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Electric Generators and their Control for Large Wind Turbines 231

The penalty component C deals mainly with overtemperature T max :
p
T (
 kT T ) ; if T > T max
−

C i
max
C Temp =  (9.27)

 0 if T < T max
where C i = C a + C f and k is the penalty coefficient.
T
Penalty components for PM demagnetization may be added for most critical situations. In addi-
tion to the initial cost, energy cost, cost of PWM converter (per KVA), and the cost of converter
losses may be added if so desired. To simplify the case, these additional terms are not included in
the study reported here.
The evolution of C during the optimization design process and power losses, efficiency, weight,
t
and material cost components are all illustrated in Figures 9.19 and 9.20 for an 8 MW, 480 rpm
PMSG.
Despite the rather large number of iterations, the total computational time was less than 100 s
on a standard dual-core 2.4 GHz desktop computer. Even with 20 of such runs, from randomly dif-
ferent initial variable vectors, to better secure a global optimum, the computational time would be
less than 2000 s (around 33 min). Finite element method (FEM) is embedded in the optimization
algorithms after each entire optimization run, in a few points, in order to check the average torque
T and inductances L and L and to correct the latter’s analytical expressions, by under relaxation
q
d
e
fudge factors until sufficient convergence is met. This would increase the computational time by up
to 30 times, meaning 6 h on a standard contemporary desktop computer. The same optimal design
methodology is applied for the same power, 8 MW, but at 1500 rpm, 3.6 kV. Selected comparative
results are shown in Table 9.4.
The initial cost, the PM, and the total weight are smaller for the 1500 rpm generator, but the
efficiency is a bit lower because the mechanical losses are notably higher than the stator electrical
losses at 1500 rpm.
For this case, the efficiency, cost, and weight of the mechanical transmission system for the two
speeds are not introduced in the optimal design.

9.4.3 Circuit Modeling and Control of PMSG

The dq model is suitable for PMSG investigation for transients and its control:

dψ
iR s − V s = − s − jωψ ; ψ = ψ d + jψ q
s
r
dt s s
ψ d = Li d + ψ PM ; ψ q = Li ; is = i d + ji q q (9.28)
qq
d
3
e T = p (ψ d q i − ψ q d i ) < 0 forgenerating
1
2
where
V , ψ , and i are voltage, flux linkage, and current space vectors, respectively, with their dq
s
s
s
components
T is the electromagnetic torque
e