Boost PFC Converters with Integral and Double Integral Sliding Mode Control F.Harirchi IUST University Tehran, Iran [email protected]

A. Rahmati, MIET IUST University Tehran, Iran [email protected]

Abstract— In this article, sliding mode controller is used to improve power factor in AC-DC boost converters. A comparison between integral sliding mode controller (ISMC) and double integral sliding mode controller (DISMC) for the current loop of the boost PFC converter is proposed in this paper. In sliding mode control method, circuit complexity is less. Also good stability and fast response versus to the changes of load, source and parameters of the circuit are obtained. In this paper by using second order Sliding Mode, we reduced chattering. The simulations have been done by MATLAB and simulations shown good performance of the DISMC compared with ISMC against load, source and parameters of the circuit changes. The results are: close to unity power factor, low THD, constant output voltage, low chattering and low steady state error. Also converter performance is shown in high frequency application.

Keywords- integral sliding mode control(ISMC); Double integral sliding mode control(DISMC); power factor correction (PFC); boost converter; Total harmonic distortion

I.

INTRODUCTION

Power supplies connected to ac mains introduce harmonic currents in the utility. It is clear that these harmonic currents cause several problems such as voltage distortion, heating, noise and reduce the capability of the line to provide energy. This fact and the need to comply with global standards have been forced us to use power factor correction systems. An ideal power factor corrector (PFC) should emulate a resistor on the supply side while maintaining a fairly regulated output voltage [1]. In the case of sinusoidal line voltage, this means that the converter must draw a sinusoidal current from the utility; in order to satisfy this problem, a suitable sinusoidal reference is generally needed and the control objective is to force the input current to follow this reference current as close as possible. Primary goal in PFC is to achieve sinusoidal input current which have the same phase as the input voltage. Also output voltage ripple is inevitable for the balance of system power. To solve this problem, the Output capacitor quantity should be set in the way which, the range of output ripple is acceptable. Many ways for improving power factor is reported in different papers. Some of these methods represented for eliminating the effect of parasitic capacitors of the switching elements to

743

A. Abrishamifar, Member, IEEE IUST University Tehran,Iran [email protected]

reduce harmonic distortion[2]. Some of researchers began to adjust the switching frequency and duty cycle of the output PWM [3] and some others use Active power factor correction (APFC) method to improve the power factor quality. Directly regulation of output voltage in boost converters introduce nonminimum phase control problem in this kinds of converters. In order to avoid this problem, in some literature, indirect controlling of the output voltage to follow the desired input current using SMC is mentioned [4],[5]. Several dc-dc converters are suitable to work as a “power factor preregulator” or “resistor emulator” in ac–dc applications [6]. The most popular topology in PFC applications is almost the boost topology, shown in Fig.1 together with a generic controller. Sliding mode controller is a kind of nonlinear controllers used for control the variable structure systems (VSS) which has been introduced before in [7],[4],[2]. The main advantage of the SMC is, its stability and robustness against the unknown load and line parameters. Moreover, in comparison with other types of nonlinear controllers, implementation of the SMC is relatively easier than other control strategies. Because of mentioned features, this control method has a very good performance in applications of nonlinear control systems such as power electronics. In practical use of a sliding mode controller in power converters-due to its limited switching frequency-we usually face steady state error in regulation. As we know from the control system properties, increasing order of the controller improve steady state error [5]. In [8] authors utilized Double Integral Sliding Mode (DISM) with assuming “input current and voltage” together as states variables, to improve steady state regulation error of DC output voltage in a DC-DC converter. In this article, we use sliding mode controller to improve power factor in AC-DC boost converters. ISMC and DISMC are proposed to control current loop of the boost power factor correction converter. We utilize input current as state variable and for reducing chattering we use equivalent control method to found control low. Analyzing boost converter and governing dynamic equations are discussed in section 2. Designing of ISMC and DISMC are drawn in section 3, and in section 4 frequency response analysis and the way of selecting sliding surface coefficients are given. Simulation results are

shown in section 5. Finally we conclude the algorithm in section 6.

