Generators - Wind Energy Systems for Electric Power Generation - page 43

voltage U 1 and the inductor voltage U p . The load angle is positive for leading in-
ductor voltage, as in generator operation.
Siusoidal voltages, currents and flux are represented by vectors in a complex
plane, either in Park's d,q-coordinates or in w,b-coordinates indicating active and
reactive components as referred to terminal voltage U 1 . The relation between the
coordinates is:

e -j ϑ e j ϑ

I dq = I d + jI q = j I wb I wb = I w + jI b = I dq · · ←→ − j · · (3.19)

3.3.2.3 Complex Locus and Vector Representation

Figure 3.16a shows complex currents with relative inductor voltage as parameter, in
the wb-plane. For the turbo-type machine the loci are concentric circles with center
M. Considering the quadrants in the figure, current vectors in the upper half-plane
(quadrants I, II) indicate motoring, while the lower half-plane (quadrants III, IV) in-
dicates generator operation. Also currents in the right half-plane (quadrants I, IV) in-
dicate reactive power consumption (“under excitation”), whereas the left half-plane
(quadrants II, III) indicates reactive power supply into the grid (“over excitation”).
Note that for a given active load the reactive power adjusts itself depending on the
inductor voltage. Reactive power can be set in machines with excitation windings.
Shown in the figure is a motor operation, over excited.
A vector diagram describing a generator performance with reactive power deliv-
ered is shown in Fig. 3.16b. Note the positive load angle of approximately 30 ◦ .

Fig. 3.16 Steady-state characteristics of a synchronous machine in grid-operation ( a ) Current locus
diagram; ( b ) vector diagram