Average Model of the Bootstrap Variable Inductance (BVI)

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Average Model of the Bootstrap Variable Inductance (BVI)

Mohammad Tavakoli Bina David C. Hamill

School of Electronic Engineering, Information Technology and Mathematics University of Surrey, Guildford GU2 7XH, United Kingdom

Abstract—The BVI is a recently introduced FACTS controller which can emulate positive and negative inductance. Here it is modelled using an averaging approach, resulting in an equivalent cir- cuit model. Theoretical considerations show that the averaged model should agree well with the ori- ginal system, and this is confirmed by MATLAB and PSpice simulations. The average model can be improved by adding a switching-ripple approxim- ation, produced by an auxiliary state-space equa- tion. The solution is expandable as a Fourier series which can be suitably truncated. The average-plus- ripple model gives good agreement with the ori- ginal system.

I. Introduction

T

HE Bootstrap Variable Inductance (BVI) was introduced recently [1]–[3] as a new Flexible AC Trans- mission System (FACTS) controller. Emulating variable positive/negative inductance, the BVI can be employed for series and parallel applications in ac power systems.

The concept of negative inductance, or reluctance, and an analysis of the pulse width modulated (PWM) BVI as a FACTS controller have been presented in previous papers [1]-[3].

The analysis of a power electronic system is complex, due to its switching behaviour. Therefore there is a need for simpler, approximate models. One common approach to the modelling of power converters is averaging. This ap- proximates the operation of the discontinuous system by a continuous-time model. The model produces waveforms that approximate the time-averaged waveforms of the original system. As well as simplifying analysis and making it easier to understand the system’s behaviour under steady state and transient conditions, averaged models have the advantage that they speed up simulation.

In this paper, we start with the time-varying state space equations of the BVI. We approximate them by averaged equations, then introduce an equivalent circuit model. Fi- nally, we extend our approximate model to incorporate ripple effects. The combined average-plus-ripple model gives good agreement with the original system, as demon- strated using MATLAB and PSpice.

II. Overview of the BVI

Negative inductance was first proposed as a power- system series compensator in 1992, in the form of a “variable active-passive reactance” (VAPAR) [4], and this circuit has subsequently been developed by its inventors. In 1999, the present authors introduced the BVI for parallel and series applications in power systems. Fig. 1(a) shows the principle. An impedance Z connected with a variable gain amplifier A(jLJ) provides a variable input impedance

2/=!= z

I 1 – A(joJ) (1)

where V and I are the input voltage and current phasors, respectively. Now suppose A(ju) = A is a positive real constant. When A < 1, Z’ has the same sign as Z but greater magnitude. When A = 1, Z’ = co so I = O.This principle is knowu in analogue electronics as bootstrapping.

When A >1, Z’ has the opposite sign to Z (negative impedance conversion). Thus if Z is an inductor (Z = jwL), the circuit will emulate a negative inductance, or reluct-

ance. The input reluctance is 17’ = –L’, and the input impedance is Z’ = –jwr. By varying A, awide range of inductance and reluctance can be emulated, as shown in Fig. l(b).

Fig. 1(c) shows a single-phase BVI circuit. The capacit- ors are fed by a voltage-doubler rectifier which is supplied by the applied voltage. (It is also possible to self-power the circuit directly.) The applied voltage also forms the reference for the P WM, whose output drives two switches. By adjusting the PWM carrier waveform, its reference voltage or the dc supply voltage, the gain A and hence the input inductance or reluctance can be varied.

III. Overview of Averaging

We first review the theoretical foundations of average modelling, then explain how it can be applied to the BVI.

A. Theoretical basis

State-space averaging (SSA) was established by Middlebrook and Cuk [5], and has been widely used for

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10 _‘t

8 . . . ..} . . . ... . . .

o

-101 1 1

0 0.5 1 1.5 2

A

(c) SWrtchingamplifier Voltage-doubler rectifier

--- --- --- --- --- . . . --- .-, z

(Inductor m.del) ~

. . . . . . . . ‘Y i

Rp; ;

. . . .^J I : . . . .

Figure 1: (a) Principle of variable inductance/reluctance; (b) range of emulated inductance L’ as a function of amplifier gain A; (c) BVI circuit schematic, showing a possible implementation with rectifier powering from the applied voltage.

modelling de-de converters. The time-piecewise state equations are averaged over a switching cycle to give a time- continuous description. Essentially SSA assumes that the averaged state equations will give waveforms that are close to the averaged exact waveforms. This is true only at zero perturbation frequency, and when perturbations approach the switching frequency the error is ill defined. Although SSA is widely used in practice, its theoretical foundations are somewhat lacking in rigour.

