W. Sauer. Dynamic Modeling of Doubly Fed Induction

8/12/2019 W. Sauer. Dynamic Modeling of Doubly Fed Induction

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Dynamic modeling of wind power generation

Hector A. Pulgar-Painemal and Peter W. Sauer.Department of Electrical and Computer Engineering

University of Illinois at Urbana-Champaign, USA

E-mail: [email protected] and [email protected]

AbstractThis paper presents a dynamic model appropriatefor power system analysis. This article shows modeling assump-tions, derivation of a third order model for a doubly-fed inductiongenerator and its controller models. Due to the detail level,it can be used as a tutorial for students and engineers thatare new in this area. A four-bus system with one synchronousmachine and one wind turbine is used to perform a small signalstability analysis. No considerable difference is observed betweenthe modal behavior of our test system and a small system with

just one synchronous generator.Index TermsPower System Dynamics, Wind Power Modeling

I. INTRODUCTION

The energy consumption over the world has been sus-

tainedly increased in the last decades due to the rate of growth

in world gross domestic productthe main driver of energy

demand. It is expected that the electricity demand will increase

at a rate of 2.6% per year during the period 2004-2030.Additionally, global energy-related carbon-dioxide emissions,

a major cause of global warming, are expected to increase by

1.7% per year during the same periodreaching 40.4 109

tonnes in 2030. Unfortunately, power generation is projected

to contribute almost50% of that increased emission [1]. Thus,the power generation sector is under scrutiny and has to be

expanded to fulfill the high-energy demand scenario but takinginto account environmental effects such as global warming.

Due to the notorious impact on environmental problems and

the depletion of fossil fuels, renewable energy sources have

been considered appealing to face the forthcoming energy sce-

nario. Hydraulic, wind, solar, biomass and geothermal sources

are the most common alternative generation systems. Hy-

draulic systems are attractive due to their robustness, reliability

and high rated power levels. However, the main drawbacks are

the scarce available locations and the negative impact on the

local ecosystem by flooding extensive areas. At present, among

the other alternatives, wind generation systems are the most

qualified to produce electricity in power systems. Although

being irregular in their electricity production, wind farms areable to provide energy: (a) without the risk of depletion of

their primary energy source, and (b) being able to comply with

operations standards. Additionally, wind generation systems

have the fastest payback period [2], less than a year; the lowest

project gestation period, due to modular concept; and low

operation-maintenance costs [3]. These characteristics make

wind power attractive for mass production which is reflected

by the increase of the worldwide installed wind-power capacity

over the last years [4].

In the 1990s, typical wind power turbines were characteri-

zed by a fixed-speed operation. Basically, they consisted of the

coupling of a wind turbine, a gearbox and an induction ma-

chine directly connected to the grid. Additionally, they used a

soft starter to energize the machine and a bank of capacitors to

compensate the machine power-reactive absorption. Although

being simple, reliable and robust, the fixed-speed wind turbines

were inefficient and power fluctuations were transmitted to the

network due to wind speed fluctuations [5].

In the mid-1990s, variable-speed wind power turbines gave

an impulse to the wind power industry. A better turbinecontrol is able to reduce power fluctuations. Besides, optimal

power extraction from the wind was possible by operating the

turbine at optimal speed. Among the different configurations

of variable-speed wind turbines, the doubly-fed induction gen-

erator (DFIG), at present, is the most used in the development

of new wind farm projects. This configuration consists of the

coupling of a turbine, a gearbox and an induction machine

doubly connected to the griddirectly connected from the

stator circuits and indirectly connected from the rotor circuits

by using converters. Its main drawbacks are the use of slip

rings and protection in case of grid disturbances [6]. The

control can be done (a) by controlling the voltage applied

to the rotor circuits, (b) by adjusting the pitch angle of theturbine bladesangle of incidence of the blade and the wind

direction, and (c) by designing aerodynamically the turbine

blades to stall when the wind speed exceeds its limit [7].

Due to the importance of wind power generation on the

current and future worldwide energetic scenario, dynamic

models are required for teaching and training purposes. One

major problem is that dynamic models of commercial wind

power turbines contain proprietary information and require

confidentiality agreements between the company and the user

[8]. In the literature, there are several articles dealing with

different issues related to wind power generators. However, a

rather general description of the dynamic model can be found

and a complete dynamic model appropriate for power systemanalysis is not directly available. Thus, this paper presents

a detailed dynamic model for wind power generation which

is suitable for power system analysis. Modeling assumptions,

derivation of a third order model for the DFIG and its

controllers are described through out the article which can be

used as a tutorial for students and engineers that are new in

this area. Using a four-bus test system with one synchronous

generator and a wind turbine, the modal behavior is similar

to a small system with just one synchronous generator. The


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paper is structured as follow: in Section II a description of the

turbine, induction machine generator and its controllers are

presented; in Section III a four-bus system is used to perform

a small signal stability analysis; and in Section IV conclusions

are presented.

