A Direct Power Control Scheme for Three-Phase PWM Rectifiers Based on Sliding-Mode Variable Structure Control Theory

A Direct Power Control Scheme for Three-Phase PWM Rectifiers Based on Sliding- Mode Variable Structure Control Theory

Wanwei Wang, Huajie Yin and Lin Guan

College of Electrical Power, South China University of Technology Guangzhou, 510640, China

[email protected]

Abstract -- A direct power control (DPC) scheme for three-

phase PWM rectifier based on sliding-mode variable structure control theory is presented. Compared with the traditional PI controller, the proposed scheme is much more robust with better transient performance, lower total harmonic distortion (THD) and simpler tuning of parameters. Modeling and simulation done in MATLAB/SIMULINK environment shows that the proposed scheme is superior to traditional PI controller.

Index Terms--Direct power control, proportional control, pulse width modulation, rectifier, simulation, variable structure systems.

I. I

NTRODUCTION

Pulse Width Modulated (PWM) voltage source rectifiers (VSR) are widely used in AC variable speed drive systems, reactive compensation and active power filtering for their high power factor, small total harmonic distortion (THD), bidirectional power flow and fast dynamic response.

In a PWM-VSR, the current control strategy can be divided into two types, direct [1]-[3] and indirect current control [1][4][5]. The direct current control is with complex control structure and difficult parameter tuning, and the indirect current control is with poor dynamic characteristics.

Hence [6] put forward a direct power control (DPC) strategy based on instantaneous power balance, which gained applause from the beginning [7][11][12]. The DPC system uses an outer voltage loop with PI controller to get the active power reference, and uses an inner power loop with a hysteresis comparator to force the actual active power to keep up with the reference [7][8]. However this strategy is still very sensitive to parameters and disturbances and is with large THD. Though many improved measurements were proposed, but their results are still unsatisfactory [7][11][12].

Here we propose a DPC scheme based on sliding-mode variable structure controller. This scheme is simple in parameter tuning and convenient in design. It is more robust with better dynamics and smaller THD compared with traditional PI controller.

II. DPC M

^{ODEL OF}

T

^HREE

P

^HASE

PWM-VSR The main circuit topology of the three phase PWM-VSR is shown in Fig.1. The grid side uses three phase symmetric connection without neutral line, and the three phase full

This work is supported by National Natural Science Foundation of China (NSFC Grant No. 50777022).

bridge rectifier uses IGBT with parallel diode as the switching devices.

C R L

ua

ub

uc

ia

ib

ic

A B

C Udc

i0

RL

Fig. 1 Circuit topology of three-phase PWM VSR

The switching function S

k

for each phase of the PWM rectifier is defined as

1 if the upper arm of phase k conducts 0 if the lower arm of phase k conducts , ,

S

k

⎧ k a b c

= ⎨ ⎩ ( = ). (1)

Based on Fig. 1, the state space equations of the PWM rectifier in the three phase abc stationary coordinate system can be written as

^[11]

, ,

, ,

, , 0

1 1 1

( )

3

1 1 1

( )

3

1 1 ( 1 )

3

1 ( )

=

=

=

⎧ = − − −

⎪ ⎪

⎪ = − − −

⎪⎪ ⎨

⎪ = − − −

⎪ ⎪

⎪ = − + + +

⎪⎩

∑

∑

∑

a a a dc a k

k a b c

b b b dc b k

k a b c

c c c dc c k

k a b c

dc a a b b c c

R

d i u i U S S

dt L L L

R

d i u i U S S

dt L L L

d i u R i U S S

dt L L L

i

d U S i S i S i

dt C C

, (2)

where u

x

and i

x

is the voltage and current of phase x(x=a, b, c), L and R are the grid side filter inductor and resistance, U

dc

is the dc voltage, and i

o

is the load current whose direction is positive when in rectifying state.

Transforming (2) from the abc three phase stationary coordinate system to the dq two-phase rotating coordinate system, we get

0

1 1

1 1

1 ( )

⎧ = − + −

⎪ ⎪

⎪ = − − −

⎨ ⎪

⎪ = − + +

⎪⎩

d d d q d dc

q q q d q dc

dc d d q q

d R

i u i i S U

dt L L L

d i u R i i S U

dt L L L

i

d U S i S i

dt C C

ω

ω , (3)

where S

d

and S

q

denote the rectifier switching function in dq

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coordinate system, u

d

, u

q

and i

d

, i

q

are the d- and q-axis source voltages and currents respectively, and ω is the angular frequency. Here we assume the source voltage is with constant frequency and amplitude, and let the source voltage space vector be on the d-axis (i.e. using source voltage orientation), hence we have u

q

=0 and du

d

/dt =0.

