,βqr≤-ηq0Vz1,z2,αqr,βqr1/2-εα1δ1α˙qr-ωq12δ1-εβ1δ2β˙qr-ωq22δ2,(52)V˙z1,z2,αqr,βqr≤-ηq0Vz1,z2,αqr,βqr1/2+ξq,where ξq=-εα((1/δ1)α˙qr-ωq1/2δ1)-εβ((1/δ2)β˙qr-ωq2/2δ2).
(C1) Provided that t≥0, we have |s1|>μq and αqr>αqrm. Then considering formula (32)(53)α˙qr=ωq1δ12,ξq=-εβ1δ2β˙qr-ωq22δ2.
To chooseεq=(ωq2/ωq1)δ2/δ1 and combine formula (42), then(54)β˙qr=ωq2δ22;then ξq=0, and(55)V˙z1,z2,αqr,βqr≤-ηq0Vz1,z2,αqr,βqr1/2.
Take note thatαqr increases according to α˙qr=ωq1δ1/2. When formula (22) is satisfied, positive definiteness of matrix Θ and formula (55) are guaranteed. Then after finite time tf, |s1|≤μq is established.
(C2) To suppose |s1|<μq, then according to formula (32), the formula(56)α˙qr=-ωq1δ12ifαqr>αqrmηqifαqr≤αqrmand formula (57) may become positive number:(57)ξq=2αqr-αqr∗ωq12δ1ifαqr>αqrm-αqrm-αar∗+ηq·tηqδ1-ωq12δ1ifαqr≤αqrm.
It is worth noting that the expression underαqr≤αqrm is effective in finite time, because as long as αqr<αqrm is valid, αqr will increase according to αqr=αqrm+ηq·t, and then the other expression of formula (57) is effective.
The symbolV˙(z1,z2,αqr,βqr) in formula (52) becomes uncertain in view of formula (57). |s1| may be greater than μq due to the diminution of control gain αqr and βqr. Once |s1| is greater than μq, C1 is established. Hence, s1 reaches |s1|≤μq in finite time during adaptive process and then leaves this range and always stays in a bigger one |s1|≤ηq1, ηq1>μq.
When|s1|≤μq is satisfied, |s˙1| can be estimated via formulas (29) and (32):(58)s˙1≤1-δ1αqrμq1/2+εq1-δ1αqr+χq3t2-t1=η-q2,where t1 and t2 are the moment entering into and leaving |s1|≤μq. When μq≤|s1|≤ηq1(59)s˙1≤1-δ1ηq11/2+εqαqrt2+ωq1ηq1δ12t3-t2+χ3t3-t2=η~q2,where t2 and t3 are the moment entering into and leaving |s1|≤μq. From formulas (58) and (59)(60)s˙1≤max⁡η-q2,η~q2=ηq2.
Namely,s1 can enter into W={s1,s˙1:|s1|≤ηq1,|s˙1|≤ηq2,ηq1>μq} in finite time and stay here. Theorem 1 is proved.


## 3.1. Control Objective Description

In the constant power zone, the objective is to adjust pitch angle to maintain wind turbine system operating at rated power, while, in the partial load zone, the objective is to track maximum wind energy. Control objectives of this paper are to achieve maximum wind power tracking and reactive power regulation according to grid demand and tolerance range of wind power system.The pitch angleβ is set as zero when wind turbine system operates at the partial load zone. Then wind turbine power coefficient Cp is only related to tip speed ratio λ. As is shown in Figure 2, there is always Cpmax related to optimal tip speed ratio λopt for certain wind turbine. In other words, wind turbine can only operate under specific revolving speed to achieve maximum wind energy conversion efficiency for specific wind speed.Figure 2
The relation between power coefficient and tip speed ratio.When wind turbine operates at maximum power point (Cp=Cpmax, λ=λopt), the corresponding torque is expressed as(13)Topt=πρR5Cpmax2kgb3λopt3ωr2=k0ωr2,where k0=πρR5Cpmax/2kgb3λopt3.The objective of maximum wind power tracking control is further presented as achievingTe=Topt for any wind speed at partial load zone. Considering power regulation ability enhancement and power system stability improvement, the other control objective is reactive power regulation. The choice of reactive power reference value mainly relies on certain evaluation function to attain the optimal one.


## 3.2. Controller Design

For doubly fed wind turbine system (10), this section firstly presents standard super-twisting second-order sliding mode design method. Then considering unknown uncertainty upper bound and system wear reduction, adaptive gain second-order sliding mode control scheme is detaily designed to achieve maximum wind power tracking and reactive power regulation. Finite time stability of the closed-loop system is also strictly analyzed and proved.


