Wind Energy

148 S. Jost et al.
to destination node f is transmitted along the links of the shortest-hop path
[i→f]hop. Out of all N(N − 1) paths between the N nodes, the betweenness
centrality
N�
Ln[hop] = path([i→f]hop; n)sif
(26.1)
i�=f=1
counts all shortest-hop paths which go over the picked node n and defines its
load during normal network operation. The index function path([i→f]hop; n)
is equal to one if n belongs to the shortest-hop path [i→f]hop, and zero else.
In case of a heterogeneous network structure, a very heterogeneous load
distribution emerges; see Fig. 26.1. A few nodes have to carry an exceptionally
large load. If some of them fail, it comes to a network-wide load redistribution.
The shortest flow paths, which were going via the failed nodes, are readjusting
and are using the other (transmission) nodes. As a consequence of this readjustment,
some nodes have to carry a larger load than before. If the new loads
exceed their capacities, then the respective nodes will also fail, triggering a
new load redistribution with possibly more overload failure.
In order to reduce the occurrence of such a cascading failure, an (N−1)
analysis is evoked. One of the N nodes, say m, is virtually removed from
the network. The shortest-hop flow paths of the reduced (N−1) network are
recalculated, which then according to (26.1) determine the readjusted loads
Ln(m) of the remaining N−1 nodes. This procedure is repeated for every
1
0.1
0.01
0.001
1e-04
1e-05
1e-06
hop metric
load metric
�
�
1 10 100 1000
load
Fig. 26.1. Distribution of node loads Ln resulting from the hop metric (vertical
crosses) and the load-dependent metric (rotated crosses). The two curves are averaged
over 50 independent random scale-free network realizations with parameters
N = 1000 and γ = 3. A scale-free network is characterized by the degree distribution
pk ∼ k −γ to find a node attached to k links. The load is given in units of the
network size