# Maximum Power Point Tracking of DFIG with DC-Based Converter System Using Coordinated Feedback Linearization Control

Authors: Yuliang Sun; Shaomin Yan; Bin Cai; Yuqiang Wu
Journal: Mathematical Problems in Engineering
 (2018)

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## Abstract

This paper presents a coordinated feedback linearization strategy (CFLS) for DC-based doubly-fed induction generator (DFIG) system to track the maximum power point. The stator and rotor of DFIG are connected to DC grid directly by two voltage source converters. Compared with a traditional DFIG system, the DC-based DFIG system has more system inputs and coupling, which increases the difficulty of vector control strategy. Accordingly, CFLS is proposed to make DFIG operate at the maximum power point (MPP), and two aspects are improved: first a single-loop control is adopted to make DFIG operate steady and accurate under coordinated the control of RSC and SSC. Second system control laws are obtained by the feedback linearization strategy that achieves DC-based DFIG system decoupling fully during the MPPT and system control. Simulations are carried out the comparison between CFLS and conventional vector control (VC), and it shows that the control performance of CFLS is superior.


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## Body

## 1. Introduction

In recent years, the technology of wind power generation has developed rapidly, due to its freely available and renewable resource. Variable speed constant frequency doubly fed induction generator (DFIG) is often selected in wind power generation [1]. Compared with the traditional AC transmission, the DC transmission is more economical and stable for the long distance high voltage transmission [2]. The traditional DFIG stator windings are connected to an AC grid by transformer, and the rotor windings is connected to the AC grid by back to back converters. Therefore, the traditional DFIG is used into the DC grid and additional converter will be needed, which is bound to increase the cost [3].Accordingly, the new converter system is adopted for DFIG in DC grid. A diode-based stator converter interfaces a DFIG with a DC grid, which has the advantages of low cost and simple structure [4–7]. But harmonic may reach from 5.97% to 11.66% [8]. An IGBT-based converters system consisting of a rotor side converter (RSC) and a stator side converter (SSC) connects rotor and stator with DC grid, respectively [9–13]. This structure has the advantages that it can reduce the current harmonic effectively and regulate the stator flux and current flexibly according to the needs of the system, not limited to AC grid.However, the above improved structure results in that the state variables and the system outputs of DFIG with DC-based converters system are twice that of the traditional system, which greatly increases the complexity and coupling of the system and also increases the control difficulty of the system. Therefore, the traditional vector control (VC), which is based on approximate linear model, is difficult to achieve the global optimal control requirements of the new system [14]. Reference [9] adopts indirect air gap flux linkage orientation strategy to control DFIG in a DC grid, but air gap flux linkage is not suitable for measurement. Model predictive control for DFIG in a DC grid is adopted by [13], which requires high precision of system model.Nowadays, Feedback linearization control (FLC) has been widely used in power electronics and power systems [15–21]. It adopts a dual-loop control strategy of FLC and PI to control DFIG in an AC grid in [22]. The advantage of this method is simple to calculate, but the disadvantage is that could not achieve full system decoupling. Article [23] adopts rotor speed and stator reactive power of DFIG in an AC grid as state variables for FLC, which computationally complex and does not select direct system parameters to controller which affects control accuracy [24].This paper designs a CFLS for DC-based DFIG to achieve maximum power point tracking (MPPT). This control strategy employs stator flux, rotor speed, and rotor current as system inputs, applies FLC to make up a single-loop control, and achieves complete decoupling of the system, which attains coordinated optimal control performance between the SSC and the RSC. This proposed control strategy has better control accuracy and tracking speed than the traditional VC strategy.The paper is organized as follows. In Section2, the model of DFIG based on DC grid is introduced. In Section 3, the maximum wind power is achieved by designing CFLS. In Section 4, simulation studies are evaluate the control performance of the proposed control strategy on a DC-based DFIG system. Finally there is conclusion of this paper.


## 2. Modeling of DFIG System

### 2.1. System Configuration

DFIG with its DC-based converters system in a DC grid is shown in Figure1 [12]. The stator and the rotor of DFIG are connected to a DC grid through SSC and RSC. The stator voltage and frequency are completely unrestricted by the power grid. SSC can adjust stator voltage and frequency to control stator current and compensate stator reactive power. RSC can adjust rotor voltage to control rotor flux. The du/dt filter inductors are connected to stator and rotor, respectively, to prevent sharp voltage caused by converters and smooth currents.Figure 1
DFIG with its DC-based converter system in a DC grid.


