Phase amplitude model for doubly fed induction generators

Transcription

1 J. Mod. Powe Syst. Clean Enegy (219) 7(2): Phase amplitude model fo doubly fed induction geneatos Hua HUANG 1,2, Ping JU 2, Xueping PAN 2, Yuqing JIN 2, Xiaoming YUAN 1, Yuan GAO 2 Abstact The doubly fed induction geneato (DFIG) is majo type of wind tubine geneato used in gid-connected wind fams. Pactical models of DFIG have been built to study the influence of wind powe geneation on powe system dynamics. Howeve, most existing pactical models of the DFIG ae based on ectangula coodinates, in which fequency vaiation is neglected. In this pape, a phase amplitude (P A) model is poposed fo a DFIG based on phase and amplitude of the intenal voltage. The model stuctue is much like that of the synchonous geneato, and the oto voltage can manipulate both the amplitude and the phase of the intenal voltage. CossCheck date: 13 July 21 Received: 3 Decembe 216 / Accepted: 13 July 21 / Published online: 16 Octobe 21 Ó The Autho(s) 21 & Ping JU pju@hhu.edu.cn Hua HUANG hua.h@hotmail.com Xueping PAN xueping_pan@163.com Yuqing JIN jyq16@hhu.edu.cn Xiaoming YUAN yuanxm@mail.hust.edu.cn Yuan GAO gaoyuan@163.com 1 2 State Key Laboatoy of Advanced Electomagnetic Engineeing and Technology, School of Electical and Electonic Engineeing, Huazhong Univesity of Science and Technology, Wuhan 4374, China College of Enegy and Electical Engineeing, Hohai Univesity, Nanjing 219, China Compaisons have been made between the new P A model of the DFIG and the synchonous geneato model, as well as the asynchonous moto model. The contibutions of the new P A model of the DFIG ae discussed and it is demonstated that the poposed model has bette ability in descibing powe system dynamic phenomena such as voltage dynamics and stuctual dynamics in geneal. Simulation esults and a field test validate these contibutions. Keywods Doubly fed induction geneato (DFIG), Model, Intenal voltage, Voltage dynamics, Fequency dynamics 1 Intoduction With the inceasing penetation of wind enegy in powe systems, thei dynamic chaacteistics need to be investigated [1, 2]. Doubly fed induction geneato (DFIG) is an impotant subsystem in wind tubine geneatos (WTGs) [3, 4], and modelling of it is essential. Analytical DFIG models, which ae epesented by flux linkages and oto speed [5 7], ae deived fom basic physical pinciples and have well founded theoy. These models ae valid fo lage excusions in fequency and voltage of the system. Consideable attention has been paid to the pactical models of DFIG. Pactical models of the DFIG ae deived fom the analytical models by defining the pactical vaiables. In powe system electo-magnetic tansient stability studies, DFIG is nomally epesented by a fifth-ode pactical model [, 9], which consides both the stato and oto tansients. Neglecting stato tansients in the fifthode model yields a thid-ode pactical model [9, 1], fequently used in powe system electo-mechanical

2 37 Hua HUANG et al. tansient stability studies. If oto electical tansients ae also neglected, the fist-ode oto speed model (o twomass dive tain model) is obtained and used in conjunction with the cicuit epesentation of the algebaic equations [11 14]. Pactical models of the DFIG ae usually expessed based on diect and quadatue axes. In [15, 16], the intenal voltage and phase of the DFIG wee calculated and used to design the contolle based on flux magnitude and angle contol (FMAC) stategy, which adjusts the magnitude of the oto voltage fo the contol of electical powe. In [17], a synchonized model of the DFIG was pesented and a moe simplified contol system, which makes the DFIG equivalent to a synchonous geneato, was poposed. In the synchonized model of the DFIG, the elation between the electical powe and the magnitude/phase of the intenal voltage, and the elation between the oto voltage and the magnitude/phase of the intenal voltage wee pesented. Howeve, the analytical expession of the dynamic model of the DFIG based on the state vaiables of magnitude/phase of the intenal voltage has not been povided. The concept of an intenal voltage motion equation was poposed ecently in [1, 19]. The elationship between phase amplitude of the intenal voltage and the unbalanced powe is analyzed based on the tansfe function fo smallsignal stability analysis. In [2], a pola angle modelling method fo DFIG is pesented. Based on [2], a new phase amplitude (P A) model fo the DFIG is poposed. The new model is achieved by edefining the pactical vaiables. Contibutions of the new P A model fo the DFIG ae as follows. 