A Fast Dynamic Phasor Model of Autotransformer Rectifier Unit for More Electric Aircraft T. Wu, S.V.Bozhko, G.M.Asher and D.W.P Thomas School of Electrical and Electronic Engineering, University of Nottingham, Nottingham NG7 2RD, UK E-mail:
[email protected] Abstract-This paper presents a dynamic phasor model of the autotransformer rectifier unit (ATRU) for the more-electric aircraft power system study. This model considerably reduces the complexity in modeling of an aircraft power system making it more practical to model the electrical power system for transient and stability analysis. The developed phasor model of the ATRU is based on the development of a non-switching fundamental component model of the ATRU, in which the switching behavior of the 12-pulse diode rectifier is represented by a dc transformer. The developed phasor model is capable of accurately modeling both ac- and dc-side transients. The computation time demanded by this phasor model is significantly reduced compared to the benchmark model. Simulation results show that the dynamic phasor model of the ATRU is nearly 500 times faster than the corresponding benchmark model.
I.
INTRODUCTION
The more-electric aircraft (MEA) concept, where the majority of the aircraft’s secondary power needs are supplied in an electrical form, has generated a major technology roadmap in modern aviation industry [1-2]. As many functions which are traditionally operated by hydraulic, pneumatic and mechanical power will be electrified, an increased number of power electronic converters (PEC) and motor drive systems expand in the MEA power system. Since increased percentage of electrical power is processed by PECs, which are non-linear and much more dynamic, it poses new challenges in terms of maintaining the system stability and ensuring the power quality of MEA power system. The particular concern is harmonic currents are generated by variable-speed drives and their interaction with other loads of the system. To ensure system power quality, 12-pulse autotransformer rectifier unit (ATRU) is typically used for the MEA power system where low total harmonic distortion (THD) of the input line current is required [3]. Since the possible number of PEC-based loads on-board can be in order of hundreds, detailed component models of the ATRU can not be simulated within practical constraints of computing time and computer storage. Hence, the development of accurate and time-efficient model for ATRU is of great importance. An averaged-value, dc-side model of the ATRU, derived using the non-linear averaging method, has already been reported in [4]. However, the limitation of that analytical model is that it can only model the dynamic performance at the dc-side of the ATRU. The dynamic behavior of the input
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current of the ATRU can not be modeled. Therefore, that analytical model is not suitable for a complete simulation study of the MEA power system. This paper is a logical continuation of our previous study reported in [5]. The aim of this paper is to present a novel dynamic phasor model of the ATRU for the MEA power system study. The proposed phasor model of the ATRU is capable of accurately modeling both ac- and dc-side behaviors of the ATRU. The advantage offered by this dynamic phasor model is that the ATRU can be represented by two simple dc circuits (real and imaginary parts). The model is further reduced by developing a dq model of the diode rectifier which simplifies the switching to a dc transformation. It is found, from comparison with a benchmark model that, good accuracy within the specified frequency range 133Hz is preserved and it is shown that proposed approach results in model that is nearly 500 times faster than the traditional benchmark model. II.
MODELING OF THE BENCHMARK MODEL OF ATRU
The benchmark model of the ATRU, shown in Fig.1, is constructed using SABER simulation package. Fig.2 shows the basic winding structure of the transformer and the electrical interconnection of windings. Considering the symmetry of the winding structure and the same number of turns on primary coils and secondary coils, there are two different values of self inductance and five different values of mutual inductance between three primary windings and six secondary windings. The inductor model used in Saber is linear and only accounts for the magnetizing inductance. To take account the winding resistance, the resistor model is in series with the inductor in the benchmark model of the autotransformer. The transformer is modeled by the inductance matrix which is represented by nine coupled inductors as follows ⎡ v1 ⎤ ⎡ LA ⎢v ⎥ ⎢ M ⎢ 2⎥ ⎢ C ⎢ v3 ⎥ ⎢ M C ⎢ ⎥ ⎢ ⎢v4 ⎥ ⎢ M A ⎢v ⎥ = ⎢ M ⎢ 5⎥ ⎢ D ⎢ v6 ⎥ ⎢ M D ⎢v ⎥ ⎢ M ⎢ 7⎥ ⎢ A ⎢ v8 ⎥ ⎢ M D ⎢ ⎥ ⎢ ⎣ v9 ⎦ ⎣ M D