The DC component of the input power in stable condition is:

P

!

"#$ I V rI % "#$

% I&'( V sin% ωt rI&'( sin% ωt "#$

1 % I A rI&'( 2 &'(

(2)

The DC component of the output power in stable condition is: P

I&'(

ANALYSIS OF THE BOOST CONVERTER

Consider the boost PFC circuit shown in fig.1. The circuit includes three main components: storage element (inductor), energy gate and output filter. The control signal u can be 1 or -1. u=1 means that the switch is on and u= -1 means the switch is off. Circuit behavior expressed with following equations:

LI u V V rI CV u I

V R

V&'( , A*

(3)

V : output voltage

V Asinωt: input AC voltage with ω frequency

r: inductor parasitic resistance(but represents the switching elements and the AC voltage source parasitic resistance) R: nominal value of the load resistance

R 8r

(4)

(5)

Using i&'( I&'( sin ωt as a reference current for input current, active power balance in steady state condition is guaranteed.

(1)

I : input inductor current

III.

DESIGN OF SLIDING MODE CONTROLLER

A. case1: Integral sliding mode control In this part, design the integral sliding mode control that force the input current to follow the calculated reference current (./01 ) which is inphase with the input voltage. The proposed controller use current error x1, the derivative of the current error x2 and the integral of the current error x3 as state variables. State variables are defined as follows: x3 i&'( i

L: nominal value of the input inductor C: nominal value of the output capacitor

x%

dx3 dt

x5 6 x3 dt

(6)

Sliding surface is defined as follows:

Control objectives are as follows:

•

% V&'( R

A A% 2V % * % &'( 2r 4r rR

The answer is real when:

Where u 1 u. The circuit parameters are defined as follows:

•

"#$

With equating equations (2) and (3) and solve based on I&'( we have:

Figure1. Boost PFC converter with SMC

II.

! !

DC component of the output voltage must be equal to constant desired value and harmonic components must be reduced even as far as possible. The input current should be in the same phase as the input voltage and should only include the first harmonic with ω frequency.

In stable condition, output power and input power are equal. The only problem is equating the signals with different waveforms (harmonic signal at the input side and DC signal at the output side). To overcome this problem, equating DC components of the both sides is considered.

744

S ax3 bx% cx5

(7)

Where a, b and c are defined as sliding mode coefficients. The first time derivative of S is: x3

S ax3 bx% cx5

d 1 i V u V rI dt &'( L

x% x3; x5 x3

Under the invariance of SM control S=0 and S 0. When S=0 called the system is in the sliding mode. In this case I i&'( . S ax3 bx% cx5

(8)

a α b

,

c β b

x3 i&'( i

(9)

The first time derivative of S is: x3

S ax3 bx% cx5

d 1 i V u V rI dt &'( L

b α a

That should be negative then: For S " 0 ( u 1, u 0 ) and S 0 we have:

a

d% i

di&'( a b dV dV &'( V V rI b dt L dt % L dt dt rdI cx3 " 0 dt

u'C

1 d i&'( dV rdI di&'( [L Lα αV αV dt % dt dt dt αrI Lβx3 ]

(18)

,

c β a

(19)

Sliding mode control low defined as: (13)

u signS

(20)

Existense of sliding mode can be determined by lyapanov stability analysis. Consider the lyapunov function defined as follows: 1 A B% 2

(14)

(21)

Derivation based on motion trajectory of the system is: V SS

Equivalent control method is based on to putting the derivative of sliding surface, zero to maintain the trajectory motion of the system on the sliding surface. In this case the discrete signal is replaced with a continuous signal. So we have: %

x% x3 x5 6 x3 dt

S ax3 bx% cx5

(12)

For S 0 ( u 0, u 1 ) and S " 0 we have:

(17)

Under the invariance of SM control S=0 and S 0. When S=0 called the system is in the sliding mode. In this case I i&'( .

