WO2004027964A1

WO2004027964A1 - Power factor precompensator

Info

Publication number: WO2004027964A1
Application number: PCT/GB2003/003989
Authority: WO
Inventors: Alessandro Astolfi; Dimitrios Karagiannis; Romeo Ortega
Original assignee: Imperial College Innovations Limited
Priority date: 2002-09-17
Filing date: 2003-09-17
Publication date: 2004-04-01
Also published as: GB0221573D0; AU2003269141A1

Abstract

A power factor precompensator uses pulse-width modulation control techniques to regulate an output voltage to a desired constant level in the presence of variations in the load, while retaining a unit power factor at the input. This is achieved by forcing the input current to follow the input voltage both in frequency and in phase while tat the same time providing an adaptation scheme which operates either on the input current amplitude, or upon the load resistance and the internal resistance.

Description

Power Factor Precompensator

The present invention relates to a particular class of AC-DC converters known as power factor precompensators. The invention relates to a precompensator using pulse- width modulation (PWM).

Power factor precompensators (PFP) are an important class of switched AC- DC converters. As their name suggests, their main function is to achieve a nearly unit power factor by drawing a sinusoidal current that is in phase with the source voltage, thus eliminating the reactive power and the harmonic interference with other equipment operating off the same source.

More precisely, the control objective is twofold. First, the input current should track a sinusoidal reference signal that is in phase with the input voltage. Second, the output voltage should be driven to a desired constant level. An additional requirement is robustness against variation of the system parameters and in particular of the load, which is usually unknown.

As the amplitude of the input current determines explicitly the DC output voltage, attempts have been made to satisfy both objectives in a single current control loop, which typically comprises a hysteresis or sliding mode controller.

The main drawback of these known controllers is that they require very high switching frequency (typically few hundred kHz) leading to high converter losses. We propose the use of pulse-width modulation (PWM) control techniques, which can be implemented using lower switching frequency (e.g. 10kHz). Furthermore, the issue of robustness against variations of the load is dealt with in the present invention by appropriate adaptive schemes. This allows us to avoid the use of an output current sensor, thus making a practical implementation of the invention more attractive.

According to a first aspect of the present invention there is provided a method of controlling a power factor precompensator circuit, the method comprising causing an input current (x₇) to the circuit to track a reference value (xj *) comprising a sinusoidal wave (sin(ωt)) of variable amplitude (I_d), the amplitude being arranged to adapt with time as a function of an output characteristic of the circuit.

According to a second aspect of the invention, a power factor precompensator circuit is operated by the method set out above.

According to a third aspect of the invention, there is provided a power factor precompensator circuit having a full bridge with complementary pairs of transistor-diode switches, pulse-width modulation being applied to the switches to control the input current according to a method as set out above.

The invention may be carried into practice in a number of ways and one specific embodiment will now be described, by way of example, with reference to the accompanying drawing, in which:

Figure 1 shows a full-bridge boost PFP circuit in accordance with the preferred embodiment of the invention.

We consider the full-bridge boost PFP circuit shown in Figure 1, which consists of two pairs of transistor-diode switches working in a complementary way. The control objective is to robustly regulate the output voltage to a desired constant level in the presence of variations in the load, while retaining a unit power factor at the input, i.e. the input current should follow in frequency and in phase the input voltage. This is to be achieved by appropriate control of a switching signal δ, applied to the switches.

The switching signal δ is generated by a PWM circuit and takes values in the finite set {-1,1}. The averaged model of the PFP can be obtained using Kirchhoff s laws and is given by the equations

Lx_j = - x₂ - rX_j+V_ι(t) (1)

where v_t(t) = Esin (ωt) is the source voltage x_t is the input (inductor) current, x₂ > 0 is the output (capacitor) voltage and u e [-1, 1] is the duty ratio of the PWM. Note that the exact model of the PFP is described by the same equations, if we replace u with δ.

x^*(t) = I_d sin(ωt), (3)

for some I_d > 0 yet to be specified. Substituting equation (3) into equation (1) yields the steady-state value for ux₂, namely

u^*(t)xl(t) = v_i(t)-rx₁(t). (4)

Substituting equations (3) and (4) into equation (2) written for x_{2 =}x₂* yields

The steady-state solution of (5) can be directly computed and is given by

{E-rI_d)RI_d

(t) - + Asin(2ωt + φ), (6)

where

_A _

E-rI. -RCωLI_dω φ = arctan - .