### 3.2.1. Design Procedure under Fixed Control Gain

For tracking maximum wind power and regulating reactive power, the sliding variables are chosen as(14)s1=Topt-Te=k0ωr2-3pLmVs2ωsLsiqr,s2=Qref-Q=Qref-3pVs2LsVsωs-Lmidr.To calculate first-order time derivative ofs1 and s2(15)s˙1=2k0ωrω˙r-3pLmVs2ωsLsi˙qr=f1t,iqr,idr,ωr+g1t,iqr,idr,ωruqr,(16)s˙2=Q˙ref+3pLmVs2ωsLsi˙dr=f2t,iqr,idr,ωr+g2t,iqr,idr,ωrudr,where(17)f1t,iqr,idr,ωr=3pLmVs2ωsLsLmVsLeq+RrLsLeqiqr+ωs-pωr-pLmVsωsLeqωr+2k0ωrJTa-Te,g1t,iqr,idr,ωr=-3pLmVs2ωsLeq,f2t,iqr,idr,ωr=Q˙ref+3pLmVs2Lsωs-pωr-RrLsLeqidr,g2t,iqr,idr,ωr=3pLmVs2Leq.Then there exist positive constantsUMi, Ci, Γmi, ΓMi, qi∈(0,1), satisfying(18)f˙it,iqr,idr,ωr+g˙it,iqr,idr,ωrUMi≤Ci,0<Γmi≤git,iqr,idr,ωr≤ΓMi,fit,iqr,idr,ωrgit,iqr,idr,ωr<qiUMi,where i=1,2. To apply standard super-twisting second-order sliding mode control algorithm(19)uqr=-α1s11/2sign⁡s1+u1qr,u1qr=-β1sign⁡s1uqr≤UM1-uqruqr≤UM1,(20)udr=-α2s21/2sign⁡s2+u1dru1dr=-β2sign⁡s2udr≤UM2-udrudr≤UM2.According to sufficient condition for finite time stability of super-twisting algorithm [25, 26], when control parameters satisfy(21)Γmiβi>Ciαi>2Γmiβi-CiΓmiβi-CiΓMi1+qiΓmi21-qisecond-order sliding mode with respect to si can be established in finite time, and optimal torque tracking and reactive power regulation are achieved.It is observed that uncertainty upper bounds are needed when designing rotor voltage second-order sliding mode controller, yet the upper bounds are usually unknown in practical wind power system and hard to be estimated. Overestimation of these upper bounds may lead to conservative choice of controller parameters which will produce more control effective, aggravate control chattering, and shorten service cycle of wind turbines.


