- Email: [email protected]
Comparison of 5th order and 3rd order machine models for doubly fed induction generator (DFIG) wind turbines
Electric Power Systems Research 67 (2003) 207 /215 www.elsevier.com/locate/epsr
Comparison of 5th order and 3rd order machine models for doubly fed induction generator (DFIG) wind turbines J.B. Ekanayake a,c,*, L. Holdsworth b, N. Jenkins b,c a
Department of Electrical and Electronics Engineering, University of Peradeniya, Peradeniya, Sri Lanka b The Manchester Center for Electrical Energy (MCEE), UMIST, Manchester, UK c Tyndall Center for Climate Change Research, Norwich, UK Received 24 September 2002; received in revised form 28 March 2003; accepted 1 April 2003
Abstract With increasing concern over climate change, a number of countries have implemented new renewable energy targets, which require significant amounts of wind generation. It is now recognized that much of this new wind generation plant will be variable speed type using doubly fed induction generators (DFIG). In order to investigate the impacts of these DFIG installations on the operation and control of the power system, accurate models are required. A fifth order and reduced order (3rd) machine models are described and the control of the wind turbine discussed. The capability of the DFIG for voltage control (VC) and its performance during a network fault is also addressed. # 2003 Elsevier B.V. All rights reserved. Keywords: Machine models; DFIG; Wind turbines
1. Introduction The exploitation of the wind as a source of renewable energy continues to increase with some 25 GW of wind turbine capacity installed worldwide. In some countries the penetration of wind energy is such that already it is a significant fraction of generation capacity and projections of future installations indicate that the number of wind turbines will increase rapidly over the next 10 years. Hence, wind farms must be included in computer simulations to study both the development and operation of the power system. At present variable speed operation of wind turbines, using doubly fed induction generators (DFIG), is emerging as the preferred technology. This is due mainly to the reduced mechanical loads on the wind turbines that arise from variable speed operation [1]. However, a secondary advantage is the increased possibilities of control of both real and reactive power to allow easier integration of wind turbines into the power system.
* Corresponding author. 0378-7796/03/$ - see front matter # 2003 Elsevier B.V. All rights reserved. doi:10.1016/S0378-7796(03)00109-3
In comparison to conventional synchronous generation, wind power is developed in relatively small units. Typical wind turbine ratings vary between 800 kW and 3 MW with wind farms ranging from 1 to 200 MW. Thus it is important to determine the simplest possible models of wind turbines that give an accurate representation in the various studies that are undertaken on the power system. In this paper a comparison is made between a 5th order model of the DFIG wind turbine and a 3rd order representation where the stator transients are neglected. The development of the models is described and it is shown that for dynamic modeling the 3rd order model is adequate. For detailed representation of fault current contribution, the 5th order model provides better resolution although it must be recognized that the behavior of the converter control systems is likely to have a dominant effect on fault currents.
2. Doubly fed induction generator (DFIG) DFIG wind turbines utilize a wound rotor induction generator, where the rotor winding is fed through backto-back variable frequency, voltage source converters. A
208
J.B. Ekanayake et al. / Electric Power Systems Research 67 (2003) 207 /215
a) The stator current was assumed positive when flowing towards the machine. b) The equations were derived on the synchronous reference. c) The q-axis was assumed to be 908 ahead of the daxis with respect to the direction of rotation. d) The q component of the stator voltage was selected as the real part of the busbar voltage and d component was selected as the imaginary part.
Fig. 1. Basic configuration of a DFIG wind turbine.
Fig. 2. Direct (d) and quadrature (q) representation of induction machine.
typical configuration of a DFIG based wind turbine is shown schematically in Fig. 1. The converter system enables variable speed operation of the wind turbine by decoupling the power system electrical frequency and the rotor mechanical frequency. A more detailed description of the DFIG system together with its control and protection circuits can be found in [2 /4].
3. Fifth order model of the DFIG Three-phase stator and rotor windings of an induction machine can be represented by two sets of orthogonal fictitious coils. Appendix B shows the transformation from three-phase to two-phase fictitious coils which are placed in direct (d) and quadrature (q) axes [5,6]. A generalized fifth order machine model was then developed by considering the following conditions and assumptions:
