Virtual-Sensor-Based Control of PWM Current Source Rectifiers

I Jose R. Espinoza Giza Jobs * Ernest0 Araya Daniel Sbiirbaro Luis A. M o r h T.

Departamento de Ingenieria ElCctrica Universidad de Concepcion

Casilla 53-C, Correo 3 Concepcion, CHILE

Tel.: +56 41 203512, Fax.: +56 41 246999 e-mail: jespinoz@manet.die.udec.cl

Abstruct - High performance control strategies applied to pulse- width modulated current source rectifiers (PWM-CSR) require sensing of the supply currents, the input capacitor voltages, and synchronization with the ac supply voltage in addition to the dc voltage and current sensors used for protection purposes. For instance, control strategies developed to provide the necessary input line current damping - thus avoiding the need for damping resistors - and decoupled control of the active and reactive instantaneous powers, require the sensing of the supply currents and input capacitor voltages. As a result, a large number of sensors is needed and the overall reliability is therefore reduced. This paper proposes a technique based on virtual sensors to provide the required ac current and voltage values without actually sensing the electrical variables. The technique takes into account the non- linear model of the PWM-CSR by using the information from the dc current and dc voltage sensors in combination with a linear state observer and a linear parameter identification algorithm. As a result, at least four sensors can be eliminated, while the features of the control strategy are preserved. The paper includes a complete formulation of the virtual sensor based algorithm and its application to the control of active and reactive instantaneous powers in a PWM-CSR. Results are presented to confirm the validity of the theoretical considerations.

I. INTRODUCTION

PWM current source rectifiers (CSR) offer an interesting alternative to thyristor and voltage source rectifiers as adjustable dc high power source [l-21. Features include low input current harmonic distortion, variable dc voltage, and a high overall power factor. Recently introduced control improvements have included: (a) the elimination of the input filter damping resistors which reduce the overall performance [ I ] , and (b) the complete control of the input displacement power factor, allowing the rectifier to operate as a variable source of leading or lagging reactive power in addition to its inherent operation as a variable high active power source [ 2 ] .

In practice, to implement these high performance control strategies, several electrical variables must be measured and processed on-line to achieve the control objectives. For instance, the aforementioned strategies require the sensing of the supply currents and input capacitor voltages, and synchronization with the ac supply voltage in addition to the dc voltage and current sensing used for protection purposes. As a result of the large number of sensors, the overall reliability is

Department of Electrical and Comp. Eng. * Concordia University

1455 de Maisonneuve B.W. Montreal, QuCbec, CANADA, H3G 1M8

Tel.: + I 514 8483080, Fax.: +I 514 8482802 e-mail: geza@ece.concordia.ca

reduced, a draw back in power electronic converters. On the other hand, identification algorithms as well as state observers have been used in electrical systems for parameter estimation [3], modeling [4], protection [ 5 ] , waveform reconstruction [6], and control purposes [7-81.

This paper proposes a technique based on virtual sensors [9-101 to derive the required ac supply input current and ac input filter voltage measurements without actually sensing the electrical variables in PWM-CSR. Thus, the hardware required to implement enhanced control strategies is reduced. The technique uses the information of the dc voltage sensor in combination with switching states and a parameter identification algorithm to reconstruct the ac input filter voltages. Then, these voltages and the dc current measurement in combination with a reduced-order observer are used to derive the ac supply input current. This combination deals naturally with the non-linear model of the topology allowing the implementation of a linear parameter identification and a linear reduced-order observer algorithm. As a result, at least two voltage and two current sensors can be removed, while the features of the on-line control strategy are preserved.

The paper includes a complete formulation of the virtual sensor based algorithm and its application to control the active and reactive instantaneous 'powers in a PWM current source rectifier. Results based on digital signal processor implementation confirm the validity of the theoretical considerations.

