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Multi-stage integer programs in power grids
Daniel BienstockAbhinav Verma
Columbia University, New York
Informs 07
Daniel Bienstock Abhinav Verma ( Columbia University, New York)Multi-stage integer programs in power grids Informs 07 1 / 21
Definition
Power flows
We are given a network G with:
A set of S of supply nodes (the “generators”); for each generatori an “operating range” 0 ≤ SL
i ≤ SUi ,
A set D of demand nodes (the “loads”); for each load i a“maximum demand” 0 ≤ Dmax
i .
For each arc (i, j) values xij and uij .
Daniel Bienstock Abhinav Verma ( Columbia University, New York)Multi-stage integer programs in power grids Informs 07 2 / 21
Definition
Power flows
We are given a network G with:
A set of S of supply nodes (the “generators”); for each generatori an “operating range” 0 ≤ SL
i ≤ SUi ,
A set D of demand nodes (the “loads”); for each load i a“maximum demand” 0 ≤ Dmax
i .
For each arc (i, j) values xij and uij .
Daniel Bienstock Abhinav Verma ( Columbia University, New York)Multi-stage integer programs in power grids Informs 07 2 / 21
Definition
Power flows
We are given a network G with:
A set of S of supply nodes (the “generators”); for each generatori an “operating range” 0 ≤ SL
i ≤ SUi ,
A set D of demand nodes (the “loads”); for each load i a“maximum demand” 0 ≤ Dmax
i .
For each arc (i, j) values xij and uij .
Daniel Bienstock Abhinav Verma ( Columbia University, New York)Multi-stage integer programs in power grids Informs 07 2 / 21
Definition
Linearized power flows
A power flow is a solution f , θ to:∑ij fij −
∑ij fji = bi , for all i , where
SLi ≤ bi ≤ SU
i OR bi = 0 for i ∈ S,
0 ≤ −bi ≤ Dmaxi for i ∈ D,
and bi = 0, otherwise.
xij fij − θi + θj = 0 for all (i, j). (Ohm’s equation)
Lemma Given a choice for b, the system has a unique solution.
The solution is feasible if |fij | ≤ uij for every (i, j).
Its throughput is∑
i∈D −bi .
Daniel Bienstock Abhinav Verma ( Columbia University, New York)Multi-stage integer programs in power grids Informs 07 3 / 21
Definition
Linearized power flows
A power flow is a solution f , θ to:∑ij fij −
∑ij fji = bi , for all i , where
SLi ≤ bi ≤ SU
i OR bi = 0 for i ∈ S,
0 ≤ −bi ≤ Dmaxi for i ∈ D,
and bi = 0, otherwise.
xij fij − θi + θj = 0 for all (i, j). (Ohm’s equation)
Lemma Given a choice for b, the system has a unique solution.
The solution is feasible if |fij | ≤ uij for every (i, j).
Its throughput is∑
i∈D −bi .
Daniel Bienstock Abhinav Verma ( Columbia University, New York)Multi-stage integer programs in power grids Informs 07 3 / 21
Definition
Linearized power flows
A power flow is a solution f , θ to:∑ij fij −
∑ij fji = bi , for all i , where
SLi ≤ bi ≤ SU
i OR bi = 0 for i ∈ S,
0 ≤ −bi ≤ Dmaxi for i ∈ D,
and bi = 0, otherwise.
xij fij − θi + θj = 0 for all (i, j). (Ohm’s equation)
Lemma Given a choice for b, the system has a unique solution.
The solution is feasible if |fij | ≤ uij for every (i, j).
Its throughput is∑
i∈D −bi .
Daniel Bienstock Abhinav Verma ( Columbia University, New York)Multi-stage integer programs in power grids Informs 07 3 / 21
Definition
Linearized power flows
A power flow is a solution f , θ to:∑ij fij −
∑ij fji = bi , for all i , where
SLi ≤ bi ≤ SU
i OR bi = 0 for i ∈ S,
0 ≤ −bi ≤ Dmaxi for i ∈ D,
and bi = 0, otherwise.
xij fij − θi + θj = 0 for all (i, j). (Ohm’s equation)
Lemma Given a choice for b, the system has a unique solution.
The solution is feasible if |fij | ≤ uij for every (i, j).
Its throughput is∑
i∈D −bi .
