Small-Signal Stability Analysis of Wind Power System Based on DFIG

doi:10.4236/epe.2013.54B081

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Energy and Power Engineering, 2013, 5, 418-422

doi:10.4236/epe.2013.54B081 Published Online July 2013 (http://www.scirp.org/journal/epe)

Small-Signal Stability Analysis of Wind Power System

Based on DFIG

Bin Sun, Zhengyou He, Yong Jia, Kai Liao

School of Electrical Engineering, Southwest Jiaotong University, Chengdu, China

Email: [email protected]

Received April, 2013

ABSTRACT

This paper focuses on the small-signal stability of power syste m integrated with DFIG-based wind farm. The model of

DFIG for small-signal stability analysis has built; the 3-generator 9-bus WECC test system is modified to investigate

the impacts of large scale integration of wind power on power system small-signal stab ility. Different oscillatory modes

are obtained with their eigenvalue, frequency and damping ratio, the results from eigenvalue analysis are presented to

demonstrate the small-signal stability of power system is enhanced with the increasing output of the wind farm.

Keywords: Small-Signal Stability; DFIG; Wind Power; Power System; Wind Farm

1. Introduction

As of the end of 2011, the installed capacity of wind

power in China has reached 62.36 GW [1], the planning

and construction of large scale wind farm will be the

inevitable trend of the development of wind power in

China. Wind power has obvious characteristics of ran-

domness, volatility and intermittent, this development

affects the small-signal stability of the traditional power

system, so it is significant to study the impacts of large

scale integration of wind power on power system

small-signal stability.

In [2], an aggregated wind farm model with DFIG

based wind turbine for small-signal stability study is

proposed to deal with the situation that the wind turbin es

operate receiving different incoming wind speeds. In [3],

the 10-generator 39-bus New England test system is ap-

plied to assess the effect of the large scale wind farm on

power system small-signal stability. In [4], the effect of

wind power on the oscillations is investigated by gradu-

ally replacing the power generated by the synchronous

generators in the system by power from either constant or

variable speed wind turbines, while observing the

movement of the eigenvalues through the complex plane.

The aim of this paper is to study the small-signal sta-

bility of wind power system based on DFIG. This paper

is structured as follows, with DFIG model in section II,

and section III describe the modified 3-generator 9-bus

WECC system applied in the simulation and analysis.

The small-signal stability analysis of DFIG based wind

farm is described in section IV, with discussions and

conclusions presented in section V.

2. Modeling of DFIG

There are mainly three kinds of wind turbine widely used

nowadays: the constant speed wind turbine with squirrel

cage induction generator, the variable speed wind turbine

with doubly fed induction generator (DFIG) and the di-

rect drive synchronous generator. The main differences

between the three schemes are the generating system and

the way in which the aerodyna mic efficiency of the rotor

is limited during wind speed variations [5, 6]. As a mat-

ter of fact, each of the wind generating systems has its

own advantages and disadvantages. Almost all newer

larger wind generating system being produced are vari-

able speed systems based on DFIGs, so DFIG based

wind farm is the main object in this study.

As is well known, the rotor terminals of a DFIG are

fed with a symmetrical three-phase voltage of variable

frequency and amplitude fed through a voltage source

converter usually equipped with IGBT based power elec-

tronic circuitry [7]. The basic topolog y of DFIG is shown

in Figure 1.

2.1. Turbine and Drive Train

The mechanical power input to the wind turbine is con-

sidered as constant, so the pitch angle do not change

during the period of study. In this paper, the two-mass

drive train model [8] is considered and the dynamics can

be expressed by the following d ifferential equations [9]:





dTT

dt H





(1)



bt r









 (2)

B. SUN ET AL. 419



msh

dTT

dt H







(3)

where ωr is the generator angular speed, ωt is the wind

turbine angular speed, ωb is the electrical base speed, θt is

the shaft twist angle, Hg is the inertia constant of the ge-

nerator, Ht is the inertia constant of turbine, Tsh is the

shaft torque, Te is the electrical torque, Tm is the me-

chanical torque.

2.2. Generator

The most common way of representing DFIG for the

purpose of simulation and control is in terms of direct

and quadrature axes (dq axes) quantities, which form a

reference frame that rotate synchronously with the stator

flux vector [10]. The various variables are defined as:

dsmrr s qr



 , qsmrr s dr



, 2

()

ssm rr

LL LL

 ,

rrrr

TLR, mrrm rr

LL, e sb







e,

and 12s. Here, ds and qs are d-axis and

q-axis voltage behind transient reactance, respectively,

2mrr r

RKR

RRRe





and qr



are d-axis and q-axis rotor fluxes, respec-

tively, Lss is the stator self-inductance, Lrr is the rotor

self-inductance, Lm is the mutual inductance between

rotor and stator, Rr is the rotor resistance, and Rs is the

stator resistance.

