Power grid vulnerability, new models, algorithms and computing

106 Pages·2009·0.87 MB·English

Power grid vulnerability, new models, algorithms
and computing
Daniel Bienstock
Abhinav Verma
ColumbiaUniversity,NewYork
February, 2009
DanielBienstockAbhinavVerma(ColumbPiaoUwneirvgerrisditvy,uNlneewraYboilrikty),newmodels,algorithmsandcompFuetbinrugary,2009 1/44
Talk Outline
Lossless power flows
Solving the N-k problem
A better model for the N-k problem
DanielBienstockAbhinavVerma(ColumbPiaoUwneirvgerrisditvy,uNlneewraYboilrikty),newmodels,algorithmsandcompFuetbinrugary,2009 2/44
Power flow model - lossless model
We are given a network with nodes N (“buses”) and arcs A (“lines”):
A set of G of supply nodes (the “generators”); each generator i
has an “operating range” 0 ≤ SL ≤ SU,
i i
A set D of demand nodes (the “loads”); for each load i a
“maximum demand” 0 ≤ Dmax.
i
For each arc (i,j) a parameter x .
ij
Basic problem: operate network so as to maximize amount of
delivered power
DanielBienstockAbhinavVerma(ColumbPiaoUwneirvgerrisditvy,uNlneewraYboilrikty),newmodels,algorithmsandcompFuetbinrugary,2009 3/44
Power flow model - lossless model
We are given a network with nodes N (“buses”) and arcs A (“lines”):
A set of G of supply nodes (the “generators”); each generator i
has an “operating range” 0 ≤ SL ≤ SU,
i i
A set D of demand nodes (the “loads”); for each load i a
“maximum demand” 0 ≤ Dmax.
i
For each arc (i,j) a parameter x .
ij
Basic problem: operate network so as to maximize amount of
delivered power
DanielBienstockAbhinavVerma(ColumbPiaoUwneirvgerrisditvy,uNlneewraYboilrikty),newmodels,algorithmsandcompFuetbinrugary,2009 3/44
Power flow model - lossless model
We are given a network with nodes N (“buses”) and arcs A (“lines”):
A set of G of supply nodes (the “generators”); each generator i
has an “operating range” 0 ≤ SL ≤ SU,
i i
A set D of demand nodes (the “loads”); for each load i a
“maximum demand” 0 ≤ Dmax.
i
For each arc (i,j) a parameter x .
ij
Basic problem: operate network so as to maximize amount of
delivered power
DanielBienstockAbhinavVerma(ColumbPiaoUwneirvgerrisditvy,uNlneewraYboilrikty),newmodels,algorithmsandcompFuetbinrugary,2009 3/44
Power flow model - lossless model
We are given a network with nodes N (“buses”) and arcs A (“lines”):
A set of G of supply nodes (the “generators”); each generator i
has an “operating range” 0 ≤ SL ≤ SU,
i i
A set D of demand nodes (the “loads”); for each load i a
“maximum demand” 0 ≤ Dmax.
i
For each arc (i,j) a parameter x .
ij
Basic problem: operate network so as to maximize amount of
delivered power
DanielBienstockAbhinavVerma(ColumbPiaoUwneirvgerrisditvy,uNlneewraYboilrikty),newmodels,algorithmsandcompFuetbinrugary,2009 3/44
Feasible power flows
Suppose we choose an output level b > 0 for each generator i, and
i
a demand level −b > 0 for each load i.
i
Write b = 0 for each other node i.
i
A power flow is a solution f, θ to:
P P P
f − f = b , for all i (so must assume b = 0)
ij ij ij ji i i i
x f − sin(θ − θ ) = 0, for all (i,j), (physics)
ij ij i j
| θ −θ | ≤ π (“stability”)
i j 2
A difficult system?
DanielBienstockAbhinavVerma(ColumbPiaoUwneirvgerrisditvy,uNlneewraYboilrikty),newmodels,algorithmsandcompFuetbinrugary,2009 4/44
Feasible power flows
Suppose we choose an output level b > 0 for each generator i, and
i
a demand level −b > 0 for each load i.
i
Write b = 0 for each other node i.
i
A power flow is a solution f, θ to:
P P P
f − f = b , for all i (so must assume b = 0)
ij ij ij ji i i i
x f − sin(θ − θ ) = 0, for all (i,j), (physics)
ij ij i j
| θ −θ | ≤ π (“stability”)
i j 2
A difficult system?
DanielBienstockAbhinavVerma(ColumbPiaoUwneirvgerrisditvy,uNlneewraYboilrikty),newmodels,algorithmsandcompFuetbinrugary,2009 4/44
Feasible power flows
Suppose we choose an output level b > 0 for each generator i, and
i
a demand level −b > 0 for each load i.
i
Write b = 0 for each other node i.
i
A power flow is a solution f, θ to:
P P P
f − f = b , for all i (so must assume b = 0)
ij ij ij ji i i i
x f − sin(θ − θ ) = 0, for all (i,j), (physics)
ij ij i j
| θ −θ | ≤ π (“stability”)
i j 2
A difficult system?
DanielBienstockAbhinavVerma(ColumbPiaoUwneirvgerrisditvy,uNlneewraYboilrikty),newmodels,algorithmsandcompFuetbinrugary,2009 4/44
Feasible power flows
Suppose we choose an output level b > 0 for each generator i, and
i
a demand level −b > 0 for each load i.
i
Write b = 0 for each other node i.
i
A power flow is a solution f, θ to:
P P P
f − f = b , for all i (so must assume b = 0)
ij ij ij ji i i i
x f − sin(θ − θ ) = 0, for all (i,j), (physics)
ij ij i j
| θ −θ | ≤ π (“stability”)
i j 2
A difficult system?
DanielBienstockAbhinavVerma(ColumbPiaoUwneirvgerrisditvy,uNlneewraYboilrikty),newmodels,algorithmsandcompFuetbinrugary,2009 4/44