### 2.2. DFIG Model Description

In this paper, DFIG model is adopted according to the motor direction in a d-q synchronous frame that can be expressed as [24, 25](1)usd=dψsddt-ω1ψsq+Rsisdusq=dψsqdt+ω1ψsd+Rsisqurd=dψrddt-ωsψrq+Rrirdurq=dψrqdt+ωsψrd+Rrirq(2)ψsd=Lsisd+Lmirdψsq=Lsisq+Lmirqψrd=Lrird+Lmisdψrq=Lrirq+Lmisq(3)Te-Tm=Jdωmdt=JnpdωrdtTe=3Lmnp2Lsψsdirq-ψsqirdwhere usd, usq, isd, isq, ψsd, and ψsq are the d-q components of stator voltage, current, and flux, respectively; urd, urq, ird, irq, ψrd, and ψrq are the d-q components of rotor voltage, current, and flux, respectively; ω1, ωs, ωr, and ωm are synchronous, slip, rotor, and mechanical angular frequency, respectively; Ls, Lr, and Lm are the stator, rotor, and mutual inductance respectively; Rr and Rs are the rotor and stator resistance, respectively; Te is electromagnetic torque; J is generator rotational inertia; and np is the number of pairs of poles.


## 2.1. System Configuration

DFIG with its DC-based converters system in a DC grid is shown in Figure1 [12]. The stator and the rotor of DFIG are connected to a DC grid through SSC and RSC. The stator voltage and frequency are completely unrestricted by the power grid. SSC can adjust stator voltage and frequency to control stator current and compensate stator reactive power. RSC can adjust rotor voltage to control rotor flux. The du/dt filter inductors are connected to stator and rotor, respectively, to prevent sharp voltage caused by converters and smooth currents.Figure 1
DFIG with its DC-based converter system in a DC grid.


## 2.2. DFIG Model Description

In this paper, DFIG model is adopted according to the motor direction in a d-q synchronous frame that can be expressed as [24, 25](1)usd=dψsddt-ω1ψsq+Rsisdusq=dψsqdt+ω1ψsd+Rsisqurd=dψrddt-ωsψrq+Rrirdurq=dψrqdt+ωsψrd+Rrirq(2)ψsd=Lsisd+Lmirdψsq=Lsisq+Lmirqψrd=Lrird+Lmisdψrq=Lrirq+Lmisq(3)Te-Tm=Jdωmdt=JnpdωrdtTe=3Lmnp2Lsψsdirq-ψsqirdwhere usd, usq, isd, isq, ψsd, and ψsq are the d-q components of stator voltage, current, and flux, respectively; urd, urq, ird, irq, ψrd, and ψrq are the d-q components of rotor voltage, current, and flux, respectively; ω1, ωs, ωr, and ωm are synchronous, slip, rotor, and mechanical angular frequency, respectively; Ls, Lr, and Lm are the stator, rotor, and mutual inductance respectively; Rr and Rs are the rotor and stator resistance, respectively; Te is electromagnetic torque; J is generator rotational inertia; and np is the number of pairs of poles.


## 3. Control Strategy Description

### 3.1. MPPT Control Strategy

Usually the maximum kinetic power captured from the wind by a wind turbine is expressed as follows [26]:(4)Pwtmax=12ρπRwtCpmaxVwind3where ρ is air density, Rwt is radius of wind turbine, Vwind is wind speed, and Cpmax is maximum power coefficient.To capture the maximum wind power, the power coefficientCp should maintain maximum Cpmax at any wind speed within the operating range. Maximum Cpmax is achieved by maintaining the tip speed ratio λ equal to optimal value λopt and the pitch angle β at a fixed value.(5)Cpmax=Cpλopt=0.5176116λi-0.4β-5e-21/λi+0.0068λopt1λi=1λopt+0.08β-0.035β3+1The tip speed ratio λ indicates the state of the wind wheel under different wind speeds, as(6)λ=ωrRwtK1VwindWhen λ is equal to λopt, the optimal reference ωr∗ is(7)ωr∗=K1λoptRwtVwindIn this paper, the pitch angle is β=0°, the optimal tip speed ratio is λopt=9.7, and the maximum power coefficient is Cpmax=0.4642. Pwtmax is shown in the Figure 2.Figure 2
The output mechanical power, optimal power, and rotor speed.