Machine equations can be represented in terms of the machine variables or in terms of arbitrary reference frame variables. However, for power system studies it is desirable to use a per unit (pu) representation. This enables the conversion of the entire system to pu quantities on a single power base. The voltage equations for the fictitious coils were normalized as shown in Appendix C. Fig. 2 shows the dq representation of the machine. The voltage equations for the induction generator are given below, where all quantities except the synchronous speed are in pu: 8 1 d > > ¯ ¯ ¯ ¯ > 1 d > ¯ ¯ ¯ ¯ > v ¯ R i l l : qs s qs ds qs vs dt 8 1 d > > l¯dr >v¯dr R¯r i¯dr s l¯qr < vs dt (2) > 1 d > ¯ ¯ ¯ ¯ > lqr :v¯qr Rr iqr s ldr vs dt where, l¯ds L¯ss i¯ds L¯m i¯dr l¯qs L¯ss i¯qs L¯m i¯qr l¯dr L¯rr i¯dr L¯m i¯ds l¯qr L¯rr i¯qr L¯m i¯qs From equation (4): ¯ l L¯m i¯ds and i¯dr dr L¯rr ¯ l L¯m i¯qs ¯iqr qr L¯rr
(3) (4)
(5)
Substituting from Eq. (5) into Eq. (3) and with X¯ 1 L¯2 ½L¯ss m : L¯rr ¯ L ¯ ¯ ¯ lds X 1 ids m l¯dr and L¯rr (6) ¯ Lm ¯ ¯ ¯ ¯ lqs X 1 iqs lqr L¯rr In order to obtain a voltage behind a transient model
J.B. Ekanayake et al. / Electric Power Systems Research 67 (2003) 207 /215
for the DFIG, the following two voltage components were defined. e¯d
L¯m l¯qr L¯rr
and
e¯q
L¯m l¯dr L¯rr
dvr
1 (Tm Te ) J dt
209
(16)
(7)
Substituting from equation (7) into equation (6): l¯ds X¯ 1 i¯ds e¯q
and
l¯qs X¯ 1 i¯qs e¯d
(8)
By substituting from equation (8) into equation (1): 8 1 d X¯ 1 d > > ¯ ¯ ¯ ¯ ¯ > d 1 X >v¯ R¯ i¯ X¯ i¯ e¯ 1 i¯ d e¯ > : qs s qs 1 ds q qs d vs dt vs dt Substituting from equation (5) into equation (2) and then substituting for l¯dr and l¯qr in terms of e¯d and e¯q from equation (7): 8 d e¯d 1 L¯2m L¯m > > ¯ i v¯qr e ¯ > d qs svs e¯q vs < @ t¯ ¯ ¯ To Lrr L¯rr > d e¯q 1 L¯2m L¯m > > : ¯ ¯ e¯q ¯ i¯ds svs e¯d vs ¯ v¯dr @t To Lrr Lrr (10) where T¯ o
L¯rr :/ vs R¯r
Substituting from equation (7) into equation (5): 8 ¯ 1 l L¯m i¯ds L¯ > > ¯dr dr e¯q m i¯ds i > < L¯m L¯rr L¯rr (11) ¯ ¯ ¯ > L i 1 l L¯m qr m qs > ¯ ¯ > e ¯ i i : qr d qs L¯m L¯rr L¯rr The electromagnetic torque is calculated using: Te l¯ds i¯qs l¯qs i¯ds
(12)
Substituting from equation (8) to equation (12) the following equation can be obtained: Te (X¯ 1 i¯ds e¯q ) i¯qs (X¯ 1 i¯qs e¯d ) i¯ds e¯q i¯qs e¯d i¯ds
(13)
Substituting for e¯d and e¯q/in terms of l¯dr and l¯qr in equation (13): Te
L¯m (l¯dr i¯qs l¯qr i¯ds ) L¯rr
4. Third order model of the DFIG For power system transient studies inclusion of the network transients and generator stator transients increases the order of the overall system model, thus limiting the size of the system that can be simulated. Further, a small time step is required for numerical integration resulting in an increased computational time. For these reasons, it has become conventional to reduce the order of the generator and neglect the network transients for stability analysis of large power systems. Different methods of reducing generator equations are discussed in [7]. In this investigation, a standard method of reducing the order of the induction generator model was considered where the rate of change of stator flux linkage is neglected. The reduced order model was derived by ignoring the differential term in equation (1), which is equivalent to neglecting the stator electric transients [5]. The reduced order voltage equations in pu: v¯ds R¯ s i¯ds l¯qs (17) v¯qs R¯ s i¯qs l¯ds 8 1 d ¯ > > ¯ ¯ ¯ > 1 d > ¯ ¯ ¯ ¯ > v ¯ R i s l l : qr r qr dr qr vs dt When the stator transient is neglected, the machine equations given by Eqs. (9) and (10) can be simplified to: v¯ds R¯ s i¯ds X¯ 1 i¯qs e¯d (19) v¯qs R¯ s i¯qs X¯ 1 i¯ds e¯q 8 d e¯d 1 L¯2m ¯ L¯m > > i v¯qr e ¯ > d qs svs e¯q vs < @ t¯ T¯ o L¯rr L¯rr > d e¯q 1 L¯2m L¯m > > : ¯ ¯ e¯q ¯ i¯ds svs e¯d vs ¯ v¯dr To @t Lrr Lrr (20)
(14) where T¯ o
Substituting from equation (4) into equation (14):
L¯rr :/ vs R¯r
(15)
The electromagnetic torque given in equation (15) can be simplified using the following steps:
Finally, if Tm is the mechanical torque, dependent upon the local wind speed:
. Using equations (17) and (3) and neglecting the resistance term
Te L¯m (i¯dr i¯qs i¯qr i¯ds )
J.B. Ekanayake et al. / Electric Power Systems Research 67 (2003) 207 /215
210
Fig. 3. Torque/speed control scheme of the DFIG.
8 1 L¯m > > ¯ ¯ > < ids ¯ v¯qs ¯ idr Lss Lss > 1 L¯m > > :i¯qs ¯ v¯ds ¯ i¯qr Lss Lss
(21)
. Substituting in equation (15): Te
L¯m (i¯dr v¯ds i¯qr v¯qs ) L¯ss
(22)
The d-component of the converter C1 voltage, vdr , was used for compensation for the generator magnetizing reactive power (No load PFC) as shown in Fig. 4. An outer loop (shown in dotted lines) was introduced for voltage control (VC) through converter C1.
