Generalized Predictive Direct Power Control for AC/DC Converters Sergio Vazquez, Leopoldo G. Franquelo
Ricardo P. Aguilera, Daniel E. Quevedo School of Electrical Engineering & Computer Science The University of Newcastle NSW, Australia e-mails: {raguilera, dquevedo} @ieee.org
Abstract-Predictive-Direct Power Control (P-DPC) is an at tractive predictive control technique for ACIDC converters. This control strategy considers a dynamic power model of converter to forecast possible future converter behaviours. Thus, P-DPC provides an optimal switching pattern in order to directly control the active and reactive powers. Nevertheless, as will be shown in this work, this control method limits the converter operation range to a small power angle. In this work, we propose an improved P-DPC strategy, which allows one to extend the power angle operation range of an ACIDC converter while improving its power quality.
I.
INTRODUCT ION
In many industrial applications AC/DC converters play an important rol. A particular class of AC/DC converter is the, so called, active front-end rectifier (AFE), which has emerged as an attractive topology. The main advantages of using an AFE instead of a traditional diode-based rectifier, come from the fact that this topology provides sinusoidal currents with low harmonic distortion while delivering a regulated rectified voltage. Additionally, reactive power can also be controlled to absorve or deliver reactive power to the electricity grid, allowing one to achieve a unitary power factor when need it, see e.g. [1]-[3]. Due to these remarkable features, AFEs are also used as active filters; see e.g. [4], where this topology is connected in parallel to a non-linear load in order to mitigate the pollution in the source. To govern AC/DC converters, different control strategies based on PWM techniques have been proposed. The most popular algorithms use Voltage-Oriented Control (VOC) , or Direct Power Control (DPC) , see [5]. The VOC approach works in the dq-coordinates using two control loops: The external loop is based on a Proportional Integral (PI) controller which seeks to compensate the DC-voltage error by generating the direct current reference. The internal loop reduces the dq current error by using two PI controllers. These generate dq voltages which are utilized to produce the associated space vector modulation. The DPC technique also requires two control loops, but in a different manner: With DPC, the external controller seeks to compensate the DC-voltage error by directly generating the power reference for an internal hysteresis control loop. The switching actions are obtained from a lookup-table, which takes into account not only the DC-voltage error, but also active and reactive power estimates.
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Electronics Engineering Department University of Seville Seville, Spain e-mails:
[email protected].
[email protected]
The main drawback of both VOC and DPC methods is the need for local linearizations for the linear control design part, see [6]. Recently, predictive control has attracted significant atten tion among power electronics researchers [7]-[10]. This comes from the fact that power converters governed by predictive controllers can, in some cases, outperform those one governed by PWM-based linear controllers [11]. Regarding the control of AC/DC converters, there exist several predictive control strategies in the literature, being one of the most popular the Predictive Direct Power Control (P-DPC) [12]. This predictive control strategy provides an optimal switching pattern in order to directly control the active and reactive power. Although this predictive control technique presents, in principle, some advantages when compared to traditional PWM-based strategies, we show in this work that this control method constraints the converter operation range to a small power angle, which limits the active power that one can transfer from the AC side to the DC side (and viceversa). In this work, we present a generalized version of the P-PDC strategy in order to extend the operation range of the converter and also improve the power quality. We focus on the AC/DC converter shown in Fig. 1. This converter is a three-phase fully controlled bridge consisting of 6 power transistors connected to a three-phase power source Vs by means of a filter. The latter is represented by an inductance Ls and the parasitic resistance r s' The control target for this power converter is to control the active power, Ps, in the AC side to obtain a
1215
Cde
Fig. 1.
Active front-end ACIDC converter
Rdc
regulated voltage, Vdc, in the DC side. Additionally, one can control the reactive power, Q s in order to adjust the power factor in the grid connection point.
Thus, it is possible to obtain the following dynamic model of the active and reactive powers:
II. PREDICTIVE DIRE C T POWER CONTROL This predictive control strategy is aimed to control directly the active and reactive power of the converter instead of the system state (three-phase currents). In the literature, there exist two main classes of Predictive Direct Power Control (P-DPC) for power converters, the Variable Switching frequency P DPC (VSF-P-DPC) [13], [14] and the Constant Switching frequency P-DPC (CSF-P-DPC) [15], [16]. For a comparative study see [12]. In this work we will focus on the latter. The key idea of this predictive control strategy is to use a predefined switching pattern based on the available switching vectors that the converter can generate. Thus, one need to find, in an optimal way, the instant when these vectors must be applied within the sampling period.
