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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: [email protected]

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.

~

~

~

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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