Synchronverters: Grid-friendly Inverters that Mimic Synchronous Generators - Control of Power Inverters in Renewable Energy and Smart Grid Integration - page 268

From simple energy considerations (see, e.g., (Ellison 1965; Fitzgerald et al . 2003)), there is

, f constant.

E
∂θ m

∂

T e =

It is not difficult to verify (using the formula for the derivative of the inverse of a matrix
function) that this is equivalent to

i , i f constant.

T e =− ∂

E
∂θ m

Since the mechanical rotor angle θ m satisfies θ = p θ m ,

i , i f constant

p ∂

E
∂θ

T e =−

pM f i f i

∂
∂θ

=− , cos θ

pM f i f i θ .

, sin = (18.7)

i 0 sin

Note that if i = ϕ (as would be the case in sinusoidal steady state), then

pM f i f i 0 sin θ =

3
2 pM f i f i 0 cos(

ϕ, sin T e = θ − ϕ ) .

Note also that if i f is constant (as is usually the case), then (18.7) with (18.4) yields

T e ˙ θ m = i , e .

18.1.3 Presence of a Neutral Line
The above analysis is based on the assumption that the neutral line is not connected. If the
neutral line is connected, then

i a + i b + i c = i N ,

where i N is the current flowing through the neutral line. Then, the formula for the stator flux
linkages (18.1) becomes

⎡ ⎤

1
1
1

⎣ ⎦ Mi N

= L s i + M f i f cos θ −