Abstract - Modeling and control of the three-phase PWM boost rectifier is presented. Current controller in rotating coordinates assures unity input power factor ...
Control of Front-End Three-phase Boost Rectifier' Silva Hiti and Dushan Boroyevich Virginia Power Electronics Center The Bradley Department of Electrical Engineering Virginia Polytechnic Inst it 11t e and State University Blacksburg, Virginia 24061
Abstract - Modeling and control of the three-phase PWM boost rectifier is presented. Current controller in rotating coordinates assures unity input power factor and fast output voltage regulation. Sensitivity of stability margins to different loads is analyzed. Two typical loads, a dc-dc converter and an inverter supplying the ac motor, are considered. Linear and nonlinear current controllers i n the direct axis are compared. With the nonlinear controller, which employs the load current feedforward, the rectifier sensitivity to load variations is greatly reduced and single optimal output voltage compensator can be designed for all load conditions. Small-signal output impedance is significantly improved, and faster transient response is obtained. The results are illustrated with simulations of a practical 10 kW rectifier.
1. INTRODUCTION
Three-phase PWM converters are increasingly being considered for use as a front-end power processing unit with power factor correction at higher power levels. The three-phase boost rectifier with closed loop control of the input currents features sinusoidal input currents, adjustable input displacement factor and dc output voltage with no low frequency harmonics [I]. In most cases a front-end converter is followed either by a dc-dc converter providing postregulation of the dc link voltage, or by a three-phase PWM voltage source inverter (VSI) supplying an ac motor [ 1,2] The load is usually connected directly to the rectifier. without an intermediate dc link filter. In this case. loading effect on the front-end convai-ter can be considerable, and the system stability margins and performance may vary significantly with the load type If the conventional current-mode control is used. the control-to-output transfer functions change by a large amount with load variations A voltage compensator designed to provide stability for all load conditions cannot provide optimum performance for all loads In this paper, typical load input impedances are analyzed, and their effect on the stability margins of
L
v, vb
SW,
'L 0
I
The work was partially supported by a fellowship from Hewlett-Packard Co., Rockaway, NJ.
927
io
+ C ---
1
/
'L
vc
-
1
L
swb 0
1
/
v
Load
-
swc 10
0-7803-1456-5194 $4.00 0 1994 IEEE
A
and d is the vector of the line-to-line duty cycles
di
4
1 I dt = - U> id - 3 L d4v.
(12.b)
Since dab
+ dbc+ d,,
=0
(13)
it can be easily shown that where dj is the average duty cycle of the switch sw,. For modeling and control design, it is very convenient to transform the three-phase variables into rotating d-q coordinates [ 5 ] . Transformed variables (2)-(4) are defined as
-T:
d
2
/=-d 3
-r
r
y
T T I .
(14)
After substituting (14) into ( l . b ) , by using ( 5 ) and simple matrix manipulation, the equation (1.b) is obtained as dv 3 1 dt - 2c (dd id d4 i4) - (15) c io .
+
- sin(cl),t) - sin(r1jrf - 21r / 3 ) - sin(o),t + 2n / 3)
cos ( U ) $) - 2n / 3 ) COS(W,~ cos(w,f 2n I 3 )
+
1'
Equations (12) and (15) represent the average model of the three-phase boost rectifier in rotating coordinates, where U ) r = w = const..
,
(6) 3. LOAD MODELING
where Cnuy is the angular frequency of the d-q rotating coordinate frame, and it can be time varying, UJOy = U ) r ( t ) . Because r(t)is arbitrary, it can be chosen so that vabsin(mr
t) + vbcs i n ( c l l r t - 2n / 3 ) + vcasin(clj, t + 2n / 3 ) E 0
(7)
for any value of the instantaneous input voltages and in that case v4 E O . (8) If the input voltages are sinusoidal Vab
= v, cos
v, vca = v, VbC =
Wt
, A
cos(ot - 2n I 3 ) , cos(mt 2n 1 3 ) ,
+
2
r=TT[t],
Gg=Tr[;].