The dc side voltage dynamic process can be described by using the active power balance relation. Let p

ac

denoting the ac grid side input active power and p

dc

the dc side absorbed active power, we have

ac d d q q d d

p = u i + u i = u i , (4) 1

2

dc dc dc dc dc dc

L

p U i CU d U U

dt R

= = + , (5)

where R

L

denotes the equivalent load resistance, CU

dc

dU

dc

/dt denotes the dc side transient energy absorbed or released by capacitance C , and U /R

_dc² L

denotes the instantaneous energy absorbed or released by load.

In accordance with the principle of power balance, the input power of the rectifier equals to the instantaneous output power, such that:

ac dc loss

p = p + p , (6) where p

loss

is the losses in the filter inductor and in the switch equivalent resistance. Neglecting losses, we have

1

2

ac dc dc dc

L

p CU d U U

dt R

= + . (7)

or

2

2

2

2

(

_dc

)

_{dc ac}

L

d U U p

dt = − R C + C (8) And the grid side reactive power is

q d d q d q

q u i = − u i = − u i . (9) From (3) and (9), we get

1

ac d q dc

dq R q p u S U

dt ⁼ L ⁺ ω ⁺ L . (10) III. D

^{ESIGN OF}

S

^LIDING

M

^ODE

C

ONTROLLER FOR

DPC Systems with changeable structure during controlling are called variable structure system. Sliding mode variable structure system is a special type of variable structure system, in which the system control is with switching action, and on the switching surface the system will slide along fixed trajectory. Because the sliding mode is very robust to external disturbance and internal parameter change, the sliding mode variable structure system is used widely.

The design of sliding mode controller mainly includes two points: (i)choose proper sliding modes based on the requirements on dynamics; (ii)determine the control rules so that the sliding modes be stable and reachable.

A. Choice of the sliding surface

Because there are two external control variables U

dc

and q in the three phase PWM-VSR, we can choose the following sliding surface according to (8) and (10),

*

1 1

*2 2 *2 2

2 2

( ) 0

( ) ( ) 0

⎧ _{= − =}

⎪ ⎨

= − + − =

⎪⎩

^{dc dc dc dc}

S k q q

S k U U d U U

dt

(11)

where U

dc*

is the reference for U

dc

, q

^*

is the reference for q, and the control coefficients k

1

and k

2

are non-zero constants.

Substitute (8) into (11) and let dU

_dc^*2

/ dt = 0 , we get

*

1 1

2

*2 2

2 2

( ) 0

( ) 0

2

dc

dc dc ac

L

S k q q

Ck U

S U U p

R

⎧ = − =

⎪ ⎨

= − + − =

⎪ ⎩

(12)

Through farther simplification we have

* 1

*2 2

2 0

0

(

_{dc dc}

)

_{dc ac}

0

S q q

S Ck U U U i p

⎧ = − =

⎪ ⎨

= − + − =

⎪⎩ (13)

Because we need the rectifier be with unity power factor, hence we have q=0 in steady state, and it is reasonable to assume

*

* *2 2

0

0

( )

ac dc dc dc

q

p Ck U U U i

⎧ =

⎪ ⎨

= − +

⎪⎩ (14)

From it we have

* 1

* 2

0

ac ac

0

S q q

S p p

⎧ = − =

⎪ ⎨

= − =

⎪⎩ (15)

B. Determining of the Control rules

According to [9], when we choose the Lyapunov function as v=S

²

/2, the sufficient condition for the existence of the sliding modes is that v is positive definite and v is negative definite, that is

0

SS  < (16) Let us consider the control rules of S

1

. When S

1

>0, we have q

^*

>q, and in order to satisfy (16), we need to increase q so that we can have S 

₁

< 0 . When S

1

<0, we have q

^*

<q, and in order to satisfy (16), we need to decrease q so that we can have S 

1

> 0 .

In the same way, we can get the control rules for S

2

. From the above analysis, we can see that the fundamental requirement is that the actual value of the control variables should follow its reference value continuously so that the deviation decrease. This requirement can be satisfied by using hysteresis control.

IV. B

UILDING OF

S

^WITCHING

T

^ABLE

From the previous analysis, we can get the DPC control block diagram based on sliding mode variable structure algorithm for three phase PWM-VSR, as shown in Fig. 2.

And the PDC control block diagram based on traditional PI algorithm is also shown in Fig. 3. Both of these two algorithms use the same logic switching table based on space vector PWM scheme to control the switches of the rectifier.