5. DFIG control For the model it was assumed that the converters are ideal and the dc link voltage between the converters is constant. The q and d components of the injected voltage through converter C1, respectively, were used for speed and power factor control of the machine (see Fig. 1). Speed control was achieved by driving the machine along an optimum torque /speed curve, which corresponds to the maximum power extraction from the wind, as shown in Fig. 3. Between the cut-in and speed limit, the optimum torque /speed curve was characterized by an optimal characteristic curve (a) given by Topt /Kopt v2r (where vr is the measured rotor speed) and a constant speed characteristic (b) up to the rated torque. Beyond the speed limit, a small margin was introduced between the pitch control and electronic control in order to prevent the two controllers fighting each other. The set point torque corresponding to the speed of the machine was then translated into the reference of iqr using equation (22). In order to linearize the controller, vds was neglected with respect to vqs . Then from equation (22): i¯qrref
Tsp v¯qs
L¯ss L¯m
Fig. 4. No load PFC and VC through C1 for the DFIG.
(23)
The current error (the difference between the desired and achieved iqr) together with a PI controller was used to obtain vqr .
6. Modeling of the DFIG protection The controller model of the DFIG system included rotor voltage and current limits. The limits were selected depending on the MW capacity of the generator and the rating of the converters. Converter C1 was protected against over-current on the rotor circuits by a ‘singleshot crowbar’, as shown in Fig. 1. The operation of the crowbar was modeled by deactivating the converters upon the detection of rotor current magnitude above the current protection limit and short-circuiting the generator rotor.
7. Simulation results Using the 5th order and 3rd order machine models the start-up of the induction generator was simulated. The rotor speed, electromagnetic torque and stator current components are shown in Fig. 5. The machine used for simulation was a 2 MW, 690 V wind turbine and machine parameters are given in Appendix D. In order to obtain the complete start-up process, both the speed and power factor/voltage controllers were frozen at zero rotor injection voltage. The standard start up of an induction motor is shown to illustrate the effect of the
J.B. Ekanayake et al. / Electric Power Systems Research 67 (2003) 207 /215
211
Fig. 5. Start-up of the DFIG with zero rotor injection.
Fig. 6. Response to a torque step (all are in pu).
stator transients upon the characteristics of the 5th order model in comparison to the 3rd order machine model. However, this would not be a typical initialization procedure for the DFIG based wind turbine. For commercial applications the generator may be magnetized from the rotor windings to speed the generator up
to an initialization speed (around synchronous speed). The generator would then be connected to the grid. Using the 5th order and 3rd order models, the response of the DFIG for a step change in mechanical torque from 0.3 to 1.0 pu at t/40 s was studied. The simulation results showing the rotor speed and torque
212
J.B. Ekanayake et al. / Electric Power Systems Research 67 (2003) 207 /215
Fig. 7. Connection of DFIG turbine to double circuit power system model.
Fig. 8. VC capability of the DFIG.
changes are shown in Fig. 6. The results illustrate the operation of the torque/speed controller within specified rotor speed limits as shown in Fig. 3. With an applied mechanical torque of 1.0 pu the controller provides operation on the optimal characteristic until the rotor speed limit is reached. The applied control scheme then enables the electromagnetic torque to approach 1 pu, which increases generated output power through the stator, whilst limiting the generator rotor speed to 1.2 pu. This is illustrated at approximately t/46 s in Fig. 6.
8. Application study In order to investigate the VC capability of the DFIG, a double circuit, double busbar power system was considered, as shown in Fig. 7. The system short circuit level was assumed to be 16 MVA (SCR /8) and X/R ratio was assumed to be 5. The VC capability of the DFIG is demonstrated in Fig. 8. The no-load power factor correction loop in Fig. 4 was activated throughout the simulation. The VC loop was introduced at t/ 40 s with the set point voltage of 1.02 pu. Simulations also carried out to investigate the performance of the DFIG during a system fault. A fault was applied at the mid point of one of the double circuit lines
at t/49.85 s and was cleared after 150 ms. Fig. 9a shows the machine terminal voltage, stator current, rotor speed and electromagnetic torque obtained from the 5th order model. A similar set of results was obtained using the 3rd order model and shown in Fig. 9b. From the simulations, it was found that the rotor current in pu during the transient is very similar to the stator current. In these cases the rotor crowbar limit was increased to maintain the machine in operation during the fault. In practice this means, increasing the shortterm current rating of components in the voltage source converter C1. Although not directly effected by the fault the rotor voltage applied to the DFIG throughout the duration of the network fault is entirely dependent upon the control strategy applied and the ratings of the converters. Therefore, it should be observed that in a practical DFIG system the converter voltage and current ratings together with the size of the dc link capacitor are all critical to ensure good performance during network disturbances.
9. Conclusions 5th and 3rd order models of a DFIG were developed and coupled to a control system of a variable speed wind
J.B. Ekanayake et al. / Electric Power Systems Research 67 (2003) 207 /215
213
Fig. 9. DFIG response to a system fault (all are in pu).
turbine. Example simulations demonstrated that both models give very similar results for machine runup and in response to changes in applied torque. The use of the generator side, voltage /source, converter for network VC was demonstrated and again, very similar results were obtained from the two machine representations. A three-phase short circuit on the network close to the generator was investigated. As, expected, the 5th order generator model represented the behavior of the generator in more detail and included the transient behavior of the stator current. The 3rd order representation gave a similar mean value of the stator current correctly but did not show any details of the transient. For classical, phasor domain electro-mechanical dynamic studies of large power systems the simplicity and reduced computation time of the 3rd order model appears attractive. For more detailed representations of fault current contribution and investigation of the required ratings of the converters then the 5th order representation may be preferred. However, other effects, such as the control system and various limits of the converters as well as the transient behavior of the generator rotor circuits are also likely to be important in more detailed studies.
Acknowledgements The authors would like to thank Dr X.G. Wu and Dr A. Arulampalam for their contributions in the preliminary stages of this work.
