A NEW CONTROL ALGORITHM FOR 3-PHASE PWM VOLTAGE SOURCE

CONVERTERS CONNECTED TO THE GRID Robert Thibault, Kamal Al-Haddad, Louis. A. Dessaint

École de technologie supérieure, Chaire de recherche du Canada CRC-CÉÉÉP

1100 Notre Dame ouest Montréal Québec, H3C 1K3 robert.thibault.1@ens.etsmtl.ca, kamal@ele.etsmtl.ca

Abstract This paper deals with the control of three-phase PWM voltage

source converters connected to the grid. With the use of “field-oriented control”, voltages and currents are usually transformed to the rotating d-q reference frame, where control becomes quite straightforward: because the current space vector in the rotating d-q reference frame is fixed, the PI controllers operate on dc, rather than sinusoidal signals. It is reported that one of the fundamental weakness of more classical control structures is the great difficulty in controlling the variables with sinusoidal references because of the limited ability of PI controllers to accurately track time varying inputs.

This paper describes a new and simpler, but otherwise equivalent control technique which solves this problem without the burden of transforming time-dependent three-phase quantities into the time invariant d-q reference frame and vice versa. This new control structure is based on a complex controller (i.e. with real and imaginary parts), which achieves zero steady-state error by controlling the current space vector directly in the stationary d-q reference frame.

The mathematical equivalence with field-oriented control is fully demonstrated. Simulation results are also presented which clearly show the indistinguishability of the two control systems.

Keywords: Voltage source converter; PWM; unity power factor.

1 IntroductionControl systems that control the active and reactive current

independently are generally based on “field-oriented control”. The control structure is comprised of an inner loop and an outer loop. The inner loop controls the transfer of power between the dc link and the grid. The outer loop provides the reference current to the inner loop and is designed to maintain a constant dc link voltage. With field-oriented control it is possible to deliver near sinusoidal currents to the grid at unity power factor. In steady state, vectors of ac current appear as constants in the synchronous reference frame, therefore, static errors can be nullified by the use of conventional PI controllers.

In this paper, we show that, by using a complex compensator operating directly in the stationary vector space, it is possible to obtain results absolutely identical to those obtained using field-oriented control. It turns out that the gains of the new complex regulator are exactly identical to those of the stationary PI controllers. The equivalence between the two control techniques is founded on the mathematical equivalence

of a conventional regulator of integral type in the synchronous reference frame with this new type of complex regulator in the stationary vector space.

The circuit diagram of the three-phase PWM voltage source converter is presented in figure 1.

Fig. 1. Three-phase voltage source converter topology.

2 Presentation of the method

Let us consider a general three-phase balanced system abcx :

( )( )( )

( ) ( )( )

( ) ( )

( ) ( )++

+−

+

==

tttX

tttX

tttX

tx

tx

tx

x

c

b

a

abc

φπω

φπω

φω

32cos

32cos

cos

0

0

0

X represents any variable such as a current or a voltage with

time varying amplitude ( )tX and phase ( )tφ . As in [1], we

define a complex space vector sx by the transformation: [ ] abc

s xx ⋅= 2,,1 αα where 32π

αj

e= and where the → symbol is used to represent complex numbers. The “s” index signifies that the space vector is expressed in the stationary reference frame. We obtain:

( ) ( ) ( ) ( ) ( )( )ttjcba

s etXtxtxtxx φωαα +⋅=++= 0

232 The system

being balanced, the inverse transformation can be written as:

⋅= sabc xx

αα 2

1Re

32

Our method can be summarized as follows: First, the current error is transformed from abc to the complex vector space:

0-7803-8886-0/05/$20.00 ©2005 IEEECCECE/CCGEI, Saskatoon, May 2005

1274

[ ] abcs ii ∆⋅=∆ 2,,1 αα . Next, the current error si∆ is processed by

the new complex regulator iG . Finally, the output vector su of the regulator is transformed back to abc via the inverse transformation:

⋅= sabc uu

αα 2

1Re

32

3 Current control in the complex vector space

Fig. 2. Representation of the inner current loop

Consider the inner current loop (Fig 2). To simplify, the PWM is represented by a unit gain transfer function. The reference current *si whose amplitude *I and phase *φ are constants and where 00 >ω corresponds to a positive sequence three-phase balanced system. The asterisks are used to indicate the reference.

( )*0** φω += tjs eIi , si is the space vector representing the

actual current, si∆ indicates the current error: sss iii −=∆ * .

iG represents the complex regulator acting directly on space vectors in the stationary reference frame which nullifies the

error si∆ in steady state. 0ωjs

KG i

i −=

, *

11 s

i

s iG

i+

=∆, s

is iGu ∆⋅=

A simple way to see that the error si∆ nullifies in steady state is the following: The error si∆ is of direct sequence as is the reference *si . When 0ωjs → we get ∞→iG .

