A repetitive-PI Current Controller for Boost Single Phase PFC Converters

doi:10.4236/epe.2011.32010

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Energy and Power En gi neering, 2011, 3, 69-78

doi:10.4236/epe.2011.32010 Published Online May 2011 (http://www.SciRP.org/journal/epe)

A Repetitive-PI Current Controller for Boost Single Phase

PFC Converters

Turki Kahawish Hassan

Electrical Engineering Department, University of AL-Mustansiriya, Baghdad, Iraq

E-mail: [email protected]

Received November 27, 2010; revised January 4, 2011; accepted January 26, 2011

Abstract

In this paper, the Author presents the theory and application of repetitive proportional integral current con-

troller for boost single phase ac-dc converter with power factor correction (PFC). A repetitive controller

which is inserted in series with the proportional integral (PI) controller shows very low crossover distortion

of input current, low total harmonic distortion and very low tracking error when is compared with the con-

ventional proportional integral controller. Full analysis of proposed controller is given and Matlab/Simulink

is used for simulation. The simulation results show the validity of the proposed control method.

Keywords: Boost Converter, Unity Power Factor Correction, Repetitive Control

1. Introduction

The developments in power factor correction technology

in the past two decades have enabled the design and im-

plementation of single phase ac-dc boost converters with

close to unity power factor and much less input current

distortion that generated by simple diode rectification

circuits.

Typical PFC schemes use two cascaded loops: the first

loop aims to regulate the boost dc output voltage and

provide the amplitude for the reference to the second

controller, which controls sinusoidal current provided

from grid. This current always contains some residual

distortion, especially in the vicinity of zero crossing of

the input voltage. The reasons of this zero crossing dis-

tortion (also referred to as cross distortion) are discussed

by authors in [1,2] but no any effective method is sug-

gested to reduce or eliminate this phenomenon. Feed

forward current control methods for boost single phase

PFC converters are suggested for robust control against

parameters variation and to reduce the total harmonic

distortion [3,4] .These methods need additional circuits

and the cost is increased .

Receptive control theory [5-7] originating from inter-

nal model principle provides a solution for eliminating

periodic errors which occur in a dynamic system. A re-

petitive controller can be viewed as a periodic waveform

generator augmented within the control loop of a control

system, which is closed loop regulated by a feed back

controller so that the periodic errors can be eliminated. A

number of repetitive control schemes have been devel-

oped and applied to eliminate periodic distortions in

PWM inverters [8,9].

In this paper, a repetitive controller is proposed and

designed to eliminate the zero crossing distortion of in-

put current of the converter and to obtain good tracking

performance. The design of the repetitive controller is

performed by analyzing the frequency domain, and Ny-

quist plot play a central role throughout the design phase.

The feasibility of the proposed design technique is

shown by simulation. The simulation results of the pro-

posed system show a highly improvement compared to

the conventional PI controlled system.

The reminder of this paper is organized as follows.

Section 2 gives operation principle of boost single phase

PFC converter and formulates the problem. In Section 3,

the principle of repetitive control is reviewed and a re-

petitive controller for PFC boost converter is proposed.

In section 4, the repetitive controller is designed and

several simulation results for proposed and conventional

systems are presented. Section 5 concludes the paper.

2. Operation Principe and Problem

Formulation

Power circuit configuration of a boost single-phase

power factor correction (PFC) converter and its control

scheme are shown in Figure 1(a) and (b) respectively.

70 T. K. HASSAN

d(t)

SW L

(a)

d(t)



ref

_ G



Peak

ABS

s(t)

ref

tri

cont

ˆin

(b)

Figure 1. Boost single phase PFC converter (a) power circuit and (b) control block diagram.

The circuit consists of a full-bridge diode rectifier and

a boost dc/dc converter. The dc output voltage of the

boost converter Vo is compared with the reference Vref

and the error ev is applied to the voltage controller Gcv(s)

which is a PI controller. A current command iref is

yielded by multiplying the output of voltage controller

Iref with a rectified unity sine wave (s(t)) which is:



sin



 (1)

where ˆsin

in in

vv t



, is the amplitude of input

voltage.

