Preliminaries - Control of Power Inverters in Renewable Energy and Smart Grid Integration - page 67

+

e

+

p

e -τ d s W(s)

Figure 2.2 Internal model for repetitive control

control scheme discussed in Chapter 16. The output voltage of an inverter is full of harmonics
and it is important to reduce the level of major harmonics in order to obtain a low THD. One
possible solution is to include a bundle of models in the form of

1

at different harmonic

s 2 + ω

2

frequencies.
Another solution is the so-called repetitive control (Chen et al. 2008; Hara et al. 1988;
Nakano et al. 1989; Weiss, 1997; Weiss and Hafele, 1999), which adopts an infinite-
dimensional internal model M ( s ) shown in Figure 2.2 to provide a series of conjugate poles at
all harmonic frequencies. The internal model consists of a local positive feedback via a delay
line cascaded with a low-pass filter W ( s ), which is added to improve the stability of a repetitive
control system because the system without it is a neutral-type delay system (Logemann and
Pandolfi, 1994; Logemann and Townley, 1996; Partington and Bonnet, 2004). As shown in the
next subsection, this internal model has a series of poles very close to the harmonic frequencies
and hence is able to improve THD.

2.2.2 Poles of the Internal Model M(s)
The transfer function of the internal model is

1

=

W ( s ) e − τ d s ,

M ( s ) (2.2)

1 −

where W ( s ) is often chosen as a low-pass filter

ω c

W ( s ) =

s + ω c

with the cut-off frequency ω c . The poles of the internal model M ( s ) from (2.2) are the solutions

of the transcendental equation

s + ω c

ω c

e − τ d s = ,

1

e − τ d s k e τ d ω c which has infinitely many roots s k , k ∈ Z . Substitute s k = s k − ω c , then

ω c s k =

, i.e.,

τ d s k e τ d s k = τ d ω c e τ d ω c = a , with a . (2.3)

The Lambert W function, whose history and properties are beautifully presented in (Corless
et al. 1996), is needed to solve this equation. The Lambert W function is a multi-valued analytic