Appendix A: List of symbols vs , v r is , ir , ig va , i a Pg , Qg , vn XGT , Xa Rs , Rr vs , vbase , vr l Lm Lss , Lrr s
stator (vs /vds /jvqs ) and rotor (vr /vdr / jvqr ) voltage stator (is /ids /jiqs ), rotor (ir /idr /jiqr ) and generated current stator side converter (C2) voltage and current generated active and reactive power, network voltage transformer reactances stator and rotor machine resistance synchronous, base and rotor angular frequency flux linkage mutual inductance between the stator and the rotor stator and rotor self-inductance rotor slip
214
X ?, X ed , eq To J Tm , Te , Tsp Topt , Kopt
J.B. Ekanayake et al. / Electric Power Systems Research 67 (2003) 207 /215
transient or short circuit reactance and open circuit reactance voltage behind transient reactance d/q components transient open circuit time constant inertia constant mechanical, electromagnetic, set point torque optimal torque and optimal constant of wind turbine superscript indicates a per unit quantity
Appendix B The voltage equation for the three-phase stator is [3,4]: 8 d > > > vas Rs ias las > > dt > > > < d vbs Rs ibs lbs (A1) > dt > > > > d > > > : vcs Rs ics lcs dt where 8 > > >var Rr iar lar > dt > > > < d vbr Rr ibr lbr > dt > > > > d > > > : vcr Rr icr lcr dt where 8
(A2)
82 3 9 2 3 2 3 vas ias < las = d vds 4lbs 5 Ts 4vbs 5 Rs Ts 4ibs 5 Ts vqs dt : l ; vcs ics cs 2 3 ias d l Ts1 ds Rs Ts 4ibs 5 Ts lqs dt ics dT 1 d lds l i Rs ds Ts s ds Ts Ts1 iqs lqs dt lqs dt dT 1 d lds l i (A6) Rs ds Ts s ds iqs l qs dt dt lqs
By taking the inverse of equation (A5) and then differentiating [6]: 2 3 vs sin(vs t) vs cos(vs t) dTs1 4 vs sin(vs t2p=3) vs cos(vs t2p=3)5 dt vs sin(vs t2p=3) vs cos(vs t2p=3) (A7) Therefore: Ts
dTs1 l l lqs 0 1 ds vs ds vs lqs lqs lds 1 0 dt
(A8)
Substituting equation (A8) into equation (A6): d lds ids vds lqs Rs vs (A9) vqs iqs lds dt lqs With reference to the synchronous reference frame as the rotor rotates at svs , where s is the slip, the transformation matrix for the rotor voltages and currents is: Tr
(A3)
2 cos(svs t) cos(svs t2p=3) cos(svs t2p=3) 3 sin(svs t) sin(svs t2p=3) sin(svs t2p=3) (A10)
The rotor voltage equation can also be transformed into the dq frame, similarly as the stator equations, and the following equation results: (A4)
The stator voltage equation can be transformed to a dq coordinate system in a synchronous reference frame using the following transformation matrix [6]: 2 cos(vs t) cos(vs t2p=3) cos(vs t2p=3) Ts 3 sin(vs t) sin(vs t2p=3) sin(vs t2p=3) (A5)
Equation (A1) can be transformed to the dq frame by:
d ldr vdr i lqr Rr dr svs vqr iqr ldr dt lqr
(A11)
Similarly the flux equations given in equation (A2) and (A4) can be transformed into the dq frame and the new flux equations are: lds Lss ids Lm idr (A12) lqs Lss iqs Lm iqr ldr Lrr idr Lm ids (A13) lqr Lrr iqr Lm iqs
J.B. Ekanayake et al. / Electric Power Systems Research 67 (2003) 207 /215
Rotor resistance (Rr)
Appendix C In order to obtain a pu representation of the voltage equations, consider equation (A9). By dividing first row V of equation (A9) by Zbase base vs Lbase : Ibase vds Zbase
Rs Zbase
ids
vs lqs Zbase
1 Zbase
d dt
lds
vds R i lqs 1 d lds s ds Vbase Zbase Ibase Ibase Lbase vs dt Ibase Lbase
(B1)
0.00549 pu
where Lss (Xm Xls )
215
Rotor leakage re- 0.09955 actance (Xlr) pu
1 1 and Lrr (Xm Xlr ) :/ 2pf 2pf
Airgap magnetizing reactance (Xm ): 3.95279 pu. Lumped inertia constant (H): 3.5 s. Control model parameters: Cut-in speed/1000 rpm; speed limit /1800 rpm; shutdown speed/2000 rpm. Kopt /0.56, KP 2 /0.5, K12 /0.5.