ˆin

The current command is compared with the inductor

current iL and the error ei is applied to the PI current con-

troller Gci(s) to obtain Vcont. A pulse width modulation

control signal d which is obtained by comparing Vcont

with the triangular wave Vtri is applied to gate of power

MOSFET transistor to achieve the sinusoidal input cur-

rent iin.

In order to make the following control system analysis

and design, some reasonable assumptions are made:

1) the losses of converter components are neglected;

2) the dc output voltage is well regulated such that Vo

equals to reference value Vref.

Accordingly one can derive the following equation

using the state-space averaging method.



dˆsin 1

in ref

Lvt dtV



 (2)

where L is the boosting inductance and d(t) is the duty

T. K. HASSAN

ratio function of the PWM modulator which can be ex-

pressed as



cont

tri

dt V

 (3)

where is the amplitude of the triangular wave.

From (2) and (3), one can find

ˆtri

ˆsin

ref ref

cont

tri

tLL



 (4)

As shown in Figure 1(b), Vcont can be written in laplace

form as

 



contrefL ci

Vs isisGs (5)

The inductor current iL can be derived from (4) and (5)

to be

 



 

ref

vs Gs

isi s

sLG ssLG s

 



 r

(6)

where

 

ref

tri

GsG s

sLV

 (7)

and



ci p

Gs K

 (8)

where Kp and Ki are proportional and integral gains re-

spectively.

According to (6), the absolute value of input voltage

can be viewed as a disturbance to the control loop. Also

Vref can be considered as a constant disturbance. The

objective of control is to make the inductor current tracks

a sinusoidal reference current, in face of these distur-

bances. In the conventional control method, a propor-

tional integral (PI) controller Gci(s) is used in design and

implementation, and also the most commonly used one

in practical applications. However, it fails to yield excel-

lent control performances in both tracking and distur-

bance rejection issues. To solve these problems, a repeti-

tive controller is proposed and applied to the current loop

of boost converter. This controller is discussed in next

section.

3. Repetitive Control

3.1. Principle of Repetitive Control

Any periodic signal with period Ts can be generated by

the free time delay system. The block diagram of time

delay system including unity positive feedback is shown



in Figure 2(a). The resulting transfer function is [5]:







us e

9)

where y(s) is the output and u(s) is the input. Due to this

ys e



 (

delay, the transfer function has infinitely many poles on

the imaginary axis: jkωs (see Figure 2(b)).The poles can

be found from 1.0

sjks





 for every k = 0, ±1,

±2,, where ωs = 2л/Ts.

e model in (9) is said to be a

he of the proposed repetitive control

A controller including th

repetitive controller and a system with such controller is

called a repetitive control system.

The basic concept of the repetitive controller origi-

nates from the internal model principle [7]. This princi-

ple state that the controlled output tracks a set of refer-

ence inputs without steady state error if the model which

generates these references is included in the stable closed

loop system. For example, no steady state error occurs

for step reference commands in type-1 stable feedback

system that has an integrator (1/s) in the loop, i.e., the

generator of step function .However ,stand-alone repeti-

tive controller cannot yield good transient performance,

therefore the repetitive controller is often used together

with another controller such as P-I controller to give

quick transient response.

.2. PFC Boost Converter with Repetitive 3

Controller

block diagramT

system is shown in Figure 3. The repetitive controller

CRP(s) is located in series with conventional PI controller

Gci(s) of current loop of boost converter. The transfer

function of the repetitive controller is:

u(s) y(s)

e

(a)

jω





2π





2jω

(b)

Figure 2. Generaf periodic signal. tor o

T. K. HASSAN

(s)





ref







qse





ref

ˆtri



ref

(s)

+ +

Figure 3. Block diagram of a repetitive control system.

 

Cs qse



 (10)

Note that CRP(s) is equivalent to the modified repeti-

tiv

ntrol system, (6) can be

e controller with a(s) = 1 as proposed by Hara et al.

[5]. q(s) is a low pass filter that should be appropriately

chosen so that good tracking performance is obtained

without resultant system instability. The choice of q(s) is

not straightforward and simulations are needed to choose

the most appropriate filter [10].

