Sensitivity analysis for the outages of nuclear power plants - SADCO ...

Introduction Study of the reference problem Sensitivity analysis
Sensitivity analysis for the outages of nuclear
power plants
SADCO: 2 nd Industrial Workshop
Laurent Pfeiffer ∗ , Frédéric Bonnans ∗ and Kengy Barty †
∗ INRIA Saclay and CMAP, Ecole Polytechnique, † EDF R&D
February 2, 2012

Introduction Study of the reference problem Sensitivity analysis
Introduction
Study of a two-level problem:
optimization of the dates of the outages of nuclear power
plants
optimization of the production of electricity.
Our approach:
1 fix a schedule τ
2 optimize the production of electricity: V (τ)
3 perform a sensitivity analysis: compute V ′ (τ)
4 improve the schedule.

Introduction Study of the reference problem Sensitivity analysis
Introduction
Study of a two-level problem:
optimization of the dates of the outages of nuclear power
plants
optimization of the production of electricity.
Our approach:
1 fix a schedule τ
2 optimize the production of electricity: V (τ)
3 perform a sensitivity analysis: compute V ′ (τ)
4 improve the schedule.

Introduction Study of the reference problem Sensitivity analysis
1 Study of the reference problem
Model
Pontryagin’s principle
Structure of optimal controls
2 Sensitivity analysis
Abstract theorem
Toy example
Application to the outages

Introduction Study of the reference problem Sensitivity analysis
1 Study of the reference problem
Model
Pontryagin’s principle
Structure of optimal controls
2 Sensitivity analysis
Abstract theorem
Toy example
Application to the outages

Introduction Study of the reference problem Sensitivity analysis
Model
General notations:
S the set of plants, of cardinal n
T the horizon of the problem
dt the demand of electricity
Control variables:
u i t the rate of production of plant i
0 ≤ u i t ≤ u i the bound on production
U=
i∈S [0, ui ]

Introduction Study of the reference problem Sensitivity analysis
State variables:
s i t the level of fuel of plant i
[τ i b , τ i e] the dates of the outages
a i the rate of refuelling
Dynamic: ˙s i t = −u i t1 t /∈[τ i b ,τ i e] + ai 1 t∈[τ i b ,τ i e ]
s i 0
= si,0
State constraints:
si τ i = 0
b
si T ≥ 0

Introduction Study of the reference problem Sensitivity analysis
Optimization criterion:
where:
T
min c dt −
0
u
i∈W (t)
i
t dt + φ(sT ),
c and φ and strongly convex and smooth and φ is decreasing
W (t) is the set of working plants at time t.
Functional spaces:
u ∈ L ∞ (0, T ; R n )
s ∈ W 1,∞ (0, T ; R n )

Introduction Study of the reference problem Sensitivity analysis
Pontryagin’s principle
The Hamiltonian is independent on the state!
H(t, u, p) = c
Proposition
dt −
i∈W (t)
u i
+
i∈S
p i
−u i 1 t /∈[τ i b ,τ i e ] +ai 1 t∈[τ i b ,τ i e ]
If (u, s) is optimal, there exists a costate t ↦→ p(t) such that
1 Each coordinate pi takes only two values over time, pi 0 on
[0, τ i b ] and pi T on [τ i b , T ] such that
pT ≤ Dsi φ(sT ) and p i T = Dsi φ(sT ) if s i T
2 For almost all t in [0, T ],
H(t, ut, pt) ≤ H(t, v, pt), ∀v ∈ U.
= 0.
.

Introduction Study of the reference problem Sensitivity analysis
Pontryagin’s principle
The Hamiltonian is independent on the state!
H(t, u, p) = c
Proposition
dt −
i∈W (t)
u i
+
i∈S
p i
−u i 1 t /∈[τ i b ,τ i e ] +ai 1 t∈[τ i b ,τ i e ]
If (u, s) is optimal, there exists a costate t ↦→ p(t) such that
1 Each coordinate pi takes only two values over time, pi 0 on
[0, τ i b ] and pi T on [τ i b , T ] such that
pT ≤ Dsi φ(sT ) and p i T = Dsi φ(sT ) if s i T
2 For almost all t in [0, T ],
H(t, ut, pt) ≤ H(t, v, pt), ∀v ∈ U.
= 0.
.