LI_dω + RCω(E-rI_d)

Hence, in steady-state x₂* (t) consists only of a DC term and a second-order harmonic, which can be neglected, since in practice its amplitude is much smaller than the DC term. The average of x₂(t), denoted by Vj, is given by

From (7), solving for I_d yields the solutions

_{7 = ±} ) E² 2V_d ² ^d 2r ^~ \ 4r² rR

which are real if and only if

E 8r Selecting the smallest solution, which corresponds to minimum power (i.e. minimum converter losses), the amplitude of the input current that drives the output voltage to the desired level V_d is given by

A similar result has been obtained by G. Escobar, D. Chevreau, R. Ortega, and E. Mendes, An adaptive passivity-based controller for a unity power factor rectifier, IEEE Transactions on Control Systems Technology, 9:637-644, 2001.

We conclude that by controlling the input current so that it tracks the signal (3), where I_d is given by equation (8), we can achieve both control objectives, namely unit power factor and output voltage regulation, provided that the closed-loop system is asymptotically stable. The input voltage V„ of course, is generally provided externally and has to be taken as 'given'.

The main drawback of this approach, taken alone, as equations (7) and (8) reveal is the sensitivity of the output voltage to the parameters r and R. The dependence on R may pose a significant problem, since in many applications the load is unknown or time-varying. In accordance with the present invention, this obstacle is overcome by adding an adaptation scheme either on I_d or on R, to be described below.

The input current in Figure 1 may be regulated by a variety of different controllers. For the simple case where all the parameters are known, any of the following controllers may be used: • Passivity-Based Controller (PB)

• Feed-Forward Controller (FF)

• Feedback Linearising Controller (FL)

• Internal Model Controller (IM) The first three of these controllers are well known, and may easily be constructed by a person skilled in the art. The IM Controller, however, is new, and hence requires some further consideration. We will now turn to the discussion of this new controller.

If instead of xj we take as output the signal z = ux₂, which corresponds to the input voltage of the transistor-diode bridge, then the system can be linearised by means of the dynamic controller

The linearised system is described by the equations

(10)

(11)

Hence, the transfer function from w to y is given by

1

G(s) = (12) s +l/RC

Note the cancellation of the stable pole at s = -r/L.

It is clear from the analysis of the circuit that we can achieve the control objectives by forcing y(t) to track the signal y^*(t) = z^"(t) = v_i -rx_I ^* -Lx₁. (13) To this end, consider the error

e = y —y = v_i —rx₁ —Kx₁ —ux₂ (14)

and the controller

where k> 0, a > 0 and b > 0 are design parameters. Then, for sufficiently large k, the closed-loop system is asymptotically stable. Moreover, by the presence of the poles at s = +jω, we conclude that z converges to z and hence xj converges to xj.

The choice of the controller (15) is motivated by the internal model principle and the fact that the reference signal y (t) is a sinusoid of frequency ω. The zeros have been inserted to ensure stability.

In practice, in order to compensate for unmodeled dynamics, it may be necessary to add to the reference signal (13) a correction term proportional to the error Xj = Xj * -Xj , i.e.

z^* (t) = v_i - rx) - Lx] - K_j (x - X_j ).

To incorporate this into our linear formulation, consider instead of (11) the output

and the reference y _'(t) = υ_i - rx*_l - Lx*_l - K₁xl.

The transfer function from to y is now given by

Clearly, for sufficiently large k > 0, the closed-loop system is asymptotically stable provided that Kj > -r.

That concludes our discussion of the Internal Model (IM) Controller.

So far we have assumed that the parameters r and R were known and so we could control the output voltage indirectly via formula (8). In practice, however, the load R is usually unknown or time- arying. To overcome this robustness problem we propose to add an adaptation scheme either to the parameter R or to the parameter I_d.

