Decoupling Control of d and q Current

Components in Three-Phase Voltage Source

Inverter

Mirjana Milosevic

Abstract

The current control is an important issue in power electronic circuits, partic-ulary in dc to ac inverters which are used in ac motor drives and continuous acpower supplies where the objective is to produce a sinusoidal ac output whosemagnitude and frequency can both be controlled.

In this paper, the current control of three-phase pulse width modulatedvoltage source inverter (PWM-VSI) has been implemented in the rotating d,qreference frame.

Introduction

Three major classes of regulators have been developed over last few decades:hysteresis regulators, linear PI regulators and predictive dead-beat regulators[1]. Although high performance control strategies have been proposed, theystill exhibit coupling problems not providing the decoupling of the active andreactive current components (not allowing the independent control of the activeand reactive power) [2]. These classes can be further divided into stationaryand synchronous d,q reference frame implementations by applying ac machinerotating field theory [3]. The relationship between stationary and synchronousframe controllers is given in [4]. A short review of the available current controltechniques for the three-phase systems is presented in [5]. The synchronousframe controller has, also, become the standard solution for current control ofPWM rectifiers [6]. The real-time control strategy based on a non-linear statevariables feedback approach that decouples the active and reactive line cur-rent components allowing the independent control of active and reactive supplypower has been proposed in [7]. The decoupling control has been also appliedon high speed operation of induction motors where its depends on the accuracyof the stator inductance and the leakage factor [8].

In this paper, the current control of PWM-VSI has been implemented inthe rotating (synchronous) d,q reference frame because the synchronous framecontroller can eliminate steady state error and has fast transient response by de-coupling control. However, synchronous frame controller is more complex thenthe stationary frame controller and requires transforming of measured station-ary frame ac current to rotating frame dc components, and transforming theresult of control back to the stationary frame for implementation.

Two methods of decoupled current control are explained here: feedforward

1

and feedback decoupling control.

Mathematical Model of the Three-Phase VSI

The power circuit of three-phase VSI is shown in figure 1.

Vsc

Vsb

Vsa+

+

+

R

R

R

L

L

L

Sa

Sb

Sc

C+

-

Vdc

R1

V

ia

ib

ic

idc

Figure 1: VSI power topology

The dc and ac side equivalent circuits of the VSI are depicted in figure 2.

C+

-

vdc

R1

V

+

-

idc

+

RL

[v]abc

abc[i]

[vs]abc

Figure 2: Equivalent circuit of the VSI

Mathematical model of the equivalent circuit is given by:

Cdvdcdt

+ idc =V vdcR1

(1)

Ld[i]abcdt

+Ri = [v]abc (2)

ia + ib + ic = 0 (3)

where [v]abc = [v]abc [vs]abc.

2

Applying the transformation method from three-phase system (abc) to ro-tating frame (dq):

[xdxq

]=

2

3

cosw1t sinw1tcos(w1t 120) sin(w1t 120)

cos(w1t+ 120) sin(w1t+ 120

)

T

xaxb

xc

(4)

to the current [i]abc we have:

id =2

3[ia cosw1t+ ib cos(w1t 120

) + ic cos(w1t+ 120)] (5)

iq = 2

3[ia sinw1t+ ib sin(w1t 120

) + ic sin(w1t+ 120)] (6)

and similarly to the voltage [v]abc we have:

vd =2

3[va cosw1t+vb cos(w1t 120

) + vc cos(w1t+ 120)] (7)

vq = 2

3[va sinw1t+vb sin(w1t 120

) + vc sin(w1t+ 120)] (8)

If we apply the derivative to equation 5:

diddt

=2

3[diadt

cosw1t+dibdt

cos(w1t 120) +

dicdt

cos(w1t+ 120)]

2

3w1[ia sinw1t+ ib sin(w1t 120

) + ic sin(w1t+ 120)]

(9)

From equation 2 we have that:

diadt

=1

Lva

R

Lia

dibdt

=1

Lvb

R

Lib

dicdt

=1

Lvc

R

Lic

(10)

3

Using equations 5, 6, 7 and 10, equation 9 can be rewritten as :

diddt

= w1iq R

Lid +

1

Lvd (11)

Similarly, if we apply the derivative to equation 6 we have:

diqdt

= w1id R

Liq +

1

Lvq (12)

Equations 11 and 12 can be transformed in the Laplace domain (s-domain)as:

(sL+R)Id = Vd + w1LIq (13)

(sL+R)Iq = Vq w1LId (14)

Multiplying equation 14 with the complex number j and adding to equation13 we have:

(sL+R)(Id + jIq) = Vd + jVq + w1L(Iq jId) (15)

which can be rewritten as:

(sL+R+ jw1L)I =

V (16)

whereI = Id + jIq and

V = Vd + jVq.

The transfer function G(s) of the circuit is given with:

G(s) =

I

V

=1

sL+R+ jw1L(17)

4

sL+R

sL+R

wL

wL

Id

Iq

+

-

+

+

Vd

Vq

Figure 3: Crosscoupling of the d and q components

It can be seen from equation 17 that there is a cross-coupling between thed and q components (because of the jw1L part) which is also depicted in thefigure 3.

However, crosscoupling can affect the dynamic performance of the regulator[9]. Therefore, it is very important to decouple the coupling term for betterperformance.

Decoupling Methods

In this paper we applied two decoupling methods: feedforward and feedbackdecoupling method for current control of VSI in rotating d,q frame.

1) Feedforward method

-

+IrefPI Decoupling

+

+

dq

dq

dq

abc

abc

abc

Switchingcontrol

V

VSIVs

IV

Figure 4: Current control of VSI (feedforward decoupling method)

The block diagram of the current control of VSI with feedforward decouplingmethod is shown in figure 4.