### 3.2.2. Adaptive Gain Second-Order Sliding Mode Control Design

Adaptive control gain can deal with unknown uncertainty upper bound problem and further restrain rotor voltage chattering. Hereuqr is redesigned as an example because the system has been decoupled, and design procedure of udr is similar to uqr.Due to the existence of parameter uncertainty and external disturbance,f1(t,iqr,idr,ωr) and g1(t,iqr,idr,ωr) may be expressed as(22)f1t,iqr,idr,ωr=f-1t,iqr,idr,ωr+Δf1,(23)g1t,iqr,idr,ωr=g-1t,iqr,idr,ωr+Δg1,where Δf1 and Δg1 mainly include fluctuation of Lm, Vs, ωs, Leq and satisfy(24)Δg1g-1t,iqr,idr,ωr=χqt,iqr,idr,ωr≤χq1<1,where χq1 is unknown upper bound.Carrying out feedback control for system (15)(25)uqr=g-1-1t,iqr,idr,ωr-f-1t,iqr,idr,ωr+vqr;then(26)σ˙1=Δf1-Δg1g-1t,iqr,idr,ωrf-1t,iqr,idr,ωr+1+Δg1g-1t,iqr,idr,ωrvqr=Δf1-χqt,iqr,idr,ωrf-1t,iqr,idr,ωr+1+χqt,iqr,idr,ωrvqr=a~q+b~qvqr,where a~q=Δf1-χq(t,iqr,idr,ωr)f-1(t,iqr,idr,ωr) and b~q=1+χq(t,iqr,idr,ωr).In practice, parametersLm, Vs, ωs, Ls, J vary slowly if they are influenced by uncertainties, and iqr, Ta, Te are also continuous, such that a~q satisfies(27)a~˙≤aq1,where aq1 exists and is unknown.Then aiming at formula (26), control law vqr is designed to establish second-order sliding mode with respect to s1(28)vqr=-αqrs1/2sign⁡s1+vsm1,v˙sm1=-βqr2sign⁡s1,where αqr and βqr are adaptive control gains.b
~
q can be regarded as an uncertain piecewise constant function. Then formulas (26) and (28) may be written as(29)s˙1=-αqrb~qs11/2sign⁡s1+ωq∗,ω˙q∗=-βqrb~q2sign⁡s1+a~˙q+b~˙qvsm1,ωq∗=a~q+b~qvsm1,ωq∗0=0.b
~
˙
q
v
s
m
1 is supposed to have uncertain upper bound χq2>0; meanwhile, control gain βqr is bounded, meaning βqr≤βqr∗, βqr∗>0; then in any finite time interval(30)b~˙qτsm1≤12b~˙q∫0tβqrds1≤12b~˙qβqr∗t≤χq2.Thus, uncertain functionγ˙q=a~˙q+b~˙qτsm1 possesses unknown upper bound; then(31)γ˙q≤aq1+χq2=χq3.Second-order sliding mode control problem of system (26) is converted to design adaptive gain super-twisting algorithm for system (29) to achieve s1→0, s˙1→0 in finite time under conditional formulas (24), (27), (30), and (31).Theorem1 provides adaptive law of control gain αqr, βqr, and guarantees establishment of second-order sliding mode with respect to s1.Theorem 1.
In order to achieve optimal torque tracking of doubly fed wind power system,q-axis rotor voltage uqr is designed as formula (25), auxiliary control law is formula (28), and adaptive control gain αqr, βqr are designed as(32)α˙qr=ωq1δq12signs1-μqαqr>αqrmηqαqr≤αqrm,βqr=2εqαqr;then actual second-order sliding mode with respect to s1 is established after finite time tf>0 and αqr>(2χ3+λq+4εq2)2/12εqλq(1-χq1)-εq(4χ3+1)/λq(1-χq1), where s1≤ηq1, s˙q1≤ηq2, αqrm, ωq1, εq are positive constant.Proof.
To bring in new state vectorz(33)z=z1z2T=s11/2sign⁡s1ωq∗T.
Then formula (29) can be expressed as(34)z˙1=12z1-αqrb~qz1+z2,z˙2=-βqrb~q2z1z1+γ˙q.
Formula (34) is rewritten as matrix form(35)z˙1z˙2=12z1-αqrb~q1-βqrb~q0z1z2+12z11002z10γ˙q.
Considering|a~˙q|≤aq1 and |γ˙q|≤aq1+χq2=χq3, then γ˙q is expressed as(36)γ˙q=ρq22sign⁡s1=ρq22z1z1,where 0<ρq2≤2χq3; then formula (35) is further denoted as(37)z˙1z˙2=A-z1z1z2,where A-(z1)=12z1-αqrb~q1-βqrb~q+ρq20.
It is observed that ifz1,z2→0 in finite time, then s1,s˙1→0 is set up in finite time, and |z1|=|s1|1/2, sign(z1)=sign(s1) are both established.
To choose Lyapunov function(38)Vz1,z2,αqr,βqr=V0+12δ1αqr-αqr∗2+12δ2βqr-βqr∗2,where V0=(λq+4εq2)z12+z22-4εqz1z2=zTPz, P=λq+4εq2-2εq-2εq1, and αqr∗, βqr∗ are positive constant. It is noted that matrix P is always positive if λq>0 and εq is arbitrary real number.