For the proposed repetitive co

written as:

 



 

ref

RP RP

ref

LGssLGs





(11)

where

  

ref

RPci RP

tri

Gs GsCs

sLV

 (12)

Let Us discuss the effectiveness of the repetitive con-

troller according to (11) and (12). The transfer function

CRP(s) is large enough for s = jkωs, k = 0, 1, 2,  If the

proportional integral controller Gci(s) of currentoop of

boost converter is chosen such that



Gjk



is suf-

ficiently large for the selective range

are multiples of frequency of reference current, then

 

of harmonics which

PsRPs

Gjk Gjk



. Since the denominator is

jected and the inductor cur-

rent tracks the sinusoidal reference current with very

small steady state error.

The stability condition t

large, the disturbances are re

hat is obtained in [5] is applied

ing proposition:

ystem without repetitive controller, i.e.,

proposed repetitive control system to give the follow-

In the repetitive control system shown in Figure 3 if

the closed loop s













1s Gs, is stable and







1qj Gj









 where



2ˆ

ref pi

tri

VKs



Gs sV L



then the system is exponentially stable.

The system without repetitive controller is stable since

e control system

G(s) has no unstable poles. The repetitiv

therefore exponentially stable if the Nyquist plot of

G(s) [11] does not encircle the (–1, j0) point and lies

outside of the circle of radius





centered at the

(–1, j0) point in the complex plane. The low pass filter

q(s) should be chosen so that tuist plot of G(s)

remains outside the circle for all frequencies.

4. Controller Design and Results

he Nyq

verter with power factor

orrection has outer voltage loop and inner current loop.

is shown in Figure 3. The

for stable system

4.1. Controller Design

The boost single phase ac-dc con

Some nominal values and circuit parameters of the con-

verter are shown in Table 1.

Our investigation is focused on the inner current loop

with repetitive controller that

rameters of P-I controller of current loop are chosen as

Kp = 0.8 and Ki = 300. The repetitive controller is in-

serted in series with the PI controller.

Figure 4 shows the Nyquist plot of G(s). The plot

does not encircle the (–1, j0) point and

e plot must be outside the circle of radius of







To ensure the stability for all frequencies the value of

T. K. HASSAN





Figure 4. Stability circle and Nyquist plot of G(s).

Table 1. Simulated converter parameters.

Input line voltage (peak) ˆin

v= 170 V

Input line frequency f =

e F

ncy

50 Hz

Smoothing capacitancC = 1000 µ

Smoothing inductance L = 1 mH

Rated power 500 W

Carrier freque25 KHz

Output voltage Vo = 300





between

must be less than unity to prevent any contact

obtain the cutoff frequency

the plot of G(s) and the circle (see the stability

condition in the previous section). The gain of low pass

filter q(s) is chosen as 0.98 to give stable system and

good tracking performance.

No systematic method to

q(s), therefore simulation is required to find the best

value from the point of view of good tracking and dis-

turbances rejection. Accordingly, the cutoff frequency is

chosen as 1000 HZ. It seems that 1000 HZ is reasonable

choice since most periodic disturbances are expected to

lie within this band. Let the filter q(s) be



0.98

1 2000π

qs s

 (13)

In our applications, we require a delay of Ts = 10 ms

function

r compensation of harmonics of fs = 100 Hz. Finally,

we get a repetitive controller CRP(s ), which has transfer



/100

0.98

e





(14)

1 2000πs

The effects that the designed repetiti

CRP(s) has on the PFC boost converter system can be

investigated by the bode plots of loop gains G(s) and

in, are clearly seen from the

ve controller

RP(s) (without and with repetitive controller respec-

tively) shown in Figure 5.

The major characteristics of the repetitive controller,

such as effective rejection of periodic disturbances and

reduction of stability marg

de plots. Note that the added repetitive controller in-

creases the loop gain at particular frequencies, integral

multiples of 100 Hz, maintaining a relatively unchanged

gain at other frequencies. Let us considerer the sensitive-

ity function



1RP

Gs which is the transfer function

from disturbance input to tracking error, which is of pri-

mary importaning the performance of feedback ce in judg

control systems. Gain increase at particular frequencies

imulation is performed by Matlab/Simulink to verify

leads to decrease in sensitivity at these frequencies. This

implies that zero crossing distortion in inductor current

which is considered as the periodic disturbances appear-

ing at multiple of 100 Hz are more strongly attenuated by

the repetitive controller than non periodic disturbances.