1) The P A model of the DFIG can be used conveniently fo powe system voltage dynamic analysis and contolle design since the state vaiables of intenal voltage and angle ae expessed diectly by the oto voltages. 2) Based on the pola fom model of the DFIG, the stuctual dynamics of the DFIG can be illustated. It indicates that the DFIG is a two time-scale system, whee the fast behavio is dominated by the electical states, and the slow is dominated by the mechanical states. 3) The effects of system fequency ae consideed in the poposed models. This pape is oganized as follows. The existing modelling method of DFIGs is biefly pesented in Section 2.In Section 3, two new thid-ode models, based on ectangula coodinates and pola coodinates espectively, ae poposed. The pola fom model of the DFIG is compaed with the synchonous geneato model and an induction machine in Section 4, and the stuctual dynamics of the DFIG ae also discussed in Section 4. Simulation esults and field tests with a low voltage ide-though expeiment ae pesented and the validation of the poposed model is shown in Section 5. 2 Existing models fo DFIG Basically, DFIG is an induction type geneato; the equivalent cicuit of DFIG is simila to that of an induction machine. In the following, the development of a fifth-ode model and a thid-ode epesentation of the DFIG ae descibed. 2.1 Fifth-ode model DFIG is geneally epesented by a set of fifth-ode diffeential equations of the flux linkages and shaft speed [9]. Accoding to the standad pe-unit notation, the diectand quadatue-axis stato and oto flux linkages ae epesented in the synchonously-otating efeence fame by: u ds ¼ R s i ds þ dw ds x s w qs u qs ¼ R s i qs þ dw qs þ x s w ds u d ¼ R i d þ dw ð1þ d ðx s x Þw q u q ¼ R i q þ dw q þðx s x Þw d whee w ds and w qs ae the d-axis and q-axis stato flux linkages; w d and w q ae the d-axis and q-axis oto flux linkages; u ds and u qs ae the d-axis and q-axis stato teminal voltages; u d and u q ae the d-axis and q-axis oto voltages; x s and x ae the synchonous and oto phase speed; R s and R ae the stato and oto esistance. The flux cuent elationships ae shown as: w ds ¼ L s i ds þ L m i d w qs ¼ L s i qs þ L m i q ð2þ w d ¼ i d L m i ds w q ¼ i q L m i qs whee i ds and i qs ae the d-axis and q-axis stato cuents; i d and i q ae the d-axis and q-axis oto cuents; L s = L s? L m, =? L m, L s is the stato leakage inductance, is the oto leakage inductance, and L m is the mutual inductance between stato and oto. The geneato oto shaft is connected to the tubine shaft flexibly via a geabox and coupling. The fifth ode diffeential equation descibes the wind tubine dive system by a one-mass model, which is shown as: 2H dx ¼ T m T e ð3þ

3 Phase amplitude model fo doubly fed induction geneatos 371 whee H = H t? H g, H t and H g ae the inetial constant of the tubine and the geneato; T m is the wind toque, which is the powe input of the wind tubine (WT); T e is the electomagnetic toque. The equation of the electomagnetic toque is: T e ¼ w ds i qs w qs i ds ¼ w d i q w q i d ð4þ The model of (2) and (3) is called a fifth-ode model. By defining the vaiables (these defined vaiables ae often called pactical vaiables) as in (5), the pactical fifth-ode models of the DFIG ae obtained [9]. This is shown in (6). 2.2 Classical thid-ode model In powe system electomechanical tansient analysis, the stato tansient of the DFIG is often neglected by setting dw ds = ¼ dw qs ¼. Substitution of (5) and (2) into (1) yields the stato and oto voltage equations. The stato voltage equations of the DFIG ae: u ds ¼ R s i ds þ X i qs þ Ed u qs ¼ R s i qs X i ds þ Eq ð7þ The oto voltage equations can be expessed as: u d ¼ 1 T Eq x s L þ L mi ds þ d E q þ ðx s x Þ Ed m x s L m x s L m u q ¼ 1 T Ed x s L L mi qs þ d E d ðx s x Þ Eq m x s L m x s L m ðþ Ed ¼ x sl m w L q Eq ¼ x sl m w L d X ¼ x s L s X ¼ x s L s L2 m T ¼ R ð5þ whee Ed and E q ae the d-axis and q-axis voltages behind the tansient eactance; X s is the stato eactance; Xs is the stato tansient eactance; T is the oto cicuit time constant. Xs di ds x s X s di qs x s de d deq ¼ v ds ½R s þ 1 x s T ðx s Xs ÞŠi ds ð1 sþed L m v d þ 1 x s T Eq þ X s i qs ¼ v qs ½R s þ 1 x s T ðx s Xs ÞŠi