# A PI-Type Sliding Mode Controller Design for PMSG-Based Wind Turbine

Authors: Jun Liu; Feihang Zhou; Chencong Zhao; Zhuoran Wang
Journal: Complexity
 (2019)

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## Abstract

Because the PMSG (permanent magnet synchronous generator)-based WECS (wind energy conversion system) has some uncertainties, the conventional control strategy with poor robustness is sometimes difficult to meet the performance requirements of control. In order to ensure efficient and stability of the system, this paper proposed a novel PI (proportional-integral)-type SMC (sliding mode control) strategy for PMSG-based WECS uncertainties and presented the detailed analysis and design process. Compared with the conventional control method, the PI-type SMC proposed in this paper not only can make the closed-loop system globally stable, but also has a better robustness and slightly reduced the current ripple and distortion. Finally, the simulation results verify the correctness and effectiveness of this algorithm.


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## Body

## 1. Introduction

Environment problems such air pollution and global warming caused by fossil fuels have drawn the world’s attention to exploration and utilization of renewable energy sources, in recent years [1–5]. At present, wind energy is the fastest growing renewable energy source and is most prevalent in coastal regions spanning temperate and boreal climates [6–8]. There is great potential for wind power development in China, USA, Denmark, and other countries, due to their high average wind velocities [9]. Therefore, the research on wind power generation technology has a significant value nowadays. Compared with the constant-speed constant-frequency wind turbine, the variable-speed constant-frequency wind turbines can obtain the maximum energy conversion due to its rotational speed could vary with wind speed to ensure that the system has the OTSR (optimal tip speed ratio) and maximum wind energy utilization coefficient. The variable-speed constant-frequency wind energy conversion system (WECS) consists of DFIG (doubly fed induction generator)-based WECS and PMSG (permanent magnet synchronous generator)-based WECS. The PMSG-based WECS was selected to study in this paper, due to the fact that PMSG has many superior characteristics such as wider speed control range, higher reliability, and more efficient performance compared with DIFG [10].In fact, the practical systems have many uncertainties. The uncertainties of PMSG or PMSM (permanent magnet synchronous motor) consist of the unmodeled converter dynamics and the parameters perturbations [11–14]. A robust control scheme for PMSM uncertainties based on an adaptive DOB (disturbance observer) was introduced in [11]. The results indicated that the controller obtained a good control performance. Reference [12] proposed a robust nonlinear predictive control strategy for PMSM uncertainties. The control system obtained a high speed tracking precision. In [13], a PFC (predictive functional control) + ESO (extended state observer) method was studied. After that, the effectiveness of this new method was verified. Reference [14] proposed a novel decoupled PI current control method for the PMSG-based WECS. This method can successfully achieve improved the transient performances and the nominal performance recovery under the model uncertainty.SMC (sliding mode control) first proposed in the early 1950s has a good robustness and powerful ability to reject the plant uncertainties and disturbances [15, 16]. References [15, 16] summarized the development of SMC and examined key technical research issues and future perspectives. Although the SMC has been widely and extensively employed in some industrial applications for a long term, it is still the researching focus for scholars and worth studying in depth. A DPC (direct power control) based on SMC for the grid-connected WECS was presented in [17]. When the grid voltage is unbalanced, the controller also can regulate the instantaneous active and reactive powers directly in stator stationary reference frame. Reference [18] enhanced the exponential reaching law to improve the control efficiency and performance of SMC used in PMSG-based WECS.An adaptive second-order SMC strategy was explored in [19]. This method can effectively deals with the presence of model uncertainties, intrinsic nonlinear behavior of WECS, and random wind. A SMC method for mismatched uncertainties via a nonlinear DOB was developed in [20]. Meanwhile, the I-SMC (integral sliding-model control) with a good steady-state precision was mentioned in [20, 21]. The PI-type SMC is derived by the I-SMC and FBL (feedback linearization) approach and has been widely used in renewable energy conversion systems and the electric motor drives [21–23]. Furthermore, compared with the I-SMC, the proportional and integral parameters of the PI-type SMC are able to be adjusted to better meet the control performance indexes such as the celerity and accuracy.Based on the above background, this paper presents a novel PI-type SMC for the PMSG-based WECS. Although the inspiration for this paper comes from literature [20–23], the whole designing thought and procedure of the proposed controller are completely different from the methods in [20–23]. Furthermore, [20, 23] just only referred to the control of PMSG without the consideration of grid-connected control. And the study subjects in [22, 23] and this paper are different. At the same time, the suppression of flexible drive-train torsional vibration also has been regarded as a control target in this paper. Then, we set up a detailed 2 MW WECS simulation test platform based on MATLAB/SIMULINK/SimPowerSysterms to verify the effectiveness and correctness of the proposed method. A large number of existing packaging modules in SimPowerSysterms are used in the simulation test platform that is relatively close to the real physical system. Finally, the simulation results indicate the proposed strategy has good control performance.