(IO)
The derivative of the input current vector in d-q coordinates is given by:
Substituting (10)-(11) into ( l . a ) ,after some matrix manipulation, two scalar equations are obtained:
+- 1
did/dt=wi 4
3 L
V,--
1 ddv 3 L
,
(123)
A
zic= v I I,
(9)
(7) is satisfied when O r = (1) = const.. In that case vd = vm. The inverse transformation, from d-q into the original three-phase variables, is given by: d =Tr[2].
Generally, the rectifier load establishes a nonlinear, dynamic relationship between the load current, io, and the rectifier output voltage v . Two typical load types are considered: buck dc-dc converter operating in continuous current mode, and the PWM VSI supplying a permanent magnet synchronous motor (PMSM) running at constant speed. The relationship between io and v is derived using simplified average models of the loads. The average model of a buck dc-dc converter is shown in Fig. 2 [6]. Definitions of the symbols and the numerical values of the parameters are given in Appendix B. Input impedance of the converter ,
where the A denotes a perturbed variable, can be easily derived from Fig. 2. However, when the load converter output voltage, voc, is controlled in a closed loop, the duty cycle, d,, is a function of voc, and io becomes nonlinearly dependent on the dc link voltage v . In that case, the model from Fig. 2 with current and voltage regulation can be linearized at an operating point, and a small-signal input impedance can be detived [7]. The small-signal input impedances for the ronverter in open-loop and with closed voltage loop It can be seen that both are shown in Fig. 3 . impedances are constant in a wide range of frequencies and can be validly approximated by a smallsignal resistance RAC. For the converter running in open-loop RAC = V / lo, and for the converter with output voltage regulation RAC = - V / lo, where the uppercase letters denote operating point values. In fact, the converter with voltage regulation acts as a constant power load, which corresponds to the negative small-signal resistance RAC. 928
I
100
~ d C * i I C 9 4 '
V
y
VOC $Rc
Fig. 2: Average model of buck dc-dc converter
The average model of a three-phase PMSM supplied by a PWM VSI is shown in Fig.4 [$]. Symbols are defined and parameter values are given in Appendix C. Because the motor peak torque capability is assumed to be three times its nominal torque, the motor power rating corresponds to one third of the front-end converter rating. Due to the load complexity, the dynamic relationship between io and v is nonlinear, even for the motor without speed regulation. Small-signal input impedances for the motor running in open-loop and for the motor with current and speed regulation are also shown in Fig. 3. The salient droop of the open-loop input impedance is due to the lowfrequency motor natural resonance. The closed loop input impedance is relatively constant only in the narrow low-frequency range bellow the speed loop crossover frequency. For these reasons, the motor input impedance cannot be approximated by a smallsignal resistance for any operating conditions. 4. PROPORTIONAL CURRENT-MODE CONTROL
The standard control scheme [3] for the three-phase boost rectifier in the d-q coordinates is shown in Fig. 5. It consists of two current loops and a superimposed output voltage loop. With the chosen rotating coordinate frame ( v q = 0) unity input displacement factor is achieved for iq = 0. However, in most high power applications an additional input filter must be added to meet the required .................................
-2000',
1
10
100
1000
10000
Freq (Hz) Fig. 3: Typical load input impedances: a) dc-dc converter in open-loop (Po=lO kW), b) dc-dc converter with current and voltage regulation (Po=lO kW), c) motor in open-loop (Po=3.3 kW), d) motor with current and speed regulation (P0~3.3kW).
attenuation of the input current switching frequency harmonics. The input filter produces phase shift between the line frequency components of the input currents and input voltages, mainly due to the reactive line frequency current through the total input filter parallel capacitance Cp [9]:
The solution suggested in [9] is to adjust the reactive component of the average converter input current to cancel the reactive input current drawn by the input filter in steady-state, i.e. to set i q r e f = - 1,. In Fig. 5 the terms (3 o L i4) / Vref and -~~ ( 3 o) L id) / V,ef, where Vref is the desired output voltage, are added to the control inputs dd and d,, respectively, in order to reduce the cross-coupling between the currents id and i, in (12). In that case the I ectifier equations are:
, ...................................................................................
i c :
V
1.5djdm'idm 1.5dqm.i
qm
ddm'
I, . PWM --VSI .............................
