Input-Output Linearization Control of Single-Phase Buck-Boost Power Factor Corrector

Erdal Şehirli

Kastamonu University Vocational School of Higher Education

esehirli@kastamonu.edu.tr

Meral Altınay Technical Education Faculty of Kocaeli

University meral.altinay@kocaeli.edu.tr

Abstract- In this paper, simulations of single – phase power factor corrector (PFC) that has buck-boost circuit topology under input – output linearization nonlinear control with current and voltage control are realized. Buck-boost converter is connected to ac grid by single phase diode bridge rectifier and it works in boundary conduction mode (BCM). For switching of single switched buck-boost converter, compared saw tooth with dc control signal (PWM) technique has been utilized and switching frequency adjusted to 100 kHz. Simulations are carried out with Matlab/Simulink. By the simulations, the effect of PFC buck-boost rectifier on unity power factor, line current harmonic distortion, the sinusoidality of line current and dc – link voltage are presented.

Index Terms—Buck-boost converter, Feedback linearization, Nonlinear control, Power factor correction (PFC), Simulation.

I. INTRODUCTION

Recently, power factor correction (PFC) has gained more attention in the field of power electronics. Before, the bulky passive and harmonic filters have been used. However, these kinds of filters are not so efficient and they increase the cost. Therefore, power converters which are more efficient and useful are used. As a single phase PFC, basic dc-dc converters that are buck, boost and buck-boost topologies can be implemented. Using these dc-dc converters, the grid connection can be obtained by a diode bridge. Thanks to the control of inductance current in the dc-dc converter, the PFC is provided.

In literature, there are lots of publications about PFC and it seems that it is everlasting topic of power electronics. Reference [1] examines the suitability of the basic dc-dc converter topologies in continuous conduction mode (CCM) and discontinuous conduction mode (DCM) as a PFC circuit. A general review of single phase improved power quality ac-dc converters are realized as in [2]. Reference [3] identifies the difference between single and two stages PFC. Furthermore, the advantages and the disadvantages of single and two stages PFC are defined as in [4]. References [5-6] realize the design analysis of PFC rectifier. In reference [7] the basic control methods of boost PFC are introduced and the differences are determined. Reference [8] realizes the unified input-output linearization control of PFC boost rectifier. In [9-10] input-output linearization control is applied to Sepic and Sheppard-Taylor topology. However, in [11-13] nonlinear control and feedback linearization of dc-dc

converters are implemented and in [14] detailed analysis of dc-dc converters are made.

In this paper, the input-output linearization nonlinear control of single phase PFC that has buck-boost topology and works in BCM is realized. In the simulation studies, both of voltage and current control are carried out. Furthermore, current controlled nonlinear controller can be used as a two-stage PFC. On the other hand, voltage controlled one can be used as a single-stage PFC. By means of the simulations that are achieved by Matlab/Simulink, the effects of the proposed controllers to the unity power factor, line current harmonic distortion, sinusoidality of line current and dc-link voltage are presented.

II. MATHEMATICAL MODEL

The topology of PFC buck-boost converter is shown in Fig.1. It is seen from this figure that the buck-boost converter is connected to the grid from a diode bridge that has four diodes.

It is known that the main disadvantage of the buck-boost converter is the reverse voltage on the load.

Grid voltage is considered as a constant at every period of time for mathematical model of PFC converter and so the mathematical model of dc-dc buck-boost converter can be used for PFC buck-boost converter.

Therefore, the mathematical model of dc-dc buck-boost converter is derived by the states of the switch that is shown by Fig.2.

Fig.2. a) S is on; b) S is off

Fig. 1. Single Phase PFC buck-boost converter.

Using Kirchhoff’s voltage and current laws, the state-space representation of buck-boost converter is obtained for BCM and CCM in (1), as in [11].

0 0 (1)

It is known that PFC converters or dc-dc converters can

work in three modes which are DCM, BCM and CCM. The CCM mode is more appropriate for boost PFC. Since, the PFC process is provided naturally by the boost converter [1]. However, if the single phase structure is chosen like in Fig.1, the power loss of the switch will increase because of the high switching frequency. Therefore, the soft switching techniques that are active or passive must be used but it may increase the cost. By the way, DCM or BCM modes which are more suitable for buck-boost derived topologies and they can be used without the requirement of soft switching techniques. It can be seen as an advantage. However, the single phase PFC which works in DCM and BCM needs an input filter because there is discontinuity in the grid current and this input filter makes the grid current continuous [1,7].

III. FEEDBACK LINEARIZATION

Systems have generally a nonlinear nature. But, designing the controller, linear control methods are chosen due to the simplicity and the powerful tools to define stability. However, linear control methods are inadequate for large operation conditions and also the hard nonlinearities cannot be derived by these methods. Furthermore, in the system there must not be uncertainty which can reduce the performance and cause the instability. Therefore, nonlinear control techniques are chosen [15].

