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Modified Direct Power Control for PWM Rectifier Under Unbalanced
Grid Voltage Conditions
CHEN Hongjun1, XING Gaoxing1, ZHOU Xin1, ZHANG Ming1, QU Yuepeng1
1. Department of Electrical Engineering, Harbin Institute of Technology, Harbin 150001
E-mail: hongjun@hit.edu.cn
Abstract: Under the unbalanced power grid voltage conditions, three-phase voltage source PWM rectifier would produce
non-sinusoidal and negative sequence current, fluctuations of active and reactive power. To deal with these problems, the
mathematical model is established in the stationary frame, then a dual power control scheme is presented to control the active
and reactive power independently. By setting reference of power inner loop, the negative sequence current can be eliminated and
grid current is sinusoidal. The proportional resonant controller in the outer loop suppresses the DC-link ripple. The PI repetitive
control is applied in the inner loop to eliminate active and reactive power oscillation. Test results show the good performance of
the proposed strategy.
Key Words: Unbalanced grid voltage, PWM rectifier, Dual power control scheme, repetitive control
1 Introduction
Three-phase voltage source PWM rectifiers (Fig. 1) have
been widely applied in wind turbine, active power filter
motor drive and so on, due to their advantages such as
bidirectional power flow, almost sinusoidal currents,
near-unity power factor, and regulation of dc-link
voltage[1,2]. Thus, the control of these power systems is
currently one of the objectives of the researchers. One of the
most efficient control strategies for this system is the direct
power control (DPC). The DPC strategy lies on the
instantaneous power theory introduced by H.Akagi [3] and
is based on the evaluation of the active and reactiveinstantaneous power with internal control loop for the active
and reactive power. Many scholars achieved good results of
the three phase voltage rectifier control study based on
direct power control [4-6]. Applications based on DPC
demonstrated that it is a simple and efficient control strategy
that achieves good dynamic performance and near-unity
power factor.
~
~
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idc
iLscs bsa
scs bsa
C
a
b
c
Rl
Rl
Rl
ia
i b
ic
ea
e b
ec
L
L
LO RL
U dc
Fig. 1: The topology of three-phase voltage source PWM rectifiers
The three-phase voltage source PWM rectifiers would
produce non-sinusoidal and negative sequence current and
fluctuations of active and reactive power under unbalanced
*This work is supported by National Natural Science Foundation (NNSF)
of China under Grant 60871036.
grid voltage conditions, a common occurrence due to the
unbalanced load/transmission impedance or grid faults. In
[7-9], DPC of PWM rectifiers under unbalanced grid
voltage conditions was proposed, which use a single inner
loop with negative sequence power compensation. In
[10,11], a control scheme was introduced under unbalanced
grid voltage conditions, where two PI controllers were
adopted to control the positive and negative sequence
currents respectively, but the design of current inner loop is
very complicated, and cannot guarantee good transient
response. In this paper, we used dual power control scheme
to control the power immediately and get good performance.
2 Model of PWM Rectifier
As shown in Fig. 1, the input inductors of the PWM rectifier
are used to suppress line current harmonics and the dc-link
capacitor is used to smooth output dc voltage. The system
parameters and variables are defined in Table 1.
Table 1: System Variables and Parameters
variable Description
e =abc a b ce e e Phase input voltage vector
i =abc a b ci i i Phase input voltage current
s =abc a b c s s s Control input vector
Grid frequency
R Input equivalent resistance
L Smoothing inductor
C Output capacitor
R L Load resistance
The system model
frame as follows
d- 0 1 0 0
= +
0d 0 1 0 0d o
i e L i R sdt
i e
R s L i ut
(1)
Proceedings of the 32nd Chinese Control Conference
July 26-28, 2013, Xi'an, China
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dc L
d= + -
dC U i s i s i
t
(2)
1 1 2 1 22
3 0 3 2 3 2
a
b
c
X X
X X
X
(3)
Equations (1) and (2) are the mathematical model of the
PWM Rectifier. T
a b c X X X are the variables and
T
X X
are their forms in the frame.
When the grid is unbalanced, the grid voltage and current
can be expressed as the sum of their respective positive and
negative sequence components-
-
e
i
j t p j t n
abc
j t p j t n
abc
e E e E
e I e I
(4)
where p E ,
n E ,
p I and
n I are the amplitudes of positive
and negative sequence components of grid voltages and
currents, respectively. The format in the stationary
coordinate is expressed as
e
i
p n
p n
e e
i i
(5)
where
+j , +j
+j , +j
p p p n n n
p p p n n n
e e e e e e
i i i i i i
(6)
The active and reactive power can be computed as [7].
e e i P
e e iQ
(7)
Then, the variations of active and reactive power can be
given as
d d d(( ) ) ( ) ( )
d d d
d d d(( ) ) ( ) ( )
d d d
T T
T T
P t t t
Qt t t
e i e i
e J i e J i (8)
where0 1
1 0J
,
dc
u sU
u s
.