A variety of adaptation schemes are already known but we describe below two novel schemes for wnich asymptotic stability can be proven. The first, wnich we call NLPI control, is based upon a development of the ideas presented first presented in R. Ortega and A. Astolli, Nonlinear PI control of uncertain systems: an alternative to parameter adaptation, in 40^u Conference on Decision and Control, pages 1749-1754, 2002. The second scheme, which we call I&I control is based upon a development of ideas presented in A. Astolfi and R. Ortega, Immersion and invariance: a new tool for stabilization and adaptive control of nonlinear systems, in IF AC Symposium on Nonlinear Control System Design, St Petersburg, Russia, pages 81-86, 2002. The operation of both of these adaptation schemes relies on the previously- mentioned novel approach of having the current track some reference alue of unknown amplitude, which depends on both R and r. In particular, the input current is constrained to take the form defined in equation (3), with I_d defined by equation (8). This allows us to achieve both control objectives, namely unit power factor and output voltage regulation even in situations where the j load R is either unknown or time-varying. The adaptation schemes described are also capable of handling variations in the internal resistance r of the voltage generator.

a. N PΪ control (adapt for I_d)

Invoking time-separation arguments, we can assume that the dynamics of the output voltage are slow compared to the (closed-loop) dynamics of the input current and hence x_} = x_}* at all times. Note that, for this assumption to be valid, the dynamics of I_d must be sufficiently slow. Furthermore, we assume r - 0.

Then, ignoring second-order harmonics, the dynamics of x₂ are given by

1 ft¹ ⁱ^ -RC^X2^ 2Cχ-₂ ^Id- (17)

Note that the assumption r = 0 ensures the above system is affine in the "control" I_d.

The equilibrium x₂ = x * is rendered asymptotically stable by the dynamic control law

where and β are positive constants and x₂ = x * - x₂. To prove this fact, consider the positive definite function

whose time derivative along the trajectories of (27)-(28) is. given by

-\ R Rx2 2x2 ; which is negative semi-definite and hence the system (27)-(28) is stable. Mόreovetj V - 0 implies that x₂ = 0 and so o^*ι, by LaSalle's invariance principle, the system is asymptotically stable.

The difference between the controller (18) and a conventional PI is that here the integral gain is not a constant, but a (nonlinear) function of x . In particular, it is proportional to the ratio E/x₂. Approximating this ratio by a constant results in the industry-standard PI controller.

The assumption r - 0 in the above designs can be replaced by the assumption rxj - e sin (cot) where e is a positive constant. Then we can incorporate the voltage drop across r into the source voltage by taking

Note that this simple modification improves the experimental results significantly. b. I&ϊ control (Adapt for R and r)

Using the adaptive I&I method we can design a parameter estimator for both R and r provided that the system (l)-(2) is bounded-input bounded-state (BIBS) stable. We riow prove that the trajectories of (l)-(2) are bounded/or any u.

Consider the positive definite function

u v= 14 ++ ?Z4

and its time derivative along the system trajectories, namely

ti

By Young's inequality, we have

hence

which shows that the trajectories remain bounded provided r > 0. Note also that the system (l)-(2) is linearly parameterised with respect to the parameters.

1 θ = T, 0.2 - _R-

Following the adaptive I&I methodology, we define the estimation errors

and the update laws

(19) θι ~ ¹ {ux₂ - Vi + Xl(θ₁ + βl)) _t

L d

The resulting error dynamics are

hence by choosing

„2

J9ι(xι) =^'-'^eiBi (21)

. fttø) = -λ^χ ₂, (22)

where,' κ^~ λ are positive constants and recalling that x₂ > 0 and x_} is not identically zero, we obtain the asymptotically stable dynamics

Λ A

Hence, θj + β] (xj) converges to θ_} and θ₂ + β₂ (x₂) converges to θ₂.

The main advantage of the I&I method is that it relies only on BIBS stability. In this case, we have seen that the trajectories of (l)-(2) are bounded regardless of the input. Hence, the I&I adaptation can be used in conjunction with any

Λ stabilising control law u(x_1: x₂,r,l/R) which can thus be replaced by u(xι,x₂βι + βj, +β^ where θ_hθ₂ and β_h β₂ are given by (19)-(20) and (21)-(22) respectively.

Claims

CLAIMS:

1. A method of controlling a power factor precompensator circuit, the method comprising causing an input current (xj) to the circuit to track a reference value (xj *) comprising a sinusoidal wave (sin(ωt)) of variable amplitude (I_d), the amplitude being arranged to adapt with time as a nonlinear function of past values representative of a time-varying output characteristic of the circuit.

2. A method as claimed in claim 1 in which the output characteristic includes the unknown value of a load (R) on the circuit.

3. A method as claimed in claim 2 including minimization of an estimation error

Z₂ = θ₂ -θ₂ +β₂(x₂)

where θ₂ = 1/R, x₂ is the output voltage, and/?₂ is a function of x₂.