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MC
MC
MA
MD
MD
MA
MD
LA
MC
MD
MA
MD
MD
MA
MC MD
LA MD
MD LB
MD ME
MA ME
MD MB
MD ME
MA
MD
ME
LB
ME
ME
MB
MD
MA
ME
ME
LB
ME
ME
MD MA
MD MD
MB ME
ME MB
ME ME
LB ME
ME LB
MD
MA
ME
ME
MB
ME
ME
MD ⎤ M D ⎥⎥ MA⎥ ⎥ ME ⎥ d ME ⎥⋅ ⎥ dt MB ⎥ ME ⎥ ⎥ ME ⎥ ⎥ LB ⎦
⎡ i1 ⎤ ⎢i ⎥ ⎢ 2⎥ ⎢ i3 ⎥ ⎢ ⎥ ⎢i4 ⎥ ⎢i ⎥ ⎢ 5⎥ ⎢i6 ⎥ ⎢i ⎥ ⎢ 7⎥ ⎢ i8 ⎥ ⎢ ⎥ ⎣i9 ⎦
(1)
ia 3
ib 3
ia
ic 3
va
ib vb
ic
idc
va 5
va 3
vc 5
vb 3 Cf
vb 5
ia 5 ib 5
vc 3
vc
iL
ic 5 Fig.1 The circuit configuration of the ATRU
Fig.2 The physical structure and the electrical connection of the autotransformer
where LA and LB are self inductance of the primary winding and the secondary winding, respectively; MA ,MB, MC, MD, ME are mutual inductances. Assuming that the coupling factor between any two coils on the same transformer limb is k, the mutual inductances and self inductances will be related by M A = k LA LB (2) M B = k ⋅ LB
(3)
LA (4) 2 M MD = − A (5) 2 M ME = − B (6) 2 The remaining components, such as the interphase reactor and the ideal diode bridge are collected from the SABER library. M C = −k
III.
DEVELOPMENT OF NON-SWITCHING FUNDAMENTAL COMPONENT MODEL OF ATRU
Due to the existence of the diode bridge, the switching behavior is therefore included within the benchmark model of the ATRU. For the study of MEA power system, the switching behavior will result in increased computational time and storage. Therefore, this section will focus on the development of a non-switching fundamental component model of the ATRU.
A. Non-Switching Fundamental Component Model of ATRU In [6-7] the idea of the circuit DQ transformation to transform the switch set within the controlled PWM rectifier to a time-invariant transformer was first adopted. However, the equivalent dq model of the six-pulse diode rectifier is not included in [7]. Therefore, the circuit DQ transformation will be applied here to derive an equivalent dq model of the 6pulse diode rectifier and to show how to extend it to a 12pulse unit. For an ideal diode rectifier shown in Fig.3, we can define the input terminal voltage of the diode rectifier as follows vin, a = vmag ,diode sin(ωt ) (7.1) vin,b = vmag , diode sin(ωt − 2π / 3)
(7.2)
vin,c = vmag ,diode sin(ωt + 2π / 3)
(7.3)
where vmag,diode is the peak magnitude of input terminal voltage of the diode rectifier. The voltage relationship between the input and output terminal of the diode rectifier can be given by vo,diode = sa vin, a + sb vin,b + sc vin,c (8) where sa, sb and sc are switching functions. The current relationship between the input and the output terminal of the diode rectifier can be given by (9) iin, j = s j io,diode where j=a,b,c The switching function of sa, sb, and sc is depicted in Fig.3. The fundamental component of switching functions can be given by [8] 2 3 sin ωt sa = (10.1)
π
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sb =
2 3
π
sin (ωt − 2π / 3)
3⎜ ⎟ 1/ 2 1/ 2 ⎝ 1/ 2 ⎠ Using (11), equation (8) will be transformed into a dq frame expressed by vo,diode = Sd vbus ,d (12)
where Sd= 6 π 2 , vbus , d = 3vmag ,diode
2
And the current relationship of (9) can be expressed in the dq frame as ibus ,d = Sd io,diode (13) 3 imag ,diode , imag, diode is the peak magnitude of 2 the input terminal current of the diode rectifier. According to (12) and (13), the ideal diode rectifier can be transformed to a dc transformer with a turn ratio of Sd in the dq rotating reference frame. It should be noted that switching functions given by (10) are only valid when the commutation overlap angle equals to zero. However, the imperfect coupling between coils of the autotransformer will result in the existence of the leakage inductance which is connected to the input terminal of the diode rectifier. In [8] such a commutation effect associated with the leakage inductance and the use of a resistor rμ to represent the voltage drop at the dc side are explained. Fig. 4 shows a general equivalent dq model of the diode rectifier which takes into account the commutation effect associated with the leakage inductance seen by the 6-pulse diode rectifier.