## 2. PMSG-Based WECS Model

The simplified PMSG-based WECS mainly consists of a wind turbine, a flexible drive-train, a PMSG, fully rated converters, and its control level shown in Figure1. In order to capture the maximum wind energy, the maximum power point tracking (MPPT) strategy is adopted to control the machine-side converter (MSC). In the loop control level, the grid-side converter (GSC) control is to regulate the reactive power and keep DC-link voltageUdc stable at 1800V. Meanwhile, in order to suppress the torsional vibration of flexible drive-train, the damping compensation torqueTDamp was introduced.Figure 1
A WECS configuration.


### 2.1. PMSG and MSC Dynamic Model

The PMSG and MSC mathematical model is [5, 20–27](1)disddt=-RsLsdisd+LsqLsdnpωisq+1LsdUsd-1Lsdεsddisqdt=-RsLsqisq-LsdLsqnpωisd-npωψLsq+1LsqUsq-1LsqεsqwhereUsd andUsq are the d-axis and q-axis stator armature voltages.isd andisq are the d-axis and q-axis components of stator currents.Lsd andLsq are the d-axis and q-axis stator inductances.ω is the rotor speed. np represents magnetic pole logarithms. Rs is the stator resistance.ψ represents the permanent magnet chain. And the disturbance vector εsd and εsq represent model uncertainties including the external disturbances and the PWM offset. εsd and εsq were generally assumed to be bounded byDsd andDsd. (2)εsd≤Dsd;εsd≤DsqwhereDsd andDsq are the boundaries of εsd and εsq.The PMSG torque is given by(3)Tg=1.5npLsd-Lsqisdisq+ψisq


### 2.2. GSC Dynamic Model

The GSC dynamic model is given by(4)digddt=-RcLcigd+ωgigq+1LcEgd-1LcUgd-1Lcεgddigqdt=-RcLcigq-ωgigd+1LcEgq-1LcUgq-1Lcεgqwhere Ugd and Ugq are the control voltages. igd and igq are the components of grid-side currents. ωg is the power grid frequency.Lc andRc are the filter inductance and resistance. Egd and Egq express the d-axis and q-axis power grid potential components (usually, Egq = 0). The ωg and Egd can be gotten by the voltage phase-locked loop. εgd and εgq are also the uncertainties and meet(5)εgd≤Dgd;εgd≤DgqDgd and Dgq are the boundaries of εgd and εgq.


## 2.1. PMSG and MSC Dynamic Model

The PMSG and MSC mathematical model is [5, 20–27](1)disddt=-RsLsdisd+LsqLsdnpωisq+1LsdUsd-1Lsdεsddisqdt=-RsLsqisq-LsdLsqnpωisd-npωψLsq+1LsqUsq-1LsqεsqwhereUsd andUsq are the d-axis and q-axis stator armature voltages.isd andisq are the d-axis and q-axis components of stator currents.Lsd andLsq are the d-axis and q-axis stator inductances.ω is the rotor speed. np represents magnetic pole logarithms. Rs is the stator resistance.ψ represents the permanent magnet chain. And the disturbance vector εsd and εsq represent model uncertainties including the external disturbances and the PWM offset. εsd and εsq were generally assumed to be bounded byDsd andDsd. (2)εsd≤Dsd;εsd≤DsqwhereDsd andDsq are the boundaries of εsd and εsq.The PMSG torque is given by(3)Tg=1.5npLsd-Lsqisdisq+ψisq