Introduction Study of the reference problem Sensitivity analysis
Stucture of optimal controls
Each stock of fuel i has two marginal prices associated:
−p i 0 ≥ 0 and − p i T
≥ 0.
At time t, the Hamiltonian is the sum of
the integral cost: c dt −
i∈W (t) ui
the cost associated to fuel:
i∈S −pi tui .
Moreover, the costate induces an ordering of the plants. If
−p i t > −p j t,
then plant i is used only if plant j produces at its maximum rate.

Introduction Study of the reference problem Sensitivity analysis
Stucture of optimal controls
Each stock of fuel i has two marginal prices associated:
−p i 0 ≥ 0 and − p i T
≥ 0.
At time t, the Hamiltonian is the sum of
the integral cost: c dt −
i∈W (t) ui
the cost associated to fuel:
i∈S −pi tui .
Moreover, the costate induces an ordering of the plants. If
−p i t > −p j t,
then plant i is used only if plant j produces at its maximum rate.

Introduction Study of the reference problem Sensitivity analysis
Total production
#
#
Demand
The bounds / # depend on: , W(t), and p(t).

Introduction Study of the reference problem Sensitivity analysis
If some plants share the same costate, then the optimal
controls are not unique.
In our model, the total production is unique.
The costate has to be considered as a dual variable,
characterized by (p0, pT ). It is not necessarily unique.

Introduction Study of the reference problem Sensitivity analysis
1 Study of the reference problem
Model
Pontryagin’s principle
Structure of optimal controls
2 Sensitivity analysis
Abstract theorem
Toy example
Application to the outages

Introduction Study of the reference problem Sensitivity analysis
Abstract theorem
Consider the abstract family of optimization problems Py
and its Lagrangian
V (y) = min f (x, y), s.t. g(x, y) ≤ 0,
x
L(x, y, λ) = f (x, y) + 〈λ, g(x, y)〉.
The functions f and g are continuously differentiable.
For a reference value y0, suppose that Py0 is convex and denote by
S(y0), the set of solutions of Py0
Λ(y0), the set of Lagrange multipliers.

Introduction Study of the reference problem Sensitivity analysis
Abstract theorem
Consider the abstract family of optimization problems Py
and its Lagrangian
V (y) = min f (x, y), s.t. g(x, y) ≤ 0,
x
L(x, y, λ) = f (x, y) + 〈λ, g(x, y)〉.
The functions f and g are continuously differentiable.
For a reference value y0, suppose that Py0 is convex and denote by
S(y0), the set of solutions of Py0
Λ(y0), the set of Lagrange multipliers.

Introduction Study of the reference problem Sensitivity analysis
Theorem
Suppose that
1 problem Py0
2 problem Py0
has solutions
is qualified, at all the solutions
3 for all sequence yn → y0, Pyn has a solution xn such that
(xn)n has a limit point x in S(y0)
Then, V is directionally differentiable at y0 in all direction h and
V ′ (y0, h) = inf sup Dy L(x, λ, y0)h.
x∈S(y0) λ∈Λ(y0)
Our goal: applying this result to V (τb, τe).

Introduction Study of the reference problem Sensitivity analysis
Theorem
Suppose that
1 problem Py0
2 problem Py0
has solutions
is qualified, at all the solutions
3 for all sequence yn → y0, Pyn has a solution xn such that
(xn)n has a limit point x in S(y0)
Then, V is directionally differentiable at y0 in all direction h and
V ′ (y0, h) = inf sup Dy L(x, λ, y0)h.
x∈S(y0) λ∈Λ(y0)
Our goal: applying this result to V (τb, τe).

Introduction Study of the reference problem Sensitivity analysis
An example of a directionally differentiable function:
f : x ∈ R ↦→ |x|. At 0, we have:
f ′
h if h ≥ 0,
(0, h) =
−h if h ≤ 0.

Introduction Study of the reference problem Sensitivity analysis
Toy example
We consider a simplified problem with parameter τ.
τ
1
V (τ) = min c1(t, xt, ut) dt + c2(t, xt, ut) dt
⎧
⎪⎨ ˙xt =
0
f1(t, xt, ut),
τ
t ∈ [0, τ],
s.t. ˙xt =
⎪⎩
f2(t, xt, ut), t ∈ [τ, 1],
x0 = x 0 .
In this framework, impossible to apply the general result and
compute V ′ (τ)!