qs ð1 sþe q L m v q 1 ¼ sx s Eq x L m s ¼ sx s Ed þ x L m s v d 1 T x s T Ed X s i ds v q 1 T Ed þðx s Xs Þi qs h i Eq ðx s Xs Þi ds ð6þ whee s is the slip, which is defined as s =(x s - x )/x s. The fifth-ode model of the DFIG consides both the stato dynamics and oto dynamics. This is often used in powe system electomagnetic tansient analysis. The oto voltage equations widely used in the liteatue ae expessed as: ded ¼ 1 T ½Ed ðx X Þi qs Šþsx s Eq x su q deq ¼ 1 ð9þ T ½Eq þðx X Þi ds Š sx s Ed þ x su d whee u d ¼ ð L m= u q ¼ ð L m= Þu d Þu q 2.3 Thid-ode model including system fequency dynamics ð1þ Duing a sevee system fault, powe systems may expeience lage excusions of fequency. The fist tem of () should be expessed as follows when the system fequency dynamics ae included. d Eq ¼ L 1 deq E q x s L m L m x s x 2 s d Ed ¼ L 1 de d E d x s L m L m x s x 2 s dx s dx s ð11þ The oto voltage equations should be expessed as follows by substituting (11) into (). ded ¼ 1 T ½Ed ðx X Þi qs Šþsx s Eq x su q þ E d dx s x s deq ¼ 1 T ½Eq þðx X Þi ds Š sx s Ed þ x su d þ E q dx s x s ð12þ Using the same method as above, the model of (6) can

4 372 Hua HUANG et al. be ewitten by including the effects of the system fequency vaiations. It can be seen fom (6) and (1) that the system fequency dynamics ae neglected in the existing pactical models of the DFIG. This is easonable when the powe system is vey lage and the system fequency is consideed to be constant duing distubances. Howeve, if the studied system is vey small, the system fequency may expeience lage excusions duing a sevee fault. In this situation, the model of (12) should be applied by including the system fequency dynamics. 3 P A models fo DFIG In this section, the pactical vaiables ae edefined, and two kinds of DFIG models ae poposed. 3.1 Poposed model of DFIG based on ectangula coodinates Redefine the pactical vaiables as in (13). Ed ¼ L m w L q Eq ¼ L m w L d L ¼ L s L ¼ L s L2 m ð13þ Substitution of (13) and (2) into (1) yields the new stato and oto voltage equations. The stato voltage equations ae: u ds ¼ R s i ds þ x s L i qs þ x s Ed u qs ¼ R s i qs x s L i ds þ x s Eq ð14þ and the oto voltage equations ae as follows, in which the dynamics of fequency ae included. ded ¼ 1 T ½Ed ðl L Þi qs Šþsx s Eq u q de q ¼ 1 T ½Eq þðl L Þi ds Š sx s Ed þ u d ð15þ Phaso elationship between diffeent voltages Convesion of the following vaiables fom ectangula coodinates to pola coodinates. _U ¼ u ds þ ju qs _E ¼ Ed þ je q ð17þ _I ¼ i ds þ ji qs Suppose the imaginay axis j ovelaps the q-axis, and the eal axis ovelaps the d-axis. The a is defined as the phase between the phaso _U and the d-axis. b is the phase between the phaso _E and the d-axis, and d is the phase between the phaso _E and _U. d ¼ b a u ds ¼ U cos a u qs ¼ U sin a ð1þ Ed ¼ E cos b Eq ¼ E sin b The phaso elationships between diffeent voltages ae shown in Fig Stato voltage equation in phaso fom Using phasos of (1), we can eplace (14) by: _U ¼ x s _E ð s þ jx s L Þ_I ¼ x s _E Z s _I ð19þ Figue 2 is the equivalent cicuit coesponding to (19) Intenal voltage equation in pola fom The deivation of intenal voltages can be expessed as follows based on (1). q/j E E q The equation of the electomagnetic toque is: T e ¼ Ed i ds þ Eq i qs 3.2 Poposed model of DFIG based on pola coodinates ð16þ In the following, the pola fom model of the DFIG is deduced fom the poposed DFIG model based on ectangula coodinates. u qs U α E d u ds Fig. 1 Phaso elationship between diffeent voltages d/

5 Phase amplitude model fo doubly fed induction geneatos 373 se ded ¼ d ð E cos bþ deq ¼ d ð E sin bþ ¼ de cos b E sin b db ¼ de sin b þ E cos b db ð2þ Substituting (2) into(15), the phaso fom of (15) can be attained by adding ded = times cosb to de q = times sinb. T de ¼ E þ L L ð Þ i qs cos b i ds sin b ð21þ þ T u q cos b þ u d sin b Similaly, (15) will be tansfomed as follows by subtacting ded = times sinb fom de q = times cosb. T db E ¼ L ð L Þ i ds cos b þ i qs sin b ð22þ T sx se þ T u d cos b þ u q sin b Since the stato esistance is vey small, the stato cuents can be expessed as in (23) by setting R s = in (14). i ds ¼ E sin b U sin a=x s L i qs ¼ U cos a=x s E ð23þ cos b L The following two expessions can be obtained fom (23). i qs cos b i ds sin b ¼ E L þ U x s L cos d i ds cos b þ i qs sin b ¼ U ð24þ x s L sin d Let C ¼ L L L, T ¼ T L L. The