## 2.2. GSC Dynamic Model

The GSC dynamic model is given by(4)digddt=-RcLcigd+ωgigq+1LcEgd-1LcUgd-1Lcεgddigqdt=-RcLcigq-ωgigd+1LcEgq-1LcUgq-1Lcεgqwhere Ugd and Ugq are the control voltages. igd and igq are the components of grid-side currents. ωg is the power grid frequency.Lc andRc are the filter inductance and resistance. Egd and Egq express the d-axis and q-axis power grid potential components (usually, Egq = 0). The ωg and Egd can be gotten by the voltage phase-locked loop. εgd and εgq are also the uncertainties and meet(5)εgd≤Dgd;εgd≤DgqDgd and Dgq are the boundaries of εgd and εgq.


## 3. PI-Type Sliding Mode Controller Design

### 3.1. Control Objectives

If the state vectorx and the reference state vector x_ref  are(6)x=isdisqigdigqT(7)xref=isd_refisq_refigd_refigq_refTthe control objectives of WECS can be expressed as(8)limt→+∞⁡e=limt→+∞⁡xref-x=0whereeis the error vector. In the above formula, usually we have(9)isd_ref=0isq_ref=2Tg_ref3npψ=2Topt+TDamp3npψigd_ref=KP+KIsUdc_ref-Udcigq_ref=Qg_refEgdwhereKP andKI are the PI (proportional-integral) parameters. The optimal torqueTopt in (9) meets [1, 26–29](10)Topt=Koptω2withKopt=πρR5CP_max2λopt3whereR is the wind wheel radius,ρ is the air density, CP_max is defined as the maximum wind energy conversion coefficient,λopt is the OTSR (optimal tip speed ratio), and the damping compensation torqueTDamp is given by [29–38](11)TDamp=KDHsωwhere KD∈R+ andH(s) is the transfer function of bandpass filter shown in [29–38].