'
I..-..
PMSM
.............................................................................
Fig. 4: Average model in rotating coordinates of PMSM supplied by PWM VSI.
929
-20
I V
phase a - 4 U
-40
-60 8 -
- -80 I
I
Fig. 5:Standard control scheme for three-phase boost rectifier in d-q coordinates.
did -=o(I--)i dt
V
Vrer 4
+-3L1
v --1 d d l v , (18.a) d 3L
dv --
3 1 . dt - 2~ (ddl i d + d41 i4) - -c io
(18.c)
Consequently, the cross terms in (18.a) and (18.b) are reduced by the factor (1 - v/Vref),and they disappear in steady-state when v = Vref. The conventional average current-mode controllers (CCMC) are implemented as proportional feedback so that the average duty cycles are:
ddl = K d ( i d - i d r e f )
,
(19.a)
d41 = K 4 ( i4 - i q r e f ) .
(19.b)
The small-signal model for vd= Vm can be obtained by linearizing (18) at the operating point v = v m / Ddl = Vref? ld = (2 lo) GQDdl), Dq1 = 0 , and l q = i q r e f . Assuming that the load input impedance is approximated by the small-signal resistance RAC, the small-signal model is: A
di
dt
-
RAC) variations. If 3 a) L I, < < Vm, assuming K d = K, = K , the transfer function can be simplified to a single-pole transfer function:
1 " 1 '4 3L ddl V - (Ddl - r o -) 3L V
A
v ,
(2O.a)
A
(20.b)
Equation (21) gives the guideline for choosing the gain I 0
j - j - - - E = 10 kW, RAC < O j . : _ _ _E = 3.3 kW, RAC > 0 : -: .
= _
_
_
_
I 5
Therefore, the voltage compensator H* will provide stable (albeit slow) response for all considered loads.
3.3 kW, RA, < 0 _
_
_
_
_
_
_
_
_
_
_
_
_
_
_
-250
4. NONLINEAR CONVERTER CONTROL
A way to design the rectifier control for optimal performance with varying load is to introduce nonlinear current-mode control (NCMC) (1,2,3,4] so that the average duty cycle ddl is:
Ideally, it is desirable to have the same gains for the linear and nonlinear terms, i.e. K = K ' . However, due to the sensor mismatches this cannot be practically achieved, and then it is preferable to have K' slightly smaller than K [4]. Assuming that V m is constant, and that the load can be modeled by its small-signal resistance RAC, the perturbed duty cycle is: 50
0 ' 100 I
I
50
x
-100 -150
-200 -25Ool1
I 1
10
1000
100
10000
Freq (Hz)
&,
Fig. 9: Control-to-outputtransfer function ;I/ with complete motor model and P = 3.3 kW: a) open loop, b) closed current and speed regulGtionloops.
931
50
3
I
40
’0
a~ 30
U
.-” c
20
0)
0.1
10
1
100
1000
Freq (Hz) Fig. 12: Comparison of rectifier output impedances ( R A C open-loop motor input impedance.
-250 0.1 I
1
10
1000
100
10000
1
Freq (Hz) Fig. 10: NCMC control-to-outputtransfer functions v / idref : a) P,= 10 kW, R A C > 0, b) p0 = 10 kW, R A C < 0, c) Po= 3.3 kW, motor in open-loop, d) P, = 3.3 kW, motor with current and speed regulation.