following will be obtained by substituting (24) into (22). db ¼ ð x x s Þ I Fig. 2 Equivalent cicuit of stato voltage T x s E sin d þ u d cos b þ u q sin b E ð25þ Because the phaso _U and the d-axis ae synchonous, a is constant. The deivation of d is: dd ¼ db ¼ x ð x s Þ T x s E sin d þ u d cos b þ u q sin b E ð26þ The electomagnetic toque can be expessed as follows by substituting (1) and (24) into (16). Z s U T e ¼ E U x s L sinðb aþ U ¼E x s L sin d ð27þ The pola fom of thid-ode DFIG model can be ewitten as: T de ¼ E þ cos d þ T u d sin b u q cos b x s dd ¼ x ð x s Þ T x s E 2H dx ¼ T m E U X sin d sin d þ u d cos b þ u q sin b E ð2þ whee X ¼ x s L. In a DFIG-based wind tubine, the u ds (o u qs ) is located on the d-axis (o q-axis) because of the flux oientation vecto contol technique [21]. Theefoe, the following conditions will be met: a ¼ b ¼ d ð29þ u ds ¼ U u qs ¼ The tems concening b in (2) can be tansfomed as: u d sin b u q cos b ¼ L m u d sin d u q cos d u d cos b þ u q sin b ¼ L m ð3þ u d cos d þ u q sin d As the phase and amplitude of the intenal voltage ae taken as the state vaiables in the model, so the model is efeed to as an intenal voltage phase amplitude model o P A model. 4 Model discussions It can be seen fom (14) and (15) that the poposed DFIG model based on ectangula coodinates has the same model stuctue as the existing models expessed by (7) and (1). Howeve, the poposed models in (14) and (15) have the advantage of taking into account the effects of system fequency without adding the tem of dx s = to the equations. The poposed P A model of DFIG has a simila model stuctue to the synchonous geneato and the induction moto. This is discussed in detail in the following. 4.1 Compaisons with synchonous geneato equation The thid-ode model of a synchonous geneato [22] can be witten as:

6 374 Hua HUANG et al. T de ¼ E þ dd ¼ x x s 2H dx ¼ T m x s E U X cos d þ E f sin d ð31þ whee E is the voltage behind the tansient eactance; E f is the voltage of excite; C is the constant. The model of the DFIG in (2) pesents some stiking similaities with a one-axis model of a synchonous geneato in (31) with the following diffeences: 1) The phase of the DFIG is not a otation phase of the shaft, but the phase of the oto flux amplitude with espect to the synchonously-otating efeence fame. 2) Thee ae two exta tems in the phase equation of the DFIG as compaed with that of a synchonous machine. The fist tem is due to the vaiable speed opeation of the induction machine, and the second tem is due to excitation effects. Theefoe, the DFIG s phase can be contolled diectly by the excitation voltage, wheeas the phase d of synchonous geneato cannot. 4.2 Compaisons with induction moto equation A pola fom model of the induction machine was poposed in [22]. This is shown in (32). T de ¼ E þ cos d dd ¼ðx x s Þ T E sin d ð32þ 2H dx ¼ E U X sin d T m It can be seen fom (2) and (32) that thee is an exta tem in the equations of E and d in the DFIG model. This is attibuted to the excitation effects in DFIG-based WTGs. 4.3 Stuctual dynamic analysis of DFIG based on poposed pola fom model The stuctual modelling method, which applied integal manifold theoy to descibe the dominant dynamic behavio of small and lage induction machines, was used in [23]. Based on the DFIG s P A model, its stuctual model is poposed fo analyzing the DFIG s dominant dynamics. Fom the pola fom model of the induction machines in [23], it was found that the paametes have the elations of p 2HX \T \ ffiffiffiffiffiffiffiffiffiffi 2HX fo small induction machines, while p 2HX \ ffiffiffiffiffiffiffiffiffiffi 2HX \T fo the lage ones. Stuctual dynamics of small and lage induction machines wee investigated based on integal manifolds, and it was shown that the dynamics of the small induction machine ae govened by the speed model, while the dynamics of the lage machines ae dominated by the voltage equation. Basically, a DFIG is an induction type geneato with its oto voltage contolled by convete contolles. Using the same analysis method as in [2], the stuctual dynamics of the DFIG based on the pola fom model ae investigated. The DFIG s paametes fom thee wind tubine manufactues in China ae listed in Appendix A Table A1, and pffiffiffiffiffiffiffiffiffiffi the values of paametes 2HX, 2HX, T and T ae calculated and listed in Table 1. It can be seen fom Table 1 that the elations of 2HX \T \ pffiffiffiffiffiffiffiffiffiffi 2HX ae available fo the DFIGs. Theefoe, a two time-scale singulaly petubed model can be obtained based on (2) by escaling the speed vaiables of X ¼ T x and X s ¼ T x s,and the small paamete is e ¼ T 2 ð2hx Þ. Equation (2) canbe ewitten as: e de ¼ T 2HX E þ e cos d X s þ e u d sin b u q cos b e dd ¼ T 2HX ðx X s Þ e þ e u d cos b þ u q sin b E dx ¼ 1 2H T m E U 2HX sin d X s E sin d ð33þ Based on singula petubation theoy, it can be concluded fom (33) that the DFIG is a two time-scale system, whee the fast states ae E and d, and the slow one is x. Theefoe, the stuctual dynamics of the DFIG can be obtained conveniently based on the pola fom model of the DFIG. Based on the stuctual dynamics of the DFIG, it can be seen that the DFIG s mechanical dynamics and the electical dynamics ae decoupling, which is diffeent fom synchonous geneatos, which ae dominated by electomechanical dynamics. This is consistent with the common knowledge that the dynamic behavio of a type-3 WTG is dominated by contolle esponse athe than physical chaacteistics [24]. Table 1 Paamete values of DFIGs fom thee diffeent manufactues Manufactue 2HX pffiffiffiffiffiffiffiffiffiffi 2HX T T

7 Phase amplitude model fo doubly fed induction geneatos Simulation esults and field test The poposed pola fom model of the DFIG is beneficial fo the stuctual dynamic analysis of the DFIG, and is also helpful fo powe system stability studies. In the following, the simulation and field test ae caied out, and the esponses ae obtained to illustate these contibutions. 5.1 Simulation esults The simulation analysis is conducted based on MATLAB /Simulink. Figue 3 shows the schematic diagam of the simulation system. It consists of a wind fam B4 B3 B2 B1 DFIG 3 km system 12 kv/25 kv f 25 kv/575 V 1.5 MW with six 1.5 MW DFIG-based WTGs connected to a 25 kv distibution system, which expots powe to a 12 kv gid though a 3 km tansmission line. The details of the system can be found in the Demo of MATLAB [25]. At t = 1 s, distubances with diffeent voltage sag depth ae applied at the teminal of the DFIG-based WTG, and the fault is cleaed at t = 1.15 s with the system back to its oiginal state. The distubed tajectoies of voltage amplitude and phase angle, active powe P and eactive powe Q unde voltage sag (voltage magnitude dops to 4%U N ) ae shown in Fig. 4. The second sub-tem ð=x s Þcos d and the thid subtem T u d sin b u q cos b on the ight side of the amplitude equation ae shown in Fig. 5a; the second subtem T x s E sin d and the thid sub-tem u d cos bþu q sin b E on the ight side of the phase equation ae shown in Fig. 5b. Fig. 3 Stuctue of test system Voltage (p.u.) ( ) P (p.u.) Q (p.u.) (a) Magnitude of teminal voltage, and the magnitude and angle of the intenal voltage.5 U (b) Active powe and eactive powe Fig. 4 Responses of DFIG unde voltage dip distubance E' Second tem (p.u.) Second tem (p.u.) Thid tem (p.u.) Thid tem (p.u.) (a) Responses of sub-tems in intenal voltage equation (b) Responses of sub-tems in angle equation Fig. 5 Responses of sub-tems on ight side of intenal voltage and phase equations

8 376 Hua HUANG et al. 5.2 Field test In the following, the field test is used to obtain the esponses of the DFIG by a low voltage ide-though (LVRT) expeiment Expeiment Many tests have been caied out to evaluate the fault ide-though capabilities of the WTG by povoking balanced and unbalanced faults unde diffeent opeational egimes. Hee the DFIG fom manufactue 1 with the paamete values shown in the Table A1 is tested and the esponses ae used to analyze the DFIG s dynamics. The expeimental set-up, to emulate a voltage dip, is shown in Fig. 6. The DFIG-based WTG is connected to the gid though a testing device. The testing device is designed to educe the voltage at the DFIG teminal to a specified level within a vey shot time. Duing nomal opeation, a seies impedance of Z1 is connected and switch S is open. When switch S is closed and the shot cicuit impedance Z2 is connected in paallel to the DFIG, this causes a voltage dip at the DFIG teminal. Afte a time, switch S is opened and the voltage at the DFIG ecoves to its oiginal level Responses unde LVRT distubance 1) Duing the voltage dip distubance, the voltage magnitude and active powe at MP decease while the eactive powe inceases. The dynamics of the intenal voltage and the teminal voltage ae vey simila, which indicates that the pola fom model of the DFIG is suitable fo analysis of the teminal voltage. 