Introduction Study of the reference problem Sensitivity analysis
A piecewise affine change of variables θ τ enables us to fix the date
of the perturbation.
1
0
0 1

Introduction Study of the reference problem Sensitivity analysis
We obtain:
τ0
V (τ) = min
s.t.
⎧
⎪⎨
⎪⎩
0
˙θ τ s c1(θ τ 1
s , xs, us) ds +
˙xs = ˙ θ τ s f1(θ τ s , xs, us), s ∈ [0, τ0],
˙xs = ˙ θ τ s f2(θ τ s , xs, us), s ∈ [τ0, 1],
x0 = x 0 .
The Lagrangian is
τ0
L(x, u, τ, p)=
0
+
τ0
˙θ τ s H1(θ τ s , xs, us, ps) ds
˙θ τ s c2(θ τ s , xs, us) ds,
1
˙θ
τ0
τ s H2(θ τ 1
s , xs, us, ps) ds −
0
where H1(t, x, u, p) = c1(t, x, u) + 〈p, f1(t, x, u)〉.
ps ˙xs ds,

Introduction Study of the reference problem Sensitivity analysis
We obtain:
τ0
V (τ) = min
s.t.
⎧
⎪⎨
⎪⎩
0
˙θ τ s c1(θ τ 1
s , xs, us) ds +
˙xs = ˙ θ τ s f1(θ τ s , xs, us), s ∈ [0, τ0],
˙xs = ˙ θ τ s f2(θ τ s , xs, us), s ∈ [τ0, 1],
x0 = x 0 .
The Lagrangian is
τ0
L(x, u, τ, p)=
0
+
τ0
˙θ τ s H1(θ τ s , xs, us, ps) ds
˙θ τ s c2(θ τ s , xs, us) ds,
1
˙θ
τ0
τ s H2(θ τ 1
s , xs, us, ps) ds −
0
where H1(t, x, u, p) = c1(t, x, u) + 〈p, f1(t, x, u)〉.
ps ˙xs ds,

Introduction Study of the reference problem Sensitivity analysis
For the reference problem with τ = τ0, we set
by a classical property,
h1[p]t = min
v∈U H1(t, x t, v, pt), for t ∈ [0, τ0],
˙h1[p]t = DtH1(t, x t, ut, pt).
We define similarly h2. These functions are called the true
Hamiltonians.

Introduction Study of the reference problem Sensitivity analysis
We obtain
Dτ L(x, u, τ0, p)
= 1
τ0
h1[p]t + t ˙h1[p]t dt +
τ0 0
1
1 − τ0
= 1
τ0 th1[p]t
τ0
0 +
1
(1 − t)h2[p]t
1 − τ0
= −(h2[p]τ0 − h1[p]τ0 ).
1
τ0
1 τ0
−h2[p]t + (1 − t) ˙h2[p]t dt

Introduction Study of the reference problem Sensitivity analysis
Application to the outages
For our application problem, we set
∆h i b [p] and ∆hi e[p], the jump of the true Hamiltonian at
times τ i b and τ i e, resp.
Π the set of costates
About the costates Π:
in ”most cases”, a singleton
described by a set of inequalities
for example, if p i 0 is not unique, then ui is bang-bang on
[0, τ i b ].

Introduction Study of the reference problem Sensitivity analysis
Application to the outages
For our application problem, we set
∆h i b [p] and ∆hi e[p], the jump of the true Hamiltonian at
times τ i b and τ i e, resp.
Π the set of costates
About the costates Π:
in ”most cases”, a singleton
described by a set of inequalities
for example, if p i 0 is not unique, then ui is bang-bang on
[0, τ i b ].

Introduction Study of the reference problem Sensitivity analysis
Theorem
If all the dates are different, the value function is directionally
differentiable and
V ′ (τb, τe), (δτb, δτe) = sup
(p0,pT )∈Π
−δτ i b∆hi b [p] − δτ i e∆h i e[p].
i∈S
The result does not depend on the optimal solution.
A different change of variable is needed if some dates are
equal.

Introduction Study of the reference problem Sensitivity analysis
Theorem
If all the dates are different, the value function is directionally
differentiable and
V ′ (τb, τe), (δτb, δτe) = sup
(p0,pT )∈Π
−δτ i b∆hi b [p] − δτ i e∆h i e[p].
i∈S
The result does not depend on the optimal solution.
A different change of variable is needed if some dates are
equal.

Introduction Study of the reference problem Sensitivity analysis
Conclusion
Our study provides a local approximation of the value
function, as long as the perturbation of the dates does not
modify the initial order of the dates.
An extension to a stochastic framework should be possible.
Reference: J.F. Bonnans and A. Shapiro, Perturbation Analysis of
Optimization Problems, Springer-Verlag, New-York, 2000.

Introduction Study of the reference problem Sensitivity analysis
Thank you for your attention!