The decoupling part from the same figure is given in figure 5 separately,

where V is the controlled voltage (output of the PI controller employed in

active and reactive current control in VSI).

In order to have d and q components decoupled we want to define transferfunction G1 such that:

5

V,

G1(s) G(s)V I

Figure 5: Feedfoward decoupling

Gdc(s) =

I

V

=1

sL+R(18)

From figure 5 we can calculate:

I = G(s)

V (19)

V = G1(s)

V (20)

combining these two equations and equation 18 and knowing, also, thatGdc(s) = G1(s)G(s)the transfer function G1(s) is defined as:

G1(s) = 1 + jw1L

sL+R(21)

2) Feedback method

-+Iref

PI

Decoupling

+

+

dq

abc

dq

abc

dq

abc

Switchingcontrol

V

VSIVs

IV

+

Figure 6: Current control of VSI (feedback decoupling method)

The block diagram of the current control of VSI with feedback decouplingmethod is shown in figure 6.

The decoupling part from the same figure is given in figure 7 separately,

where V is the controlled voltage (like in the feedforward decoupling method).

6

G(s)

G2(s)

+

+

V,

V I

Figure 7: Feedback decoupling

Similarly as in the previous method we have to define the transfer functionG2(s) such that the equation 18 is valid.

From figure 7 can be seen that:

V = G2(s)

I +

V (22)

Equation 19 is valid here, also.

From equation 19 and 22 we have:

(1G(s)G2(s))I = G(s)

V (23)

Finally, we have that:

Gdc(s) =

I

V

=G(s)

1G(s)G2(s)(24)

Combining this equation with equations 18 and 17 it can be seen that de-coupling transfer function G2(s) in the case of feedback method is given with:

G2(s) = jw1L (25)

7

Simulation Results

The VSI shown in figure 1 has been simulated using both Simulink andPLECS. Simulation results for steady state are presented in figure 8 (feedfor-ward method) and in figure 10 (feedback method). Also, simulation resultsfor the step change in reference current are given in figures 9 and 11 for bothmethods. The circuit parameters are L=2mH, C=100F, f=50Hz, V=400V,R1 = 1, Idref = 10A, Iqref = 0A (after 0.01 sec Idref = 15A).

0 0.005 0.01 0.015 0.02 0.025 0.0310

5

0

5

10

15

20

time [sec]

id,iq

Feedforward decoupling (id,iq)

Figure 8: d and q current components using feedforward decoupling method

0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.0510

5

0

5

10

15

20

25Feedforward decoupling method (id, iq)

id,iq

time[sec]

Figure 9: d and q current components using feedforward decoupling methodwith step change in reference current

8

0 0.005 0.01 0.015 0.02 0.025 0.034

2

0

2

4

6

8

10

12

14

time [sec]

id,iq

Feedback decoupling (id,iq)

Figure 10: d and q current components using feedback decoupling method

0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.055

0

5

10

15

20Feedback decoupling method

time[sec]

id,iq

Figure 11: d and q current components using feedback decoupling method withstep change in reference current

9

Comments

From equation 21 it can be concluded that when R 0 the value of theinductance is not necessary to be known and that could be an advantage ofthe feedforward method, which is not the case with the feedback method wherewe have to know value of inductance (equation 25). On the other hand, underthe same conditions (when R 0) we have an oscillatory behavior in the feed-forward method due to the integration (we have the integrator part 1/s), andwe do not have oscillations in the feedback method. It can be seen, as well, infigures 8 and 10.

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References

1. S. Buso, L. Malesani, P. Mattavelli, Comparison of Current Control Tech-niques for Active Filter Applications, IEEE Transaction on Industrial Elec-tronics, Vol. 45, No.5, October 1998., pp.722-729

2. J. Choi, S. Sul, Fast current controller in three-phase ac/dc boost converterusing d-q axis crosscoupling, IEEE Transaction on Power Electronics, Vol. 13,pp. 179-185, Jan. 1998.

3. Y.Sato, T. Ishiuka, K. Nezu, T.Kataoka, A New Control Strategy forVoltage-Type PWM Rectifiers to Realise Zero Steady-State Control Error inInput Current, IEEE Transaction on Industry Applications, Vol. 34, No.3, pp.480-486, 1998.

4. D. N. Zmood, D. G. Holmes, Stationary Frame Current Regulation of PWMInverters with Zero Steady State Error, PESC 99., pp. 1185 -1190, Vol.2

5. M.P. Kazmierkowski, L. Malesani, Current control techniques for three-phase voltage-source PWM converters: a survey, Industrial Electronics, IEEETransactions on Industrial Electronics, Vol. 45, No. 5, Oct.1998., pp. 691 -703

6. M.P. Kazmierkowski, M. Cichowlas, Comparison of current control tech-niques for PWM rectifiers , Proceedings of the 2002 IEEE International Sym-posium on Industrial Electronics, Vol. 4, pp. 1259 -1263

7. J.R. Espinoza, G. Joos, L. Moran, Decoupled control of the active and reac-tive power in three-phase PWM rectifiers based on non-linear control strategiesPESC 99., pp. 131 -136, Vol.1

8. J. Jung, K. Nam, A dynamic decoupling control scheme for high-speed oper-ation of induction motors, IEEE Transactions on Industrial Electronics, Vol.46, No. 1 , Feb. 1999., pp. 100 -110

9. D. N. Zmood, D. G. Holmes,G.H. Bode,Frequency-domain analysis of three-phase linear current regulators, IEEE Transactions on Industry Applications,Vol. 37, No. 2, March-April 2001., pp. 601 -610

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