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O

R

+ +

_ _

Power and Voltage estim ator

Switching table

ua

ub

uc

L ia

ib

ic

ua

uß

SaSbSc C

i0

Udc*

Udc

q^*=0 q θn

Lim iter

Load

*ac

p

Sq′ Sp′

pa c

Sliding M ode Control + _

X

X X

X

X

X C

k

X

++ _X

i0

Fig. 2 Block diagram of sliding-mode DPC PWM rectifier

O

+ _ R

+ X +

_ _

Power and Voltage estim ator

Switching table

PI

ua

ub

uc

L ia

ib

ic

ua

uß

SaSbSc C

i0

Udc*

Udc

q^*=0 q θn

Limiter

Load

*ac

p

Sq′ Sp′

pa c

Fig. 3 Block diagram of PI algorithm DPC PWM rectifier

In Fig. 2 and Fig. 3, the output signal S′ and

_p

S′ of the

_q

comparators are defined as

* '

*

* '

*

1 ( )

0 ( )

1 ( )

0 ( )

ac ac p

p

ac ac p

q q

q

when p p H

S when p p H

when q q H

S when q q H

⎧ < −

= ⎨ ⎪

> +

⎪⎩

⎧ < −

= ⎨ ⎪

> +

⎪⎩

, (17)

where H

p

is the width of the active power hysteresis operator, and H

q

is the width of the reactive power hysteresis operator.

The θ

ⁿ

in Fig.2 and Fig.3 is the phase angle of the estimated source voltage vector u, and can be calculated in the αβ two phase stationary coordinate system by the following equation [13],

arctan ( 0)

arctan ( 0)

n

u when u

u

u when u

u

β

α α

β α

α

θ π

⎧ ≥

⎪ ⎪

= ⎨ ⎪ + ⎪⎩ <

, (18)

and we divide θ

n

into twelve equal intervals as shown in Fig.4, which tells us where the estimated voltage vector u is located.

u I

U

2

(110)

U

1

(100)

U

6

(101) = u

r

U

5

(001) U

4

(011)

U

3

(010)

i

r

i I

r

− θ

1

θ

7

θ

6

θ

8

θ

9

θ

10

θ

₁₁

θ

5

θ

⁴

θ

3

θ

2

θ

12

U

0

(000) U

7

(111)

u

_β

u

_α

u u −

r

Fig. 4. Twelve equal intervals of

θ

_nand the voltage vector selection for space vector PWM scheme

From (17) and (18), we can determine the values of the switching function S

a

, S

b

and S

c

, whose combination corresponds to the desired voltage vector u

r

, which is among the eight voltage vectors U

0

, U

1

, …, and U

7

as shown in Fig.

4. For example, consider a case where the estimated u is in the region of θ

¹

, the desired current vector for generating p

^* ac

is i

r

, and the estimated current is i=I which lags behind and less than i

r

, as shown in Fig.3. Because p

ac

= i

_α

u

_α

+ i

_β

u

_β

and q=i

α

u

β

−i

β

u

α

, we have p

ac

<p

^* ac

and q>q

^*

. If we further have p

ac

<p

^* ac

−H

p

and q>q

^*

+H

q

, then according to (17) we will have S′ =1 and

p

S′ =0 for this case. To make i approach to i

q r

, or in other words, to make p

ac

approach to p

^* ac

and q approach to q

^*

, we need to choose a proper u

r

. From (2), when neglecting the grid side resistance R, we have

0

( ) (0) 1

^t

(

_r

)

i t i u u dt

= + L ∫ − , (19) where i(0)=I. From Fig. 4 we can see U

6

(101) is a proper choice for u

_r

because i can approach to i

_r

along the direction of u −u

r

certainly. Hence we have S

a

S

b

S

c

=101 for S′ =1,

_p

S

q

′ =0 and θ

n

in θ

1

. In similar way we can get all the other switching function values for other cases, and they are listed in Table I [8][10][11][14].

