CROSS-REFERENCE TO RELATED APPLICATIONS

The present application is a continuation of International Application No. PCT/CN2021/089735, filed on Apr. 25, 2021, the content of which is incorporated herein by reference in its entirety.

TECHNICAL FIELD

The present application relates to a method for analyzing the stability of a permanent magnet synchronous generator-based wind turbine connected to a weak power grid considering the influence of power control, and belongs to the field of stability analysis of a new energy power generation system in an electrical power system.

BACKGROUND

The proportion of renewable energy power generation continues to increase, the installed capacity of wind power generation continues to grow, and the stable operation of a grid-connected wind power system has become an important issue. A permanent magnet synchronous generator (PMSG) based wind turbines (WTs) has the advantages of high energy conversion efficiency, good operational reliability and strong controllability, and the proportion of installed capacity continues to increase. In order to make better use of wind energy, a suitable maximum power point tracking algorithm must be adopted. A perturbation and observation (P&O) method does not need to measure the wind speed, and thus has a better practical value. However, the P&O method implies a nonlinear discontinuous link, which makes the traditional small-signal stability analysis method difficult to apply. The conventional stability analysis method generally ignores a power loop based on the P&O method. A describing function method can well complete the modeling of a nonlinear link and the stability analysis of the system, the basic idea is that when the system meets certain assumed conditions, under the action of sinusoidal input, the output of the nonlinear link of the system can be approximately expressed by its first harmonic component, so as to obtain the approximate equivalent frequency characteristics of the nonlinear link. Therefore, based on the describing function method, the present application proposes a method for analyzing the stability of a permanent magnet synchronous generator-based wind turbines (PMSG-WT) connected to a weak power grid considering the influence of power control, which fully considers the nonlinear link in the power loop, and improves the accuracy of stability analysis.

SUMMARY

The present application aims to provide a method for analyzing the stability of a PMSG-WT connected to a weak power grid considering the influence of power control. The PMSG-WT includes a wind turbine, a generator, a machine-side converter (MSC), a DC capacitor, a grid-side converter (GSC), a filter, an MSC controller, and a GSC controller. The MSC controller includes a power loop, a rotating speed loop, and a machine-side current loop; and the GSC controller includes a voltage loop and a grid-side current loop.

The present application adopts the following technical solutions:

Step 1: acquiring main parameters of the PMSG-WT, establishing mathematical models of the wind turbine, the generator and the MSC, the MSC controller, the DC capacitor, the GSC and the filter, and the GSC controller, respectively, performing linearization in a dq coordinate system, and calculating steady-state operation parameters to obtain small-signal models of the wind turbine, the generator and the MSC, the MSC controller, the DC capacitor, the GSC and the filter, and the GSC controller.

Step 2: modeling the power loop in the MSC controller based on a describing function method, wherein its mathematical expression is:

ω
g
ref

=

ε

T
Ρ

⁢

∫

sgn
⁡
(

P
ref

-

P
n

)

⁢
sgn
⁢

(

P
n

-

P

n
-
1

)

⁢

sgn
⁡
(

ω

g
,
n

-

ω

g
,

n
-
1

)

⁢
dt

where ε represents a perturbation step length of the power loop, T_prepresents a control period of the power loop, P^refrepresents a reference value of the output power of the PMSG-WT, P represents an output power of the PMSG-WT, P_nrepresents an output power of the PMSG-WT at the current sampling moment n, P_n−1represents an output power of the PMSG-WT at the previous sampling moment, ω_grepresents a rotating speed of the generator, ω_g^refrepresents a reference value of the rotating speed of the generator, ω_g,nrepresents a rotating speed of the generator at the current sampling moment, and ω_g,n−1represents a rotating speed of the generator at the previous sampling moment. sgn(x) is a sign function, when x≥0, sgn(x)=1, and when x<0, sgn(x)=−1. Considering an actual power-rotating speed curve of the PMSG-WT, it can be deemed that:

sgn
⁡
(

P
n

-

P

n
-
1

)

⁢

sgn
⁡
(

ω

g
,
n

-

ω

g
,

n
-
1

)

=

{

1

(

ω
g

≤

ω
mpp

)

-
1

(

ω
g

>

ω
mpp

)

where ω_mpprepresents the rotating speed of the generator at the maximum power point. Then, the power loon model can be simplified as:

ω
g
ref

=

{

ε

T
Ρ

⁢

∫

sgn
⁡
(

P
ref

-

P
n

)

⁢
dt

,

(

ω
g

≤

ω
mpp

)

-

ε

T
Ρ

⁢

∫

sgn
⁡
(

P
ref

-

P
n

)

⁢
dt

,

(

ω
g

>

ω
mpp

)

the sign function in the formula can be modeled by a describing function, and the describing function is:

N
⁡
(
A
)

=

4

π
⁢
A

in the formula, A represents an amplitude of an input signal.

Step 3: considering the influence of the weak AC power grid, combining the linear parts of the small-signal models of the weak power grid and the power loop with the small-signal model of the PMSG-WT established in step 1, and deriving a transfer function G(s) of the linear part of the system.