Dividing flux linkage equation (A12) by Ibase Lbase : lds Lss ids Lm idr
References
lds L ids Lm idr ss Lbase Ibase Lbase Ibase Lbase Ibase
(B2)
If all the pu quantities are represented by an over-line, v¯ds R¯s i¯ds l¯qs
1 d l¯ds vs dt
l¯ds L¯ss i¯ds L¯m i¯dr
(B3)
Appendix D Vbase /690 V, Sbase /2 MW, vbase /2pfbase, fbase / 50 Hz 2 MW induction wind turbine model parameters (star equivalent circuit): Stator resistance (Rs)
0.00488 pu
Stator leakage re- 0.09241 actance (Xls) pu
[1] S. Muller, M. Deicke, W. Rik, De Doncker, ‘‘Doubly fed induction generator systems for wind turbines’’, IEEE Ind. Appl. Magazine 8 (3) (May/June 2002) 26 /33. [2] CIGRE, ‘‘CIGRE Technical Brochure on Modeling New Forms of Generation and Storage’’, TF 38.01.10, June 2000, contributions by N. Hatziargyriou, M. Donnelly, S. Papathanassiou, J.A. Pecas Lopes, M. Takasaki, H. Chao, J. Usaola, R. Lasseter, A. Efthymiadis. [3] R. Pena, J.C. Clare, G.M. Asher, ‘‘Doubly fed induction generator using back-to-back PWM converters and its application to variable speed wind-energy generation’’, IEE Proc. */Electr. Power Appl. 143 (3) (1996) 231 /241. [4] L. Holdsworth, X.G. Wu, J.B. Ekanayake, N. Jenkins, ‘‘Steady state and transient behavior of induction generators (including doubly fed machines) connected to distribution networks’’, IEE Tutorial on ‘‘Principles and Modeling of Distributed Generators’’, 4 July 2002, UMIST, UK, http://www.iee.org/OnComms/pn/ powertrading. [5] P.C. Krause, Analysis of Electric Machinery, McGraw-Hill, 1986. [6] P. Kundur, Power System Stability and Control, McGraw-Hill, 1994. [7] O. Wasynezuk, D. Yi-Min, P.C. Krause, ‘‘Theory and comparison of reduced order models of induction machines’’, IEEE Trans. Power Apparatus Syst. PAS-104 (3) (1985) 598 /606.
Comparison of 5th order and 3rd order machine models for doubly fed induction generator (DFIG) wind turbines J.B. Ekanayake a,c,*, L. Holdsworth b, N. Jenkins b,c a
Department of Electrical and Electronics Engineering, University of Peradeniya, Peradeniya, Sri Lanka b The Manchester Center for Electrical Energy (MCEE), UMIST, Manchester, UK c Tyndall Center for Climate Change Research, Norwich, UK Received 24 September 2002; received in revised form 28 March 2003; accepted 1 April 2003
Abstract With increasing concern over climate change, a number of countries have implemented new renewable energy targets, which require significant amounts of wind generation. It is now recognized that much of this new wind generation plant will be variable speed type using doubly fed induction generators (DFIG). In order to investigate the impacts of these DFIG installations on the operation and control of the power system, accurate models are required. A fifth order and reduced order (3rd) machine models are described and the control of the wind turbine discussed. The capability of the DFIG for voltage control (VC) and its performance during a network fault is also addressed. # 2003 Elsevier B.V. All rights reserved. Keywords: Machine models; DFIG; Wind turbines
1. Introduction The exploitation of the wind as a source of renewable energy continues to increase with some 25 GW of wind turbine capacity installed worldwide. In some countries the penetration of wind energy is such that already it is a significant fraction of generation capacity and projections of future installations indicate that the number of wind turbines will increase rapidly over the next 10 years. Hence, wind farms must be included in computer simulations to study both the development and operation of the power system. At present variable speed operation of wind turbines, using doubly fed induction generators (DFIG), is emerging as the preferred technology. This is due mainly to the reduced mechanical loads on the wind turbines that arise from variable speed operation [1]. However, a secondary advantage is the increased possibilities of control of both real and reactive power to allow easier integration of wind turbines into the power system.
* Corresponding author. 0378-7796/03/$ - see front matter # 2003 Elsevier B.V. All rights reserved. doi:10.1016/S0378-7796(03)00109-3
In comparison to conventional synchronous generation, wind power is developed in relatively small units. Typical wind turbine ratings vary between 800 kW and 3 MW with wind farms ranging from 1 to 200 MW. Thus it is important to determine the simplest possible models of wind turbines that give an accurate representation in the various studies that are undertaken on the power system. In this paper a comparison is made between a 5th order model of the DFIG wind turbine and a 3rd order representation where the stator transients are neglected. The development of the models is described and it is shown that for dynamic modeling the 3rd order model is adequate. For detailed representation of fault current contribution, the 5th order model provides better resolution although it must be recognized that the behavior of the converter control systems is likely to have a dominant effect on fault currents.
2. Doubly fed induction generator (DFIG) DFIG wind turbines utilize a wound rotor induction generator, where the rotor winding is fed through backto-back variable frequency, voltage source converters. A
208
J.B. Ekanayake et al. / Electric Power Systems Research 67 (2003) 207 /215
a) The stator current was assumed positive when flowing towards the machine. b) The equations were derived on the synchronous reference. c) The q-axis was assumed to be 908 ahead of the daxis with respect to the direction of rotation. d) The q component of the stator voltage was selected as the real part of the busbar voltage and d component was selected as the imaginary part.
Fig. 1. Basic configuration of a DFIG wind turbine.
Fig. 2. Direct (d) and quadrature (q) representation of induction machine.
typical configuration of a DFIG based wind turbine is shown schematically in Fig. 1. The converter system enables variable speed operation of the wind turbine by decoupling the power system electrical frequency and the rotor mechanical frequency. A more detailed description of the DFIG system together with its control and protection circuits can be found in [2 /4].