### 3.2. CFLS of DFIG

From (1)-(3), the DFIG model can be derived in the form(8)x˙=fx+gxuy=hxwhere (9)
x
=
x
1
x
2
x
3
x
4
x
5
T
=
ψ
s
d
ψ
s
q
i
r
d
i
r
q
ω
r
T
u
=
u
1
u
2
u
3
u
4
T
=
u
s
d
u
s
q
u
r
d
u
r
q
T
y
=
h
1
x
h
2
x
h
3
x
h
4
x
T
=
x
1
x
2
x
4
x
5
T
=
ψ
s
d
ψ
s
q
i
r
q
ω
r
T
f
x
=
f
1
x
f
2
x
f
3
x
f
4
x
f
5
x
=
-
R
s
L
s
x
1
+
ω
1
x
2
+
β
R
s
x
3
-
ω
1
x
1
-
R
s
L
s
x
2
+
β
R
s
x
4
β
R
s
α
L
s
x
1
-
β
α
x
2
x
5
-
R
r
+
β
2
R
s
α
x
3
+
ω
s
x
4
β
α
x
1
x
5
+
β
R
s
α
L
s
x
2
-
ω
s
x
3
-
R
r
+
β
2
R
s
α
x
4
3
L
m
n
p
2
2
J
L
s
x
1
x
4
-
x
2
x
3
-
n
p
J
T
m
g
x
=
g
1
x
g
2
x
g
3
x
g
4
x
T
=
1
0
0
0
0
1
0
0
-
β
α
0
1
α
0
0
-
β
α
0
1
α
0
0
0
0
α
=
L
r
L
s
-
L
m
2
L
s
β
=
L
m
L
sThis is a multi-input multioutput (MIMO) system. An approach to obtain the inputCoutput linearization of the MIMO system is to differentiate the output yi of the system until the inputs uj appear, assuming that the corresponding relation degree ri is the smallest integer such that at least one of the inputs explicitly appears in [27](10)yiri=Lfrihi+∑j=1mLgjLfri-1hiujwhere yi(ri) denotes the rith-order derivative of yi.In (8), x is 5 dimensional state phase quantity, fi(x)(i=1,2,…,5) is 5d smooth vector field, and gi(x)(i=1,2,3,4) is 4d smooth vector field. Each output yi has a ri, and by calculating that is 1, 1, 1 and 2. The system relation degree is r=1+1+1+2=5=n; therefore, the system has no nontrivial zero dynamics, which can be linearized by feedback linearization. According to (10) and ri of each output yi, the Lie Derivative of h with along f and the Lie Derivative of Lfh(x) with along g(x) are obtained, and they are (11)y˙1y˙2y˙3y¨4=Lfh1xLfh2xLfh3xLf2h4x+Lg1Lf0h1xLg2Lf0h1xLg3Lf0h1xLg4Lf0h1xLg1Lf0h2xLg2Lf0h2xLg-3Lf0h2xLg4Lf0h2xLg1Lf0h3xLg2Lf0h3xLg3Lf0h3xLg4Lf0h3xLg1Lf1h4xLg2Lf1h4xLg3Lf1h4xLg4Lf1h4xu1u2u3u4According to (8) and (11), the system can be described in the following matrix form:(12)y˙1y˙2y˙3y¨4T=Ax+Exuwhere (13)
A
x
=
-
R
s
L
s
x
1
+
ω
1
x
2
+
β
R
s
x
3
-
ω
1
x
1
-
R
s
L
s
x
2
+
β
R
s
x
4
β
α
x
1
x
5
+
β
R
s
α
L
s
x
2
-
ω
s
x
3
-
R
r
+
β
2
R
s
α
x
4
ξ
-
R
s
L
s
-
R
r
+
β
2
R
s
α
x
1
x
4
+
ω
1
-
ω
s
x
2
x
4
+
x
1
x
3
+
R
s
L
s
+
R
r
+
β
2
R
s
α
x
2
x
3
+
β
α
x
5
x
1
2
+
x
2
2
E
x
=
1
0
0
0
0
1
0
0
0
-
β
α
0
1
α
2
ξ
x
4
+
β
α
x
2
-
2
ξ
x
3
+
β
α
x
1
-
2
ξ
1
α
x
2
2
ξ
1
α
x
1
ξ
=
3
L
m
n
p
2
2
J
L
sThere is det(E(x))=2ξx2/α2≠0, so the inverse matrix E(x)-1 exists. A new input variable is defined for input-output feedback linearization, as v=v1v2v3v4. The conversion relation between the original input variable and the new input variable is as follows:(14)u=Ex-1-Ax+vwhere y1˙y2˙y3˙y4¨T=v1v2v3v4T. Thus, the input-output mapping of (8) can be simplified as (14), which realizes the linear decoupling between the system output and input variables. According to the MTTP control strategy, and the output is stabilized at σ∗=ψsd∗ψsq∗irq∗ωr∗T, where ψsd∗ψsq∗irq∗ωr∗ are the MPP reference value of the system. Through variable substitution e=σ∗-y the equilibrium point is moved to the origin, and the input variable can be redesigned into (14)(15)
v
1
v
2
v
3
v
4
=
ψ
˙
s
d
∗
-
k
p
1
e
1
-
k
i
1
∫
e
1
d
t
ψ
˙
s
q
∗
-
k
p
2
e
2
-
k
i
2
∫
e
2
d
t
i
˙
r
q
∗
-
k
p
3
e
3
-
k
i
3
∫
e
3
d
t
ω
¨
r
∗
-
k
p
4
e
4
-
k
i
4
∫
e
4
d
t
(16)
e
¨
i
+
k
p
i
e
˙
i
+
k
i
i
e
i
=
0In the controller design we can choice the parameters kpi and kii(i=1,2,3,4) in (15) and (16) to ensure the convergence and stability of the ei(i=1,2,3,4). Now we have derived that the partial states x1x2x3x4 track the setting reference points.Therefore, from the above analysis, the whole control scheme is shown in Figure3.Figure 3
Control scheme for RSC and SSC.