Step 4: drawing G(s) and −1/N(A) curves in a complex plane, and analyzing the stability of the system based on the describing function method, wherein the method is specifically that if G(s) contains a right-half-plane (RHP) pole, the system must be unstable; if the G(s) does not contain any RHP pole, then judging the stability of the system through the relationship between the G(s) curve and the −1/N (A) curve:

a. if the G(s) curve does not surround the −1/N(A) curve, the system is stable and does not oscillate;

b. if the G(s) curve and the −1/N(A) curve intersect, the system is critically stable, at this time, the system oscillates at a constant amplitude and a constant frequency, and the frequency and amplitude of the oscillation can be calculated by the following formula:

{

G

I
⁢
m

(

ω
0

)

=
0

N
⁡
(

A
0

)

=

-
1

/

G
Re

(

ω
0

)

where G(jω)=G_Re(ω)+jG_Im(ω), G_Remeans to find the real part of a complex number, G_Immeans to find the imaginary part of the complex number, ω₀represents an angular frequency of the oscillation, and A₀represents the amplitude of the oscillation; and

c. if the G(s) curve surrounds the −1/N(A) curve, the system is unstable.

Further, in step 1, the small-signal models of the wind turbine, the generator and the MSC, and the MSC controller are established as follows:

the mathematical model of the wind turbine is established as:

sJω_g=T_m−T_e−Bω_g

in the formula, J represents rotational inertia of an equivalent concentrated mass block of the wind turbine and the generator, T_mrepresents a mechanical torque of the generator, T_erepresents an electromagnetic torque of the generator, B represents a self-damping coefficient, it is deemed that B=0 herein, and s represents a parameter introduced by the Laplace transform. Linearization is performed on the model to obtain:

sJΔω_g=ΔT_m−ΔT_e

the electromagnetic torque of the generator is:

T_e= 3/2n_pψ_fi_qr

n_prepresents the number of pole pairs of the generator, i_qrrepresents stator current of the q-axis generator, and ψ_frepresents a permanent magnet flux linkage of the generator. This formula is linearized to obtain:

ΔT_e= 3/2n_pψ_fΔi_qr

the mechanical torque of the generator is:

T_m=B_tω_g

in the formula,

B
t

=

dT
m

d
⁢

ω
g

|

ω
g

=

ω
g
*

represents a linearization constant of the mechanical torque of the wind turbine, and ω_g* represents a steady-state value of the rotating speed of the generator, which is a specific constant at an equilibrium point. This formula is linearized to obtain:

ΔT_m=B_tΔω_g

from this, the small-signal model of the wind turbine can be obtained,

Δω
g

=

3
⁢

n
p

⁢
ψ
⁢
f

2
⁢

(

B
t

-
sJ

)

⁢
Δ
⁢

i
qr

=

[

0

3
⁢

n
p

⁢
ψ
⁢
f

2
⁢

(

B
t

-
sJ

)

]

·
Δ

⁢

i
dqr

setting

G

iq
⁢
ω

=

[

0

3
⁢

n
p

⁢
ψ
⁢
f

2
⁢

(

B
t

-
sJ

)

]

,

then

Δω_g=G_iqω·Δi_dqr, Δω_e=n_pG_iqω·Δi_dqr.

The mathematical model of the generator and the MSC is:

[

sL
s

+

R
s

-

ω
e

⁢

L
s

ω
e

⁢

L
s

sL
s

+

R
s

]

·

[

i
dr

i

q
⁢
r

]

=

[

0

ω
e

⁢

ψ
f

]

-

[

d

d
⁢
r

d

q
⁢
r

]

·

u

d
⁢
c

in the formula, R_sand L_srepresent rotor resistance and armature inductance of the generator, respectively, ω_erepresents an electrical angular speed of the rotor, and ω_e=n_pω_g. i_drand i_qrrepresent the stator current of the generator in the dq coordinate system, d_drand d_qrrepresent output duty ratios of the MSC controller in the dq coordinate system, u_dcrepresents a DC voltage. By linearizing the model, the small-signal model of the generator and MSC can be obtained as:

Z_dqr·Δi_dqr=−U*_dc·Δd_dqr−D*_dqr·Δu_dc+G_ωe·Δω_e

where

Z

d
⁢
q
⁢
r

=

[

sL
s

+

R
s

-

ω
e

⁢

L
s

ω
e

⁢

L
s

sL
s

+

R
s

]

,

Δ
⁢

i

d
⁢
q
⁢
r

=

[

Δ
⁢

i

d
⁢
r

Δ
⁢

i

q
⁢
r

]

,

Δ
⁢

d

d
⁢
q
⁢
r

=

[

Δ
⁢

d

d
⁢
r

Δ
⁢

d

q
⁢
r

]

,

D

d
⁢
q
⁢
r

*

=

[

D

d
⁢
r

*

D

q
⁢
r

*

]

,

G

ω
⁢
e

=

[

L
s

⁢

I

q
⁢
r

*

ψ
f

-

L
s

⁢

I

d
⁢
r

*

]

;

and capital letters and superscripts * represent steady-state components of corresponding lowercase variables, and Δ represents a small-signal component of the corresponding variable.

The mathematical model of the MSC controller is established as:

{

d
dr
c

⁢

U

d
⁢
c

*

=

-

H
cr

(

0
-

i
dr
c

)

+

ω
e

⁢

L
s

d
qr
c

⁢

U

d
⁢
c

*

=

-

H
cr

(

-

H
ω

(

ω
g
ref

-

ω
g

)

-

i
qr
c

)

-

ω
e

⁢

L
s

⁢

i
dr
c

+

ω
e

⁢

ψ
f

where

H
cr

=

K
cpr

+

K
cir

s

,

K_cprand K_cirrepresent a proportional parameter and an integral parameter of machine-side current loop PI control, respectively,

H
ω

=

K

ω
⁢
p

+

K

ω
⁢
i

s

,

and K_ωpand K_ωirepresent the proportional parameter and the integral parameter of rotating speed loop PI control, respectively. Superscript c represents a dq coordinate system of the MSC controller. ω_g^refrepresents the reference value of the rotating speed of the generator. By linearizing the model, the small-signal model of the MSC controller can be obtained as:

Δ
⁢

d
dqr
c

=

1

U

d
⁢
c

*

⁢

(

G
cr

·
Δ

⁢

i
dqr
c

+

G

ω
⁢
g

(

ω
g
ref

-

ω
g

)

+

G

ω
⁢
e

·
Δ

⁢

ω
e

)

⁢

where

G
cr

=

[

H
cr

ω
e
*

⁢

L
s

-

ω
e
*

⁢

L
s

H
cr

]

,

G

ω
⁢
g

=

[

0

H
cr

⁢

H
ω

]

.

