Derivative-based solution of the optimization …...Ahmed Attia Supervised by: Mihai Anitescu August...
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Problem FormulationDerivative Information
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Derivative-based solution of the optimization problem(s)in DeMarco’s model
Ahmed AttiaSupervised by: Mihai Anitescu
August 5, 2014
Ahmed Attia Supervised by: Mihai Anitescu Derivative-based solution of the optimization problem(s) in DeMarco’s model
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Problem FormulationDerivative Information
ResultsQuestions
1 Problem Formulation
2 Derivative Information
3 Results
4 Questions
Ahmed Attia Supervised by: Mihai Anitescu Derivative-based solution of the optimization problem(s) in DeMarco’s model
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Problem FormulationDerivative Information
ResultsQuestions
Cascading Network Failure
To predict cascading failure in large-scale networks, solid understanding ofpropagations of failures in small-scale networks is vital.
This allows optimal redistribution of loads and network design.
Ahmed Attia Supervised by: Mihai Anitescu Derivative-based solution of the optimization problem(s) in DeMarco’s model
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Problem FormulationDerivative Information
ResultsQuestions
Cascading Network Failure
To predict cascading failure in large-scale networks, solid understanding ofpropagations of failures in small-scale networks is vital.
This allows optimal redistribution of loads and network design.
Ahmed Attia Supervised by: Mihai Anitescu Derivative-based solution of the optimization problem(s) in DeMarco’s model
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Problem FormulationDerivative Information
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Network Model(s): I
Eight-node, Eleven-branch circuit is used as a toy model.
Model state
x =
φqγ
∈ R26 (1)
where
1) φ is a vector of nodal flux differences ( φ ∈ R7),2) q is the vector of nodal charges on capacitors ( q ∈ R8),3) γ is the failure state of branches ( γ ∈ R11),
Ahmed Attia Supervised by: Mihai Anitescu Derivative-based solution of the optimization problem(s) in DeMarco’s model
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Network Model(s): II
DeMarco’s original model equations
dφ = ETr C−1q dt (2)
dq =(−ErHrL
−1HTr φ− GC−1q + iin
)dt, (3)
Stochastic Version of the model[dφdq
]= M + P + U (4)
=
[ETr C−1q(t)dt
0
]+
[0
−ErHrL−1HT
r φ(t)dt
]+
[0
−GC−1q(t)dt +√
2Gτ dWt
].
- iin : the current input, τ : system’s temperature,- C , Er , Hr , L, G : constant matrices,- M, P, U : system’s mass, potential, Ornstein Uhlenbeck process.
Ahmed Attia Supervised by: Mihai Anitescu Derivative-based solution of the optimization problem(s) in DeMarco’s model
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Problem FormulationDerivative Information
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Network Model(s): II
DeMarco’s original model equations
dφ = ETr C−1q dt (2)
dq =(−ErHrL
−1HTr φ− GC−1q + iin
)dt, (3)
Stochastic Version of the model[dφdq
]= M + P + U (4)
=
[ETr C−1q(t)dt
0
]+
[0
−ErHrL−1HT
r φ(t)dt
]+
[0
−GC−1q(t)dt +√
2Gτ dWt
].
- iin : the current input, τ : system’s temperature,- C , Er , Hr , L, G : constant matrices,- M, P, U : system’s mass, potential, Ornstein Uhlenbeck process.
Ahmed Attia Supervised by: Mihai Anitescu Derivative-based solution of the optimization problem(s) in DeMarco’s model
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Problem FormulationDerivative Information
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Time stepping: A Splitting Solver
The integrator is a composition of maps[φ(t)q(t)
]= P t
2◦M t
2◦Ut ◦M t
2◦ P t
2
([φ(0)q(0)
]); (5)
Mt
([φ(0)q(0)
])=
[φ(0) + ET
r C−1q(0)tq(0)
], (6)
Pt
([φ(0)q(0)
])=
[φ(0)
q(0)− ErHrL−1HT
r φ(0)t
], (7)
Ut
([φ(0)q(0)
])=
[φ(0)
e−GC−1tq(0) +√τC(I − e−2GC−1t
)d
], (8)
Where t is the step size, and d is the stochastic force.Ahmed Attia Supervised by: Mihai Anitescu Derivative-based solution of the optimization problem(s) in DeMarco’s model
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Problem FormulationDerivative Information
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Optimization problems
Since we are interested in failure, we ask how the white noise might steer the systemtowards increasing energy.
The exact problem:
mind1,d2,...,dN
J (d1,d2, . . . ,dN) =N∑i=1
dTi di (9)
subject tomax (EN) > ε. (10)
Solving the exact problem requires solving one optimization problem for each line.
A proxy problem: replace constraint with:
ITNEN > ε, (Or ETN EN > ε) (11)
where EN is the energy function, and IN is a vector of all ones.
Ahmed Attia Supervised by: Mihai Anitescu Derivative-based solution of the optimization problem(s) in DeMarco’s model
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Problem FormulationDerivative Information
ResultsQuestions
Optimization problems
Since we are interested in failure, we ask how the white noise might steer the systemtowards increasing energy.
The exact problem:
mind1,d2,...,dN
J (d1,d2, . . . ,dN) =N∑i=1
dTi di (9)
subject tomax (EN) > ε. (10)
Solving the exact problem requires solving one optimization problem for each line.