Daniel Bienstock Abhinav Verma ( Columbia University, New York)Multi-stage integer programs in power grids Informs 07 3 / 21
Definition
Linearized power flows
A power flow is a solution f , θ to:∑ij fij −
∑ij fji = bi , for all i , where
SLi ≤ bi ≤ SU
i OR bi = 0 for i ∈ S,
0 ≤ −bi ≤ Dmaxi for i ∈ D,
and bi = 0, otherwise.
xij fij − θi + θj = 0 for all (i, j). (Ohm’s equation)
Lemma Given a choice for b, the system has a unique solution.
The solution is feasible if |fij | ≤ uij for every (i, j).
Its throughput is∑
i∈D −bi .
Daniel Bienstock Abhinav Verma ( Columbia University, New York)Multi-stage integer programs in power grids Informs 07 3 / 21
Definition
The attack problem
Delete a small set of arcs, such that in the resulting network allfeasible flows have small throughput
Used to model “natural” blackouts
“Small” throughput: we satisfy less than some amount Dmin oftotal demand
“Small” set of arcs = very small
Delete 1 arc = the “N-1” problem
Of interest: delete k = 2, 3, 4, . . . edges
Naive enumeration blows up
Daniel Bienstock Abhinav Verma ( Columbia University, New York)Multi-stage integer programs in power grids Informs 07 4 / 21
Definition
The attack problem
Delete a small set of arcs, such that in the resulting network allfeasible flows have small throughput
Used to model “natural” blackouts
“Small” throughput: we satisfy less than some amount Dmin oftotal demand
“Small” set of arcs = very small
Delete 1 arc = the “N-1” problem
Of interest: delete k = 2, 3, 4, . . . edges
Naive enumeration blows up
Daniel Bienstock Abhinav Verma ( Columbia University, New York)Multi-stage integer programs in power grids Informs 07 4 / 21
Definition
The attack problem
Delete a small set of arcs, such that in the resulting network allfeasible flows have small throughput
Used to model “natural” blackouts
“Small” throughput: we satisfy less than some amount Dmin oftotal demand
“Small” set of arcs = very small
Delete 1 arc = the “N-1” problem
Of interest: delete k = 2, 3, 4, . . . edges
Naive enumeration blows up
Daniel Bienstock Abhinav Verma ( Columbia University, New York)Multi-stage integer programs in power grids Informs 07 4 / 21
Definition
The attack problem
Delete a small set of arcs, such that in the resulting network allfeasible flows have small throughput
Used to model “natural” blackouts
“Small” throughput: we satisfy less than some amount Dmin oftotal demand
“Small” set of arcs = very small
Delete 1 arc = the “N-1” problem
Of interest: delete k = 2, 3, 4, . . . edges
Naive enumeration blows up
Daniel Bienstock Abhinav Verma ( Columbia University, New York)Multi-stage integer programs in power grids Informs 07 4 / 21
Definition
The attack problem
Delete a small set of arcs, such that in the resulting network allfeasible flows have small throughput
Used to model “natural” blackouts
“Small” throughput: we satisfy less than some amount Dmin oftotal demand
“Small” set of arcs = very small
Delete 1 arc = the “N-1” problem
Of interest: delete k = 2, 3, 4, . . . edges
Naive enumeration blows up
Daniel Bienstock Abhinav Verma ( Columbia University, New York)Multi-stage integer programs in power grids Informs 07 4 / 21
Definition
The attack problem
Delete a small set of arcs, such that in the resulting network allfeasible flows have small throughput
Used to model “natural” blackouts
“Small” throughput: we satisfy less than some amount Dmin oftotal demand
“Small” set of arcs = very small
Delete 1 arc = the “N-1” problem
Of interest: delete k = 2, 3, 4, . . . edges
Naive enumeration blows up
Daniel Bienstock Abhinav Verma ( Columbia University, New York)Multi-stage integer programs in power grids Informs 07 4 / 21
Definition
The attack problem
Delete a small set of arcs, such that in the resulting network allfeasible flows have small throughput
Used to model “natural” blackouts
“Small” throughput: we satisfy less than some amount Dmin oftotal demand
“Small” set of arcs = very small
Delete 1 arc = the “N-1” problem
Of interest: delete k = 2, 3, 4, . . . edges
Naive enumeration blows up
Daniel Bienstock Abhinav Verma ( Columbia University, New York)Multi-stage integer programs in power grids Informs 07 4 / 21
Definition
Two types of successful attacks
Type 1: Network becomes disconnected with a mismatch of supplyand demand.