2) The magnitudes of the two sub-tems on the ight side of the intenal voltage equation decease duing the voltage dip distubance. This indicates that the intenal voltage is affected by the joint influence of voltage, angle, and excitation (contolled by oto voltage). 3) The value of the thid sub-tem (which coesponds to the oto voltage) in both the intenal voltage and angle equations is much bigge than the values of the othe two sub-tems. This indicates that the oto voltage (o excitation) dominates the dynamics of the intenal voltage and angle. Voltage (p.u.) U E' The WTG teminal voltage is shown in Fig. 7a. As S is closed, the voltage dops to 2%U n. Afte S is opened, it etuns to its nomal value quickly. Responses of active powe and eactive powe at MP ae shown in Fig. 7b, and the amplitude and phase of the intenal voltage ae illustated in Fig. 7a. The second sub-tem and the thid sub-tem on the ight side of the amplitude equation ae shown in Fig. a; and the second sub-tem and the thid sub-tem on the ight side of the phase equation ae shown in Fig. b. δ/( ) (a) Magnitude of teminal voltage, and the magnitude and angle of the intenal voltage 5.3 Contibutions of P A model to voltage dynamic analysis It can be seen fom the simulation and field test that: G Gid Z1 Testing device Z2 S Tansfome 35 kv/.69 kv MP DFIG Fig. 6 Schematic diagam to emulate a voltage dip on DFIG inteconnected to powe system P/(p.u.) Q (p.u.) (b) Active powe and eactive powe Fig. 7 Responses of DFIG at MP

9 Phase amplitude model fo doubly fed induction geneatos 377 Second sub-tem (p.u.) Thid sub-tem (p.u.) Second sub-tem (p.u.) Thid sub-tem (p.u.) (a) Responses of sub-tems in intenal voltage equation (b) Responses of sub-tems in angle equation Fig. Responses of sub-tems on ight side of intenal voltage and phase equations The P A model of the DFIG eflects the elationships between the teminal voltage and the intenal voltage of the DFIG. The DFIG s dynamics ae mainly contolled by the oto voltage. Theefoe, the pola fom model of the DFIG can be used conveniently fo powe system voltage poblem analysis. 5.4 Contibution of P A model to stuctual dynamic analysis It can be seen fom Fig. 4 that afte the fault is cleaed, thee ae two dominant oscillation modes in active powe tajectoies; one is a slow mode, which is attibuted to the dive system, and the othe is a fast mode, which coesponds to the oto electical tansients. The measuements ae in ageement with the stuctual analysis of (33). This indicates that the DFIG is a two-timescale model dominated by the electical dynamics and mechanical behavio, espectively. 6 Conclusion In this study, the classical models of the DFIG ae illustated and it is shown that all the existing models ae expessed based on ectangula coodinates. In FMAC, the magnitude and angle of the intenal voltage ae often calculated based on DFIG models of ectangula coodinates. To obtain the pola fom model of the DFIG, a new DFIG model based on ectangula coodinates is pesented by defining the pactical vaiables in a new manne, based on which, the pola fom model is deived. The P A model of the DFIG demonstates some stiking similaities with the model of synchonous geneatos; it shows clea elations between oto voltage and the magnitude/angle of the intenal voltage. The P A model of the DFIG is compaed with the model of the synchonous geneato and the pola fom model of the induction machine. The stuctual dynamics of the DFIG ae also consideed, and it is appaent that the DFIG is a two time-scale system, whee the fast states ae E and d, and the slow one is x. Simulation and a field test ae used to validate the contibutions of the P A model to the DFIG. It shows that the poposed pola fom of the DFIG model has bette ability in descibing powe system dynamic phenomena, such as voltage stability and stuctual dynamics. Open Access This aticle is distibuted unde the tems of the Ceative Commons Attibution 4. Intenational License ( ceativecommons.og/licenses/by/4./), which pemits unesticted use, distibution, and epoduction in any medium, povided you give appopiate cedit to the oiginal autho(s) and the souce, povide a link to the Ceative Commons license, and indicate if changes wee made. Appendix A Paamete values of DFIG fom thee manufactues ae shown in Table A1.