It is assumed ( ) sin( ) cos( )eT E t E t , the
variations of instantaneous grid voltages can be obtained as
d
( ) -d e
T
e et
(9)
Substituting (1) and (9) into (8) yields2
( ) ud
d
( ) ud-
d
T
T
R P Q P t L L L
RQ P Q
t L L
e e
e J
(10)
where2
e is the amplitude of grid voltage.
Substituting (5) and (6) into (7), the instantaneous active and
reactive power during network unbalance can be expressed
as
0 2
0 2
p n p n p n
p n p n p n
P P e e e e i i P
Q Qe e e e i iQ
(11)
where P 0 and Q0 are the respective average components of
active and reactive power, and P 2 and Q2 are the oscillating
components at twice the grid frequency of active and
reactive power, respectively. For a clear illustration, they
can be represented as
0
2
0
2
p p n n p
n n p p p
p p n n n
n n p p n
e e e e i P
e e e e i P
e e e e iQ
e e e e iQ
(12)
The control objective for the three-phase power converter is
to eliminate negative current components and to obtain
sinusoidal and symmetrical current. As shown in (12), in
order to eliminate negative sequence current, the ni and
ni
must be zero. When the system works on the steady state,
the grid current should be sinusoidal and is proportional to
voltage. The relationship can be expressed as follows * *
* *
( ) ( )
( ) ( )
p
n
T T j p p p p
p
T T jn n n n
n
i i k e e e
i i k e e e
(13)
p j
pk e
and n j
nk e
denote proportional relationships
between the positive and negative sequence component. Making p j
p p K k e
, n j
n n K k e , then substituting (13)
into (12) yields*
0
2 2( ) ( ) p p p
P K
e e
(14) Then
* *
0
*
*
*
( )
( ) 0
( ) 0
( ) 0
p
p
n
n
P P
Q
P
Q
(15)
However,*
2 0 P ,*
2 0Q , and it means that active and
reactive power have oscillation. 3 The Control Strategy
3.1 Dual inner loop Controller
Equation (10) shows that the system is a MIMO nonlinear
system. The system can be expressed as follows.
1
1 1 1 2 2 2
d( ) ( )
d
( ) , ( )
m
i i
i
x f x g x ut
y h x x y h x x
(16)
where =T x P Q .
2
1 2
1 2
( )
e p
R x x
L L f x R
x x L
(17)
1
1( )
m
i i
i
e e u
g x u e e u L
(18)
Its state feedback is (19)
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11
22
2
1 2
1 2
( )
e
1
f
f
L hv D x u
L hv
R x x e e u
L Le e u L R
x x L
(19)
Considering 1
( ) 0e e
D xe e L
, the system can be
partially feedback linearization, then the positive sequence
model can be controlled as (20) with the method of
feedback linearization.
11 1
2 2
2
*
*2
( )
(s)( ( ) )
(s)( ( ) )
p f
p f
p
p p p p p p
p p p p p p
L hu v D x
L hu v
e R P Q K P P L L L
K Q Qe R P Q
L
E
(20)
Similarly, the negative sequence model shows as (21)2
*
2 *
e(s)( ( ) )
(s)( ( ) )e
n
n nn n nnn
n n nnnn n
R P Qu K P P L L L
u K Q Q R P Q
L
E (21)
Here E p
=
p p
p p
e e
e e
, E
n=
n n
n n
e e
e e
. (s) p K and (s)n K are
respectively the function of inner loop positive sequence
power controller and negative sequence power controller.To suppress the active power oscillation and the reactive
power oscillation of the power loop, the power inner loop
adopts the repetitive controller [13]. The repetitive control is
a strategy based on internal model, which could suppress the
influence of the periodic disturbance effectively. The
frequency doubling fluctuation of the active and reactive
power could be regarded as periodic disturbance under
unbalanced grid condition.
Although the repetitive control has the good steady state
performance and is not sensitive to the system parameter, it
has a static differential. This paper mainly adopts the
complex control with PI controller and repetitive controller.
The real system mostly uses the improved repeated signalgenerator as shown in Fig. 2.
N z
1( )S z
1( )Q z
c
1
T
I p
1
K z K
z
( ) P z p p*
( ) p K s
Fig. 2: PI repetitive control scheme
where1( )Q z is a band-limit filter . It could be a low-pass
filter or a constant less than 1.1( )Q z is often shown as(22)
when it is a low-pass filter 1
1 2( )
4
z z Q z
(22)
When1( )Q z is a constant, it is usually taken as
0.95[14].1( )S z is an a post-delay filter , it is used for
compensating both the phase delay of the basic servo plant
and the main delay loop such that the repetitive control
signal can keep the same phase with the cyclic disturbance
to be eliminated. The ordinary structure of1
( )S z is (23)
1 ( )( ) N M
g S z k z (23)
here, s c N T T , T s is the fundamental wave period of output,
T c is switching period, M is the equivalent object delay
number, k g is the gain of repetitive controller, when k g is
smaller, the system’s stability becomes better, but its
convergence speed becomes slower and the steady stateerror rises. k g is often taken by experiments. In fact, the
system has a delay of a switching period [14]. Finally,1
( ) g S z k z (24)
In Fig. (2), K p and K I are respectively the proportionality
coefficient and the integral coefficient of PI controller. The
PI controller could produce adjustment to the tracking error
instantly.