4 Error response in dynamic mode

*

0

0*

0

*

1

11

1 s

i

s

i

s

i

s iKjs

jsi

jsK

iG

i+−

−=

−+

=+

=∆ω

ω

ωExpressed in the form of differential equations, we have:

*00

ssi ij

dtd

iKjdtd −=∆+− ωω .

The right member of the above expression cancels out for a reference current *si with pulsation 0ω :

( )( ) 0*

0*0

*0 =−=− +φωωω tjs eIj

dtd

ijdtd

We get: 00 =∆+− si iKj

dtd ω

This last equation constitutes a homogenous differential system of equations. There are no ‘sources’ in the right side, thus

si∆ will decay naturally towards zero.

Let us separate real and imaginary parts :

21 ijii s ∆+∆=∆ ,∆∆

−−−

=∆

ƥ

•

2

1

0

0

2i

i

KK

i

i

i

ii

ωω

We can write: ∆∆

=∆2

1

i

ii ,

−−−

=i

i

K

KA

0

0

ωω , iAi ∆⋅=∆

•

Applying Laplace transform allows to solve the system algebraically: ( ) ( ) ( )sIAisIs ∆⋅=∆−∆ 0 . Let us isolate I∆ :

( ) ( ) ( )01 iAsIsI ∆−=∆ − . Let us apply the inverse Laplace transform: ( ) ( ){ } ( ) ( )0011 ieiAsIti Kt∆=∆⋅−=∆ −−

( ) ( )( ) ( )

−=

+−+

+++= −−

tt

tte

Ks

Ks

KsKse tK

i

i

ii

Kt i

00

00

0

022

02

1

cossinsincos

21

ωωωω

ωω

ω

We get: ( ) ( ) ( )( ) ( ) ( )0

cossinsincos

00

00 itt

tteti tKi ∆

−=∆ −

ωωωω

Let us pose: ( ) 000 φjs eIi ∆=∆ I.e. : ( ) ( )

( )∆∆

=∆00

00

sincos

0φφ

I

Ii

We get:

( ) ( ) ( )( ) ( )

( )( )

( )( )+∆

+∆=

∆∆−

=∆ −−

000

000

00

00

00

00

sincos

sincos

cossinsincos

φωφω

φφ

ωωωω

tI

tIe

I

I

tt

tteti tKtK ii

Expressed under complex form: ( ) ( ) tKtjs ieeIti −+∆=∆ 000

φω

Finally: ( ) ( ) tKtjss ieeiti −∆=∆ 00 ω

Therefore, the current error si∆ rotates in vector space with the same angular pulsation 0ω as the reference current *si while its module undergoes an exponential decay.

5 Complex regulator expressed in system abc

Fig. 3. Complex regulator in coordinate system abc.

Let the complex transfer function iG acting on space vectors be:

sisi

s ijs

KiGu ∆

−=∆⋅=

0ω We have the equations relating phase voltages or currents to space vectors:

[ ] cbaabcs iiiii ∆+∆+∆=∆⋅=∆ 22,,1 αααα

[ ] cbaabcs uuuuu 22,,1 αααα ++=⋅=

For more generality, let us include a homopolar component which does not appear in the representation by space vectors:

[ ] cbaabc iiiii ∆+∆+∆=∆⋅=∆ 1,1,10

[ ] cbaabc uuuuu ++=⋅= 1,1,10

1275

The transfer function for the homopolar component is pure real [3]:

020

20 is

sKu i ∆

+=

ωBy simple substitution, we can write:

[ ] [ ]∆∆∆

⋅−

=⋅

c

b

ai

c

b

a

i

i

i

jsK

u

u

u2

0

2 ,,1,,1 ααω

αα

[ ] [ ]∆∆∆

⋅+

=⋅

c

b

ai

c

b

a

i

i

i

ssK

u

u

u

1,1,11,1,1 20

2 ω

By extraction of real and imaginary parts, we can solve for

each component of abcu :

∆

∆

∆

−

−

−

+=

c

b

a

i

c

b

a

i

i

i

s

s

s

sK

u

u

u

33

33

33

00

00

00

20

2

ωω

ωω

ωω

ω

6 Inverter block diagram in system abc

Fig. 4. Inverter schematics in coordinate system abc.

( )−

−

−

+=

s

s

s

sK

sG ii

33

33

33

00

00

00

20

2

ωω

ωω

ωω

ω

( )( ) abcpiabc iKsGu ∆⋅+=

7 Inverter block diagram in the D-Q synchronous reference frame

According to reference [2].