(6)
These equations can be compactly rewritten as:
dP dt = A (t) isafJ + B (t)vrafJ + E, where
and
]
VSa , vsfJ - VSfJ , Vsa
A. System Model
Considering an a(3 framework, the active and reactive power can be expressed by [17]: (1)
From the latter, one can derive the following dynamic equations for the active and reactive powers
(2)
Neglecting the effect of rs, the dynamic model of the ac currents can be represented via
]
B. Vector Selection
It is well known that a 2-level converter can generate 8 vectors in the a-(3 plane, Vri with i E {O, ... , 7} where Vo and V7 are redundant vectors (see Fig. 2). As proposed in [15], the vector sequence applied to the converter is selected in order to minimize the switching losses. To do this, the a-(3 plane is divided in 12 sectors as depicted in Fig. 2. Each partition is assigned with a vector sequence, Vj with j E {I, ... , 12} as presented in Table I.
(3
(3)
VrfJ]T, vsafJ = [vsa VsfJ]T, and isfJ]T. Here, [] X T denotes the transpose of x.
where vrafJ = [vra
isafJ = [isa
Considering a balanced sinusoidal three-phase source volt age, represented in a(3 framework, we have that
Vsa =Vs sin(wt), vsfJ = - Vs cos(wt).
(4)
where its derivative is expressed by:
dVsa dt = -wvsfJ dVsfJ dt = WVsa'
(7)
(5) Fig. 2.
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Converter voltage vectors
Vri.
1
1 0 I 0 I 0
2
I I I
I I I
I I
7
2
I I I
I I I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I I I
I
t2
I
I
I I
t3
1
I I
I
I I I
I I
T
k Fig. 3.
I
I
I
tI
7
I
k+l
Modulation obtained when applying vector sequence iiI in Sector 1.
The sequence to be applied to the converter during a sampling period, T, is chosen depending on which sector the source voltage, Vs, lies. For example, in Fig. 2, Vs lies inside Sector 1. Thus, the sequence to be applied in a sampling period will be ih {Vrl' Vr2, Vr7 , Vr7 , Vr2, Vrl}' The resulting voltage pattern is depicted in Fig. 3. Now, it is necessary to determine the commutation instants tl, t2, and t3.
T
k Fig. 4.
k +I
Active and reactive power behaviour during the intersampling.
=
represented by:
Psk+l
C. Application Times
Based on the dynamic model presented of the converter power presented in (6), one can consider that active and reactive power increments remain constant during the time that a vector Vri is applied. Thus, these increments can be expressed via:
Qsk+l
=
=
3 PsO + 2 L jpi ti , i=l 3 Qso + 2 L jqi ti , i=l
(9)
�
where Psk+l Ps6 , Qsk+l Qs6 , and t3 - tl - t2' This situation is illustrated in Fig. 4. Now, a quadratic cost function which includes only the active and reactive power errors at the next sampling instant is considered =
=
=
(8) where Considering an initial value of Psk Pso and Qsk within a sampling period we obtain that =
Psi Q si
=
=
=
Q so ,
Psi-l + jpi ti , Qsi-l + jqi ti ,
epk+l eQk+l
for all i E {l , . . . , 6}. Thus, the predicted values of the active and reactive power for the next sampling instant, k + 1, are
8 9
10 11 12
Vri
=
3 epk - 2 L jpi ti , i=l 3 eQk - 2 L jqi ti , i=l Q; - Qsk.
(11)
=
(ePk + 2 ( (fP3 - jpdh + (fp3 - jp2)t2 + ( eQk + 2 ( (fq3 - jqdtl + (fq3 - jq2)t2
sequence
127721 012210 032230 327723 347743 034430 054450 547745 567765 056650 016610 16776 I
Q; - Qsk+l
=
=
=
Jk (h, t2)
1 2 3 4 5 6 7
=
P; - Psk+l
with epk P; - Psk and eQk The cost function can be rewritten as a function of the times tl and t2 as:
TABLE I VECTOR SEQUENCE TO BE APPLIED Sector
=
2
_
_
�)) 2 jq3 �)) jp3
(12)
Now, one can obtain the times by performing the following partial derivatives:
aJk (h, t2) atl aJk (tl, t2) at2
1217
=
0' (13)
=
O.
