Sensitivity analysis for the outages of nuclear power plants Kengy - - PowerPoint PPT Presentation

Introduction Study of the reference problem Sensitivity analysis

Sensitivity analysis for the outages of nuclear power plants

Kengy Barty†, Fr´ ed´ eric Bonnans∗, and Laurent Pfeiffer∗

∗ INRIA Saclay and CMAP, Ecole Polytechnique, †EDF R&D

February 2012

Introduction Study of the reference problem Sensitivity analysis

Introduction

Study of a two-level problem:

• ptimization of the dates of the outages of nuclear power

plants

• ptimization 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:

• ptimization of the dates of the outages of nuclear power

plants

• ptimization 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: ui

t the rate of production of plant i

0 ≤ ui

t ≤ ui the bound on production

U=

i∈S[0, ui]

Introduction Study of the reference problem Sensitivity analysis

State variables: si

t the level of fuel of plant i

[τ i

b, τ i e] the dates of the outages

ai the rate of refuelling Dynamic:

• ˙

si

t =

−ui

t1t / ∈[τ i

b,τ i e] + ai1t∈[τ i b,τ i e]

si

0 =

si,0 State constraints:

• si

τ i

b = 0

si

T ≥ 0

Introduction Study of the reference problem Sensitivity analysis

Optimization criterion: min T c

• dt −
• i∈W (t)

ui

t

• dt + φ(sT),

where: 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; Rn) s ∈ W 1,∞(0, T; Rn)

Introduction Study of the reference problem Sensitivity analysis

Pontryagin’s principle

The Hamiltonian is independent on the state! H(t, u, p) = c

• dt −
• i∈W (t)

ui +

• i∈S

pi −ui1t /

∈[τ i

b,τ i e]+ai1t∈[τ i b,τ i e]

• .

Proposition 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 ≤ Dsiφ(sT) and pi

T = Dsiφ(sT) if si T = 0. 2 For almost all t in [0, T],

H(t, ut, pt) ≤ H(t, v, pt), ∀v ∈ U.

Introduction Study of the reference problem Sensitivity analysis

Pontryagin’s principle

The Hamiltonian is independent on the state! H(t, u, p) = c

• dt −
• i∈W (t)

ui +

• i∈S

pi −ui1t /

∈[τ i

b,τ i e]+ai1t∈[τ i b,τ i e]

• .

Proposition 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 ≤ Dsiφ(sT) and pi

T = Dsiφ(sT) if si T = 0. 2 For almost all t in [0, T],

H(t, ut, pt) ≤ H(t, v, pt), ∀v ∈ U.

Introduction Study of the reference problem Sensitivity analysis

Stucture of optimal controls

Each stock of fuel i has two marginal prices associated: −pi

0 ≥ 0

and − pi

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 −pi

t > −pj 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: −pi

0 ≥ 0

and − pi

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 −pi

t > −pj t,

then plant i is used only if plant j produces at its maximum rate.

Introduction Study of the reference problem Sensitivity analysis

• #
• #

Demand

• Total production

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 V (y) = min

x f (x, y),

s.t. g(x, y) ≤ 0, and its Lagrangian 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 V (y) = min

x f (x, y),

s.t. g(x, y) ≤ 0, and its Lagrangian 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 has solutions 2 problem Py0 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 DyL(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 has solutions 2 problem Py0 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 DyL(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 ′(0, h) =

• h

if h ≥ 0, −h if h ≤ 0.

Introduction Study of the reference problem Sensitivity analysis

Toy example

We consider a simplified problem with parameter τ. V (τ) = min τ c1(t, xt, ut) dt + 1

τ

c2(t, xt, ut) dt s.t.      ˙ xt = f1(t, xt, ut), t ∈ [0, τ], ˙ xt = f2(t, xt, ut), t ∈ [τ, 1], x0 = x0. 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 θτ enables us to fix the date

• f the perturbation.

1 1

Introduction Study of the reference problem Sensitivity analysis

We obtain: V (τ) = min τ0 ˙ θτ

s c1(θτ s , xs, us) ds +

1

τ0

˙ θτ

s c2(θτ s , xs, us) ds,

s.t.      ˙ xs = ˙ θτ

s f1(θτ s , xs, us),

s ∈ [0, τ0], ˙ xs = ˙ θτ

s f2(θτ s , xs, us),

s ∈ [τ0, 1], x0 = x0. The Lagrangian is L(x, u, τ, p)= τ0 ˙ θτ

s H1(θτ s , xs, us, ps) ds

+ 1

τ0

˙ θτ

s H2(θτ s , xs, us, ps) ds −

1 ps ˙ xs ds, where H1(t, x, u, p) = c1(t, x, u) + p, f1(t, x, u).

Introduction Study of the reference problem Sensitivity analysis

We obtain: V (τ) = min τ0 ˙ θτ

s c1(θτ s , xs, us) ds +

1

τ0

˙ θτ

s c2(θτ s , xs, us) ds,

s.t.      ˙ xs = ˙ θτ

s f1(θτ s , xs, us),

s ∈ [0, τ0], ˙ xs = ˙ θτ

s f2(θτ s , xs, us),

s ∈ [τ0, 1], x0 = x0. The Lagrangian is L(x, u, τ, p)= τ0 ˙ θτ

s H1(θτ s , xs, us, ps) ds

+ 1

τ0

˙ θτ

s H2(θτ s , xs, us, ps) ds −

1 ps ˙ xs ds, where H1(t, x, u, p) = c1(t, x, u) + p, f1(t, x, u).

Introduction Study of the reference problem Sensitivity analysis

For the reference problem with τ = τ0, we set h1[p]t = min

v∈U H1(t, xt, v, pt), for t ∈ [0, τ0],

by a classical property, ˙ h1[p]t = DtH1(t, xt, ut, pt). We define similarly h2. These functions are called the true Hamiltonians.

Introduction Study of the reference problem Sensitivity analysis

We obtain DτL(x, u, τ0, p) = 1 τ0 τ0 h1[p]t + t ˙ h1[p]t dt + 1 1 − τ0 1

τ0

−h2[p]t + (1 − t)˙ h2[p]t dt = 1 τ0

• th1[p]t

τ0 + 1 1 − τ0

• (1 − t)h2[p]t

1

τ0

= −(h2[p]τ0 − h1[p]τ0).

Introduction Study of the reference problem Sensitivity analysis

Application to the outages

For our application problem, we set ∆hi

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 pi

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 ∆hi

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 pi

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

• i∈S

−δτ i

b∆hi b[p] − δτ i e∆hi e[p].

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

• i∈S

−δτ i

b∆hi b[p] − δτ i e∆hi e[p].

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 more sophisticated 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

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