Daniel Bienstock Abhinav Verma ( Columbia University, New York)Multi-stage integer programs in power grids Informs 07 5 / 21
Definition
Two types of successful attacks
Type 2: Uniqueness of power flows means exceeded capacities orinsufficient supply.
Daniel Bienstock Abhinav Verma ( Columbia University, New York)Multi-stage integer programs in power grids Informs 07 6 / 21
Definition
A little game:
The controller’s problem: Given a set A of arcs that has beendeleted by the attacker, choose a set G of generators to operate, so asto feasibly meet demand (at least) Dmin.
The attacker’s problem: Choose a set A of arcs to delete, so as todefeat the controller, no matter how the controller chooses G.
Daniel Bienstock Abhinav Verma ( Columbia University, New York)Multi-stage integer programs in power grids Informs 07 7 / 21
Definition
A little game:
The controller’s problem: Given a set A of arcs that has beendeleted by the attacker, choose a set G of generators to operate, so asto feasibly meet demand (at least) Dmin.
The attacker’s problem: Choose a set A of arcs to delete, so as todefeat the controller, no matter how the controller chooses G.
Daniel Bienstock Abhinav Verma ( Columbia University, New York)Multi-stage integer programs in power grids Informs 07 7 / 21
Definition
A little game:
The controller’s problem: Given a set A of arcs that has beendeleted by the attacker, choose a set G of generators to operate, so asto feasibly meet demand (at least) Dmin.
The attacker’s problem: Choose a set A of arcs to delete, so as todefeat the controller, no matter how the controller chooses G.
Daniel Bienstock Abhinav Verma ( Columbia University, New York)Multi-stage integer programs in power grids Informs 07 7 / 21
Definition
The controller’s problem for a given choice of generators
Given a set A of arcs that has been deleted by the attacker, and achoice G of which generators to operate, set demands and suppliesso as to feasibly meet total demand (at least) Dmin.
This a linear program:
Daniel Bienstock Abhinav Verma ( Columbia University, New York)Multi-stage integer programs in power grids Informs 07 8 / 21
Definition
The controller’s problem for a given choice of generators
Given a set A of arcs that has been deleted by the attacker, and achoice G of which generators to operate, set demands and suppliesso as to feasibly meet total demand (at least) Dmin.
This a linear program:
Daniel Bienstock Abhinav Verma ( Columbia University, New York)Multi-stage integer programs in power grids Informs 07 8 / 21
Definition
tA(G).= min t
Subject to:∑ij fij −
∑ij fji − bi = 0, for all nodes i ,
Smini ≤ bi ≤ Smax
i for i ∈ G, 0 ≤ −bi ≤ Dmaxi for i ∈ D
bi = 0 otherwise.
xij fij − θi + θj = 0 for all (i, j) /∈ A
−∑
i∈D bi + Dmin t ≥ 2Dmin
uij t ≥ |fij | for all (i, j) /∈ A
fij = 0 for all (i, j) ∈ A
Lemma: tA(G) > 1 iff the attack is successful against the choice G.
Daniel Bienstock Abhinav Verma ( Columbia University, New York)Multi-stage integer programs in power grids Informs 07 9 / 21
Definition
tA(G).= min t
Subject to:∑ij fij −
∑ij fji − bi = 0, for all nodes i ,
Smini ≤ bi ≤ Smax
i for i ∈ G, 0 ≤ −bi ≤ Dmaxi for i ∈ D
bi = 0 otherwise.
xij fij − θi + θj = 0 for all (i, j) /∈ A
−∑
i∈D bi + Dmin t ≥ 2Dmin
uij t ≥ |fij | for all (i, j) /∈ A
fij = 0 for all (i, j) ∈ A
Lemma: tA(G) > 1 iff the attack is successful against the choice G.
Daniel Bienstock Abhinav Verma ( Columbia University, New York)Multi-stage integer programs in power grids Informs 07 9 / 21
Definition
tA(G).= min t
Subject to:∑ij fij −
∑ij fji − bi = 0, for all nodes i ,
Smini ≤ bi ≤ Smax
i for i ∈ G, 0 ≤ −bi ≤ Dmaxi for i ∈ D
bi = 0 otherwise.
xij fij − θi + θj = 0 for all (i, j) /∈ A
−∑
i∈D bi + Dmin t ≥ 2Dmin
uij t ≥ |fij | for all (i, j) /∈ A
fij = 0 for all (i, j) ∈ A
Lemma: tA(G) > 1 iff the attack is successful against the choice G.