To calculate time derivative of Lyapunov function (38)(39)V˙z1,z2,αqr,βqr=zTA-Tz1P+PA-z1z+1δ1αqr-αqr∗α˙qr+1δ2βqr-βqr∗β˙qr.
Considering formulas (35) and (37)(40)V˙0=zTA-Tz1P+PA-z1z≤-12z1zTΘz.
Considering|a~˙q|≤aq1 and formula (36), Symmetric matrix Θ is calculated as(41)Θ=Θ11Θ12Θ214εq,where Θ11=2λqαqrb~q+4εqb~q(2εqαqr-βqr)+4εqρq2, Θ12=Θ21=(βqrb~q-2εqαqrb~q-λq-4εq2)-ρq2.
To guarantee positive definiteness of matrixΘ, βqr is designed as(42)βqr=2εqαqr.
Then, ifαqr satisfies(43)αqr>2χ3+λq+4εq2212εqλq1-χq1-εq4χ3+1λq1-χq1matrix Θ is positive definite, and minimum eigenvalue λmin(Θ)≥2εq is satisfied.
In view of formulas (40), (42), and (43) being established, then(44)V˙0≤-12z1zTΘz≤-2εq2z1zTz=-εqz1z2,λminPz2≤zTPz≤λmaxPz2,where z2=z12+z22=s1+z22, and(45)z1=s11/2≤z≤V01/2zλmin1/2P.
Then(46)V˙0≤-rqV01/2,where rq=εqλmin1/2(P)/λmax(P).
To combine formulas (39) and (46), then(47)V˙z,α,β=z˙TPz+zTPz˙+1δ1εαα˙qr+1δ2εββ˙qr≤-1z1zTΘz+1δ1εαα˙qr+1δ2εββ˙qr≤-rqV01/2+1δ1εαα˙qr+1δ2εββ˙qr=-rqV01/2-ωq12δ1εα-ωq22δ2εβ+1δ1εαα˙qr+1δ2εββ˙qr+ωq12δ1εα+ωq22δ2εβ,where εα=αqr-αqr∗εβ=βqr-βqr∗.
Considering inequation(a2+b2+c2)1/2≤a+b+c and formula (38), then(48)-rqV01/2-ωq12δ1εα-ωq22δ2εβ≤-ηq0Vz,αqr,βqr,where ηq0=min(rq,ωq1,ωq2). Then(49)V˙z1,z2,αqr,βqr≤-ηq0Vz1,z2,αqr,βqr+1δ1εαα˙qr+1δ2εββ˙qr+ωq12δ1εα+ωq22δ2εβ.
One solution of formula (32) is(50)αqr=αqr0+ωq1δ12·t0≤t≤tf.
Hence,αqr, βqr are both bounded; namely, αqr-αqr∗<0, βqr-βqr∗<0. Then formula (49) is expressed as(51)V˙z1,z2,αqr,βqr≤-ηq0Vz1,z2,αqr,βqr1/2-εα1δ1α˙qr-ωq12δ1-εβ1δ2β˙qr-ωq22δ2,(52)V˙z1,z2,αqr,βqr≤-ηq0Vz1,z2,αqr,βqr1/2+ξq,where ξq=-εα((1/δ1)α˙qr-ωq1/2δ1)-εβ((1/δ2)β˙qr-ωq2/2δ2).
(C1) Provided that t≥0, we have |s1|>μq and αqr>αqrm. Then considering formula (32)(53)α˙qr=ωq1δ12,ξq=-εβ1δ2β˙qr-ωq22δ2.
To chooseεq=(ωq2/ωq1)δ2/δ1 and combine formula (42), then(54)β˙qr=ωq2δ22;then ξq=0, and(55)V˙z1,z2,αqr,βqr≤-ηq0Vz1,z2,αqr,βqr1/2.
Take note thatαqr increases according to α˙qr=ωq1δ1/2. When formula (22) is satisfied, positive definiteness of matrix Θ and formula (55) are guaranteed. Then after finite time tf, |s1|≤μq is established.
(C2) To suppose |s1|<μq, then according to formula (32), the formula(56)α˙qr=-ωq1δ12ifαqr>αqrmηqifαqr≤αqrmand formula (57) may become positive number:(57)ξq=2αqr-αqr∗ωq12δ1ifαqr>αqrm-αqrm-αar∗+ηq·tηqδ1-ωq12δ1ifαqr≤αqrm.
It is worth noting that the expression underαqr≤αqrm is effective in finite time, because as long as αqr<αqrm is valid, αqr will increase according to αqr=αqrm+ηq·t, and then the other expression of formula (57) is effective.
The symbolV˙(z1,z2,αqr,βqr) in formula (52) becomes uncertain in view of formula (57). |s1| may be greater than μq due to the diminution of control gain αqr and βqr. Once |s1| is greater than μq, C1 is established. Hence, s1 reaches |s1|≤μq in finite time during adaptive process and then leaves this range and always stays in a bigger one |s1|≤ηq1, ηq1>μq.
When|s1|≤μq is satisfied, |s˙1| can be estimated via formulas (29) and (32):(58)s˙1≤1-δ1αqrμq1/2+εq1-δ1αqr+χq3t2-t1=η-q2,where t1 and t2 are the moment entering into and leaving |s1|≤μq. When μq≤|s1|≤ηq1(59)s˙1≤1-δ1ηq11/2+εqαqrt2+ωq1ηq1δ12t3-t2+χ3t3-t2=η~q2,where t2 and t3 are the moment entering into and leaving |s1|≤μq. From formulas (58) and (59)(60)s˙1≤max⁡η-q2,η~q2=ηq2.
Namely,s1 can enter into W={s1,s˙1:|s1|≤ηq1,|s˙1|≤ηq2,ηq1>μq} in finite time and stay here. Theorem 1 is proved.


