Etienne Goddet

Nicolas Retière
email: nicolas.retiere@univ-grenoble-alpes.fr

Vojislav Stojanović

Anca Dieudonné

Jérôme Genoulaz

Jean-Michel Guichon

Maximizing the algebraic connectivity of meshed electrical pathways used as current return network

Keywords: Graph theory, Spectral analysis, Optimization, Algebraic connectivity, Current return networks

This paper proposes an innovative method for optimizing the connectivity of meshed electrical pathways used to carry return current and provide grounding in more composite aircrafts. In normal operation, these networks should ensure a low voltage drop between power sources and electrical loads. The networks are modelled as graphs and spectral graph analysis is used to study their complexity. Thanks to the insight into graph's spectrum, a multi-scale reading of meshed networks topology is proposed and an optimization problem is defined to decrease the DC resistance of the network while keeping constant its total weight. It is based on the maximization of the second eigenvalue of the graph's Laplacian matrix. The optimization problem is applied to a mock-up of a real aircraft current return network. The paper is concluded by checking the DC voltage drops in steady state conditions.

Introduction

In recent years, the conventional metallic materials used for aircrafts structure and fuselage have been replaced by new composite materials. This has been justified by the excellent mechanical properties of composites (e.g. their high resistance to fatigue and corrosion) and the reduction of aircraft weight they provide. The "standard" aircrafts benefit from metallic bodies which are commonly used as conductive pathways to carry the return of direct and alternating currents, faults currents, lightning currents and also to provide other functions related to grounding, electrostatic charge draining, electromagnetic shielding etc. Unfortunately, such benefits are not available on aircrafts made of Upper bound of the branch sections (m) x i ith-entry of the nodal vector x relative to the nodal basis ⟨x⟩ i ith-entry of the nodal vector x relative to the eigenvectors basis y branch Branch series admittance (S) y u,v

Series admittance of branch between nodes u and v (S) Eigenvector associated to the eigenvalue θ i ψ u,i uth-entry of the eigenvector related to mode i ρ Branch material resistivity (Ω .m) θ

D
Eigenvalues vector of the Laplacian matrix θ i Eigenvalue related to mode i µ mat Branch material density (kg/m 3 ) composite materials because of their low conductivity and bad thermal properties. Hence, composite aircrafts require an additional electrical network to provide the functions that the composite structures are not anymore able to ensure. The additional electrical system can be a metallic network used to create a highly conductive electrical pathway for the return of currents. Such a network is practically realized by exploiting both structural metallic parts of the aircraft (beams, seats rails, etc.) and/or dedicated paths. Boeing has opted for a network made by the addition of long cables, bars and cable shielding for the current return network of the 787. Airbus, for its A350 airplane, has decided to reuse parts of the metallic structure of the aircraft to build a new Electrical Structure Network (ESN) [1,[START_REF] Gutierrez | On the design of aircraft electrical structure networks[END_REF][START_REF] Marsh | Airbus takes on boeing with reinforced plastic A350 XWB[END_REF]. This structure is composed of circular frames which support the composite structure, a cabin floor made by the assemblage of crossbeams and longitudinal elements (seat tracks, false tracks and raceways), a cargo floor constituted by crossbeams and roller tracks (longitudinal elements), several vertical elements which link the cabin floor to the cargo floor and the metallic cockpit (cf. Fig. 1).

Both Airbus and Boeing supplementary networks are meshed and embed many elements. This brings a lot of complexity in terms of construction. That is why a special attention has to be paid to the pre-sizing process. Predimensioning must ensure that the conditions for a safe operation of the aircraft electrical system are met while handling weight reduction. This paper focuses on the direct current return function of the additional network. A good ground connection shall be made between every electrical loads and power sources. Actually, the additional network shall guarantee a voltage drop between loads and sources lower than 0.5 V in steady state conditions. A low impedance electrical pathway will facilitate the respect of this constraint. To decrease impedance, critical elements can be oversized. On the other hand, to cope with the objectives of fuel efficiency, the mass of the additional network must be sufficiently low to not cancel the advantages provided by the use of a composite structure. To find the best compromise between the weight reduction and the low impedance specification becomes therefore very challenging. This challenge is addressed in this paper.