### 3.3. Reference Points of Controller

The rotor flux and stator current are the control targets; the method determined in this paper is that, when a stator-oriented flux frame is adopted with its vector direction aligned with the q-axis, from (2), the stator flux reference value (ψs∗) and its d-q components are given by(17)ψsd∗=0ψsq∗=ψs∗=-VSω1=-1where Vs is generator rated voltage amplitude, ω1 is synchronous speed. In this paper, Vs=1.0 pu, ω1=1.0 pu. Substituting (17) into (2), the rotor current are obtained:(18)irq∗=-ψs∗Lm=-0.34


### 3.4. Stability Analysis of State Variablex3

From the error equation (16), we obtain that the partial states x1x2x4x5 are bounded are converge to the reference point. Next we will analyze the boundedness of the state x3. From (8), the equation about x3 is extracted(19)x˙3=βRsαLsx1-βαx2x5-Rr+β2Rsαx3+ωsx4-βαusd+1αurdSubstituting (14) into (19), that can be derived as(20)x˙3=3β2Rs-Rr2α+2Rs2Ls+2ω1x1-βRs+1x4+v2x2-x5x12x2x3-βx2x52α+RsLs+Rr+β2Rsαx1x42x2-x5x42-βx5x122αx2+x4v1+x1v3x2-12ξx2v4From (15), the states x1x2x4x5 are convergent to ψsd∗ψsq∗irq∗ωr∗, and further from (17), (18), and (7) know that their values are 0-1-0.340.09Vwind, so that dynamics (20) can be approximated as(21)x˙3=-0.0588x3+0.14355Vwind+0.0243V¨windIn this paper, the rated wind speed of wind turbine is Vwind_nom=13 m/s, and the variable wind speed range is focus on 6-14 m/s. And the high frequency components in the wind speed measurement Vwind may cause undesired noise; therefore the measured wind speed is passed through low-pass filter to attenuate its effect [28]. It is assumed that the magnitude of its derivative |V¨wind| is bounded as |V¨wind|≤5; it becomes(22)x˙3=-0.0588x3+M0.7385≤M≤2.1215The above can be known, the state variable x3(ird) is bounded, when t→∞.


## 3.1. MPPT Control Strategy

Usually the maximum kinetic power captured from the wind by a wind turbine is expressed as follows [26]:(4)Pwtmax=12ρπRwtCpmaxVwind3where ρ is air density, Rwt is radius of wind turbine, Vwind is wind speed, and Cpmax is maximum power coefficient.To capture the maximum wind power, the power coefficientCp should maintain maximum Cpmax at any wind speed within the operating range. Maximum Cpmax is achieved by maintaining the tip speed ratio λ equal to optimal value λopt and the pitch angle β at a fixed value.(5)Cpmax=Cpλopt=0.5176116λi-0.4β-5e-21/λi+0.0068λopt1λi=1λopt+0.08β-0.035β3+1The tip speed ratio λ indicates the state of the wind wheel under different wind speeds, as(6)λ=ωrRwtK1VwindWhen λ is equal to λopt, the optimal reference ωr∗ is(7)ωr∗=K1λoptRwtVwindIn this paper, the pitch angle is β=0°, the optimal tip speed ratio is λopt=9.7, and the maximum power coefficient is Cpmax=0.4642. Pwtmax is shown in the Figure 2.Figure 2
The output mechanical power, optimal power, and rotor speed.


