Page 410 - Decision Making Applications in Modern Power Systems

Integration of fixed-speed wind Chapter | 14 371

The electromagnetic torque (T e ) and its relation with the mechanical tor-
que (T m ) of the shaft can be expressed as follows:

T e 5 1:5p ϕ i qs 2 ϕ i ds ð14:13Þ
ds qs
d
T e 2 T m 5 2H ω m ð14:14Þ
dt
where p, H, and ω m are pole pairs of the machine, inertia constant, and rotor
angular velocity, respectively.
The relationships among the line voltages and the d q voltage compo-
nents are given in (14.15) and (14.16), thus
p
1 2cos θ 3sin θ
ffiffiffi
V qs cos θ 1 V abs
5 p ffiffiffi ð14:15Þ
V ds 3 2sin θ sin θ 2 3cos θ V bcs
p
ffiffiffi
V 0 qr 5 1 2cos β cos β 1 p 3sin β V 0 abr
ffiffiffi
V 0 dr 3 2sin β sin β 2 3cos β V 0 bcr ð14:16Þ
where θ is the angular position of the reference frame, and β is the difference
between θ and the angular position of the rotor (θ r ).
Hence, by using the d q current components, the phase currents of the
induction machine can be calculated as follows:

2 sin θ 3
cos θ ffiffiffip
p ffiffiffi
I as 2 cos θ 1 3sin θ 2 I qs
5 4 3cos θ 2 sin θ 5 ð14:17Þ
I bs I ds
2 2
cos β sin β
2 3
p p
I 0 3sin β 3cos β 2 sin β 7 I 0
ffiffiffi
ffiffiffi
ar 6 2 cos β 1 2 qr ð14:18Þ
I 0 5 4 5 I 0
br 2 2 dr
I cs 52 I as 2 I bs ð14:19Þ
0
I 52 I 2 I 0 ð14:20Þ
0
cr ar br
14.2.2 Determination of the permissible penetration level
Determination of the maximum loading ratio or PPL for induction machines
is based on one of the following procedures [14]:
1. Use of a standardized DF curve (or expression) in terms of voltage unbal-
ance level based on the National Electrical Manufacturers Association.
This is more common for induction motors.