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Problem 1:
Power Grid Load Shedding
Definitions, Notations, and Assumptions
Let N denote the set of all natural numbers (i.e., nonnegative integers).
Regarding the basic terminology in graph theory,
refer to ICS241 handout
"An Introduction to Graph Theory."
Assume that a power grid network is represented by
a loop-free, weighted, directed graph G=(V,E) in which
the following four weights are associated with each vertex v ∈ V
- Maximum Generation m(v) ∈ N
representing the maximum amount [W] of electric power the vertex v can genetate:
m(v) > 0 if v represents an electricity generator and
m(v) = 0 otherwise.
- Current Generation g(v) ∈ N
that is the amount [W] of power currently being genetated by v:
g(v) ≤ m(v) holds for every v ∈ V and
g(v) > 0 holds only when v represents an electricity generator.
- Demand d(v) ∈ N
that is the amount [W] of electric power consumed by v
- Priority p(v) ∈ N
representing a relative priority for vertex v to receive
electric power demanded by v
(higher the value, higher the priority)
and the following weight is associated with each edge e ∈ E.
- Transmission Capacity c(e) ∈ N
representing the maximum amount [W] of electric power
that can be delivered through a transmission line e
Without loss of generality, we can assume that a power grid network G=(V,E)
is a simple, weighted digraph.
Note that parallel edges (i.e., multiple transmission lines)
with the common head u and tail v can be converted to a single edge (u,v).
A stable flow f in a power grid network G=(V,E) is
a function f : E → N satisfying
the transmission capacity constraint
f(u,v) ≤ c(u,v)
for every edge (u,v) ∈ E
and
the following preservation property required by Kirchhoff's laws.
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g(v)
+
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=
d(v)
+
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for each vertex v ∈ V
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Problem Formulation
Input:
A simple, weighted digraph G=(V,E) representing a power grid network
with the following weights on vertices and edges
- Maximum Generation m(v) ∈ N
for each vertex v ∈ V
- Current Generation g(v) ∈ N
for each vertex v ∈ V
- Demand d(v) ∈ N
for each vertex v ∈ V
- Priority p(v) ∈ N
for each vertex v ∈ V
- Transmission Capacity c(u,v) ∈ N
for each edge (u,v) ∈ E
Output:
- A new value of g'(v) for each vertex v ∈ V
- A subset S ⊆ V that represents a set of victims cut off for load shedding
Constraints:
- g'(v) ≤ m(v) for each vertex v ∈ V
- There exists a stable flow f on the new power grid network G'
that is a subgraph of G induced by V−S,
i.e., G' = (V−S, E ∩ (V−S)×(V−S))
with new g'(v).
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