4.2. Results

74 T. K. HASSAN

Figure 5. Bode plots of G(s) with (dash) and GRP(s) with solid.

the proposed PFC boost con

oller. The converter parameters in Table 1 and the de-

r current iL waveforms with 100 W (Rl = 900 Ω)

put voltage and input current

for Rl = 450 Ω (output power equals 200 W) with repeti-

t current is canceled compared with Figure

current when load

racking is achieved with repetitive controller (very

verter with repetitive con-Figure 8(a) shows the in

signed repetitive controller in the previous section are

used in simulation. For all simulation results, the output

voltage of the converter is constant and equals to 300 V.

Figure 6(a) and (b) show the input current iin and the

input voltage vin waveforms with and without repetitive

ntroller respectively for 100 W output power and load

resistance (Rl) of 900 Ω. The input current is in phase

with input voltage and has an amplitude of 1.18 A with

no zero switching distortion when repetitive controller is

used.

Figure 7(a) shows the reference current iref and the

inducto

hen the repetitive controller is applied. Comparison

with waveforms in Figure 7(b) when only PI-controller

is used indicates that current tracking performance is

significantly improved by the proposed repetitive con-

troller (reference and inductor current waveforms are

coincident.

tive controller. It is noticed that the zero crossing distor-

tion in inpu

b) when only P-I controller is used.

Zero crossing distortion in input current shown in Fig-

ure 6(b) is more than the distortion shown in Figure 8(b)

because of increment in load current with load resistance

of 450 Ω.More leading phase of input

sistance is increased (less load current) which causes

more zero crossing distortion [1]. This leading phase is

not appeared when the proposed repetitive controller is

used.

Figure 9(a) and (b) show the reference and inductor

current waveforms for load resistance of 450 Ω with and

without repetitive controller respectively. It is clear that a

good t

all steady state error between reference and inductor

currents).

Figure 10(a) and (b) show the input voltage and input

T. K. HASSAN

(a)

(b)

Figure 6. vin and iin waveforms for Rl = 900 Ω (a) With re-

petitive controller; (b) Without repetitive controller. Hori-

zontal axis 10 ms/div), vertical axes: voltage (50 V/div),

current (2.5 A/div).

(a)

(b)

Figure 8. vin and iin waveforms for Rl = 450 Ω. (a) With re-

petitive controller; (b) Without repetitive controller. Hori

zontal axis (10 ms/div).Vertical axes: voltage (50 V/div),

current (2.5 A/div).

(a)

(b)

Figure 7. iref and iL waveforms for Rl = 900 Ω. (a) With re-

petitive controller; (b) Without repetitive controller. Hori-

zontal axis (5 ms/div).Vertical axis (0.2 A/div).

(b)

Figure 9. iref and iL waveforms for Rl = 450 Ω. (a) With re-

petitive controller; (b) Without repetitive controller. Hori

zontal axis (5 ms/div). Vertical axis (0.5 A/div).

76 T. K. HASSAN

(a)

(b)

Figure 10. vin and iin waveforms for Rl = 225 Ω. (a) With

repetitive controller; (b) Without repetitive controller.

Horizontal axis (10 ms/div). Vertical axes: voltage (5

V/div), current (2.5 A/div).

Figure 11(a) shows the reference current and inductor

ident.

factor (PF) and the total harmonic distor-

tio

900 Ω to 180 Ω. The input cur-

current waveforms with and without repetitive controller

respectively with Rl = 225 Ω (output power equals 400

W).

current waveforms for Rl = 225 Ω when the repetitive

controller is applied. It is noticed that the waveforms are

coinc

Figure 11(b) is similar to Figure 11(a) but no repeti-

tive controller is used. No good tracking between the

reference and inductor current is observed.

