modeling of active front-end rectifier using dynamic phasors concept
TRANSCRIPT
Abstract—This paper extends the dynamic phasor approach to modeling pulse modulation (PWM) active front-end rectifier for more electric aircraft (MEA) power system study. The higher order harmonics due to switching behaviour are neglected for accelerating simulation speed. The accuracy and effectiveness of dynamic phasor model have been revealed by comparing with time-domain model in both ABC and DQ0 frame under both balanced and unbalanced condition.
Keywords - Active rectifier; aircraft power system; controlled rectifier unit; dynamic phasor; unbalanced system
I. INTRODUCTION he More Electric Aircraft (MEA) has recently became feasible due to the advances in power electronics, motor
drives and control systems. The aim is to replace traditional mechanical, hydraulic and pneumatic power systems with a larger electrical power system (EPS) offering reduced overall system weight, as well as increased reliability, efficiency and performance of the aircraft [1],[2].
The MEA concept results in a number of potential electrical power system (EPS) architectures, such as AC fixed frequency, AC variable frequency, DC or hybrid systems. The EPS architecture development requires extensive simulations of the system, under both normal and faulty operating conditions, to assess the power quality, transient behaviour and to address the system protection and availability issues. The loads of an EPS in MEA are normally driven by power electronic converters, such as electromechanical (EMA) or electrohydrostatic (EHA) actuators for flight surface actuation, wing de-icing systems, landing gear systems and flight control systems. The simulations based on the detailed traditional models of such loads would be complex, impractical and computationally intensive.
The primary challenge for large-scale power system simulation is to balance the simulation speed and model accuracy. The functional modelling techniques based on the state-space vector representations of three-phase quantities for MEA applications presented in [3-5], offer improved performance and a reduction in simulation time, with minimal loss of accuracy during the balanced conditions, however, for unbalanced and faulty conditions, this technique loses its time efficiency due to the introduction of additional harmonic content [6].
Dynamic phasor approach is based on the time-varying coefficients of dominant terms in the Fourier series. This approach provides a middle ground between the sinusoidal quasi-steady-state representation and the time-domain representation, for electric power system modelling [7]. Truncating the unimportant higher order harmonic components and only considering the significant components, the dynamic phasor model is capable of retaining the dominant dynamic features of the EPS and is suitable for transient stability study. The slow variation of the dynamic phasor allows for larger simulation time steps, resulting in faster simulations.
The dynamic phasor method has been used successfully to analyze synchronous machines, induction machines [8], doubly-fed induction generator (DFIG) [9], flexible AC transmission system (FACTS) devices such as USSC, and HVDC in balanced operating conditions [10], [11]. When operating in the unbalanced condition the dynamic phasor has been implemented with positive and negative sequences theory and the models for synchronous generators, DFIGs and transmission lines have been established [12], [13]. A fault injector was introduced to cause the unbalanced condition and the efficiency of dynamic phasor models has been revealed through comparing with time-domain models in both ABC and DQ0 frames [14].
This paper presents a method which extends the previous works on dynamic phasor modeling and applies this method to a controlled rectifier unit (CRU). The developed dynamic phasor model is compared with CRU time-domain model in both ABC and DQ0 frames under both balanced and unbalanced conditions. The comparison of computation time between the three models has shown the significant time efficiency of the dynamic phasor model.
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Modeling of Active Front-End Rectifier Using Dynamic Phasors Concept
Tao Yang, Elisabetta Lavopa, Serhiy Bozhko, Greg Asher University of Nottingham, NG7 2RD, United Kingdom
T
Fig.1. PWM active filter simulation scheme
II. DYNAMIC PHASORS The generalized averaging method used in this paper is based on the fact that the time-domain waveform x(t) can be represented in the interval [t-T, t] using a Fourier series of the form [15].
tjkw
kk
setXtx ∑∞
−∞== )()( (1)
where ωs=2π/T. Xk(t) is the kth Fourier coefficient in complex form, also called the dynamic phasor. These Fourier coefficients are time-varying as the window slides as a function of time. The kth coefficient is determined as follows,
k
t
Tt
tjkwk xdtetx
TtX s == ∫
−
−)(1)( (2)
where k is the dynamic phasor index and the selected set of k will define the accuracy of the approximation. In particular, for constant signals the index set K={0} and for sinusoidal signals, K={1}. When the original x(t) is real, the phasor X-k(t) equals the conjugate of Xk(t) and expressed as . One important property for our development is the derivative of kth Fourier coefficients, given by equation below.
ksk
k xjkdtdx
dtxd
ω−= (3)
This formula can be verified using (1), (2) and will be used in evaluating the kth phasor of time-domain model dx/dt on the right hand side. Another key fact is that the dynamic phasor of product of two time-domain variables equals the convolution of corresponding phasor of each component i.e.:
∑ −=
iikkk
yxxy (4)
The above two equations (3) and (4) play key roles when transforming the time-domain models to dynamic phasor models.