The feedback linearization technique is a kind of nonlinear design methodology. The basic idea is firstly to transform nonlinear system into linear system and then use powerful linear design techniques to complete the control design. This method is different from conventional linearization. In this technique, the process is achieved by exact state transformation and feedback instead of linear approximations. Furthermore, it is considered that the original system is transformed into an equivalent simpler form. There are two kinds of feedback linearization techniques that are input-state and input-output linearization [15].

The procedure of input-output feedback linearization can be summarized by three heading. • Derivation of system output is made until the system

input appears. • New control variable which cancels the nonlinearity and

reduces the tracking error is chosen. • The stability of internal dynamic that is the part of

system dynamics cannot be observed in input-output linearization is analysed.

Input-output linearization technique can be explained as follows. Firstly, the system is thought in the form of (2)-(3) as in [15-18].

(2) (3) To obtain input-output linearization of this system, the

outputs y must be differentiated until inputs u appear. By differentiating (3), (4) is obtained.

(4) In (4), and are the Lie derivatives of f(x) and h(x),

respectively. If is equal to ‘0’, inputs are independent from outputs and derivation process continues. After r times derivation, if the condition of (5) is provided, inputs appears in outputs and (6) is obtained.

0 (5)

∑ (6) Applying (6) for all n outputs, (7) is derived.

(7)

E(x) in (8) is a decoupling matrix, if it is invertible and new

control variable is chosen, feedback transformation is obtained, as in (9).

(8)

(9)

Equation (10) shows the relation between the new inputs v

and the outputs y. The input-output relation is decoupled and linear [17].

(10)

If the closed loop error dynamics is considered, as in (11) – (12), (13) defines new inputs for tracking control.

00 (11)

(12)

(13)

k values in equations show the constant values for stability

of systems and tracking of y references, as in [18].

IV. APPLICATION OF FEEDBACK LINEARIZATION TO PFC BUCK-BOOT CONVERTER

Applying feedback linearization to PFC buck-boost converter, the state-space representation in (1), is used. It is required to use the form that states in (2)-(3). As a system output, the inductance current and the capacitor voltage can be used. Therefore, there are two kinds of applications on the basis of system output. In this part, it is shown the use of both outputs. Furthermore, equations in this chapter, the system input u is the switching function d.

A. Current Control In current control, the system output is chosen as

inductance current . If inductance current is chosen directly, it is meaningful to use this controller as a two-stage power factor corrector.

Firstly, (1), must be reordered for in the form of (2) and the switching function must be set as a system input u. So, the state-space representation can be written in (14). And the system output y is shown in (15).

(14)

(15)

Differentiating output of (15), (16) is obtained. The order

of derivation process, finding a relation between y output and u input, is called as relative degree. It is also seen that the relative degree of current control system is ‘1’.

(16)

After organizing (16) as in (9) and adding new control

input (17) is obtained.

(17)

The current control algorithm is shown in Fig.3.

The main purpose is to control the inductance current

instead of output voltage. Therefore, this controller can be used the first part of two-stage PFC whose main task is to provide tracking grid current in phase with grid voltage. So, the unity power factor can be realized.

The two-stage PFC structure is shown in Fig.4. It is understood by this figure that the first switch is responsible for power factor correction and the mission of other switch is to regulate output voltage.

B. Voltage Control In voltage control, the system output is chosen as output

voltage . If output voltage is chosen directly, it is meaningful to use this controller as a single-stage PFC.

It is applied the same procedure that is used for current control and the system output is chosen as in (18).

(18)

Differentiating output of (18), (19) is obtained. It is also seen that the relative degree of voltage control system is ‘1’.

(19)

After organizing (19) as in (9) and adding new control

input (20) is obtained. 1 (20)

Fig. 3. Current control algorithm of single phase PFC buck-boost converter.

Va C

PFCcontroller

DC/DCcontroller

PFCstage

seperate switch

DC/DCstage

seperate switch

Fig. 4. Two-stage PFC structure.

The voltage control algorithm is shown in Fig.5.

Since, the main purpose is to control the output voltage, power factor correction is realized naturally. Therefore, this controller can be used as a single-stage PFC whose main task is to provide tracking reference voltage while realizing PFC naturally.

The single-stage PFC structure is shown in Fig.6.

V. SIMULATIONS

Simulations are carried out by Matlab/Simulink software. Line voltage is taken 220V, 50 Hz and switching frequency is defined as 100 kHz, for switching DC-PWM technique is used. The values of parameters are shown in Table.I.

TABLE I PARAMETER VALUES

Current Control Voltage Control L(H) C(F) R(Ω) k L(H) C(F) R(Ω) k 13e-5 220e-6 100 160000 15e-5 47e-5 100 14000

A. Results of Current Control Simulation diagram of current control is shown in Fig.7.

It is shown in Fig.8, that the inductor current has reached its reference values which change 2A to 4A after 0.5 second. If this converter works in CCM instead of BCM, the tracking of current reference will be seen explicitly as in Fig.9. From the same figure, settling time is nearly 0.001s.

It is seen by the Fig.10, that the power factor is higher than 0.995 even in the reference change.