I I . DYNAMIC MODEL

In Figure 1, the scheme of a variable-speed wind turbine

based on doubly-fed induction generator is presented. Typi-

cally, the DC/ACconverter on the grid side is controlled tohave a constant Vdc and a unity power factor. The AC/DCconverter on the rotor side is usually controlled to have (a)

optimal power extraction from the wind and (b) a specified

reactive power at the generator terminal. Note that this con-

verter provides sinusoidal three-phase voltages at the slip rotor

frequency.

A. General considerations

A gearbox required to increase the angular speed from the

wind turbine to the induction-machine shaft is shown in Figure

1. It is characterized by its gearbox ratio, k, which is the ratiobetween the wind turbine speed wt and the machine shaftspeedms. Assuming a machine ofp poles, these speeds andthe electrical rotor speed of the machine, r, are related as

r =p

2ms=

p

2kwt (1)

Another important definition is the tip speed ratio, , whichis the ratio between the speed of a blade tip vtip [

ms

] and thewind speed vwind [

ms

]. IfR is the radius of the turbine, thetip speed ratio is

= vtipvwind

=wtR

vwind=

2k

p

rR

vwind(2)

Additionally, a crowbaran electronic deviceis some-times considered to protect the machine and converters from

grid disturbances. When the protection is activated, the

machine-rotor phases are short-circuited protecting them from

over-voltages and protecting the rotor-side converter from

over-currents. This crowbar protection is activated whenever

the rotor-current limit is surpassed [9]. Note that there are wind

turbines equipped withfault ride-throughcapability which can

withstand a fault during some cycles without using the crowbar

protection [6].

B. Assumptions

Ideal converters are considered and the dc voltage between

converters is assumed to be constant. Therefore, the conversionis lossless and the converters on the rotor- and grid-side are

modelled as a controlled-voltage source and a controlled-

current source, respectively. By power balance, the active-

power absorbed by the grid-side converter is the same than

the power injected by the rotor-side converter. Using this

active-power and the grid voltage, the current on the grid-side

converter is calculated. Other premises can be found in [10].

A scheme of the wind-power generation model is depicted in

Figure 2.

Fig. 1. Wind power generator scheme

Fig. 2. Schematic of the wind turbine generator model

C. Wind turbine model

The wind turbine model basically represents the relation

between the mechanical power extracted from the turbine,

Pwind, and the wind speed as [11]

Pwind=

1

2AwtCp(, )v

3

wind [W] (3)

where is the air density [ kgm3

],Awt is the wind turbine sweptarea [m2], vwind is the wind speed [

ms

] and Cp is the powercoefficient. Cp is dimensionless and depends on both the tipspeed ratio, , and the pitch angle, [degrees]. Using a fixedpitch angle, typical power curves in function of the wind speed

and the electrical rotor speed is depicted in Figure 3.

Considering manufacturer data and optimization techniques,

a power coefficient function has been derived for a variable-

speed wind turbine [12]. Using an intermediate parameter i

Cp(i, ) = 0.22

116

i 0.4 5

e

12.5i (4)

wherei =

1

+ 0.08

0.035

3 + 1

1

(5)

A torque expression is required to model the motion of the

rotatory massturbine, gearbox and machine shaft. Assume

that the power transmission from the wind turbine to the

machine shaft is lossless. Then, the torque in [Nm] at themachine shaft is

TM=Pwindms

= k

2R3Cp(, )v

2wind [Nm] (6)


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Fig. 3. Extracted power from the wind turbine for various wind speeds

Use the per-unit base of the induction-machine to define power

base, Sb, voltage base, Vb, and electrical speed base, b. Thetorque base can be calculated as Tb = Sb

p2b

. More details

about per unit system in electrical machines can be found in

[13][15]. As in the next sections all expressions are in per

unit, the torque equation in per-unit becomes

Tm=TMTb

=1

2

R2bSbr

Cp(, )v3wind [pu] (7)

D. DFIG model

A DFIG is simply a wound rotor induction machine where

the stator- and rotor-circuits are energized. Both stator and

rotor windings participate in the electromechanical energy

conversion. Consequently, aDFIGmodel is basically the same

than the model of an induction machine in which the rotor

voltages are supplied by an electric source, i.e., rotor side

AC/DC converter. This configuration allows the machine tooperate in a wide speed range.