III. ACTIVE RECTIFIER IN ABC AND DQ0 DOMAINS A typical structure of active front-end converter used as a controlled rectifier unit (CRU) in the AC bus fed EMA is well documented in previous publications [16] as shown in Fig.1. The d-axis in synchronous frame is aligned with voltage vector, expressed in (5).
( )ππ 3/43/2
32 j
cj
ba evevvv ++= (5)
The voltage vector angle θ here is derived from synchronous reference frame phasor locked loop (SRF-PLL)[17]. In the SRF-PLL, the voltage vector is translated from ABC natural reference frame to DQ0 synchronously rotating frame. The angular position of the DQ0 frame is controlled by a feedback loop which regulates q component to zero. Therefore, under steady state conditions, the voltage vector will be aligned with the d axis. Referring current vector to this synchronized frame, the current components in the d and q axes are responsible for active and reactive power respectively. The unity power factor is achieved by regulating the q axis current component to zero. The three
phase model for the CRU is built in Dymola/Modelica v7.4. The exploited DQ0 model for CRU has been considered in details [18].
IV. DYNAMIC PHASOR MODEL In this section, the three-phase dynamic phasor model (DP-ABC) for the CRU will be developed. The main idea is to translate all the variables in Fig.1 from time domain into dynamic phasors. The non-switching model for the power converter has been used in DP-ABC model. The whole system shown in Fig.1 can be divided into several segments and the dynamic phasor models will be developed separately.
A. Electrical segment The non-switching model for the controlled converter in time domain is given by equations (6) and (7).
jdcj mVv21= (6)
∑=j
jjdc imi21 (7)
Where Vdc is the dc-link voltage, mj is the modulation index of phase j(j=a,b,c), ij is the ac phase current into the converter. Transforming equations (6) and (7) to dynamic phasor with the properties shown in (3) and (4) yields:
kjdckj mVv02
1= (8)
∑ −=
jkjkjdc imi
21
0 (9)
Here, only the fundamental component is considered on the ac side, so the dynamic phasor index is chosen as K={1}. For dc side variables, only the dc component is considered, so K={0}. The dynamic phasor models for other electrical elements were well developed in previous work [14] and will not be detailed here.
B. Control segment
1) PI controller The state-space equation for the PI controller is:
,ukx i= (10.a)
xuky p += (10.b)
Where u is the input of the PI controller; kp and ki are the proportional and integral gains of the PI controller. The dynamic phasor model of the PI controller shown below:
kikk ukxjk
dtxd
=+ ω (11.a)
kkpkxuky += (11.b)
2) Voltage and Current Vector In the ABC model of CRU, the angle of voltage vector is derived from a SRF-PLL. In this session, the relation
between the voltage vector and its dynamic phasors will be given. In addition, the dynamics of the SRF-PLL in ABC model will be neglected in the dynamic phasor model, which means the angle is directly calculated using the real and imaginary parts of the vector. Considering a general three-phase AC voltage sources with arbitrary magnitude and phase shift as
cbaitVv iii ,, ),cos( =+= ϕω (12)
With the dynamic phasor definition in the equation (2) and the Euler’s Formula below
2cos
tjtj eetωω
ω−+= (13)
The dynamic phasors for the voltage above can be derived as follows:
cbaieVv ijii ,, ,
21
1== ϕ (14)
Combining the equations (5), (12)-(14) yields the relation between voltage vector and the dynamic phasors:
( )
( )3/2*
13/2*
1
*
1
3/21
3/211
3232
ππω
ππω
jc
jba
tj
jc
