Doublyfed induction generator wind turbine modelling for detailed electromagnetic system studies

This PSCAD/EMTDC model can be used to evaluate the DFIG WT performances under different operating modes, control schemes and the grid integration capabilities.. 2.1 Aerodynamic modelling

Published in IET Renewable Power Generation

Received on 10th August 2012

Revised on 21st November 2012

Accepted on 17th December 2012

doi: 10.1049/iet-rpg.2012.0222

ISSN 1752-1416 Doubly-fed induction generator wind turbine

modelling for detailed electromagnetic system

studies

Ting Lei, Mike Barnes, Meliksah Ozakturk

Power Conversion Group, University of Manchester, M1 0QD, UK

E-mail: Ting.lei-2@postgrad.manchester.ac.uk

Abstract: Wind turbine (WT) technology is currently driven by offshore development, which requires more reliable, multi-megawatt turbines Models with different levels of detail have been continuously explored but tend to focus either on the electrical system or the mechanical system This study presents a 4.5 MW doubly-fed induction generator (DFIG) WT model with pitch control The model is developed in a simulation package, which has two control levels, the WT control and the DFIG control Both a detailed and a simplified converter model are presented Mathematical system block diagrams of the closed-loop control systems are derived and verified against the simulation model This includes a detailed model of the DC-link voltage control– a component which is usually only presented in abstract form Simulation results show that the output responses from the two models have good agreement The grid-side converter control with several disturbance inputs has been evaluated for three cases and its dynamic stiffness affected by operating points are presented In addition, the relation of pitch controller bandwidth and torsional oscillation mode has been investigated using a two-mass shaft model This model can be employed to evaluate the control scheme, mechanical and electrical dynamics and the fault ride-through capability for the turbine

1 Introduction

The trend of future wind turbine (WT) installations moving

offshore is stimulating the need for high reliability and ever

larger WTs in order to minimise cost Two WT concepts

currently used are considered to be suitable for the

multi-megawatt offshore installations – the doubly-fed

induction generator (DFIG) WT and the permanent magnet

synchronous generator WT [1] Currently the former with a

capacity up to 5 MW has the largest market share

Increasingly comprehensive studies need to be carried out

to evaluate the control strategies and system dynamic

behaviour These studies require accurate models

DFIG WTs have been investigated for many years [2–10]

In most of these, the converters are simplified as controllable

voltage or current sources with only fundamental frequency

components, which makes it impossible to implement a

detailed study of power converter dynamics The drive-train

is often treated as a lumped-mass system, which means the

induced torsional oscillations are neglected

In this paper, a PSCAD/EMTDC-based DFIG WT model

with two control levels is provided The DFIG control level

involves the rotor-side converter (RSC) control and the

grid-side converter (GSC) control The WT control level

involves the pitch control and the optimum torque tracking

[4] Mathematical analysis of each individual control system

is carried out and the mathematical blocks (MB) have been

verified against PSCAD/EMTDC simulation results Two

converter representations, with and without IGBT switches are used here, which are referred to as the full switched model (FSM) and the switch-averaged model (SAM) They both have been implemented for observing GSC control performances where the DC-link dynamic is involved The latter is used to demonstrate the WT mechanical responses such as rotor speed and pitch angle to wind speed changes Finally, a multi-mass shaft model is added to investigate the correlation of the pitch mechanism and shaft torsional modes This PSCAD/EMTDC model can be used to evaluate the DFIG WT performances under different operating modes, control schemes and the grid integration capabilities

The paper is structured as follows Section 2 presents the component modelling and equations Sections 3 and 4 then elaborate the DFIG-level and WT-level control systems, respectively, as well as the mathematical analysis and simulation results Section 5 concludes the main results

2 System modelling

A schematic diagram of the DFIG WT and its overall control systems are illustrated in Fig 1 The turbine rotor is connected to the DFIG through a shaft system The generator rotor is fed from the grid through a back-to-back converter which handles only the slip power (up to 30% of the rated power)

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2.1 Aerodynamic modelling

The aerodynamic model can be described by the equation of

aerodynamic power or torque generated as the wind passes the

turbine blades [11]

Pa=r

2pR2

aCp(l, b)v3w (1)

Ta=r

2pR3 a

Cp(l, b)

withρ is the air density (kg/m3

), Rais the radius of the rotor (m), vwis the wind speed upstream the rotor (m/s) andωris

the rotor speed (rad/s) The power coefficient Cp is a

function of the tip speed ratio λ and the pitch angle β

(deg.), for which the numerical approximation in [9] is used

l =vrRa

1

li=l + 0.08b1 −b03.035

Cp(l, b)= 0.22 116l

i

− 0.4b − 5

e− 12.5/ l( ( )i ) (5)