10 37 Hua HUANG et al. Table A1 Paamete values of DFIG fom thee manufactues Manufactue Paamete values 1 P N = 1.5 MW U N = 69 V p = 2 S N = 1.65 MVA k = 1.4 H g = 7 kg/m 2 H t = kg/m 2 R s =.3471 X R =.2694 X L s = H L m =.576 H = H 2 P N = 2MW U N = 69 V p = 2 S N = 2.1 MVA k = H g = kg/m 2 H t = kg/m 2 R s =.2 X R =.1567 X L s = H L m = H = H 3 P N = 1.5 MW U N = 69 V p = 2 S N = 1.65 MVA k = 15.6 H g = 69 kg/m 2 H t = kg/m 2 R s =.265 X R =.263 X L s =.53 H L m = 1.72 H =.42 H Refeences [1] Abhilash KG, Kusum V, Khaleequ RN (217) Dynamic impact analysis of DFIG-based wind tubine geneatos on low-fequency oscillations in powe system. IET Renew Powe Gene 11(1): [2] Shi LB, Su JL, Yao LZ (217) SS esonance analysis of complex powe system incopoating wind powe. IET Renew Powe Gene 11(3): [3] Tsili M, Papathanassiou S (29) A eview of gid code technical equiements fo wind fams. IET Renew Powe Gene 3(3):3 332 [4] Fan Z, Enslin JHR (26) Challenges, pinciples and issues elating to the development of wind powe in China. In: Poceedings of the 26 IEEE powe systems confeence and exposition, Atlanta, USA, 29 Octobe 1 Novembe 26, 7 pp [5] Ekanayake JB, Holdswoth L, Wu XG et al (23) Dynamic modeling of doubly fed induction geneato wind tubines. IEEE Tans Powe Syst 1(2):3 9 [6] Mei F, Pal BC (25) Modelling and small-signal analysis of a gid connected doubly-fed induction geneato. In: Poceedings of the 25 IEEE powe engineeing society geneal meeting, San Fancisco, USA, 16 June 25, pp [7] Luna A, Lima FKA, Santos D et al (211) Simplified modeling of a DFIG fo tansient studies in wind powe applications. IEEE Tans Ind Electon 5(1):9 2 [] Pal BC, Mei F (2) Modelling adequacy of the doubly fed induction geneato fo small-signal stability studies in powe systems. IET Renew Powe Gene 2(3):11 19 [9] Ekanayake JB, Holdswoth L, Jenkins N (23) Compaison of 5th ode and 3d ode machine models fo doubly fed induction geneato (DFIG) wind tubines. Elect Powe Syst Res 67(3): [1] Feijóo A, Cidás J, Caillo C (2) A thid ode model fo the doubly-fed induction machine. Elect Powe Syst Res 56(2): [11] Ellis A, Kazachkov Y, Muljadi E et al (211) Desciption and technical specifications fo geneic WTG models a status epot. In: Poceedings of the 211 IEEE PES PSCE, Phoenix, USA, 2 23 Mach 211, pp [12] Søensen P, Andesen B, Fotmann J et al (214) Oveview, status and outline of the new IEC electical simulation models fo wind powe geneation. In: Poceedings of 1th intenational wokshop on lage-scale integation of wind powe into powe systems as well as on tansmission netwoks fo offshoe wind fams, Aahus, Denmak, Octobe 211, 6 pp [13] Poubeik P (213) Poposed changes to the WECC WT3 geneic model fo type 3 wind tubine geneatos. powewold.com/webhelp/content/othe_documents/wecc- Type-3-Wind-Tubine-Geneato-Model-Phase-II-14.pdf. Accessed 3 August 215 [14] Poubeik P, Ellis A, Sanchez-Gasca J et al (213) Geneic stability models fo type 3 & 4 wind tubine geneatos fo WECC. In: Poceedings of 213 powe and enegy society geneal meeting, Vancouve, Canada, July 213, 5 pp [15] Hughes FM, Anaya-Laa O, Jenkins N et al (25) Contol of DFIG-based wind geneation fo powe netwok suppot. IEEE Tans Powe Syst 2(4): [16] Anaya-Laa O, Hughes FM, Jenkins N et al (27) Povision of a synchonising powe chaacteistic on DFIG-based wind fams. IET Gene Tansm Distib 1(1): [17] Wang Z, Sun Y, Li G et al (21) Magnitude and fequency contol of gid connected doubly fed induction geneato based on synchonised model fo wind powe geneation. IET Renew Powe Gene 4(3): [1] Yuan