Daniel Bienstock Abhinav Verma ( Columbia University, New York)Multi-stage integer programs in power grids Informs 07 9 / 21
Definition
tA(G).= min t
Subject to:∑ij fij −
∑ij fji − bi = 0, for all nodes i ,
Smini ≤ bi ≤ Smax
i for i ∈ G, 0 ≤ −bi ≤ Dmaxi for i ∈ D
bi = 0 otherwise.
xij fij − θi + θj = 0 for all (i, j) /∈ A
−∑
i∈D bi + Dmin t ≥ 2Dmin
uij t ≥ |fij | for all (i, j) /∈ A
fij = 0 for all (i, j) ∈ A
Lemma: tA(G) > 1 iff the attack is successful against the choice G.
Daniel Bienstock Abhinav Verma ( Columbia University, New York)Multi-stage integer programs in power grids Informs 07 9 / 21
Definition
tA(G).= min t
Subject to:∑ij fij −
∑ij fji − bi = 0, for all nodes i ,
Smini ≤ bi ≤ Smax
i for i ∈ G, 0 ≤ −bi ≤ Dmaxi for i ∈ D
bi = 0 otherwise.
xij fij − θi + θj = 0 for all (i, j) /∈ A
−∑
i∈D bi + Dmin t ≥ 2Dmin
uij t ≥ |fij | for all (i, j) /∈ A
fij = 0 for all (i, j) ∈ A
Lemma: tA(G) > 1 iff the attack is successful against the choice G.
Daniel Bienstock Abhinav Verma ( Columbia University, New York)Multi-stage integer programs in power grids Informs 07 9 / 21
Definition
tA(G).= min t
Subject to:∑ij fij −
∑ij fji − bi = 0, for all nodes i ,
Smini ≤ bi ≤ Smax
i for i ∈ G, 0 ≤ −bi ≤ Dmaxi for i ∈ D
bi = 0 otherwise.
xij fij − θi + θj = 0 for all (i, j) /∈ A
−∑
i∈D bi + Dmin t ≥ 2Dmin
uij t ≥ |fij | for all (i, j) /∈ A
fij = 0 for all (i, j) ∈ A
Lemma: tA(G) > 1 iff the attack is successful against the choice G.
Daniel Bienstock Abhinav Verma ( Columbia University, New York)Multi-stage integer programs in power grids Informs 07 9 / 21
Definition
tA(G).= min t
Subject to:∑ij fij −
∑ij fji − bi = 0, for all nodes i ,
Smini ≤ bi ≤ Smax
i for i ∈ G, 0 ≤ −bi ≤ Dmaxi for i ∈ D
bi = 0 otherwise.
xij fij − θi + θj = 0 for all (i, j) /∈ A
−∑
i∈D bi + Dmin t ≥ 2Dmin
uij t ≥ |fij | for all (i, j) /∈ A
fij = 0 for all (i, j) ∈ A
Lemma: tA(G) > 1 iff the attack is successful against the choice G.
Daniel Bienstock Abhinav Verma ( Columbia University, New York)Multi-stage integer programs in power grids Informs 07 9 / 21
Definition
tA(G).= min t
Subject to:∑ij fij −
∑ij fji − bi = 0, for all nodes i ,
Smini ≤ bi ≤ Smax
i for i ∈ G, 0 ≤ −bi ≤ Dmaxi for i ∈ D
bi = 0 otherwise.
xij fij − θi + θj = 0 for all (i, j) /∈ A
−∑
i∈D bi + Dmin t ≥ 2Dmin
uij t ≥ |fij | for all (i, j) /∈ A
for all (i, j) ∈ A, t ≥ 1 + |fij |/uij
Lemma: tA(G) > 1 iff the attack is successful against the choice G.