11. ADVANCED CONTROL STRATEGIES IN PWM-CSRs

The power circuit topology is composed of a three-phase PWM-CSR, a dc link inductor LdL. and a three-phase LC input filter, Fig. 1. The PWM-CSR generates a dc voltage v, that is regulated by properly switching the power devices &,,.6

throughout the gating signals which are generated by a PWM technique (e.g. space vector modulation). The LC input components filter out the current harmonics allowing the supply current [i,y]abc to become near sinusoidal. Finally, the dc link inductor L d c smoothes the dc link current id, which assures a PWM type input current [i,labc and a current source operating mode on the dc side of the converter. This mode is appropriate to medium voltage drives based on current source inverters, where PWM-CSRs can allow the overall topology to either

0-7803-5589-X/99/$10.00 0 1999 IEEE 220 1

I I I I I Load I Rectifier ; Filter !

ac Input PWM I DC I Supply Filter

I I

v., Space Vector Modulation a

GI-6 lm1dq

Non-linear Controller Non-linear Controller

Modulation

,labc

I

t t ' I I 1

Pre/ 4re/

V, =22OV fw =900 z =50Hz LJ, = 2 0 m H

Li = 3 m H hC = 2 0 n C, = 150 pF

Fig. 1 PWM current source rectifier.

absorb /generate reactive power frodinto the distribution system [2 ] , an important consideration in medium to high power applications.

This last feature is obtained by combining an appropriate modulating technique with a high performance control strategy. In this paper, a space vector modulation (SVM) technique in combination with a non-linear controller are used. The space vector modulation block, Fig. 1, generates low switching frequency ( 2 1 kHz ) gating patterns GI-6 based on the modulating signals and the supply voltages [v,]abc to assure perfect synchronization with the ac mains. The modulating signals are sinusoidal waveforms for near sinusoidal ac supply currents. The non-linear controller block, Fig. 1, generates the modulating signals [m]dq such that the resulting ac supply currents [irlabc draw from the ac mains pWf active power and qre, reactive power [2] . This is achieved by proper processing of the ac currents [&labc and capacitor voltages [Vr]abc as shown in (1). Such expression is in practice implemented by a digital system, where the modulating

technique is also executed. Thus, the power switch states (either ON or OFF) are known by the digital system. Fig. 2 shows the characteristics voltage and current waveforms in the P W M - CSR for leading power factor.

As expected, the supply current and the capacitor voltage measurements are required on-line. In general, high performance control strategies require these variables to effectively achieve their goals. In this work, a virtual sensor provides these values on a instantaneous basis by properly manipulating the dc link voltage, dc link current, supply voltages and the power switch states.

111. PROPOSED VIRTUAL SENSOR FOR THE SUPPLY CURRENTS AND CAPACITOR VOLTAGES

The virtual sensor is composed of a parameter identification algorithm and a reduced order observer algorithm that reconstruct the capacitor voltages [v,],bc and produce the supply line currents [i.Jabc, respectively. It is implemented in

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j 2 ( t k = +cp ( t k = a2 ( t k >vsa ( t k -k b2 ( t k > v , ~ p ( t k > (4)

where, iCa ( tk ), i,, (tk ) are the reconstructed capacitor voltages; and a,(tk), b,(tk) are the coefficients to be identified.

The sampled dc bus voltage at the instant f k v,.(tk) corresponds to one of the actual capacitor line-to-line voltages in the abc fiame according to the status (ON or OFF) of the rectifier power switches. This assignation, which is also implemented by the algorithm, is as follows: ~

if((G2(tk) = ON) AND (G3(tk) = ON)) Vbc(fk) = V,.(tk);

if ((Gl(tk) = ON) AND (G2(tk) = ON)) Vcu(fJ = - ~ , . ( t k ) ;

if ((G3(tk) = ON) AND (G4(tk) = ON)) vub(tk) = -V,.(fk);

if((G4(fk) = ON) AND (G5(tk) = 0" VCU(tk) = V X t k h if ((G~(tk) = ON) AND (G6(tk) = ON)) Vbc(tk) = -V,.(tk);

if((Gdtk) = ON) AND (Gi(tk) = ON)) vub( tk)

where vub(tk), vbc(tk), and vc,(tk) are the actual portions of the capacitor line-to-line voltages. Thus, the actual phase voltages in the ap fiame at the instant tk are given by,

v,.(tk);

Fig. 2 Characteristics steady. state waveforms for leading power factor. (a) Supply phase voltage, v,, supply line current, b, line capacitor voltage, v,, and ac rectifier current, i,o. (b) Rectifier dc link voltage, v,, dc link current, idc, and gating signal for switch I , GI.

conjunction the control strategy (e.g., non-linear controller) and the modulation technique (e.g., space vector) in the same digital contro 1 ler . A. Capacitor voltage reconstruction