## 3.2.1. Design Procedure under Fixed Control Gain

For tracking maximum wind power and regulating reactive power, the sliding variables are chosen as(14)s1=Topt-Te=k0ωr2-3pLmVs2ωsLsiqr,s2=Qref-Q=Qref-3pVs2LsVsωs-Lmidr.To calculate first-order time derivative ofs1 and s2(15)s˙1=2k0ωrω˙r-3pLmVs2ωsLsi˙qr=f1t,iqr,idr,ωr+g1t,iqr,idr,ωruqr,(16)s˙2=Q˙ref+3pLmVs2ωsLsi˙dr=f2t,iqr,idr,ωr+g2t,iqr,idr,ωrudr,where(17)f1t,iqr,idr,ωr=3pLmVs2ωsLsLmVsLeq+RrLsLeqiqr+ωs-pωr-pLmVsωsLeqωr+2k0ωrJTa-Te,g1t,iqr,idr,ωr=-3pLmVs2ωsLeq,f2t,iqr,idr,ωr=Q˙ref+3pLmVs2Lsωs-pωr-RrLsLeqidr,g2t,iqr,idr,ωr=3pLmVs2Leq.Then there exist positive constantsUMi, Ci, Γmi, ΓMi, qi∈(0,1), satisfying(18)f˙it,iqr,idr,ωr+g˙it,iqr,idr,ωrUMi≤Ci,0<Γmi≤git,iqr,idr,ωr≤ΓMi,fit,iqr,idr,ωrgit,iqr,idr,ωr<qiUMi,where i=1,2. To apply standard super-twisting second-order sliding mode control algorithm(19)uqr=-α1s11/2sign⁡s1+u1qr,u1qr=-β1sign⁡s1uqr≤UM1-uqruqr≤UM1,(20)udr=-α2s21/2sign⁡s2+u1dru1dr=-β2sign⁡s2udr≤UM2-udrudr≤UM2.According to sufficient condition for finite time stability of super-twisting algorithm [25, 26], when control parameters satisfy(21)Γmiβi>Ciαi>2Γmiβi-CiΓmiβi-CiΓMi1+qiΓmi21-qisecond-order sliding mode with respect to si can be established in finite time, and optimal torque tracking and reactive power regulation are achieved.It is observed that uncertainty upper bounds are needed when designing rotor voltage second-order sliding mode controller, yet the upper bounds are usually unknown in practical wind power system and hard to be estimated. Overestimation of these upper bounds may lead to conservative choice of controller parameters which will produce more control effective, aggravate control chattering, and shorten service cycle of wind turbines.