Finally, the optimal instant in which the predefined vectors will be applied are:
tl
=
t2
=
t3
=
(fq2 - !q3)epk + (fp3 - !p2)eQk + (fp2!q3 - !p3!q2)T 2 ((fq2 - !q3)!pl + (fq3 - !qd!p2 + (fql - !q2)!p3) ) (fq3 - !ql)epk + (fpl - !p3)eQk + (fqI!p3 - !q3!pI)T �����----������--���--���-2 ((fq2 - !q3)!pl + (fq3 - !qd!p2 + (fql - !q2)!p3) ) T 2" - tl - t2'
�����----������--���--���--
(14)
It is important to emphasize that to transfer active power,
Ps, from the power supply to the converter there must exist an angle 0 between the vectors vs and v,.., i,e, 0 i- 0, Thus, the required converter vector, v,.., may not lie in the same sector than the voltage source vector, Vs' Moreover, the separation between both vectors becomes larger as the active power, Ps, is increased. In a steady state condition where Ps = P; and Qs = Q; we have that
It is important to emphasize that the above analysis does not depend on a particular sector. It is only limited to the application of three vectors in a synunetrical way. Therefore, one could use a different vector sequence than the proposed in Table I. III.
GENER ALIZED
P-DPC
In this section, a vector analysis of the system is carried out to establish the limitation of standard P-DPC in terms of the power angle. Then, based on this analysis, an improved P-DPC is proposed. A. Vector Analysis
As mentioned above, P-DPC relies on the source voltage, vs, to choose one of the twelve sectors; see Fig. 2. However, since this predictive strategy aims to synthesize the inverter voltage v,.., using v s may not be appropriate. A simplified single-phase representation of the system in a steady-state is shown in Fig. 5-a). In this case, the active and reactive power can be expressed via: p
,
. .
i
s
_
Qs
_
-
VsV,.. sin(0) XL
'
Vs (Vs - V,.. cos(o))
-
XL
(15) )
..............................................................................................................................., ' X
,
'.jG",
I
�.............................................................. .,...............................................................-i
(3
(3
, ..............................�l ..:.��
....................... �
Fig. 5.
For the particular case when a unity power factor is required, i.e., Q'd = 0 , we obtain A
V,..IQ�=O oIQ�=O
=
Vs cos(o)
=tan-1
(Xts:S) .
(17)
To illustrate this problem, in Fig. 5-b) a phasor represen tation of the system in a steady-state is depicted. Here, one can notice that the source voltage, vs, lies in Sector I, while the required inverter voltage, v,.., which provides Ps = P* and Qs = 0, lies in Sector 2. Since P-DPC constraints v,.. to belong to the same sector than vs, i.e. Sector I, the optimization carried out in (10)-(14) adjusts v,.. in order to achieve the desired active power reference. However, this action affects the tracking of the reactive power. This situation is depicted in Fig. 5-c). Consequently, the times obtained by solving (14) are only a local optimum in such partition. This problem is further compounded during power transients. On the other hand, P-DPC does not take into account the continuous-time dynamics during the intersampling. As can be noticed in (10), only the instantaneous power error at the next sampling instant is considered in the optimization, i.e., epk+l and eQk+l' As shown in [18], neglecting the continuous-time dynamics may reduce the performance during the steady-state behaviour. To overcome these issues, in the next section we present a generalized P-DPC (GP-DPC) strategy. B. GP-DPC Algorithm
..............................�l..::.:.......................
Vectorial representation of the steady state
(16)
As previously stated, the required inverter voltage, v,.., to track a desired active and reactive power, P; and Q;, may lie in a different sector than the source voltage vector, vs, see Fig. 5-b). Thus, one can directly use v,.. to determine the suitable sector. However, its magnitude may be larger than the hexagonal bound, e.g., during power reference changes. Thus, the simple magnitud saturation of v,.. may not be the optimal inverter voltage. To address this problem, in this work we propose to optimally obtain the sector in which the inverter voltage, v,..,
1218
Fio ,Qso
IV.