where ibus ,d =
vin,a
vin,b vin,c
iin,a iin,b iin,c
+ iin, d
2 3
sin (ωt + 2π / 3) (10.3) π It can be understood from (8), (9) and (10) that the input terminal voltages, the fundamental component of switching functions and input terminal currents are in phase with each other. Therefore, the d-axis of the rotating dq reference frame is aligned with the vector of the input terminal voltage to simplify the analysis. The transformation matrix is given as follows f dq 0 = Tf abc (11) where ⎛ cos θ cos(θ − 2π / 3) cos(θ + 2π / 3) ⎞ ⎟ 2⎜ T= ⎜ − sin θ − sin(θ − 2π / 3) − sin(θ + 2π / 3) ⎟ sc =
rμ = 6 f line leq
(10.2)
io,diode +
sa
1
-1
vo,diode
−
π /6
θ
5π / 6
sb
θ
sc
θ
π Fig.3 The ideal diode rectifier and switching functions
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2π
io, diode
vbus ,d
−
1:
6
+ vo,diode −
π 2
Fig.4 Generalized dq model of the diode rectifier
The voltage relationship between the input voltage and the terminal voltage of two diode rectifiers can be given by [3] va 3 = va − vbc N AT (14.1) vb3 = vb − vca N AT (14.2) vc 3 = vc − vab N AT (14.3) va 5 = va + vbc N AT (15.1) vb5 = vb + vca N AT (15.2) vc 5 = vc + vab N AT (15.3) where NAT is turns ratio between primary and secondary windings of the autotransformer. From equations (14) and (15) the input voltages of two diode rectifiers are symmetrical but displaced from each other by a phase angle of 30°. Assuming that the total output current is equally shared between two diode rectifiers, two sets of input currents will be symmetrical too, displaced from each other by 30°. Therefore, input currents of the ATRU can be represented by the input terminal current of one diode rectifier as [3] ia = 2 ⎡⎣ia 3 + ( ib3 − ic3 ) / N AT ⎤⎦ (16.1) ib = 2 ⎡⎣ib3 + ( ic 3 − ia 3 ) / N AT ⎤⎦ ic = 2 ⎡⎣ic 3 + ( ia 3 − ib3 ) / N AT ⎤⎦
(16.2) (16.3)
Employing the symmetry of two diode rectifiers, the 12pulse diode rectifier can be modeled by an equivalent 6-pulse diode rectifier. Fig.5 shows the developed non-switching fundamental component model of ATRU. The voltage source va3, vb3 and vc3 are obtained by using (14). The 3-phase input currents of this non-switching model are functions of the input terminal current of one diode rectifier, determined by (16). The 12-pulse diode rectifier is represented by the equivalent transformer model shown in Fig.4. The equivalent winding resistance and the leakage inductance seen by one diode rectifier are labeled as Req and Leq, respectively. The value of Req can be given by ⎞ 1 ⎛ RLp (17) Req = ⎜ 2 + RLs ⎟ ⎜ ⎟ 2 ⎝ N AT ⎠ The value of Leq can be given by ⎞ 1 ⎛ lp (18) Leq = ⎜ 2 + ls ⎟ ⎜ ⎟ 2 ⎝ N AT ⎠ where lp and ls are the primary and secondary leakage inductances of the autotransformer. It should be noted that the factor of 1/2 appears in (17) and (18) because two diode rectifiers are in parallel operation.
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The interface between three-phase R-L circuit and the dq model of 12-pulse diode rectifier is modeled by a set of controlled voltage and current sources in Fig. 5, which can be given by 0 0] = T ⎡⎣ vbus , a T
ib 3 Req
vbus ,c ⎤⎦
vbus ,b
ic 3 ] = 0.5T−1 [id 1
T
20
(19)
T
(20)
-60 0.015
vbus , a
ia 3 −
Req
+ v − b3
ib
Leq
idc , ATRU
vbus ,b −
Req
ib 3
+ v − c3
ic
+ − v d1
0.019
0.02 0.021 Time(sec)
0.022
0.023
0.024
0.025
0.018
0.019
0.02 0.021 Time(sec)
0.022
0.023
0.024
0.025
540 535
Cf
530 0.015
iL
vbus ,c
1:
Benchmark Fundemental 0.016
0.017
(a) step change of the load current
6
π 2 100
ic 3 − ia(A)
B. Validation of Fundamental Component Model of ATRU The effectiveness of the fundamental component model of the ATRU is verified though a comparison with the corresponding benchmark model. The parameters of the benchmark model are given in appendix. Both the input current at phase A and the dc-link voltage of the ATRU from two models are shown overlaid in Fig. 6. Fig. 6(a) shows transients with the step change of the load current from 4A to 40A. Fig. 6(b) shows transients with the variation of the amplitude of the input voltage from 230VRMS to 180VRMS. Another transient with the step change of the frequency of the input voltage from 400Hz to 800Hz is shown in Fig. 6(c). As seen in Fig.6, the developed fundamental component model of the ATRU can accurately model both input current at the ac-side of the ATRU and dc-link voltage. The excellent agreement between two models confirms the accuracy of the developed fundamental component model.
