F Fossati

D Medhi
email: dmedhi@umkc.edu

S Moretti

Abstract-In resource allocation systems, inaccurate information sharing situations are such that users can be aware, up to a small error, about the other users' demands and the available global resource (which can be insufficient to meet the overall demand). Consequently, given an allocation rule, users can predict an allocation that will not necessarily coincide with the actual one. In this work, we provide an estimation of the error for a number of allocation rules and compare their robustness in inaccurate information sharing settings.

I. BACKGROUND

Resource allocation protocols in communication networks and computing systems are commonly used to ensure fairness and efficiency when dividing resources among different agents or users. A challenging situation is the choice of the allocation rule when the amount of available resource is limited and not sufficiently high to fully satisfy the entire demand.

Classically, the network setting is such that users have little information about the available resources and demands of other users. Nonetheless, with the emergence of new networking features such as 5G infrastructure sharing and programmability in SDN, and for auditability requirements (i.e., to ensure tenants fair sharing), network setting is evolving toward a complete information sharing situation so that all users can be aware of the demands of the other users and of the available resources for resource allocation systems [START_REF] Fossati | A Mood Value for Fair Resource Allocations[END_REF], [START_REF] Fossati | Fair Resource Allocation in Systems With Complete Information Sharing[END_REF].

As an intermediate context between the classical no information sharing and the complete information sharing, we consider the scenario in which the information is known to have inaccuracy. In this work, we study the behavior of resource allocation rules under this scenario that we call inaccurate information sharing context, where the amount of available resource is known up to a constant (Fig. 1); we also highlight the impact of inaccurate information sharing on the demand of the other users. Indeed, in certain practical situations, such as in radio resource availability or in systems over/under provisioned by the infrastructure provider, it is likely to suffer from inaccurate information on the available resources to be shared. Furthermore, complete sharing may not be possible since this may require a lot of exchanges of updates causing a large overhead. While the problem of Fig. 1: Information sharing contexts in resource allocation. inaccurate information in networks has been studied before [START_REF] Guerin | QoS routing in networks with inaccurate information: theory and algorithms[END_REF], a formal treatment on the error estimate and fairness has not been studied for different allocations schemes.

Formally, a resource allocation problem is characterized by a pair (c, E), where c is the demand vector for users in N = {1, ..., n}, with n > 1, and E is the resource to split among them. Resource allocation is challenging when E is less than the global demand of users (

n i=1 c i ≥ E).
An allocation x ∈ R N is a n-dimensional vector where x i represents the quantity of resource E assigned to each user i ∈ N , and an allocation rule is a function that associates a unique allocation x to each pair (c, E). In this work, we consider three allocation rules discussed below.

Weighted proportional allocation rule: this is the result of the maximization of n i=1 w i log x i subject to demand boundedness (x i ≤ c i , ∀i ∈ N ) and efficiency constraints ( [START_REF] Kelly | Rate control for communication networks: shadow prices, proportional fairness and stability[END_REF]. The weights w i can be chosen equal to 1 to get the proportional allocation rule or equal to c i to obtain the allocation that assigns the same proportion of demand to all users. We only consider the second case with w i = c i , that is the allocation that maximizes the Jain's index of fairness [START_REF] Jain | A quantitative measure of fairness and discrimination for resource allocation in shared computer system[END_REF].

n i=1 x i = E)
Max-Min Fair (MMF) allocation rule: This is an egalitarian solution that privileges users with small demands [START_REF] Bertsekas | Data networks[END_REF]. After ordering the users according to their increasing demand, i.e., c 1 ≤ c 2 ≤ • • • ≤ c n , the MMF allocation for user i is given by: M M F i (c, E) = min c i ,

E-i-1 j=1 M M Fj (c,E) n-i+1
.

Mood value: It revisits user satisfaction taking into account the awareness of other users' demands and the amount of available resources [START_REF] Fossati | A Mood Value for Fair Resource Allocations[END_REF], [START_REF] Fossati | Fair Resource Allocation in Systems With Complete Information Sharing[END_REF]. Each user i ∈ N is assigned min i + m(max i -min i ) as its resource share, where min i is what remains if all the other users in N \ {i} are fully satisfied (min i = max{0, E -j∈N \{i} c j }), max i is the maximum user i can get, i.e., its own demand or the available resource if the demand overcomes it, and m = (E -

n i=1 min i )/( n i=1 max i - n i=1 min i ) takes value in [0, 1].

II. ERROR ASSESSMENT

We are interested in evaluating the error on the allocation when users are in a inaccurate information context (Fig. 1). We treat first the case where there is error on the available resource. For each user i ∈ N the error ERR i is defined as |x i -x i |, where x i is the share obtained by i when the resource is E and xi is the allocation for user i when resource is E ±ε.

In the following, we calculate the value of the error, considering the allocations described in the previous section.

Weighted proportional allocation: This allocation coincides with the allocation that assigns the resource proportionally to the demand, i.e., x p i = c i E/ n i=1 c i when the resource is E.

If users believe that the available resource is E ± ε then the allocation is

xp i = c i (E ± ε)/ n i=1
c i , which implies that the error on the allocation for each user is:

ERR i = ± c i n i=1 c i ε (1)
The error ε is divided between the users proportionally to their demands.

MMF allocation: We consider the hypothesis that ε is small enough not to change the nature of the user. This means that if c i < E n , it holds also that c i < E-ε n . It follows that the users with small demands receive the same amount of resource (i.e., their demand), while the excess ε is equally divided between the users that receive less than their demand. When the first k users receive their demands, the error is:

ERR i = 0, if i = 1, .., k ±ε n-k , if i = k + 1, ..., n. (2)
Mood value: For the mood value allocation, we need to consider that we have four types of users when we take into account the minimum and the maximum value they can get. Table I shows that they are categorized checking (i) what remains if the other users received what they demand and (ii) the maximum quantity of resource they can get, i.e., c i if they ask less than the available resource, E otherwise. In [START_REF] Fossati | A Mood Value for Fair Resource Allocations[END_REF], [START_REF] Fossati | Fair Resource Allocation in Systems With Complete Information Sharing[END_REF] it was shown that there exists just six types of scenarios combining the different type of users. The six combinations of users are:

• GM: All the players are in scenario GM.

• GG: All the players are in scenario GG.

• MM: All the players are in scenario MM.

• GM-GG: Some players of type GM, some of type GG.

• GM-MM: Some players of type GM, some of type MM.

• GM-MG: One player of type MG, the others of type GM. We again consider the hypothesis that ε is small enough not to change the nature of the user. This means that, e.g., if for a user i ∈ N it also holds c i ≥ E + ε, if j =i c j < E it holds also j =i c j < E -ε, and so on. 1) Case GM: This case coincides with the weighted proportional allocation. For each user i the error is given by (1).

c i ≥ E max i = c i max i = E (c i < E) (c i ≥ E) min i = 0 ( j =i c j > E) GM GG min i = 0 ( j =i c j < E) MM MG
2) Case GG: In this case, if the resource is E it holds that min i = 0, max i = E for each user i. The value of the mood is m = E-0 nE-0 = 1 n , and the mood value is x m i = E n . If the value of the available resource is E ± ε the value of the mood m is again equal to m = 1 n and the mood value is

xm i = E±ε n .
It follows that for each user i the error is equal to: ERR i = ±ε/n

(3). In this case, the error is divided equally between the users without considering the value of their demands.