In classical SSA, the switching frequency is absent from the average model, while it is clearly an important para- meter of the real system. However, in [6] a switching- frequency dependent average model for de-de converters was proposed, giving more accurate results than standard SSA.

To put averaging on a firmer basis, a review in [7] sur- veyed the Bogoliubov theorem for finding a bound for approximation of the time-varying state equation by averaging. It also described the KBM method for finding the approximation error.

The averaging theory discussed in [7] has been extended to cases in which state discontinuities occur [8]. A general feedback PWM was discussed, based on an integral form of the state equations rather than the standard differential form. A theorem was proved to this effect: for an arbitrarily large but bounded time interval and a sufficiently small

switching period, the exact system and its average model can remain arbitrarily close to each other [8]. Moreover, if the average model tends to an asymptotically stable equilibrium point, the theorem can be extended to an infinite time interval [9].

B. Application to the B VI

There are two main periods involved in the BVI: the reference period TR (16. 7ms or 20ms) and the swit thing period Tc, the period of the PWM carrier, typically a few kilohertz. As is usual in PWM, TR >> TC. Further, it is assumed that TRis an integer multiple of Tc: TR = MTC.

When operated open-loop, the switching transitions occur at predetermined times. However, when the BVI is embedded in a control loop, the transitions will occur at times determined by the state variables themselves. With this in mind, the open-loop average equations are obtained in a standard form that can later be modified for closed- loop control:

x(t) = f(x(t), s(t), u(t)) (2) where x(t) is the state vector [i~, Vcl, –VC2J, u(t) is the input vector (in general; here it is a scalar, u(t), the input volt age represent ed elsewhere as phasor V) and s(t) is the

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PWM switching function, which consists of a sequence of L1 pulses (M = TR/Tc, an integer). We define

and

s(t) = ~ s.(t)

n=l

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{

1 (n – l)TC < t < (n – 1 + D(nTc))Tc Sri(t) = –1 (n – 1 + D(nTc))Tc < t ~ nTc

O otherwise

(4) where Sn(t)is the switching function within the nth switching period and D(nTc) is the duty ratio for the nth switching period. Note that at this stage we do not consider D to be a continuous function of time, but rather a discrete value associated with an individual switching period. For S~(t) weadopt this convention: ‘ 1’ means the upper switch of the BVI is closed, and ‘–1’ means the lower switch is closed.

We now apply the averaging operator t

~.(t) = average ~(t) ~ & ~ ~(~) dr (5) t–TC

to (2) over [t –TC, t], to get an average model described by

xo(t) = g(x. (t), D(t), u(t)) (6) where X.(t) is the average state vector, and D(t) is the approximate continuous duty ratio. This new D(t) is a continuous function of time.

From (2) and (6) it can easily be shown that

]Ix(tj - Xa(t)l[ < [lx(o) - Xa(o)[l

+11

j’[f(x(~)> s(~)>u(r)) – g(x~(~), D(T), u(~))lddl ‘7) o

Starting from (7), a theorem in [8] describes the closeness of x(t) and X.(t). For any small 6 > 0 and large M > to (we take to = Ohere), there exists a TO (a function of J andM) and a positive constant K such that for switching period Tc G [0, To]

Ilx(t) - xa(t)tl ~ (11x(O)- XO(0)[I + c$)e~~ (8) Now let x(0) = X.(0). Equation (8) says that for any bounded time interval, x(t) and X.(t) can remain close to each other, provided the switching period is small enough.

Another theorem in [8] states if X.(t)approaches an asymptotically stable equilibrium point, then there exists a sufficiently small TO(a function of 6) such that for switching period Tc c [0,To], Ilz(t) – r~(t)ll S d. Therefore, if the averaged model is asymptotically stable, which is generally true, x(t) will be very close to x.(t). This validates the averaging approach to modelling the BVI.

IV. Average Model of the BVI

In this section we apply the foregoing methodology to develop an average model of the PWM BVI.

