J O Ël

Thoi Nhan Ngo

Joël Blot
email: blot@univ-paris1.fr

LIGHTENINGS OF ASSUMPTIONS FOR PONTRYAGIN PRINCIPLES IN INFINITE HORIZON AND DISCRETE TIME

Keywords: MSC 2010: 49J21, 65K05, 39A99 infinite-horizon optimal control, discrete time

In the infinite-horizon and discrete-time framework we establish maximum principles of Pontryagin under assumptions which weaker than these ones of existing results. We avoid several assumptions of continuity and of Fréchet-differentiability and of linear independence.

Introduction

The aim of this paper is to establish maximum principles of Pontryagin under assumptions which are weaker than those of existing results. Now we state the considered problems. For all t ∈ N let X t be a nonempty open subset of R n , U t be a nonempty subset of R d , and f t : X t × U t → X t+1 be a mapping. We introduce the two following dynamical systems.

(Di) x t+1 ≤ f t (x t , u t ), t ∈ N.

(De) x t+1 = f t (x t , u t ), t ∈ N. The order in (Di) is the usual order of R n : when x = (x 1 , ..., x n ) and y = (y 1 , ..., y n ) belong to R n , x ≤ y means x i ≤ y i for all i ∈ {1, ..., n}. We fix σ ∈ X 0 , and when k ∈ {i, e}, we define Adm k as the set of the (x, u) = ((x t ) t∈N , (u t ) t∈N ) ∈ t∈N X t × t∈N U t such that (x, u) satisfies (Dk) for all t ∈ N and such that x 0 = σ.

For all t ∈ N, we consider the function φ t : X t × U t → R. When k ∈ {i, e}, we define Dom k as the set of the (x, u) ∈ Adm k such that the series +∞ t=0 φ t (x t , u t ) is convergent in R. We define the functional J : Dom k → R by setting J(x, u) := +∞ t=0 φ t (x t , u t ). When k ∈ {i, e}, we consider the following list of problems. (P 1 k ) Maximize J(x, u) when (x, u) ∈ Dom k . (P 2 k ) Find (x, û) ∈ Adm k such that, for all (x, u) ∈ Adm k , lim sup h→+∞ ( h t=0 φ t (x t , ût ) -h t=0 φ t (x t , u t )) ≥ 0. (P 3 k ) Find (x, û) ∈ Adm k such that, for all (x, u) ∈ Adm k , lim inf h→+∞ ( h t=0 φ t (x t , ût ) -h t=0 φ t (x t , u t )) ≥ 0. Now we describe the contents of the paper.

In Section 2 we specify notions of differentiability and their notation, and we recall the method of reduction to finite horizon (Theorem 2.1).

Date: January 22, 2016.

1

In Section 3 we establish weak maximum principles where the values of the optimal control belong to the interior of the sets of controls for system which governed by difference inequations (Theorem 3.1 and Theorem 3.2). These results are new and only use the Gâteaux differentiability of the criterion, of the vector field and of the inequality constraints. Neither continuity nor Fréchet differentiability is necessary. These principles use recent results on multipliers rules in static optimization which are established in [START_REF] Blot | On the multiplier rules[END_REF] In Section 4, we establish a weak maximum principle when the sets of controls are defined by inequalities (Theorem 4.3) when the system is governed by difference inequations. This result also only uses the Gâteaux differentiability of the criterion, of the vector field and of the inequality constraints and a condition of separation of the origine and of the convex hull of the Gâteaux differentials of the inequelities constraints in the spirit of a Mangarasian-Fromowitz condition. Secondly we establish a weak maximum principle when the sets of controls are define by equalities and inequalities (Theorem 4.7) when the system is governed by a difference inequation. Such a case is treated in [START_REF] Blot | Infinite-horizon discrete-time Pontryagin principles via results of Michel[END_REF] (Theorem 3.1 and Theorem 3.2). In comparison with the result of [START_REF] Blot | Infinite-horizon discrete-time Pontryagin principles via results of Michel[END_REF], the improvements are the following ones: we avoid a condition of continuity for the saturated inequality constraints and for the vector field, we avoid a condition of linear independence of all the differentials of the constraints. A similar result is Theorem 4.8 for which the system is governed by a difference equation.

Notation and Recall

When E and F are finite-dimensional real normed vector spaces, when A ⊂ E, when Φ : A → F is a mapping, and a ∈ A, Φ is said Gâteaux differentiable at a when, for all v ∈ E, DΦ(a, v) := lim s→0 1 s (Φ(a + sv) -Φ(a)) exists for all v ∈ E and when v → DΦ(a, v) is linear, then its Gâteaux differential of Φ at a is D G Φ(a) ∈ L(E, F ) defined by D G Φ(a)v := DΦ(a, v). When it exists, the Fréchet differential of φ at a is denoted by Dφ(a).

When E = E 1 × E 2 is a product of normed vector spaces, a = (a 1 , a 2 ), D G,1 Φ(a 1 , a 2 ) (respectively D G,2 Φ(a 1 , a 2 )) is the Gâteaux differential of Φ(•, a 2 ) at a 1 (respectively Φ(a 1 , •) at a 2 ) and when Φ is Fréchet differentiable at (a 1 , a 2 ), D 1 Φ(a 1 , a 2 ) (respectively D 2 φ(a 1 , a 2 ) denotes the Fréchet differential of Φ(•, a 2 ) at a 1 (respectively D 2 Φ(a 1 , •) at a 2 ).
The method of reduction to finite horizon, which comes from [START_REF] Blot | Discrete time Pontryagin principle in infinite horizon[END_REF], is contained in the following result.

Theorem 2.1. The two assertions hold.

(a) Let (x, û) be a solution of (P j i ) when j ∈ {1, 2, 3}. Then, for all h ∈ N * , (x 0 , ..., xh+1 , û0 , ..., ûh ) is a solution of the following finite-horizon problem

(F h i )        Maximize h t=0 φ t (x t , u t ) when (x 0 , ..., x h+1 , u 0 , ..., u h ) ∈ h+1 t=0 X t × h t=0 U t x t+1 ≤ f t (x t , u t ) when t ∈ {0, ..., h} x 0 = σ, x h+1 = xh+1 .
(b) Let (x, û) be a solution of (P j e ) when j ∈ {1, 2, 3}. Then, for all h ∈ N * , (x 0 , ..., xh+1 , û0 , ..., ûh ) is a solution of the following finite-horizon problem

(F h e )        Maximize h t=0 φ t (x t , u t ) when (x 0 , ..., x h+1 , u 0 , ..., u h ) ∈ h+1 t=0 X t × h t=0 U t x t+1 = f t (x t , u t ) when t ∈ {0, ..., h} x 0 = σ, x h+1 = xh+1 .
The proof of this theorem is given in [START_REF] Blot | Discrete time Pontryagin principle in infinite horizon[END_REF] and in [START_REF] Blot | Infinite-horizon optimal control in the discrete-time framework[END_REF]. Note that this result does not require any special assumption.

Weak Pontryagin principles with interior optimal controls

In this section we consider the case where values of the optimal control sequence belong to the topological interior of the set U t of the considered controls at each time t, and where the system is governed by the difference inequation (Di). Theorem 3.1. Let (x, û) be a solution of (P j i ) when j ∈ {1, 2, 3}. We assume that the following assumptions are fulfilled.

(i) For all t ∈ N, ût ∈ intU t .

(ii) For all t ∈ N, φ t and f t are Gâteaux differentiable at (x t , ût ).

(iii) For all t ∈ N, for all α ∈ {1, ..., n}, f α t is lower semicontinuous at (x t , ût )

when f α t (x t , ût ) > xα t+1 . (iv) For all t ∈ N, D G,1 f t (x t , ût ) is invertible.
Then there exist λ 0 ∈ R and (p t ) t∈N * ∈ (R n * ) N * which satisfy the following properties.

(NN) (λ 0 , p 1 ) = (0, 0). (Si) λ 0 ≥ 0 and, for all t ∈ N * , p t ≥ 0.

(Sℓ) For all t ∈ N, for all α ∈ {1, ..., n}, p α t+1

• (f α t (x t , ût ) -xα t+1 ) = 0. (AE) For all t ∈ N * , p t = p t+1 • D G,1 f t (x t , ût ) + λ 0 D G,1 φ t (x t , ût ). (WM) For all t ∈ N, p t+1 • D G,2 f t (x t , ût ) + λ 0 D G,2 φ t (x t , ût ) = 0.
Proof. Using Theorem 2.1 we can assert that, for all h ∈ N * , (x 0 , ..., xh+1 , û0 , ..., ûh ) is a solution of (F h i ). We introduce the function ψ : h t=1 X t × h t=0 U t → R by setting

ψ(x 1 , ..., x h , u 0 , ..., u h ) := φ 0 (σ, u 0 ) + h t=1 φ t (x t , u t ).
We introduce the mapping ψ 0 : h t=1 X t × h t=0 U t → R n by setting ψ 0 (x 1 , ..., x h , u 0 , ..., u h ) := f 0 (σ, u 0 )x 1 . For all t ∈ {1, ..., h -1} we introduce the mapping ψ t :

h t=1 X t × h t=0 U t → R n by setting ψ t (x 1 , ..., x h , u 0 , ..., u h ) := f t (x t , u t ) -x t+1 .
We introduce the mapping ψ h :

h t=1 X t × h t=0 U t → R n by setting ψ h (x 1 , ..., x h , u 0 , ..., u h ) := f h (x h , u h ) -xh+1 .
Then we can formulate (F h i ) in the following form. Maximize ψ(x 1 , ..., x h , u 0 , ..., u h ) when ∀t ∈ {1, ..., h}, x t ∈ X t , u t ∈ U t ∀t ∈ {0, ..., h}, ∀α ∈ {1, ..., n}, ψ α t (x 1 , ..., x h , u 0 , ..., u h ) ≥ 0

   (3.1)
where the ψ α t are the coordinates of ψ t . Our assumptions (i, ii, iii) imply that the assumptions of Theorem 3.1 in [START_REF] Blot | On the multiplier rules[END_REF] are fulfilled and so we know that, for all h ∈ N * , there exists (λ h 0 , p h 1 , ..., p h h+1 ) ∈ R × (R n * ) h+1 which satisfies the following conditions.

(λ h 0 , p h 1 , ..., p h h+1 ) = (0, 0, ..., 0). (3.2)

λ h 0 ≥ 0,
= p h t+1 • D G,1 f t (x t , ût ) + λ h 0 D G,1 φ t (x t , ût ). (3.5) ∀t ∈ {1, ..., h}, p h t+1 • D G,2 f t (x t , ût ) + λ h 0 D G,2 φ t (x t , ût ) = 0. (3.6)
Using assumption (iv) we can formulate (3.5) as follows.