3) Case MM: In this case, if the resource is E it holds that min i = 0, max i = c i for each user i. The value of the

mood is m = E-n(E)+(n-1) n i=1 ci n i=1 ci-n(E)+(n-1) n i=1 ci = n-1
n , and the mood value is

x m i = E - j =i c j + n-1 n ( n i=1 c i -E). If the value
of the available resource is E ± ε, the value of the mood m is given by: m =

E±ε-n(E±ε)+(n-1) n i=1 ci n i=1 ci-n(E±ε)+(n-1) n i=1 ci = n-1 n and the mood value is xm i = E ± ε - j =i c j + n-1 n ( n i=1 c i -E ∓ ε).
It follows that for each user i the error is:

ERR i = ±ε/n (4)
The error is equally divided between the users also here. 4) Case GM-GG: Let N = N 1 ∪ N 2 be partitioned into two disjoint sets N 1 and N 2 representing the set of user of type GM and GG, respectively. When the resource is E, the value of the mood is m = E i∈N 1 ci+n2E , and the mood value is

x m i = E i∈N 1 ci+n2E c i if i ∈ N 1 and x m i = E 2 i∈N 1 ci+n2E if i ∈ N 2 .
If the value of the available resource is E ± ε, only the maximum value for the user GG is changing. The value of the mood is m = E±ε i∈N 1 ci+n2E±n2ε , and the mood value is

xm i = E±ε i∈N 1 ci+n2E±n2ε c i if i ∈ N 1 and xm i = (E±ε) 2 i∈N 1 ci+n2E±n2ε if i ∈ N 2 .
Called D the denominator of m, the error is :

ERR i =          ± i∈N 1 ciεci ( i∈N 1 ci+n2E)( i∈N 1 ci+n2E±n2ε) , i ∈ N 1 ε±2E D±n2ε ∓ E 2 n2 D(D±n2ε) ε, i ∈ N 2 (5)
5) Case GM-MM: Let N = N 1 ∪ N 2 be partitioned into two disjoint sets N 1 and N 2 representing the set of user of type GM and MM, respectively. When the resource is E, the value of the mood is m =

(n2-1)( i∈N 2 ci-E)+n2( i∈N 1 ci) n2( i∈N 2 ci-E)+(n2+1)( i∈N 1
ci) , and the mood value is

x m i = mc i if i ∈ N 1 and x m i = E - j =i c j + m( i∈N c i -E) if i ∈ N 2 .
When the available resource is E ± ε the mood and the mood value are, respectively: m =

(n2-1)( i∈N 2 ci-E∓ε)+n2( i∈N 1 ci) n2( i∈N 2 ci-E∓ε)+(n2+1)( i∈N 1 ci) , xm i = mc i if i ∈ N 1 , xm i = E ± ε - j =i c + m( i∈N c i -E ∓ ε) if i ∈ N 2 .
Called A the denominator of m and B the one of m, the error is:

ERR i =          ± i∈N1 c i εc i /(AB), i ∈ N 1 ± n i=1 ci-E∓ε B + ( n i=1 ci-E) i∈N 1 ci AB ε, i ∈ N 2 (6
) 6) Case GM-MG: Let N = N 1 ∪N 2 be partitioned into two disjoint sets N 1 and N 2 representing the set of user of type GM and the only one MG user, respectively. When the resource is

E, the value of the mood is m = E-E+ i∈N 1 ci i∈N 1 ci+E-E+ i∈N 1 ci = 1 2 ,
and the mood value is

x m i = 1 2 c i if i ∈ N 1 and x m i = E - i∈N1 c i + 1 2 (E -E + i∈N1 c i ) = E -1 2 ( i∈N1 c i ) if i ∈ N 2 .
If the value of the available resource is E ± ε, due to the hypothesis that we consider, only the minimum value for the user MG is changing. The value of the mood m is again equal to 1 2 and the mood value

is xi = 1 2 c i if i ∈ N 1 and xi = E ± ε -1 2 ( i∈N1 c i ) if i ∈ N 2 .
It follows:

ERR i = 0, i ∈ N 1 ±ε, i ∈ N 2 . ( 7
)
Concerning the boundness of the error in case of the three allocation policy we can state the following proposition.

Proposition 1. If the allocation rule is proportional, MMF or mood value, the error to the users is less than or equal to ε.

Proof. The error boundness in case of proportional and MMF allocation is easily proof from the error formulas (1), [START_REF] Fossati | Fair Resource Allocation in Systems With Complete Information Sharing[END_REF]. The mood value corresponds to the τ -value solution of bankruptcy games as proved in [START_REF] Fossati | Fair Resource Allocation in Systems With Complete Information Sharing[END_REF] and satisfies the monotonicity property as proved in [START_REF] Curiel | Bankruptcy games[END_REF]. We show that |x i -x i | ≤ ε. When the resource is E + ε due to the monotonicity it holds:

x m i (E, c) ≤ x m i (E + ε, c), ∀i ∈ N (8
) and due to the efficiency it holds:

n i=1 x m i (E, c) = E, n i=1 x m i (E + ε, c) = E + ε (9)
From ( 8) and [START_REF] Ghodsi | Dominant Resource Fairness: Fair Allocation of Multiple Resource Types[END_REF] follows that

x m i (E + ε, c) -x m i (E, c) ≤ ǫ, ∀i ∈ N .
In similar way when the resource is E -ε due to the monotonicity it holds:

x m i (E -ε, c) ≤ x m i (E, c), ∀i ∈ N (10
) and due to the efficiency it holds:

n i=1 x m i (E, c) = E, n i=1 x m i (E -ε, c) = E -ε (11)
Given ( 10), (11) then

x m i (E-ε, c)-x m i (E, c) ≤ ǫ, ∀i ∈ N .

III. ALLOCATED ERROR AND FAIRNESS CONSIDERATIONS

When each user has the same misknowledge of the available resource (i.e., the same ε) equations ( 1)-( 7) explain how the error ε is distributed among them. In this section, we are interested in the analysis of the fairness in the error sharing for the allocation rules presented in section II.

As already noticed, the error, for each allocation, is bounded by ε, i.e., the error is split between the users without anyone being severely disadvantaged. Furthermore, considering the fairness policy behind each error allocation, we can notice that it is close to the one of the resource allocation. In fact:

• the weighted proportional allocation rule splits the error proportionally to the users demands; • the MMF allocation protects weak users, i.e., users with a smaller demand compared to the other users, not allocating them the error. No differences exist between the other users, receiving the same proportion of the error; • the mood value takes into account the nature of each user and of the others. In particular the mood value in the GM case allocates the error as the proportional rule does; if the users are all of type GG, or all of type MM, it does not make difference between the user and that is a good property due to the fact that they have close demands; in the case of mixed users, it assigns the error considering the group to which a user belongs.

We now look at the variation of the user satisfaction between the two scenarios with and without misknowledge on the available resource value using the three different allocation rules 1 . The user satisfaction, when users can collect information about other users' demands and the available resource, has to be measured as S i = xi-mini maxi-mini where min i and max i are equal to are the smallest and the biggest possible allocation for the user i [1], [START_REF] Fossati | Fair Resource Allocation in Systems With Complete Information Sharing[END_REF]. We can state the following: Theorem 1. If each user has a full knowledge of the other user demands and the same misknowledge on the available resource (E ± ε instead of E), the mood value is the only scheme that:

1) equalizes the satisfaction of the users, 2) equalizes the error on the user satisfaction.

Proof. The proof of the first part is in [START_REF] Fossati | A Mood Value for Fair Resource Allocations[END_REF], [START_REF] Fossati | Fair Resource Allocation in Systems With Complete Information Sharing[END_REF]. Due to the fact that the value of the satisfaction for all the user is the same for both the case in which the resource is E and E ± ε, the error on the satisfaction, i.e., the difference between the satisfaction in case without and with the misknowledge on the available resource, is the same.

Tables II andIII show two counterexamples where proportional allocation and MMF allocation do not allocate the same satisfaction and the same error on the user satisfaction.