2.2 Induction generator modelling

The DFIG machine equations have been described in [5,9]

Note that in Fig 1, the currents are set as outputs from the

stator and rotor If generation convention is considered, the

set of machine equations can be derived as

vs= −Rsis+dcs

dt + jvscs (6)

vr = −Rrir+dcr

dt + j vs− vr

cs= −Lsis− Lmir (8)

cr = −Lrir− Lmis (9)

with v is the voltage (kV), R is the resistance (Ω), i is the current (kA), ωsis the synchronous electric speed (rad/s), ψ

is the flux linkage (Wb), Lm is the mutual inductance between stator and rotor windings (H ) The subscripts s and

r denote the stator and rotor quantities

2.3 Back-to-back converter modelling Two voltage source inverters (VSI) are connected back-to-back via a DC-link to comprise the converter This enables bidirectional power flow In Fig 2, the FSM is presented with all the IGBT switches and a PWM frequency of 4.5 kHz This model provides a deeper insight

of the converter dynamics over a short time scale

In the SAM, the converter is presented as two current-controlled voltage sources coupled through a DC-link The DC-dynamics are based on the power balance between the RSC and GSC, which generates two current disturbances from the VSIs feeding into the DC-link The SAM is suitable for inspecting the mechanical dynamics over a longer time scale without the disturbance from the switching noise

2.4 Shaft system modelling The shaft system has been presented as six, three, two and lumped-mass models in other research [12], among which the lumped and two-mass shaft models are often used to study the electric behaviours of the DFIG It is suggested in [11] that for a generator with shaft stiffness lower than 3 pu/el rad, a two- mass shaft model should be considered PSCAD/EMTDC provides the standard models of the wound rotor induction machine and the multi-mass shaft They can be interfaced with WT aerodynamic model and pitch controller as shown in Fig.3 The performances of the two-mass shaft model are investigated in Section 4 If not specified, the analysis refers to the lumped shaft model with

a SAM converter in the paper

The model performances during single or three-phase fault conditions are presented in Fig.4, where the stator and rotor voltages subject to significant changes when the grid voltages drop to 0 for 100 ms

Fig 1 DFIG WT model and its overall control systems

3 DFIG control

The control system has been shown in Fig.1, in which two

control levels are identified based on different bandwidths

The DFIG control consists of the RSC control and the GSC

control The former is used to provide decoupled control of

the active and reactive power while the latter is mainly used

to ensure a constant voltage on the DC-link [5,7]

3.1 RSC control

Stator-flux orientation is used for the RSC control in which

the stator flux is collinear with the d-axis and the other

rotor quantities are converted to this frame The equations

of the electric torque and the stator reactive power can be

found in [5], which are modified here using generator

convention,

Te=3 2

Lm

Lsscsir q (10)

Qs= −3

2



2

√

Vs

Lss cs−3

2



2

√

VsLm

Lss ir d (11) where Vsis the stator phase voltage in rms

Resolving (6)–(9), splitting the rotor voltage in d–

q-components and neglecting the statorflux transients (dψs/

dt = 0) gives

vr d∗ = −Rrir d− Lc

dir d

dt + vslipLcir q (12)

v∗r q= −Rrir q− Lc

dir q

dt − vslipLcir d+ vslip

Lm

Lsscs (13) where

vslip= vs− vr, Lc= Lrr−L2m

Lss

The MB diagram of RSC control is depicted in Fig 5, where the decoupling of d–q control-loops is achieved by adding the feed-forward compensation after the inner-loop

PI controller Reactive power control is cascaded with d-current control loop and electric torque control is cascaded with q-current control loop The estimated quantities are marked by ‘^’ to distinguish them from the real quantities

In PSCAD/EMTDC, the DFIG WT is connected to the grid through a step-up transformer Parameters used in the model are tabulated in appendix The DFIG is set to the speed control mode with a nominal rotor speed

Responses from the MB are compared with PSCAD/ EMTDC simulations (‘PCD’), applying the same reference signals (Ref) In Fig 6, the plots are resulted from a step input of Q∗s, Te∗, i∗r d, i∗r q at the four control loops separately Note that in the lower plots, the time scale is smaller since the bandwidth of the current-loop is much higher than the power loops It can be observed that the curves ‘MB’ and ‘PCD’ have a good agreement, indicating that the mathematical model can describe the software model accurately for this case

3.2 GSC control of the DC-link

In GSC control, the q-component controls the DC-link voltage and the d-component controls the reactive power Positive current is considered from the grid to the converter