A. State-space model

Consider the BVI of Fig. 1(c). There are two topological modes. In the first, the upper switch is closed and the lower switch is open, and in second, the upper switch is open and the lower switch is closed. The state equations for the two modes can be obtained separately then, introducing the PWM switching function s(t) E {– 1, 1}, combined into a single state equation:

x(t)= (Al + s(t) A2)x(t) + bu(t)

[

~ 2L

Al= O

0

1[

^–1

& %

2CI ,A2= O

J-_

2C2 o

0

_l_

2_cj -12C2

X(t) = [ iL VC, Vc, ]T, b=[+ O O]T (9) Then (9) is averaged over a switching period to develop a time-continuous model, in the form of (6). Applying the averaging operator of (5), let X.(t), the averaged state vector, be defined as

+ x.(t) = &

/

X(T) dr

t–Tig

Now, t

(lo)

x(t) = X(t – Tc) +_t–TC

I

^[dx(~)/dr] ^dr ⁽¹¹⁾

and differentiating (10) gives us X.(t) = (x(t) – x(t – Tc))/Tc; comparing these equations yields

1 t dx(~) ^&

x.(t)= ~

/ dr

t–TC

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Integrating (9) over [t– Tc, t]and applying (10) and (12), we get

/ Xa(t) = Alxa+A2—

1-

Tc s(T)x(@d~+b&

/

U(7) dr

t–TC t–TC

(13) The waveform of s(r) during [t – TC, t] has four possible forms, as shown in Fig. 2. Each of these forms can be substituted as the second term in the right hand side of (13). Taking the worst case leads us to

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to+t&

QP t

¹^-1-^t-Tc ^to ^to+tm^. ^t

Figure 2: The four possible forms of s(-r) over [t - Z’C, t]. Figure 3: Equivalent circuit average model of the BVI, suitable for circuit simulators such as SPICE.

It -/^Tc t–Tc

S(T)X(T) dr = (2D(t) – l)xa(t) + * (14)

Here D(t) = tOn(t)/Tc, where tOn(t)is the sub-interval of [t– Tc, t] during which the upper switch is closed. If the average of x(~) taken over Tc is close to its average taken over the sub-interval, the error term in (14), x(t) Tc/2, will be negligible. The slower the variation of x(t) and the higher the PWM carrier frequency, the smaller the error.

Integrating the PWM switching functions in Fig. 2 over [t– Tc, t],the switching function average is

t Sa(t) =

J

S(r) dr = 2D(t) – 1 (15) t–TC

defining the continuous duty-ratio function D(t). The res- ultant averaged state equation is

.00

..0

..o.L ^I

0 (.)002 004 006 008 01 0.12 .,4 ..,, ‘<a 02

t !s..1

4.0 }“ i

x.(t) = (Al + (2D(t) – l) Ap)xa(t) + bu(t) (16) In practice, D(t) has a sinusoidal waveform. We therefore consider the special case where the PWM has a carrier waveform that ramps between –1 and 1, and a sinusoidal reference of m sin(wt – 7r/2), m c [0, 1] being the modulation index. This leads to a Fourier series for s(t)

co

s(t) = ~ Ske~’”kt (17)

k=–cc

Its fundamental phasor S1 = m e–~~12 is substituted in (14) (the dc term and higher harmonics being neglected), to find the continuous D(t) as

D(t) R ~[1 + msij~!jM) sin(tit - ~)] (18) A MATLAB program was written to calculate the error between the exact duty ratio (at t = nTc ,n = 1,2,..., M) and the continuous approximation in (18). For M = 40 (e.g. 2kHz carrier, 50Hz reference), the worst-case error is less than 2%.

Figure 4: MATLAB simulation results for the average model, two cases: (a) an emulated inductance with amplifier gain A = 0.5 (m = 0.25); (b) an emulated reluctance with amplifier gain A = 2 (m = 1).

V. Average Circuit Model

Three state equations (16) describe the average inductor current and capacitor voltages. The first equation can be interpreted as meaning that the average inductor current depends on the voltage difference between the input u(t) and a voltage-controlled voltage source (VCVS) D (t)VCI + (1 – D(t))Vcp. The second and third equations show that the average current of capacitor Cl isz~D(t) and of Cp is iL (1 – D(t)); these can each be represented by a current- controlled current source (C CC S). The resulting equivalent circuit model is shown in Fig. 3, and is suitable for circuit simulators such as SPICE. The current 2A preserves KCL at node A.

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Figure 5: The state variables of the average-plus-ripple model, simulated with MATLAB for two carrier frequencies: (a) 400 Hz; (b) 1000 Hz.