Figure 2 shows the variation of the satisfaction for the 6 possible cases. We can notice, as the theorem states, that the value of the satisfaction is the same for each user and for each value of E because it coincides with the mood m and 1 In this analysis we consider the fairness concept linked to the users satisfaction but other fairness properties can be analyzed, such as the envyfreeness or other generalized measure of fairness, not strictly linked to the concept of satisfaction [START_REF] Lan | An axiomatic theory of fairness in network resource allocation[END_REF] m. Furthermore, in Figures 2a,2e, 2d, we clearly see that the gap between the satisfaction value when the resource is E (i.e., m) and when the resource is Ê (i.e., m) is the same for each user. In addition to the two properties stated in the theorem, we can see that in Figures 2b,2c, 2f the value of the satisfaction does not increase or decrease when we consider the error on the available resource. In this case the satisfaction of the users, called also mood, depends only by the number of users and not by the value of the demands. That are in fact situations (i) in which each single user has the same nature of the coalition of the other ones, or (ii) in which there is only one greedy user. As already explained, in the first case the the error is split uniformly between the users and in second one the greedy users keeps all the error. Another interesting fact is that the slope of the satisfaction line for users with smaller demands is not smaller than the one of users with bigger demands. This imply that for these users the allocated error cannot be bigger than the other ones. We now look at a global measure of fairness called Jain index [START_REF] Jain | A quantitative measure of fairness and discrimination for resource allocation in shared computer system[END_REF]. The fairness is maximized using the mood value allocation, when we define the index as follows [START_REF] Fossati | A Mood Value for Fair Resource Allocations[END_REF], [START_REF] Fossati | Fair Resource Allocation in Systems With Complete Information Sharing[END_REF]:

x M M F i S xMMF i Ŝ ∆S ( Ŝ -S)
J S = n i=1 S i 2 n n i=1 S i 2 (12)
where S i = xi-mini maxi-mini . We are then interested into evaluate the global fairness on the error. In particular we can re-define Jain index as follows:

J ∆S = n i=1 ∆S i 2 n n i=1 ∆S i 2 (13)
where ∆S i is the difference of the satisfaction calculated when the resource is E and when the resource is E ± ε. The redefinition of the index is necessary to evaluate the fairness on the satisfaction error. We can derive the following theorem.

Theorem 2. J S and J ∆S are maximized when the resource allocation is based on the mood value.

Proof. The maximization of J S happens when each user receives the same satisfaction S i , and the maximization of J ∆S happens when each user receives the same ∆S i [1], [START_REF] Fossati | Fair Resource Allocation in Systems With Complete Information Sharing[END_REF]. From Theorem 1, it follows that the mood value is the allocation that maximizes the two indices of fairness.

We now test the behavior of the three considered allocation schemes in term of J ∆S by simulating 100 resource allocation problems with random demands belonging to [START_REF] Fossati | A Mood Value for Fair Resource Allocations[END_REF][START_REF] Fossati | Multi-Resource Allocation for Network Slicing[END_REF] while varying the value of E. We set the error equal to 10 -2 . Fig. 3 shows the boxplot of J ∆S when E is the 20%, 50% and 80% of the sum of users demands. We note that the mood value maximizes the fairness on the satisfaction error. In fact, the index takes value in [0, 1], and the higher is its value, the higher is the fairness. The proportional and the MMF allocation can produce inequality between users, presenting median values different from one and high variability.

In summary, when users are in the analyzed inaccurate information scenario, they prefer the mood value allocation because:

• it equalizes the user satisfaction and the satisfaction error and it maximizes the Jain index of fairness on the allocation and on the error; • it takes into account the user nature in error splitting;

• in some cases, it allocates a portion of resource that provides exactly the expected satisfaction.

IV. OTHER CASES OF INACCURATE INFORMATION

SHARING

Additionally, we consider (i) the case in which each user has misknowledge on the available resource but the error is weighted proportional

∓ (n-1)c i δE ( n i=1 c i ±(n-1)δ)( n i=1 c i ) MMF 0, = 1, .., k ∓kδ n-k , i = k + 1, ..., n Mood value GM ∓ (n-1)c i δE ( n i=1 c i ±(n-1)δ)( n i=1 c i ) GG 0 MM ∓ n-1 n δ
TABLE IV: Evaluation errors for errors on users demands.

not equal among the users and (ii) the case in which users has misknowledge of the other users' demands.

When we have misknowledge on the available resources, interestingly (1)-( 7) still provide the evaluation of the error for an user i, but clearly it depends on ε i . We can notice that, for each allocation and for each group of user, again the error depends linearly on the value of the error, but compared to the case analyzed in Section II, in which the error ε is shared between the users so that the sum of the users error is equal to ε, here it does not happen. The coefficient of dependency varies between the users, taking into account the nature of the user, i.e., the absolute value of the demand and the demand compared to the other users. Because of the error of each user depends on different variables, we can not compare in general the allocations errors from ( 1)-( 7) we can see that the error is always limited by ε i , so that each of the allocation considered does not strongly advantages/disadvantages an user.

In contrast, for scenario (ii) the error is not always limited by ε and most of the time increases with the number of users. Table IV shows the value of the error in case (ii) when the error for each user is the same and equal to δ (for the mood value we do not report mixed scenarios GM-GG, GM-MM, GM-MG). For example, in the MMF case, the error can be greater than δ but bounded by kδ. Furthermore, we can notice one more time that the mood value assigns an error that depends on the nature of the problem: it assigns the same error to users belonging to cases GG and MM, while it differentiates users belonging the group GM.

Looking at the satisfaction S, we can state the following theorem.

Theorem 3. If each user has a full knowledge of the available resource and the same misknowledge on the other users demand, the mood value is the only scheme that equalizes the error on the satisfaction for the same type of user.

Proof. We calculate the value of the ∆S in each case for each type of user. We report the evaluation in Table V, where we can see that for users of same type the error on the satisfaction, i.e., ∆S ( Ŝ -S) is the same.

From Table V we can also see that in some resource allocation problem types (e.g. GG, MM,GM-MG), as it was happening in case of misknowledge on the available resource, the users receive an allocation that satisfies them at the same level of the case of complete information.

V. CONCLUSION AND OPEN QUESTIONS

In this work, we analyze resource allocation with inaccurate information sharing. We present three theorems that show that the mood value allocation is superior to the proportional and MMF allocation in terms of fairness.

A number of questions and research paths remain open for further work. Consider the analysis of the scenario in which the error is on both the resource and users' demand and the multi-resource scenario, i.e., when the resources to share are more than one and there exists a dependency between resources. In this case allocation rules, as the DRF [START_REF] Ghodsi | Dominant Resource Fairness: Fair Allocation of Multiple Resource Types[END_REF], are obtained as results of an optimization problem [START_REF] Fossati | Multi-Resource Allocation for Network Slicing[END_REF]. Differently from the case single-resource in which there are rules with a direct formula to calculate the allocation, the estimation of the error becomes complex and will be studied separately.

  (a) GM, c = (5, 6, 9), = 10 (b) GG, c = (12, 13), E = 10 (c) MM, c = (2, 3, 4), E = 8 (d) GM-GG, c = (6, 8, 12), E = 10 (e) GM-MM, c = (2, 6, 7) , E = 10 (f) GM-MG, c = (2, 6, 12), E = 10

Fig. 2 :

 2 Fig. 2: Users satisfaction with and without misknowledge on the available resource -mood value case.

  x i and xi are the allocation when E = 10 and E = 10 + ε, S and Ŝ are the satisfaction when the allocations are x i and xi , ε = 10 -3 , c = (2, 6, 7).

Fig. 3 :

 3 Fig. 3: J ∆S for three congestion levels as a function of E.

  Error Estimate and Fairness in Resource Allocation with Inaccurate Information Sharing

Francesca Fossati, Deep Medhi, Fellow, IEEE, Stefano Moretti, Stefano Secci, Senior, IEEE

  can be used.

		x i	S	xi	Ŝ	∆S ( Ŝ -S)
	User 1	1.3333	0.66665	1.3335	0.66675	10 -4
	User 2	4	0.6	4.0004	0.6	0
	User 3	4.6667	0.53334	4.6671	0.53333	-10 -5

TABLE II

 II

: Error on user satisfaction with the proportional allocation -x i and xi are the allocation when E = 10 and E = 10 + ε, S and Ŝ are the satisfaction when the allocations are x i and xi , ε = 10 -3 , c = (2, 6, 7).

