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Unformatted text preview: er supply network connects water supplies (such
as reservoirs) to consumers via a network of pipes. Water ﬂow in the network is due to gravity
(as opposed to pumps, which could also be added to the formulation). The network is composed
of a set of n nodes and m directed edges between pairs of nodes. The ﬁrst k nodes are supply or
reservoir nodes, and the remaining n − k are consumer nodes. The edges correspond to the pipes
in the water supply network.
We let fj ≥ 0 denote the water ﬂow in pipe (edge) j , and hi denote the (known) altitude or height
of node i (say, above sea level). At nodes i = 1, . . . , k , we let si ≥ 0 denote the ﬂow into the network
from the supply. For i = 1, . . . , n − k , we let ci ≥ 0 denote the water ﬂow taken out of the network
(by consumers) at node k + i. Conservation of ﬂow can be expressed as
Af = −s
c , where A ∈ Rn×m is the incidence matrix for the supply network, given by −1 if edge j leaves node i
+1 if edge j enters node i
Aij = 0
otherwise. We assume that each edge is oriented from a node of higher altitude to a node of lower altitude; if
edge j goes from node i to node l, we have hi > hl . The pipe ﬂows are determined by
fj = 2
αθj Rj (hi − hl )
,
Lj where edge j goes from node i to node l, α > 0 is a known constant, Lj > 0 is the (known) length
of pipe j , Rj > 0 is the...
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This note was uploaded on 09/10/2013 for the course C 231 taught by Professor F.borrelli during the Fall '13 term at University of California, Berkeley.
 Fall '13
 F.Borrelli
 The Aeneid

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