3. Fifth order model of the DFIG Three-phase stator and rotor windings of an induction machine can be represented by two sets of orthogonal fictitious coils. Appendix B shows the transformation from three-phase to two-phase fictitious coils which are placed in direct (d) and quadrature (q) axes [5,6]. A generalized fifth order machine model was then developed by considering the following conditions and assumptions:
Machine equations can be represented in terms of the machine variables or in terms of arbitrary reference frame variables. However, for power system studies it is desirable to use a per unit (pu) representation. This enables the conversion of the entire system to pu quantities on a single power base. The voltage equations for the fictitious coils were normalized as shown in Appendix C. Fig. 2 shows the dq representation of the machine. The voltage equations for the induction generator are given below, where all quantities except the synchronous speed are in pu: 8 1 d > > ¯ ¯ ¯ ¯ >
(3) (4)
(5)
Substituting from Eq. (5) into Eq. (3) and with X¯ 1 L¯2 ½L¯ss m : L¯rr ¯ L ¯ ¯ ¯ lds X 1 ids m l¯dr and L¯rr (6) ¯ Lm ¯ ¯ ¯ ¯ lqs X 1 iqs lqr L¯rr In order to obtain a voltage behind a transient model
J.B. Ekanayake et al. / Electric Power Systems Research 67 (2003) 207 /215
for the DFIG, the following two voltage components were defined. e¯d
L¯m l¯qr L¯rr
and
e¯q
L¯m l¯dr L¯rr
dvr
1 (Tm Te ) J dt
209
(16)
(7)
Substituting from equation (7) into equation (6): l¯ds X¯ 1 i¯ds e¯q
and
l¯qs X¯ 1 i¯qs e¯d
(8)
By substituting from equation (8) into equation (1): 8 1 d X¯ 1 d > > ¯ ¯ ¯ ¯ ¯ >
L¯rr :/ vs R¯r
Substituting from equation (7) into equation (5): 8 ¯ 1 l L¯m i¯ds L¯ > > ¯dr dr e¯q m i¯ds i > < L¯m L¯rr L¯rr (11) ¯ ¯ ¯ > L i 1 l L¯m qr m qs > ¯ ¯ > e ¯ i i : qr d qs L¯m L¯rr L¯rr The electromagnetic torque is calculated using: Te l¯ds i¯qs l¯qs i¯ds
(12)
Substituting from equation (8) to equation (12) the following equation can be obtained: Te (X¯ 1 i¯ds e¯q ) i¯qs (X¯ 1 i¯qs e¯d ) i¯ds e¯q i¯qs e¯d i¯ds
(13)
Substituting for e¯d and e¯q/in terms of l¯dr and l¯qr in equation (13): Te
L¯m (l¯dr i¯qs l¯qr i¯ds ) L¯rr
4. Third order model of the DFIG For power system transient studies inclusion of the network transients and generator stator transients increases the order of the overall system model, thus limiting the size of the system that can be simulated. Further, a small time step is required for numerical integration resulting in an increased computational time. For these reasons, it has become conventional to reduce the order of the generator and neglect the network transients for stability analysis of large power systems. Different methods of reducing generator equations are discussed in [7]. In this investigation, a standard method of reducing the order of the induction generator model was considered where the rate of change of stator flux linkage is neglected. The reduced order model was derived by ignoring the differential term in equation (1), which is equivalent to neglecting the stator electric transients [5]. The reduced order voltage equations in pu: v¯ds R¯ s i¯ds l¯qs (17) v¯qs R¯ s i¯qs l¯ds 8 1 d ¯ > > ¯ ¯ ¯ >
(14) where T¯ o
Substituting from equation (4) into equation (14):
L¯rr :/ vs R¯r
(15)
The electromagnetic torque given in equation (15) can be simplified using the following steps:
Finally, if Tm is the mechanical torque, dependent upon the local wind speed:
. Using equations (17) and (3) and neglecting the resistance term
Te L¯m (i¯dr i¯qs i¯qr i¯ds )
J.B. Ekanayake et al. / Electric Power Systems Research 67 (2003) 207 /215
210
Fig. 3. Torque/speed control scheme of the DFIG.
8 1 L¯m > > ¯ ¯ > < ids ¯ v¯qs ¯ idr Lss Lss > 1 L¯m > > :i¯qs ¯ v¯ds ¯ i¯qr Lss Lss
(21)
. Substituting in equation (15): Te
L¯m (i¯dr v¯ds i¯qr v¯qs ) L¯ss
(22)
The d-component of the converter C1 voltage, vdr , was used for compensation for the generator magnetizing reactive power (No load PFC) as shown in Fig. 4. An outer loop (shown in dotted lines) was introduced for voltage control (VC) through converter C1.
5. DFIG control For the model it was assumed that the converters are ideal and the dc link voltage between the converters is constant. The q and d components of the injected voltage through converter C1, respectively, were used for speed and power factor control of the machine (see Fig. 1). Speed control was achieved by driving the machine along an optimum torque /speed curve, which corresponds to the maximum power extraction from the wind, as shown in Fig. 3. Between the cut-in and speed limit, the optimum torque /speed curve was characterized by an optimal characteristic curve (a) given by Topt /Kopt v2r (where vr is the measured rotor speed) and a constant speed characteristic (b) up to the rated torque. Beyond the speed limit, a small margin was introduced between the pitch control and electronic control in order to prevent the two controllers fighting each other. The set point torque corresponding to the speed of the machine was then translated into the reference of iqr using equation (22). In order to linearize the controller, vds was neglected with respect to vqs . Then from equation (22): i¯qrref
Tsp v¯qs
L¯ss L¯m
Fig. 4. No load PFC and VC through C1 for the DFIG.
(23)
The current error (the difference between the desired and achieved iqr) together with a PI controller was used to obtain vqr .
6. Modeling of the DFIG protection The controller model of the DFIG system included rotor voltage and current limits. The limits were selected depending on the MW capacity of the generator and the rating of the converters. Converter C1 was protected against over-current on the rotor circuits by a ‘singleshot crowbar’, as shown in Fig. 1. The operation of the crowbar was modeled by deactivating the converters upon the detection of rotor current magnitude above the current protection limit and short-circuiting the generator rotor.