TABLE III :

 III Error on user satisfaction with the MMF allocation -

  -1)(a±n 2 δ)+n 2 (b±(n 1 -1)δ) n 2 (a±n 2 δ)+(n 2 +1)(b±(n 1 -1)δ) MM (n 2 -1)a+n 2 b n 2 a+(n 2 +1)b (n 2 -1)(a±(n 2 -1)δ)+n 2 (b±n 1 δ) n 2 (a±(n 2 -1)δ)+(n 2 +1)(b±n 1 δ)

	User type	S			Ŝ
	GM	E n i=1 c i	E i=1 c i ±(n-1)δ n
	GG	1/n			1/n
	MM	(n -1)/n		(n -1)/n
	GM	E i∈N 1 c i +n 2 E	i∈N 1	E c i +n 2 E±(n 1 -1)δ
	GG	E i∈N 1 c i +n 2 E	i∈N 1	E c i +n 2 E±n 1 δ
	GM (n 2 GM (n 2 -1)a+n 2 b n 2 a+(n 2 +1)b 1/2			1/2
	MG	1/2			1/2

TABLE V :

 V Evaluation of Ŝ and S in case of full knowledge of the available resource and the same misknowledge on the other users demand. N 1 =set of users GM, N 2 =set of users of the other type. i∈N2 c i -E = a, i∈N1 c i = b.

ACKNOWLEDGEMENT

work was partially supported by NSF grant no. CNS-1526299 and ANR MAESTRO-5G (https://maestro5g.roc.cnam.fr) project (ANR-18-CE25-0012).


Georg Hanke
email: georg.hanke@ec.europa.eu

Abstract

The JRC exploratory project RIMMEL provides information about litter, mainly plastic waste, entering the European Seas through river systems. RIMMEL has collected data on riverine floating macro litter inputs to the sea. Data acquisition was based on the Riverine Litter Observation Network (RiLON) activities, which collected data from rivers in the European marine basins over a period of one year (September 2016 -September 2017). Data was collected by visual observations and documented with the JRC Floating Litter Monitoring Application for mobile devices, allowing a harmonized reporting, compatible with the MSFD Master List of Categories for Litter Items. This report includes the Top Items lists of riverine floating macro litter, based on the total amount of litter items identified during RiLON activities and ranked by abundance. Top Items lists have been elaborated considering the whole database for the European Seas and further detailed for each individual European regional sea: Baltic Sea, Black Sea, Mediterranean Sea and North-East Atlantic. The North-East Atlantic and the Mediterranean Sea regions showed similar litter categories in their Top 20 Items. These two regions provided most of the available data, influencing the general Top Items list. In the Black Sea and Baltic Sea regions, where data availability was limited, the Top Items lists showed more differences among the different regions. Overall, the general Top Items list for the European Seas showed a predominance of plastic item categories (artificial polymer materials). As a whole, plastic items made up to 80.8% of all objects, with plastic and polystyrene fragments comprising 45% of the identified items in the database. Additionally, Single Use Plastics such as bottles, cover/packaging and bags were also ranked among the most frequently found floating litter. The similarities in the Top 10 and Top 20 items for the different regions, and the appearance of Single Use Plastics scoring high in the ranking, support the need for common actions against plastic pollution at EU level.

Introduction

Marine pollution by anthropogenic litter has become a global environmental concern. The political agenda is setting priorities on tackling plastics from land-based sources, which contribute greatly to the marine litter issue. This has been stated internationally in the United Nations Sustainable Development Goal 14 (https://sustainabledevelopment.un.org/sdg14), the G7 Action Plan to Combat Marine Litter (Germany, 2015;and Japan, 2016) and the G20 Action Plan on Marine Litter (Germany, 2017). In Europe, Member States have committed through the Marine Strategy Framework Directive (MSFD, 2008/56/EC) to ensure that there is no harm caused by marine litter. At EU level the European Strategy for Plastics in a Circular Economy has recently introduced the plan for key actions to limit plastic pollution [START_REF] Ec | European Commission[END_REF].

In Europe, the MSFD Technical Group on Marine Litter (TG Litter), co-chaired by France, Germany and the DG Joint Research Centre (JRC), is providing advice to the Marine Strategy Framework Directive (MSFD, 2008/56/EC) regarding the implementation of Descriptor 10 (Marine Litter) and the EU external activities against Marine Litter, and supporting the Strategy for Plastics mentioned before. Within a series of technical reports, the TG Litter has also highlighted the importance of riverine litter input to the sea. The TG Litter has published the Riverine Litter Monitoring -Options and Recommendations -report [START_REF] González | Riverine Litter Monitoring -Options and Recommendations[END_REF]), reviewing the existing information and providing direction for further monitoring. Currently, characteristics and quantities of plastic input from rivers to the sea remain mostly unknown. In fact, the existing modelling exercises are based on limited field data (Lebreton et al., 2017;Schmidt et al., 2017), and therefore subject to major uncertainties. The implementation of regulations to reduce marine litter requires further information on the subject to prioritize prevention and reduction measures. Thus, monitoring field data is needed in order to e.g. identify hotspots, quantify loads and characterize sources.

Data acquisition

The JRC exploratory research project RIMMEL (Riverine and Marine floating macro litter Monitoring and Modelling of Environmental Loading), provides observation data on floating macro litter items entering the European Seas through rivers. RIMMEL proposed a harmonized approach based on visual observations, providing a common tool for monitoring and data reporting: the JRC Floating Litter Monitoring App [START_REF] González-Fernández | Toward a Harmonized Approach for Monitoring of Riverine Floating Macro Litter Inputs to the Marine Environment[END_REF]. The study of the macro litter fraction input was a pioneering activity in the topic of riverine litter at European level.

Within RIMMEL, a Riverine Litter Observation Network (RiLON) was set up for acquisition of field data on floating macro litter inputs to the sea. The network was a collaborative activity of 32 institutions covering rivers from 15 EU and non-EU countries in the four European Regional Seas: Baltic Sea, Black Sea, Mediterranean Sea and North-East Atlantic. Additional details on the participants in the network can be consulted in Annex I.

The RiLON participants acquired field data through short visual observation sessions (approx. 30 minutes) from bridges or other vantage points on rivers. Positions were selected close to the sea in order to monitor inputs to the marine environment as far as possible. In the case of larger rivers, observers defined their observation track width according to the distance from which they could identify litter > 2.5 cm size, thus monitoring only a part of the river section. Observers documented only litter floating or suspended in the river surface layer.

By using the JRC Floating Litter Monitoring App, the observers collected and reported data according to the MSFD Master List of Categories of Litter Items [START_REF] Galgani | Guidance on Monitoring of Marine Litter in European Seas[END_REF]. Consequently, the use of a common list allowed a harmonized data processing and analysis, facilitating the ranking of the most frequent litter items observed by the RiLON.

The RIMMEL database is the first of its kind, containing details on riverine macro litter items entering the European seas. The Top Items list provides abundance information linked to the identity of floating macro litter in European rivers. This information can be used to prioritize prevention and reduction measures, supporting the implementation of environmental regulations against marine litter, e.g. contributing to the monitoring and assessment of Descriptor 10 under the MSFD, and the EU proposal for action on 'Single Use Plastics' (SUP) on the reduction of the impact of certain plastic products on the environment (EC, 2018b). This report reviews and synthesises the data collected by the RiLON under the JRC exploratory project RIMMEL, presenting the Top Items list for the whole database as well as for the individual regional seas. Top Items lists are described along with supporting data on number of rivers, monitoring sessions, observation time, and number of items considered in the calculations.