jba
tj
evevve
evevvev
−−
−
+++
++= (15)
The first item on the right side of the equation (15) gives the positive sequence vector, and the second shows the negative sequence vector. The negative sequence will only appear in the unbalanced situation. In the static αβ frame, the voltage vector in the equation (15) can be express as
βα vjvv ⋅+= (16)
The rotating frame is continually leading this static αβ frame by an angle θ and assuming θ =ωt. Thus the voltage vector in a synchronous rotating frame can be expressed as . Multiplying e-jωt on both side of the equation (15) and rewriting the first and second terms on the right side, yields:
( ) ( )222
00 qdtj
qdtj jVVejVVev +++=⋅ −− ωω (17)
Where
( )⎥⎦⎤
⎢⎣⎡ ++ℜ= − 3/2
13/2
110 32 ππ j
cj
bad evevveV (18.a)
( )⎥⎦⎤
⎢⎣⎡ ++ℑ= − 3/2
13/2
110 32 ππ j
cj
baq evevvmV (18.b)
( )⎥⎦⎤
⎢⎣⎡ ++ℜ= − 3/2*
13/2*
1
*
12 32 ππ j
cj
bad evevveV (18.c)
( )⎥⎦⎤
⎢⎣⎡ ++ℑ= − 3/2*
13/2*
1
*
12 32 ππ j
cj
baq evevvmV (18.d)
Recalling the equation (15), it can be seen that in the equation (18), the variables Vd0 and Vq0 are from the positive sequence vector, and the Vd2 and Vq2 are from the negative sequence vector. Collecting the real part and imaginary part of (17) gives:
( )( )tVtVVj
tVtVVev
dqq
qddtj
ωωωωω
2sin2cos
2sin2cos
220
220
−++
++=⋅ − (19)
Please be noted that the d axis of this synchronous rotating frame, denoted as ds-qs frame, is not aligned with the voltage vector but lag it by an angle φ. Choosing this frame makes it more convenient to derive the dynamic phasors v
d,q k, and that will be shown later. The real and imaginary parts of the vector in this ds-qs frame are denoted as vd and vq. The voltage vector in ds-qs frame thus can be expressed as:
qdtj vjvev ⋅+=⋅ − ω (20)
Combining the equations (19) and (20), It can be concluded that during the unbalanced condition, as Vd2 and Vq2 are not equal to zero, the d and q components of the voltage vector in ds-qs synchronous frame, i.e. vd and vq, include both DC and the 2nd harmonic components, which means K={0,2} for dynamic phasor v
d,q k. In addition, According to the equation (18), the variables Vd0, Vq0, Vd2 and Vq2 can all be calculated from v
j , , , . Thus the dynamic phasors , for vd and vq in the equation (20) can also be derived from vj , , , , as shown in table I. The equations (12)-(20)
can also be used to derive the dynamic phasor of the current
vector i and its components in ds-qs frame are denoted as id and iq respectively.
TABLE I dynamic phasors for voltage vector in rotating frame
Variable Dynamic phasor {k=0}
Dynamic phasor {k=2}
vd Vd0 0.5Vd2-0.5jVq2
id Id0 0.5Id2-0.5jIq2 vq Vq0 0.5Vq2+0.5jVd2
iq Iq0 0.5Iq2+0.5jId2
3) Active and Reactive Current Component The active power and reactive power control in the time
domain CRU model is achieved by decomposition of the current vector along the voltage vector and then controls the d component (idv) and q component (iqv) of current vector separately. When referring the current vector to the voltage vector as shown in Fig. 3, the current components idv and iqv can be expressed as:
ϕϕ sincos qddv iii += (21)
ϕϕ cossin qdqv iii +−= (22)
where
221 ),(cosqd
dqd
vvvvvf+
==ϕ (23)
222 ),(sinqd
qqd
vv
vvvf
+==ϕ (24)
dvi
ϕ ϕ
qvidi
qi
v
iqv
dv
Fig.3. Referring the current vector to voltage vector in synchronous DQ0
frame. Before transforming the equations (21) and (22) into frequency domain, the dynamic phasors for sinφ and cosφ should be derived firstly and will be discussed below. In the balanced situation, as there is no 2nd harmonics in both d and q axes, the functions cosφ and sinφ will keep at some constant values as vd=Vd0 and vq=Vq0 in the equations (23) and (24), which means K={0} for and