### 3.2. Design of Proportional-Integral (PI)-Type Sliding Mode Controller

The PI-type sliding surfaceScan be defined as(12)S=K+1sIewhere(13)S=S1S2S3S4T(14)K=diag⁡K1,K2,K3,K4(15)I=diag⁡I1,I2,I3,I4s is Laplace variable. Ki∈R+ and Ii∈R+ (i=1,2,3,4). Obviously, we also can get(16)S1εsd≤S1εsd=DsdS1sgn⁡S1S2εsq≤S2εsq=DsqS2sgn⁡S2S4εgd≤S4εgd=DgdS4sgn⁡S4S5εgq≤S5εgq=DgqS5sgn⁡S5if the Lyapunov function is defined as(17)V=12STS=∑i=14Viwhere(18)Vi=12Si2Equations (19)-(22) can be gotten by taking the derivative of (17).(19)
V
˙
1
=
S
1
S
˙
1
=
S
1
K
1
e
˙
s
d
+
I
1
e
s
d
=
S
1
K
1
i
˙
s
d
_
r
e
f
+
R
s
L
s
d
i
s
d
-
L
s
q
L
s
d
n
p
ω
i
s
q
-
1
L
s
d
U
s
d
+
1
L
s
d
ε
s
d
+
I
1
e
s
d
≤
S
1
K
1
i
˙
s
d
_
r
e
f
+
R
s
L
s
d
i
s
d
-
L
s
q
L
s
d
n
p
ω
i
s
q
-
1
L
s
d
U
s
d
+
1
L
s
d
D
s
d
sgn
⁡
S
1
+
I
1
e
s
d
(20)
V
˙
2
=
S
2
S
˙
2
=
S
2
K
2
e
˙
s
q
+
I
2
e
s
q
=
S
2
K
2
i
˙
s
q
_
r
e
f
+
R
s
L
s
q
i
s
q
+
L
s
d
L
s
q
n
p
ω
i
s
q
+
n
p
ω
ψ
L
s
q
-
1
L
s
q
U
s
q
+
1
L
s
q
ε
s
q
+
I
2
e
s
q
≤
S
2
K
2
i
˙
s
q
_
r
e
f
+
R
s
L
s
q
i
s
q
+
L
s
d
L
s
q
n
p
ω
i
s
q
+
n
p
ω
ψ
L
s
q
-
1
L
s
q
U
s
q
+
1
L
s
q
D
s
q
sgn
⁡
S
2
+
I
2
e
s
q
(21)
V
˙
3
=
S
3
S
˙
3
=
S
3
K
3
e
˙
g
d
+
I
3
e
g
d
=
S
3
K
3
i
˙
g
d
_
r
e
f
+
R
c
L
c
i
g
d
-
ω
g
i
g
q
-
1
L
c
E
g
d
+
1
L
c
U
g
d
+
1
L
c
ε
g
d
+
I
3
e
g
d
≤
S
3
K
3
i
˙
g
d
_
r
e
f
+
R
c
L
c
i
g
d
-
ω
g
i
g
q
-
1
L
c
E
g
d
+
1
L
c
U
g
d
+
1
L
c
D
g
d
sgn
⁡
S
4
+
I
3
e
g
d
(22)
V
˙
4
=
S
4
S
˙
4
=
S
4
K
4
e
˙
g
q
+
I
4
e
g
q
=
S
4
K
4
i
˙
g
q
_
r
e
f
+
R
c
L
c
i
g
q
+
ω
g
i
g
d
-
1
L
c
E
g
q
+
1
L
c
U
g
q
+
1
L
c
ε
g
q
+
I
4
e
g
q
≤
S
4
K
4
i
˙
g
q
_
r
e
f
+
R
c
L
c
i
g
q
+
ω
g
i
g
d
-
1
L
c
E
g
q
+
1
L
c
U
g
q
+
1
L
c
D
g
q
sgn
⁡
S
3
+
I
4
e
g
qLet(23)K1i˙sd_ref+RsLsdisd-LsqLsdnpωisq-1LsdUsd+1LsdDsdsgn⁡S1+I1esd=-γ1S1(24)K2i˙sq_ref+RsLsqisq+LsdLsqnpωisq+npωψLsq-1LsqUsq+1LsqDsqsgn⁡S2+I2esq=-γ2S2(25)K3i˙gd_ref+RcLcigd-ωgigq-1LcEgd+1LcUgd+1LcDgdsgn⁡S4+I3egd=-γ3S3(26)K4i˙gq_ref+RcLcigq+ωgigd-1LcEgq+1LcUgq+1LcDgqsgn⁡S4+I4egq=-γ4S4Equation (27) can be gotten.(27)V˙≤-∑i=14γiSi2≤0where γi>0. Therefore, the control laws are given by (28) -(31) and the control structure diagram is shown in Figure 2.(28)Usd=LsdK1γ1S1+I1esd+Lsdi˙sd_ref+Rsisd-Lsqnpωisq+Dsdsgn⁡S1(29)Usq=LsqK2γ2S2+I2esq+Lsqi˙sq_ref+Rsisd+Lsdnpωisd+npωψ+Dsqsgn⁡S2(30)Ugd=-LcK3γ3S3+I3egd-Lci˙gd_ref-Rcigd+Lcωgigq+Egd-Dgdsgn⁡S3(31)Ugq=-LcK4γ4S4+I4egq-Lci˙gq_ref-Rcigq-Lcωgigd+Egq-Dgqsgn⁡S4From (17) and (27), it is clear that the Lyapunov energy functionV is greater than or equal to 0 and the derivative ofV is less than or equal to 0. Hence, the whole system is asymptotically stable based on Lyapunov stability theorem.Figure 2
Control structure diagram of WECS.
(a) (b) (c) (d)