Fig 3 Turbine rotor, two-mass shaft and DFIG model

arrangements in PSCAD/EMTDC

Fig 2 Two converter models

Upper: FSM, lower: SAM

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Thus the voltage equations in the d–q frame are expressed as

vg d= Rgscig d+Lgscdig d

dt − vsLgscig q+ eg d (14)

vg q= Rgscig q+ Lgsc

dig q

dt + vsLgscig d+ eg q (15)

As shown in Fig.7, the inner-loop control of the GSC consists

of d–q loops with similar structure The DC control loop

Fig 4 Model performances under fault conditions

Left: single-phase fault, right: three-phase fault

Fig 5 Decoupled control block diagrams of the RSC

a Inner-loop control system

b Outer-loop control system

should be cascaded with q-current control loop, the small

signal model of which will be illustrated in the following

paragraphs

Aligning the grid voltage to the q-axis (vg_d= 0) and

applying power invariant principle to the DC and ac points

Pc=3

2vg qig q= Vdcidc g= Vdcidc r (16)

The dynamics equation of capacitor (Fig 2) can be

described as

CdVdc

dt = idc g+ idc r (17)

Combining (16) into (17) and eliminating term idc_g

dVdc

dt =idc r

C +3vg qig q

Taking partial derivatives of all variables

D ˙Vdc= 1

CDidc r+ 3

2CKVDig q+ 3

2CKGDvg q

− 3

KV=Vg q0

Vdc0 and KG=ig q0

Vdc0

For steady state

Vdc0idc r0= −1.5vg q0ig q0 (20)

The power change of the converter is then

DPc= Vdc0Didc r+ DVdcidc r0

= Vdc0Didc r− 1.5KVKGDVdc

(21)

Fig 6 Mathematical veri fication of the RSC control

Upper: outer-loop step response, lower: inner-loop step response

Fig 7 GSC control block diagrams

Upper: current control loops, lower: small signal model of DC-link control loop

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Equation (19) is the small signal model of DC-link The

subscript‘0’ denotes a particular operating point From the

equation, DC-voltage dynamics are affected by several

components: the first two terms are the injected currents at

the DC-link, which will charge or discharge the capacitor;

the last two terms reflect the impact of grid voltage and

DC-voltage changes In the closed-loop control system,

[Fig 7and (19)], the first and third terms can be treated as

external disturbances that cause the change of DC-voltage,

while the second term is the controlled current obtained

from the negative feed-back loop, trying to return

DC-voltage to its operating point On the other hand, the

first and last terms constitute the converter power change,

as shown in (21) With KG being negative, the last term is

actually a positive feedback term that counteracts with the

controlling term, introducing an energy buffer effect

respectively

The plots of SAM and FSM show that the mean value of the FSM matches with the SAM PWM switching is presented in the current waveform but not apparent in the DC-voltage waveform because of the smoothing effect All the plots from MBs of DC-voltage control can match with the software simulations except the MB_full Since there is

no change in the converter (ΔPc= 0), the terms ΔVdc and

Δidc_r will counteract each other and must be considered together in this case, as shown in (21) The effect of

Δvg_qcan be ignored if the grid is assumed to be stiff Case B and C: The DC-voltage responses to a Te-step and grid voltage sag are shown in the lower two plots of Fig.9 In both cases, the DC reference value is set to be constant The disturbance signal idc_ris the main factor that causes the output DC-voltage change, which is illustrated in the upper plots It can be observed that the mean value of FSM is lower than the SAM because of the switching losses in the IGBTs The contributions of two other disturbance inputs are evaluated

as shown in the lower plots Three curves from MBs are shown, which are the MB with all the three disturbance terms, the one neglecting the term ΔVdc and the one neglecting the term Δvg_q In case B, the MB response will slightly deviate from the software simulation if ΔVdcis not considered, which plays a more important role thanΔvg_qin the DC-dynamics However, the term Δvg_q shows more

Fig 8 Mathematical veri fication of the GSC control

Upper: case A – DC-loop step response, lower: current-loop step response

Fig 9 DC-control disturbance factor analyses for different cases

Left: case B – Te steps from 1.2 to 1.3 pu, right: case C – Grid voltage drops to 80% of nominal value

impact in case C, because of significant grid voltage

oscillations

In normal WT operating condition, case B is the most

common situation, where the reference electric torque or

power changes because of a wind speed change The grid

disturbance-term Δvg_q can be ignored Now the dynamic

stiffness (DS) [13] of DC-link system can be evaluated for

varying operating points This describes the resistance of

the system to a certain disturbance Considering only the

key disturbance input Δidc_r, resulting from the change of

input powerΔP

DS=DVDP

dc

=Vdc0Didc r

DVdc

= Vdc0Cs+ 1.5vg q0 KG+ KP

+1.5vg q0Ki s (22)