The data available are the sampled dc rectifier voltage vr( tk) ;

the power switch state information Gk(tk), k = 1, ..., 6; and the sinusoidal templates ~ , ~ , ( t k ) , which are obtained fiom the supply voltages (Fig. l), where the abc to ap transformation is,

The actual capacitor phase voltages in the ap fiame at the

( 3 )

instant t k are assumed to be of the form:

j 1 ( r k ) = ( t k ) = al ( t k ) v y a ( l k -k b, ( t k >v.$ ( t k 9

which are conveniently represented by,

Y1 ( l k = V c a ( t k 1, Y2 ( r k ) = vcp ( t k . (6)

On the other hand, the reconstructed capacitor voltages ( 3 ) to (4) can be written as,

j l ( f k ) = N T ( t k ) i r ( t k ) , i = 1, 2 (7)

where ( [ k ) = [vm ( t k v s p ( t k )]9 ( t k ) = [a, ( t k b, ( [ k I]' . After k samples, the reconstructed capacitor voltages (7)

can be written as,

? , ( l k ) = M ( t k ) 6 i ( t k ) Y ( 8 )

where i ! , ( t k ) = [ i , ( t , ) ... j , ( tk ) lT and ~ ( t ~ ) = [ ~ * ( t , ) ... NT(tk)IT. Also, after k samples the actual capacitor voltages can be expressed as,

y, ( t k ) = [M,) ... Y , ( tk 1 1 ~ ' . (9) Applying the LSE method to (8) after k samples [ l l ] ,

including a weighting matrix W(tk), results in the expression for the coefficients 6, ( tk ) ,

The weighting matrix W(tk) defines the dynamic response or tracking capability of the identification algorithm [l 1 3 . An alternative form to allow fast response is to give more weight to more recent data. Thus, W(&) is chosen of the form,

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hk-1 ... 0 0 1 : .. . . . . . . . w ( t k ) = 0 ... 0 I 0 ... 0 1

where h is the forgetting factor and must ,e 0 < h 2 1 . Note that h = 1 implies that all the sampled data are equally weighted, and h < 1 that only the more recent data are used.

In real-time applications, the expression for the coefficients (10) can be firther simplified if M(&) and W(tk) are fixed. Under these conditions, the factor { MT(tk)W(tk)M(tk)}- 'MT(tk)W(tk) becomes constant and therefore could be pre- calculated and stored off-line. Thus, (10) could be solved on- line with a minimum number of operations. Unfortunately, in this application the factor M(tk) is time dependant. Therefore, a Recursive LSE (FUSE) algorithm is preferred to solve ( 1 0) on- line with a minimum number of operations and without requiring pre-calculated and stored data.

In order to fmd a recursive algorithm to solve (IO), the matrices M(tk), Yl(tk), and W(tk) are arranged as,

M(tk-1) y, h-1) M(tk) = ......... y, ( tk ) = . . . . . . . . .

[ N J hW(tk-1) j -0 I Y , ( t k ) 1. (12)

W(tk) =[....o;...../ ....;...I. Replacing (12) in the coefficients expression (10) yields,

6, ( t k ) = @M7 ( t k - 1 IW(tk-1 )M(4+, ) +

N ( t k INT (tk >>-I {hMT (lk-1)

W(tk-1 >y, ( 4 - 1 1 + N(t, >Y, (tk I> . (13)

Additionally, the propagation matrix P(tk) is defmed as P(tk) = (MT(fk)W(tk)M(rk))-'. Thus, the coefficients (10) can also be written as,

e, (tk 1 = P(tk > M T ( t k >W(tk >y, (tk 1 >

P-l(tk) = h P - ' ( t k - , ) + N ( t , ) N 7 ' ( t k ) . (15)

e, ( t k 1 = k 0 k - I ) +

~

(14)

and from ( 1 3) it is found that,

Thus, using (14) and (15) the coefficients (13) can be expressed as,

(16) "k 1 x

{Y , (tk 1 - N T (tk 16, (b-1 I> . The resulting expression (1 6), is a recursive expression on

0, ; however, it requires the matrix P(tk) and thereby, the inverse of the matrix M'(fk)W(tk)M(tk) at every sample instant. Fortunately, this operation is avoided since P(tk) can also be expressed recursively as,

1

1

1

I I

Fig. 3 Parameter identification algorithm flow chart for capacitor voltages

Finally, using (1 7), the coefficients (1 6 ) are given by,

The forgetting factor h defmes the weighting matrix W(tJ and thereby the stability and the speed of the identification algorithm [ I 1 1 . In fact, a high value (h = 1) provides good stability; however, the algorithm has poor tracking capabilities. Therefore, stability and dynamic response as a hnction of h are important design issues which are experimentally addressed in the next section.