The NCMC transfer functions Gc, which can be derived by substituting (25) and (19.b) into (20), are shown in Fig. 10 for different loads. It can be seen that the transfer functions do not change appreciably at low and mid-frequencies for different loads, and that they are very close to the CCMC characteristics with RAC < 0. The NCMC transfer function dc gain and the low frequency pole are almost constant for different load conditions. Also shown in Fig. 10 are the control-to-output transfer functions Gc obtained with NCMC (24) and the complete WE4 and motor model from Fig. 4. The transfer function for the motor with current and speed regulation is almost indistinguishable from the transfer functions which were obtained by approximating the load with the small-signal resistance RAC. Even the transfer function for the motor in open-loop has the form which does not significantly influence design of the boost voltage compensator.
760
745
1: 730 0
1
2
3
Time (ms) Fig, 11 : Small-loadstep response for NCMC: a) RA, (Po= 9 k W -1Okw).
4
5
0, b) RAC < 0
932
= CO) with
the
The improvement provided by the NCMC is best appreciated by comparing Figs. 7 and 9 with Fig. I O . Since all the NCMC transfer functions in Fig. 10 are very close to the worst case CCMC transfer function ( P o = 10 kW, R A C < O in Fig. 7), the same voltage compensator should be used for both current control algorithms. In the NCMC case, however, the same voltage compensator does not result in slow load step response. The small step load response is shown in Fig. 11. By comparing the CCMC responses c and d in Fig, 8 with the responses in Fig. 11, it can be seen that the NCMC response is significantly faster, and the voltage overshoot is much smaller. The reason for faster response is that, unlike with CCMC, the output impedance of the rectifier with NCMC has a low frequency left-half-plane zero due to the load current feed forwa rd : 1
(K - K‘) 1 +iK ‘ i f 3
v -
O
vg’
This zero reduces the effect of the low-frequency dominant closed-loop pole, and improves the speed significantly. Even though it was demonstrated in the preceding analysis that for the successful control design, the rectifier load can be approximated with the worst case small-signal resistance (maximal load, RAC < 0), the strong interaction between the rectifier and the motor load running in open loop is still evident. A measure of the rectifier performance invariance to the loading effect is the separation between the rectifier output impedance for R A C = c n and the load input impedance. A s can be seen from Fig. 12, the output impedance of the rectifier with NCMC is much more separated from the input impedance of the VSI with the motor in open loop than the output impedance with CCMC. As a result, the rectifier-load interaction is much smaller in the NCMC case (Fig. 10) then in the CCMC case (Fig. 9). Due to the nonlinearity of the boost rectifier, the large signal behavior cannot be reliably predicted from the small-signal characteristics. The 100°/o to
I
I
10000
The motor load was analyzed only in the constant speed operation. Because the motor speed varies relatively slowly compared to the rectifier bandwidth, this is a valid approximation. However, if the change in motor speed results in regenerative operation, the power flow will reverse in the VSI and the rectifier. This case may be a subject of future research.
840 1
APPENDIX A
0
1
2
3
Time (ms) Fig. 13: Large step load response with buck converter.
- THREE-PHASE BOOST RECTIFIER PARAMETERS
Power stage parameters: V, = 537 V, L = 0.5 mH, C = 10 pF, V = 750 V, Po,,, = 10 kW, and f, = 50 kHz. Current loop parameters: Kd = 0.2, K, = 0.2, and K’ = 0.19. Voltage compensator: HVr= 383 (1 s/9500) /(s (1 s/6.8e4)), and H V 2 = 126 (1 ~/1100)/ ( s (1 ~/6.8€4)).
+
+
+
+
APPENDIX B - DC-DC BUCK CONVERTER PARAMETERS
60% step load response was, therefore, simulated for the dc-dc converter with output voltage regulation, for both types of the rectifier current control. The output voltage response is shown in Fig. 13. Approximately equal voltage overshoot is obtained with both types of control because duty cycle entered saturation in both cases. However, the NCMC control still provides significantly faster response. Similar improvements were observed in large signal responses with other types of load.