To meet the low impedance and mass requirements, we present an optimization method to improve network connectivity of complex meshed networks. Our optimization strategy is based on the spectral theory of graphs instead of a circuit-oriented approach [START_REF] Hien | Location and sizing of distributed generation units for loadabilty enhancement in primary feeder[END_REF]. Spectral theory is already commonly used for complex graphs because it gives a multiscale reading of properties which cannot be visible by a simple visual inspection [START_REF] Liu | Coupling strength allocation for synchronization in complex networks using spectral graph theory[END_REF]. For instance, the complexity of meshed electrical circuits makes difficult for human brains to perceive which parameters drive the system properties (conductance, voltage drops, robustness, weight etc.) from the circuit-oriented representation. It is impossible to catch the most critical parameters without a long-last campaign of numerical simulations. On the other hand, the spectral graph theory enables to link the graph properties to the eigen modes, expressing the graph behaviour as a sum of modal behaviours. So, the spectral theory is expected to provide an easy reading of the properties of complex network and how they are linked with the topology and the design choices performed by engineers. Very few works based on spectral graph have been published in the field of power engineering. It is worth mentioning the use of spectral analysis to link eigenvalues and synchronizability of Central Italy HV power grid [START_REF] Fioriti | Spectral analysis of a real power network[END_REF], the comprehensive investigation of energy transmission and synchronization by [START_REF] Rubido | Energy Transmission and Synchronization in Complex Networks: Mathematical Principles[END_REF] or the works about graph spectra and dynamics in complex network of oscillators in [START_REF] Ayazifar | Graph Spectra and Modal Dynamics of Oscillatory Networks[END_REF] and [START_REF] Dorfler | Spectral analysis of synchronization in a lossless structure-preserving power network model[END_REF].

Only DC operating conditions of current return networks will be investigated in this paper as more and more future aircraft electrical systems are designed to operate with DC systems. The underlying graph of the current return network will hence be described by its Laplacian matrix [START_REF] Ayazifar | Graph Spectra and Modal Dynamics of Oscillatory Networks[END_REF]. The optimization algorithm will lower network's DC resistance by maximizing the second eigenvalue of the Laplacian matrix [START_REF] Shafi | Designing node and edge weights of a graph to meet Laplacian eigenvalue constraints[END_REF].

The paper is organized as follows. Section 2 deals with the mathematical representation of electrical circuits by graphs. Section 3 outlines the spectral graph theory and the links with electrical and structural properties of networks. In Section 4, the definition of the optimization problem is explained. Section 5 presents the optimization of the mock-up of a meshed current return network called DEMLAB and developed by Safran Electrical & Power.

Graph representation of electrical networks

In order to analyse the connectivity of a network and to provide some topological drivers for its design, a mathematical representation is required that will properly describe the interconnections between the network components. In this paper, the graph representation is adopted. It will be used to link the structural properties of the underlying graph of the network and its electrical properties such as DC conductance or voltage drops. Many different natural or human-engineered interconnected systems have already been studied using graph theory. In particular, power grids have been successfully investigated as complex graphs [START_REF] Pagani | The power grid as a complex network: a survey[END_REF]. In our case, the graphs are undirected which means that all edges are bidirectional, since current flow is naturally bidirectional too. Moreover, graph edges are weighted by the corresponding branch series admittance.

The graph theory uses matrices to catch topological information of graphs. In this paper, the Laplacian matrix is used. Given a network with n nodes and m branches, the Laplacian matrix L is a (n × n) matrix defined by:

L u,v = ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ -y u,v if u and v are neighbours ∑ t∈nei(u) y u,t if u = v 0 otherwise. (1)
Two nodes are neighbours if they are connected by a branch and nei(u) is the set of neighbours of node u. y u,t is the series admittance of the branch between u and t. Laplacian matrix is symmetric and the sum of its columns and rows are equal to zero. It is a singular matrix.

Electrical networks are often described by the primitive impedance matrix. It is a matrix whose elements are the self and mutual impedances of the network. If the network is only made of series impedances, the matrix elements referred by (u, v) are equal to the admittance value y u,v connecting nodes u and v. Of course, the matrix entry is equal to zero if there is no connection. The primitive impedance can give birth to new matrices that are more suitable for circuit-oriented analysis purposes. For instance, it is possible to build a bus admittance matrix Y bus that represents the electrical behaviour of the nodal variables of a meshed network such that [START_REF] Kundur | Power System Stability and Control[END_REF]:

Y bus V = I (2)
I and V are the nodal current and voltage vectors. This matrix is used to perform electrical circuit calculations such as load-flows. It can be easily shown that the bus admittance matrix is also the Laplacian matrix of the underlying graph of the electrical network. Some interesting properties of electrical networks can be caught by their graph-oriented mathematical representation. For instance, the node degree distribution of the underlying graph is often used as an indicator of performance [START_REF] Boccaletti | Complex networks: Structure and dynamics[END_REF][START_REF] Cotilla-Sanchez | Comparing the topological and electrical structure of the north american electric power infrastructure[END_REF]. Centrality measures based on betweenness properties are also useful to identify what are the central links and nodes [START_REF] Bompard | Extended topological metrics for the analysis of power grid vulnerability[END_REF][START_REF] Newman | A measure of betweenness centrality based on random walks[END_REF][START_REF] Segarra | Stability and continuity of centrality measures in weighted graphs[END_REF]. Flow network problems allow to find the maximum current that can flow through an electrical grid while respecting thermal constraints of lines [START_REF] Gomory | Multi-terminal network flow[END_REF]. The total mass of a network can be estimated from the degree matrix. In this paper, the choice has been done to investigate the Laplacian matrix spectral properties, meaning the analysis of its eigenvalues and eigenvectors.