### 3.3. Tracking Accuracy Analysis

In order to analyze the tracking accuracy, we need to introduce Theorem1 here.Theorem 1.
For any dynamic system with state vectorX=x1x2…xnT and assuming L=fX≥0 is a Lyapunov function of system, if L˙≤-∑i=1nkixi2 or L˙≤-∑i=1nkixi where ki∈R+, then limt→∞xi=0.Proof.
There are three cases.
① IfL≡constant≥0, then 0≡L˙≤-∑i=1nkixi2≤0, and further kixi2=0. Due to ki≠0, thus xi=0. This means that there are not continuous and stable equilibrium points fX≡constant>0 such as stable limit cycle in this system.
② AssumingL0=constant, due to L˙≤-∑i=1nkixi2, then(32)L0-L+∞≥∫0+∞kixi2dt≥0.
∀ x i, if limt→∞xi≠0, then ∫0+∞kixi2dt→+∞, and furthermore L+∞→-∞. Obviously, this is in contradiction with L≥0. Thus limt→∞xi=0.
③ GivenL0→+∞, according to Lyapunov stability criterion, the system is stable. So, there must bet1 meeting Lt1=constant. Otherwise L+∞→+∞. This is clearly in contradiction with system stability. If we redefinedt1 as 0, we can get limt→∞xi=0 by ②.
WhenL˙≤-∑i=1nkixi, we have the same conclusion in the same way.
Based on Theorem1, we have(33)limt→+∞⁡S=0000T
From (12) and (33), we can get(34)limt→+∞⁡e=lims→0⁡s2Ks+I-1S=0000TThe above formula shows that the steady-state error of the system is always 0 regardless of the input. Therefore, the proposed strategy can obtain a good tracking accuracy.


## 3.1. Control Objectives

If the state vectorx and the reference state vector x_ref  are(6)x=isdisqigdigqT(7)xref=isd_refisq_refigd_refigq_refTthe control objectives of WECS can be expressed as(8)limt→+∞⁡e=limt→+∞⁡xref-x=0whereeis the error vector. In the above formula, usually we have(9)isd_ref=0isq_ref=2Tg_ref3npψ=2Topt+TDamp3npψigd_ref=KP+KIsUdc_ref-Udcigq_ref=Qg_refEgdwhereKP andKI are the PI (proportional-integral) parameters. The optimal torqueTopt in (9) meets [1, 26–29](10)Topt=Koptω2withKopt=πρR5CP_max2λopt3whereR is the wind wheel radius,ρ is the air density, CP_max is defined as the maximum wind energy conversion coefficient,λopt is the OTSR (optimal tip speed ratio), and the damping compensation torqueTDamp is given by [29–38](11)TDamp=KDHsωwhere KD∈R+ andH(s) is the transfer function of bandpass filter shown in [29–38].