VI. Ripple Estimation

The difference between the state vectors of the exact and average models is

r(t) = x(t) – x.(t) (19)

Vector r(t) is the switching ripple (plus any residual error in the average model). We can consider r(t) to be the state vector of a ripple model. Taking (9), (16), and the derivative of (19), the state equation of the ripple model is obtained:

r(t) = (Al +(sr(t)+211(t)- l) Az)r(t)+sr(t)Azx~ (t) (20) where Sr(t) = s(t) – (211(t) – 1). The ripple model can be solved to provide a perfect correction to the average model.

However, this is computationally equivalent to solving the exact model itself. In practice, some useful economies can be made. If the Fourier series for s(t), (17), is suitably truncated, an approximation to r(t) can be found. The more harmonics are taken into account, the better the approximation. Adding the approximate r(t) to the average waveforms x~(t)gives waveforms that are usefully close to the exact ones. We call this the average-plus-ripple model.

VII. MATLAB Simulation Results

In this section we compare various simulation results for the BVI and our models, performed with MATLAB.

First, the average model of the BVI was simulated. The input voltage was u(t) = 320 sin(ut + 7r/2), and L = 12mH, Cl = C’z = 81OOPF. The initial state vectors were x(0) = Xa (0) = [640, –640, O]T. Figs. 3(a) and (b) show the state variables of the BVI together with its input voltage for two cases: m = 1 (negative inductance or reluctance) and m = 0.25 (positiveinductance).

These results show the capacitor voltages remaining close to their initial values, i.e. this is the steady state. The

inductor current variation is reductive or inductive, as ex- pected.

Next we simulated the average-plus-ripple model. Nu- merical investigation of (17) reveals that the harmonics at the PWM carrier frequency and its sidebands are large compared with the fundamental. Harmonics M, M + 1, Tl + 2 and &l + 3 dominate. The 2Mth harmonic and its sidebands are also import ant, as they affect the shape of the ripple waveforms. As a rule-of-thumb, it is usually adequate to include harmonics up to about 2.25M.

The ripple model was programmed with MATLAB and added to the output of the average model. The BVI parameters used here are based on those of a published high- voltage thyristor controlled series compensator (TCSC) [10]. The reference frequency is 50Hz, and the modulation index is m = 1. Two cases are presented: with a PWM carrier frequencies of 400Hz (M = 8), taking harmonics up to the 18th (2.25 M), and lkHz (M = 20), taking harmonics up to the 45th (again 2.25 M). Figs. 5 (a) and (b) show the state vector of the averaged model and the ripple for two different cases.

VIII. PSpice Simulation Results

The equivalent circuit model of Fig. 3 was simulated with PSPICE. The parameters are the same as for the MATLAB simulations of Fig. 4. Figs. 6 (a) and (b) depict the state variables for comparison with with Fig. 4. The agreement is good, validating the equivalent circuit.

The average-plus-ripple model was converted into an equivalent circuit and simulated with PSpice, using the same parameters. Fig. 7 shows the simulation results, for comparison with Fig. 5(b). Again the agreement is good.

Finally, the average model was simulated together with the exact switched-system model [1]. Fig. 8 shows the results. Apart from the ripple, which was not added to the average model, The agreement is good. But the average model ran 3.6 times faster than the exact model, offering

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Figure 6: Results of the average equivalent circuit model simulated with PSpice: (a) inductance emulation (m = 0.25); (b) reluctance emulation (m = 1).

useful savings in situations where accurate waveforms are not import ant (e.g. investigating system transients).

IX. Conclusion

A theoretically sound averaging method has been applied to approximate the behaviour of the BVI. Starting with the exact state equations, an average model was developed.

An equivalent circuit model was derived from the resulting equations. The exact system (simulated with PSpice) and the approximate model (simulated with MATLAB and PSpice) are all in good agreement, verifying that the ne- cessary and sufficient conditions of the averaging theorems in [8] are satisfied by the proposed models. Moreover, an improved approximation to the exact waveforms was obtained from a ripple :model whose output is added to that of the average model. It is generally sufficient to include harmonics up to about 2.25i14 for acceptable accuracy of this average-plus-ripple model.

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ductance: a new FACTS control element”, Power Electronics Specialists Conf., vol. 2, pp. 619-625, June 1999

WIe

Figure 7: pspice simulation results for an average-plus-ripple circuit model. Parameters as for Fig. 5(b).

Figure 8: PSpice simulation results for the exact system and the equivalent circuit average model.

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