H, Yuan X, Hu J (217) Modeling of gid-connected VSCs fo powe system small-signal stability analysis in DClink voltage contol timescale. IEEE Tans Powe Syst 32(5): [19] Zhao M, Yuan X, Hu J (21) Modeling of DFIG wind tubine based on intenal voltage motion equation in powe systems phase amplitude dynamics analysis. IEEE Tans Powe Syst 33(2): [2] Ju P, Huang H, Jin YQ et al (215) Potential angle model of doubly fed induction geneato (DFIG) and deivation method of potential angle model. CN A [21] Li S, Challoo R, Nemmes MJ (29) Compaative study of DFIG powe contol using stato-voltage and stato-flux oiented fames. In: Poceedings of the IEEE powe and enegy society geneal meeting, Calgay, Canada, 26 3 July 29, pp [22] Kundu P (1994) Powe system stability and contol. McGaw Hill, New Yok [23] Ahmed-Zaid S, Taleb M (1991) Stuctual modeling of small and lage induction machines using integal manifolds. IEEE Tans Enegy Conves 6(3): [24] Hiskens IA (212) Dynamics of type-3 wind tubine geneato models. IEEE Tans Powe Syst 27(1): [25] MATLAB use s guide dynamic system simulation fo MATLAB. The Math Woks Inc

11 Phase amplitude model fo doubly fed induction geneatos 379 Hua HUANG eceived the B.E. degee in electical engineeing fom Hohai Univesity, Nanjing, China, in 211 and the M.S. degee in electical engineeing fom the Univesity of Stathclyde, Glasgow, UK, in 212. Cuently she is pusuing the Ph.D. degee in Huazhong Univesity of Science and Technology, Wuhan, China. He eseach inteests include modeling and contol of enewable powe geneation systems. Ping JU eceived the B. E. and M.S. degees fom Southeast Univesity, Nanjing, China in 192 and 195, and the Ph.D. degee fom Zhejiang Univesity, Hangzhou, China in 19. Fom 1994 to 1995, he was an Alexande von Humbol Fellow at the Univesity of Dotmund, Gemany. He is cuently a Pofesso of Electical Engineeing at Hohai Univesity, Nanjing, China. He has published five eseach books and authoed and coauthoed ove 2 jounal papes. He eceived the Scientific Funds fo Outstanding Young Scientists of China. His eseach inteests include modeling and contol of powe systems. Xueping PAN eceived the Ph.D. degee fom Zhejiang Univesity, Hangzhou, China, in 2. She is pesently a pofesso in the College of Enegy and Electical Engineeing, Hohai Univesity, Nanjing, China. He eseach inteests include modeling of enewable powe geneation system, powe system dynamic analysis. Yuqing JIN eceived the B.E., M.S. and Ph.D. degees in electical engineeing fom Hohai Univesity, China, in 22, 26 and 212, espectively. Cuently, he is an Associate Pofesso in the College of Enegy and Electical Engineeing at Hohai Univesity, China. His eseach inteests include modeling and contol of enewable powe geneation. Xiaoming YUAN eceived the B.E. degee fom Shandong Univesity, China, the M.E. degee fom Zhejiang Univesity, China, and the Ph.D. degee fom Fedeal Univesity of Santa Cataina, Bazil, in 196, 1993, and 199 espectively, all in electical engineeing. He is a Distinguished Expet of National Thousand Talents Pogam of China, and Chief Scientist of National Basic Reseach Pogam of China (973 Pogam). His eseach inteests include stability and contol of powe system with multi-machines multi-convetes, contol and gid-integation of enewable enegy geneations, and contol of HVDC tansmission systems. Yuan GAO eceived the M.E. degee fom Hohai Univesity, Nanjng, China, in 215. He is pesently an enginee in State Gid Jiangsu Electic Powe Maintenance Company. His eseach inteests include modeling of wind powe geneation system.