ω_e* represents the steady-state value of the electrical angular speed of the rotor. Affected by the disturbance of the rotating speed, the dq coordinate system of the MSC controller will have a phase angle difference with the dq coordinate system of the generator during the dynamic process. The electrical angle of the generator is:

θ_e=n_pω_g/s

the formula is linearized to obtain:

Δ
⁢

θ
e

=

n
p

⁢
Δ
⁢

ω
g

s

=

3
⁢

n
p
2

⁢

ψ
f

2
⁢

s
⁡
(

B
t

-
sJ

)

⁢
Δ
⁢

i
qr

=

H

θ
⁢
e

⁢
Δ
⁢

i
qr

in the formula,

H

θ
⁢
e

=

3
⁢

n
p
2

⁢

ψ
f

2
⁢

s
⁡
(

B
t

-

s
⁢
J

)

.

Therefore, the variable conversion relationship between the dq coordinate system of the MSC controller and the dq coordinate system of the generator is:

{

Δ
⁢

i
dqr
c

=

G

θ
⁢
e

i

·
Δ

⁢

i
dqr

Δ
⁢

d
dqr
c

=

G

θ
⁢
e

d

·
Δ

⁢

i
dqr

+

Δ
⁢

d
dqr

⁢

where

G

θ
⁢
e

i

=

[

1

H

θ
⁢
e

⁢

I
qr
*

0

1
-

H

θ
⁢
e

⁢

I
dr
*

]

,

G

θ
⁢
e

d

=

[

0

H

θ
⁢
e

⁢

D
qr
*

0

-

H

θ
⁢
e

⁢

D
dr
*

]

.

Then, the small-signal model of the MSC controller is:

G

θ
⁢
e

d

·
Δ

⁢

i
dqr

+

Δ
⁢

d
dqr

=

1

U

d
⁢
c

*

⁢

(

G
cr

·

G

θ
⁢
e

i

·
Δ

⁢

i
dqr

+

G

ω
⁢
g

(

ω
g
ref

-

ω
g

)

+

G

ω
⁢
e

·

Δω
e

)

.

Further, in step 1, the establishment process of the small-signal model is as follows:

the mathematical model of the DC capacitor is established as:

sC_dcu_dc=i_dc2−i_dc1=1.5(d_dri_dr+d_qri_qr)−1.5(d_dgi_dg+d_qgi_qg)

in the formula, C_dcrepresents the DC capacitor, i_dc1represents grid-side DC current, i_dc2represents machine-side DC current, i_dgand i_qgrepresent d-axis current and q-axis current of the AC port of the GSC, respectively, and d_dgand d_qgrepresent the output duty ratios of the GSC controller in the dq coordinate system, respectively. u_dcrepresents the DC voltage, i_drand i_qrrepresent the stator current of the generator in the dq coordinate system, respectively, d_drand d_qrrepresent the output duty ratios of the MSC controller in the dq coordinate system, respectively, and by linearizing the model, the small-signal model of the DC capacitor can be obtained as:

sC_dcΔu_dc=1.5(D*_dqr^T·Δi_dqr+I*_dqr^T·Δd_dqr)−1.5(D*_dqg^T·Δi_dqg+I*_dqg^T·Δd_dqg)

in the formula,

Δ
⁢

i
dqg

=

[

Δ
⁢

i

d
⁢
g

Δ
⁢

i
qg

]

,

Δ
⁢

d
dqg

=

[

Δ
⁢

d

d
⁢
g

Δ
⁢

d
qg

]

,

D
dqg
*

=

[

D

d
⁢
g

*

D
qg
*

]

,

I
dqg
*

=

[

I

d
⁢
g

*

I
qg
*

]

.

The capital letters and the superscripts * represent the steady-state components of corresponding lowercase variables, and Δ represents the small-signal component of the corresponding variable.

Further, in step 1, the establishment process of the small-signal models of the GSC and the filter, and the GSC controller is as follows:

the mathematical model of the GSC and the filter is established as:

[

sL
f

-
ω

⁢

L
f

ω
⁢

L
f

sL
f

]

·

[

i

d
⁢
g

i
qg

]

=

[

d

d
⁢
g

d
qg

]

·

u

d
⁢
c

-

[

u

d
⁢
g

u
qg

]

in the formula, L_frepresents a filtering inductance, ω represents a power frequency angular frequency, ω=100π rad/s, i_dgand i_qgrepresent the d-axis current and the q-axis current of the AC port of the GSC, respectively, d_dgand d_qgrepresent the output duty ratios of the GSC controller in the dq coordinate system, respectively, u_dcrepresents the DC voltage, and u_dgand u_qgrepresent a d-axis voltage and a q-axis voltage of a grid-connected point, respectively. By linearizing the model, the small-signal model of the GSC and the filter can be obtained as:

Z_f·Δi_dqg=U*_dc·Δd_dqg+D*_dqg·Δu_dc−Δu_dqg

where

Z
f

=

[

sL
f

-
ω

⁢

L
f

ω
⁢

L
f

sL
f

]

,

Δ
⁢

i
dqg

=

[

Δ
⁢

i

d
⁢
g

Δ
⁢

i
qg

]

,

Δ
⁢

d
dqg

=

[

Δ
⁢

d

d
⁢
g

Δ
⁢

d
qg

]

,

D
dqg
*

=

[

D

d
⁢
g

*

D
qg
*

]

,

The capital letters and the superscripts * represent the steady-state components of corresponding lowercase variables, and Δ represents the small-signal component of the corresponding variable.