4. A method as claimed in claim 3 in which β₂ is given by

where λ is a constant.

5. A method as claimed in claim 2 including updating an estimated value of θ₂ = 1/R using the update law θ (20)

where: Cis the output capacitance of the circuit xj is the input current x₂ is the output voltage β₂ is a function of x₂, and u is the duty ratio of the circuit

6. A method as claimed in claim 2 in which the output characteristic further includes the unknown value of an internal resistance (r) associated with the input current (x ).

7. A method a claimed in claim 6 including minimization of an estimation error

Z^ θ_j -θ_t + β/x,)

where θ = r, x_} is the input current, and ?_/ is a function of x_}

8. A method as claimed in claim 7 in which β_} is given by

βι(xι) = -κχ² (21)

1

where K is a constant.

9. A method as claimed in claim 6 including updating an estimated value of θj = r using the update law a.

where :L is the input inductance xi is the input current x₂ is the output voltage Vj is the input voltage βj is a function of x_l5 and u is the duty ratio of the circuit.

10. A method as claimed in any one of the preceding claims in which the input current (xi) is caused to track the reference value (x_} *) by a passivity-based controller.

11. A method as claimed in any one of claims 1 to 9 in which the input current (xi) is caused to track the reference value (xj *) by a feed-forward controller.

12. A method as claimed in any one of claims 1 to 9 in which the input current (xi) is caused to track the reference value (xj *) by a feedback- linearising controller.

13. A method as claimed in any one of claims 1 to 9 in which the input current (xi) is caused to track the reference value (xj *) by a controller that is described by the equation

where: u is the duty ratio of the circuit Xi is the input current x₂ is the output voltage C is the output capacitance, and w is a new external signal.

14. A method as claimed in claim 13 in which the controller is defined by

where :ω is the input voltage frequency, k, a and b are constants, and s is the Laplace operator

15. A method as claimed in claim 2 including minimization of an estimation error which is a function of variable R and variable x₂.

16. A method as claimed in claim 2 including dynamically updating an estimated value of R, or a function of R, according to an update rule which is a function of x (the input current), x₂ (the output voltage) and u (the duty ratio of the circuit).

17. A power factor precompensator circuit operated by a method as claimed in any one of claims 1 to 14.

18. A power factor precompensator circuit having a full bridge with complementary pairs of transistor-diode switches, pulse-width modulation being applied to the switches to control the input current (xj) according to a method as claimed in any one of claims 1 to 14.

19. A method of controlling a power factor precompensator circuit, the method comprising causing an input current (x_}) to the circuit to track a reference value (xj *) comprising a sinusoidal wave (sin(ωt)) of variable amplitude (I_d), the amplitude being arranged to adapt with time as a function taking two variables, namely: (a) the actual output voltage (x₂), and (b) a difference (x₂*) between a desired output voltage (x₂*) and the actual output voltage (x₂).

20. A method as claimed in claim 19 in which the function is non-linear in χ₂-

21. A method as claimed in claim 19 in which I_d is controlled by the equation

Where and β are constants and E is the amplitude of an input voltage (V or a mathematical equivalent thereof.

22. A method as claimed in any one of claims 19 to 21 in which the input current (xi) is caused to track the reference value (x *) by a passivity- based controller.

23. A method as claimed in any one of claims 19 to 21 in which the input current (x_}) is caused to track the reference value (xj *) by a feed-forward controller.

24. A method as claimed in any one of claims 19 to 21 in which the input current (x_}) is caused to track the reference value (x_} *) by a feedback- linearising controller.

25. A method as claimed in any one of claims 19 to 21 in which the input current (xj) is caused to track the reference value (x_} *) by a controller that is described by the equation:

where: xj is the input voltage x₂ is the output voltage C is the output capacitance, and w is a new external signal.

26. A method as claimed in claim 25 in which the controller is defined by

K(s) = ^ = k-^S ^ - (15) e(s) s +ω" where: ω is the input voltage frequency, k, a and b are constants, and s is the Laplace operator

27. A power factor precompensator circuit operated by a method as claimed in any one of claims 19 to 26.

8. A power factor precompensator circuit having a full bridge with complementary pairs of transistor-diode switches, pulse-width modulation being applied to the switches to control the input current (x_}) according to a method as claimed in any one of claims 19 to 26.