0 -50 -100 0.015
0.016
0.017
0.018
0.019
0.02 0.021 Time(sec)
0.022
550
0.023
0.024
0.025
Benchmark Fundamental
Vdc(V)
500
450
400 0.015
0.016
0.017
0.018
0.019
0.02 0.021 Time(sec)
0.022
0.023
0.024
0.025
(b) step change of the amplitude of the input voltage 100 Benchmark Fundamental
ia(A)
50
DYNAMIC PHASOR MODEL OF ATRU
0 -50
This section will focus on the development of the dynamic phasor model of the ATRU. The concept and the key properties of the dynamic phasor can be found in [10-11]. The purpose of the extension of the dynamic phasor to the ATRU is to develop a computationally efficient model without existence of any ac variables.
A. Model Development The development of the dynamic phasor model of the ATRU is built on the basis of the non-switching fundamental component model of the ATRU shown in Fig.5. Since threephase R-L circuits are connected to the equivalent transformer model of 12-pulse diode rectifier, the dynamic phasor model of the ATRU can be obtained by combing the dynamic phasor model of the R-L circuit and the transformer model of 12-pulse diode rectifier together. The dynamic phasor model of the R-L circuit has been reported in [11]. Section III has derived the voltage and current relationship
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Benchmark Fundamental
50
Fig.5 The non-switching fundamental component model of ATRU
IV.
0.018
545
Leq +
vc
0.017
550
rμ = 6 fline Leq
id 1
+ vb
0.016
555
Vdc(V)
+ v a3 −
ia
Benchmark Fundemental
Leq +
va
0 -20 -40
0 0]
T
40
ia(A)
[vd 1 [ia3
60
-100 0.015
0.016
0.017
0.018
0.019
0.02 0.021 Time(sec)
0.022
538
0.024
0.025
Benchmark Fundamental
536 Vdc(V)
0.023
534 532 530 528 0.015
0.016
0.017
0.018
0.019
0.02 0.021 Time(sec)
0.022
0.023
0.024
0.025
(c) step change of the frequency of the frequency Fig. 6 Comparison of the benchmark model and the non-switching fundamental component model
of the equivalent transformer model, which can be written by 2 2 3 3 R I vdc , ATRU = vbus + vbus − 6 fline Leq idc, ATRU (21)
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π
( ) ( ) (
)
imag ,a 3 = R
where vbus and
3
π
(22)
idc, ATRU
I are the real and imaginary parts of the vbus
controlled voltage source (vbus), respectively; imag,a3 is the peak magnitude of the controlled current source (ia3, ib3, ic3). As mentioned before, the input terminal voltage and current of one diode rectifier are in phase with each other with the assumption that switching harmonics can be ignored. This property can be taken advantage of to obtain the real and imaginary parts of the current flowing through the dynamic phasor model of the leakage inductor (R-L circuit). As shown in Fig. 5, the angle of the current phasor (ia3, ib3, ic3) can be given by ⎛ vI ⎞ θia 3 = tan −1 ⎜ bus (23) ⎜ v R ⎟⎟ ⎝ bus ⎠ Hence, the real and imaginary part of the current phasor (ia3, ib3, ic3) can be given by iaR3 = imag ,a 3 cos (θia 3 ) (24.1) iaI 3 = imag , a 3 sin (θia 3 )
(24.2)
It can be understood from (14) that the relationship between the input voltage va and the terminal voltage va3 can be given by [3] o
o
va 3 = 1 + tan 2 15o va e j15 = 1.035va e j15 (25) As seen in Fig.5, the circuit branch of the leakage inductor is connected to the terminal voltage va3, rather than the input voltage va. Thus, the real and imaginary part of input voltages feeding the dynamic phasor model of the leakage inductor branch should be vaR3 = 1.035 (vaR ) 2 + (vaI ) 2 cos (θ va + Δθ )