The mathematical model of the GSC controller is established as:

{

d
dr
c

⁢

U

d
⁢
c

*

=

u

d
⁢
g

c

+

H
cg

(

-

H
v

(

U
dcref

-

u

d
⁢
c

)

-

i

d
⁢
g

c

)

-

ω
⁢

L
f

⁢

i
qg
c

d
qg
c

⁢

U

d
⁢
c

*

=

u

d
⁢
g

c

+

H
cg

(

0
-

i
qg
c

)

+

ω
⁢

L
f

⁢

i

d
⁢
g

c

where

H

c
⁢
g

=

K

c
⁢
p
⁢
g

+

K

c
⁢
i
⁢
g

s

,

K_cpgand K_cigrepresent a proportional parameter and an integral parameter of grid-side current loop PI control, respectively,

H
v

=

K
vp

+

K
vi

s

,

and K_vpand K_virepresent the proportional parameter and the integral parameter of grid-side voltage loop PI control, respectively. U_dcrefrepresents the reference value of the DC voltage. In the GSC controller, a phase-locked loop is used to keep the PMSG-WT synchronous with the power grid. The superscript c represents the dq coordinate system of the GSC controller. By linearizing the model, the small-signal model of the GSC controller can be obtained as:

Δd_dqg^c·U*_dc=Δu_dqg^c+G_cgΔi_dqg^c+G_udcΔu_dc

where

G
cg

=

[

-

H
cg

-
ω
⁢

L
f

ω
⁢

L
f

-

H
cg

]

,

G
udc

=

[

H
cg

⁢

H
v

0

]

.

In addition, the dynamics of the phase-locked loop should also be considered in the GSC, and its mathematical model is:

θ
=

H
pll

·

1
s

·

u
qg
c

where

H
pll

=

K
ppll

+

K
ipll

s

,

K_plland K_ipllrepresent the proportional parameter and the integral parameter of a phase-locked loop PI controller, respectively, and u_qg^crepresents the q-axis voltage of the grid-connected point in the dq coordinate system of the GSC controller. The model is linearized to obtain:

Δ
⁢
θ

=

H
pll

·

1
s

·
Δ

⁢

u
qg
c

wherein there is a certain deviation between the dq coordinate system of the system and the dq coordinate system of the controller, and the two coordinate systems can be converted to each other through the following equation:

{

Δ
⁢

x
d
c

=

Δ
⁢

x
d

+

Δθ
·

X
q
*

Δ
⁢

x
q
c

=

Δ
⁢

x
q

+

Δθ
·

X
d
*

in the formula, the variables Δx_dand Δx_qmay represent the output current Δi_dgand Δi_qgof the GSC, the voltages Δu_dgand Δu_qgof the grid-connected point, or the output duty ratios Δd_dgand Δd_qgof the grid-side controller, and X*_qand X*_drepresent steady-state components I*_qg, I*_dg, U*_qg, U*_dg, D*_qg, D*_dg.

From this, the small-signal model of the phase-locked loop can be derived, namely:

Δθ=G_pll·Δu_qg

where

G
pll

=

H
pll

s
+

U
dg
*

⁢

H
pll

,

and then the relationship between the dq coordinate system of the controller and the dq coordinate system of the system can be obtained as:

{

Δ
⁢

u
dqg
c

=

G
pll
u

·
Δ

⁢

u
dqg

Δ
⁢

i
dqg
c

=

G
pll
i

·
Δ

⁢

u
dqg

+

Δ
⁢

i
dqg

Δ
⁢

d
dqg
c

=

G
pll
d

·
Δ

⁢

u
dqg

+

Δ
⁢

d
dqg

where

G
pll
u

=

[

1

G
pll

⁢

U
qg
*

0

1
-

G
pll

⁢

U
dg
*

]

,

G
pll
i

=

[

0

G
pll

⁢

I
qg
*

0

-

G
pll

⁢

I
dg
*

]

,

G
pll
d

=

[

0

G
pll

⁢

D
qg
*

0

-

G
pll

⁢

D
dg
*

]

.

Then, the small-signal model of the GSC controller is converted into:

(G_pll^d·Δu_dqg+Δd_dqg)·U*_dc=G_pll^u·Δu_dqg+G_cg(G_pllⁱ·Δu_dqg+Δi_dqg)+G_udcΔu_dc

Further, the step 3 is specifically:

the AC weak power grid is expressed by series equivalent inductance of an ideal voltage source, and its mathematical model is established as:

[

sL
g

-
ω
⁢

L
g

ω
⁢

L
g

sL
g

]

·

[

i
dg

i
qg

]

=

[

u
dg

u
qg

]

-

[

u
ds

u
qs

]

in the formula, L_grepresents the equivalent inductance of the weak grid, u_dsand u_qsrepresent ideal voltage source voltages of the d axis and the q axis, respectively, and i_dgand i_qgrepresent the d axis current and the q axis current of the AC port of the GSC, respectively. The formula is linearized to obtain:

Z_g·Δi_dqg=Δu_dqg

in the formula,

Z
g

=

[

sL
g

-
ω
⁢

L
g

ω
⁢

L
g

sL
g

]

.