The induction machine model requires the use of reference

frame theory. Basically, a model using a-b-c stator- and rotor-

phases is referred to a particular reference frame with twoorthogonal axis, namely, quadrature axis (qaxis) and directaxis (daxis). A stationary reference frame, a rotor referenceframe or a synchronously rotating reference frame can be

used [15]. The last is adopted in this article. Note that a

generator convention is used, i.e., the stator and rotor currents

are positive when they are leaving and entering the machine,

respectively. Likewise, a frame in which the q axis leadsthe d axis is considered.

1) Fifth order model: The stator and rotor circuits as well

as the equation of motion of the rotatory mass define the

following set of differential equations

1

s

dqsdt =Vqs+ RsIqs ds (8)

1

s

ddsdt

=Vds+ RsIds+ qs (9)

1

s

dqrdt

=Vqr RrIqr (s r)

sdr (10)

1

s

ddrdt

=Vdr RrIdr+(s r)

sqr (11)

2HDs

drdt

=Tm Te (12)

whereV,I,R and correspond to the voltages[pu], currents[pu], resistances[pu]and flux linkages[pu], respectively.Te=drIqr qrIdr is the electrical torque at the machine shaft[pu]. All variables and parameters are in per unit except r,s and HDD stands for DFIG.

The stator and rotor flux equations define the following set

of algebraic equations

qs = XsIqs + XmIqr ds= XsIds+ XmIdr (13)qr = XmIqs+ XrIqr dr= XmIds+ XrIdr (14)

whereXm is the mutual reactance between the stator and therotor, Xs = Xs + Xm is the stator reactance and Xr =Xr+ Xm is the rotor reactance. Xs and Xr are the statorand rotor leakage-reactance, respectively. All reactances are in

per unit.2) Third order model: In dynamics simulations, it has been

observed that stator dynamics are faster than rotor dynamics

[15][17]. In order to visualize it, solve for Idr and Iqr fromequation (14)

Iqr =Xm

XrIqs+

qr

XrIdr=

Xm

XrIds+

dr

Xr(15)

Multiply equations (10)(11) by XmRr

and replace Iqr andIdrto obtain

XrsRr

dXmXr

dr

dt =

XmRr

Vdr XmXr

dr X2mXr

Ids

+(s r)

s

XrRr

XmXr

qr

(16)

XrsRr

dXmXr

qr

dt =

XmRr

Vqr XmXr

qr X2mXr

Iqs

(s r)

s

XrRr

XmXr

dr

(17)

Define

T0 = XrsRr

Xs= Xs X2mXr

(18)

EqD =XmXr

dr E

dD= XmXr

qr (19)

whereT0 is the transient open-circuit time constant,X

s is the

transient reactance,EqD and E

dDare the quadrature and direct

axis rotor-voltages, respectively. For large machinesT0 1s

,

thus, stator dynamics are faster than rotor dynamics [16], [17].

Using a zero-order integral manifold to represent the stator

dynamics, the reduced-order machine model becomes

Vqs = RsIqs X

sIds+ E

qD (20)

Vds= RsIds+ XsIqs+ EdD (21)

T0dEqDdt

=EqD + (Xs X

s)Ids

+ T0

s

XmXr

Vdr (s r)E

dD

(22)

T0dEdDdt

= (EdD (Xs X

s)Iqs)

+ T0

s

XmXr

Vqr+ (s r)E

qD

(23)


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Fig. 4. Equivalent circuit of a DFIG connected to a grid

To obtain the stator algebraic equations (20)(21), use equa-

tions (13), (15) and (19). Rotor flux equations can be stated

in terms of the new variables EqD and E

dD as,

Idr=EqDXm

+XmXr

Ids Iqr = EdDXm

+XmXr

Iqs (24)

The third-order model of theDFIG is completely defined byequations (20)(24) plus the equation of motion (Equation 12).

Notice that multiplying Equation (21) by ej2 and adding

Equation (20), a phasor machine representation to calculate

stator algebraic-variables, given E

qD andE

dD, is obtained.