Derivation based on motion trajectory of the system is:

di&'( a d% i&'( b dV rdI a V rI b cx3 dt L dt % L dt dt 0

(16)

Where a, b and c are defined as sliding mode coefficients.

(11)

A BB

S ax3 bx% cx5

(10)

Existense of sliding mode can be determined by lyapanov stability analysis. Consider the lyapunov function defined as follows: 1 A B% 2

x5 6 6 x3 dt dt

Sliding surface is defined as follows:

Sliding mode control low defined as: u signS

x% 6 x3 dt

(15)

That should be negative then: For S " 0 ( u 1, u 0 ) and S 0 we have: a

di&'( a V rI bx3 c 6 x3 dt 0 dt L

For S 0 ( u 0, u 1 ) and S " 0 we have: a

It should be noted that 0 ueq 1. Equation (15) is a public control rule for boost PFC converter.

(22)

di&'( a V V rI bx3 c 6 x3 dt " 0 dt L

(23)

(24)

By combining equations (13) and (14) we have:

B. case2: Double integral sliding mode control In this section, design the double integral sliding mode control that force the input current to follow the calculated reference current (i&'( ) which is inphase with the input voltage. The proposed controller use current error x1, the integral of the current error x2 and the double integral of the current error x3 as state variables. State variables are defined as follows:

745

di&'( rI V di&'( rI αx3 β 6 x3 dt dt L L dt L

(25)

Equivalent control method is based on setting the derivative of sliding surface to zero, to maintain the trajectory motion of the system on the sliding surface. In this case the discrete signal is replaced with a continuous signal. So we have: u'C

1 di&'( [L V rI Lαx3 Lβ 6 x3 dt] V dt

(26)

It should be noted that 0 u eq 1. Equation (26) is a public control rule for boost PFC converter. IV.

Block diagram of the above equation is shown in fig.2.

FREQUENCY RESPONSE ANALYSIS AND SELECT SM COEFFICIENTS

A. Selection of Sliding Coefficients Figure 2. Block diagram of the current loop under ISMC and DISMC

The equation relating sliding coefficients to the characteristic response of the converter during sliding mode operation can be easily found by substituting S=0 into (7):

ax3 b

dx3 c 6 x3 dt 0 dt

Open loop gain of fig.2 under ISM and DISM controller is: 1 1 Gs α β s s

(34)

(27)

Rearranging the time differentiation of (16) into a standard second-order system form, we have: d% x3 dx3 2ξω ω % x3 0 dt % dt

(28)

Where ω IK is the undamped natural frequency and ξ

L

%√KJ

J

is the damping ratio. Recall that there are three possible types of response in a linear second-order system: under-damped 0 , ξ 1, critically-damped ξ 0, and overdamped ξ " 1. For ease of discussion, we choose to design the controller for critically-damped response thus: x3 t A3 A% teNOP!

(29)

where Q3 and Q% are determined by the initial conditions of the system. In a critically-damped system, the bandwidth of the controller’s response fST is: fST

ω 1 c I 2π 2π b

Figure 3. Bode plot of the current loop under ISMC and DISMC

By bode plots and equation (31) and the existence condition, the SM surface coefficients can be selected by desired phase margin. In this paper the desired value is 45degree. The bode plot is shown in fig.3 with α 5 X 10Y , β 4 X 10Z . To compare both methods performance we chose same values for sliding coefficients.

(30)

By rearranging (19) and substituting ξ 1 into the damping ratio, the following design equations are obtained: a 4πfST b

c 4π% fST % b

(31)

I.