∀t ∈ {1, ..., h}, p

h t+1 = (p h t -λ h 0 D G,1 φ t (x t , ût )) • D G,1 f t (x t , ût ) -1 . (3.7)
From this last equation we easily see that (λ h 0 , p h 1 ) = (0, 0) =⇒ (λ h 0 , p h i , ..., p h h+1 ) = (0, 0, ..., 0) and then from (3.2) we can assert that

(λ h 0 , p h 1 ) = (0, 0). (3.8)
Since the set of the lists of multipliers of Problem (3.1) is a cone, we can normalize the multipliers by setting

|λ h 0 | + p h 1 = 1. (3.9)
Since the values of the sequence (λ h 0 , p h 1 ) h∈N * belong to the unit sphere of R × R n * which is compact, using the Bolzano-Weierstrass theorem we can say that there exist an increasing function ϕ :

N * → N * and (λ 0 , p 1 ) ∈ R×R n * such that |λ 0 |+ p 1 = 1, lim h→+∞ λ ϕ(h) 0 = λ 0 and lim h→+∞ p ϕ(h) 1 = p 1 . Note that p ϕ(h) 2 = (p ϕ(h) 1 -λ ϕ(h) 0 D G,1 φ 1 (x 1 , û1 )) • D G,1 f 1 (x 1 , û1 ) -1 for all h ≥ t -1, which implies that p 2 := lim h→+∞ p ϕ(h) 2 = (p 1 -λ 0 D G,1 φ 1 (x 1 , û1 )) • D G,1 f 1 (x 1 , û1 ) -1 .
Proceeding recursively we define, for all t ∈ N * , p t+1 := lim h→+∞ p

ϕ(h) t+1 = lim h→+∞ (p ϕ(h) t -λ ϕ(h) 0 D G,1 φ t (x t , ût ))• D G,1 f t (x t , ût ) -1 = (p t -λ 0 D G,1 φ t (x t , ût ))• D G,1 f t (x t , ût ) -1
. And so we have built λ 0 ∈ R and a sequence (p t ) t∈N * ∈ (R n * ) N * which satisfies (AE).

We have yet seen that (NN) is satisfied. From (3.3) we obtain (Si). From (3.4) we obtain (Sℓ). From (3.6) we obtain (WM). Theorem 3.2. Let (x, û) be a solution of (P j i ) with j ∈ {1, 2, 3}. We assume that the assumption (i,ii,iii) of Theorem 3.1 are fulfilled. Moreover we assume that the following assumption is fulfilled.

(v) For all t ∈ N * , for all α, β ∈ {1, ..., n},

∂f α t (xt,ût) ∂x β
≥ 0 and for all α ∈ {1, ..., n},

∂f α t (xt,ût) ∂x α > 0.
Then the conclusions of Theorem 3.1 hold.

Proof. Proceeding as in the proof of Theorem 3.1 we obtain that, for all h ∈ N * , (x 0 , ..., xh+1 , û0 , ..., ûh ) is a solution of problem (3.1) and conditions (3.2 -3.7) are fulfilled. For all t ∈ N * we define γ t := min 1≤α≤n

∂f α t (xt,ût) ∂x α ∈ (0, +∞).
Under assumption (v), when v ∈ R n , for all α ∈ {1, ..., n}, we have

(D G,1 f t (x t , ût ) • v) α = n β=1 ∂f α t (x t , ût ) ∂x β v β ≥ ∂f α t (x t , ût ) ∂x α v α ≥ γ t v α which implies D G,1 f t (x t , ût ) • v ≥ γ t v. Then using Lemma 2.3 in [6] (p. 37) we can assert that, for all π ∈ (R n * ) + , π • D G,1 f t (x t , ût ) ≥ γ t π.
Then we have, for all t ∈ {1, ..., h},

p h t = p h t+1 • D G,1 f t (x t , ût ) + λ h 0 D G,1 φ t (x t , ût ) ≥ γ t p h t+1 + λ h 0 D G,1 φ t (x t , ût ) which implies 0 ≤ p h t+1 ≤ 1 γt (p h t -λ h 0 D G,1 φ t (x t , ût ) which implies p h t+1 ≤ 1 γ t ( p h t + λ h 0 D G,1 φ t (x t , ût ) )
which implies, since from (3.3) and (3.9) we have λ h 0 = |λ h 0 | ≤ 1, the following relation holds for all t ∈ N * and for all h ≥ t -1.

p h t+1 ≤ 1 γ t ( p h t + D G,1 φ t (x t , ût ) ). (3.10)
Now we want to prove the following assertion.

∀t ∈ N * , ∃ζ t ∈ (0, +∞), ∀h ≥ t -1, p h t ≤ ζ t . (3.11)
We proceed by induction. When t = 1, from (3.9) we know that p h t ≤ 1, and so it suffices to take ζ 1 := 1. We assume that (3.11) holds for t, then for t + 1, from (3.10) we obtain

p h t+1 ≤ 1 γ t (ζ t + D G,1 φ t (x t , ût ) ) =: ζ t+1
and so (3.11) is proven. Using (3.11) and the diagonal process of Cantor as it is formulated in [START_REF] Blot | Infinite-horizon optimal control in the discrete-time framework[END_REF] (Theorem A.1, p. 94), we can assert that there exists an increasing function ρ : N * → N * and a sequence (p t ) t∈N * ∈ (R n * + ) N * such that, for all t ∈ N * , lim h→+∞ p ρ(h) t = p t . Now we conclude as in the proof of Theorem 3.1.

Weak Pontryagin principles with constrained controls

In this section we first consider the case where the sets of controls are defined by inequalities for each t ∈ N.

U t = 1≤k≤m {u ∈ R d : g k t (u) ≥ 0} (4.1)
where g k t : R d → R. Lemma 4.1. Let E be a finite-dimensional real normed vector space and I be a nonempty finite set. Let (ϕ i ) i∈I ∈ (E * ) I . The three following assertions are equivalent.

(i) 0 / ∈ co{ϕ i : i ∈ I}.

(ii) For all (λ i ) i∈I ∈ (R + ) I , i∈I λ i ϕ i = 0 =⇒ λ i = 0 for all i ∈ I.

(iii) There exists w ∈ E such that, ϕ i , w > 0 for all i ∈ I.

Proof. First we prove that non(ii) implies non(i). From non(ii) we deduce that there exists (λ i ) i∈I ∈ (R + ) I such that (λ i ) i∈I = 0 and i∈I ( λi j∈I λj )ϕ i = 0 which implies non(i). Secondly we prove that non(i) implies non(ii). From non(i) there exists (α i ) i∈I ∈ R + such that i∈I α i = 1 and 0 = i∈I α i ϕ i , and since (α i ) i∈I is non zero, non(ii) is fulfilled. And so we have proven that non(i) and non(ii) are equivalent.

To prove that (i) implies (iii), note that 0 / ∈ co{ϕ i : i ∈ I} =: K, and K is a nonempty convex compact set. Using the theorem of separation of Hahn-Banach, we can assert that there exist ξ ∈ R n * * and a ∈ (0, +∞) such that ξ, ϕ ≥ a for all ϕ ∈ K, and ξ, 0 = 0 < a. Since R n is reflexive, there exists w ∈ R n such ξ, ϕ = ϕ, w for all ϕ ∈ R n * . Therefore for all i ∈ I, we have ϕ i , w ≥ a > 0 that is (iii).

To prove that (iii) implies (i) we set γ := min i∈I ϕ i , w > 0. When ϕ ∈ co{ϕ i : i ∈ I}, there exists (α i ) i∈I ∈ R + such that i∈I α i = 1 and ϕ = i∈I α i ϕ i . Then we have ϕ, w = i∈I α i ϕ i , w ≥ i∈I α i γ = γ > 0 which implies ϕ = 0, and so (i) is satisfied. Lemma 4.2. Let E be a finite-dimensional real normed vector space and I be a nonempty finite set. Let

(ϕ i ) i∈I ∈ (E * ) I such that 0 / ∈ co{ϕ i : i ∈ I}. For all i ∈ I, let (r h i ) h∈N * ∈ R N * + . We assume that the sequence (ψ h ) h∈N * := ( i∈I r h i ϕ i ) h∈N * is bounded in E * .
Then there exists an increasing function ρ : N * → N * such that, for all i ∈ I, the sequence (r ρ(h) i

) h∈N * is convergent in R + .
Proof. First we prove that lim inf h→+∞ i∈I r h i < +∞. We proceed by contradiction: we assume that lim inf h→+∞ i∈I r h i = +∞. Therefore we have lim h→+∞ i∈I r h i = +∞. We set s h i :=

r h i j∈I r h j ∈ R + . We have i∈I s h i = 1 and therefore i∈I s h i ϕ i ∈ co{ϕ i : i ∈ I}. Note that i∈I s h i ϕ i = 1 j∈I r h j ψ h
converges to 0 when h → +∞ since (ψ h ) h∈N * is bounded. Therefore we have lim h→+∞ i∈I s h i ϕ i = 0 which implies that 0 ∈ co{ϕ i : i ∈ I} that is a contradiction with one assumption. And so we have proven that s := lim inf h→+∞ i∈I r h i < +∞.

Now we can assert that there exists an increasing function τ :

N * → N * such that lim h→+∞ i∈I r τ (h) i = s. Therefore there exists M ∈ R + such that 0 ≤ i∈I r τ (h) i ≤ M for all h ∈ N * . Since for all i ∈ I, we have 0 ≤ r τ (h) i ≤ j∈I r τ (h) j ≤ M , i.e. the sequence (r τ (h) i
) h∈N * is bounded in R + . Using several times the Bolzano-Weierstrass theorem we can assert that there exist an increasing function τ 1 : N * → N * and r * i ∈ R + for all i ∈ I, such that lim h→+∞ r

τ •τ1(h) i = r * i . It suffices to take ρ := τ • τ 1 .
Theorem 4.3. Let (x, û) be a solution of (P j i ) where j ∈ {1, 2, 3} and where the sets U t are defined by (4.1). We assume that the following assumptions are fulfilled.

(i) For all t ∈ N, φ t and f t are Gâteaux differentiable at (x t , ût ).

(ii) For all t ∈ N, for all k ∈ {1, ..., m}, g k t is Gâteaux differentiable at ût .

(iii) For all t ∈ N, for all α ∈ {1, ..., n}, f α t is lower semicontinuous at (x t , ût ) when f α t (x t , ût ) > xα t+1 . (iv) For all t ∈ N, for all k ∈ {1, ..., m}, g k t is lower semicontinuous at ût when

g k t (û t ) > 0. (v) For all t ∈ N, 0 / ∈ co{D G g k t (û t ) : k ∈ I s t }
where I s t := {k ∈ {1, ..., m} :

g k t (û t ) = 0}. (vi) For all t ∈ N * , D G,1 f t (x t , ût ) is invertible. (vii) For all t ∈ N * for all α, β ∈ {1, ..., n}, ∂f α t (xt,ût) ∂x β
≥ 0 and for all α ∈ {1, ..., n},

∂f α t (xt,ût) ∂x α > 0.
Then, under (i-vi) or under (i-v) and (vii), there exist

λ 0 ∈ R, (p t ) t∈N * ∈ (R n * ) N * , (µ 1 t ) t∈N ∈ R N , .
.., and (µ m t ) t∈N ∈ R N which satisfy the following conditions. (NN) (λ 0 , p 1 ) = (0, 0).