Fig 10 illustrates the DS when varying two parameters’

operating points, Vdc0and vg_q0, respectively, both of which

are changed from 0.7 to 1.3 times of their nominal values

The DS is only sensitive to Vdc0 for higher frequencies

because of the first term in (22) and it is sensitive to vg_q0

for middle and low frequencies because of the other two

terms As is shown in thefigure, the higher the DC-voltage

or the grid voltage is, the higher the system DS will be

Software simulations have been presented in Fig.11, where

the DS of the system has been evaluated for three different

Vdc0 and the electric torque is oscillating at 150 Hz with

0.2 pu amplitude

4 WT control

4.1 Controller development

For the WT control, the system measures the rotor speed and

uses it to generate references both to the pitch system of the

WT and to the DFIG control level (Fig 1) There are two different control algorithms for this control level At lower wind speeds, the pitch angle remains at the optimum value (0 degrees) and the optimum torque will be tracked according to a pre-defined curve [11]

Tgopt= Koptv2

At higher wind speeds, the pitch control is activated to remove excessive power extracted from wind The two control modes sometimes work together to regulate the WT

in the high wind region [4, 12, 14] The model presented here considers the case where the two controllers operate independently for their respective regions A non-linear transfer function is employed to generate the torque-speed look-up table

The pitch controller with full-non-linear plant model is illustrated as in Fig 12a The actuator introduces a lag between the actual pitch angle and the commanded pitch from the PI controller This controller is designed with a bandwidth of one-magnitude-order smaller than the electric torque control loop The actuator time constant is τ = 0.2 s

Jt is the total rotational inertia The wind speed vw acts as the external disturbance experienced by turbine rotor and the generator torque Tg is treated as an internal disturbance signal on the shaft system and it is fixed at the rated value for the higher wind speed region

The system model should be linearised in order to evaluate its control performance The linearisation process of different mass systems has been performed in [15] and this methodology is applied in [16] to design the PI controller Here the method is extended to lower wind speed region and will be justified by software simulations

At a particular operating pointωr0,β0, vw0, the aerodynamic torque can be expressed as

Ta= Ta vr0, u0, vw0

+ gDvr+ zDb + hDvw (24)

Fig 10 DS at different operating points

Fig 11 Software simulation of DC-link DS in response to Te oscillations

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g =∂Ta

∂vr , z =∂Ta

∂b, h =∂v∂Twa The rotor dynamics can be described as

Jtdvr

For the higher wind speed region Tg= Ta0, therefore

D ˙vr = ADvr+ BDb + BdDvw (26)

where A =γ/Jt, B =ζ/Jt, C =η/Jt,

For the lower wind speed region

Tg= Tg0+ 2Koptvr0△vr (27) where Tg0= Ta0, eliminating the pitch controller

D ˙vr= (A − C)Dvr+ BdDvw (28) with C = 2Koptωr_0/J

These two linear models are depicted in Figs.12b and c For the WT-level control simulation, the induction machine should be held in torque control mode after the initial transient to interface with the aerodynamic model The system responses at two operating points (op) are shown in Fig.13, where

Op1: vw0= 15 m/s,ωr0= 1.2 pu,β = 10°

Op2: vw0= 9 m/s,ωr0= 0.78 pu,β = 0°

On the left plots, as wind rises, the pitch angle increases in order to maintain the rotor speed within the threshold (denoted

as‘Thr’) On the right plots, the pitch angle is maintained at 0 degrees despite of the wind speed change The rotor speed increases proportionally with the wind speed, to maintain the optimum tip-speed ratio The very low response is because

of the large system inertia In both cases, the results from

‘PCD’ and ‘MB’ match very well With the lumped-mass shaft model, no torsional oscillations are presented even though the pitch controller is relatively fast

a Pitch control with full-non-linear plant model

b Linearised control model for higher wind speed region

c Linearised control model for lower wind speed region

Fig 13 Mathematical veri fication of the WT-level control

left: 0.5 m/s v w step at op 1, right: 0.5 m/s v w step at op 2

4.2 Controller coordination

To illustrate the relation of pitch controller bandwidth region and the turbine torsional mode, the responses of a two-mass

WT model to a wind step is simulated for different pitch controller bandwidths Results are obtained by stepping the wind speed from 13 to 14 m/s, as shown in Fig.14