## 3.2.2. Adaptive Gain Second-Order Sliding Mode Control Design

Adaptive control gain can deal with unknown uncertainty upper bound problem and further restrain rotor voltage chattering. Hereuqr is redesigned as an example because the system has been decoupled, and design procedure of udr is similar to uqr.Due to the existence of parameter uncertainty and external disturbance,f1(t,iqr,idr,ωr) and g1(t,iqr,idr,ωr) may be expressed as(22)f1t,iqr,idr,ωr=f-1t,iqr,idr,ωr+Δf1,(23)g1t,iqr,idr,ωr=g-1t,iqr,idr,ωr+Δg1,where Δf1 and Δg1 mainly include fluctuation of Lm, Vs, ωs, Leq and satisfy(24)Δg1g-1t,iqr,idr,ωr=χqt,iqr,idr,ωr≤χq1<1,where χq1 is unknown upper bound.Carrying out feedback control for system (15)(25)uqr=g-1-1t,iqr,idr,ωr-f-1t,iqr,idr,ωr+vqr;then(26)σ˙1=Δf1-Δg1g-1t,iqr,idr,ωrf-1t,iqr,idr,ωr+1+Δg1g-1t,iqr,idr,ωrvqr=Δf1-χqt,iqr,idr,ωrf-1t,iqr,idr,ωr+1+χqt,iqr,idr,ωrvqr=a~q+b~qvqr,where a~q=Δf1-χq(t,iqr,idr,ωr)f-1(t,iqr,idr,ωr) and b~q=1+χq(t,iqr,idr,ωr).In practice, parametersLm, Vs, ωs, Ls, J vary slowly if they are influenced by uncertainties, and iqr, Ta, Te are also continuous, such that a~q satisfies(27)a~˙≤aq1,where aq1 exists and is unknown.Then aiming at formula (26), control law vqr is designed to establish second-order sliding mode with respect to s1(28)vqr=-αqrs1/2sign⁡s1+vsm1,v˙sm1=-βqr2sign⁡s1,where αqr and βqr are adaptive control gains.b
~
q can be regarded as an uncertain piecewise constant function. Then formulas (26) and (28) may be written as(29)s˙1=-αqrb~qs11/2sign⁡s1+ωq∗,ω˙q∗=-βqrb~q2sign⁡s1+a~˙q+b~˙qvsm1,ωq∗=a~q+b~qvsm1,ωq∗0=0.b
~
˙
q
v
s
m
1 is supposed to have uncertain upper bound χq2>0; meanwhile, control gain βqr is bounded, meaning βqr≤βqr∗, βqr∗>0; then in any finite time interval(30)b~˙qτsm1≤12b~˙q∫0tβqrds1≤12b~˙qβqr∗t≤χq2.Thus, uncertain functionγ˙q=a~˙q+b~˙qτsm1 possesses unknown upper bound; then(31)γ˙q≤aq1+χq2=χq3.Second-order sliding mode control problem of system (26) is converted to design adaptive gain super-twisting algorithm for system (29) to achieve s1→0, s˙1→0 in finite time under conditional formulas (24), (27), (30), and (31).Theorem1 provides adaptive law of control gain αqr, βqr, and guarantees establishment of second-order sliding mode with respect to s1.Theorem 1.
In order to achieve optimal torque tracking of doubly fed wind power system,q-axis rotor voltage uqr is designed as formula (25), auxiliary control law is formula (28), and adaptive control gain αqr, βqr are designed as(32)α˙qr=ωq1δq12signs1-μqαqr>αqrmηqαqr≤αqrm,βqr=2εqαqr;then actual second-order sliding mode with respect to s1 is established after finite time tf>0 and αqr>(2χ3+λq+4εq2)2/12εqλq(1-χq1)-εq(4χ3+1)/λq(1-χq1), where s1≤ηq1, s˙q1≤ηq2, αqrm, ωq1, εq are positive constant.Proof.
To bring in new state vectorz(33)z=z1z2T=s11/2sign⁡s1ωq∗T.
Then formula (29) can be expressed as(34)z˙1=12z1-αqrb~qz1+z2,z˙2=-βqrb~q2z1z1+γ˙q.
Formula (34) is rewritten as matrix form(35)z˙1z˙2=12z1-αqrb~q1-βqrb~q0z1z2+12z11002z10γ˙q.
Considering|a~˙q|≤aq1 and |γ˙q|≤aq1+χq2=χq3, then γ˙q is expressed as(36)γ˙q=ρq22sign⁡s1=ρq22z1z1,where 0<ρq2≤2χq3; then formula (35) is further denoted as(37)z˙1z˙2=A-z1z1z2,where A-(z1)=12z1-αqrb~q1-βqrb~q+ρq20.
It is observed that ifz1,z2→0 in finite time, then s1,s˙1→0 is set up in finite time, and |z1|=|s1|1/2, sign(z1)=sign(s1) are both established.
To choose Lyapunov function(38)Vz1,z2,αqr,βqr=V0+12δ1αqr-αqr∗2+12δ2βqr-βqr∗2,where V0=(λq+4εq2)z12+z22-4εqz1z2=zTPz, P=λq+4εq2-2εq-2εq1, and αqr∗, βqr∗ are positive constant. It is noted that matrix P is always positive if λq>0 and εq is arbitrary real number.