For balanced and unsaturated conditions, the corre-

sponding DFIG model can be expressed as [10]:

ss dsr

dsssqsds

qsdsmrr dr

Ldi Ri Lie

evKv





 







(4)

ss r

qsss dsqs

dsqsmrr qr

LRi Lie

evKv











 





(5)

ds r

qsdsqsmrr qr

erss

de RieeK v

dt T













 (6)

qs r

dsqsdsmrr dr

erss

de RieeK v

dt T









 



 (7)

where ids and iqs are d-axis and q-axis stator currents,

respectively.

2.3. Converter

The converter model in DFIG comprises of two pulse-

width modulation invertors connected back to back via a

dc link. The rotor side converter (RSC) and the grid side

converter (GSC) act as a controlled voltage sources. The

RSC injects an ac voltage at slip frequency to the rotor,

whereas the GSC injects an ac voltage at power fre-

quency to the gr id and maintains the dc link vo ltage con-

stant. The power balance equation for the converter

model can be written as follows:

rgd

PPP



 (8)

where Pr, Pg and Pdc are the active power at RSC, GSC

and dc link, respectively, which can be expressed as fol-

lows:

rdrdrqrq

Pvivi



 (9)

dg dgqg qg

Pvi vi



 (10)

dcdc dcdc

Pvi Cv

 (11)

where vdr and vqr are d-axis and q-axis rotor voltages,

respectively, idr and iqr are d-axis and q-axis rotor cur-

rents, respectively, vdg and vqg are d-axis and q-axis volt-

age of the GSC, respectively, idg and iqg are d-aixs and

q-axis of the GSC, respectively, vdc and idc are the voltage

and current of the dc link capacitor, respectively, and C

is the capacitance of the dc capacitor. The direction of

the currents and power flow are demonstrated in Figure

2.4. Controllers for DFIG

As mentioned above, there are two back to back convert-

ers named RSC and GSC in DFIG system, some meas-

ures should be taken to control the tw o converters so that

the DFIG can works in the appropriate condition. There

are many kinds of control strategy we can choose to con-

trol the RSC and GSC, in this paper, we adopt the model

of controllers built in [1 1], in which a decou pling control

for the active power and reactive power of DFIG was

developed.

3. Generic Test System

In this study, the 3-generator 9-bus WECC test system

has been modified to assess the impact of a wind power

plant based on DFIG on small-signal stab ility. The single

line diagram is shown in Figure 2. The system contains 3

synchronous generators and 3 constant impedance loads.

The large scale wind farm selected for simulation and

Figure 1. The basic topology of DFIG.

B. SUN ET AL.

420

Figure 2. Single line diagram of the analyzed case network.

analysis consists of 50 DFIG-based wind turbines and the

capacity of each turbine is 1.5 MW, so the total capacity

of this wind farm is 75 MW. Here, a single equivalent

model was used to represent all individual units within

the wind power plant in order to avoid increasing com-

putation time, so the modified WECC test system con-

tains 3 synchronous generators and a wind turbine repre-

sented by a DFIG.

The base capacity and frequency of the system are 100

MVA and 50 Hz. The active power and reactive power of

3 loads are adjusted, the active power and reactive power

of bus 5 are 1.200 and 0.075 pu respectively, the active

power and reactive power of bus 6 are 0.700 and 0.070

pu respectively, the active power and reactive power of

bus 8 are 2.250 and 0.350 pu respectively. In order to

ensure the power flow of the system unchanged, the out-

put of synchronous generator G3 should be adjusted at

the same time when increasing the output of wind farm.

The single wind turbine parameters are as follows:

UN=0.69 kV, SN=1.5 MW, f=50 Hz, Rs=0.01 pu, Xs=0.10

pu, Rr=0.01 pu, Xr=0.08 pu, Xm=4 pu, Hm=3 kWs/kVA,

Kp=10, Tp=3 s, Kv=10, TE=0.01s.

4. Small-Signal Stability Analysis

Small-signal stability is the ability of the power system to

maintain synchronism when subjected to small distur-

bances [12]. The disturbance is considered small enough

to permit the equations that describe the resulting re-

sponse of the system to be linearized and expressed in

state-space form. Then, by calculating the eigenvalues of

the linearized model, the small-signal stability character-

istics of the system can be evaluated. In today’s practical

power systems, the small-signal stability problem is usu-

ally on of insufficient damping of system oscillations.