(Si) λ 0 ≥ 0, , p t ≥ 0 for all t ∈ N * , and µ k t ≥ 0 for all t ∈ N and for all k ∈ {1, ..., m}. (Sℓ) For all t ∈ N, for all α ∈ {1, ..., n}, p α t+1 • (f α t (x t , ût )xα t+1 ) = 0, and for all k ∈ {1, ..., m},

µ k t • g k t (û t ) = 0. (AE) For all t ∈ N * , p t = p t+1 • D G,1 f t (x t , ût ) + λ 0 D G,1 φ t (x t , ût ). (WM) For all t ∈ N, p t+1 • D G,2 f t (x t , ût ) + λ 0 D G,2 φ t (x t , ût ) + m k=1 µ k t D G g k t (û t ) = 0.
Proof. From Theorem 2.1 (a) we know that, for all h ∈ N * , (x 0 , ..., xh+1 , û0 , ..., ûh ) is a solution of the following finite-horizon problem.

          
Maximize J(x 0 , ..., x h+1 , u 0 , ..., u h ) = h t=0 φ t (x t , u t ) when ∀t ∈ {0, ..., h}, f t (x t , u t )x t+1 ≥ 0 ∀t ∈ {0, ..., h}, x t ∈ X t x 0 = σ, x h+1 = xh+1 ∀t ∈ {0, ..., h}, ∀k ∈ {1, ..., m}, g k t (u t ) ≥ 0. From Theorem 3.1 in [START_REF] Blot | On the multiplier rules[END_REF] we can assert that there exists (λ h 0 , p h 1 , ..., p h h+1 , µ 1,h 1 , ..., µ m,h h ) ∈ R × (R n * ) h × R mh which satisfies the following assertions.

(λ h 0 , p h 1 , ..., p h h+1 , µ

= p h t+1 • D G,1 f t (x t , ût ) + λ h 0 D G,1 φ t (x t , ût ). (4.5) ∀t ∈ {0, ..., h}, p h t+1 • D G,2 f t (x t , ût ) + λ h 0 D G,2 φ t (x t , ût ) + m k=1 µ k,h t D G g k t (û t ) = 0. (4.6)
Using (4.5) under (vi) or (vii) and working as in the proof of Theorem 3.1 or Theorem 3.2, we obtain (λ h 0 , p h 1 ) = (0, 0) =⇒ (λ h 0 , p h 1 , ..., p h h+1 ) = (0, 0, ..., 0). Proceeding by contradiction, assuming that (λ h 0 , p h 1 ) = (0, 0), from the previous implication and (4.6) we obtain m k=1 µ k,h t D G g k t (û t ) = 0, and then using Lemma 4.1 we obtain that the µ k,h t = 0. Therefore we obtain a contradiction with (4.2). And so we have proven that (λ h 0 , p h 1 ) = (0, 0). Under (vi) proceeding as in the proof of Theorem 3.1 and under (vii) proceeding as in the proof of Theorem 3.2 we obtain the existence of an increasing function ρ : N * → N * and of λ 0 ∈ R + and of (p t ) t∈N * ∈ (R n * + ) N * such that λ 0 = lim h→+∞ λ ρ(h) 0

, p t = lim h→+∞ p ρ(h) t

, (λ 0 , p 1 ) = (0, 0), and p α t • (f α t (x t , ût )xα t+1 ) = 0 for all t ∈ N * and for all α ∈ {1, ..., n}. We fix t ∈ N and we consider, for all h ∈ N * ,

ϕ h := k∈I s t µ k,ρ(h) t D G g k t (û t ) = m k=1 µ k,ρ(h) t D G g k t (û t ) = -(p ρ(h) t+1 • D G,2 f t (x t , ût ) + λ ρ(h) 0 D G,2 φ t (x t , ût )).

Therefore we have lim

h→+∞ ϕ h = -(p t+1 • D G,2 f t (x t , ût ) + λ 0 D G,2 φ t (x t , ût ))
, and consequently the sequence (ϕ h ) h∈N * is bounded in R n * . Using Lemma 4.2 we can assert that exist an increasing function ρ 1 : N * → N * and µ 1 t , ...,

µ m t ∈ R + such that lim h→+∞ µ k,ρ•ρ1(h) t = µ k t ∈ R + .
And then the assertions (NN), (Si), (Sℓ), (AE) and (WM) are satisfied. Now we consider the case where the sets of controls are defined by equalities and inequalities for each t ∈ N,

U t = 1≤k≤mi {u ∈ R d : g k t (u) ≥ 0} ∩ ( 1≤k≤me {u ∈ R d : e k t (u) = 0}) (4.7)
where e k t : R d → R. Lemma 4.4. Let E be a real finite-dimensional normed vector space; let J and K be two nonempty finite sets, and let (ψ j ) j∈J and (ϕ k ) k∈K be two families of elements of the dual E * . Then the two following assertions are equivalent.

(i) span{ψ j : j ∈ J} ∩ co{ϕ k : k ∈ K} = ∅.

(ii) There exists w ∈ E such that ψ j , w = 0 for all j ∈ J and ϕ k , w > 0 for all k ∈ K.

Proof. We set S := span{ψ j : j ∈ J} and C := co{ϕ k : k ∈ K}.

[i =⇒ ii] Under (i) using the theorem of separation of Hahn-Banach, there exist ξ ∈ E * * and a ∈ (0, +∞) such that ξ, ψ ≤ a for all ψ ∈ S, and ξ, ϕ > a for all ϕ ∈ C. When ψ ∈ S is non zero, we have | ξ, ψ | ≤ a since -ψ ∈ S, and therefore, for all λ ∈ R, we have |λ| • | ξ, ψ | ≤ a which is impossible if | ξ, ψ | = 0, therefore we have ξ, ψ = 0 for all ψ ∈ S. Since E * * is isomorphic to E there exists w ∈ E such ξ, χ = χ, w for all χ ∈ E * , and then we obtain (ii).

[ii =⇒ i] Under (ii) we define a := min k∈K ϕ k , w > 0. When ϕ ∈ C there exists

(θ k ) k∈K ∈ R K + such that k∈K θ k = 1 and k∈K θ k ϕ k = ϕ. Then ϕ, w = k∈K θ k ϕ k , w ≥ k∈K θ k • a > 0. When ψ ∈ S there exists (ζ j ) j∈J ∈ R J such that j∈J ζ j ψ j = ψ.
Therefore we have ψ, w = j∈J ζ j ψ j , w = 0. We have proven that ψ, w = 0 for all ψ ∈ S and ϕ, w > 0 for all ϕ ∈ C, which implies (i).

Lemma 4.5. In the framework of Lemma 4.4, under condition (i) of Lemma 4.4, when (λ j ) j∈J ∈ R J and (µ k ) k∈K ∈ R K + , we have

j∈J λ j ψ j + k∈K µ k ϕ k = 0 =⇒ (∀k ∈ K, µ k = 0).
Proof. We proceed by contraposition, we assume that there exists k ∈ K such that µ k = 0. Then µ := k∈K µ k > 0 and so k∈K µ k µ ϕ k ∈ co{ϕ k : k ∈ K} and k∈K µ k µ ϕ k = -j∈J λj µ ψ j ∈ span{ψ j : j ∈ J} which provides a contradiction with condition (i).

Lemma 4.6. Let E be a real finite-dimensional normed vector space; let J and K be two nonempty finite sets, and let (ψ j ) j∈J and (ϕ k ) k∈K be two families of elements of the dual E * . We assume that the following assumptions are fulfilled.

(a) The family (ψ j ) j∈J is linearly independent.

(b) span{ψ j : j ∈ J} ∩ co{ϕ k : k ∈ K} = ∅. Let (λ h j ) j∈J ∈ R J and (µ h k ) k∈K ∈ R K + for all h ∈ N * such that the sequence (χ h ) h∈N * := ( j∈J λ h j ψ j + k∈K µ h k ϕ k ) h∈N * is bounded in E * .
Then there exists an increasing function ρ : N * → N * such that the sequences (λ ρ(h) j

) h∈N * are convergent in R for all j ∈ J and the sequences and (µ

ρ(h) k ) h∈N * are convergent in R + for all k ∈ K.
Proof. We set S := span{ψ j : j ∈ J} and C := co{ϕ k : k ∈ K}. First we prove that lim inf h→+∞ k∈K µ h k < +∞. We proceed by contradiction, we assume that lim inf h→+∞ k∈K µ h k = +∞. Therefore we have s := lim h→+∞ k∈K µ h k = +∞. We set π h k :=

µ h k k ′ ∈K µ h k ′ ∈ R + . We have k∈K π h k = 1, and therefore k∈K π h k ϕ k ∈ C. Note that j∈J λ h j k ′ ∈K µ h k ′ ψ j + k∈K π h k ϕ k = 1 k ′ ∈K µ h k ′ χ h → 0 when h → +∞, therefore lim h→+∞ ( j∈J λ h j k ′ ∈K µ h k ′ ψ j + k∈K π h k ϕ k ) = 0.
Since C is compact there exists an increasing function τ :

N * → N * and ϕ * ∈ C such that lim h→+∞ k∈K π τ (h) k ϕ k = ϕ * . Consequently lim h→+∞ j∈J -λ τ (h) j k ′ ∈K µ τ (h) k ′ ψ j = ϕ * .
Since a finite-dimensional normed vector space is complete, S is closed in E * , and consequently we have ϕ * ∈ S, and then ϕ * ∈ S ∩ C which is a contradiction with assumption (b). And so we have proven that lim inf h→+∞ k∈K µ h k < +∞. Therefore there there exists an increasing function r :

N * → N * such that lim h→+∞ k∈K µ r(h) k = lim inf h→+∞ k∈K µ h k . Therefore the sequence ( k∈K µ r(h) k ) h∈N * ) h∈N * is bounded in R + . Since 0 ≤ µ r(h) k ≤ k∈K µ r(h) k , we ob- tain that (µ r(h) k ) h∈N * is bounded in R + for all k ∈ K. Therefore ( k∈K µ r(h) k ϕ k ) h∈N * is bounded in E * . Therefore ( j∈J λ r(h) j ψ j ) h∈N * = (χ r(h) -k∈K µ r(h) k ϕ k ) h∈N *
is bounded as a difference of two bounded sequences. Under assumption (a) we can use Lemma 5.5 in [START_REF] Blot | Infinite-horizon discrete-time Pontryagin principles via results of Michel[END_REF] and assert that there exists an increasing function r 1 :

N * → N * such (λ r•r1(h) j
) h∈N * is convergent in R for all j ∈ J. Using cardK times the Bolzano-Weierstrass theorem, there exists an increasing function r 2 :

N * → N * such that (µ r•r1•r2(h) k ) h∈N * is convergent in R + . Taking ρ := r • r 1 • r 2 we have proven the lemma.
Theorem 4.7. Let (x, û) be a solution of (P j i ) where j ∈ {1, 2, 3} and where the sets U t are defined in (4.7). We assume that the following assumptions are fulfilled for all t ∈ N.