The output power of the PMSG-WT is:

P=1.5(i_dgu_dg+i_qgu_qg)

The formula is linearized to obtain:

ΔP=1.5(I*_dqg^T·Z_g+U*_dqg^T)·Δi_dqg

in the formula,

U
dqg
*

=

[

U
dg
*

U
qg
*

]

.

By combining the linear parts of various part models of the aforementioned PMSG-WT grid-connected system, the transfer function G(s) of the linear part of the system can be obtained as:

G
⁢

(
s
)

=

-

1.5
ε

T

p
S

⁢

(

I
dqg
*

T

⁢

Z
g

+

U
dqg
*

T

)

⁢

M
5

-
1

⁢

(

D
dqg
*

+

G
udc

)

⁢

M
7

-
1

⁢

M
8

·

1

1
+

T

f
S

·

1

1
+

1.5

T

p
S

M
8

=

-

D
dqr
*

T

·

M
1

-
1

·

G

ω
⁢
g

+

I
dqr
*

T

·

M
3

M
7

=

(

2
⁢

sC
dc

3

+

I
dqg
*

T

⁢

M
6

-

I
dqr
*

T

⁢

M
4

+

D
dqg
*

T

⁢

M
5

-
1

(

D
dqg
*

+

G
udc

)

+

D
dqr
*

T

⁢

M
1

-
1

⁢

D
dqr
*

)

M
6

=

(

(

(

G
cg

⁢

G
pll
i

-

G
pll
d

⁢

U
dc
*

+

G
pll
u

)

⁢

Z
g

+

G
cg

)

⁢

M
5

-
1

(

D
dqg
*

+

G
udc

)

+

G
udc

)

/

U
dc
*

M
5

=

-

G
cg

+

Z
f

-

(

(

G
cg

⁢

G
pll
i

-

G
pll
d

⁢

U
dc
*

+

G
pll
u

)

⁢

Z
g

+

Z
g

M
4

=

-

M
2

⁢

M
1

-
1

⁢

D
dqr
*

/

U
dc
*

M
3

=

(

-

M
2

⁢

M
1

-
1

⁢

G

ω
⁢
g

+

G

ω
⁢
g

)

/

U
dc
*

M
2

=

G

ω
⁢
e

⁢

G

iq
⁢
ω

⁢

n
p

+

G
cr

⁢

G

θ
⁢
e

i

-

G

ω
⁢
g

⁢

G

iq
⁢
ω

-

U
dc
*

⁢

G

θ
⁢
e

d

M
1

=

G
cr

⁢

G

θ
⁢
e

i

-

G

ω
⁢
g

⁢

G

iq
⁢
ω

-

U
dc
*

⁢

G

θ
⁢
e

d

+

Z
dqr

in the formula, T_frepresents the period of a power sampling filter, 1/(1+T_fs) represents the delay of the power sampling filter, and 1/(1+1.5T_ps) represents the delay of the controller and the PWM.

Compared with the prior art, the present application has the advantages that:

(1) the present application applies the describing function method to the stability analysis of the grid-connected PMSG-WT system, fully considers the nonlinear link in the power loop based on the P&O method, and overcomes the problem that the traditional small-signal analysis method cannot be applied to discontinuous nonlinear links.

(2) When the system is in a critically stable state, oscillation occurs at constant amplitude and constant frequency. The describing function method can quantitatively calculate the frequency and amplitude of the oscillation, which provides an important basis for the prevention and suppression of the oscillation.

BRIEF DESCRIPTION OF DRAWINGS

FIG. 1 shows a topological structure (A) of a PMSG-WT and its controller structure (B and D represent coordinate conversion, C represents a phase-locked loop, E represents an MSC controller, and F represents a GSC controller);

FIG. 2 shows a distribution diagram of G(s) poles;

FIG. 3 shows G(s) and −1/N(A) curves when an L_gvalue is changed;

FIG. 4a shows a simulation verification waveform (a) when Lg=0.1 mH;

FIG. 4b shows FFT spectrum analysis (b) when Lg=0.1 mH;

FIG. 5a shows the simulation verification waveform (a) when Lg=0.4 mH; and

FIG. 5b shows FFT spectrum analysis (b) when Lg=0.4 mH.

DESCRIPTION OF EMBODIMENTS

The topological structure of the PMSG-WT and its controller according to the present application are shown in FIG. 1. The PMSG-WT includes a wind turbine, a generator, an MSC, a DC capacitor, a GSC, and a filter. The wind turbine captures wind energy and converts the wind energy into mechanical energy, both the MSC and the GSC are two-level voltage source converters, wherein the MSC converts the alternating current output by the permanent magnet synchronous generator into direct current, and the GSC inverts the direct current into power frequency alternating current and integrates the power frequency alternating current into the power grid. Both the MSC and the GSC adopt a vector control method in a dq coordinate system, and convert three-phase voltage and current under the abc frame into the voltage and current under d-axis and q-axis through dq transformation. The MSC controller includes a power loop (P&O), a rotating speed loop (H_ω), and a machine-side current loop (H_cr); and the GSC controller includes a voltage loop (H_v) and a grid-side current loop (H_cg). In addition, the PMSG-WT also includes a phase-locked loop (PLL) and coordinate conversion links (abc/dq and dq/abc). Hereinafter, the present application will be further described in combination with specific embodiments:

In one embodiment of the present application, main parameters of the system are shown in Table 1.

