Wind Energy

26 Network Perspective of Wind-Power Production 151
extension to include link removals and link capacities, the main conclusions
from before are confirmed. These are important findings for critical infrastructures
like communication networks and power grids.
26.3 Two Wind-Power Related Model Extensions
The critical infrastructure network model of Sect. 26.2 can be extended in
many directions. The picture we have in mind for the first generalization is
that of a grid consisting of only volatile wind-powered sources. A simple, but
adequate model extension is to replace the path strengths sif in (26.1) by
random (source) node strengths si, which are independently and identically
drawn from for example a log-normal distribution
�
1
p(s) = √ exp −
2πσ2s (ln s + σ2 /2) 2
2σ2 �
(26.7)
with fluctuation strength σ. Due to the conserved mean 〈s〉 = 1, the average
load 〈Ln〉 of a node is independent of the fluctuation strength; consult again
(26.1). A capacity layout Cn =(1+α)〈Ln〉 based on these averaged loads
is not able to prevent the occasional overloading of nodes due to the source
fluctuations. As a function of the fluctuation strength and for various tolerance
parameters, Fig. 26.3a illustrates the relative size of the giant component
resulting from a fluctuation-driven cascading overload failure within an initial
N gc /N
(a) (b)
1 1
0.95
0.9
0.8
0.9
0.7
0.85
0.6
0.8
0.5
0.75
0.4
0.3
0.7
0.2
0.65
0 0.6368 0.8326 0.9572 1.0481 1.1193 1.1774
0.1
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Fluctuation strength Source strength
Fig. 26.3. The relative size of the giant network component as a function of
(a) the fluctuation strength σ of (26.7) and (b) the source strength swind of (26.8).
All curves are based on the hop metric. From bottom to top the solid curves correspond
to tolerance parameters α =0.1 (a),0.0 (b),0.2, 0.5, 1.0 andthedotted
one to the (N − 1) analysis (26.2). For (a) all curves have been averaged over 50
independent source realizations. One Poisson network realization with degree distribution
pk =(λ k /k!)e −λ has been employed; parameters are N = 100 (a), 200 (b)
and λ = 〈k〉 = 5 (a), 7 (b)
N gc /N