. In the unbalanced situation, since the 2nd harmonics are introduced into the variables vd and vq, the functions cosφ and sinφ will be distorted and some higher harmonics will be involved. In this case, the dynamic phasor index k for
and need to be carefully selected. As have discussed, the index set for the dynamic phasors , and , is chosen as K={0,2}, it would be a good idea to select the index for and , to be the same as that for , and , . It’s also worth to notice that, since the index set for , is K={0,2} and according to the equation (23) and (24), the index for the dynamic phasors and can only be the even numbers, i.e. K={0,2,4,…,2n, n is positive integer}. The higher harmonics could be included in the dynamic phasor model, however, by doing that will only trivially improve accuracy while increase the complexity of the model, the dynamic phasor sets for and are both chosen as K={0,2}, The nonlinearity of sinusoidal function, shown in the equation (23) and (24), makes the direct calculation of
and very difficult. In order to use the convolution property given by (4), the Taylor expansion has been proposed in this paper. By choosing the work point at Vd0 and Vq0, the Taylor expansions of equation (18) and (19) are:
( )
( ) ( )
( ) ( )( )00001
22
02001
2
202
0012
0001
0001
001
),( ),( 21
),( 21),(
),( ),(cos
qqddqd
qdqq
q
qd
ddd
qdqq
q
qd
ddd
qdqd
VvVvvv
VVfVv
vVVf
Vvv
VVfVv
vVVf
Vvv
VVf VVf
−−∂∂
∂+−
∂∂
+
−∂
∂+−
∂∂
+
−∂
∂+≈ϕ (25)
( )
( ) ( )
( ) ( )( )00002
22
02002
2
202
0022
0002
0002
002
),( ),( 21
),( 21),(
),( ),(sin
qqddqd
qdqq
q
qd
ddd
qdqq
q
qd
ddd
qdqd
VvVvvv
VVfVv
vVVf
Vvv
VVfVv
vVVf
Vvv
VVf VVf
−−∂∂
∂+−
∂∂
+
−∂
∂+−
∂∂
+
−∂
∂+≈ϕ (26)
Transforming equations (25) and (26) into dynamic phasor and choosing the phasor index set K={0,2} yields:
22001
2
22001
2
222001
2
222001
2
0010
),(
),( ),(
),( ),(cos
qdqd
qd
qdqd
qdqq
q
qd
ddd
qdqd
vvvv
VVf
vvvv
VVfvv
vVVf
vvv
VVf VVf
−
−−
−
∂∂∂
+
∂∂∂
+∂
∂+
∂∂
+=ϕ (27a)
2
0012
0012
),( ),( cos q
q
qdd
d
qd vv
VVfv
vVVf
∂
∂+
∂∂
=ϕ (27b)
22002
2
22002
2
222002
2
222002
2
0020
),(
),( ),(
),( ),(sin
qdqd
qd
qdqd
qdqq
q
qd
ddd
qdqd
vvvv
VVf
vvvv
VVfvv
vVVf
vvv
VVf VVf
−
−−
−
∂∂∂
+
∂∂∂
+∂
∂+
∂∂
+=ϕ (28a)
2
0022
0022
),( ),( sin q
q
qdd
d
qd vv
VVfv
vVVf
∂
∂+
∂∂
=ϕ (28b)
With the equations (27) and (28), we can transform the equations (21) and (22) into dynamic phasors
222200
2222000
sinsinsin
coscoscos
−−
−−
+++
++=
ϕϕϕ
ϕϕϕ
qqq
ddddv
iii
iiii (29a)
0220
02202
sinsin
coscos
ϕϕ
ϕϕ
dddv
ii
iii
++
+= (29b)
222200
2222000
coscoscos
sinsinsin
−−
−−
+++
−−−=
ϕϕϕ
ϕϕϕ
qqq
dddqv
iii
iiii (30a)
0220
02202
coscos
sinsin
ϕϕ
ϕϕ
ddqv
ii
iii
++
−−= (30b)
4) Modulation index In the controller, the modulation index mabc in abc frame is
transformed from index mdq in ds-qs frame. The relation between mdq to mabc is
[ ] [ ]TqdsT
cba mmKmmm = (31)
where the transformation matrix Ks from ds-qs frame to abc frame is
⎥⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢⎢
⎣
⎡
++−++
−+−−+
+−+
=
)32sin( )
32cos(
)32sin( )
32cos(
)sin( )cos(
πϕωπϕω
πϕωπϕω
ϕωϕω
tt
tt
tt
Ks
(32)
In the unbalanced situation, as has mentioned before, the voltage vector angle φ in ds-qs frame is distorted. In order to transform the equation (32) to dynamic phasor, the dynamic phasors of Ks should be derived firstly. Expanding all the elements in the matrix and combining the equations (27) and (28), the dynamic phasors of Ks can be derived. The dynamic phasor for cosωt and sinωt are constant and shown in following table.