To calculate time derivative of Lyapunov function (38)(39)V˙z1,z2,αqr,βqr=zTA-Tz1P+PA-z1z+1δ1αqr-αqr∗α˙qr+1δ2βqr-βqr∗β˙qr.
Considering formulas (35) and (37)(40)V˙0=zTA-Tz1P+PA-z1z≤-12z1zTΘz.
Considering|a~˙q|≤aq1 and formula (36), Symmetric matrix Θ is calculated as(41)Θ=Θ11Θ12Θ214εq,where Θ11=2λqαqrb~q+4εqb~q(2εqαqr-βqr)+4εqρq2, Θ12=Θ21=(βqrb~q-2εqαqrb~q-λq-4εq2)-ρq2.
To guarantee positive definiteness of matrixΘ, βqr is designed as(42)βqr=2εqαqr.
Then, ifαqr satisfies(43)αqr>2χ3+λq+4εq2212εqλq1-χq1-εq4χ3+1λq1-χq1matrix Θ is positive definite, and minimum eigenvalue λmin(Θ)≥2εq is satisfied.
In view of formulas (40), (42), and (43) being established, then(44)V˙0≤-12z1zTΘz≤-2εq2z1zTz=-εqz1z2,λminPz2≤zTPz≤λmaxPz2,where z2=z12+z22=s1+z22, and(45)z1=s11/2≤z≤V01/2zλmin1/2P.
Then(46)V˙0≤-rqV01/2,where rq=εqλmin1/2(P)/λmax(P).
To combine formulas (39) and (46), then(47)V˙z,α,β=z˙TPz+zTPz˙+1δ1εαα˙qr+1δ2εββ˙qr≤-1z1zTΘz+1δ1εαα˙qr+1δ2εββ˙qr≤-rqV01/2+1δ1εαα˙qr+1δ2εββ˙qr=-rqV01/2-ωq12δ1εα-ωq22δ2εβ+1δ1εαα˙qr+1δ2εββ˙qr+ωq12δ1εα+ωq22δ2εβ,where εα=αqr-αqr∗εβ=βqr-βqr∗.
Considering inequation(a2+b2+c2)1/2≤a+b+c and formula (38), then(48)-rqV01/2-ωq12δ1εα-ωq22δ2εβ≤-ηq0Vz,αqr,βqr,where ηq0=min(rq,ωq1,ωq2). Then(49)V˙z1,z2,αqr,βqr≤-ηq0Vz1,z2,αqr,βqr+1δ1εαα˙qr+1δ2εββ˙qr+ωq12δ1εα+ωq22δ2εβ.
One solution of formula (32) is(50)αqr=αqr0+ωq1δ12·t0≤t≤tf.
Hence,αqr, βqr are both bounded; namely, αqr-αqr∗<0, βqr-βqr∗<0. Then formula (49) is expressed as(51)V˙z1,z2,αqr,βqr≤-ηq0Vz1,z2,αqr,βqr1/2-εα1δ1α˙qr-ωq12δ1-εβ1δ2β˙qr-ωq22δ2,(52)V˙z1,z2,αqr,βqr≤-ηq0Vz1,z2,αqr,βqr1/2+ξq,where ξq=-εα((1/δ1)α˙qr-ωq1/2δ1)-εβ((1/δ2)β˙qr-ωq2/2δ2).
(C1) Provided that t≥0, we have |s1|>μq and αqr>αqrm. Then considering formula (32)(53)α˙qr=ωq1δ12,ξq=-εβ1δ2β˙qr-ωq22δ2.
To chooseεq=(ωq2/ωq1)δ2/δ1 and combine formula (42), then(54)β˙qr=ωq2δ22;then ξq=0, and(55)V˙z1,z2,αqr,βqr≤-ηq0Vz1,z2,αqr,βqr1/2.
Take note thatαqr increases according to α˙qr=ωq1δ1/2. When formula (22) is satisfied, positive definiteness of matrix Θ and formula (55) are guaranteed. Then after finite time tf, |s1|≤μq is established.
(C2) To suppose |s1|<μq, then according to formula (32), the formula(56)α˙qr=-ωq1δ12ifαqr>αqrmηqifαqr≤αqrmand formula (57) may become positive number:(57)ξq=2αqr-αqr∗ωq12δ1ifαqr>αqrm-αqrm-αar∗+ηq·tηqδ1-ωq12δ1ifαqr≤αqrm.
It is worth noting that the expression underαqr≤αqrm is effective in finite time, because as long as αqr<αqrm is valid, αqr will increase according to αqr=αqrm+ηq·t, and then the other expression of formula (57) is effective.
The symbolV˙(z1,z2,αqr,βqr) in formula (52) becomes uncertain in view of formula (57). |s1| may be greater than μq due to the diminution of control gain αqr and βqr. Once |s1| is greater than μq, C1 is established. Hence, s1 reaches |s1|≤μq in finite time during adaptive process and then leaves this range and always stays in a bigger one |s1|≤ηq1, ηq1>μq.
When|s1|≤μq is satisfied, |s˙1| can be estimated via formulas (29) and (32):(58)s˙1≤1-δ1αqrμq1/2+εq1-δ1αqr+χq3t2-t1=η-q2,where t1 and t2 are the moment entering into and leaving |s1|≤μq. When μq≤|s1|≤ηq1(59)s˙1≤1-δ1ηq11/2+εqαqrt2+ωq1ηq1δ12t3-t2+χ3t3-t2=η~q2,where t2 and t3 are the moment entering into and leaving |s1|≤μq. From formulas (58) and (59)(60)s˙1≤max⁡η-q2,η~q2=ηq2.
Namely,s1 can enter into W={s1,s˙1:|s1|≤ηq1,|s˙1|≤ηq2,ηq1>μq} in finite time and stay here. Theorem 1 is proved.