B. Frequency response analysis Considering that the dynamics of the system is obtained from B 0 thus: α

di&'( di d% i i βi&'( i 0 α dt dt dt % &'(

(32)

We can rewritten equation above in S-domain as follows: s % i s % i&'( i&'( i αs β

(33)

746

SIMULATION

simulations have been done by MATLAB. Parameters for simulation are shown in table 1. Case1(ISMC): simulation was performed at input frequency 60HZ and input AC voltage and input inductor current and the output voltage are shown in fig.4 and fig.5. As can be seen in fig.4, load resistor at half a second suddenly doubled, the good performance and fast response of the controller in this case are shown. Also fast response and less steady state error in output voltage have been illustrated in fig.5.

TABLE I.

PARAMETERS FOR SIMULATION

Table 1 Parameters for simulation A=110 Magnitude of input voltage

Frequency of input voltage

F=60HZ

Input inductor

V&'( 300V

Parasitic resistance

r =0.1Ω

Output capacitor

R=1KΩ 2KΩ

Reference output voltage

Nominal load resistor

L=1mH

C=220uF

Figure 6. input voltage and current

Figure 4. input voltage and current Figure 7.

chattering in DISMC boost PFC converter

Fig.7 shows chattering in DISMC. Close to the unit power factor even at 800HZ frequency is achievable. Total harmonic distortion of input current at 60HZ is shown in figure 8. The THD of input current is 0.21% at 60HZ and THD is 2.16% at 800HZ. Power factor at 60HZ-800HZ is about 99.999%. Obviously, it is satisfied to IEC 61000-3-2 standard.

Figure 5. output voltage

The THD of input current is 6.61% in 60HZ and Power factor is about 99.997%. Derivation of sliding surface has big changes in 0.5s when the load resistor suddenly doubled. Because of high THD and big changes in ds and less chattering we use DISMC for PFC converter. Case2(DISMC): simulation was performed at frequency 60HZ and input AC voltage and input inductor current is shown in fig.6. As can be seen in fig.6, load resistor at half a second suddenly doubled, the good performance and fast response of the controller in this case are shown. Also fast response and less steady state error in output voltage is achieved.

747

Figure 8. the THD of input current in DISMC

CONCLUSION

In this paper boost PFC converter with ISMC and DISMC are proposed. By using both controllers, in the presence of unknown parameters such as parasitic resistance of the inductor and suddenly changes of the load resistance, it is possible to achieve close to unity power factor and adjustable voltage in

output with less steady state error. As we know, chattering is a big problem in SM controllers. In this paper by using second order Sliding Mode (equivalent method), we reduced chattering phenomenon. The simulation results show the good performance of DISMC compared with ISMC in boost PFC converter such as low THD, less changes in derivation of sliding surface in load changes and less chattering. The simulation results shown the current waveform has the sinusoidal shape and inphase with line voltage, the tested power factor in DISMC is 99.999% and the input current THD is 0.21%.

REFERENCES [1]

[2]

[3]

C. Qiao and K. M. Smedley, “A topology survey of single stage power factor corrector with a boost type input current shaper,” in Proc. IEEE Appl. Power Electron. Conf. (APEC), 2000, pp. 460–467. K. D. Gusseme, D. M. Van de Sype, A. P. M. Van den Bossche, and J. A. Melkebeek, “Input-current distortion of CCM boost PFC converters operated in DCM,” IEEE Trans. Ind. Electron., vol. 54, no. 2, pp. 858– 865, Apr. 2007. Y.-K. Lo, J.-Y. Lin, and S.-Y. Ou, “Switching-frequency control for regulated discontinuous-conduction-mode boost rectifiers,” IEEE Trans. Ind. Electron., vol. 54, no. 2, pp. 760–768, Apr. 2007.