TABLE 1

Main parameters of the system

Parameter
Value

Equivalent inductance of a weak
0.1
mH

power grid L_g

Voltage amplitude of an ideal
690{square root over (2)}/{square root over (3)}

voltage source of the weak

power grid U_s

Filtering inductance L_f
1.2
mH

DC capacitor C_dc
2
mF

Armature inductance of the
1.5028
mH

generator L_s

Rotor resistance of generator R_s
0.0224
Ω

Permanent magnet flux
5.3445
Wb

linkage ψ_f

Power frequency angular
100π
rad/s

frequency ω

The number of pole pairs of the
48

generator n_p

Equivalent rotational inertia J
1417769.4

Perturbation step length of a
0.0001
rad/s

power loop ε

Control period of the power
0.1
ms

loop T_p

Reference value of output power
0.688
MW

of the PMSG-WT P^ref

Proportional parameter of a
10902

rotating speed loop K_ωp

Integral parameter of the rotating
17875

speed loop K_ωi

Proportional parameter of a
3.13

machine-side current K_cpr

Integral parameter of the
109.8

machine-side current loop K_cir

Proportional parameter of a
19.05

grid-side current loop K_cpg

Integral parameter of the
176

grid-side current loop K_cig

Proportional parameter of a
1.6

grid-side voltage loop K_vp

Integral parameter of the
28

grid-side voltage loop K_vi

Proportional parameter of the
2.1

phase-locked loop K_ppll

Integral parameter of the
36.6

phase-locked loop K_ipll

Reference value of DC voltage
1400
V

U_dcref

Steady-state value of rotating
3
rad/s

speed of the generator ω_g*

Linearization constant of a
−2.742 × 10⁵

mechanical torque of the wind

turbine B_t

Period of a power sampling
0.01

filter T_f

In the embodiment of the present application, step 1, acquiring main parameters of the PMSG-WT, wherein the main parameters as shown in Table 1, establishing mathematical models of the wind turbine, the generator and the MSC, the MSC controller, the DC capacitor, the GSC and the filter, and the GSC controller, respectively, performing linearization in a dq coordinate system, and calculating steady-state operation parameters to obtain small-signal models of the wind turbine, the generator and the MSC, the MSC controller, the DC capacitor, the GSC and the filter, and the GSC controller:

sJ
⁢

Δω
g

=

Δ
⁢

T
m

-

Δ
⁢

T
e

Δ
⁢

T
e

=

-

3
2

⁢

n
p

⁢

ψ
f

⁢
Δ
⁢

i
qr

Δ
⁢

T
m

=

B
t

⁢

Δω
g

Z
dqr

·
Δ

⁢

i
dqr

=

-

U
dc
*

·
Δ

⁢

d
dqr

-

D
dqr
*

·
Δ

⁢

u
dc

+

G

ω
⁢
e

·

Δω
e

Δ
⁢

d
dqr
c

=

1

U
dc
*

⁢

(

G
cr

·
Δ

⁢

i
dqr
c

+

G

ω
⁢
g

(

ω
g
ref

-

ω
g

)

+

G

ω
⁢
e

·

Δω
e

)

{

Δ
⁢

i
dqr
c

=

G

θ
⁢
e

i

·
Δ

⁢

i
dqr

Δ
⁢

d
dqg
c

=

G

θ
⁢
e

d

·
Δ

⁢

i
dqr

+

Δ
⁢

d
dqg

sC
dc

⁢
Δ
⁢

u
dc

=

1.5

(

D
dqr
*

T

·
Δ

⁢

i
dqr

+

I
dqr
*

T

·
Δ

⁢

d
dqr

)

-

1.5

(

D
dqg
*

T

·
Δ

⁢

i
dqg

+

I
dqg
*

T

·
Δ

⁢

d
dqg

)

Z
f

·
Δ

⁢

i
dqg

=

U
dc
*

·
Δ

⁢

d
dqg

+

D
dqg
*

·
Δ

⁢

u
dc

-

Δ
⁢

u
dqg

Δ
⁢

d
dqg
c

·

U
dc
*

=

Δ
⁢

u
dqg
c

+

G
cg

⁢
Δ
⁢

i
dqg
c

+

G
udc

⁢
Δ
⁢

u
dc

{

Δ
⁢

u
dqg
c

=

G
pll
u

·
Δ

⁢

u
dqg

Δ
⁢

i
dqg
c

=

G
pll
i

·
Δ

⁢

u
dqg

+

Δ
⁢

i
dqg

Δ
⁢

d
dqg
c

=

G
pll
d

·
Δ

⁢

u
dqg

+

Δ
⁢

d
dqg

s represents a parameter introduced by the Laplace transform, J represents rotational inertia of an equivalent concentrated mass block of the wind turbine and the generator, ω_grepresents a rotating speed of the generator, T_mrepresents a mechanical torque of the generator, and T_erepresents an electromagnetic torque of the generator. n_prepresents the number of pole pairs of the generator, ψ_frepresents a permanent magnet flux linkage of the generator, and B_trepresents a linearization constant of the mechanical torque of the wind turbine.