Daniel Bienstock Abhinav Verma ( Columbia University, New York)Multi-stage integer programs in power grids Informs 07 10 / 21
Definition
tA(G).= min t
Subject to:∑ij fij −
∑ij fji − bi = 0, for all nodes i ,
Smini ≤ bi ≤ Smax
i for i ∈ G, 0 ≤ −bi ≤ Dmaxi for i ∈ D
bi = 0 otherwise.
xij fij − θi + θj = 0 for all (i, j) /∈ A
−∑
i∈D bi + Dmin t ≥ 2Dmin
uij t ≥ |fij | for all (i, j) /∈ A
for all (i, j) ∈ A, t ≥ 1 + |fij |/uij
Lemma: tA(G) > 1 iff the attack is successful against the choice G.
Daniel Bienstock Abhinav Verma ( Columbia University, New York)Multi-stage integer programs in power grids Informs 07 10 / 21
Definition
Attack problem
min∑
ij zij
Subject to:
zij = 0 or 1, for all arcs (i, j), (choose which arcs to delete)
tsuppt(z)(G) > 1, for every subset G of generators.
[ suppt(v) = support of v]
→ Use dual to represent tsuppt(z)(G)
Daniel Bienstock Abhinav Verma ( Columbia University, New York)Multi-stage integer programs in power grids Informs 07 11 / 21
Definition
Attack problem
min∑
ij zij
Subject to:
zij = 0 or 1, for all arcs (i, j), (choose which arcs to delete)
tsuppt(z)(G) > 1, for every subset G of generators.
[ suppt(v) = support of v]
→ Use dual to represent tsuppt(z)(G)
Daniel Bienstock Abhinav Verma ( Columbia University, New York)Multi-stage integer programs in power grids Informs 07 11 / 21
Definition
Building the dual
tA(G).= min t
Subject to:∑ij fij −
∑ij fji − bi = 0, for all nodes i , (αi)
Smini ≤ bi ≤ Smax
i for i ∈ G,
0 ≤ −bi ≤ Dmaxi for i ∈ D
bi = 0 otherwise.
xij fij − θi + θj = 0 for all (i, j) /∈ A (βij)
−(∑
i∈D bi)/Dmin + t ≥ 2
uij t ≥ |fij | for all (i, j) /∈ A (pij , qij)
uij t ≥ uij + |fij | for all (i, j) ∈ A (r+ij , r−
ij )
Daniel Bienstock Abhinav Verma ( Columbia University, New York)Multi-stage integer programs in power grids Informs 07 12 / 21
Definition
Part of the dual
∑ij βij −
∑ji βji = 0 ∀i
αi − αj + xijβij = pij − qij + r+ij − r−
ij ∀(i, j)
pij + qij ≤ M(1 − zij)
r+ij + r−
ij ≤ M ′zij
→ “big M” formulation
Daniel Bienstock Abhinav Verma ( Columbia University, New York)Multi-stage integer programs in power grids Informs 07 13 / 21
Definition
Part of the dual
∑ij βij −
∑ji βji = 0 ∀i
αi − αj + xijβij = pij − qij + r+ij − r−
ij ∀(i, j)
pij + qij ≤ M(1 − zij)
r+ij + r−
ij ≤ M ′zij
→ “big M” formulation
Daniel Bienstock Abhinav Verma ( Columbia University, New York)Multi-stage integer programs in power grids Informs 07 13 / 21
Definition
Part of the dual
∑ij βij −
∑ji βji = 0 ∀i
αi − αj + xijβij = pij − qij + r+ij − r−
ij ∀(i, j)
pij + qij ≤ M(1 − zij)
r+ij + r−
ij ≤ M ′zij
→ “big M” formulation
Daniel Bienstock Abhinav Verma ( Columbia University, New York)Multi-stage integer programs in power grids Informs 07 13 / 21
Definition
Part of the dual
∑ij βij −
∑ji βji = 0 ∀i
αi − αj + xijβij = pij − qij + r+ij − r−
ij ∀(i, j)
pij + qij ≤ M(1 − zij)
r+ij + r−
ij ≤ M ′zij
→ “big M” formulation
Daniel Bienstock Abhinav Verma ( Columbia University, New York)Multi-stage integer programs in power grids Informs 07 13 / 21
Definition
Part of the dual
∑ij βij −
∑ji βji = 0 ∀i
αi − αj + xijβij = pij − qij + r+ij − r−
ij ∀(i, j)
Lemma - Given p, q, r+, r−, the above system has a unique solutionin αi − αj , β.