(i) φ t is Fréchet differentiable at (x t , ût ). (ii) For all α ∈ {1, ..., n}, f α t is Fréchet differentiable at (x t , ût ) when f α t (x t , ût ) = x α t+1 . (iii) For all α ∈ {1, ..., n}, f α t is lower semicontinuous and Gâteaux differentiable at (x t , ût ) when ≥ 0 and for all α ∈ {1, ..., n},

f α t (x t , ût ) > x α t+1 . (iv) For all k ∈ {1, ..., m i }, g k t is Fréchet differentiable at ût when g k t (û t ) = 0. (v) For all k ∈ {1, ..., m i }, g k t is
∂f α t (xt,ût) ∂x α > 0.
Then under (i-ix) or under (i-viii) and (x) there exist λ 0 ∈ R, (p t ) t∈N * ∈ (R n * ) N * , (λ 1,t ) t∈N ∈ R N , ..., (λ me,t ) t∈N ∈ R N , (µ 1,t ) t∈N ∈ R N , ..., (µ mi,t ) t∈N ∈ R N which satisfy the following conditions.

(NN) (λ 0 , p 1 ) = (0, 0). (Si) λ 0 ≥ 0, p t ≥ 0 for all t ∈ N * , µ k,t ≥ 0 for all t ∈ N and for all k ∈ {1, ..., m i }. (Sℓ) For all t ∈ N, for all α ∈ {1, ..., n}, p α t+1 • (f α t (x t , ût )x α t+1 ) = 0, and for all k ∈ {1, ...,

m i }, µ k,t • g k t (û t ) = 0. (AE) For all t ∈ N * , p t = p t+1 • D G,1 f t (x t , ût ) + λ 0 D 1 φ t (x t , ût ). (WM) For all t ∈ N, p t+1 •D G,2 f t (x t , ût )+λ 0 D 2 φ t (x t , ût )+ me j=1 λ j,t De j (û t )+ mi k=1 µ k,t D G g k t (û t ) = 0.
Proof. From Theorem 2.1 (a) we know that, for all h ∈ N * , (x 0 , ..., xh+1 , û0 , ..., ûh ) is a solution of the following finite-horizon problem.

               Maximize h t=0 φ t (x t , u t ) when ∀t ∈ {0, ..., h}, f t (x t , u t ) -x t+1 ≥ 0 ∀t ∈ {0, ..., h + 1}, x t ∈ X t x 0 = σ, x h+1 = xh+1
∀t ∈ {0, ..., h}, ∀j ∈ {1, ..., m e }, e j t (u t ) = 0 ∀t ∈ {0, ..., h}, ∀k ∈ {1, ..., m i }, g k t (u t ) ≥ 0.

We introduce the following elements

z := (x 1 , ..., x h , u 0 , ..., u h ) Φ(z) := h t=0 φ t (x t , u t ) F α 0 (z) := f α 0 (σ, u 0 ) -x α 1 , 1 ≤ α ≤ n F α t (z) := f α t (x t , u t ) -x α t , 1 ≤ t ≤ h -1, 1 ≤ α ≤ n F α h (z) := f α h (x h , u h ) -xα h+1 , 1 ≤ α ≤ n E j t (z) := e j t (u t ), 0 ≤ t ≤ h, 1 ≤ j ≤ m e G k t (z) := g k t (u t ), 0 ≤ t ≤ h, 1 ≤ k ≤ m i .
Then the previous optimization problem can be written as follows.

Maximize Φ(z) when ∀t ∈ {0, ..., h}, ∀α ∈ {1, ..., n}, F α t (z) ≥ 0 ∀t ∈ {0, ..., h}, ∀j ∈ {1, ..., m e }, E j t (z) = 0 ∀t ∈ {0, ..., h}, ∀k ∈ {1, ..., m i }, G k t (z) ≥ 0.

       (4.8)
We see that our assumptions (i-vi) imply that the assumptions of Theorem 3.2 in [START_REF] Blot | On the multiplier rules[END_REF] From (4.13), using the partial differentiations with respect to x t we obtain, for all δx t ∈ R n ,

λ h 0 D 1 φ t (x t , ût )δx t + n α=1 p h t+1,α D G,1 f α t (x t , ût )δx t - n α=1
p h t+1,α • δx t + 0 + 0 = 0 which implies, denoting by p h t the elements of R n * whose the coordinates are the p h t,α , we obtain the following relation.

λ h 0 D 1 φ t (x t , ût ) + p h t+1 • D G,1 f t (x t , ût ) = p h t . (4.16)
From (4.13) using the partial differentiations with respect tou t we obtain the following relation.

λ h 0 D 2 φ t (x t , ût ) + p h t+1 • D G,2 f t (x t , ût ) + me j=1 λ h t,j De j t (û t ) + mi k=1 µ h t,k D G g k t (û t ) = 0.
(4.17)

Using (ix) and working as in the proof in the proof of Theorem 3.1 or using (x) and working as in the proof of Theorem 3.2, from (4.16) we obtain the following condition.

(λ h 0 , p h 1 ) = (0, 0) =⇒ (λ h 0 , p h 1 , ..., p h h+1 ) = (0, 0, ..., 0). (4.18)

If (λ h 0 , p h 1 ) = (0, 0), using (4.18), (4.17) implies t,k D G g k t (û t )) h∈N * is bounded for all t ∈ N. Using (vii), (viii) and Lemma 4.6 we can assert that there exist an increasing function r 1 : N * → N * , λ t,j ∈ R (for all t ∈ N and for all j ∈ {1, ..., m e }), µ t,k ∈ R (for all t ∈ N and for all k ∈ {1, ..., m i }) such that Theorem 4.8. Let (x, û) be a solution of (P j e ) where j ∈ {1, 2, 3} and where the sets U t are defined in (4.7). We assume that the following assumptions are fulfilled for all t ∈ N.

(i) φ t is Fréchet differentiable at (x t , ût ).

(ii) f t is continuous on a neighborhood of (x t , ût ) and Fréchet differentiable at (x t , ût ). (iii) For all k ∈ {1, ..., m i }, g k t is Fréchet differentiable at ût when g k t (û t ) = 0. (iv) For all k ∈ {1, ..., m i }, g k t is lower semicontinuous and Gâteaux differentiable at ût when g k t (û t ) > 0.

(v) For all j ∈ {1, ..., m e }, e j t is continuous on a neighborhood of ût and Fréchet differentiable at ût . (vi) span{De j t (û t ) : j ∈ {1, ..., m e }} ∩ co{D G g k t (û t ) : k ∈ I s t } = ∅, where I s t := {k ∈ {1, ..., m i } : g k t (û t ) = 0}. (vii) De 1 t (û t ), ..., De me t (û t ) are linearly independent. (viii) For all t ∈ N * , D G,1 f t (x t , ût ) is invertible. Then under (i-viii) there exist λ 0 ∈ R, (p t ) t∈N * ∈ (R n * ) N * , (λ t,1 ) t∈N ∈ R N ,..., (λ t,me ) t∈N ∈ R N , (µ t,1 ) t∈N ∈ R N , ..., (µ t,mi ) t∈N ∈ R N which satisfy the following conditions.

(NN) (λ 0 , p 1 ) = (0, 0). (Si) λ 0 ≥ 0, µ k,t ≥ 0 for all t ∈ N and for all k ∈ {1, ..., m i }.

(Sℓ) For all t ∈ N, for all k ∈ {1, ..., m i }, µ k,t The proof of this theorem is similar to the this one of Theorem 4.7. The difference is the replacement of inequality constraints by equality constraints in the problem issued from the reduction to finite horizon, the consequence of this difference is the lost of the sign of the adjoint variable p t .

Finally ( 4 .

 4 20) implies (NN), (4.20), (4.21) and (4.10) imply (Si), (4.20), (4.21), (4.14) and (4.15) imply (Sℓ), (4.20) and (4.16) imply (AE), and (4.20), (4.21) and (4.17) imply (WM).

  lower semicontinuous and Gâteaux differentiable at ût when g k t (û t ) > 0. (vi) For all j ∈ {1, ..., m e }, e j t is continuous on a neighborhood of ût and Fréchet differentiable at ût . (vii) span{De j t (û t ) : j ∈ {1, ..., m e }} ∩ co{D G g k t (û t ) : k ∈ I s t } = ∅, where I s t := {k ∈ {1, ..., m i } : g k t (û t ) = 0}. (viii) De 1 t (û t ), ..., De me (û t ) are linearly independent. (ix) For all t ∈ N * , D G,1 f t (x t , ût ) is invertible. (x) For all t ∈ N * for all α, β ∈ {1, ..., n},

	∂f α t (xt,ût)
	∂x β

  are fulfilled and consequently we obtain the existence of real numbers λ h 0 , p h t,α (for t ∈ {1, ..., h + 1} and α ∈ {1, ..., n}), λ h t,j (for t ∈ {0, ..., h} and j ∈ {1, ..., m e }), µ h t,k (for t ∈ {0, ..., h} and k ∈ {1, ..., m i }) which satisfy the following conditions:

	(λ h 0 , p h 1,1 , ..., p h h+1,n , λ h 1,0 , ..., λ h me,h , µ h 1,0 , ..., µ h mi,h ) = 0	(4.9)
	λ h 0 ≥ 0, (∀t ∈ {1, ..., h + 1}, ∀α ∈ {1, ..., n}, p h t,α ≥ 0) (∀t ∈ {0, ..., h}, ∀k ∈ {1, ..., m i }, µ h t,j ≥ 0)	(4.10)
	∀t ∈ {0, ..., h}, ∀α ∈ {1, ..., n}, p h t+1,α • F α t (ẑ) = 0	(4.11)
	∀t ∈ {0, ..., h}, ∀k ∈ {1, ..., m i }, µ h t,k • G h t (ẑ) = 0	(4.12)
	+	λ h 0 DΦ(ẑ) + n α=1 p h t,α D G F α t (ẑ) t,j DE j j=1 λ h me t (ẑ) + mi k=1 µ h t,k D G G k t (ẑ) = 0.	(4.13)
	Note that (4.11) is translated by
	∀t ∈ {0, ..., h}, ∀α ∈ {1, ..., n}, p h t+1,α • (f α t (x t , ût ) -xα t+1 ) = 0.	(4.14)
	The condition (4.12) is translated by
	∀t ∈ {0, ..., h}, ∀k ∈ {1, ..., mi}, µ h t,k • g h t (û t ) = 0.	(4.15)

  • g k t (û t ) = 0. (AE) For all t ∈ N, p t = p t+1 • D G,1 f t (x t , ût ) + λ 0 D 1 φ t (x t , ût ). (WM) For all t ∈ N, p t+1 •D G,2 f t (x t , ût )+λ 0 D 2 φ t (x t , ût )+ De j (û t )+

	me	λ t,j mi	µ t,k D G g k t (û t ) =
	j=1	k=1
	0.