(26.1)
vaI 3 = 1.035 (vaR ) 2 + (vaI ) 2 sin (θ va + Δθ )
(26.2) represent real and imaginary parts of the input voltage vector of the ATRU, respectively, θva = tan −1 va I va R , Δθ = 15π 180 . where vaR and vaI
(
)
According to (16), the relationship between the input current ia and the current ia3 can be given by ia = 2 1 + tan 2 15o ia 3e j −15 = 2.0711ia 3e j −15 (27) As seen in (27), the peak magnitude of the input current is 2.0711 times as big as that of the current ia3. And the input current ia lags the current ia3 by 15 degree. Hence, the real and imaginary part of three-phase input current can be given by o
( ) + (i )
iaR = 2.0711 iaR3
2
I a3
2
( ) + (i ) 2
2
o
(
)
(
)
cos ⎡ tan −1 iaI 3 iaR3 − 15π 180 ⎤ ⎣ ⎦ (28.1)
sin ⎡ tan −1 iaI 3 iaR3 − 15π 180 ⎤ ⎣ ⎦ (28.2) Fig.7 shows the developed dynamic phasor model of the ATRU in which controlled voltage and current sources are determined by the equations derived above. As seen in Fig.7, all signals in the dynamic phasor model have dc characteristic. iaI = 2.0711 iaR3
I a3
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Compare with the fundamental component model in Fig.5, ac variables are all eliminated. Therefore, dc functionalization has been achieved by extending the concept of the dynamic phasor into the ATRU. real part
+ R − va
iaR
Req
+ −
Leq
−ωr Leq iaI 3 +−
iaR3
+ I − va
iaI
Req
+ −
vaI 3
+
idc, ATRU
R vbus
vaR3
−
imaginary part
6 fline Leq
+
Leq iaI 3
ωr Leq iaR3 +−
+
+ −
3 3
π
(v ) + (v ) R bus
2
I bus
vdc, ATRU 2
iL
I vbus
−
−
Fig.7. The full dynamic phasor model of the ATRU
B. Model Validation The developed dynamic phasor model is verified through a comparison with the benchmark model of the ATRU loaded with an ideal current source. The circuit parameters are the same as that listed in the Table 1. Simulation results of the peak magnitude of the input current and the dc-link voltage of the ATRU from two models are shown in Fig. 8. Fig. 8 (a) compares the transient when the load current is suddenly increased from 4A to 40A. Due to the existence of harmonics within the input current, the waveform of the peak magnitude of the input current contains the ripple associated with harmonics. The result of the phasor model of ATRU can accurately reproduces the transient of the peak magnitude of the input current and the dc-link voltage of ATRU. Fig. 8 (b) shows another transient with the variation of the peak magnitude of the input voltage from 230V RMS to180VRMS. Fig. 8 (c) shows the transient with the step change of the frequency of three-phase input voltages from 400Hz to 800Hz. C. Discussion of the Computation Time The computation time of the benchmark model and the developed phasor model of the ATRU are compared in Table 1. The computation time is obtained with different truncation errors used in the simulation of 0.1-second steady state operation. The recommend value for the setting of the truncation error is 0.001m to obtain accurate results for the benchmark simulation. The truncation error can be reduced to speed up the simulation with the price of accuracy. However, good simulation results (i.e. when accuracy meets the specified 5% compared to the benchmark model) can be obtained for the truncation error up to 1m for the functional model. Thus, using Table 1, one can assess that the functional model can run 13.6/0.0286=476 times faster than the benchmark model. V.