j=l (Sector)
RESULT S
To verify the performance of the methodology describe above, simulation studies were carried out for an ACIDC converter. For our study, the same system parameters used in [19] were considered, see Table II. In Fig. 7, a step-up in the active power is depicted. Initially, the system starts with an active power of P * 1000 W and a reactive power of Q* 0 VA. For this condition, the standard P-DPC shown in Fig 7-a) presents a higher ripple in the active power tracking than the proposed GP-DPC depicted in Fig. 7b). This is due to the fact that GP-DPC takes into account the intersampling dynamic. Then, a step-up in the active power reference, P* 2 000 W, is performed at t 25 ms; the same case was analyzed in [19]. From (17), this power change implies an ideal angle change, in the steady-state, from (j 6.2 2 ° to (j 12.31°. Since P-DPC limits Vr to belong to the same sector than VS, the resulting sector sequence used to obtain the optimum times does not change, as can be noticed in Fig 7c). Thus, an increment in the power ripple can be observed in Fig. 7-a), resulting in a slight distortion in the source current, is, see Fig. 7-e). On the other hand, as can be seen in Fig. 7-d), GP DPC changes the sector patron in order to obtain the global optimum. Thus, a reduction in the power ripple is observed in Fig. 7-b) when compared to Fig. 7-a). Consequently, less
J*=oo
=
=
=
=
=
=
Parameters
Vs fa Ls
Cdc Rdc
fs
Values 120 V rms 50 Hz 10 mH 470 J.LF 1000 5000 Hz
TABLE II SYSTEM PARAMETERS
Fig. 6.
Flow diagram of the proposed GP-DPC strategy. 2000 «
will be synthesized. To do this, the local optimal times in (14) are calculated for every partition, i.e., Tj [tl, t2 1 for all j E {1, ... , 12}. Then, these times are used to evaluate in each sector j the following cost function: =
�. �.
=
L ( P; - Ps ) + (Q; - Qs ) ' i
i
(18)
Notice that this proposed cost function evaluates the pre dictions for the active and reactive power in 6 different instants within the sampling period to improve the continuous time behaviour. Thus, the optimal sector, jop, is that one which produces the minimum value in the cost function. Consequently, the optimal times for this partition, Top, are the global optimum. In Fig. 6, a flow diagram of the proposed GP-DPC algorithm is presented.
0
.
2000
.
--1 20
«
�. �.
'
•
�_
o
6
Jj(tl, t2)
1000
1 : : :.j 1 �
••
a)
40
60
.
1000_ 0
N
: I
J : ...
.
!••••4e•••�•••••
0
20
b)
40
60
IllZ1I1I1 llvlZk1 : C?\A1 : C?\A1
�'
o
20
o
20
c)
Time [ms] e)
40
60
40
60
�'
0
20
0
20
d)
Timelmsj f)
40
60
40
60
Fig. 7. Converter behaviour under both standard P-DPC and GP-DPC when a step-up power change from Ps = 1000 W to Ps = 2000 W is applied
1219
r�l.:.:.D ;:�l .:L..: J ....
20
40
a)
60
....
20
c)
20
b)
40
60
40
60
d)
Time [ms] e)
40
60
20
Timelmsj f)
Fig. 8. Converter behaviour under both standard P-DPC and GP-DPC when a step-up power change from Ps = 1000 W to Ps = 3000 W is applied
distortion is induced in the source current, is, see Fig. 7-t). To emphasize the advantages of using GP-DPC, in Fig. 8 a step-up change from P; 1000 W to P; 3000 W is depicted. This implies an angle change from 15 6.2 2 ° to 15 18.12 °. In this case, it is clear from Fig. 8-a) and 8-b) that P-DPC can not track perfectly both the active and reactive power for a reference of P; 3000 [WJ. GP-DPC presents a better performance with a faster dynamic during the transient and less ripple in the stead-state. The benefits of using GP-DPC can also be appreciated in the source currents waveform in Fig. 8-e) and 8-f). Here, it is clear that with GP-DPC a reduced distortion is obtained. =
=
=
=
=
V.
CONCLUSIONS
In this work, an improved P-DPC strategy, in terms of transient dynamic and power quality, has been proposed. As a result of our analysis, it has been shown that, using the source voltage vector to determine the optimal sector limits the power operation range of the AC/DC converter when governed by standard P-DPC, achieving only a local optimum. Thus, the key idea of our proposal is to explore all sectors in order to obtain a global optimal vector sequence. Additionally, considering more instants, within the sampling period, to compare the predicted power with the references, allows one to improve the steady state performance. Future works will be focused on incorporating the DC voltage control in the optimal problem formulation, in order to address both control targets, the tracking of the AC active and reactive power, and DC-voltage, as proposed in [14J. Experimental validation will be also carried out.
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