Vincent Grelet
email: vincent.grelet@volvo.com

Thomas Reiche
email: thomas.reiche@volvo.com

Vincent Lemort
email: vincent.lemort@ulg.ac.be

Madiha Nadri
email: nadri@lagep.univ-lyon1.fr

Pascal Dufour
email: dufour@lagep.univ-lyon1.fr

TRANSIENT PERFORMANCE EVALUATION OF WASTE HEAT RECOVERY RANKINE CYCLE BASED SYSTEM FOR HEAVY DUTY TRUCKS

Keywords: Waste heat recovery system, Modeling, Thermodynamic

The study presented in this paper aims to evaluate the transient performance of a waste heat recovery Rankine cycle based system for a heavy duty truck and compare it to steady state evaluation. Assuming some conditions to hold, simple thermodynamic simulations are carried out for the comparison of several fluids. Then a detailed first principle based model is also presented. Last part is focused on the Rankine cycle arrangement choice by means of model based evaluation of fuel economy for each concept where the fuels savings are computed using two methodologies. Fluid choice and concept optimization are conducted taking into account integration constraints (heat rejection, packaging . . . ). This paper shows the importance of the modeling phase when designing Rankine cycle based heat recovery systems and yields a better understanding when it comes to a vehicle integration of a Rankine cycle in a truck.

Rankine cycle, Heavy duty trucks

INTRODUCTION

Even in nowadays heavy duty (HD) engines, which can reach 45% of efficiency, a high amount of the chemical energy contained in the fuel is released as heat to the ambient. Driven by future emissions legislation and increase in fuel prices, engine gas heat recovering has recently attracted a lot of interest. Over the last decades, most of the research has focused on waste heat recovery systems (WHRS) based on the Rankine cycle [START_REF] Patel | Compounding the truck diesel engine with an organic rankine-cycle system[END_REF][START_REF] Oomori | Waste heat recovery of passenger car using a combination of rankine bottoming cycle and evaporative cooling system[END_REF][START_REF] Boretti | Recovery of exhaust and coolant heat with R245fa organic rankine cycles in a hybrid passenger car with a naturally aspirated gasoline engine[END_REF]. These systems can lead to a decrease in fuel consumption and lower engine emissions [START_REF] Willems | Integrated powertrain control for optimal CO2-NOx tradeoff in an euro-vi diesel engine with waste heat recovery system[END_REF][START_REF] Freymann | The turbosteamer: A system introducing the principle of cogeneration in automotive applications[END_REF]. Recent studies have brought a significant potential for such systems in a HD vehicle [START_REF] Wang | A review of researches on thermal exhaust heat recovery with rankine cycle[END_REF][START_REF] Sprouse | Review of organic rankine cycles for internal combustion engine exhaust waste heat recovery[END_REF]. However, before the Rankine cycle based system can be applied to commercial vehicles, the challenges of its integration have to be faced. The work done in [START_REF] Espinosa | Contribution to the study of waste heat recovery systems on commercial truck diesel engines[END_REF] and [START_REF] Dickson | Validation and design of heavy vehicle cooling system with waste heat recovery condenser[END_REF] show that one of the main limitation is the cooling capacity of the vehicle. But other drawbacks, such as the back pressure, weight penalty or transient operation should not be minimized [START_REF] Wang | Performance analysis of a novel system combining a dual loop organic rankine cycle (orc) with a gasoline engine[END_REF][START_REF] Boretti | Transient operation of internal combustion engines with rankine waste heat recovery systems[END_REF]. Before tackling the problem of the control strategy of this system [START_REF] Peralez | Improving the control performance of an organic rankine cycle system for waste heat recovery from a heavy-duty diesel engine using a model-based approach[END_REF][START_REF] Willems | Integrated powertrain control for optimal CO2-NOx tradeoff in an euro-vi diesel engine with waste heat recovery system[END_REF], the architecture and components need to be selected to achieve a certain objective that could be to maximize the fuel savings or minimize the impact on the vehicle. This study focuses more on maximizing the system performance by taking into account the different penalties induced by the integration of the system on a heavy duty truck. Technical challenges and optimization of stationary organic Rankine cycles (ORC) are well adressed [START_REF] Quoilin | Technoeconomic survey of organic rankine cycle (orc) systems[END_REF][START_REF] Lecompte | Part load based thermo-economic optimization of the organic rankine cycle (orc) applied to a combined heat and power (chp) system[END_REF] but for mobile applications only few studies deal with fuel saving potential of WHRS on dynamic driving cycles [START_REF] Horst | Prediction of dynamic rankine cycle waste heat recovery performance and fuel saving potential 34 in passenger car applications considering interactions with vehicles' energy management[END_REF][START_REF] Legros | Comparison and Impact of Waste Heat Recovery Technologies on Passenger Car Fuel Consumption in a Normalized Driving Cycle[END_REF] and the latter is generally reduced to a certain number of steady state engine operating points [START_REF] Bredel | Waste heat recovery in drive systems of today and tomorrow[END_REF][START_REF] Freymann | The second generation turbosteamer[END_REF]. This last approach leads to an overestimation of the WHRS performance [START_REF] Boretti | Recovery of exhaust and coolant heat with R245fa organic rankine cycles in a hybrid passenger car with a naturally aspirated gasoline engine[END_REF] and therefore of the fuel economy. In [START_REF] Flik | Emissions reduction in commercial vehicles via thermomanagement[END_REF] different concepts are analyzed taking into account the system integration into the vehicle cooling module.

The concepts differ in the number of heat sources used and the temperature level of the cooling fluid. Each is simulated on different steady state engine operating points and the fuel economy is calculated taking into account the increase in cooling fan consumption, exhaust back pressure or intake manifold temperature. Depending on the Rankine configuration and the location of the condenser, improvements from 2.2% (recovering heat only from exhaust gases and condenser placed in front of the cooling package) to 6.9% (exhaust gas recirculation and exhaust heat are recovered and condenser is fed with engine coolant) are achieved. In [START_REF] Horst | Prediction of dynamic rankine cycle waste heat recovery performance and fuel saving potential 34 in passenger car applications considering interactions with vehicles' energy management[END_REF], dynamic fuel economy is evaluated on a light duty vehicle taking into account the main penalties induced by the integration of the WHRS. Fuel savings from 3.4% to 1.3% are presented depending on the level of integration of the system into the vehicle architecture. However, no optimization is proposed either on the system architecture or on the condenser integration into the cooling package. This paper is organized as follows. The second section explains the different considerations to take when designing a Rankine cycle for a HD application. In the third section, the different models used in the rest of the study are explained. In the fourth section, the scope of the study and the different methodologies are explained. In the fifth section, simulation results are analyzed and possible improvements are proposed. Finally, conclusions are drawn and directions for future research work are discussed.

DESIGN ASPECTS TO CONSIDER

Figure 1 shows a simple waste heat recovery system mounted on a 6 cylinder heavy duty engine. Working fluid flows through four basic components which are: the pump, the evaporator linked to the heat source, the expansion machine and the condenser linked to the heat sink. For sake of clarity, the link between the expander and the engine driveline is represented by a dashed line since it can be either mechanical or electrical (by coupling a generator to the expansion machine and reinject the electricity on the on board network).

Working fluid choice

There are several aspects to take into account when choosing a working fluid for this application. Unlike stationary power plants where the main consideration is the output power or the efficiency, here other aspects have to be considered such as fluid deterioration, environmental aspects or freezing.

Up to now, several studies have tried to identify the ideal fluid for WHRS [START_REF] Mago | Performance analysis of different working fluids for use in organic rankine cycles[END_REF][START_REF] Stijepovic | On the role of working fluid properties in organic rankine cycle performance[END_REF][START_REF] Papadopoulos | On the systematic design and selection of optimal working fluids for organic rankine cycles[END_REF] but no single fluid has been found. Recently, new performance indicators have been introduced [START_REF] Lecompte | Part load based thermo-economic optimization of the organic rankine cycle (orc) applied to a combined heat and power (chp) system[END_REF][START_REF] Cayer | Parametric study and optimization of a transcritical power cycle using a low temperature source[END_REF], where cost and design issues enter into consideration.

Heat sources

On a commercial vehicle, a certain number of heat sources can be found such as exhaust gases, cooling water or engine oil. These ones have several grade of quality (temperature level) and quantity (amount of energy). If the number of heat sources often yields higher fuel savings, it also brings more complexity and more challenges for the design of the system (fluid, expansion machine, control).

Heat sink

On a HD Truck, the only heat sink available is the vehicle cooling package which is a module including radiators for the compressed air and the engine coolant and cooled down by means of the ram air effect and the cooling fan. Integration of a WHRS into the cooling module results on a higher load on the latter and limits the amount of waste heat that can be converted into useful work. As such, complete system analysis is necessary to find the optimal way of recovering heat from a vehicle.

Subsystem interaction

The engine operation is influenced by the introduction of a WHRS. For example, as the WHRS shares the cooling system of the vehicle, the charge air cooling capacity can be lower, which has a negative behavior on the engine performance. Another example is the use of exhaust gas recirculation (EGR) as heat source. This leads to a trade-off between EGR cooling and Rankine cycle performance, which could impact negatively the engine emissions.

Several other interactions such as the exhaust back pressure or the weight penalty could be cited.

The WHRS performance, and so the fuel economy induced by this later, is then dependent on all these aspects. It is therefore critical to model the complete system and its environment in order to optimize its architecture. It helps to select the best design and reduce the number of experimental tests to carry out.

RANKINE MODELING

Rankine process

The Temperature Entropy (T-s) diagram represented in figure 2 shows the associated state changes of the working fluid through the Rankine cycle:

• The pressure of the liquid is increased by the pump work up to the evaporating pressure (1 → 2).

• The pressurized working fluid is pre-heated (2 → 3a), vaporized (3a → 3b) and superheated (3b → 3c) in a heat exchanger, by recovering heat ( Qin ) from the heat source.

• The superheated vapor expands from evaporating pressure to condensing pressure (3c → 4) in an expansion device creating mechanical power ( Ẇout ).

• The expanded vapor condenses (4 → 1) through a condenser (linked to the heat sink) releasing heat ( Qout ).

In this process the changes of states in both the pump and the expander are irreversible and increase the fluid entropy to a certain extent. To correctly assess the performance of a system based on the Rankine cycle, two different dimension. This is required to represent real performance of the components constituting the system either in steady state or in transient.

0D steady state modeling of a Rankine cycle

In order to simulate a high number of working fluids, a 0D model of a Rankine cycle using one heat source is developed. It does not represent a real system but it allows a fast assessment of a various number of working fluids. It helps to select the suitable working fluids for the studied application. This model is based on the enthalpy changes in the process described in section 3.1.