Elaboration of Top Items list

The data collection for the RiLON took place from September 2016 to September 2017. Additional datasets were collected previously (June-2016 to August-2016) during the setup of the observation network to test the methodological approach.

The RIMMEL database contains datasets received from 32 Scientific Institutes, Authorities, SMEs and NGOs, with spatial coverage in 15 countries (Annex I). The data analysed in this report corresponded to visual observations of floating macro litter in the final reach of 53 rivers and streams selected for the monitoring of riverine litter input to the sea * . A map with the monitoring sites is included in Figure 1.

A total of 817 valid datasets was considered in the elaboration of the Top Item lists. Each individual datasets corresponded to one observation session. Floating litter items were identified following the MSFD Official Master List [START_REF] Galgani | Guidance on Monitoring of Marine Litter in European Seas[END_REF] for harmonization purposes. Sections 3 and 4 include further details on the data used in the elaboration of each Top Items list: European Seas (general list including all datasets) and individual Regional Seas (Baltic Sea, Black Sea, Mediterranean Sea and North-East Atlantic).

Litter items were ranked in descending order according to abundance. Results are shown as percentages, considering the total number of items for each list. Item names, materials and MSFD Codes have been extracted from the MSFD Master List of Categories of Litter Items [START_REF] Galgani | Guidance on Monitoring of Marine Litter in European Seas[END_REF]. In the tables, the 'Artificial polymer materials' category is shown as 'Plastic' (under Material). Additionally, pie charts include the distribution of items by material for the Top 10 and Top 20 items (expressed as percentages of total items).

Riverine Floating Macro Litter -General Top Items list in the European Seas

The general Top Items list in the European Seas has been elaborated using the whole RIMMEL database with data from the 53 rivers and streams included in Figure 1. The selected data corresponds to 817 datasets and 470.7 hours of monitoring. The observers identified 8599 litter items. The Top 10 items comprised up to 86.0% of total items and included the following material categories: 7 plastic, 2 paper/cardboard and 1 metal. The Top 20 items contributed up to 96.8% of total items, including the following material categories: 10 plastic, 3 paper/cardboard, 3 processed/worked wood, 2 metal, 1 rubber and 1 cloth/textile.

Riverine Floating Macro Litter -Top Items list by regional sea

North-East Atlantic

The Top Items list in the North-East Atlantic region has been compiled using data from 27 rivers and streams (see Figure 1).

Mediterranean Sea

The Top Items list in the Mediterranean Sea region has been collated using data from 17 rivers and streams (see Figure 1). The selected data corresponds to 272 datasets and 170.6 hours of monitoring. The observers identified 3486 litter items. The Top 10 items made up to 82.8% of total items and included the following material categories: 7 plastic, 2 paper/cardboard and 1 metal. The Top 20 items comprised up to 96.3% of total items, including the following material categories: 10 plastic, 3 paper/cardboard, 2 processed/worked wood, 2 metal, 2 rubber and 1 cloth/textile. Plastic items in the Top 20 made up to 73.7% of total items. Plastic and polystyrene fragments reached 31.6% of the total items.

Mediterranean Sea -

Black Sea

The Top Items list in the Black Sea region has been created using data from eight rivers and streams (see Figure 1). The selected data corresponds to 58 datasets and 45 hours of monitoring. The observers identified 604 litter items. The Top 10 items made up to 92.4% of total items and included the following material categories: 7 artificial polymer materials, 2 paper/cardboard and 1 processed/worked wood. The Top 20 items made up to 99.7% of total items, including the following material categories: 11 artificial polymer materials, 3 rubber, 2 paper/cardboard, 2 processed/worked wood, 1 metal and 1 cloth/textile. In the Top 20, Plastic items made up to 79.3% of total items. Plastic and polystyrene fragments accounted for 18.2% of the total items.

Ranking Items

Baltic Sea

The Top Items list in the Baltic Sea region has been based on data from one river (see Figure 1).

Summary -outcome

The RIMMEL initiative for data collection on riverine litter, through its observation network (RiLON), has provided the first database of its kind in Europe. The use of a harmonized approach for visual observations allowed the monitoring and identification of floating macro litter (>2.5 cm) flowing down the rivers to the seas. Floating litter was classified following the MSFD Master List of Categories of Litter Items [START_REF] Galgani | Guidance on Monitoring of Marine Litter in European Seas[END_REF], facilitating the ranking of the most frequent items observed by the network across Europe.

Data from the North-East Atlantic and the Mediterranean Sea regions prevailed in the database, and therefore influenced the general Top Items list for the European Seas. These two regions together considered data from 44 rivers in 11 countries, and presented similar litter categories in their respective Top 20 Items, but in different ranking positions. Also, the diversity in the Top 20 Items matched to a great extent with the general Top Items list, as a result of the predominant data availability from these regions. In contrast, the Baltic Sea and Black Sea regions collected a smaller amount of data from 1 and 8 rivers, as well as 1 and 4 countries, respectively. In these regions, some differences were seen in the composition of the Top 20 Items in comparison to the overall Top Items list and also among the different regions. Yet the data still managed to display the most relevant plastic items, including plastic pieces, bottles, bags and cover/packaging.

Despite the imbalance in data availability from the different regions, the general Top Items list provides valuable information on the nature of litter items entering European seas via rivers. Overall, the general Top Items list for all monitored waterways showed predominance of plastic (artificial polymer materials), with 7 plastic items in the Top 10, and 10 plastic items in the Top 20. In general, plastic items comprised 80.8% of all items, with plastic and polystyrene fragments reaching 45% of the items identified in the database. Additionally, Single Use Plastics such as bottles, cover/packaging items and bags were also ranked among the most frequently identified objects.

The similarities in the Top 10 and Top 20 items among the different regions, and the appearance of iconic Single Use Plastics scoring high in the rankings indicate the need for implementation of common actions at EU level, such as the Plastic Strategy, and in coordination with non-EU countries.

The data on Top riverine litter Items from the RIMMEL project have been provided as information in support to the implementation of the EU Plastics Strategy and been used in part 2/3 of the EC Staff Working Document (page 51) on Impact Assessment of legislation for reducing marine litter (EC 2018c).

Quantitative results and analysis of the riverine litter data will be published in a peer reviewed publication (González-Fernández et al., in preparation).

The monitoring of riverine litter input is under development. Harmonization of approaches is recommended, both for resource efficiency in developing and obtaining comparable datasets, and for prioritization of efforts between different river basins as well as with the receiving marine waters.

Figure 1 .

 1 Figure 1. RiLON data collection -rivers and monitoring sites (colour coded by Regional Sea).

  The distribution of items by material showed a clear predominance of plastic (up to 80.8 % in the Top 20 items). Plastic and polystyrene fragments reached 45% of the total items.