As a result of the capacitor voltage reconstruction algorithm, the phase capacitor voltages itca ( t k ) and itcp ( t k )

are obtained at every sample instant tk in the crp fiame. A simplified algorithm flow chart is shown in Fig. 3.

B. SuppQ line currents derivation

The supply line currents are derived by proper manipulation of the sampled dc link current id'(tk); the power switch state information Gk(tk), k = I , ..., 6; and the supply voltages [v,]abc.

The model of the ac side of the rectifier is given by,

(19)

(20)

where [vJap, [iJap, [illap, and [v,Iap are the ap components of

d 1 1 2 [V' l ap = - [ i \ lap - -[if l a p 9

C, C,

- [[\ l a p = - [V' lap + - [v \ l ap , df L, L I d 1 1

[ v ~ l a b c , [ idabc, [ irlabo and [Vslabc, respectively. Note that [irlabc =

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........... . ................................... ...... ...................................... . ................ 1 &I-&

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Fig. 4 Reduced order observer block diagram for line currents

[G1-G4 G3-G6 G5-G2l7'.idc, is the PWM rectifier line current vector as shown in Fig. 1. In order to defme the reduced order observer, the model is written as:

1x1 = N x I + B1 [U] + B2[wI2 (21) [Yl = C[XI? (22)

where [U] = [illap is considered the input and [w] = [vJap is considered as a perturbation, both are available to be processed by the observer. Additionally, b] = [vclap is considered the output, which is provided by the capacitor voltage reconstruction algorithm, therefore [y] = [$,Iae and,

0 1/c, 0 1

l/:il> 0 O - I / L , 0 0 - I / L , 0 0

.=[I O O 01. 0 1 0 0

0 0 1

0 I I L , ]

The observer assumes that the vector [XI should be approximated by [?I and therefore the observer's law is defined by,

[?I = A[?] + B, [U1 + B, [wl + KO ( [ Y l - [?I> > (24) [?I = W I , (25)

where K,, is chosen to properly defme the observers dynamic performance. Fortunately, the first two components of [F] are provided by the capacitor voltage algorithm and therefore, they do not need to be approximated by the observer. Thus, a reduced order observer is preferred to obtain the supply

currents. Hence, the model given by [21] - [22] can be written as,

where IXrn 1 = [;clap > [Xdl = [ (~ lap , and Amm, Amd, Adm, Add,

Blm, Bld, Bzm, and BZd are derived fiom A, B1 and Bz. Expanding (26) we obtain,

[ i d 1 = Admix, 1 + Ad, [ x d 1 + B1d [U] +Bzd [wI 9 (28) [Xm 1 = Amm[xm 1 + A m , [ ~ d l + B I ~ [U] + B ~ m [ w l , (29)

~ x d ~ ~ A d d ~ X d ~ ~ A d m ~ X ~ ~ ~ B l d ~ U ~ ~ B 2 d ~ ~ ~ ~ (30) Am, [xd 1 = CXm ] - A m , [xm I - B 1 m [ u I - B z ~ [wl . (3 1)

that can be arranged as,

Considering that only [xd] should be provided by the observer and that [xm] is provided by the capacitor voltage reconstruction algorithm, the observer's law becomes,

(32) [.d 1 = A d d [ z d I + Ad, [Xm I + B1d [U] + B,, [w]+

(A md i X d 1- [?d I), where [Fd] = [<lap is an estimation of the supply currents

[islap in the ap frame and I<od is chosen to properly define the observers dynamic performance. Using (31) to reorganize the observer's law (32) and noting that [xm] = b] we obtain,

[%d 1- [j] = ( A d d - )([%d 1 - [ Y l ) +

(33) ({Add - A md IKod + - A mm ) [Yl+ (Bid - +

(BZd - K o d B 2 m ) [ W l .