This model is able to perform either subcritical or supercritical cycle, which avoids the vaporization process and leads to a smaller system and a better heat recovery process [START_REF] Karellas | Influence of supercritical ORC parameters on plate heat exchanger design[END_REF]. Those relations are verified as long as the heat losses in the system and in the components are neglected. The 0D model used is given by the system of equations (1):

                                                                           P cond = P sat (T cond ), P f in,pump = P cond , T f in,pump = T sat (P f in,pump ) -∆T subcooling , h f in,pump = h(T f in,pump , P f in,pump ), s f in,pump = s(h f in,pump , P f in,pump ), P fout,pump = P evap , h fout,pump = h f in,pump + (h f out,pump is -h f in,pump )
ηpump is , T fout,pump = T (h fout,pump , P fout,pump ), s fout,pump = s(h fout,pump , P fout,pump ), P f out,boiler = P fout,pump , h f out,boiler = h fout,pump + Qgas ṁf , T f out,boiler = T (h f out,boiler , P f out,boiler ), s f out,boiler = s(h f out,boiler , P f out,boiler ), P fout,exp = P cond , h fout,exp = h f out,boiler + (h f out,boiler -h fout,exp is )η exp is , s fout,exp = s(h fout,exp , P fout,exp ), P f out,cond = P fout,exp , T f out,cond = T sat (P f out,cond ) -∆T subcooling , h f out,cond = h(T f out,cond , P f out,cond ), s f out,cond = s(h f out,cond , P f out,cond ).

(

) 1
where

       h fout,pump is = h (P fout,pump , s f in,pump ),
Qgas = ṁgas c pgas * (T gas in,boiler -T gas out,boiler ), h fout,exp is = h (P fout,exp , s f out,boiler ), h fout,exp ≥ h (x fout,exp min , P fout,exp ).

(2)

In table 1 one can find the simulation model parameters, and the abbreviations are given in the appendix. In addition to that, a routine verifying that the pinch point (P P ) is respected during the evaporation process. Refprop database [START_REF] Eric | Refprop nist standard reference database 23 (version 9.0)), thermophysical properties division, national institute of standards and technology, boulder[END_REF] is used to compute the following quantity: h, s, T , P sat and T sat . Input variables of the model (1) are the gas mass flow and temperature (denoted by ṁgas and T gas in,boiler ) entering in the system and the condensing temperature (T cond ). Outputs of the model are the power produced by the

Model parameters

Variable in [START_REF] Patel | Compounding the truck diesel engine with an organic rankine-cycle system[END_REF]

   N OP = Ẇexp -Ẇpump , Ẇexp = ṁf * (h f in,exp -h fout,exp ), Ẇpump = ṁf * (h f in,pump -h fout,pump ). (3)
The model 1 is not dynamic and does not represent any real components performance. A dynamic 1D model is therefore developed to evaluate the system performance on more realistic dynamic driving conditions.

1D dynamic modeling of a Rankine cycle

Tank

The reservoir is modeled by a fixed volume, which can be either vented to the atmosphere or be hermetic (depending on the condensing pressure) in order to avoid sub atmospheric conditions. Mass and energy conservation equations are:

ṁf out,tank -ṁf in,tank = ∂m f tank ∂t , ṁf in,tank h f in,tank -ṁf out,tank h f out,tank = m f tank ∂h f tank ∂t . (4)

Working fluid pump

The working fluid pump is simply represented by a fixed displacement and isentropic efficiency. The volumetric efficiency is a function of the outlet pressure. This law is identified thanks to experimental data:

ṁfout,pump = ρ f in,pump N pump 60 C cpump η pump vol . (5)
The outlet enthalpy is calculated as shown in the equation for h fout,pump in the 0D model (1).

Heat exchangers: Evaporator(s) and condenser(s)

The models are developed to dynamically predict temperature and enthalpy of transfer and working fluid at the outlet of each heat exchanger (HEX).

When coming to dynamic models of those components, two methodologies can be found in the literature: moving boundary (MB) and finite volume (FV) models. Usually more complex in terms of computational capacity needed due to the high number of system states, the FV approach has the advantage to be more powerful and robust concerning the prediction. Both approaches have been widely used in large power recovery system and control system design [START_REF] Feru | Experimental validation of a dynamic waste heat recovery system model for control purposes[END_REF][START_REF] Horst | Dynamic heat exchanger model for performance prediction and control system design of automotive waste heat recovery systems[END_REF][START_REF] Willatzen | A general dynamic simulation model for evaporators and condensers in refrigeration. part i: moving-boundary formulation of two-phase flows with heat exchange: Modèle général dynamique pour évaporateurs et condenseurs frigorifiques. partic i: Formulation des conditions aux limites variables de flux biphasiques avec échange de chaleur[END_REF][START_REF] Judge | A heat exchanger model for mixtures and pure refrigerant cycle simulations[END_REF] and results in a simplification of the heat recovery boiler/condenser geometry in a great extent (i.e. by representing the boiler by a straight pipe in pipe counterflow heat exchanger). In this study, the FV approach is preferred since it easily handles starting and shut down phases [START_REF] Bendapudi | A comparison of movingboundary and finite-volume formulations for transients in centrifugal chillers[END_REF] when only few papers adressed those cases with a MB approach [START_REF] Bell | A generalized moving-boundary algorithm to predict the heat transfer rate of counterflow heat exchangers for any phase configuration[END_REF].

Model assumptions

Several assumptions are done to simplify the problem in a great extent. These ones are usually admitted when coming to heat exchanger modeling [START_REF] Vaja | Definition of an object oriented library for the dynamic simulation of advanced energy systems: Methodologies, tools and applications to combined ice-orc power plants[END_REF][START_REF] Feru | Experimental validation of a dynamic waste heat recovery system model for control purposes[END_REF]:

• The transfer fluid is always considered in single phase, i.e. no condensation in the EGR/exhaust gases is taken into account.

• The conductive heat fluxes are neglected since the predominant phenomenon is the convection.

• All HEX are represented by a straight pipe in pipe counterflow heat exchanger of length L.

• Fluid properties are considered homogeneous in a volume.

• Pressure dynamics is neglected since it is very fast compared to those of heat exchanger.

Governing equations

Boiler(s) and condenser(s) models are based on mass and energy conservation principles.

• Working fluid (internal pipe):

     A cross f ∂ρ f ∂t + ∂ ṁf ∂z = 0, A cross f ∂ρ f h f ∂t + ∂ ṁf h f ∂z + qconv f int = 0, qconv f int = α f P e exch f (T f -T wall int ). (6)
• Internal pipe wall: An energy balance is expressed at the wall between the working fluid and the gas and is expressed as follows:

Qconv f int + Qconvg int = ρ wall c p wall V wall int ∂T wall int ∂t . (7)
• Gas side (external pipe): The energy conservation is then formulated under the following form:

ρ g A crossg c pg ∂T g ∂t + c pg ṁg ∂T g ∂z + qconvg int + qconvg ext = 0, (8)
where the convection on the external side is used to represent the heat losses to the ambient.

• External pipe wall: As for the internal pipe an energy balance is expressed between the gas and the ambient:

Qconvg ext + Qconv amb ext = ρ wall c p wall V wallext ∂T wallext ∂t . (9)
In equation ( 7) and ( 9) the convection heat flow rate ( Qconv ) is expressed as:

Qconv j k = α j A exch j k (T wall k -T j ), (10
) where j = g, f, amb and k = int, ext.

Furthermore, to complete the system, one need boundary and initial conditions. Time-dependent boundary conditions are used at z = 0 and z = L

(t > 0): ṁf (t, 0) = ṁf 0 (t), ( 11
) h f (t, 0) = h f 0 (t), ( 12
) ṁg (t, L) = ṁg L (t), ( 13
) Ṫg (t, L) = T g L (t). ( 14
)
The initial conditions for the gas and wall temperatures and working fluid enthalpy are given by (z ∈ [0, L]):

h f (0, z) = h f init (z), (15)
T wall int (0, z) = T wall int init (z), ( 16
) T g (0, z) = T g init (z), (17)
T wallext (0, z) = T wallext init (z), (18)

Heat transfer and pressure drop

To model the convection from the transfer fluid to the pipe walls and from the internal pipe to the working fluid, a heat transfer coefficient (α) is needed.

The convection from a boundary to a moving fluid is usually represented by the dimensionless number Nusselt (N u) which is the ratio of convective to conductive heat transfer.

N u(α) = αl λ , (19)
where l represents a characteristic length and is, in this case, the hydraulic diameter. Numerous correlations to approach this number can be found in the literature and are usually derived from experiments, see for example [START_REF] Thome | Wolverine Tube Inc Engineering Data Book III[END_REF].

In single phase, the Gnielinski correlation is chosen for both fluids. In two phase, Chen (for evaporation) and Shah (for condensation) correlations are used. Pressure drop in both fluids are taken into account in order to simulate the real performance of the system. The pressure drop can be split into three main contributors:

∆P = ∆P static + ∆P momentum + ∆P f riction , (20)
where the static pressure drop (∆P static ) is function of the change in static head (i.e. the height), the momentum pressure drop (∆P momentum ) depends on the change on density during phase change and the friction contribution (∆P f riction ) is function of the speed of the fluid and the considered geometry.

Table 2 shows the different correlations used depending on flow conditions.

In laminar single phase, the assumption of a constant heat flux at the wall is made. The fluid flow ṁ through the valve is modeled using a compressible valve equation of the form:

ṁf in,v = C dv S ef fv ρ f in,v P f in,v φ, (21)
where the compressibility coefficient φ is defined as:

φ = 2γ f γ f -1 (ϕ 2 γ f -ϕ γ f +1 γ f ), (22)
with

ϕ =      P f out,v P f in,v if P f out,v P f in,v > 2 γ f +1 γ f γ f -1 2 γ f +1 γ f γ f -1 if P f out,v P f in,v ≤ 2 γ f +1 γ f γ f -1 , ( 23
)
where γ f is the ratio of the specific heats of the working fluid and depends on the temperature and the pressure. Equation [START_REF] Cayer | Parametric study and optimization of a transcritical power cycle using a low temperature source[END_REF] means that the parameter ϕ is either the pressure ratio if the flow is subsonic or the critical pressure ratio when the flow is supersonic.

Expansion machine

Several studies have been carried out in order to choose the correct expansion machine for Rankine based recovery system [START_REF] Bao | A review of working fluid and expander selections for organic rankine cycle[END_REF][START_REF] Badr | Performances of rankine-cycle engines as functions of their expanders' efficiencies[END_REF]. In most of them where vehicle installation is considered, turbine expanders are preferred for their compactness and their good performance [START_REF] Espinosa | Contribution to the study of waste heat recovery systems on commercial truck diesel engines[END_REF][START_REF] Costall | Design methodology for radial turbo expanders in mobile organic rankine cycle applications[END_REF] since the major advantage of volumetric expander such as piston machines is the expansion ratio [START_REF] Latz | Selecting an expansion machine for vehicle waste-heat recovery systems based on the rankine cycle[END_REF].