	General Top Items list
	Ranking Item	Material	MSFD Code	% of Total Items
	1	Plastic pieces 2.5 cm > < 50cm, > 50 cm	Plastic	G79 + G80	38.59%
	2	Bottles	Plastic	G6	9.55%
	3	Cover / packaging	Plastic	G38	8.42%
	4	Bags	Plastic	G2	7.77%
	5	Polystyrene pieces 2.5 cm > < 50cm, > 50 cm Plastic	G82 + G83	6.35%
	6	Other paper items	Paper/Cardboard	G158	4.01%
	7	Paper packaging	Paper/Cardboard	G149	3.61%
	8	Sheets, industrial packaging, plastic sheeting Plastic	G67	2.93%
	9	Foam packaging/insulation/polyurethane	Plastic	G74	2.70%
	10	Cans (beverage)	Metal	G175	2.05%
	11	Other plastic/polystyrene items (identifiable) Plastic	G124	1.98%
	12	Crates and containers / baskets	Plastic	G18	1.81%
	13	Other (metal)	Metal	G197	1.48%
	14	Newspapers & magazines	Paper/Cardboard	G154	1.19%
	15	Beams / Dunnage	Processed wood	G169	0.99%
	16	Other rubber pieces	Rubber	G134	0.95%
	17	Wood boards	Processed wood	G168	0.72%
	18	Synthetic rope	Plastic	G48	0.71%
	19	Other textiles (incl. rags)	Cloth/textile	G145	0.57%
	20	Pallets	Processed wood	G160	0.48%

  The selected data corresponds to 51 datasets and 24.8 hours of monitoring. The observers identified 190 litter items. The Top 10 items made up to 90.5% of total items and included the following material categories: 4 artificial polymer materials, 2

	metal, 2 paper/cardboard and 2 processed/worked wood. The Top 20 items made up to
	98.4% of total items, including the following material categories: 8 artificial polymer
	materials, 3 metal, 3 paper/cardboard, 2 processed/worked wood, 3 cloth/textile and 1
	rubber. Plastic items made up to 55.8% of total items. Plastic and polystyrene fragments
	amounted to 4.2% of the total items.
	Ranking Items	Material	MSFD Code	% of Total Items
	1	Bottles	Plastic	G6	33.68%
	2	Other (metal)	Metal	G197	12.63%
	3	Synthetic rope	Plastic	G48	10.00%
	4	Paper packaging	Paper/Cardboard	G149	6.84%
	5	Bags	Plastic	G2	5.79%
	6	Other paper items	Paper/Cardboard	G158	5.79%
	7	Cans (beverage)	Metal	G175	4.74%
	8	Beams / Dunnage	Processed wood	G169	4.21%
	9	Plastic pieces 2.5 cm > < 50cm, and > 50 cm	Plastic	G79 + G80	4.21%
	10	Pallets	Processed wood	G160	2.63%
	11	Newspapers & magazines	Paper/Cardboard	G154	1.58%
	12	Barrels	Metal	G192	1.05%
	13	Clothing (clothes, shoes)	Cloth/textile	G135	1.05%
	14	Other textiles (incl. rags)	Cloth/textile	G145	1.05%
	15	Other plastic/polystyrene items (identifiable) Plastic	G124	0.53%
	16	Balls	Rubber	G126	0.53%
	17	Foam packaging/insulation/polyurethane	Plastic	G74	0.53%
	18	Cover / packaging	Plastic	G38	0.53%
	19	Rope, string and nets	Cloth/textile	G142	0.53%
	20	Fish boxes -plastic	Plastic	G57	0.53%

Acknowledgements

Commission's science and knowledge service. It aims to provide evidence-based scientific support to the European policymaking process. The scientific output expressed does not imply a policy position of the European Commission. Neither the European Commission nor any person acting on behalf of the Commission is responsible for the use that might be made of this publication. Riverine Litter Observation Network (RiLON) in the field data collection and their feedback in the preparation of results.

Collaboration with the EMBLAS II project (Angela Bularga, DG MARE + Jaroslav Slobodnik, Slovak Environmental Institute) and colleagues involved is appreciated. This report was funded by the JRC institutional exploratory project RIMMEL (272346), and finalized with support from the PLASTREND project (BBVA Foundation) at the University of Cádiz (Spain).

* Except River Meuse: occasional data collected in Maastricht (Netherlands), at more than 100 km from the sea.

Authors

Main authors: Daniel González-Fernández (University of Cádiz, Spain) and Georg Hanke (European Commission, Joint Research Centre).

Annex 1. RiLON Institutions and rivers involved in data collection

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T El Hajji

B Kabalan

Y Cheng

E Vinot

C Dumand

Sensitivity Analysis on the Sizing Parameters of a Series-Parallel HEV

Keywords: Sensitivity Analysis, Hybrid Vehicles, Modelling, Dynamic Programming, Genetic Algorithms. 

As an alternative to power-split hybrid architectures, a simple series-parallel architecture named SPHEV 2 can be realized. In this paper, the sizing process of this architecture is briefly presented and a deeper analysis is made. Mathematical sensitivity analysis studies are conducted on the sizing parameters of the architecture in order to bring more understanding to the optimization results. Local and global sensitivities are performed to understand the influence of the sizing variables on the fuel consumption. An analysis of the sensitivities is also made.

INTRODUCTION

The automotive sector is undergoing significant changes into new forms of mobility. For instance, car manufacturers are proposing more and more electric and hybrid electric models on the market (Groupe PSA, 2016) (Renault-Nissan-Mitsubishi, 2017). These changes aim to meet the stringent regulations, the global fleet CO2 targets and the clients' new needs and expectations.

Regarding Hybrid Electric Vehicle (HEV), different powertrain architectures exist, different components technologies can be selected, and different sizing can be made. The powertrain operation and fuel consumption on a selected driving cycle will depend on the architecture chosen, the components and their sizing, and lastly on the energy management during the vehicle operation.

Series, parallel and series-parallel (power-split and nonpower-split) are the main categories of existing hybrid architectures. Series-parallel combine the advantages of series and parallel, but they can have a relatively costly design and complicated control [START_REF] Chan | The state of the art of electric, hybrid, and fuel cell vehicles[END_REF]. The simplest series-parallel architecture that can be realised is a configuration where the engine and the electric machines are mounted on a same shaft with clutches in between [START_REF] Trigui | Backward Modeling and Energy Management Optimization of a Two Clutches Series-Parallel HEV for Efficiency Assessment[END_REF]. This powertrain was proven to be less efficient than the power-split ones [START_REF] Vinot | Comparison of different power-split hybrid architectures using a global optimization design method[END_REF]). Yet, with the addition of gears or gearbox and with an adequate sizing and control, this architecture can be improved.

In this framework of improving the efficiency of simple series-parallel hybrid powertrains, an architecture named SPHEV 2 was proposed and studied in (B. [START_REF] Kabalan | Improvement of a Series-Parallel Hybrid Electric Vehicle Architecture[END_REF]. This architecture is presented below in Fig. 1 Based on the clutches' states, this architecture can operate in pure electric, series hybrid or parallel hybrid mode. In contrast, powersplit architecures that include a planetary gear (PG) operate always in a power split between both the series path and the parallel path. This continuous split of power allows a speed degree of freedom for the engine for more operation optimization, but involves a continuous use of the series path that is less efficient than the parallel one. This power split is not present in SPHEV 2.

An energetic model was developped using VEHLIB [START_REF] Vinot | Model simulation, validation and case study of the 2004 THS of Toyota Prius[END_REF] In previous work, these variables are optimized and SPHEV2 is compared to other architectures, including a reference power-split [START_REF] Kabalan | Improvement of a Series-Parallel Hybrid Electric Vehicle Architecture[END_REF]. Many works have been conducted on sensitivity analysis using different approaches based on a quantitative method [START_REF] Yanping | Sensitivity analysis of the hybrid electrical vehicle power system parameters[END_REF] or a qualitative method [START_REF] Stroe | A two-layer predictive control for hybrid electric vehicles energy management[END_REF]. In this paper, the authors make deeper analysis of this architecture. Mathematical sensitivity analysis studies are conducted to bring more understanding to the optimization results. Local and global sensitivities are performed to understand the influence of the sizing variables on the fuel consumption results. This can lead to dismiss some variables or to put more attention on others in future optimization work.

In the following section, the methods that can be generally used to perform local and global sensitivities on a system or a model are screened. In section 3, the model is presented with the methodology that was used for its optimization. In section 4, the sensitivity methods that are going to be applied on our model are selected and explained. Finally, the results are shown in section 5 with an analysis of the design parameters influence.

SENSITIVITY CALCULATION

This section presents the methods that are generally used to calculate the local and global sensitivities.