TABLE I

SWITCHING TABLE FOR DPCPWMRECTIFIER

S′p S′_{q θ}1 θ² θ³ θ⁴ θ^{5 S}θ^{a 6 Sb S}θ^c7 θ⁸ θ⁹ θ¹⁰ θ¹¹ θ¹²

1 0 101 111 100 000 110 111 010 000 011 111 001 000

1 1 111 111 000 000 111 111 000 000 111 111 000 000

0 0 101 100 100 110 110 010 010 011 011 001 001 101

0 1 100 110 110 010 010 011 011 001 001 101 101 100

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V. S

IMULATION AND

A

^NALYSIS

In order to see the validity and advantages of the sliding mode variable structure algorithm, we have done some simulations for both of the algorithms for comparison propose. The simulation parameters are as follows: the ac side is with 220Vrms phase voltage, 50Hz frequency, 0.1 Ω line resistance, and 1.6mH line inductance; and the dc side is with 700Vdc voltage, 3000 μF capacitance, and 22.3kW rated power output; the control parameters are K

p

= 3 and K

i

= 15 for the PI algorithm, and k = 200 for the sliding mode control algorithm. The simulation results are shown in Fig. 5 to Fig.

10.

Fig. 5 is the waveforms for the sliding mode algorithm.

Fig. 5(a) is the dc output voltage waveform for rectifying operation. Clearly the dc voltage overshoot is small, the rising edge is very smooth, the settling time is less than 0.02s, and the steady-state error is zero with very small ripples. Fig.

5(b) is the ac grid side voltage and current waveforms for both rectifying operation and inverting operation. Clearly the current is sinusoidal with less ripples, and the power factor is unity in both operation modes.

(a)DC voltage for rectifying operation.

(b)AC grid side voltage and current for both rectifying and inverting operations.

Fig. 6 is a comparison on the total harmonic distortions(THD) of the ac grid side current for the both the sliding mode algorithm and the PI algorithm. Clearly the sliding mode variable algorithm is superior to the PI algorithm on THD aspect.

Fig. 7 and Fig. 8 are the simulation results for the both control algorithms when the load experiences a sudden change from 11.15kW to 22.3kW at 0.05s. Fig. 7(a) and (b) tell the load change has very limited effects on the dc voltage and the ac current has no distortion for the sliding mode control. On the other side, from Fig. 8(a) and (b) we can see the load change has large effects on the dc voltage which can not follow its reference, and the ac side current needs long time to settle down.

Fig. 9 and Fig. 10 are the simulation results for the both control algorithms when the ac side voltage experiences a 10 percent rising at 0.05s. Fig. 9(a) and (b) tell for the sliding mode algorithm, the dc voltage has no significant change, the ac side current experiences only very small distortion and it will settle down very fast when the ac side voltage experiences change. On the other side, from Fig.

10(a) and (b) we can see that for the PI algorithm, the dc voltage will have non-zero steady-state error, and the ac side current will experience large distortion for a long time when the ac side voltage experiences a change.

(a)THD of PI algorithm.

(b)THD of sliding mode algorithm.

Rectification

Inversion

Time (s) Phase voltage ua(V) and current ia (A)DC voltage (V)

Time (s)

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(a)DC voltage waveform.

(b)AC side phase voltage and current.

Fig. 7 Sliding-mode DPC PWM rectifier experiences load change.

(a)DC voltage waveform.

(b)AC side phase voltage and current.

Fig. 8 PI DPC PWM rectifier experiences load change.

(a)DC voltage waveform

(b) AC side voltage and current.

Fig. 9 Sliding-mode DPC PWM rectifier experiences ac side voltage change

Phase voltage ua(V) and current ia (A)

Time (s)

DC voltage (V)

Time (s) Phase voltage ua(V) and current ia (A)

Time (s)

DC voltage (V)

Time (s) Time (s) Phase voltage ua(V) and current ia (A) DC voltage (V)

Time (s)

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(a)DC voltage waveform.

(b)AC side voltage and waveform

Fig. 10 PI DPC PWM rectifier experiences ac side voltage change

For Fig. 7 to Fig. 10, we can see clearly that the sliding mode algorithm has very good anti-interference ability and is very robust compared with the PI algorithm.

VI. C

ONCLUSION

The proposed DPC scheme based on sliding-mode variable structure algorithm for three-phase VSR is modeled and simulated, and its performances are compared with the conventional PI controller. The

simulation results show that this algorithm is rather simple, robust, not sensitive to disturbance, and with very good dynamic characteristics.

R

EFERENCES

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[11] M. Malinowski, F. Blaabjerg, “Virtual-flux-based Direct Power Control of Three-phase PWM Tectifiers,” IEEE Trans. Industry Applications, vol.37, no.6, pp.1019-1026, 2001.

[12] Mariusz Malinowski, Marian P. Kazmierkowski, “Direct Power Control of Three-phase PWM Rectifier Using Space Vector Modulation - Simulation Study,” in Proc. IEEE ISIE’02, vol.4, pp.1114-1118, July 2002.

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Phase voltage ua(V) and current ia (A)

Time (s)

DC voltage (V)

Time (s)

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