EqD jE

dD= (Rs+jX

s)(Iqs jIds) + Vqs jVds (25)

3) Machine and grid connection: Consider aDFIGsend-ing active power, through a short-length transmission line, to

an infinite busthe line and the infinite bus may represent

the Thevenins equivalent of a more complex grid. In order to

integrate the generator model in power system modeling, it is

necessary to relate the synchronously rotating reference-frame

variables to a phasor. Applying the proper transformation to

the machine stator voltage, it is obtained that the terminal-

voltage phasor at phase-a is VD = VDejD =Vq jVd [15].Consequently, the machine model to be used in power system

modeling is defined by equation (25) (see Figure 4). The grid-

side converter is represented by a controlled-current source. As

this converter does not absorb reactive power, the controlled-

currentIGC GC stands for grid-side converter is,

IGC=Protor

V

D

=VqrIqr+ VdrIdr

VDejD (26)

4) Decoupled control of active and reactive power:Assume

that thed-axis is oriented along the stator flux axis, i.e., s=ds withqs = 0. Besides, neglectRs and use equations (8)(9) in steady-state to get Vds= qs = 0and Vqs = ds= VD.Consider the flux equation (13) to obtain

Iqs =XmXs

Iqr Ids=XmXs

Idr VDXs

(27)

Then, the complex power leaving the generators stator is

P+jQ = (VdsIds+ VqsIqs) +j(VqsIds VdsIqs) (28)

=

XmXs

VsIqr

+j

VD

XmIdr VDXs

(29)

It turns out that the control of active and reactive power can

be performed independently varyingIqr andIdr, respectively.

Fig. 5. Power reference for the rotor speed controller

Fig. 6. Rotor speed controller

E. Controllers

1) Rotor speed controller: The rotor speed controller is

designed to extract maximum power from the wind turbine.As stated in Section II-C, the power extraction depends on

both the wind speed, which is uncontrollable, and the tip

speed ratio, which is controllable. Actually, depends onr as stated in equation (2). Therefore, controlling r wecan move along the power curve for a given wind speed to

maximize the power. Joining the maximum-power points for

every given wind speed of figure 3, we can obtain a one-

to-one correspondence between the optimal power and the

rotor speed. Additionally, cut-in and maximum speed due

to converter ratings [18] must be taken into account. The

cut-in point corresponds to the minimum speedtypically

0.7 ratedrequired to produce power. If the speed is

below this point, theDFIGis shutdown. The maximum speedpoint corresponds to the maximum power that the DFIGcan produce. If the speed is above the maximumtypically

1.2 ratedthe turbine aerodynamic must be modified toreduce the power extracted from the wind turbine. To this

end, pitch angle control must be performed. It turns out that

the power reference is piece-wise defined as in Figure 5. A

tracking control capable to follow Pref is used. Based onequation (29), a PI controller with an internal Iqr-controlloop is considered [19] (Figure 6).

2) Pitch angle controller:The pitch angle control modifies

the aerodynamic efficiency of the turbine in order to limit

the power production to the rated power. Note that the pitch-

angle time-constant is quite high depending on both the sizeof the turbine blades and economical/practical limitations of

the blade drives [10]. A P I controller with an internal pitch-angle control loop is considered [19] (see figure 7). ref ischosen such that the pitch-angle is around zero when the speed

is located at the optimal tracking curve and it is positive and

less than 90o when the maximum speed is binding.

3) Reactive power controller: According to the Federal

Energy Regulatory Commission, USA, [20] large wind plants

seeking to interconnect to the grid need to maintain a power


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Fig. 7. Pitch angle controller

Fig. 8. Reactive power controller

factor within the range of 0.95 leading to 0.95 lagging,

measured at the high voltage side of the substation trans-

formers. Although, this requirement is mandatory only in

the case that a power factor out of this range jeopardizes the

security and reliability of the system, it makes a precedent

for control capability requirements. In this article, a reactive-

power tracking control is used (see Figure 8). Based onequation (29), a PI controller with an internal Idr-currentcontrol loop is considered [19].

III. TEST SYSTEM SIMULATION

Consider the four-bus system of Figure 9. Assume an expo-

nential model for the load asPL= P0VkpL andQL= Q0V

kqL .

The synchronous generator (SG) is represented by a two-axis

model [14] and the transient reactance at both quadrature

and direct axis are equal, i.e., Xd = X

q. An IEEE Type 1

Exciter and a linear speed governor without droop [21] are

considered. Besides, assume that the wind speed is such that

the rotor speed is above and below of the cut-in and maximum

speed, respectively. Consequently, the rotor speed controller isoperating in the optimal tracking curve and no pitch control is

required. The optimal curve is defined as Pref=C3r [pu].The system is modelled by a set of differential algebraic

equations (DAE) of the Hessenberg index-1 form [22]. Ingeneral, this set ofDAE can be written as dx

dt = f(x,y,),

0 = g(x,y,) where x Rnd1 is the vector of differentialvariables, y Rna1 is the vector of algebraic variables and Rnp1 is the vector of parametersnd, na and np arethe number of differential variables, algebraic variables and

parameters, respectively. gy = g(x,y,)

y is nonsingular along

the solution of the DAE.In order to perform a small signal stability analysis, a linear

approximation of the set ofDAEhave to be obtained. Then,

Fig. 9. Four-bus test system

Fig. 10. Trajectory of dominant eigenvalues withvwind = 10 [m/s] and0.32 P0 1

Fig. 11. Trajectory of dominant eigenvalues withvwind = 12 [m/s] and0.32 P0 1

around an equilibrium point,

x= Ax + By (30)

