Sinusoid-locked Loops - Control of Power Inverters in Renewable Energy and Smart Grid Integration - page 368

The time constant of the frequency loop is

J
D p .

τ f = (23.10)

The choice of τ f determines the dynamic response of the loop. It is proportional to the moment

of inertia J . A large τ f is equivalent to having a large J , which makes the SLL less sensitive to

variations in the grid frequency and also makes the system more stable. However, the response
is slow. A small

τ f is equivalent to having a small J , which leads to fast frequency tracking.
As a general rule of thumb, τ f can be chosen to be much smaller than the period of the voltage

v so that the frequency can be tracked very quickly.

The time constant of the amplitude loop is proportional to

K
˙

τ q = (23.11)

θ n

and the amplitude loop generates M f i f , which directly affects the amplitude of e in (23.6).
Hence the choice of

τ q affects the dynamic response of the amplitude tracking. Generally, the
frequency loop should be tuned much faster than the amplitude loop, which is normally the
case because . This

allows the voltage to be established. Otherwise M f i f and eventually the voltage amplitude E
would be driven to zero, which is also an equilibrium point of the system, before the frequency
and phase could be synchronised. However, if a very large

τ f is often chosen to be much smaller than the period of the voltage v

τ q is chosen, it would take a long

time for the voltage amplitude E to track v m .

The inductance L and resistance R of the virtual synchronous reactance X s can be chosen
to be small to enable a large transient current i , which helps speed up the tracking process.
However, too small L and R may cause oscillations in the frequency estimated. Moreover, the
ratio

R
L

1

is the cut-off frequency of the filter

sL + R , which determines the capability of filtering

out the harmonics from the voltage .

The loop to generate the reference frequency ˙

v

θ r is an outer loop for the frequency loop so
it should be tuned much slower than the frequency loop. Its time constant is

1
D p K i

τ fn =

and can be tuned as τ fn = (10 ∼ 100) τ f .

23.6 Equivalent Structure
The SLL shown in Figure 23.2(a) can be redrawn as shown in Figure 23.2(b) to demonstrate
the differences from conventional PLLs. The hold filter

e − Ts
Ts

1 −

H ( s ) =

2 ˙
θ

with T = is adopted to take the average value of an incoming signal.