Daniel Bienstock Abhinav Verma ( Columbia University, New York)Multi-stage integer programs in power grids Informs 07 14 / 21
Definition
Part of the dual-mip
∑ij βij −
∑ji βji = 0 ∀i
αi − αj + xijβij = pij − qij + r+ij − r−
ij ∀(i, j)
pij + qij ≤ M(1 − zij)
r+ij + r−
ij ≤ M ′zij
x−1/2ij |αi − αj | + x1/2
ij |βij | ≤∑
kl x−1/2kl (pkl + qkl) + r+
ij + r−ij
valid
M = max(i,j)(√xij uij
)−1
Daniel Bienstock Abhinav Verma ( Columbia University, New York)Multi-stage integer programs in power grids Informs 07 15 / 21
Definition
Part of the dual-mip
∑ij βij −
∑ji βji = 0 ∀i
αi − αj + xijβij = pij − qij + r+ij − r−
ij ∀(i, j)
pij + qij ≤ M(1 − zij)
r+ij + r−
ij ≤ M ′zij
x−1/2ij |αi − αj | + x1/2
ij |βij | ≤∑
kl x−1/2kl (pkl + qkl) + r+
ij + r−ij
valid
M = max(i,j)(√xij uij
)−1
Daniel Bienstock Abhinav Verma ( Columbia University, New York)Multi-stage integer programs in power grids Informs 07 15 / 21
Definition
A formulation for the attack problem
min∑
ij zij
Subject to:
zij = 0 or 1, for all arcs (i, j), (choose which arcs to delete)
tsuppt(z)(G) > 1, for every subset G of generators.
Daniel Bienstock Abhinav Verma ( Columbia University, New York)Multi-stage integer programs in power grids Informs 07 16 / 21
Definition
A formulation for the attack problem
min∑
ij zij
Subject to:
zij = 0 or 1, for all arcs (i, j), (choose which arcs to delete)
value of dual mip (G) > 1, for every subset G of generators.
→ very large
Daniel Bienstock Abhinav Verma ( Columbia University, New York)Multi-stage integer programs in power grids Informs 07 17 / 21
Definition
A formulation for the attack problem
min∑
ij zij
Subject to:
zij = 0 or 1, for all arcs (i, j), (choose which arcs to delete)
value of dual mip (G) > 1, for every subset G of generators.
→ very large
Daniel Bienstock Abhinav Verma ( Columbia University, New York)Multi-stage integer programs in power grids Informs 07 17 / 21
Definition
Algorithm
Initially, the “master (attacker) MIP” is empty.
Iterate:
1. Solve master MIP, obtain 0 − 1 vector z∗.
2. Solve controller problem to test whether supp(z∗) is a successfulattack:
If successful, then z∗ is an optimal solution
If not, then for some set of generators G, tsupp(z∗)(G) ≤ 1.
3. Augment the formulation for the master MIP by adding all theformulation for the dual mip corresponding to G, and go to 1.
Daniel Bienstock Abhinav Verma ( Columbia University, New York)Multi-stage integer programs in power grids Informs 07 18 / 21
Definition
Algorithm
Initially, the “master (attacker) MIP” is empty.
Iterate:
1. Solve master MIP, obtain 0 − 1 vector z∗.
2. Solve controller problem to test whether supp(z∗) is a successfulattack:
If successful, then z∗ is an optimal solution
If not, then for some set of generators G, tsupp(z∗)(G) ≤ 1.
3. Augment the formulation for the master MIP by adding all theformulation for the dual mip corresponding to G, and go to 1.
Daniel Bienstock Abhinav Verma ( Columbia University, New York)Multi-stage integer programs in power grids Informs 07 18 / 21
Definition
Algorithm
Initially, the “master (attacker) MIP” is empty.
Iterate:
1. Solve master MIP, obtain 0 − 1 vector z∗.
2. Solve controller problem to test whether supp(z∗) is a successfulattack:
If successful, then z∗ is an optimal solution
If not, then for some set of generators G, tsupp(z∗)(G) ≤ 1.
3. Augment the formulation for the master MIP by adding all theformulation for the dual mip corresponding to G, and go to 1.
Daniel Bienstock Abhinav Verma ( Columbia University, New York)Multi-stage integer programs in power grids Informs 07 18 / 21
Definition
Algorithm
Initially, the “master (attacker) MIP” is empty.
Iterate:
1. Solve master MIP, obtain 0 − 1 vector z∗.
2. Solve controller problem to test whether supp(z∗) is a successfulattack:
If successful, then z∗ is an optimal solution
If not, then for some set of generators G, tsupp(z∗)(G) ≤ 1.