The power

n (THD) of input current with and without repetitive

controller are calculated under various load and are listed

in Table 2 and Table 3.

As shown in Table 2, a highly reduction in total har-

monic distortion of input current is shown when a repeti-

tive controller is used compared with Table 3 (without

repetitive controller).

The waveforms of input voltage and input current for

the proposed repetitive controlled PFC boost converter

are plotted in Figure 12 when the load resistance Rl is

suddenly changed from

nt magnitude is increased from 1.18 A to 5.8 A.

We can find that the input current iin always in phase

with the input voltage vin even though under transient

response. Consequently, the proposed repetitive controller

(a)

(b)

Figure 11. (a) iref and iL waveforms for Rl = 225 Ω with

repetitive controller. Horizontal axis (5ms/div).Vertical axis

(0.5A/div); (b) iref and iL waveforms for Rl = 225 Ω withou

repetitive controller. Horizl axis (5ms/div). Vertical

All the previous results are obtained for input voltage

ontroller.

Rl (Ω) Output power (w) THD (%) PF

onta

axis (0.5A/div).

can keep good performance under the condition of load

change.

Table 2. Calculated THD and PF for various loads with

repetitive c

1800 50 2.1 0.9992

900 100 0.9 0.9998

0.9999 450 200 0.41

225 400 0.22 1

Table 3. Calculated THD and PF arious loads with

repetit controller.

Rl (Ω) Output power (w) THD (%) PF

for v

ive

1800 50 34.16 0.9965

900 100 14.99 0.9977

0.9992 450 200 6.8

225 400 3.5 0.9998

T. K. HASSAN

of 1(peak). der to valie repeon-

troller performance foinput voltagariation, u-

lated steady state eforms of inductor currLnd

fee current or the case nput voith

70 V In ordate thtitive c

wav

e vthe sim

ent i a

re renciref fof iltage w

20% larger than the nominal value and Rl = 450 Ω (out-

put power equals 200 W) are plotted in Figure 13. The

reference current and inductor current are coincident and

good tracking is also maintained and is not affected with

the variation of input voltage.

Figure 14 shows the input voltage and input current

Figure 12. vin and iin waveforms with repetitive controller.

Rl is changed from 450 Ω to 180 Ω. Horizontal axis (50

ms/div).Vertical axes: voltage (50 V/div), current (5 A/div).

Figure 13. iref and iL waveforms for Rl = 450 Ω without re-

petitive controller. Horizontal axis (5 ms/div). Vertical axis

(0.2 A/div). Amplitude of input voltage equals 203 V.

Figure 14. vin and iin waveforms for Rl = 450 Ω. with repeti-

tive controller. Horizontal axis (5 ms/div). Vertical ax

voltage (50 V/div), current (5 A/div). Amplitude of input

voltage equals 203 V.

waveforms for the same conditions in Figure 13. No

zero crossing distortion in input current is observed and

the input current is in phase with the input voltage.

5. Conclusions

In this paper, the control approach to solve the problem

of zero crossing distortion of input current of PFC boost

converter has been presented. To achieve this goal, a

repetitive controller is inserted in series with the PI con

y. Because our approach is based on

bility

ndwidth is included. Several simula-

to verify the validity of the proposed

es:

troller of current loop. We presented a graphical design

technique based on the frequency domain analysis of

linear system to achieve a repetitive controller that pre-

serves system stabilit

graphical inspection of the Nyquist envelop, the design

procedure was simple and intuitive. A low pass filter

with gain lower than one to ensure the system sta

and to limit the ba

tions are performed

repetitive controller. The results obtained with and with-

out repetitive controller are compared. The results with

repetitive controller shows very low total harmonic dis-

tortion of input current, good tracking of reference cur-

rent and inductor current (very low steady state error)

and no zero crossing distortion of input current. Transit

responses to step change in load are presented to exhibit

the robustness of the proposed repetitive controller

against load variations. The performance of the system

with the proposed controller has not affected with the

variation of input voltage.