Small-signal stability analysis using linear techniques

provides valuable information about the inherent dy-

namic charateristic of the power system and assists in its

design.

The most direct way to assess small-signal stability of

the model of power system is via eigenvalue analysis.

When the wind farm output is 60% of its total capacity,

i.e., the output of wind farm is 45 MW, at the same time,

adjust the output of G3 to 40 MW in order to keep the

power flow constant. The eigenvalues associated with

this case are given in Table 1. We can get 8 oscillatory

modes, most of their oscillatory frequencies are between

0.1 Hz and 2.5 Hz, so most of them are belong to low-

frequency modes. As a matter of fact, the low-frequency

oscillation is of great harm to the safe and stable opera-

tion of the power system, so these low-frequency oscilla-

tion modes must be suppressed or eliminate by taking

some efficient measures.

The eigenvalues will change with the increase of the

output of wind farm. Since the damping ratios of mode I

and mode II are relatively small, take them as an example,

the eigenvalue, and frequency and damping ratio of mode

I and mode II shifting with the output of wind farm va-

rying are given in Table 2 and Table 3, the eigenvalue of

mode I and mode II shifting with the output of wind farm

increasing are shown in Figure 3 and Figure 4.

We can learn from two tables and two figures above,

with the increase of the output of wind farm, the eigen-

values move to the left on the complex plane, the damp-

ing ratios become bigger and bigger, i.e., the small-signal

stability of the power system is constantly enhanced.

When the wind far m output at rated power, the system is

the most stable. When the wind farm output below rated

power, the larger the difference between output power

and rated power, the more unstable the system is.

5. Conclusion

In this paper, we build the model of DFIG for small-

signal stability an alysis and modify the 3-generator 9-bus

WECC test system to study the impact of large scale in-

tegration of wind power on power system small-signal

stability first. Then, different oscillatory modes are got

by us with their eigenvalue, frequency and damping ratio.

Table 1. Main eigenvalues, frequenc ies and damping Ratios

of the system.

Oscillatory

Mode No.Eigenvalue Frequency

( Hz ) Damping ratio

( % )

1 －1.168±j10.764 1.713 10.788

2 －0.268±j7.628 1.214 3.511

3 －5.472±j7.935 1.263 56.770

4 －5.171±j7.794 1.240 55.285

5 －5.240±j7.891 1.256 55.319

6 －3.647±j0.633 0.101 98.527

7 －0.492±j1.082 0.174 41.393

8 －0.463±j0.717 0.114 54.247

B. SUN ET AL. 421

Table 2. The eigenvalue, frequency and damping ratio of

mode I shifting with the output of wind farm varying.

Wind Farm

Output ( % ) Eigenvalue Frequency of oscilla-

tion ( Hz ) Damping

Ratio ( % )

0 －0.667±j11.159 1.776 5.967

10 －0.726±j11.105 1.767 6.524

20 －0.795±j11.045 1.758 7.179

30 －0.875±j10.979 1.747 7.945

40 －0.965±j10.908 1.736 8.812

50 －1.064±j10.835 1.724 9.773

60 －1.168±j10.764 1.713 10.788

70 －1.271±j10.700 1.703 11.796

80 －1.364±j10.647 1.695 12.707

90 －1.440±j10.609 1.688 13.450

100 －1.528±j10.565 1.681 14.314

Table 3. The eigenvalue, frequency and damping ratio of

Mode II shifting with the output of wind farm varyi ng.

Wind Farm

Output ( % ) Eigenvalue Frequency of

oscillation ( Hz ) Damping

Ratio (%)

0 －0.223±j7.648 1.217 2.915

10 －0.226±j7.643 1.216 2.956

20 －0.231±j7.639 1.216 3.023

30 －0.238±j7.635 1.215 3.116

40 －0.247±j7.631 1.215 3.235

50 －0.257±j7.629 1.214 3.367

60 －0.268±j7.628 1.214 3.511

70 －0.279±j7.629 1.214 3.655

80 －0.288±j7.632 1.215 3.771

90 －0.296±j7.636 1.215 3.873

100 －0.306±j7.640 1.216 4.002

Figure 3. The eigenvalue of mode I shifting with the output

of wind farm incre asing.

Figure 4. The eigenvalue of mode II shifting with the output

of wind Farm varying.

At last, we study the relationship between the output of

wind farm and the small-signal stability of power system,

and get some preliminary conclusions. Since the 3-gen-

erator 9-bus WECC test system is just a small system,

further research should be done in some larger systems.

6. Acknowledgements

The authors would like to thank the National High-tech

Research and Development Program of China. (Grant No.

2012AA050208) for financial support.

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