Z
dqr

=

[

sL
s

+

R
s

-

ω
e
*

⁢

L
s

ω
e
*

⁢

L
s

sL
s

+

R
s

]

,

Δ
⁢

i
dqr

=

[

Δ
⁢

i
dr

Δ
⁢

i
qr

]

,

Δ
⁢

d
dqr

=

[

Δ
⁢

d
dr

Δ
⁢

d
qr

]

,

D
dqr
*

=

[

D
dr
*

D
qr
*

]

,

G

ω
⁢
e

=

[

L
s

⁢

I
qr
*

ψ
f

-

L
s

⁢

I
dr
*

]

,

R_sand L_srepresent rotor resistance and armature inductance of the generator, respectively, ω_erepresents an electrical angular speed of the rotor, and ω_e=n_pω_g. i_drand i_qrrepresent the stator current of the generator in the dq coordinate system, d_drand d_qrrepresent output duty ratios of the MSC controller in the dq coordinate system, and u_dcrepresents a DC voltage.

G
cr

=

[

H
cr

ω
e
*

⁢

L
s

-

ω
e
*

⁢

L
s

H
cr

]

,

G

ω
⁢
g

=

[

0

H
cr

⁢

H
ω

]

,

H
cr

=

K
cpr

+

K
cir

s

,

K_cprand K_cirrepresent a proportional parameter and an integral parameter of machine-side current loop PI control, respectively,

H
ω

=

K

ω
⁢
p

+

K

ω
⁢
i

s

,

K_ωpand K_ωirepresent the proportional parameter and the integral parameter of rotating speed loop PI control, respectively, ω_g^refrepresents the reference value of the rotating speed of the generator, ω*_erepresents the steady-state value of the electrical angular speed of the rotor, and the superscript c represents a dq coordinate system of the MSC controller.

G

θ
⁢
e

i

=

[

1

H

θ
⁢
e

⁢

I
qr
*

0

1
-

H

θ
⁢
e

⁢

I
dr
*

]

,

G

θ
⁢
e

d

=

[

0

H

θ
⁢
e

⁢

D
qr
*

0

-

H

θ
⁢
e

⁢

D
dr
*

]

,

H

θ
⁢
e

=

3
⁢

n
p
2

⁢

ψ
f

2
⁢

s
⁡
(

B
t

-
sJ

)

.

I
dqr
*

[

I
dr
*

I
qr
*

]

,

Δ
⁢

i
dqg

=

[

Δ
⁢

i
dg

Δ
⁢

i
qg

]

,

Δ
⁢

d
dqg

=

[

Δ
⁢

d
dg

Δ
⁢

d
qg

]

,

D
dqg
*

=

[

D
dg
*

D
qg
*

]

,

I
dqg
*

=

[

I
dg
*

I
qg
*

]

,

C_dcrepresents the DC capacitor, i_dgand i_qgrepresent d-axis current and q-axis current of an AC port of the GSC, respectively, and d_dgand d_qgrepresent the output duty ratios of the GSC controller in the dq coordinate system, respectively.

Z
f

=

[

sL
f

-
ω

⁢

L
f

ω
⁢

L
f

sL
f

]

,

Δ
⁢

i
dqg

=

[

Δ
⁢

i
dg

Δ
⁢

i
qg

]

,

Δ
⁢

d
dqg

=

[

Δ
⁢

d
dg

Δ
⁢

d
qg

]

,

D
dqg
*

=

[

D
dg
*

D
qg
*

]

,

Δ
⁢

u
dqg

=

[

Δ
⁢

u
dg

Δ
⁢

u
qg

]

,

L_frepresents filtering inductance, ω represents power frequency angular frequency, ω=100π rad/s, i_dgand i_qgrepresent the d-axis current and the q-axis current of the AC port of the GSC, respectively, d_dgand d_qgrepresent the output duty ratios of the GSC controller in the dq coordinate system, respectively, and u_dgand u_qgrepresent a d-axis voltage and a q-axis voltage of a grid-connected point, respectively.

G
cg

=

[

-

H
cg

-
ω

⁢

L
f

ω
⁢

L
f

-

H
cg

]

,

G
udc

=

[

H
cg

⁢

H
v

0

]

,

H
cg

=

K
cpg

+

K
cig

s

,

K_cpgand K_cigrepresent the proportional parameter and the integral parameter of grid-side current loop PI control, respectively,

H
v

=

K
vp

+

K
vi

s

,

and K_vpand K_virepresent the proportional parameter and the integral parameter of grid-side voltage loop PI control, respectively.

G
pll
u

=

[

1

G
pll

⁢

U
qg
*

0

1
-

G
pll

⁢

U
dg
*

]

,

G
pll
i

=

[

0

G
pll

⁢

I
qg
*

0

-

G
pll

⁢

I
dg
*

]

,

G
pll
d

=

[

0

G
pll

⁢

D
qg
*

0

-

G
pll

⁢

D
dg
*

]

,

G
pll

=

H
pll

s
+

U
dg
*

⁢

H
pll

,

H
pll

=

K
ppll

+

K
ipll

s

,

K_pplland K_ipllrepresent the proportional parameter and the integral parameter of a phase-locked loop PI controller, respectively.