## 3.2. Design of Proportional-Integral (PI)-Type Sliding Mode Controller

The PI-type sliding surfaceScan be defined as(12)S=K+1sIewhere(13)S=S1S2S3S4T(14)K=diag⁡K1,K2,K3,K4(15)I=diag⁡I1,I2,I3,I4s is Laplace variable. Ki∈R+ and Ii∈R+ (i=1,2,3,4). Obviously, we also can get(16)S1εsd≤S1εsd=DsdS1sgn⁡S1S2εsq≤S2εsq=DsqS2sgn⁡S2S4εgd≤S4εgd=DgdS4sgn⁡S4S5εgq≤S5εgq=DgqS5sgn⁡S5if the Lyapunov function is defined as(17)V=12STS=∑i=14Viwhere(18)Vi=12Si2Equations (19)-(22) can be gotten by taking the derivative of (17).(19)
V
˙
1
=
S
1
S
˙
1
=
S
1
K
1
e
˙
s
d
+
I
1
e
s
d
=
S
1
K
1
i
˙
s
d
_
r
e
f
+
R
s
L
s
d
i
s
d
-
L
s
q
L
s
d
n
p
ω
i
s
q
-
1
L
s
d
U
s
d
+
1
L
s
d
ε
s
d
+
I
1
e
s
d
≤
S
1
K
1
i
˙
s
d
_
r
e
f
+
R
s
L
s
d
i
s
d
-
L
s
q
L
s
d
n
p
ω
i
s
q
-
1
L
s
d
U
s
d
+
1
L
s
d
D
s
d
sgn
⁡
S
1
+
I
1
e
s
d
(20)
V
˙
2
=
S
2
S
˙
2
=
S
2
K
2
e
˙
s
q
+
I
2
e
s
q
=
S
2
K
2
i
˙
s
q
_
r
e
f
+
R
s
L
s
q
i
s
q
+
L
s
d
L
s
q
n
p
ω
i
s
q
+
n
p
ω
ψ
L
s
q
-
1
L
s
q
U
s
q
+
1
L
s
q
ε
s
q
+
I
2
e
s
q
≤
S
2
K
2
i
˙
s
q
_
r
e
f
+
R
s
L
s
q
i
s
q
+
L
s
d
L
s
q
n
p
ω
i
s
q
+
n
p
ω
ψ
L
s
q
-
1
L
s
q
U
s
q
+
1
L
s
q
D
s
q
sgn
⁡
S
2
+
I
2
e
s
q
(21)
V
˙
3
=
S
3
S
˙
3
=
S
3
K
3
e
˙
g
d
+
I
3
e
g
d
=
S
3
K
3
i
˙
g
d
_
r
e
f
+
R
c
L
c
i
g
d
-
ω
g
i
g
q
-
1
L
c
E
g
d
+
1
L
c
U
g
d
+
1
L
c
ε
g
d
+
I
3
e
g
d
≤
S
3
K
3
i
˙
g
d
_
r
e
f
+
R
c
L
c
i
g
d
-
ω
g
i
g
q
-
1
L
c
E
g
d
+
1
L
c
U
g
d
+
1
L
c
D
g
d
sgn
⁡
S
4
+
I
3
e
g
d
(22)
V
˙
4
=
S
4
S
˙
4
=
S
4
K
4
e
˙
g
q
+
I
4
e
g
q
=
S
4
K
4
i
˙
g
q
_
r
e
f
+
R
c
L
c
i
g
q
+
ω
g
i
g
d
-
1
L
c
E
g
q
+
1
L
c
U
g
q
+
1
L
c
ε
g
q
+
I
4
e
g
q
≤
S
4
K
4
i
˙
g
q
_
r
e
f
+
R
c
L
c
i
g
q
+
ω
g
i
g
d
-
1
L
c
E
g
q
+
1
L
c
U
g
q
+
1
L
c
D
g
q
sgn
⁡
S
3
+
I
4
e
g
qLet(23)K1i˙sd_ref+RsLsdisd-LsqLsdnpωisq-1LsdUsd+1LsdDsdsgn⁡S1+I1esd=-γ1S1(24)K2i˙sq_ref+RsLsqisq+LsdLsqnpωisq+npωψLsq-1LsqUsq+1LsqDsqsgn⁡S2+I2esq=-γ2S2(25)K3i˙gd_ref+RcLcigd-ωgigq-1LcEgd+1LcUgd+1LcDgdsgn⁡S4+I3egd=-γ3S3(26)K4i˙gq_ref+RcLcigq+ωgigd-1LcEgq+1LcUgq+1LcDgqsgn⁡S4+I4egq=-γ4S4Equation (27) can be gotten.(27)V˙≤-∑i=14γiSi2≤0where γi>0. Therefore, the control laws are given by (28) -(31) and the control structure diagram is shown in Figure 2.(28)Usd=LsdK1γ1S1+I1esd+Lsdi˙sd_ref+Rsisd-Lsqnpωisq+Dsdsgn⁡S1(29)Usq=LsqK2γ2S2+I2esq+Lsqi˙sq_ref+Rsisd+Lsdnpωisd+npωψ+Dsqsgn⁡S2(30)Ugd=-LcK3γ3S3+I3egd-Lci˙gd_ref-Rcigd+Lcωgigq+Egd-Dgdsgn⁡S3(31)Ugq=-LcK4γ4S4+I4egq-Lci˙gq_ref-Rcigq-Lcωgigd+Egq-Dgqsgn⁡S4From (17) and (27), it is clear that the Lyapunov energy functionV is greater than or equal to 0 and the derivative ofV is less than or equal to 0. Hence, the whole system is asymptotically stable based on Lyapunov stability theorem.Figure 2
Control structure diagram of WECS.
(a) (b) (c) (d)