0 = Cx + Dy (31)

Using Krons reduction, the systems equations can be reduced

tox= (ABD1C)x= Asysx. Eigenvalues ofAsysdetermine the stability of the operating point. It is desired to

study the system stability as the load is increased. Then, the

eigenvalues ofAsys are calculated at every equilibrium point.Note that if all eigenvalues are located on the left half of the

complex plane, i.e., all eigenvalues have a negative real part,

then the equilibrium point is stable. If some eigenvalue has

a positive real part, then the equilibrium point is unstable.

The point at which a complex pair of eigenvalues crosses the

imaginary axis to the right half plane, while the other remain


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on the left half plane, is called a Hopf Bifurcation (HB)point. The system becomes critically stable. The eigenvalues

trajectories are shown in Figures 10 and 11 for a wind speed

of 10 [m/s] and 12 [m/s], respectively. The load is variedfrom P0 = 0.32 [pu] to P0 = 1 [pu].

The incidence of eigenvalues on state variables can be

estimated by using participation factors [14]. For the two

presented cases, the pathway of eigenvalues are notably differ-

ent. This difference may be related to parameter sensibilities

on system stability [23] due to the increase of generated

power of the DFIG. Note that the eigenvalues associatedto r, and Pm do not move when load and wind speedare varied. The eigenvalues that cross the imaginary axis are

associated to states (Eq,E

d,Efd ,VR) and (E

q ,Efd,VR) whenthe wind speed is 10 [m/s] and 12 [m/s], respectively. TheHB points for these two cases happen at P0 = 0.941 [pu]andP0 = 0.971 [pu], respectively. This does not considerablydiffer from the modal behavior of a system with a conventional

synchronous generator [14], [24]. For all wind speeds and

loading simulated, up to the HB point, eigenvalues associated

to theDFIG state-variables are stable. From these results, itseems that the dynamic of the wind-power generator does nothave a serious impact on the system behavior. Probably the

major interaction between the machines is the interchange of

active-power due to wind variations. Thus, as future work, a

reduced-order equivalent model that captures the active-power

dynamic of the wind-power generator is going to be derived.

IV. CONCLUSION

A complete dynamic model appropriate for power system

analysis is presented. This article contains several details such

as modeling assumptions, derivation of a third order model for

theDFIGand its controller descriptions, thus, it can be used as

a tutorial for students and engineers that are new in this area.A four-bus system with one DFIG and one SG is considered

to perform a small signal stability analysis. The results reveal

that the state-variables associated to the unstable modes around

theHB point are those associated to the SG voltage controller.

Based on the eigenvalue pathways, the modal behavior does

not considerably differ from the modal behavior of a system

with only one SG.

APPENDIX

Parameters are in per unit unless otherwise is specified.

Synchronous machine

Xd = 2.2, Xq = 1.76, Xd = 0.2, Xq = 0.2, Td0 = 8 [s],Tq0 = 1 [s], H= 10 [s], KE= 1, TE= 0.7 [s], KF = 0.03,TF = 1 [s], KA = 200, TA = 0.04 [s], TG = 5 [s],Vref= 1.0078

Network

R1 = 0.03 and X1 = 0.10 (Line 1); R2 = 0.10 andX2 = 0.10 (Line 2); pv = 0 and qv = 0 (Load Parameters),XT = 0.07 (Transformer)

Doubly-fed induction generator

Xm = 3.5092, Xs = 3.5547, Xr = 3.5859, s =120 [rad/s], Rs = 0.01015, Rr = 0.0088, H = 2 [s],p = 4, = 1.225 [kg/m3], R = 15 [m], Sb = 1 [MVA],C = 3.2397 109 [s3/rad3], k = 1/45, KP1 = KP2 =KP3 = KP4 = 1, KI1 = KI2 = KI3 = KI4 = 5, Qref= 0

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W. Sauer. Dynamic Modeling of Doubly Fed Induction

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