Finally, defming [Td] - KO, [ y ] = [GI . The observer's law becomes,

[GI = A,Ji i l+ A,,[Yl+ B,l[UI + Bq2[W1 7 (34)

where Aql = A d d - I<odAmd, Aq2 = (Add - I<odAmd)&d + A m d - I<adAmm, Bq1

the observer provides the vector [GI which is then used to obtain the vector [Xd] as,

(35 )

Bid - I<odBi,, and Bqz = Bzd - I<odBZm. Thus,

[.dl = G + K,d[Yl~

which corresponds to the supply line currents in the ap fiame, Fig. 4. This vector is finally transformed into the abc fiame by using the inverse transform as required by the non-linear controller Fig. 1. This is,

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Fig. 5 Virtual sensor characteristics waveforms for h = 0.75 and &d = 2000. (a) Actual capacitor phase voltage, v,, and reconstructed capacitor phase voltage. (b) Actual supply line current, b, and observed line current.

IV. EXPERIMENTAL RESULTS

A. Implementation Aspects

The virtual sensor approach is used in the control of a PWM-CSR as shown in Fig. 1. A digital system based on the TMS320C30 DSP has been used to implement the various algorithms corresponding to (a) the virtual sensor, (b) the space vector modulation technique, and (c) a discrete model of the topology. In addition, a PC has been used to on-line modi@ the various parameters and conditions used by the DSP system in a user-6iendly fashion. Experimental tests showed that a minimum sample period of 220 ps provide enough time to run the totality of algorithms which allows a maximum switching 6equency of 4.5 kHz. In this work, a reduced switching frequency of about 1 kHz is used to test in extreme conditions the proposed approach.

B. Virtual Sensor Performance

Several steady state (Fig. 5 & Fig. 6) and transient tests

Fig. 6 Virtual sensor static performance. (a) Reconstructed capacitor phase voltage as a hnction of h (&d = 2000). a): h = 0.95, b): h = 0.75, c): h = 0.55. (b) Observed supply line current as a function of &d (h = 0.75). a): &d = 100, b): &d = 2000, C): K,,d = 3500. .

(Fig. 7) have been performed to verify the feasibility of the virtual sensor. They also show the dependence of the virtual sensor on (a) the parameter identification forgetting factor ( I ) , (b) the reduced order observer gain (I<od), and (c) changes in the ac mains voltage.

I ) The parameter identfication forgetting factor: (h) defines the stability and tracking capabilities of the capacitor voltage v,, reconstruction algorithm. In fact, values close to h = 1 .O improve the stability but present poor tracking capabilities and small values, h = 0.55, improve the tracking capabilities but compromise the stability of the algorithm Fig. 6(a). h = 0.75 has been experimentally found to provide good results.

2) The reduced order observer gain: (Kd) defines the stability and tracking capabilities of the line current is, reduced order observer algorithm. In fact, small values, I<od= 100, improve the stability but present poor tracking capabilities and large values, I<od = 3500, improve the

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Fig. 7 Virtual sensor transient performance (a) 50% step up in the ac supply voltage Actual capacitor phase voltage, vco, and reconstructed capacitor phase voltage (b) 50% step down in the ac supply voltage Actual supply line current, I,”, and observed supply line current

tracking capabilities but compromise the stability of the algorithm Fig. 6(b). I(od = 2000 has shown the best results for both steady state and transient conditions.

3) AC supply voltage changes: Step-up Fig. 7(a) and step- down Fig. 7(b) voltage changes are applied at the ac source to investigate the dynamic performance of the virtual sensor. The results show that in both cases the reconstructed capacitor voltage and observed line current can track the actual waveforms within’a quarter of a cycle, which confirms the good overall performance and robustness of the proposed virtual sensor.

V. CONCLUSIONS

A virtual sensor that provides the line current and capacitor voltage values without measuring these variables is proposed. The technique takes into account the non-linear model of the PWM-CSR by using the information fiom the dc current and dc

voltage sensors in combination with a linear state observer and a linear parameter identification algorithm. As a result, at least four sensors can be eliminated, while the features of the control strategy are preserved. Steady state and transient tests show that the performance of the virtual sensor depends upon the identlfication parameter forgetting factor h and the reduced order observer gain Go, which can be easily tuned. The virtual sensor is used in the control of a PWM-CSR showing an acceptable overall performance

ACKNOWLEDGMENTS

The authors wish to thank to the Chilean Fund for Scientific and Technological Development (FONDECYT), for financial support through project 1990401. The authors also wish to thank the financial support provided by Fundacidn Andes through Programa de Insercidn de Cientificos Chilenos - 1998,

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