## 3.2. CFLS of DFIG

From (1)-(3), the DFIG model can be derived in the form(8)x˙=fx+gxuy=hxwhere (9)
x
=
x
1
x
2
x
3
x
4
x
5
T
=
ψ
s
d
ψ
s
q
i
r
d
i
r
q
ω
r
T
u
=
u
1
u
2
u
3
u
4
T
=
u
s
d
u
s
q
u
r
d
u
r
q
T
y
=
h
1
x
h
2
x
h
3
x
h
4
x
T
=
x
1
x
2
x
4
x
5
T
=
ψ
s
d
ψ
s
q
i
r
q
ω
r
T
f
x
=
f
1
x
f
2
x
f
3
x
f
4
x
f
5
x
=
-
R
s
L
s
x
1
+
ω
1
x
2
+
β
R
s
x
3
-
ω
1
x
1
-
R
s
L
s
x
2
+
β
R
s
x
4
β
R
s
α
L
s
x
1
-
β
α
x
2
x
5
-
R
r
+
β
2
R
s
α
x
3
+
ω
s
x
4
β
α
x
1
x
5
+
β
R
s
α
L
s
x
2
-
ω
s
x
3
-
R
r
+
β
2
R
s
α
x
4
3
L
m
n
p
2
2
J
L
s
x
1
x
4
-
x
2
x
3
-
n
p
J
T
m
g
x
=
g
1
x
g
2
x
g
3
x
g
4
x
T
=
1
0
0
0
0
1
0
0
-
β
α
0
1
α
0
0
-
β
α
0
1
α
0
0
0
0
α
=
L
r
L
s
-
L
m
2
L
s
β
=
L
m
L
sThis is a multi-input multioutput (MIMO) system. An approach to obtain the inputCoutput linearization of the MIMO system is to differentiate the output yi of the system until the inputs uj appear, assuming that the corresponding relation degree ri is the smallest integer such that at least one of the inputs explicitly appears in [27](10)yiri=Lfrihi+∑j=1mLgjLfri-1hiujwhere yi(ri) denotes the rith-order derivative of yi.In (8), x is 5 dimensional state phase quantity, fi(x)(i=1,2,…,5) is 5d smooth vector field, and gi(x)(i=1,2,3,4) is 4d smooth vector field. Each output yi has a ri, and by calculating that is 1, 1, 1 and 2. The system relation degree is r=1+1+1+2=5=n; therefore, the system has no nontrivial zero dynamics, which can be linearized by feedback linearization. According to (10) and ri of each output yi, the Lie Derivative of h with along f and the Lie Derivative of Lfh(x) with along g(x) are obtained, and they are (11)y˙1y˙2y˙3y¨4=Lfh1xLfh2xLfh3xLf2h4x+Lg1Lf0h1xLg2Lf0h1xLg3Lf0h1xLg4Lf0h1xLg1Lf0h2xLg2Lf0h2xLg-3Lf0h2xLg4Lf0h2xLg1Lf0h3xLg2Lf0h3xLg3Lf0h3xLg4Lf0h3xLg1Lf1h4xLg2Lf1h4xLg3Lf1h4xLg4Lf1h4xu1u2u3u4According to (8) and (11), the system can be described in the following matrix form:(12)y˙1y˙2y˙3y¨4T=Ax+Exuwhere (13)
A
x
=
-
R
s
L
s
x
1
+
ω
1
x
2
+
β
R
s
x
3
-
ω
1
x
1
-
R
s
L
s
x
2
+
β
R
s
x
4
β
α
x
1
x
5
+
β
R
s
α
L
s
x
2
-
ω
s
x
3
-
R
r
+
β
2
R
s
α
x
4
ξ
-
R
s
L
s
-
R
r
+
β
2
R
s
α
x
1
x
4
+
ω
1
-
ω
s
x
2
x
4
+
x
1
x
3
+
R
s
L
s
+
R
r
+
β
2
R
s
α
x
2
x
3
+
β
α
x
5
x
1
2
+
x
2
2
E
x
=
1
0
0
0
0
1
0
0
0
-
β
α
0
1
α
2
ξ
x
4
+
β
α
x
2
-
2
ξ
x
3
+
β
α
x
1
-
2
ξ
1
α
x
2
2
ξ
1
α
x
1
ξ
=
3
L
m
n
p
2
2
J
L
sThere is det(E(x))=2ξx2/α2≠0, so the inverse matrix E(x)-1 exists. A new input variable is defined for input-output feedback linearization, as v=v1v2v3v4. The conversion relation between the original input variable and the new input variable is as follows:(14)u=Ex-1-Ax+vwhere y1˙y2˙y3˙y4¨T=v1v2v3v4T. Thus, the input-output mapping of (8) can be simplified as (14), which realizes the linear decoupling between the system output and input variables. According to the MTTP control strategy, and the output is stabilized at σ∗=ψsd∗ψsq∗irq∗ωr∗T, where ψsd∗ψsq∗irq∗ωr∗ are the MPP reference value of the system. Through variable substitution e=σ∗-y the equilibrium point is moved to the origin, and the input variable can be redesigned into (14)(15)
v
1
v
2
v
3
v
4
=
ψ
˙
s
d
∗
-
k
p
1
e
1
-
k
i
1
∫
e
1
d
t
ψ
˙
s
q
∗
-
k
p
2
e
2
-
k
i
2
∫
e
2
d
t
i
˙
r
q
∗
-
k
p
3
e
3
-
k
i
3
∫
e
3
d
t
ω
¨
r
∗
-
k
p
4
e
4
-
k
i
4
∫
e
4
d
t
(16)
e
¨
i
+
k
p
i
e
˙
i
+
k
i
i
e
i
=
0In the controller design we can choice the parameters kpi and kii(i=1,2,3,4) in (15) and (16) to ensure the convergence and stability of the ei(i=1,2,3,4). Now we have derived that the partial states x1x2x3x4 track the setting reference points.Therefore, from the above analysis, the whole control scheme is shown in Figure3.Figure 3
Control scheme for RSC and SSC.