6. References

[1] J. Sun, “On the Zero-Crossing Distortion in Single-Phase

PFC Converters,” IEEE Transactions on Power Elec-

tronics, Vol. 19, No. 3, 2004, pp. 685-692.

doi:10.1109/TPEL.2004.826491

[2] H. C. Chen, “Duty Phase Control for single-Phase Boost-

Type SMR,” IEEE Transactions on Power Electronics,

Vol. 23, No. 4, 2008, pp. 1927-1934.

doi:10.1109/TPEL.2008.924627

[3] M. Chen and J. Sun, “Feedforward Current Control of

Boost Single-Phase PFC Converters,” IEEE Transactions

on Power Electronics, Vol. 21, No. 2, 2006, pp. 338-345.

doi:10.1109/TPEL.2005.869746

. H. Li and C. M. Liaw, “Switch-Mode

Digital Robust Ripple Compensation and

Current Waveform Controls,” IEEE Transactions on

[4] H. C. Chen, S

Rectifier with

Power Electronics, Vol. 19, No. 2, 2004, pp. 560-566.

doi:10.1109/TPEL.2003.823200

[5] S. Hara, Y. Yamamoto, T. Omata and M. Nak

petitive Control System: A New

ano, “Re-

Type Servo System for

Periodic Exogenous Signals,” IEEE Transaction on

Automatic Control, Vol. 33, No. 7, 1988, pp. 659-668.

doi:10.1109/9.1274

T. K. HASSAN

ontrol System Design,” [9] G. Escobar, A. A. Valdez, J. L. Ramos and P. Mattavelli,

“Repetitive-Based Controller for a UP

[6] T. Inoue, “Practical Repetitive C

IEEE Proceedings of 29th Conference on Decision and

control, Honolulu, 5-7 December1990, pp. 1673-1678.

doi: 10.1109/CDC.1990.203906

[7] B. A. France and W. M. Wonha

S Inverter to Com-

pensate Unbalance and Harmonic Distortion,” IEEE

Transactions on Industrial Electronics, Vol. 54, No. 1,

2007, pp. 504-510. doi:10.1109/TIE.2006.888803

[10] J. H. Moon, M. N. Lee and M. J. Chung, “Repetitive

Control for the Track-Following Serv

m, “The Internal Model

Principle for Linear Multivariable Regulators,” Applied

Mathematics and Optimization, Vol. 2, No. 2, pp. 170-

194, 1975. doi: 10.1007/BF01447855

[8] Y. Y. Tzou, S. L. Jung, and H. C. Yeh, “Adaptive Repe

tive Control of PWM Inverter

o System of an Op-

tical Disk Drive,” IEEE Transactions on Control Systems

Technology, Vol. 6, No. 5, 1998, pp. 663-670.

doi:10.1109/87.709501

[11] K. Ogata, “Modern Control Engineering,” 4th Edition,

Prentice Hall, New Jersey, 2002.

ti-

s for Very Low THD

Ac-Voltage Regulation with Unknown Loads,” IEEE

Transactions on Power Electronics, Vol. 14, No. 5, 1999,

pp. 973-981. doi:10.1109/63.788503

Nomenclature

o Dc output voltage of boost converter.

ref Reference voltage for voltage loop.

v Voltage error (the difference between refer-

ence and output voltage).

cv(s) Voltage controller transfer function.

voltage controller.

e wave.

t)).

n current

ction.

Vcont Output of current controller.

V Triangular wave.

Amplitude of triangular wave.

in Ac input current to the rectifier.

L Boosting inductance.

d(t) Duty ratio.

Kp Proportional gain of current controller.

ller.

on.

troller.

out repetitive

unction with repetitive

tri

ˆin

Iref Output of

(t) Unity sin

Ki Integral gain of current contro

snsfer functis q() Low pass filter tra

CRP siref Current command (Iref multiplied by s(

() Transfer function of repetitive con

vin Ac input voltage to the rectifier

G(s) Open loop transfer function with

ˆin

vAmplitude of ac input voltage.

iL Inductor current.

controller).

transfer f

GRP(s) Open loop

ei Current error (the difference betwee

t).

controller.

/sec)

ωs Double of input line frequency (radcommand and inductor curren

s) Cu transfer funGci(rrent controller