CONCLUSION
The modeling methodology of the benchmark of the ATRU has been presented. The derivation of a dynamic phasor model of the ATRU is also given in detail. The main advantages offered by this phasor model are that : The computation time is 476 times faster than the corresponding benchmark model of the ATRU; direct
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modeling of the switching behavior is avoided in the developed phasor model and DC functionalization has been achieved by extending the concept of the dynamic phasor into the ATRU; steady-state and dynamic transients both at ac side and dc side of the ATRU are well preserved; the phasor model of the ATRU is suitable for the simulation study of a more complete representation of the power system for the large more electric aircraft. Table 1 Comparison of the computation time Truncation Benchmark Phasor error Model (s) Model (s) 1m Fails 0.0286 0.1m 4.23 0.0312 0.01m 7.53 0.0469 0.001m 13.6 0.0625 0.0001m 23 0.093
This research is being conducted in the frame of the MOET project (More-Open Electrical Technologies), a FP6 European Integrated Project. (www.moetproject.eu) 60 50 ia mag(A)
40 30 20 10
Benchmark Phasor 0.016
0.017
0.018
0.019
0.02 0.021 Time (sec)
0.022
0.023
0.024
0.025
555 Benchmark Phasor
Vdc (V)
550 545 540 535 530 0.015
0.016
0.017
0.018
0.019
0.02 0.021 Time (sec)
0.022
0.023
0.024
0.025
(a) step change of the load current 60
ia mag(A)
50 40 30 20 Benchmark Phasor
10 0 0.015
0.016
0.017
0.018
0.019
0.02 0.021 Time (sec)
0.022
0.023
550
vdc (V)
ia mag(A)
50 40 30 20 0.015
Benchmark Phasor 0.016
0.017
0.018
0.019
0.02 0.021 Time (sec)
0.022
0.023
538
0.025
Benchmark Phasor
536 vdc (V)
0.024
534 532 530 528 0.015
0.016
0.017
0.018
0.019
0.02 0.021 Time (sec)
0.022
0.023
0.024
0.025
(c) step change of the frequency of the input voltage Fig. 8 Comparison of the benchmark model and the dynamic phasor model
ACKNOWLEDGEMENTS
0.015
60
0.024
0.025
Benchmark Phasor
500
REFERENCES [1]
J. Weimer, “Electrical power technology for the more electric aircraft,” in Conference Proceedings of IEEE DASC’ 1993, vol. 3, pp.445-450, 1993 [2] K.J. Karimi, “The role of power electronics in more-electric airplanes (MEA),” presentation at 2006 IEEE Workshop on Computers in Power Electronics, July 2006 [3] S. Choi, P.N. Enjeti and I.J. Pitel, “Polyphase transformer arrangements with reduced kva capacities for harmonics current reduction in rectifiertype utility interface,” IEEE Trans. on Power Electronics, vol. 11, no. 5, pp. 680-690, Sept. 1996 [4] A. Baghramian and A.J. Forsyth, “Averaged-value models of twelvepulse rectifiers for aerospace applications,” PEMD’ 2004, vol. 1, pp.220-225, Mar. 2004 [5] T.Wu, S.V, Bozhko, G.M. Asher and P.W. Wheeler, “Fast Reduced Functional Models of Electromechanical Actuators for More-Electric Aircraft Power System Study,” in Proceedings of 2008 SAE Power System Conference, (Paper #2008-01-2859) [6] C.T. Rim, D.Y. Hu, and G.H. Cho, “Transformers as equivalent circuits for switches: general proofs and d-q transformation-based analyses,” IEEE Trans. on Industry Applications, vol. 26, no. 4, pp. 777-785., July/August 1990 [7] C.T. Rim, N.S. Choi, G.C. Cho, and G.H. Cho, “A complete dc and ac analysis of three-phase controlled-current pwm rectifier using circuit dq transformation,” IEEE Trans. on Power Electronics, vol. 9, no. 4, pp. 390-396, July 1994 [8] M. Sakui, H. Fujita, and M. Shioya, “A method for calculating harmonic currents of a three-phase bridge uncontrolled rectifier with dc filter,” IEEE Trans. on Industrial Electronics, vol. 36, no. 3, pp. 434440, August 1989 [9] N. Mohan, T.M. Underland, and W.P. Robbins, Power Electronics: Converters, Applications, and Design, John Wiley & Son, USA, 2003, pp. 106-108 [10] S.R. Sanders, J.M. Noworolski, X.Z. Liu and G.C. Verghese, “Generalized averaging method for power conversion circuits,” IEEE Trans. on Power Electronics, vol. 6, pp. 251-259, April 1991 [11] Z.J. E, K. W. Chan and D.Z. Fang, “A practical dynamic phasor model of static var compensator,” ICPESA’ 2006, vol. 3, pp.23-27, Nov. 2006
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APPENDIX
400 0.015
0.016
0.017
0.018
0.019
0.02 0.021 Time (sec)
0.022
0.023
0.024
(b) step change of the amplitude of the input voltage
0.025
Primary wining inductance and resistance: LA =1.273H, Rlp =1Ω, Secondary wining inductance and resistance: LB =0.03046H, Rls =0.36Ω, Dc-link capacitance: Cf=600μF, Turns ratio: NAT=6.464
Coupling factor: k=0.998
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