## 3.3. Reference Points of Controller

The rotor flux and stator current are the control targets; the method determined in this paper is that, when a stator-oriented flux frame is adopted with its vector direction aligned with the q-axis, from (2), the stator flux reference value (ψs∗) and its d-q components are given by(17)ψsd∗=0ψsq∗=ψs∗=-VSω1=-1where Vs is generator rated voltage amplitude, ω1 is synchronous speed. In this paper, Vs=1.0 pu, ω1=1.0 pu. Substituting (17) into (2), the rotor current are obtained:(18)irq∗=-ψs∗Lm=-0.34


## 3.4. Stability Analysis of State Variablex3

From the error equation (16), we obtain that the partial states x1x2x4x5 are bounded are converge to the reference point. Next we will analyze the boundedness of the state x3. From (8), the equation about x3 is extracted(19)x˙3=βRsαLsx1-βαx2x5-Rr+β2Rsαx3+ωsx4-βαusd+1αurdSubstituting (14) into (19), that can be derived as(20)x˙3=3β2Rs-Rr2α+2Rs2Ls+2ω1x1-βRs+1x4+v2x2-x5x12x2x3-βx2x52α+RsLs+Rr+β2Rsαx1x42x2-x5x42-βx5x122αx2+x4v1+x1v3x2-12ξx2v4From (15), the states x1x2x4x5 are convergent to ψsd∗ψsq∗irq∗ωr∗, and further from (17), (18), and (7) know that their values are 0-1-0.340.09Vwind, so that dynamics (20) can be approximated as(21)x˙3=-0.0588x3+0.14355Vwind+0.0243V¨windIn this paper, the rated wind speed of wind turbine is Vwind_nom=13 m/s, and the variable wind speed range is focus on 6-14 m/s. And the high frequency components in the wind speed measurement Vwind may cause undesired noise; therefore the measured wind speed is passed through low-pass filter to attenuate its effect [28]. It is assumed that the magnitude of its derivative |V¨wind| is bounded as |V¨wind|≤5; it becomes(22)x˙3=-0.0588x3+M0.7385≤M≤2.1215The above can be known, the state variable x3(ird) is bounded, when t→∞.


## 4. Simulation Results

In this section, simulation results are carried out in MATLAB/SIMULINK, to verify the coordinated optimal control performance in a wide range operating conditions. There are two tests performed: case one shows the accuracy of CFLS compare with the VC under the ramp-change wind and case two is the system that operates at random-change wind to show the tracking speed of controller. The DFIG system parameters in the following simulation are listed in AppendixA.