3. Augment the formulation for the master MIP by adding all theformulation for the dual mip corresponding to G, and go to 1.
Daniel Bienstock Abhinav Verma ( Columbia University, New York)Multi-stage integer programs in power grids Informs 07 18 / 21
Definition
Algorithm
Initially, the “master (attacker) MIP” is empty.
Iterate:
1. Solve master MIP, obtain 0 − 1 vector z∗.
2. Solve controller problem to test whether supp(z∗) is a successfulattack:
If successful, then z∗ is an optimal solution
If not, then for some set of generators G, tsupp(z∗)(G) ≤ 1.
3. Augment the formulation for the master MIP by adding all theformulation for the dual mip corresponding to G, and go to 1.
Daniel Bienstock Abhinav Verma ( Columbia University, New York)Multi-stage integer programs in power grids Informs 07 18 / 21
Definition
Algorithm
Initially, the “master (attacker) MIP” is empty.
Iterate:
1. Solve master MIP, obtain 0 − 1 vector z∗.
2. Solve controller problem to test whether supp(z∗) is a successfulattack:
If successful, then z∗ is an optimal solution
If not, then for some set of generators G, tsupp(z∗)(G) ≤ 1.
3. Augment the formulation for the master MIP by adding all theformulation for the dual mip corresponding to G, and go to 1.
Daniel Bienstock Abhinav Verma ( Columbia University, New York)Multi-stage integer programs in power grids Informs 07 18 / 21
Definition
Algorithm
Initially, the “master (attacker) MIP” is empty.
Iterate:
1. Solve master MIP, obtain 0 − 1 vector z∗.
2. Solve controller problem to test whether supp(z∗) is a successfulattack:
If successful, then z∗ is an optimal solution
If not, then for some set of generators G, tsupp(z∗)(G) ≤ 1.
3. Augment the formulation for the master MIP by adding all theformulation for the dual mip corresponding to G, and go to 1.
Daniel Bienstock Abhinav Verma ( Columbia University, New York)Multi-stage integer programs in power grids Informs 07 18 / 21
Definition
Algorithm
1. Solve master MIP, obtain 0 − 1 vector z∗.
2. Solve controller problem to test whether supp(z∗) is a successfulattack:
If successful, then z∗ is an optimal solution
If not, then for some set of generators G, tsupp(z∗)(G) ≤ 1.
3a. Augment the formulation for the master MIP by adding all theformulation for the dual mip corresponding to G, and go to 1. – do this“at the beginning”, else:3b. Separate z∗ using a Benders’ cut, go to 1.
Daniel Bienstock Abhinav Verma ( Columbia University, New York)Multi-stage integer programs in power grids Informs 07 19 / 21
Definition
Algorithm
1. Solve master MIP, obtain 0 − 1 vector z∗.
2. Solve controller problem to test whether supp(z∗) is a successfulattack:
If successful, then z∗ is an optimal solution
If not, then for some set of generators G, tsupp(z∗)(G) ≤ 1.
3a. Augment the formulation for the master MIP by adding all theformulation for the dual mip corresponding to G, and go to 1. – do this“at the beginning”, else:3b. Separate z∗ using a Benders’ cut, go to 1.
Daniel Bienstock Abhinav Verma ( Columbia University, New York)Multi-stage integer programs in power grids Informs 07 19 / 21
Definition
98 nodes 203 arcs, 15 generators
Surviving |A|Demand Fraction 2 3 4
(2) 223 (11) 6540.94 Not Enough Kill
(2) 212 (16) 9817 (21) 135400.92 Not Enough Not Enough Kill
(2) 181 (11) 6912 –0.90 Not Enough Not Enough
Daniel Bienstock Abhinav Verma ( Columbia University, New York)Multi-stage integer programs in power grids Informs 07 20 / 21
Definition
98 nodes 203 arcs, 10 generators
Surviving |A|Demand Fraction 2 3 4
(2) 177 (30) 555 –0.89 Not Enough Kill
(2) 233 (13) 5931 (73) 82730.86 Not Enough Not Enough Kill
(2) 152 (41) 15211 –0.84 Not Enough Not Enough
Daniel Bienstock Abhinav Verma ( Columbia University, New York)Multi-stage integer programs in power grids Informs 07 21 / 21