748

[4]

V. Utkin, J. Gulder, and M. Shijun, Sliding Mode Control in Electromechanical Systems, 2nd ed. New York: Taylor & Francis, 1999 [5] Chu, G., Siew-Chong Tan, Tse, C.K. and Siu Chung Wong, “General control for boost PFC converter from a sliding mode viewpoint.” In proc. PESC’08, June 2008, pp.4452 – 4456. [6] L. H. Dixon, “High power factor pre-regulators for off-line power supplies,” in Unitrode Power Supply Design Sem., 1990, SEM-700, pp. I2.1–I2.6. [7] Y. Shtessel, S. Baev, H. Biglari, "Unity Power Factor Control in ThreePhase AC/DC Boost Converter Using Sliding Modes," IEEE Trans. on Industrial Electronics, vol. 55, no. 11, pp. 3874-3882, Nov 2008. [8] S. C. Tan, Y. M. Lai and C. K. Tse, “Indirect sliding mode control of power converters via double integral sliding surface,” IEEE Trans.Power. Electron., vol. 23, no. 2, pp.600 611, Mar. 2008. [9] S.-C. Tan, Y. M. Lai, C. K. Tse, L. Martinez-Salamero, and C.-K. Wu, “A fast-response sliding-mode controller for boost-type converters with a wide range of operating conditions,” IEEE Trans. Ind. Electron., vol. 54, no. 6, pp. 3276–3286, Dec. 2007. [10] O. Garcia, J. A. Cobos, R. Prieto, P. Alou, and J. Uceda, “Single phase power factor correction: A survey,” IEEE Trans. Power Electron., vol. 18, no. 3, pp. 749–755, May 2003. [11] Jia-You Lee, Yu-Ming Chang, Fang-Yu Liu, "A new UPS topology employing a PFC boost rectifier cascaded high-frequency tri-port converter," IEEE Trans. on Industrial Electronics, vol. 46, no. 4, pp. 803-813, August 1999

A. Rahmati, MIET IUST University Tehran, Iran [email protected]

Abstract— In this article, sliding mode controller is used to improve power factor in AC-DC boost converters. A comparison between integral sliding mode controller (ISMC) and double integral sliding mode controller (DISMC) for the current loop of the boost PFC converter is proposed in this paper. In sliding mode control method, circuit complexity is less. Also good stability and fast response versus to the changes of load, source and parameters of the circuit are obtained. In this paper by using second order Sliding Mode, we reduced chattering. The simulations have been done by MATLAB and simulations shown good performance of the DISMC compared with ISMC against load, source and parameters of the circuit changes. The results are: close to unity power factor, low THD, constant output voltage, low chattering and low steady state error. Also converter performance is shown in high frequency application.

Keywords- integral sliding mode control(ISMC); Double integral sliding mode control(DISMC); power factor correction (PFC); boost converter; Total harmonic distortion

I.

INTRODUCTION

Power supplies connected to ac mains introduce harmonic currents in the utility. It is clear that these harmonic currents cause several problems such as voltage distortion, heating, noise and reduce the capability of the line to provide energy. This fact and the need to comply with global standards have been forced us to use power factor correction systems. An ideal power factor corrector (PFC) should emulate a resistor on the supply side while maintaining a fairly regulated output voltage [1]. In the case of sinusoidal line voltage, this means that the converter must draw a sinusoidal current from the utility; in order to satisfy this problem, a suitable sinusoidal reference is generally needed and the control objective is to force the input current to follow this reference current as close as possible. Primary goal in PFC is to achieve sinusoidal input current which have the same phase as the input voltage. Also output voltage ripple is inevitable for the balance of system power. To solve this problem, the Output capacitor quantity should be set in the way which, the range of output ripple is acceptable. Many ways for improving power factor is reported in different papers. Some of these methods represented for eliminating the effect of parasitic capacitors of the switching elements to