Fig. 5. Voltage control algorithm of single phase PFC buck-boost converter.

Fig. 7. Simulation diagram of current control.

Vo

voltage measurement

Continuous

pow ergui

Iref _I

IL

Vcon

nonlinearcurrent_control ler

current reference

IL

current measurement

v+- v+

-

v contpwm

PWM generation

g m

d s

Mosfet

Ig

Vg

signal ref

Measurement

Diode

i+ -

i+

-

Fig. 8. Inductance current in BCM.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-2

0

2

4

6

8

10

time (s)

curr

ent

(A)

inductance current

Fig. 10. Power factor.

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.985

0.99

0.995

1

time (s)

pow

er f

acto

r (P

/S)

power factor

Fig. 9. Inductance current in CCM.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-0.5

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

time (s)

curr

ent

(A)

inductance current

Fig. 6. Single-stage PFC structure.

The output voltage is shown by Fig.11.

It is shown by Fig.12, which grid voltage and current are in same phase.

The grid current is shown in Fig.13 with reference change.

The total harmonic distortion (THD) of grid current is shown in Fig.14.

B. Results of Voltage Control Simulation diagram of voltage control is shown in Fig.15.

It is shown in Fig.16, which the output voltage of converter is tracking the reference voltage that changes at 0.5 second from 100V to 150V. Furthermore, the steady-state errors of dc voltages are ±2V and ±5V respectively. Besides, the settling time is nearly 0.05s.

Power factor is shown in Fig.17. It is seen by this figure

that the power factor is higher than 0.995 even in the reference change.

Fig. 13. Grid current.

0.48 0.485 0.49 0.495 0.5 0.505 0.51 0.515 0.52

-6

-4

-2

0

2

4

6

time (s)

curr

ent

(A)

grid current

Fig. 11. Output voltage.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-50

0

50

100

150

200

time (s)

volta

ge (

V)

d.c voltage

Fig. 12. Grid voltage and current.

0.44 0.46 0.48 0.5 0.52 0.54 0.56

-300

-200

-100

0

100

200

300

time (s)

volta

ge,

curr

ent

(V,A

)

voltage

current

Fig. 14. Total harmonic distortion of grid current.

Fig. 15. Simulation diagram of voltage control.

Continuous

pow ergui

IL

Vo

Vref -V

Vc

nonl inearvol tage control ler

v+- v+

-

Step

Scope1

VconPWM

PWMgenaration

g m

d s

Mosfet

V1

Measurement1

I

V

Measurement

Diode

i+ -

i+

-

Fig. 16. Output voltage.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.90

20

40

60

80

100

120

140

160

time (s)

volta

ge (

V)

d.c voltage

Fig. 17. Power factor.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

0.96

0.965

0.97

0.975

0.98

0.985

0.99

0.995

1

time (s)

pow

er f

acto

r

power factor

The grid voltage and current are shown in Fig.18. It is seen by this figure, the current and voltage are in phase.

The grid current is shown in Fig.19.

The total harmonic distortion (THD) of grid current is shown in Fig.20, that the THD is %19.81.

VI. CONCLUSION

In this paper, the input-output linearization nonlinear control that use both voltage and current separately as system output of single phase buck-boost PFC is realized. Because of reducing the soft switching requirement for buck-boost topology, the BCM is chosen.

It is shown by simulations which are carried out by Matlab/Simulink, that the power factor is higher than 0.99 and so grid voltage and current are in phase both voltage and current controlled input-output linearization technique.

Besides, the inductor current in current control and the output voltage in voltage control have a quite low settling time approximately, 0.001s and 0.05s respectively. However, THD of current control is %7.47 and voltage control is %19.81. The THD of current control is in allowable limit but the THD of voltage control must be reduced. The reason of higher THD is the requirement of input filter of the buck-boost PFC topology that works in BCM. Furthermore, designing the proper input filter, THD`s will be reduced. Due to the THD`s the shapes of current is not as desired completely.

The future work is to design a proper input filter and to make an experimental set up of these simulations.

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[17] D. C. Lee, G.M. Lee, K.D.Lee, “Dc – bus voltage control of three – phase Ac/Dc PWM converters using feedback linearization,” IEEE Transactions on Industrial Applications, vol.36, pp. 826 – 833,2000.

[18] T. S. Lee, “Input – output linearizing and zero – dynamics control of three – phase Ac/Dc voltage source converters,” IEEE Transactions on Power Electronics, vol.18, pp. 11 – 22,2003.

Fig. 18. Grid voltage and current.

0.44 0.46 0.48 0.5 0.52 0.54

-300

-200

-100

0

100

200

300

time (s)

volta

ge,c

urre

nt (

V,A

)

voltage

current

Fig. 19. Grid current.

0.48 0.485 0.49 0.495 0.5 0.505 0.51 0.515 0.52

-6

-4

-2

0

2

4

6

8

time (s)

curr

ent

(A)

grid current

Fig. 20. Total harmonic distortion of grid current.