Local Sensitivity

The local sensitivity of 𝑛 input variables (𝑥 1 , … , 𝑥 𝑛 ) on an output function 𝑓 is the sensitivity of the input variables on the function at a point of study of these input variables. This point is noted (𝑥 1 ̃,…,𝑥 𝑛 ̃). For more understanding, one can refer to [START_REF] Saltelli | Sensitivity Analysis in Practice[END_REF]. The analytical method used to calculate this local sensitivity is presented hereafter [START_REF] Yanping | Sensitivity analysis of the hybrid electrical vehicle power system parameters[END_REF].

 Derivative Formulas

They are commonly used in physics for sensitivity calculation. The first formula is given with direct differential method and is expressed as follows:

𝑆 𝑥 𝑖 = 𝑓 ′ (𝑥 𝑖 ) , 𝑖 = 1. . 𝑛 (1) This needs the expression of the function or its approximation using the Response Surface. If 𝑆 𝑥 𝑖 is high it means that the slope of the tangent is high and hence a small variation in the parameter 𝑥 𝑖 leads to a high variation in the output function. The second formula uses an approximation of the derivative formula and is expressed as follows:

𝑆 𝑥 𝑖 = 𝑓(𝑥 𝑖 +𝛿𝑥 𝑖 )-𝑓(𝑥 𝑖 ) 𝛿𝑥 𝑖 , 𝑖 = 1. . 𝑛 (2)
However, variables' sensitivity should be normalized before comparison in order to have the same range of variation. Thus, a fair comparison needs a third formula of sensitivity that is expressed as follows:

𝑆 𝑥 𝑖 = [𝑓(𝑥 𝑖 +𝛿𝑥 𝑖 )-𝑓(𝑥 𝑖 )].𝑥 𝑖 𝛿𝑥 𝑖 .𝑓(𝑥 𝑖 ) 𝑖 = 1. . 𝑛 (3)
The function 𝑓 corresponds to the vehicle's fuel consumption of the driving cycle. Thus, there is no case of division by zero because the function is always strictly positive. Both formulas ( 2) and (3) provide the same ranks of variables' sensitivity at a point (𝑥 1 ̃,…,𝑥 𝑛 ̃) because the value of 𝑓 at this point is the same when calculating sensitivity of variables. Instance analysis shows that the sensitivity 𝑆 𝑥 𝑖 calculated using this third formula could lead to good engineering understanding. In addition, in the case of this work, the fuel consumption is not expressed as analytical formula in terms of the input variables. The formula (3) is therefore the most relevant.

Global Sensitivity

It is the study of how the uncertainty in the model output (numerical or otherwise) can be apportioned to different sources of uncertainty in the model input. The mainly used analytical methods are presented below. More information can be found in [START_REF] Saltelli | Global Sensitivity Analysis[END_REF].

 Linear Regression

In order to calculate global sensitivity, we use Taylor-Mac Laurin series. Three models are presented: Degree 2 (4), Degree 1 with interactions (5), and Degree 1 (6). The coefficients (𝛽 1 , … , 𝛽 𝑛 ) represent the sensitivity of the normalized variables (𝑥 1 , … , 𝑥 𝑛 ). The calculation of the coefficients (𝛽 1 , … , 𝛽 𝑛 ) is done by Least Squares Method [START_REF] Saltelli | Global Sensitivity Analysis[END_REF] with either analytical or with numerical method. In this work, we used Degree 1 model ( 6). The computation time of Degree 1 model is the least among the three models though its accuracy is the worst. The choice of combinations (points) is done with Monte Carlo Method. This method consists of generating all possible combinations of the n variables' value by mixing different parameter's value range.

𝑦 = 𝛽 0 + 𝛽 1 . 𝑥 1 + 𝛽 2 . 𝑥 2 + ⋯ +
Another method of choosing combinations [START_REF] Goupy | Introduction aux plans d'expériences. Groupe PSA[END_REF] consists on using combinations with extreme values of each variable (after normalization). For n variables, there will be 2 𝑛 combinations. These combinations are used to determine the coefficients of the formulas above that correspond to variables' sensitivity.

MODELS AND OPTIMIZATION

When designing a hybrid vehicle powertrain, three interdependent dimensions should be taken into account. In fact, the powertrain architecture, the components choice and sizing, and the related energy management, strongly interfere in the design process [START_REF] Silvas | Review of Optimization Strategies for System-Level Design in Hybrid Electric Vehicles[END_REF]. To address this problem, a bi-level framework (Fig. 2) was developed in [START_REF] Vinot | Global optimized design of an Electric Variable Transmission for HEVs[END_REF]. Once the choice of the architecture is done, a genetic algorithm (NSGA-2, [START_REF] Deb | A Fast and Elitist Multiobjective Genetic Algorithm : NSGA-II[END_REF] will act on different parameters of the vehicle. An explanation of the entire bi-level optimization can be found in [START_REF] Kabalan | Improvement of a Series-Parallel Hybrid Electric Vehicle Architecture[END_REF].

One of the optimization objectives of this algorithm is the fuel consumption. It is assessed on a known-in-advance driving cycle using optimal energy management (dynamic programming method in this paper) to guarantee the minimum fuel consumption while respecting a constraint on the state of charge variation. An energetic systemic model is used to assess, on a backward manner, the fuel consumption going upstream from wheel to engine and battery. The gears model consists of a constant ratio and efficiency, the engine modeled by a Brake Specific Fuel Consumption map (BSFC) (Fig. 3), the electrical machines by efficiency maps (Fig. 3) and the battery by an equivalent electrical circuit. This circuit consists of an open circuit voltage, an internal resistance, and a faradic efficiency. These three parameters depend on the battery state of charge, on the current, and on the temperature.

The model or system in this study is the entire vehicle model which has eight optimization variables: the Number of Battery Modules, the powers of engine and electrical machines, and the gear ratios (final drive FG, engine gear G3, EM1 gear G1, and EM2 gear G2). The sizes of the ICE and EMs change according to scaling factors applied on the torque. For EMs, the maximum torque, the losses, and weight are multiplied by a scaling factor kEM. The inertia is multiplied by (kEM) 5/3 . For the ICE, the maximum torque, the fuel consumption, and the weight are multiplied by kICE. The inertia is multiplied by (kICE) 5/3 . The shape of the BSFC maps does not change; it is only the torque scale that is changed. Such scaling technique is a classical method used for the components sizing in the context of powertrain optimization [START_REF] Vinot | Comparison of different power-split hybrid architectures using a global optimization design method[END_REF]. For more accurate sizing, large experimental data of component maps is needed. Another alternative is to use precise models which takes into account the size of the component [START_REF] Reinbold | Optimal sizing of an electrical machine using a magnetic circuit model: application to a hybrid electrical vehicle[END_REF][START_REF] Zhao | Design and Control Co-Optimization for Advanced Vehicle Propulsion Systems[END_REF]. A compromise is to be done the accuracy of the models and the calculation time. For the calculation of the local sensitivity, the derivative formula requires the evaluation of 20 combinations for each variable. In fact, to calculate local sensitivity of each variable 𝑥 𝑖 , the step 𝛿𝑥 𝑖 is taken equal to a variation from -10% to 10% of 𝑥 𝑖 ̃ (with a step of 1%), 𝑥 𝑖 ̃ being the local value of 𝑥 𝑖 . Local sensitivity is calculated as the mean (in absolute values) of the sensitivities obtained for each value of 𝛿𝑥 𝑖 .

For the calculation of the global sensitivity, a choice is to be made for the sizing combinations that need to be evaluated. This can be usually done using Design of Experiments (DOE). In the case of this application, the design space is constrained by the performance requirements. For instance, if the combinations of extreme sizing values are considered (2 8 = 256 combinations), only 9 combinations among the 256 are able to pass the performance requirement constraints (maximum speed, acceleration time from 0 to 100km/h and overtaking capability at 80km/h) and to be evaluated. This is why the choice of combinations is done using Monte-Carlo method instead of DOE. First, the variables are normalized, then the [0,1] interval of each variable is divided in three intervals [0, Each of the 8 variables can take three possible values; hence, 3 8 = 6561 combinations can be generated. Not knowing a priori the design space, the performance constraints are checked for all those combinations. The fuel consumption is calculated only for the feasible combinations (in this case 842 out of 6561). Linear Regression's coefficients are then calculated either numerically or analytically.