743

A. Abrishamifar, Member, IEEE IUST University Tehran,Iran [email protected]

reduce harmonic distortion[2]. Some of researchers began to adjust the switching frequency and duty cycle of the output PWM [3] and some others use Active power factor correction (APFC) method to improve the power factor quality. Directly regulation of output voltage in boost converters introduce nonminimum phase control problem in this kinds of converters. In order to avoid this problem, in some literature, indirect controlling of the output voltage to follow the desired input current using SMC is mentioned [4],[5]. Several dc-dc converters are suitable to work as a “power factor preregulator” or “resistor emulator” in ac–dc applications [6]. The most popular topology in PFC applications is almost the boost topology, shown in Fig.1 together with a generic controller. Sliding mode controller is a kind of nonlinear controllers used for control the variable structure systems (VSS) which has been introduced before in [7],[4],[2]. The main advantage of the SMC is, its stability and robustness against the unknown load and line parameters. Moreover, in comparison with other types of nonlinear controllers, implementation of the SMC is relatively easier than other control strategies. Because of mentioned features, this control method has a very good performance in applications of nonlinear control systems such as power electronics. In practical use of a sliding mode controller in power converters-due to its limited switching frequency-we usually face steady state error in regulation. As we know from the control system properties, increasing order of the controller improve steady state error [5]. In [8] authors utilized Double Integral Sliding Mode (DISM) with assuming “input current and voltage” together as states variables, to improve steady state regulation error of DC output voltage in a DC-DC converter. In this article, we use sliding mode controller to improve power factor in AC-DC boost converters. ISMC and DISMC are proposed to control current loop of the boost power factor correction converter. We utilize input current as state variable and for reducing chattering we use equivalent control method to found control low. Analyzing boost converter and governing dynamic equations are discussed in section 2. Designing of ISMC and DISMC are drawn in section 3, and in section 4 frequency response analysis and the way of selecting sliding surface coefficients are given. Simulation results are

shown in section 5. Finally we conclude the algorithm in section 6.

The DC component of the input power in stable condition is:

P

!

"#$ I V rI % "#$

% I&'( V sin% ωt rI&'( sin% ωt "#$

1 % I A rI&'( 2 &'(

(2)

The DC component of the output power in stable condition is: P

I&'(

ANALYSIS OF THE BOOST CONVERTER

Consider the boost PFC circuit shown in fig.1. The circuit includes three main components: storage element (inductor), energy gate and output filter. The control signal u can be 1 or -1. u=1 means that the switch is on and u= -1 means the switch is off. Circuit behavior expressed with following equations:

LI u V V rI CV u I

V R

V&'( , A*

(3)

V : output voltage

V Asinωt: input AC voltage with ω frequency

r: inductor parasitic resistance(but represents the switching elements and the AC voltage source parasitic resistance) R: nominal value of the load resistance

R 8r

(4)

(5)

Using i&'( I&'( sin ωt as a reference current for input current, active power balance in steady state condition is guaranteed.

(1)

I : input inductor current

III.

DESIGN OF SLIDING MODE CONTROLLER

A. case1: Integral sliding mode control In this part, design the integral sliding mode control that force the input current to follow the calculated reference current (./01 ) which is inphase with the input voltage. The proposed controller use current error x1, the derivative of the current error x2 and the integral of the current error x3 as state variables. State variables are defined as follows: x3 i&'( i

L: nominal value of the input inductor C: nominal value of the output capacitor

x%

dx3 dt

x5 6 x3 dt

(6)

Sliding surface is defined as follows:

Control objectives are as follows:

•

% V&'( R

A A% 2V % * % &'( 2r 4r rR

The answer is real when:

Where u 1 u. The circuit parameters are defined as follows:

•

"#$

With equating equations (2) and (3) and solve based on I&'( we have:

Figure1. Boost PFC converter with SMC

II.

! !

DC component of the output voltage must be equal to constant desired value and harmonic components must be reduced even as far as possible. The input current should be in the same phase as the input voltage and should only include the first harmonic with ω frequency.

In stable condition, output power and input power are equal. The only problem is equating the signals with different waveforms (harmonic signal at the input side and DC signal at the output side). To overcome this problem, equating DC components of the both sides is considered.