### 4.1. Ramp-Change Wind

Figures4–7 shows the performance of a DFIG in DC-grid controlled by the VC and CFLS under the ramp-change wind. The wind speed condition is depicted in Figure 4(a). The wind speed rises from 7 m/s to 14 m/s and then remains 14 m/s. It can be seen from Figure 4(b) that the optimal reference rotor speed ωr∗ is well tracked by the CFLS, while the response of the VC with larger overshoots is slower. Figures 4(c), 4(e), and 4(f) show that the tracking references performance of VC and CFLS and the tracking accuracy of CFLS are better than that of VC. The state variable x3(ird) is shown in Figure 4(d), as it keep stable in both VC and CFLS under ramp-change wind.Figure 4
Performance response to the ramp-change wind speed.
(a) (b) (c) (d) (e) (f)Figure 5
C p at the ramp-change wind speed.Figure 6
V d c at the ramp-change wind speed.Figure 7
Performance response to the ramp-change wind speed.
(a)
Rotor currentsirabc under CFLS (b)
Rotor voltagesurabc under CFLS (c)
Stator currentsisabc under CFLS (d)
Stator voltagesusabc under CFLSAs shown in Figure5, the CFLS can well capture the maximum power coefficient Cpmax value, but the VC needs long time to catch Cpmax. It means that the wind turbine maintains the maximum output power under the proposed control strategy.Figure6 shows the system output DC-voltage Vdc that connected to a stable DC-grid. It can be seen that the DC voltage remains stable under the control of CFLS and VC, when the wind ramp changes.The rotor currents and voltages(irabc,urabc), and the stator currents and voltages (isabc,usabc) can be seen that the DFIG is operated at the rated value while under the CFLS in Figure 7. isabc and usabc are sine wave with frequency of 60 Hz, and their amplitudes are 1.0 pu and 0.8 pu. irabc is sine wave with amplitude of 0.85 pu. urabc is the RSC output pulse wave, and its amplitude is 0.7 pu.


### 4.2. Random-Change Wind

The performance of a DC-based DFIG system controlled by the VC and CFLS under random-change wind is shown in Figures8–11. The wind speed condition is depicted in Figure 8(a), which fluctuates between 6.8 m/s and 13.4 m/s. As shown in Figures 8(b), 8(c), 8(d), and 8(e), the tracking speed and accuracy of CFLS are obviously better than VC when the wind speed is time-varying. The state variable x3(ird) in the CFLS is shown in Figure 8(f), as it can stay mostly stable under the influence of random-change wind.Figure 8
Performance response to the random-change wind speed.
(a)
Wind speed (b)
Rotor speedωr (c)
ψ s d (d)
ψ s q (e)
i r q ( x 3 ) (f)
i r d under the CFLSFigure 9
C p at the random-change wind speed.Figure 10
V d c at the random-change wind speed.Figure 11
The output control signals response to the random-change wind speed.
(a)
u r d under the CFLS (b)
u r q under the CFLS (c)
u s d under the CFLS (d)
u s q under the CFLSWhen the wind speed changes at any time, the power coefficientCp should be kept at the maximum value in order to extract the maximum wind power. As shown in Figure 9, the maximum power coefficient Cpmax can be well captured by the CFLS. However, the change of the wind speed will affect the maximum power coefficient Cpmax obtained by the VC strategy. This means that the CFLS maximum power tracking performance is much better than VC in a wide range operating conditions.System output DC-voltageVdc connected to a stable DC-grid in the random-change wind speed is shown in Figure 10; it remains stable under the control of CFLS and VC.The output control signalsurdurqusdusq of controllers are listed in Figure 11. Under the CFLS, because of the rotor speed ωr tracking the optimal reference rotor speed ωr∗ that fluctuates follow the random-change wind, so the rotor side output signals urdurq are fluctuated near the working point. But the stator side output signals usdusq less fluctuant and operate steadily at working point. This shows that the proposed control strategy achieves the complete decoupling of DC-based DFIG system and that the stator side and rotor side have each performs its own functions, and coordinate the work.