Though, recent study [START_REF] Kunte | Partial admission impulse turbine for automotive orc application[END_REF] has shown turbine with expansion ratio over 40:1 on a single stage with really good performance at tolerable speed for a vehicle installation. In this study, only a kinetic expander is modeled. The turbine nozzle is represented by the following equation:

ṁf in,exp = K eq ρ f in,exp P f in,exp 1 - P f in,exp P fout,exp -2 . ( 24
)
And the isentropic efficiency is calculated according to the following relation:

η exp is = η exp ismax 2c us c usmax - c us c usmax 2 , (25)
where

c us = u c s = ω exp R exp 2 h f in,exp -h f in,exp is . ( 26
)
Model parameters are fitted using data from supplier and similarity relation [START_REF] Baljé | A study on design criteria and matching of turbomachines: Part a-similarity relations and design criteria of turbines[END_REF].

Other heat exchanger(s)

In order to describe the vehicle cooling system, the number of transfer unit (NTU) approach is used. It is commonly adopted when it comes to single phase heat exchanger modeling. For an air cooled radiator the following relations are used:

Qair = ṁair c p air ε(T coolant in -T air in ). ( 27
)
For a given geometry, ε can be calculated using correlations based on the heat capacity ratio. By considering parallel flow configuration for the radiators, the effectiveness can be written:

ε = 1 -e -N T U 1+ ( ṁcp) min ( ṁcp)max 1 + ( ṁcp) min ( ṁcp)max , (28)
with

N T U = U A ( ṁc p ) min . ( 29
)

Coolant pump and fan

The coolant pump model used is a map-based model function of engine speed and pressure rise. This one is sized to deliver enough subcooling even at high engine load. The engine fan is also a map-based model delivering a given mass flow at a given speed. The fan consumption is calculated according to:

Ẇfan = C f an ρ air N eng G ratio N 2 f an , (30)
where Sr air is the ratio between the vehicle speed and the air speed in front of the cooling package and is either calculated via computational fluid dynamics (CFD) or measured in a wind tunnel.

SYSTEM OPTIMIZATION

Key aspects

In this section, the degrees of freedom used to optimize the WHRS are detailed:

1. Fluid choice: the fluid selection is a critical part of the system optimization. The correct fluid choice has to match both heat source and cold sink in order to generate as much power as possible [START_REF] Maraver | Systematic optimization of subcritical and transcritical organic rankine cycles (orcs) constrained by technical parameters in multiple applications[END_REF][START_REF] Lecompte | Review of organic rankine cycle (orc) architectures for waste heat recovery[END_REF]. From environmental and legal points of view, the working fluid has to respect:

• Its chemical class: chlorofluorocarbons (CFCs) have been banished by the Montreal Protocol and hydrochlorofluorocarbons (HCFCs)

production is planned to be phased out by 2030.

• Its presence on the global automotive declarable substance list (GADSL).

• Its chemical properties such as the global warming potential (GWP), the ozone depletion potential (ODP) or the risk phrases (R-phrases).

• Its classification according the national fire protection agency (NFPA) 704 classification (ranking above 1 in Health or Instability class)

In top of that, the freezing point which has to be below 0 ℃.

Components choice and design: the correct choice of components and

particularly the expansion machine have an important impact on the system performance and the control design. Indeed, a volumetric expander is less stringent in terms of degree of superheat and tolerate a given amount of liquid during the expansion process whereas a kinetic expander requires a higher degree of superheat in order to have a full vapor expansion (liquid droplets can cause blade erosion and broke the machine). The design of all other components of the Rankine system is also critical to maximize its potential. For example, too big heat exchangers show higher performance but also inertia which could be a disadvantage when coming to highly dynamic driving cycle since the more interesting points (i.e. high load engine operating points) are not lasting for long. A heavy evaporator is therefore not catching up the maximum potential of this high heat flow rate.

3. Heat sources and sinks arrangement: the architecture of sources and sinks has to be adapted to increase overall performance. Heat sources choice and arrangement impact a lot the system performance by changing the heat input to the system. The cold sinks choice is influencing the condensing pressure so the overall pressure ratio (and therefore the power generated by the expander).

4. Other system interactions: as the final goal is to implement the system in a heavy duty vehicle, the WHRS must be considered not as a standalone system but as a connected sub system of the complete vehicle.

The interactions of the Rankine system on the other sub-systems have to be taken into account (e.g. increase in fan consumption due to the heat rejection coming from the condenser).

Investigated architectures and components

Several studies have been conducted in the field of waste heat recovery Rankine based systems for mobile applications. A screening of the different heat sources available is reported in [START_REF] Freymann | The turbosteamer: A system introducing the principle of cogeneration in automotive applications[END_REF] and shows that the most promising ones are the EGR and the Exhaust streams. In the present study, only these two heat sources are considered since they present the higher grade of temperatures among other sources. Therefore four different Rankine layout are studied:

1. Exhaust recovery only where the only heat source are the exhaust gases. Concerning the components, as previously said, the study is limited to one expansion machine technology: turbine expander. For the heat exchangers (evaporator(s) and condenser), only counter-current configurations which are usually used in this kind of applications [START_REF] Mavridou | Comparative design study of a diesel exhaust gas heat exchanger for truck applications with conventional and state of the art heat transfer enhancements[END_REF] are considered.

Duty cycles

Driving conditions are acting as input disturbances and therefore, their impact on the target performance must be studied with care.

Steady state evaluation

Under steady state driving conditions, the performance is evaluated by expressing the weighted average net output power of the 1D model (3.3) (the NOP, which is the additional power that the engine receives, therefore corresponds to the fuel economy) on [START_REF] Quoilin | Technoeconomic survey of organic rankine cycle (orc) systems[END_REF]

Dynamic evaluation

In order to accurately assess the potential of the WHRS, dynamic driving cycles are also used to complete the study and check whether the performance found with the previous method is correct. This is really important when coming to thermal systems performance estimation since they generally have long response time [START_REF] Stobart | The controllability of vapour based thermal recovery systems in vehicles[END_REF]. a number from 1 to 7) and weighted according to their real life importance (for a long haul truck highway is predominant).

System performance evaluation

The criterion used for the performance evaluation under steady state and dynamic driving conditions, is the total net reinjected power to the conventional driveline. This is done by taking into account the producer (WHRS expander) and different consumers (cooling fan, WHRS pump and WHRS coolant pump) and assuming them to be mechanically driven (this is not always true for the pumps but efficiencies are detuned to take into account the mechanical to electrical conversion). A complete vehicle model integrating engine, exhaust after treatment system (EATS), transmission, cooling package, WHRS and road environment is used to simulate the total vehicle approach and calculate the power needed to drive the vehicle. The performance criterion (P C) is then calculated as the ratio of this reinjected power to the engine power:

P C i = t f inal t init Ẇexp -Ẇpump -Ẇcool,pump -Ẇfan Ẇeng , (32)
where the engine power ( Ẇeng ) taking into account the mechanical auxiliaries consumption mounted on it and the increase in exhaust backpressure (due to the exhaust evaporator). The vehicle gross weight is assumed constant and equal to 36 tons which intends to represent the average load on a long haul truck. The performance criterion (P C) over the different steady state operating points or driving cycles is then calculated by summing the weighted P C on each points/cycles:

P C = k i=1 w i P C i , (33)
where k ∈ [1 13] for steady state evaluation (presented in section 4.2.1) and k ∈ [1 7] for evaluation on dynamic driving cycle (presented in section 4.2.2).

RESULTS AND DISCUSSION

1D model validation

In this section some component models, judged as critical for the overall system performance evaluation, are first validated thanks to supplier or experimental data. The different studied configurations being made of the same components model it has been decided to validate the models components by components. The validation is done by comparing experimental to the modeling results. A model is further considered as valid if the average modeling error is below 5% of the predicted quantity. Since the main dynamic of the system is contained in the evaporators [START_REF] Stobart | The controllability of vapour based thermal recovery systems in vehicles[END_REF] and the final aim is to predict the power generated by the system only validation figures are presented for the evaporators and the expansion machine. It should be said that a more detailed validation on the whole system mounted on the vehicle should be carried out but this requires the system to be built. Unfortunately this technology is still under investigation at the truck makers level and no figures

are available yet. This study intends then to compare the architecture and analyze their impact on the truck fuel consumption.

Heat exchangers

A high attention is paid to the evaporators in order to accurately predict the steady state and dynamic performance of those components (corresponding to the model presented in section 3.3.3). In this paper, a finite volume approach has been chosen to implement the continuous set of equations (equations 6, 7, 8, 9). Figure 9 shows the schematic of the discretized model.

Table 5 and 6 show respectively steady state and dynamic prediction errors.

Note that in both cases the relative error is computed according to the maximum temperature difference between the heat exchanger bounds (usually

T g L -T f 0 ).
The steady state validation is conducted on a lot of operating conditions supposed to represent the complete range of operation for those components. The dynamic behavior is evaluated on different load point variations but obviously need further validation especially on fast change that can

T wext 0 T g 0 T w int 0 h f 0 h f 1 T w int 1 T g 1 T wext 1 T wext 2 T g 2 T w int 2 h f 2 T g L T w int L h f L T wext L T g i-1
T w int i-1

h f i-1 T wext i-1 T g i T w int i h f i T wext i T g n-1 T w int n-1 h f n-1 T wext n-1 ṁf 0 ṁg L ∆z L
Figure 9: HEX model schematic take place on real driving conditions. However, the mean relative modeling error remains lower than 3.5%, which is considered acceptable.

T f L EGRB T f L

Expansion machine

The turbine expander model presented in 3.3.5 is fitted thanks to supplier data. Figure 10

Model analysis

The whole 1D models are built from the same component models. The inertial effect of the pipes are neglected since their effect are negligible compared to the other components dynamics (namely the evaporators) [START_REF] Stobart | The controllability of vapour based thermal recovery systems in vehicles[END_REF]. The full model is then a combination of validated detailed models (e.g. heat exchangers) and quasi-static models (pumps expansion machine and fan) used for comparison purpose. This study then intends to compare the different heat sources and sinks configurations possible on a heavy duty vehicle to select the best system in terms of performance.

Optimization of the WHRS

Working fluid selection

From an exhaustive fluid list [START_REF] Eric | Refprop nist standard reference database 23 (version 9.0)), thermophysical properties division, national institute of standards and technology, boulder[END_REF], all those that do not respect the different criteria mentioned in 4.1 have been removed. However, as water is a good reference fluid since it is generally used in power plant [START_REF] Fischer | Comparison of trilateral cycles and organic rankine cycles[END_REF], it has been kept.