## 3.3. Tracking Accuracy Analysis

In order to analyze the tracking accuracy, we need to introduce Theorem1 here.Theorem 1.
For any dynamic system with state vectorX=x1x2…xnT and assuming L=fX≥0 is a Lyapunov function of system, if L˙≤-∑i=1nkixi2 or L˙≤-∑i=1nkixi where ki∈R+, then limt→∞xi=0.Proof.
There are three cases.
① IfL≡constant≥0, then 0≡L˙≤-∑i=1nkixi2≤0, and further kixi2=0. Due to ki≠0, thus xi=0. This means that there are not continuous and stable equilibrium points fX≡constant>0 such as stable limit cycle in this system.
② AssumingL0=constant, due to L˙≤-∑i=1nkixi2, then(32)L0-L+∞≥∫0+∞kixi2dt≥0.
∀ x i, if limt→∞xi≠0, then ∫0+∞kixi2dt→+∞, and furthermore L+∞→-∞. Obviously, this is in contradiction with L≥0. Thus limt→∞xi=0.
③ GivenL0→+∞, according to Lyapunov stability criterion, the system is stable. So, there must bet1 meeting Lt1=constant. Otherwise L+∞→+∞. This is clearly in contradiction with system stability. If we redefinedt1 as 0, we can get limt→∞xi=0 by ②.
WhenL˙≤-∑i=1nkixi, we have the same conclusion in the same way.
Based on Theorem1, we have(33)limt→+∞⁡S=0000T
From (12) and (33), we can get(34)limt→+∞⁡e=lims→0⁡s2Ks+I-1S=0000TThe above formula shows that the steady-state error of the system is always 0 regardless of the input. Therefore, the proposed strategy can obtain a good tracking accuracy.