744

S ax3 bx% cx5

(7)

Where a, b and c are defined as sliding mode coefficients. The first time derivative of S is: x3

S ax3 bx% cx5

d 1 i V u V rI dt &'( L

x% x3; x5 x3

Under the invariance of SM control S=0 and S 0. When S=0 called the system is in the sliding mode. In this case I i&'( . S ax3 bx% cx5

(8)

a α b

,

c β b

x3 i&'( i

(9)

The first time derivative of S is: x3

S ax3 bx% cx5

d 1 i V u V rI dt &'( L

b α a

That should be negative then: For S " 0 ( u 1, u 0 ) and S 0 we have:

a

d% i

di&'( a b dV dV &'( V V rI b dt L dt % L dt dt rdI cx3 " 0 dt

u'C

1 d i&'( dV rdI di&'( [L Lα αV αV dt % dt dt dt αrI Lβx3 ]

(18)

,

c β a

(19)

Sliding mode control low defined as: (13)

u signS

(20)

Existense of sliding mode can be determined by lyapanov stability analysis. Consider the lyapunov function defined as follows: 1 A B% 2

(14)

(21)

Derivation based on motion trajectory of the system is: V SS

Equivalent control method is based on to putting the derivative of sliding surface, zero to maintain the trajectory motion of the system on the sliding surface. In this case the discrete signal is replaced with a continuous signal. So we have: %

x% x3 x5 6 x3 dt

S ax3 bx% cx5

(12)

For S 0 ( u 0, u 1 ) and S " 0 we have:

(17)

Under the invariance of SM control S=0 and S 0. When S=0 called the system is in the sliding mode. In this case I i&'( .

Derivation based on motion trajectory of the system is:

di&'( a d% i&'( b dV rdI a V rI b cx3 dt L dt % L dt dt 0

(16)

Where a, b and c are defined as sliding mode coefficients.

(11)

A BB

S ax3 bx% cx5

(10)

Existense of sliding mode can be determined by lyapanov stability analysis. Consider the lyapunov function defined as follows: 1 A B% 2

x5 6 6 x3 dt dt

Sliding surface is defined as follows:

Sliding mode control low defined as: u signS

x% 6 x3 dt

(15)

That should be negative then: For S " 0 ( u 1, u 0 ) and S 0 we have: a

di&'( a V rI bx3 c 6 x3 dt 0 dt L

For S 0 ( u 0, u 1 ) and S " 0 we have: a

It should be noted that 0 ueq 1. Equation (15) is a public control rule for boost PFC converter.

(22)

di&'( a V V rI bx3 c 6 x3 dt " 0 dt L

(23)

(24)

By combining equations (13) and (14) we have:

B. case2: Double integral sliding mode control In this section, design the double integral sliding mode control that force the input current to follow the calculated reference current (i&'( ) which is inphase with the input voltage. The proposed controller use current error x1, the integral of the current error x2 and the double integral of the current error x3 as state variables. State variables are defined as follows:

745

di&'( rI V di&'( rI αx3 β 6 x3 dt dt L L dt L

(25)

Equivalent control method is based on setting the derivative of sliding surface to zero, to maintain the trajectory motion of the system on the sliding surface. In this case the discrete signal is replaced with a continuous signal. So we have: u'C

1 di&'( [L V rI Lαx3 Lβ 6 x3 dt] V dt

(26)

It should be noted that 0 u eq 1. Equation (26) is a public control rule for boost PFC converter. IV.

Block diagram of the above equation is shown in fig.2.

FREQUENCY RESPONSE ANALYSIS AND SELECT SM COEFFICIENTS

A. Selection of Sliding Coefficients Figure 2. Block diagram of the current loop under ISMC and DISMC

The equation relating sliding coefficients to the characteristic response of the converter during sliding mode operation can be easily found by substituting S=0 into (7):

ax3 b

dx3 c 6 x3 dt 0 dt

Open loop gain of fig.2 under ISM and DISM controller is: 1 1 Gs α β s s

(34)

(27)

Rearranging the time differentiation of (16) into a standard second-order system form, we have: d% x3 dx3 2ξω ω % x3 0 dt % dt

(28)

Where ω IK is the undamped natural frequency and ξ

L

%√KJ

J

is the damping ratio. Recall that there are three possible types of response in a linear second-order system: under-damped 0 , ξ 1, critically-damped ξ 0, and overdamped ξ " 1. For ease of discussion, we choose to design the controller for critically-damped response thus: x3 t A3 A% teNOP!