3.2 Outer loop Controller
The DC voltage regulator’s output is the reference of inner
power loop *
0 P .Making both sides of (2) multiplied by dcU
and substituting (7), then the instantaneous active power can be express as
2
dc dcdc
d= +
d
U U P CU
t R (25)
Using the small-signal modeling method, namely*
*
ˆ
ˆ
dc dc dcU U U
P P P
(26)
Substituting (26) into (25), ˆ P is obtained2
*
*
ˆˆdˆˆˆ ( ) (2 )/d
dc dcdc dc dc
dc
U U P C U U U R
t U (27)
Since*ˆ
dc dcU U ,
2
*
ˆdc
dc
U
U can be ignored, then (27) is simplified
as follows:
*ˆ
ˆˆˆ ( ) 2 /dcdc dc dc
dU P C U U U R
dt (28)
Then, the transfer function between DC bus voltage and
PWM rectifier instantaneous active power could be shown
as*ˆ (s) 0.5
( )= =ˆ 0.5 +1(s)
dc dc p
U R U G s
RCs P (29)
Equation (29) is a one-order inertial element. In previouscontrol schemes, the voltage loop adopts a PI controller to
eliminate tracking error. However, if voltage ripples appear
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on the dc bus, the PI controller is unable to work properly
owing to the lack of a gain high enough at the ripple
frequency. To suppress voltage ripples on the DC side, a
resonant regulator link is brought in composing a
proportional–resonant (PR) control [15], and the formula of
the regulator is as (30)
pR 2 2
( )= +2 s+(2 )
I r P
c
K K sG s K
s s
(30)
where =100 is the grid angular frequency and r K is the
resonant coefficient. Its control parameters are designed
as =0.02 p K , =0.4 I K , =20r K and =c .
Finally, the overall control structure of the paper is shown
as in Fig. 3.
*
0 p Eq.
(15)
pi
pe Eq.
(7)
p P
pQ
*( )
p P +-
+-*
( ) pQ
ni
ne
Eq.
(7)n P
nQ
*( )n
Q +
-
*( )n P +-
p s
p s
n s
n s
+
+
s
++ s
pe
ne
Eq.
(20)
Eq.
(21)
PR ( )G s
*
dcU +-abce abci
a s
b s
c s
L RC R L
R L
R L
ae
ce
be
Fig. 3: the proposedcontrol structure based on DPC for three-phase
PWM voltage-source rectifier
4 Simulation
To verify the effectiveness and validity of the proposed
control strategy, the simulation has been implemented using
MATLAB/SIMULINK firstly. The smoothing inductor is
4mH, the DC bus voltage is 250V, the peak value of
unbalanced three-phase AC voltage is respectively 60V,
80V, 80V, the frequency is 50Hz, the load RLis100, the
output capacitor is 990F, SVPWM has a frequency of10kHz.
0.8 0.85 0.9 0.95
-50
50
0
t/s
8 0 8 0 9 0
ea
ia
i a / A
e a
/ V
(a)
0 0.3 0.6 0.9t/s
0
200
100
250
U d c
/ V
0 0 3 0 6
0.6 0.65 0.7
250.0
250.2
249.8
(b)
0 0.04 0.08 0.12t/s
-10
10
0 i a b c
/ A
(c)
0 0.3 0.6 0.9t/s
-10
-5
0
i n / A
(d)
Fig. 4: Simulation results under unbalanced situation
Fig. 4 (a) shows the grid voltage and input current of
A-phase, ia has the same phase with ea, and ,the power
factor is nearly 1. Fig. 4(b) shows the three-phase rectifiercould achieve given voltage very well under the grid
unbalance. Although the DC side voltage has second
harmonic, the oscillation range is only 0.2V, the second
harmonic gets very good suppression. Fig. 4(c) shows the
three-phase current is nearly symmetrical, and the current is
close to sinusoidal. Fig. 4(d) shows the negative sequence
current is almost zero.
The material object experiment is carried on with a
1600w prototype. Fig. 5(a) shows the output voltage and
input current when the PWM rectifier first starts. From Fig.
5 (b), we can see that during steady time, the phase between
input current and output voltage are basically same, the
current has a sinusoidal wave. Fig. 5 (c) shows the dynamicfeatures of output voltage and input current when the load
changing from 100 to 75 suddenly.
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Fig. 5: The waves of practicality experiment results
5 Conclusion
This paper presents a novel control scheme for PWM
rectifier under unbalanced grid voltage conditions to
eliminate the negative sequence current and fluctuations of
active and reactive power. A dual power control scheme is
presented and the active and reactive power can be
decoupled and controlled independently. The independent
control strategy made it possible to regulate the real power
completely, thus achieving the constant dc-link voltage. The
inner current loop consists of a PI controller and a repetitive
controller concerns about the power tracking performance
and suppress the power oscillation. The outer voltage loop is
used to regulate the dc output voltage.
The result shows that the proposed strategy could keep
grid current the same phase as grid voltage and realize the
grid side can function under unity power factor, the current
is nearly sinusoidal, the system could suppress the second
harmonic oscillation effectively and get good performance.
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