7. Simulation results Using the 5th order and 3rd order machine models the start-up of the induction generator was simulated. The rotor speed, electromagnetic torque and stator current components are shown in Fig. 5. The machine used for simulation was a 2 MW, 690 V wind turbine and machine parameters are given in Appendix D. In order to obtain the complete start-up process, both the speed and power factor/voltage controllers were frozen at zero rotor injection voltage. The standard start up of an induction motor is shown to illustrate the effect of the
J.B. Ekanayake et al. / Electric Power Systems Research 67 (2003) 207 /215
211
Fig. 5. Start-up of the DFIG with zero rotor injection.
Fig. 6. Response to a torque step (all are in pu).
stator transients upon the characteristics of the 5th order model in comparison to the 3rd order machine model. However, this would not be a typical initialization procedure for the DFIG based wind turbine. For commercial applications the generator may be magnetized from the rotor windings to speed the generator up
to an initialization speed (around synchronous speed). The generator would then be connected to the grid. Using the 5th order and 3rd order models, the response of the DFIG for a step change in mechanical torque from 0.3 to 1.0 pu at t/40 s was studied. The simulation results showing the rotor speed and torque
212
J.B. Ekanayake et al. / Electric Power Systems Research 67 (2003) 207 /215
Fig. 7. Connection of DFIG turbine to double circuit power system model.
Fig. 8. VC capability of the DFIG.
changes are shown in Fig. 6. The results illustrate the operation of the torque/speed controller within specified rotor speed limits as shown in Fig. 3. With an applied mechanical torque of 1.0 pu the controller provides operation on the optimal characteristic until the rotor speed limit is reached. The applied control scheme then enables the electromagnetic torque to approach 1 pu, which increases generated output power through the stator, whilst limiting the generator rotor speed to 1.2 pu. This is illustrated at approximately t/46 s in Fig. 6.
8. Application study In order to investigate the VC capability of the DFIG, a double circuit, double busbar power system was considered, as shown in Fig. 7. The system short circuit level was assumed to be 16 MVA (SCR /8) and X/R ratio was assumed to be 5. The VC capability of the DFIG is demonstrated in Fig. 8. The no-load power factor correction loop in Fig. 4 was activated throughout the simulation. The VC loop was introduced at t/ 40 s with the set point voltage of 1.02 pu. Simulations also carried out to investigate the performance of the DFIG during a system fault. A fault was applied at the mid point of one of the double circuit lines
at t/49.85 s and was cleared after 150 ms. Fig. 9a shows the machine terminal voltage, stator current, rotor speed and electromagnetic torque obtained from the 5th order model. A similar set of results was obtained using the 3rd order model and shown in Fig. 9b. From the simulations, it was found that the rotor current in pu during the transient is very similar to the stator current. In these cases the rotor crowbar limit was increased to maintain the machine in operation during the fault. In practice this means, increasing the shortterm current rating of components in the voltage source converter C1. Although not directly effected by the fault the rotor voltage applied to the DFIG throughout the duration of the network fault is entirely dependent upon the control strategy applied and the ratings of the converters. Therefore, it should be observed that in a practical DFIG system the converter voltage and current ratings together with the size of the dc link capacitor are all critical to ensure good performance during network disturbances.
9. Conclusions 5th and 3rd order models of a DFIG were developed and coupled to a control system of a variable speed wind
J.B. Ekanayake et al. / Electric Power Systems Research 67 (2003) 207 /215
213
Fig. 9. DFIG response to a system fault (all are in pu).
turbine. Example simulations demonstrated that both models give very similar results for machine runup and in response to changes in applied torque. The use of the generator side, voltage /source, converter for network VC was demonstrated and again, very similar results were obtained from the two machine representations. A three-phase short circuit on the network close to the generator was investigated. As, expected, the 5th order generator model represented the behavior of the generator in more detail and included the transient behavior of the stator current. The 3rd order representation gave a similar mean value of the stator current correctly but did not show any details of the transient. For classical, phasor domain electro-mechanical dynamic studies of large power systems the simplicity and reduced computation time of the 3rd order model appears attractive. For more detailed representations of fault current contribution and investigation of the required ratings of the converters then the 5th order representation may be preferred. However, other effects, such as the control system and various limits of the converters as well as the transient behavior of the generator rotor circuits are also likely to be important in more detailed studies.
Acknowledgements The authors would like to thank Dr X.G. Wu and Dr A. Arulampalam for their contributions in the preliminary stages of this work.