## 4.1. Ramp-Change Wind

Figures4–7 shows the performance of a DFIG in DC-grid controlled by the VC and CFLS under the ramp-change wind. The wind speed condition is depicted in Figure 4(a). The wind speed rises from 7 m/s to 14 m/s and then remains 14 m/s. It can be seen from Figure 4(b) that the optimal reference rotor speed ωr∗ is well tracked by the CFLS, while the response of the VC with larger overshoots is slower. Figures 4(c), 4(e), and 4(f) show that the tracking references performance of VC and CFLS and the tracking accuracy of CFLS are better than that of VC. The state variable x3(ird) is shown in Figure 4(d), as it keep stable in both VC and CFLS under ramp-change wind.Figure 4
Performance response to the ramp-change wind speed.
(a) (b) (c) (d) (e) (f)Figure 5
C p at the ramp-change wind speed.Figure 6
V d c at the ramp-change wind speed.Figure 7
Performance response to the ramp-change wind speed.
(a)
Rotor currentsirabc under CFLS (b)
Rotor voltagesurabc under CFLS (c)
Stator currentsisabc under CFLS (d)
Stator voltagesusabc under CFLSAs shown in Figure5, the CFLS can well capture the maximum power coefficient Cpmax value, but the VC needs long time to catch Cpmax. It means that the wind turbine maintains the maximum output power under the proposed control strategy.Figure6 shows the system output DC-voltage Vdc that connected to a stable DC-grid. It can be seen that the DC voltage remains stable under the control of CFLS and VC, when the wind ramp changes.The rotor currents and voltages(irabc,urabc), and the stator currents and voltages (isabc,usabc) can be seen that the DFIG is operated at the rated value while under the CFLS in Figure 7. isabc and usabc are sine wave with frequency of 60 Hz, and their amplitudes are 1.0 pu and 0.8 pu. irabc is sine wave with amplitude of 0.85 pu. urabc is the RSC output pulse wave, and its amplitude is 0.7 pu.


## 4.2. Random-Change Wind

The performance of a DC-based DFIG system controlled by the VC and CFLS under random-change wind is shown in Figures8–11. The wind speed condition is depicted in Figure 8(a), which fluctuates between 6.8 m/s and 13.4 m/s. As shown in Figures 8(b), 8(c), 8(d), and 8(e), the tracking speed and accuracy of CFLS are obviously better than VC when the wind speed is time-varying. The state variable x3(ird) in the CFLS is shown in Figure 8(f), as it can stay mostly stable under the influence of random-change wind.Figure 8
Performance response to the random-change wind speed.
(a)
Wind speed (b)
Rotor speedωr (c)
ψ s d (d)
ψ s q (e)
i r q ( x 3 ) (f)
i r d under the CFLSFigure 9
C p at the random-change wind speed.Figure 10
V d c at the random-change wind speed.Figure 11
The output control signals response to the random-change wind speed.
(a)
u r d under the CFLS (b)
u r q under the CFLS (c)
u s d under the CFLS (d)
u s q under the CFLSWhen the wind speed changes at any time, the power coefficientCp should be kept at the maximum value in order to extract the maximum wind power. As shown in Figure 9, the maximum power coefficient Cpmax can be well captured by the CFLS. However, the change of the wind speed will affect the maximum power coefficient Cpmax obtained by the VC strategy. This means that the CFLS maximum power tracking performance is much better than VC in a wide range operating conditions.System output DC-voltageVdc connected to a stable DC-grid in the random-change wind speed is shown in Figure 10; it remains stable under the control of CFLS and VC.The output control signalsurdurqusdusq of controllers are listed in Figure 11. Under the CFLS, because of the rotor speed ωr tracking the optimal reference rotor speed ωr∗ that fluctuates follow the random-change wind, so the rotor side output signals urdurq are fluctuated near the working point. But the stator side output signals usdusq less fluctuant and operate steadily at working point. This shows that the proposed control strategy achieves the complete decoupling of DC-based DFIG system and that the stator side and rotor side have each performs its own functions, and coordinate the work.


## 5. Conclusion

From the simulation results and analysis, the following conclusions may be drawn out:(1) A coordinated feedback linearization strategy is applied to achieve the complete decoupling and linearization for the DFIG with DC-based converter system.(2) This paper gives up the traditional indirect dual-loop control method of power and current and selects stator flux (ψsdψsq), rotor speed (ωr), and rotor current (irq) as state variables. A direct single-loop coordinated control method is implemented, which achieves the division cooperation between stator side and rotor side, and enhances the robustness of the system.(3) The maximum power point tracking performance of the proposed control strategy is better than traditional VC. The CFLS kept power coefficient (Cp) at its maximum, to make DFIG operate at the maximum power point (MPP).In further study, this experiment will be extended to verify the effectiveness of the system in case of failure.



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*Source: 9642123-2018-11-11.xml*