A proxy problem: replace constraint with:
ITNEN > ε, (Or ETN EN > ε) (11)
where EN is the energy function, and IN is a vector of all ones.
Ahmed Attia Supervised by: Mihai Anitescu Derivative-based solution of the optimization problem(s) in DeMarco’s model
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Problem FormulationDerivative Information
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Energy Function
EN(φN) = EN
(φN)1(φN)2(φN)3(φN)4(φN)5(φN)6(φN)7
=
12L
−11 γ1 (−(φN)1)2
12L
−12 γ2 (−(φN)2)2
12L
−13 γ3 ((φN)2 − (φN)3)2
12L
−14 γ4 ((φN)1 − (φN)3)2
12L
−15 γ5 ((φN)3 − (φN)4)2
12L
−16 γ6 ((φN)3 − (φN)5)2
12L
−17 γ7 ((φN)1 − (φN)5)2
12L
−18 γ8 ((φN)4 − (φN)5)2
12L
−19 γ9 ((φN)4 − (φN)7)2
12L
−110 γ10 ((φN)5 − (φN)6)2
12L
−111 γ11 ((φN)6 − (φN)7)2
(12)
Ahmed Attia Supervised by: Mihai Anitescu Derivative-based solution of the optimization problem(s) in DeMarco’s model
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Problem FormulationDerivative Information
ResultsQuestions
Probability of Failure(s)
We can compute the probability of one failure (at time tN) from optimal noise vectorsd1,d1, . . .dN via
P(Failure|d1,d1, . . .dN) = e−∑N
i=1 dTi diτ (13)
Multiple failure is an enumeration problem solved by exhaustive search.
Ahmed Attia Supervised by: Mihai Anitescu Derivative-based solution of the optimization problem(s) in DeMarco’s model
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Problem FormulationDerivative Information
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Derivative Information: I
We have reformulated the splitting solver as linear discrete map:
xi = Axi−1 + Bdi , (14)
The blocks of A are:
A1,1 = I + (ETr C−1)(I + e−GC−1h)(−ErHrL
−1HTr )(
h2
4) (15)
A1,2 = (ETr C−1)(I + e−GC−1h)(
h
2) (16)
A2,1 =
((−ErHrL
−1HTr )(ET
r C−1)(h2
4) + I
)(I + e−GC−1h)(−ErHrL
−1HTr )(
h
2) (17)
A2,2 = (−ErHrL−1HT
r )(ETr C−1)(I + e−GC−1h)(
h2
4) + (e−GC−1h) (18)
A1,3 = A2,3 = A3,1 = A3,2 = 0; A3,3 = I (19)
Ahmed Attia Supervised by: Mihai Anitescu Derivative-based solution of the optimization problem(s) in DeMarco’s model
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Derivative Information: II
B reads
B =
(ETr C−1)
√τC (I − e−2GC−1h)(h2 )(
(−ErHrL−1HT
r )(ETr C−1)(h
2
4 ) + I)√
τC (I − e−2GC−1h)
0
(20)
the derivatives read
∇xi−kxi = Ak ∀k = 1, 2, . . . , i − 1 (21)
∇di−kxi = AkB ∀k = 0, 1, . . . , i − 1 (22)
∇diEN = (∇φNEN)(∇diφN)
= (ENφ)(∇diφN) ∀i = 1, 2, . . . ,N (23)
ENφ = dENdφN∈ R11×7 is the Jacobian of the energy functional w.r.t flux differences.
Ahmed Attia Supervised by: Mihai Anitescu Derivative-based solution of the optimization problem(s) in DeMarco’s model
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Problem FormulationDerivative Information
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Derivative Information: III
Gradient of the cost function:
∇[dT1 ,dT2 ,...,d
TN ]
TJ = 2
d1d2d3...dN
. (24)
Gradient of the constraint(s):
∇φN
(ITNEN
)= ET
NφIN ; ∇φN
(ETN EN
)= 2
(∇d1EN)T EN
(∇d2EN)T EN
...
(∇dNEN)T EN
(25)
Ahmed Attia Supervised by: Mihai Anitescu Derivative-based solution of the optimization problem(s) in DeMarco’s model
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Problem FormulationDerivative Information
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Results I: Probability of failure(s)
(a) One failure (b) Two failures (c) Three failures (d) Four failures
Figure: Probability of line failure(s). One, two, three, and four failures are plotted.
Ahmed Attia Supervised by: Mihai Anitescu Derivative-based solution of the optimization problem(s) in DeMarco’s model
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Results III: Probability of failure(s)
(a) One failure (b) Two failures
Figure: Probability of line failure(s) on higher resolution grid
Ahmed Attia Supervised by: Mihai Anitescu Derivative-based solution of the optimization problem(s) in DeMarco’s model
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Results IV: Probability of failure(s)
(a) Simulation time = 0.5 (b) Simulation time = 1
Figure: Relation between probability of branch failures and system’s temperature.
Ahmed Attia Supervised by: Mihai Anitescu Derivative-based solution of the optimization problem(s) in DeMarco’s model
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Results V: Computational Time
Figure: CPU-time of the optimization step for one failure case with and without derivativeinformation.
Ahmed Attia Supervised by: Mihai Anitescu Derivative-based solution of the optimization problem(s) in DeMarco’s model
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Thanks
Questions?
Ahmed Attia Supervised by: Mihai Anitescu Derivative-based solution of the optimization problem(s) in DeMarco’s model