The results presented hereafter are coming from the ideal thermodynamic 0D model presented in section 3.2 where all 13 operating points are simulated for two condensing temperatures 60℃ and 90℃, which intends to represent the two cooling configurations presented in the previous section. The parameters of the cycle, P fout,pump and ṁf are optimized to reach the highest performance (i.e. maximize the N OP ). Here, each hot stream is simulated separately in order to see the impact of the heat source on the Rankine fluid selection. The simulation matrix contains 13 operating points (listed in section 5.2.2) and 13 selected working fluids. For the sake of simplicity, the results presented in figure 12 show the number of occurrences where the fluid is in the top five1 regarding the N OP . When analyzing each operating point and configuration separately among the 13x13 simulations, water is the best fluid for heavily loaded operating points. For low and medium engine load, as gases temperatures are lower and due to the large enthalpy of vaporization of water and the high level of superheating required, it is not recommended to use it.

Acetone and ethanol show good performance at mid and high engine load no matter of the cold sink temperature. Refrigerants such as R1233zd or Novec 649 show good results for heat source temperature under 280 ℃ for the lowest condensing temperature. More exotic fluids such as cis-butene or MM (silicon oil) could be attractive for low and medium engine load respectively at 60℃ condensing temperature for the first one and 90℃ for the second one. ethanol is then considered suitable for a mobile application.

Steady state performance analysis

Now, the performance criterion is analyzed on the 13 operating points and the 2 cooling architectures (Cooling config 1 and 2) for the previously chosen working fluids. The savings are computed thanks to the weight factors presented in table 3. Figure 13 presents the NOP to engine power ratio evaluated for the 2 cooling configurations. It can be observed that the decrease due to higher condensing temperatures induced by cooling configuration 2 is somehow constant (between 11 and 15 %) no matter of the Rankine cycle arrangement. This drop in performance is due to the increase in condensing pressure which affects the overall pressure ratio through the expansion machine and therefore its performance. This could be partially balanced by a specific design of the expansion machine in order to have a variable nozzle geometry that keeps the pressure ratio constant when the condensing pressure increases. A similar approach is done in [START_REF] Kunte | Partial admission impulse turbine for automotive orc application[END_REF] to adapt the nozzle geometry to the mass flow entering in the turbine. With those components, the parallel arrangement of the heat sources gives the best P C, followed by the serial one, the exhaust only and the EGR recovery. However the difference between series and parallel layout is not so important and the lower number of valves needed by the first one could compensate this drop in performance.

Moreover, in this configuration, as the working fluid mass flow is controlled to get a superheated vapor state at the outlet of the tailpipe boiler the mass flow rate is then higher than in any other configurations. It results into lower EGR temperature which could be a benefit in terms of engine performance and pollutant emission control [START_REF] Willems | Integrated powertrain control for optimal CO2-NOx tradeoff in an euro-vi diesel engine with waste heat recovery system[END_REF]. Last but not least, with the EGR only solution even if the weight and installation impact is low (the heaviest component is the EGR evaporator that replaces the traditional EGR cooler), the P C seems too low for a vehicle installation. This obviously needs further analysis taking into account also the cost impact of each solution on the total cost of ownership. spectively for cooling configuration 1 and 2. As shown is 5.2.2 the decrease in performance using cooling configuration 2 rather than cooling configuration 1 is more or less constant and around 11%. The main information brought by this study remains the lower fuel savings when simulating the system in dynamic instead of steady state, which can be as big as 50% for the systems using exhaust as heat source. This is due to two main reasons:

• the exhaust after treatment system, which has a very important constant time, causes big temperature drop during fast highly loaded engine conditions where a lot of heat is supposed to be available.

• the non optimal design of the tailpipe boiler used in the simulation model. Indeed the validation of the model shown in section 5.1 is based on prototypes components that do not represent the optimum in terms of size and transient performance.

• the constraint implemented on the EGR temperature at the evaporator outlet not to derate the emission control. The maximum EGR temperature is set to 150 ℃ which on some phases is not going hand in hand with the superheat level control. The EGR temperature becomes the control objective and when the superheat is not sufficient the expander is not fed with working fluid and therefore the power production is null.

Table 7 resumes the time (relative to the duty cylce time) where the superheat is sufficient to feed the expansion vapor. Systems recovering heat from the exhaust stream mainly suffer from a long start-up phase but then the system never lose the superheat level needed to expand the working fluid in the kinetic turbine. This long start-up is due to the boundary conditions used where all the sub-systems initial temperatures are set to ambient. For the system recovering heat only from the EGR the start-up is not significant since on some cycles the system is generating power more than 99% of the time (the EGR gets its normal operation temperature after few seconds whereas the thermal inertia of the EATS makes the temperature rise very slow).

Nevertheless on highly loaded cycles (namely 3 and 7) the high engine load results into high EGR temperature and to not interact too much with the engine emissions system, the superheat is dropped to the detriment of the EGR temperature. Superheated vapor is no longer generated by the boiler and the fluid goes back to a diphasic state and the expansion machine is bypassed.

Anyway, similarly to the previous results in steady state, the best system in terms of fuel savings remains the EGR and exhaust in parallel with cooling configuration 1 that brings 2.2% savings on the overall weighted driving cycle.

In addition to that, it can be seen that the relative performance are kept from arrangement to arrangement (compared to section 5.2.2).

over dynamic driving cycles h h h h h h h h h h h h h h h h h h

Driving cycle

Configuration

Optimal WHRS

The low performance figures presented in the previous sections are mainly due to non optimized components for the considered application. In order to evaluate what could be the economy brought by an optimized system, the different components constituting the WHRS are redesigned to perfectly match the targeted application. In addition to that, a perfect insulation of these different components is then considered. In this section, only cooling configuration 1 is evaluated since it has been shown that it leads to larger savings.

Both approaches previously used (steady state and dynamic analysis) are presented in figure 17. Optimization has been done on boilers and condenser size with respect with the additional weight. Pump and expansion machine performance are increased to reach standards in power plant Rankine cycles (η pump is = 70% and η exp is,max = 78%). More acceptable results are reached for a vehicle implementation of such a system, especially when considering a system recovering from both EGR and exhaust in parallel. Again, a big step is observed between the two evaluation methodologies which tends to prove that the cycle division into a certain number of steady state engine operating points is not adapted for performance evaluation of thermal systems which generally have a long response time. representing what could be a mass-produced system. However, the absolute numbers should not be interpreted as the maximal potential for WHRS in HD trucks, since transient control of the system and components are not optimal. Different systems layout have been analyzed to maximize the system performance over a broad variety of driving cycles. However the results presented in this paper need to be treated carefully and further completed with the cost and the packaging effort for each configurations. An optimized scenario is also presented where a specific attention has been paid to the components size and performance in order to perfectly match the application. However fuel savings are rather low compared to what can be found in the literature [START_REF] Edwards | Waste heat recovery: The next challenge for commercial vehicle thermomanagement[END_REF][START_REF] Boretti | Recovery of exhaust and coolant heat with R245fa organic rankine cycles in a hybrid passenger car with a naturally aspirated gasoline engine[END_REF] and need to be further validated by experimental results on a system mounted onto a vehicle. In addition to that, control issues are not approached in this paper but remain a big part of the system performance maximization. In this study, perfect sensors and actuators are used, which reduce the control effort. Moreover, actual mass-produced control units are not as powerful as current laptop central processing unit and reduce considerably the possibility in terms of advanced control algorithm development. Recent studies have brought significant advances in this field [START_REF] Luong | Linear quadratic integral control of an organic rankine cycle for waste heat recovery in heavy-duty diesel powertrain[END_REF][START_REF] Peralez | Improving the control performance of an organic rankine cycle system for waste heat recovery from a heavy-duty diesel engine using a model-based approach[END_REF][START_REF] Quoilin | Dynamic modeling and optimal control strategy of waste heat recovery organic rankine cycles[END_REF] but this still needs to be addressed when vehicle implementation is touched.

Figure 1 :

 1 Figure 1: Simple waste heat recovery Rankine based system

Figure 2 :

 2 Figure 2: Temperature-Entropy diagram of the Rankine cycle

  where the coefficients C f an and G ratio are dependent on the fan model and vehicle. The mass flow rate blown by the fan is mapped according to data from supplier and depends on the fan speed and atmospheric conditions. The air mass flow rate going through the cooling package ( ṁair ) is a combination of the natural air mass flow rate (corresponding to a fraction of the vehicle speed) and the forced mass flow rate (corresponding to the mass flow blown by the fan). ṁair = ρ air A cool pack Sr air V vehicle + ṁfan (N f an , ρ air ),

Figure 3 :

 3 Figure 3: Exhaust only system schematic

Figure 4 :

 4 Figure 4: EGR only system schematic

Figure 5 :

 5 Figure 5: Exhaust and EGR in parallel system schematic

Figure 6 :

 6 Figure 6: Exhaust and EGR in series system schematic

Figure 7 :config 1 •

 71 Figure 7: Cooling config 1

Figure 8 :

 8 Figure 8: Cooling config 2

  and 11 respectively show the working fluid inlet pressure and the generated power predicted by the model versus the normalized working fluid mass flow entering in the turbine. Those two quantities are well fitted and this model is further considered validated.

Figure 10 :

 10 Figure 10: Turbine pressure model validation

Figure 11 :

 11 Figure 11: Turbine power model validation

Figure 12 :

 12 Figure 12: Number of occurrences of each fluid in top five 1 for different boundary conditions

Figure 13 :

 13 Figure 13: Steady state P C assessment

Figure 14 :

 14 Figure 14: Road profiles examples

Figures 15 and 16

 16 Figures 15 and 16 show the P C reached by each Rankine configuration re-

Figure 15 :

 15 Figure 15: P C for cooling configuration 1 over dynamic driving cycles

Figure 16 :

 16 Figure 16: P C for cooling configuration 2 over dynamic driving cycles

Figure 17 :

 17 Figure 17: P C for optimal sizing of the components

Table 2 :

 2 Correlations used in HEX

	Laminar Turbulent

3.3.4. Valve(s)

Table 3 :

 3

Steady state evaluation: Driving conditions and weight for 13 engine operating points in table

3

) These operating points are chosen to represent a classical long haul driving cycle and weighted according to the percentage of energy used on each operating point. Operating point number 5 is identified as the designing point whereas the operating points 3 and 11 are considered critical due to the high engine load and the low vehicle speed.

Table 4 :

 4 Dynamic evaluation: Driving conditions and weight for the 7 phases

	The driving cycle used is split into 7 phases (sum-
	marized in table 4) supposed to represent all conditions of a long haul truck
	usage. In the previous 1D model (3.3), each phase is considered, in the rest
	of the study, as a driving cycle of approximately the same length (denoted by

Table 5 :

 5 Evaporators steady state validation

		T f L EGRB	T f L ExhB	T egr 0 EGRB	T exh 0 ExhB
	Error	max mean max mean max mean max mean
	Absolute (K) 4.5	1.5	25.9	2.3	7.9	2.8	20	4.2
	Relative (%) 1.38 0.46 14.37 1.28 2.43 0.86 11.1 2.33

Table 6 :

 6 Evaporators dynamic validation

Table 7 :

 7 Vapor creation time ratios summary

top five means the N OP related to the fluid is ranked in one of the five first position