RESULTS AND ANALYSIS

Sensitivity results

The local sensitivity is calculated on highway cycle at two points of sizing that correspond to the minimum and maximum of fuel consumption in the Pareto front (Fig. 5) of SPHEV 2 [START_REF] Kabalan | Improvement of a Series-Parallel Hybrid Electric Vehicle Architecture[END_REF]. This Pareto front corresponds to an optimization on mixed driving condition. Fig. 6 and Fig. 7 show the results of sensitivity for the 8 variables at the 2 points shown in red arrow in Fig. 5.

The global sensitivity of the variables is calculated on highway cycle using the linear regression method. Fig. 8 shows the influence and the sign of the influence of the 8 variables. A negative sensitivity means that the consumption and the variable have an opposite sign of variation. The Final Drive Ratio and the gear for EM1 appear to be the 2 most significant variables. For a small number of battery modules, the size of the battery does not allow to recover all the braking energy and constraint the operation in electric mode. Therefore, increasing the Number of Battery Modules here will have an important influence on the fuel consumption. On the other hand, for high Number of Battery Modules, the electric operations of the powertrain are not affected by the size of the battery pack. Since the size of the battery greatly influences the price of the powertrain, a trade off needs to be find between cost and fuel consumption improvement.

 The final drive ratio

It is the most sensitive variable locally and globally. This is due to the fact that it is linked to all other components thus a variation in final drive ratio leads to speed variation and will affect all other components' operating points. Globally, an increase in final drive ratio leads to a decrease in fuel consumption.

 The power of ICE

It is very sensitive locally and globally. In fact, a variation in the power of ICE leads to a change of the BSFC areas in which the engine operating points are located. Similarly, a variation in the gear of ICE leads to a change in the ICE operating points, what explains the sensitivity of ICE gear to fuel consumption locally and globally.

 The powers and gears of the electrical machines

Their impacts on fuel consumption is different from local to global study. As for the ICE, those variables have an impact on the location of the operating points inside the efficiency maps. EM1 power and gear are more sensitive than those of EM2, because EM1 is more involved in the electric mode than EM2.

Analysis on powertrain operation

For a better understanding of the influence of these variables on the fuel consumption, a closer look should be made on the system operating modes and the global efficiencies of the components. Three modes are present:

 Pure Electric: The components involved are EM1 only, EM2 only or EM1 and EM2.

 Series: ICE and EM2 are used as electric generator, only EM1 provides mechanical power to the wheels.

 Parallel: ICE is mechanically connected to the wheels. EM1 and/or EM2 are also connected mechanically to provide or absorb power.

Table 1 presents the split between these modes and the global efficiencies of the EMs when the variables are changed. A reference sizing is shown in the first column ('Sizing of Minimum Fuel Consumption'). It corresponds to the minimum fuel consumption point (Maximum Number of Battery Modules). The other columns correspond to the component sizing where only one variable at a time is changed compared to reference one. For example, the column 'FDR' is the sizing similar to the reference, with only variation on the FDR. In the rows, the percentage of time and consumed energy in each of the three modes (Electric, Series, and Parallel) are shown, in addition to the electrical losses and average efficiency of both electrical machines over the entire driving cycle. For example, the column 'FDR' shows that 1.68% of the time is spent in the 'Series' mode. This corresponds to 1.72% of the global traction energy transferred to the wheels during the driving cycle. The total losses of EM1 on all the driving cycle are 11.58 Wh/km and its average efficiency is 79.64%.

The red cells correspond to the noticeable variation from the reference. It shows that the final drive ratio and the power of ICE are the variables that have important impacts on electrical losses (11.58 vs 11.74 and 1.89 vs 1.72) and on time and energy sharing between different operating modes (power of ICE increases the series mode from 1.31% to 1.96% of the time). This can be explained by the fact that the change in the power of ICE results in more efficient series operation (improvement of mean BSFC in series).

CONCLUSIONS

Through a sensitivity analysis conducted on the sizing variables of the SPHEV 2 architecture, this work allows more understanding of the optimization results previously done on this architecture. Moreover, this paper allows the identification of the most influential variables on the fuel consumption of SPHEV 2, like the final drive ratio for instance. Such information can be used later on to exclude some variables from the optimization for a gain in optimization time, or to put more efforts on other variables because they are more influential. This work can be extended to any other architecture. Future work will also include conducting such sensitivity analysis on the energy management variables that are decided by Dynamic Programming in the case of this work. This can give more understanding to the decisions made by Dynamic Programming and can also lead to a detection of the most influential variables that should be given more attention in any online simplified energy management technique.

  Fig.1. SPHEV 2 architecture

  to calculate the performance and fuel comsumption of SPHEV 2. This model has 8 sizing variables: -Number of Battery Modules -ICE gear (G3) -Power of ICE -EM1 gear (G1) -Power of EM1 -EM2 gear (G2) -Power of EM2 -Final Drive Ratio (FDR)

  Fig. 2. Bi-level optimization framework

  to choose a value in each of the three intervals. Then, all possible combinations that mix different parameter's value range (low, medium, high) are generated.

Fig. 5 .

 5 Fig. 5. Pareto front of SPHEV2

  𝛽 𝑛 . 𝑥 𝑛 + ∑𝛽 𝑖 . 𝛽 𝑗 . 𝑥 𝑖 . 𝑥 𝑗 + ∑𝛽 𝑖 . 𝑥 𝑖

2

(4) 𝑦 = 𝛽 0 + 𝛽 1 . 𝑥 1 + 𝛽 2 . 𝑥 2 + ⋯ + 𝛽 𝑛 . 𝑥 𝑛 + ∑𝛽 𝑖 . 𝛽 𝑗 . 𝑥 𝑖 . 𝑥 𝑗 (5) 𝑦 = 𝛽 0 + 𝛽 1 . 𝑥 1 + 𝛽 2 . 𝑥 2 + ⋯ + 𝛽 𝑛 . 𝑥 𝑛

(6)

Table 1 . EMs losses and time/energy sharing between modes for different sizings

 1

			Sizing of	FDR			Power
			Minimum of		Power	Power	of ICE
			Fuel		of EM1	of EM2
			Consumption
		Electric	29.09% 3.13%	27.78% 2.56%	29.37% 3.29%	29% 3.07%	26.38% 2%
	%Time %Energy	Series	1.31% 1.34%	1.68% 1.72%	1.12% 1.18%	1.22% 1.18%	1.96% 1.8%
		Parallel	69.6% 95.53%	70.53% 95.73%	69.5% 95.53%	69.78% 95.75%	71.66% 96.18%
	EM1 Losses (Wh/km)	11.74	11.58	10.9	11.7	11.78%
	Average Efficiency	78.64%	79.64%	79.95%	78.67%	79.21%
	EM2 Losses (Wh/km)	11.72	1.8	1.77	1.89	1.89
	Average Efficiency	84.35%	83.85%	82.9%	84.55%	85.38%
			Gear of EM1	Gear of EM2	Gear of ICE		Battery
		Electric	29.09% 3.13%	29% 3.1%	27.78% 2.55%		29.65% 3.41%
	%Time %Energy	Series	1.4% 1.48%	1.12% 1.18%	1.68% 1.72%		1.5% 1.69%
		Parallel	69.5% 95.39%	69.88% 95.72%	70.53% 95.73%		68.85% 94.9%
	EM1 Losses (Wh/km)	11.92	11.63	11.8		11.72%
	Average Efficiency	78.33%	78.67%	79.27%		78.46%
	EM2 Losses (Wh/km)	1.76	1.91	1.797		1.75
	Average Efficiency	84.81%	83.61%	84.97%		84.7%