=ed

eqe

qdu

uu

,=

ed

eqe

qdi

ii

, =*

**

ed

eqe

qdi

ii , e

qdeqd

eqd iii −=∆ *

eqdp

ieqd iK

sK

u ∆⋅+= , =0E

eeqd

E represents peak grid phase voltage. To achieve a unit power-factor, we must have: 0* =e

di . The active power

supplied by the grid is : ( ) eq

ed

ed

eq

eq Eiieie

23

23 =+

Fig. 5. Inverter controllers in synchronous d-q frame

8 Equivalence of transfer functions in stationary coordinate system ABC and synchronous

frame D-Q Let us show the equivalence of the transfer functions

( ) pi KsG + and p

i Ks

K+ in the coordinate systems ABC and

D-Q respectively:

( )( ) abcpiabc iKsGu ∆⋅+= , eqdp

ieqd iK

sK

u ∆⋅+=

Let us start by carrying out the passage from the synchronous D-Q reference frame to the stationary D-Q reference frame: Let us express the transfer function in the synchronous reference frame in the form of a differential equation:

eqdpi

eqd i

dtd

KKudtd ∆⋅+=

The passage to the stationary reference frame is carried out by the reverse Park transformation:

( ) ( )( ) ( )

−=

tt

ttR

00

00

cossinsincos

ωωωω ,

qdeqd xRx ⋅=

Note that for convenience, Park’s transformation has been separated into two operations. After substitution, multiplication by the inverse Park transformation, yield to:

( ) ( ) ( )qdpqdiqd iRdtd

KiRKuRdtd ∆⋅+∆⋅=⋅

( ) ( ) ( )∆⋅+∆⋅⋅=⋅⋅ −−qdpqdiqd iR

dtd

KiRKRuRdtd

R 11

We get:

( ) ( ) ∆−

−∆+∆=⋅−

− qdqdpqdiqdqd iidtd

KiKuudtd

00

00

0

0

0

0

ωω

ωω

Finally, let us apply the Laplace transform:

qdpqdi

qd iKis

s

sK

u ∆+∆⋅−+

=0

020

2 ωω

ωFor more generality, the homopolar component is included:

qdpqdi

qd iKi

s

ss

sK

u 00

0

020

20

00

00∆+∆

−+

=ω

ωω

1276

=

d

qqd

u

u

u

u0

0,

∆∆∆

=∆

d

qqd

i

i

i

i0

0,

−

−−=

23

230

21

211

21

21

21

32T

,abcqd xTx ⋅=0

Let us carry out now the passage from the stationary reference frame D-Q to the stationary system ABC by using the inverse Clarke transformation. By substitution, then multiplication by the inverse operator, one obtains:

abcpabci

abc iKiTs

ss

Ts

Ku ∆+∆⋅

−⋅

+= −

0

01

20

2

00

00

ωω

ω

Finally, we obtain the transfer function of the compensator in the coordinate system ABC:

abcpabci

abc iKi

s

s

s

sK

u ∆+∆

−

−

−

+=

33

33

33

00

00

00

20

2

ωω

ωω

ωω

ω

9 Equivalence of current references in systems ABC and synchronous D-Q

We have: 0* =edi

( ) ( )( ) ( )

( )( )−

=−

== −

titii

tt

ttiRi e

q

eq

eqe

qdqd0

*0

**

00

00*1*

sincos

0cossinsincos

ωω

ωωωω

( )( )

( )

+

−=−

= −

32cos

32cos

cos

sincos

0

0

0

0

*

0*

0*1*

πω

πω

ω

ωω

t

t

t

i

ti

tiPi eq

eq

eqabc

10 Passage from synchronous D-Q frame to system ABC including the decoupling matrix

The feed-forward (not shown on the block diagram) being used to uncouple the axes D and Q expressed in the synchronous reference frame D-Q:

eqd

eqdp

ieqd i

LL

iKs

Ku ⋅

−+∆⋅+=

00

0

0

ωω

In the ABC coordinate system, one finds:

( )( ) abcabcpiabc i

LL

LL

LL

iKsGu ⋅

−

−

−

+∆⋅+=

033

30

3

330

00

00

00

ωω

ωω

ωω

11 Simulation results

A step change in dcI has been applied at t = 0.2. The values of the parameters used for the simulation are: L = 1 mH, R = 0.1 ohms, ω = 377, C = 5000 Fµ , E = 165 Volts, PWM carrier frequency = 1 kHz. As noted from the identical current

responses, there is no observable difference between the two control algorithms.

Fig. 6. Step change in dcI at t = 0.2.

Fig. 7. Current response after a step change in dcI at t = 0.2 with F.O.C. and with the new complex controller respectively.

12 Conclusions In this paper, it was shown that a complex controller in the

stationary reference frame acting on space vectors of current is mathematically identical to a conventional integral controller operating in the synchronous d-q reference frame.

References [1] Joseph Vithayathil, POWER ELECTRONICS PRINCIPLES AND

APPLICATIONS. New York: McGraw-Hill, 1995. [2] V. Blasko and V. Kaura, “A new mathematical model and control of a

three-phase ac–dc voltage source converter,” IEEE Trans. Power Electron., vol. 12, pp. 116-123, Jan. 1997

[3] S. Fukuda and R. Imamura, “Application of a sinusoidal internal model to current control of three phase utility-interface-converters,” Power Electronics Specialist, 2003. PESC 34th Annual Conference on , Volume: 3 , 15-19 June 2003, Pages:1301 - 1306 vol.3

1277