Significant mechanical oscillations are presented in the rotor speed and pitch angle, which are known as torsional oscillations The torsional frequency of the WT can be calculated from [11]

fT = 1 2p



Ksv0(Hturbine+ Hgenerator) 2HturbineHgenerator



= 2.55 Hz (29)

where Ks is the shaft stiffness (pu/el rad), ω0 is the synchronous electric speed (rad/s), H is the moment of inertia (s)

In Fig.14, lower pitch bandwidth can produce smoother but slower response and can result in longer over speed Torsional oscillations occur as the pitch bandwidth approaches 0.2 Hz, where the controller bandwidth begins to interfer the shaft natural frequency The responses will become unstable if a

Fig 14 Torsional oscillations observed for a two-mass drive-train

system at different controller bandwidths

Fig 15 Electrical oscillations induced by torsional modes – a comparison of both shaft models

Fig 16 Controller adjustment and coordination

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be four to ten times faster than the outer-loop so that they

can be treated separately The speed of GSC DC-link control

fGSC_DCneeds to be no smaller than the RSC current control

loop fRSC_i in order achieve instant power transfer and

minimise the voltage transients on the capacitor Owing to

very large inertia of the WT rotor, the electrical controller

system will not excite the shaft torsional mode even when

their bandwidths are very close The controllers are adjusted

as in Fig 16, from which their constants can therefore be

obtained based on the MB diagrams

5 Conclusions

With future WTs moving towards offshore, very large turbines

will be installed making the reliability more critical

Comprehensive studies on the WT behaviours and control

systems are needed in order to improve their design and

operation This paper presents a complete DFIG WT model

and its overall control systems The interaction of the WT

control level with the DFIG control level has been presented

in the paper A FSM with IGBTs and a SAM are

implemented The former is suitable for detailed studies of

short-time transient behaviours while the latter is more

sensible for investigating mechanical responses over a longer

time scale Mathematical models of individual control-loops

are developed and tested The results are consistent with the

PSCAD/EMTDC simulations These are used to adjust the

controller bandwidth and damping in the software model DS

of DC-link to power changes is analysed for different

operating points Analysis shows that higher stiffness can be

achieved by increasing the grid voltage or the DC-link

voltage It can be observed that torsional oscillations may be

excited by a fast pitch controller This indicates the two-mass

shaft model is necessary for studying the controller

coordination especially when pitch control is involved What

is more, it can induce electrical oscillations and may further

impose stresses on power electronics or grid stability

6 Acknowledgment

The authors thank the Engineering and Physical Sciences

Research Council for supporting this work through grant

no EP/H018662/1– Supergen ‘Wind Energy Technologies’

7 Peña, R., Clare, J.C., Asher, G.M.: ‘Doubly fed induction generator using back-to-back PWM converters and its application to variable speed wind-energy generation ’, IEEE Proc Elecr Power Appl., 1996,

143, (3), pp 231–241

8 Slootweg, J.G., de Haan, S.W.H., Polinder, H., Kling, W.L.: ‘General model for representing variable speed wind turbines in power system dynamics simulation ’, IEEE Trans Power Syst., 2003, 18, (1),

pp 144 –151

9 Slootweg, J.G., Polinder, H., Kling, W.L.: ‘Dynamic modelling of a wind turbine with doubly fed induction generator ’ IEEE Proc on Power Engineering Society, Vancouver, BC, Canada, 2001

10 Todd, R.: ‘High power wind energy conversion systems’ EngD thesis, University of Manchester, 2006

11 Akhmatov, V.: ‘Analysis of dynamic behaviour of electrical power systems with large amount of wind power ’ PhD thesis, Ørsted DTU, Technical University of Denmark, 2003

12 Muyeen, S.M., Ali, M.H., Takahashi, R., et al.: ‘Comparative study on transient stability analysis of wind turbine generator system using different drive train models ’, IET Renew Power Gener., 2007, 1, (2),

pp 131 –141

13 Lorenz, R.D., Schmidt, P.B.: ‘Synchoronized motion control for process automation ’ Industry Applications Society Annual Meeting, October

1989, pp 1693 –1698

14 Bossany, E.A.: ‘The design of closed loop controllers for wind turbines’, Wind Energy, 2000, 3, (3), pp 149–163

15 Wright, A.D.: ‘Modern control design for flexible wind wurbines’ Report no TP-500-35816, NREL, 2004

16 Wright, A.D., Fingersh, L.J.: ‘Advanced control des1ign for wind turbines-part 1: Control design, implementation, and initial tests ’ Report no TP-500-42437, NREL, 2008

8 Appendix

Parameters that are used in the DFIG WT model

spring constant (K s ) 0.7 pu/el.rad generator self-damping 0.032 pu

stator voltage (L –L, RMS) 1 kV stator/rator turns ratio 1