Appendix A: List of symbols vs , v r is , ir , ig va , i a Pg , Qg , vn XGT , Xa Rs , Rr vs , vbase , vr l Lm Lss , Lrr s
stator (vs /vds /jvqs ) and rotor (vr /vdr / jvqr ) voltage stator (is /ids /jiqs ), rotor (ir /idr /jiqr ) and generated current stator side converter (C2) voltage and current generated active and reactive power, network voltage transformer reactances stator and rotor machine resistance synchronous, base and rotor angular frequency flux linkage mutual inductance between the stator and the rotor stator and rotor self-inductance rotor slip
214
X ?, X ed , eq To J Tm , Te , Tsp Topt , Kopt
J.B. Ekanayake et al. / Electric Power Systems Research 67 (2003) 207 /215
transient or short circuit reactance and open circuit reactance voltage behind transient reactance d/q components transient open circuit time constant inertia constant mechanical, electromagnetic, set point torque optimal torque and optimal constant of wind turbine superscript indicates a per unit quantity
Appendix B The voltage equation for the three-phase stator is [3,4]: 8 d > > > vas Rs ias las > > dt > > > < d vbs Rs ibs lbs (A1) > dt > > > > d > > > : vcs Rs ics lcs dt where 8
(A2)
82 3 9 2 3 2 3 vas ias < las = d vds 4lbs 5 Ts 4vbs 5 Rs Ts 4ibs 5 Ts vqs dt : l ; vcs ics cs 2 3 ias d l Ts1 ds Rs Ts 4ibs 5 Ts lqs dt ics dT 1 d lds l i Rs ds Ts s ds Ts Ts1 iqs lqs dt lqs dt dT 1 d lds l i (A6) Rs ds Ts s ds iqs l qs dt dt lqs
By taking the inverse of equation (A5) and then differentiating [6]: 2 3 vs sin(vs t) vs cos(vs t) dTs1 4 vs sin(vs t2p=3) vs cos(vs t2p=3)5 dt vs sin(vs t2p=3) vs cos(vs t2p=3) (A7) Therefore: Ts
dTs1 l l lqs 0 1 ds vs ds vs lqs lqs lds 1 0 dt
(A8)
Substituting equation (A8) into equation (A6): d lds ids vds lqs Rs vs (A9) vqs iqs lds dt lqs With reference to the synchronous reference frame as the rotor rotates at svs , where s is the slip, the transformation matrix for the rotor voltages and currents is: Tr
(A3)
2 cos(svs t) cos(svs t2p=3) cos(svs t2p=3) 3 sin(svs t) sin(svs t2p=3) sin(svs t2p=3) (A10)
The rotor voltage equation can also be transformed into the dq frame, similarly as the stator equations, and the following equation results: (A4)
The stator voltage equation can be transformed to a dq coordinate system in a synchronous reference frame using the following transformation matrix [6]: 2 cos(vs t) cos(vs t2p=3) cos(vs t2p=3) Ts 3 sin(vs t) sin(vs t2p=3) sin(vs t2p=3) (A5)
Equation (A1) can be transformed to the dq frame by:
d ldr vdr i lqr Rr dr svs vqr iqr ldr dt lqr
(A11)
Similarly the flux equations given in equation (A2) and (A4) can be transformed into the dq frame and the new flux equations are: lds Lss ids Lm idr (A12) lqs Lss iqs Lm iqr ldr Lrr idr Lm ids (A13) lqr Lrr iqr Lm iqs
J.B. Ekanayake et al. / Electric Power Systems Research 67 (2003) 207 /215
Rotor resistance (Rr)
Appendix C In order to obtain a pu representation of the voltage equations, consider equation (A9). By dividing first row V of equation (A9) by Zbase base vs Lbase : Ibase vds Zbase
Rs Zbase
ids
vs lqs Zbase
1 Zbase
d dt
lds
vds R i lqs 1 d lds s ds Vbase Zbase Ibase Ibase Lbase vs dt Ibase Lbase
(B1)
0.00549 pu
where Lss (Xm Xls )
215
Rotor leakage re- 0.09955 actance (Xlr) pu
1 1 and Lrr (Xm Xlr ) :/ 2pf 2pf
Airgap magnetizing reactance (Xm ): 3.95279 pu. Lumped inertia constant (H): 3.5 s. Control model parameters: Cut-in speed/1000 rpm; speed limit /1800 rpm; shutdown speed/2000 rpm. Kopt /0.56, KP 2 /0.5, K12 /0.5.
Dividing flux linkage equation (A12) by Ibase Lbase : lds Lss ids Lm idr
References
lds L ids Lm idr ss Lbase Ibase Lbase Ibase Lbase Ibase
(B2)
If all the pu quantities are represented by an over-line, v¯ds R¯s i¯ds l¯qs
1 d l¯ds vs dt
l¯ds L¯ss i¯ds L¯m i¯dr
(B3)
Appendix D Vbase /690 V, Sbase /2 MW, vbase /2pfbase, fbase / 50 Hz 2 MW induction wind turbine model parameters (star equivalent circuit): Stator resistance (Rs)
0.00488 pu
Stator leakage re- 0.09241 actance (Xls) pu
[1] S. Muller, M. Deicke, W. Rik, De Doncker, ‘‘Doubly fed induction generator systems for wind turbines’’, IEEE Ind. Appl. Magazine 8 (3) (May/June 2002) 26 /33. [2] CIGRE, ‘‘CIGRE Technical Brochure on Modeling New Forms of Generation and Storage’’, TF 38.01.10, June 2000, contributions by N. Hatziargyriou, M. Donnelly, S. Papathanassiou, J.A. Pecas Lopes, M. Takasaki, H. Chao, J. Usaola, R. Lasseter, A. Efthymiadis. [3] R. Pena, J.C. Clare, G.M. Asher, ‘‘Doubly fed induction generator using back-to-back PWM converters and its application to variable speed wind-energy generation’’, IEE Proc. */Electr. Power Appl. 143 (3) (1996) 231 /241. [4] L. Holdsworth, X.G. Wu, J.B. Ekanayake, N. Jenkins, ‘‘Steady state and transient behavior of induction generators (including doubly fed machines) connected to distribution networks’’, IEE Tutorial on ‘‘Principles and Modeling of Distributed Generators’’, 4 July 2002, UMIST, UK, http://www.iee.org/OnComms/pn/ powertrading. [5] P.C. Krause, Analysis of Electric Machinery, McGraw-Hill, 1986. [6] P. Kundur, Power System Stability and Control, McGraw-Hill, 1994. [7] O. Wasynezuk, D. Yi-Min, P.C. Krause, ‘‘Theory and comparison of reduced order models of induction machines’’, IEEE Trans. Power Apparatus Syst. PAS-104 (3) (1985) 598 /606.