## 4. Case Studies

Simulation verification of the proposed control scheme is carried out based on MATLAB platform. Nominal parameters of doubly fed wind turbine system areωs=2π60rad/s, Vs=4602/3V, Ls=35.5mH, Lr=35.5mH, Lm=35.7mH, Rr=228mΩ, Rs=82mΩ, p=2, J=3.662Kgm2, Kgb=25, R=7.3m, Cpmax=0.4, and λopt=7.5. The choice of control parameters is according to formulas (19) to (21) and formula (32) and combines actual simulation experiment. The related fixed gain control parameters are α1=4.5, β1=40, α2=0.1, β2=5, Um1=300, Um2=300, and the adaptive gain parameters are ωq1=2, δ1=2, μq=0.2, ηq1=1, εq1=5, αqrm=0.5, ωd1=2, δd2=2, μd=0.2, ηd=1, εd1=10, αdrm=0.2.Case 1 (gradient wind).
Gradient wind speed model can be described by formulas (61). To choose t1G=6s, t2G=12s and max⁡R=4(61)vw=vbt<t1Gvrampt1G<t<t1G+t2Gmax⁡Rt>t2Gvramp=max⁡R1-t-t2Rt1R-t2R.Figure3 shows gradient wind speed variation. The curves of optimal tracking torque and reactive power regulation under fixed control gain scheme are shown in Figures 4 and 5. It is observed that the two control objectives can be achieved, but the uncertainty upper bound is not easy to measure in actual wind power system. Therefore, adaptive gain second-order sliding mode control scheme is put into effect. Maximum wind power tracking and reactive power regulation effect can also be achieved under adaptive control gain scheme as shown in Figures 6 and 7. Figure 8 shows rotor control voltages which indicate chattering reduction. Figure 9 is d-axis rotor voltage and Figure 10 shows d-q axis rotor current. Control gain regulation according to wind speed variation is shown in Figure 11.Figure 3
Gradient wind speed.Figure 4
Optimal torque tracking under fixed gain.Figure 5
Reactive power regulation under fixed gain.Figure 6
Optimal torque tracking under adaptive gain.Figure 7
Reactive power regulation under adaptive gain.Figure 8
Rotor voltage inq-axis.Figure 9
Rotor voltage ind-axis.Figure 10
Rotor current indq-axis under adaptive gain.Figure 11
Control gain under adaptive gain.Case 2 (random wind).
The random wind speed variation is shown in Figure12. Considering system uncertainties, coefficients of mutual induction Lm and rotor resistance fluctuate ± 10%. Grid voltage and grid frequency fluctuation are ±10 and ±20%, respectively. For highlighting the controller performances, the proposed AGSOSM (adaptive gain second-order sliding mode) method and ASOSM (adaptive second-order sliding mode) controller [25] are, respectively, executed on the wind turbine system. The related ASOSM controller parameters are referred to in [25]. Figure 13 shows that the actual electromagnetic torque can rapidly track desired value, and Figure 14 shows that reactive power regulation can still satisfy the demand under random wind under the two control schemes. However, the proposed scheme can achieve higher control accuracy. As is shown in Figure 15, the rotor control voltage chattering is greatly alleviated under the proposed control method. Rotor current under dq-axis is shown in Figure 16.Figure 12
Random wind speed.Figure 13
Optimal torque tracking.Figure 14
Reactive power regulation.Figure 15
Rotor voltage indq-axis.Figure 16
Rotor current indq-axis.


## 5. Conclusions

For achieving maximum wind power tracking and reactive power regulation of doubly fed wind turbine system, this paper proposed an adaptive gain second-order sliding mode control scheme. Firstly, the system model is established. Then second-order sliding mode rotor voltage controllers are designed after detailed description of the two control objectives. Considering the circumstance of unknown uncertainty upper bound, adaptive gain second-order sliding mode rotor voltages controllers are designed, and finite time stability is proved in detail. Finally, simulation experiments under gradient wind and random wind verify that the proposed control strategy can achieve optimal torque tracking and reactive power regulation and is robust to parameter uncertainties such as network voltage, network frequency, rotor resistance, and mutual inductance. The proposed control method also makes rotor control voltage continuous and the control chattering is greatly restrained.



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*Source: 5342971-2018-01-29.xml*