(29)

where Q3 and Q% are determined by the initial conditions of the system. In a critically-damped system, the bandwidth of the controller’s response fST is: fST

ω 1 c I 2π 2π b

Figure 3. Bode plot of the current loop under ISMC and DISMC

By bode plots and equation (31) and the existence condition, the SM surface coefficients can be selected by desired phase margin. In this paper the desired value is 45degree. The bode plot is shown in fig.3 with α 5 X 10Y , β 4 X 10Z . To compare both methods performance we chose same values for sliding coefficients.

(30)

By rearranging (19) and substituting ξ 1 into the damping ratio, the following design equations are obtained: a 4πfST b

c 4π% fST % b

(31)

I.

B. Frequency response analysis Considering that the dynamics of the system is obtained from B 0 thus: α

di&'( di d% i i βi&'( i 0 α dt dt dt % &'(

(32)

We can rewritten equation above in S-domain as follows: s % i s % i&'( i&'( i αs β

(33)

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SIMULATION

simulations have been done by MATLAB. Parameters for simulation are shown in table 1. Case1(ISMC): simulation was performed at input frequency 60HZ and input AC voltage and input inductor current and the output voltage are shown in fig.4 and fig.5. As can be seen in fig.4, load resistor at half a second suddenly doubled, the good performance and fast response of the controller in this case are shown. Also fast response and less steady state error in output voltage have been illustrated in fig.5.

TABLE I.

PARAMETERS FOR SIMULATION

Table 1 Parameters for simulation A=110 Magnitude of input voltage

Frequency of input voltage

F=60HZ

Input inductor

V&'( 300V

Parasitic resistance

r =0.1Ω

Output capacitor

R=1KΩ 2KΩ

Reference output voltage

Nominal load resistor

L=1mH

C=220uF

Figure 6. input voltage and current

Figure 4. input voltage and current Figure 7.

chattering in DISMC boost PFC converter

Fig.7 shows chattering in DISMC. Close to the unit power factor even at 800HZ frequency is achievable. Total harmonic distortion of input current at 60HZ is shown in figure 8. The THD of input current is 0.21% at 60HZ and THD is 2.16% at 800HZ. Power factor at 60HZ-800HZ is about 99.999%. Obviously, it is satisfied to IEC 61000-3-2 standard.

Figure 5. output voltage

The THD of input current is 6.61% in 60HZ and Power factor is about 99.997%. Derivation of sliding surface has big changes in 0.5s when the load resistor suddenly doubled. Because of high THD and big changes in ds and less chattering we use DISMC for PFC converter. Case2(DISMC): simulation was performed at frequency 60HZ and input AC voltage and input inductor current is shown in fig.6. As can be seen in fig.6, load resistor at half a second suddenly doubled, the good performance and fast response of the controller in this case are shown. Also fast response and less steady state error in output voltage is achieved.

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Figure 8. the THD of input current in DISMC

CONCLUSION

In this paper boost PFC converter with ISMC and DISMC are proposed. By using both controllers, in the presence of unknown parameters such as parasitic resistance of the inductor and suddenly changes of the load resistance, it is possible to achieve close to unity power factor and adjustable voltage in

output with less steady state error. As we know, chattering is a big problem in SM controllers. In this paper by using second order Sliding Mode (equivalent method), we reduced chattering phenomenon. The simulation results show the good performance of DISMC compared with ISMC in boost PFC converter such as low THD, less changes in derivation of sliding surface in load changes and less chattering. The simulation results shown the current waveform has the sinusoidal shape and inphase with line voltage, the tested power factor in DISMC is 99.999% and the input current THD is 0.21%.

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