TABLE II DYNAMIC PHASOR FOR SINUSOIDAL FUNCTIONS
cosωt 5.0coscos11
==−
tt ωω
sinωt jt 5.0sin1
−=ω , jt 5.0sin1
=−
ω
The dynamic phasors for other elements in the matrix Ks can be calculated in the same way. Using the properties (3) and (4), the dynamic phasor transformation of equation (31) thus can be written below,
[ ] [ ] [ ]21011
Tqds
Tqds
Tcba mmKmmKmmm
−+= (36)
The sum of all the above leads to the full transformation of CRU from ABC model to DP-ABC model and the structure is shown as following,
1,, cbav1av 1bv
1cv2,0
sinϕ
2,0cosϕ
)1/(1 +sτ
÷
÷
2,0cosϕ 2,0
sinϕ
*
0dvi
0dvi
2dvi
0*
2=dvi
0qvi
0*
0=qvi
0*
2=qvi
2qvi
2,0dm
2,0qm
1,, cbai
dcV2,0dvi
1am
1bm
1cm
1,, cbam
Fig.4. Controlling scheme for CRU in DP-ABC model
V. MODEL VALIDATION
This section will assess the performance and effectiveness of the dynamic phasor model. The simulation scheme is shown in Fig.1. Three modelling techniques, i.e. ABC, DQ0 and DP-ABC, are compared in both balanced and unbalanced condition. The consumed CPU time has been taken to evaluate the computation efficiency of these modelling techniques. The evaluation of simulation accuracy has been done by comparing plots of the AC currents flowing into active rectifier and voltage at dc-link side in the figures.
A. Balanced condition
In the balanced conditions, the ideal three-phase voltage source are balanced with phase voltage Vrms=115Vol and f=400Hz. The dc-link voltage reference has been set at Vdc=580Vol. The system transient dynamics have been studied through the impact of load current from 5A to 10A at
t=0.2s. Fig.5 below shows the dc-link voltage response and Fig.6 shows the AC terminal currents flowing in to CRU. It can be seen that results from different modelling techniques match very well.
Fig.5 Vdc comparison between different modelling techniques under
balanced situation
Fig.6. AC current comparison between different modelling techniques under
balanced situation
The computation time consumed by each model has been detailed in Table III. As can be seen, DP-ABC is the most time-efficient modelling technique compared with ABC and DQ0 models.
TABLE III COMPUTATION TIME COMPASION UNDER BALANCED OPERATION
Model ABC
DQ0 DP-ABC CPU Time(s) 10.031 0.202 0.125
B. Unbalanced condition To illustrate the performance of different models under
unbalanced operation, the voltage source in phase B has been set to zero, i.e. Vb≡0, with the step change in load current source from 5A to 10A occurring at t=0.2s. The dc-link voltage and AC current flowing into CRU have been compared and a very good agreement among different modelling methods can be seen in Fig.7 and Fig.8.
0.19 0.2 0.21 0.22 0.23 0.24 0.25572
574
576
578
580
582
Time(s)
Vdc
(Vol
)
ABC DQ0 DP-ABC
0.195 0.2 0.205 0.21 0.215 0.22-50
0
50
Ia(A
)
0.195 0.2 0.205 0.21 0.215 0.22-50
0
50
Ib(A
)
0.195 0.2 0.205 0.21 0.215 0.22-50
0
50
Time(s)
Ic(A
)ABC DQ0 DP-ABC
Fig.7 Vdc comparison between different modelling techniques under
unbalanced situation
Fig.8. AC current comparison between different modelling techniques under
unbalanced situation The CPU time consumed is compared in table IV and it’s
can be seen that the DP-ABC model is the fastest model and hold roughly the same simulation speed as it is in balanced condition. This is due to the slowly varying or even time-invariant of the dynamic phasors in both balanced and unbalanced conditions, which makes the big simulation steps available and thus accelerates the simulation speed.
TABLE IV COMPUTATION TIME COMPARISON UNDER UNBALANCED OPERATION
Model ABC
DQ0 DP-ABC CPU Time(s) 16.901 0.484 0.171
VI. CONCLUSION
In this paper the dynamic phasor technique was successfully applied for modeling a PWM-based active front-end rectifier of CRU. The developed model shows very good performance under both balanced and unbalanced operation. Compared to well-established ABC- and DQ0-domain models the reported model is much more time-efficient for unbalanced conditions. The developed CRU model has been added to the dynamic phasor based
modeling library for the studies of future EMA EPS architectures. A test rig for models validation is currently under construction and will be subject of our future publications.
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0.19 0.2 0.21 0.22 0.23 0.24 0.25565
570
575
580
585
Time(s)
Vdc
(Vol
)
ABC DQ0 DP-ABC
0.19 0.195 0.2 0.205 0.21 0.215 0.22-60
0
60
Ia(A
)
0.19 0.195 0.2 0.205 0.21 0.215 0.22-60
0
60
Ib(A
)
0.19 0.195 0.2 0.205 0.21 0.215 0.22-60
0
60
Time(s)
Ic(A
)
ABC DQ0 DP-ABC