Capital letters and superscripts * represent steady-state components of corresponding lowercase variables, and the specific calculation method is as follows:

{

I
qr
*

=

3
⁢

P
ref

/

n
p

⁢

Ψ
f

⁢

ω
g
*

I
dr
*

=
0

U
dc
*

=

U
dcref

D
dr
*

=

(

ω
e

⁢

I
qr
*

⁢

L
s

-

R
s

⁢

I
dr
*

)

/

U
dc
*

D
qr
*

=

(

-

ω
e

⁢

L
s

⁢

I
dr
*

+

ω
e

⁢

Ψ
f

-

R
s

⁢

I
qr
*

)

/

U
dc
*

U
dg
*

=

(

(

1.5

U
s

)

2

+

(

1.5

U
s

)

4

-

(

3
⁢

P
ref

⁢
ω
⁢

L
g

)

2

)

/
4.5

U
qg
*

=
0

I
dg
*

=

P
ref

/

(

1.5

U
dg
*

)

I
qg
*

=
0

D
dg
*

=

(

-

I
qg
*

⁢
ω
⁢

L
f

+

U
dg
*

)

/

U
dc
*

D
dg
*

=

(

I
dg
*

⁢
ω
⁢

L
f

+

U
dg
*

)

/

U
dc
*

Step 2: modeling the power loop in the MSC controller based on a describing function method, wherein its mathematical expression is:

ω
g
ref

=

{

ε

s
⁢

T
P

⁢

sgn
⁡
(

P
ref

-

P
n

)

 
,

(

ω
g

≤

ω
mpp

)

-

ε

s
⁢

T
P

⁢

sgn
⁡
(

P
ref

-

P
n

)

 
,

(

ω
g

>

ω
mpp

)

the sign function in the formula can be modeled by using a describing function, and the describing function is:

N
⁡
(
A
)

=

4

π
⁢
A

Step 3: considering the influence of the weak AC power grid, the weak AC power grid model is:

Z_g·Δi_dqg=Δu_dqg

in the formula

,

Z
g

=

[

sL
g

-
ω

⁢

L
g

ω
⁢

L
g

sL
g

]

.

The small-signal model of the output power of the PMSG-WT is:

ΔP=1.5(I*_dqg^T·Z_g+U*_dqg^T)·Δi_dqg

combing the linear parts of the weak power grid model and the power loop model with the small-signal model of the PMSG-WT in step 1, and deriving a transfer function G(s) of the linear part of the system:

G
⁡
(
s
)

=

-

1
.
5

⁢
ε

T
P

⁢
s

⁢

(

I
dqg

*
T

⁢

Z
g

+

U
dqg

*
T

)

⁢

M
5

-
1

(

D
dqg
*

+

G
udc

)

⁢

M
7

-
1

⁢

M
8

·

1

1
+

T
f

⁢
s

·

1

1
+

T
P

⁢
s

where

M
8

=

-

D
dqr

*
T

·

M
1

-
1

·

G

ω
⁢
g

+

I
dqr

*
T

·

M
3

M
7

=

(

2
⁢
s
⁢

C
dc

3

+

I
dqg

*
T

⁢

M
6

-

I
dqr

*
T

⁢

M
4

+

D
dqg

*
T

⁢

M
5

-
1

(

D
dqg
*

+

G
udc

)

+

D
dqr

*
T

⁢

M
1

-
1

⁢

D
dqr
*

)

M
6

=

(

(

(

G
cg

⁢

G
plli

-

G
plld

⁢

U
dc
*

+

G
pllu

)

⁢

Z
g

+

G
cg

)

⁢

M
5

-
1

(

D
dqg
*

+

G
udc

)

+

G
udc

)

/

U
dc
*

M
5

=

-

G
cg

+

Z
f

-

(

G
cg

⁢

G
plli

-

G
plld

⁢

U
dc
*

+

G
pllu

)

⁢

Z
g

+

Z
g

M
4

=

-

M
2

⁢

M
1

-
1

⁢

D
dqr
*

/

U
dc
*

M
3

=

(

-

M
2

⁢

M
1

-
1

⁢

G

ω
⁢
g

+

G

ω
⁢
g

)

/

U
dc
*

M
2

=

G

ω
⁢
e

⁢

G

i
⁢
q
⁢
ω

⁢

n
p

+

G
cr

⁢

G

θ
⁢
e

i

-

G

ω
⁢
g

⁢

G

i
⁢
q
⁢
ω

-

U
dc
*

⁢

G

θ
⁢
e

d

M
1

=

G
cr

⁢

G

θ
⁢
e

i

-

G

ω
⁢
g

⁢

G

i
⁢
q
⁢
ω

-

U
dc
*

⁢

G

θ
⁢
e

d

+

z
dqr

Step 4: analyzing the stability of the system. Firstly, a G(s) pole diagram is drawn, as shown in FIG. 2, it can be seen that G(s) does not contain the pole of the right half plane (the real part is greater than 0), so the first condition for the stability of the system is satisfied. G(s) and −1/N(A) images are drawn in a complex plane, as shown in FIG. 3, G(s) and −1/N(A) intersect, indicating that the system is in a critically stable state at this time. It can be seen from calculation that, when Lg=0.1 mH, the oscillation frequency of the system is about 129 rad/s (20.5 Hz), and the oscillation amplitude is about 61 kW; and when Lg=0.4 m, the oscillation frequency of the system is about 131 rad/s (20.9 Hz), the oscillation amplitude is about 56 kW. When the power grid strength decreases (Lg increases), the oscillation amplitude of the system decreases, indicating that under certain conditions, the increase of the equivalent impedance of the power grid is conducive to maintaining the stability of the system.

FIGS. 4a, 4b, 5a and 5b show system simulation and FFT spectrum analysis results when Lg=0.1 mH and Lg=0.4 mH, respectively. In FIGS. 4a and 4b, the DC component is 0.6889 MW, the corresponding oscillation frequency is 19.5 Hz, and the amplitude is 58 kW, which are basically consistent with the theoretical analysis results; and in FIGS. 5a and 5b, the DC component is 0.6889 MW, the corresponding oscillation frequency is 19.5 Hz, and the amplitude is 53 kW, which are basically consistent with the theoretical analysis results. The simulation results verify the effectiveness and accuracy of the analysis method.