Parallel Operation of Inverters - Control of Power Inverters in Renewable Energy and Smart Grid Integration - page 296

Alternatively, according to (19.4), there is

m 1 E 1 V o
R o 1

m 2 E 2 V o
R o 2

sin δ 1 = sin δ 2 . (19.14)

If δ 1 = δ 2 and E 1 = E 2 then

m 1
R o 1 =

m 2
R o 2 .

(19.15)

For inverters designed to have resistive output impedances, if the system is stable, then the
following two sets of conditions are equivalent:
C 1 : E 1 =

C 2 :

E 2 δ 1 = δ 2

m 1
R o 1 =

n 1
R o 1 =

n 2
R o 2

m 2
R o 2

⇐⇒ .

If C 1 holds, then proportional real power sharing is achieved according to (19.5). As a
result, (19.12) holds, according to (19.10) and (19.14). Furthermore, reactive power sharing
proportional to their ratings is achieved and (19.15) holds. Conversely, if C 2 holds, then
E 1 =

E 2 according to (19.14). Furthermore, (19.11) holds according to (19.10).
A by-product from this is that n i and m i should be proportional. In other words, it is
questionable for the conventional droop strategy to achieve different sharing ratios for real
power and reactive power. It also indicates that if R-inverters achieve accurate proportional real
power sharing under condition C 1 , then they also achieve proportional reactive power sharing.
If they achieve proportional reactive power sharing under condition C 2 , then they also achieve
proportional real power sharing. However, this is almost impossible in reality. It is difficult
to maintain E 1 = δ 1 = δ 2 because there are always numerical computational errors,

disturbances and noises. It is also difficult to maintain

E 2 or

γ 1 = γ 2 because of different feeder
impedances, parameter drifts and component mismatches. The reality is that none of these
conditions would be met although the reactive power sharing is accurate (note that conditions
C 1 and C 2 are only sufficient but not necessary). A mechanism is needed to guarantee that
accurate proportional load sharing can be achieved when these uncertain factors appear.

19.6 Robust Droop Control of R-inverters
19.6.1 Control Strategy
As a matter of fact, the voltage droop (19.5) can be re-written as

E ∗ =− E i = E i − n i P i ,

and the voltage E i can be implemented via integrating E i , that is,

t
0

E i = E i d t .

This works for the grid-connected mode where E i is eventually 0 (so that the desired power

is sent to the grid without error), as in (Dai et al. 2008a; Li et al. 2004; Marwali et al. 2004).