5. CONCLUSION
Modeling and control of the front-end three-phase boost rectifier loaded by the dc-dc buck postregulator or by the VSI with PMSM, has been presented. It is shown that the input impedance of the dc-dc converter can be validly approximated by a small-signal resistance, which is not the case for the motor load. With the boost rectifier current regulator in rotating coordinates, the quadrature component of the input current can be controlled to compensate the input displacement factor produced by an EM1 filter. Use of the proportional control for the direct component of the input currents results in strong interaction between the rectifier and the load. A voltage compensator which assures desired stability margin for all loads results in slow transient response. The interaction can be significantly reduced and much faster transient response obtained with the nonlinear current control which uses load current feedforward Because the current controller in rotating coordinates is always implemented digitally, the nonlinear regulator can be easily incorporated into the existing software and requires only additional sensing of the load current . The presented analysis demonstrated effectiveness of the average modeling. However, influence of sampling and filtering of the voltages and currents used in feedback has not been included. These effects will reduce the maximum achievable speed of response.
Power stage parameters: V = 750 V, L, = 295 pU, C, = 20 pF, V,, = 200 V, R, = 4 LA, and ,f = 100 kHz. Current loop parameters: current sensor gain Ri = 0.1 SZ, modulator gain FM = 0.48 1/V ~ l , and external ramp Se = 21600V I S. Voltage compensator: H,, = 4876 (1 s/15000) / s.
+
APPENDIX C
- PMSM PARAMETERS
a,
Motor parameters: stator resistance R, = 0.18 stator inductance L, = 1.9 mH, stator-to-rotor mutual linkage ‘Ir, = 1.21 V / (rad/s), damping constant D = 0.0044 Nm / (rad/s). rnoment of inertia J = 0.0232 kgm2, number of pole pairs P = 1, synchronous speed a), = 157 rad/s, and load torque TL = 20 Nm. Current loop parameters: d i, controller Kd, = 0.02, and I,, controller Hi,, = (1 s/55.55) / s. Speed compensator: H,L,, = 666 (1 s/72) / s. VSI is considered ideal.
+
+
REFERENCES
T. G. Habetler, “A space vector based rectifier regulator for AClDClAC converters,” /E€€ Trans. on Power Electronics, Vol. 8, no. 1. pp. 30-36, 1993. F. Blaabjerg, J. K. Pederson, “An integrated high power factor three-phase AC-DC-AC converter for AC machines implemented in one microcontroller,” \E€€ Power ElectronC I S Specialists Con[ ’93 Rec., pp. 285-292, 1993. -1. W. Kolar, H. Ertl, K. Edelmoser, F. C. Zach “Analysis of the control behavior of a bidirectional three-phase PWM rectifier system,” Proc. of 4th European Conference on Power Electronics and Applications, pp. 2-095 - 2-100, 1991. S. Hiti, D. Borojevic, “Robust nonlinear control for boost converter,” /€€E Power flectronics Specialists Conf. ’93 R ~ c .pp. . 191-196, 1993. C. T. Rim. D. Y. Hu, and G. H. Cho, “Transformers as equivalent circuits for switches: General proofs and d-q transfot tiration-based analysis,” / € E € Trans. on Industry Applrcat/ons, Vo1.26, n0.4, pp. 777-785, 1990. V. Vorperian, “Simplified analysis of pwm converters using the tnodel of the pwm switch Part I: Continuous conduction mode,” Proceedings o f the VPEC Seminar, Blacksburg, VA, pp. 1-9, 1989. R. D. Middlebrook, Input filter considerations in design and application of switching regulators,” /E€€ lndustry Application Society Annual Meeting ’76 Rec.. pp. 366-382, 1976. P. Pillay, R. Krishnan, “Modeling, simulation and analysis of permatlent magnet motor drives, Part I: The permanent magnet synchronous motor drive,” / € E € Trans. on Industry App//cation, Vol. 25, no. 2, pp. 265-273, 1989. S. Hiti, V. Vlatkovic, D. Borojevic. and F. C. Lee, “A new control algorithm for three-phase PWM buck rectifier with input displacement factor compensation,” / € € E Power Electronics Specialists Conf. ’93 Rec., pp. 648-654, 1993. IC