## 4. Algorithm Analysis and Verification

In this section, a detailed simulation test platform is constructed to verify the correctness and effectiveness of the new algorithm, shown in Figure3. The simulation test platform is based on the MATLAB/Simulink environment and the packaged modules in SimPowerSysterms are used in this simulation test platform, including the wind turbine, PMSG, and VSCs. Meanwhile, the two mass block spring damping model mentioned in [21–31] is adopted to describe the dynamics of flexible drive chain. Therefore, the simulation test platform is relatively close to the actual physical system, due to the adoption of many packaged modules. The controller of WECS consists of MSC controller and GSC controller which are mentioned in Figure 3. The system parameters are shown in Table 1. In general, the actual wind speed is time-varying and random; however we also believe the wind speed is constant for a short period of time. Hence, we assume that the wind speed meetsv = 12 + 0.4 × rand(t). The DC-link currentsi1 andi2 are shown in Figure 4 and the other response curves of WECS are shown in Figure 5.Table 1
WECS parameters.
Parameters Value Wind turbine inertia momentJtur[kg·m2] 2×104 Air densityρ[kg/m3] 1.225 Blade lengthR[m] 31 Maximum utilization coefficient of wind energyCP_max 0.476 Rated wind speedvrate[m/s] 14 Rated powerPrate[MW] 2 Rated torqueTrate[kN·m] 4×102 Rate rotor speedωrate[rad/s] 5 Generator pole logarithmnp 102 Permanent fluxφ[Wb] 1.25 Stator-resistanceRS[Ω] 0.001 stator-reluctanceL[mH] 8.35 DC-Link voltage[V] 1800 DC-Link capacitorC[F] 0.22 System resistanceRc[Ω] 0.001 System inductorLc[mH] 6 Power grid potentialEgd[V] 380 Generator inertia momentJgen[kg·m2] 700Figure 3
Simulation test platform.Figure 4
DC-link currents.
(a) (b)Figure 5
Response curves of WECS.
(a) (b) (c) (d) (e) (f)From Figure4, we know that the DC-link currentsi1 andi2 are bidirectional. This means that the MSC or GSC is able to switch between rectifier and inverter state. However, because of the mean values ofi1 andi2 are positive, therefore the MSC is basically working as a rectifier and the GSC is mostly in the state of inverter. This is in accordance with the actual accident situation and can verify the accuracy of the model in a sense.Figure5 shows the response curves of wind energy utilization coefficientCP, DC-link voltageUdc, d-axis, and q-axis components of stator current and grid current. The disturbance component of the wind speed is the main causes ofCP,Udc, current, and power fluctuations. Figure 5(a) depicts the wind energy utilization coefficientCP is always in the vicinity of the maximum value CP_max (or 0.4763) under the proposed control strategy. This implies the WECS is operating in MPPT mode to capture the maximum wind energy under the rated wind speed. It is clear that the DC-link voltageUdc is very close to its reference Udc_ref (or 1800V) and the deviation does not exceed 0.8V by Figure 5(b). Figures 5(b)–5(f) show that the proposed control can reduce the current ripple of stator currents and grid currents compared with the conventional control, because the uncertainty of the unmodeled dynamics is taken into account in the design of the proposed controller. In one word, the proposed control strategy has a better robustness than the conventional control strategy.The electromagnetic torque and sliding mode surfaces curves are shown in Figures6 and 7. It is clear that the electromagnetic torque fluctuation under the proposed control method is smaller, compared with the conventional control method. Meanwhile, Figure 7 shows all steady-state values of sliding mode surfaces are tend to zero. Thus, the correctness of the analysis in Section 3.3 is verified by Figure 7. Furthermore, the proposed scheme was further verified for the suppression of current harmonics. Here, we take the A-phase stator current of generator as an example. The A-phase stator current waveform is shown in Figure 8. Obviously, the proposed method also can reduce the current distortion.Figure 6
Electromagnetic torque curve.
(a) (b)Figure 7
Sliding mode surface curve.
(a) (b) (c) (d)Figure 8
A-phase stator current waveform.To further verify the effectiveness of the proposed method, the system responses under variable wind speed must be observed and analyzed. As Figure9(a) shows, the wind speed varied from 10 m/s to 12.5m/s. At this time, the system response curves under the variable wind speed are shown in Figures 9(b)–9(h). It is clear that the results demonstrated the correctness and effectiveness of the proposed approach again. Figures 9(c) and 9(f) show that the wind energy utilization coefficientCP is always maintained at its maximum value CP_max (0.476) and the DC-link voltageUdc always fluctuates aroundUdc_ref (1800V), whether the wind speed changes. From Figure 9(d), the grid-connected active power is always less than the electromagnetic power due to the copper loss.Figure 9
Variable wind speed and system responses.
(a) (b) (c) (d) (e) (f) (g) (h)


## 5. Conclusion

A novel PI-type SMC was proposed to ensure efficient and stability of PMSG-based WECS, in this paper. The presented strategy with a strong robustness for the uncertainties and disturbances of the system can make the closed-loop system globally stable and more performant, compared with the conventional control method. Finally, a 2MW WECS simulation test platform which is based on the MATLAB/MATLAB/SIMULINK/SimPowerSysterms environment was built to verify the maturity and effectiveness of this presented control algorithm. The simulation results indicated that the new method is able to reduce the current distortion and torque fluctuation, which has important actual significance for the control of practical wind turbine.



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*Source: 2538206-2019-06-19.xml*