Spectral analysis

Basic concepts

Only the most relevant results for our study will now be presented. For a complete review of the spectral graph theory, the reader may refer to [START_REF] Chung | Laplacians of graphs and Cheeger's inequalities[END_REF][START_REF] Dekel | Eigenvectors of random graphs: Nodal domains[END_REF].

The spectral graph theory is the study of the properties of the characteristic polynomials, eigenvalues and eigenvectors associated to the matrices describing the graph. For instance, the eigenvalues θ of the Laplacian matrix L are the non-trivial roots of the characteristic equation:

det (L + θ E) = 0 ( 3
)
where E is the identity matrix. For a (n×n) matrix, solving Eq. ( 3) provides n eigenvalues. Algebraic multiplicity of an eigenvalue is the multiplicity of its associated root. Since the Laplacian matrix is symmetric, semi-definite and positive, all its eigenvalues are positive real numbers. Furthermore, Laplacian matrix of any graph has at least an eigenvalue equal to zero because of its singularity. Algebraic multiplicity of the zero eigenvalue gives the number of connected components of the graph. By convention, eigenvalues are sorted in increasing order

θ 1 = 0 ≤ θ 2 ≤ • • • ≤ θ n .
For a single connected component graph, the second smallest eigenvalue θ 2 is called algebraic connectivity and the largest eigenvalue θ n is called spectral radius. The algebraic connectivity is the core of our work because the higher it is, the better connected is the graph. Indeed, according to Cheeger's inequality, for two subgraphs 1 and 2 as shown in Fig. 2, high values of θ 2 increase the minimal number of branches that are connecting the subgraphs [START_REF] Chung | Laplacians of graphs and Cheeger's inequalities[END_REF]. To define the Cheeger's inequality, a parameter h G is introduced. It is the minimal ratio between the number of branches that have to be removed to divide the graph into two disjoint subgroups and the number of nodes of the smallest subgraph. The inequality of Cheeger is expressed by:

θ 2 ≤ 2h G . (4)
It means that the algebraic connectivity gives the inferior bound of the number of branches required to disconnect a subgroup of nodes from the main graph. Or, for a given number of removed branches, the algebraic connectivity gives the superior bound of the size of the smallest disconnected subgraph. A high algebraic connectivity should therefore correspond to a well-connected network.

To each eigenvalue θ i is associated with an eigenvector ψ i defined by:

Lψ i = θ i ψ i . (5)
For a real symmetric matrix, eigenvectors are orthogonal, meaning that ψ t i .ψ j = 0 if i ̸ = j. And, every eigenvectors are normalized such that ψ t i .ψ i = 1. All the components of the first eigenvector associated to the zero eigenvalue are therefore equal to 1/ √ n. All the other eigenvectors being orthogonal to the first eigenvector, it results in that the sum of their entries is equal to zero.

Each eigenvalue θ i and its associated eigenvectors ψ i define the mode i.

Spectral solving of the DC network equations

A meshed DC electrical network is considered. It is supplied by constant current sources located at some nodes. The loads are modelled by constant current sources too. The network is balanced meaning that the total supply current is equal to the total load current.

The nodal currents vector I is projected onto the orthogonal eigenvector basis such that:

I = n ∑ i=1 ⟨I ⟩ i ψ i (6)
n is the number of nodes. ⟨I ⟩ i is the ith-component of the nodal currents vector relative to the eigenvectors basis. It is given by:

⟨I ⟩ i = ψ t i .I. (7)
The modal components ⟨I ⟩ i of the current vector are an image of the contribution of each mode to the nodal excitation of the electrical system. It shall be noted that ⟨I ⟩ 1 is equal to zero as the network is balanced.

According to Eq. ( 2), the nodal voltages vector V is given by:

I = L V (8)
L is the weighted Laplacian matrix of the graph related to the network. The weights are given by the conductance of the lines. The projection onto the eigen basis is therefore given by:

n ∑ i=1 ⟨I ⟩ i ψ i = L n ∑ i=1 ⟨V ⟩ i ψ i . (9)
It results from the definition and orthogonality of the eigenvectors that for every i except 1:

⟨V ⟩ i = ⟨I ⟩ i θ i (10)
⟨V ⟩ 1 is undetermined because ⟨I ⟩ 1 and θ 1 are both equal to zero. Its value depends on the choice of the reference voltage. For simplification purposes, we choose a zero value [START_REF] Rubido | Energy Transmission and Synchronization in Complex Networks: Mathematical Principles[END_REF]. Hence, the nodal voltages vector is expressed as:

V = n ∑ i=2 ⟨I ⟩ i θ i ψ i . (11)
Considering Eqs. ( 6) and ( 11), the spectral analysis expresses the network's electrical behaviour as a sum of simple behaviours related to the modes.

Voltage drops and nodal domains

The voltage drops across the branches are defined as the gradient of the voltage vector, meaning the voltage difference between two neighbouring nodes. It is noted ∇V where the gradient ∇ is defined by the transpose of the incidence matrix [START_REF] Caputo | Oscillations of networks: the role of soft nodes[END_REF]. The voltage drops are therefore given by:

∇V = n ∑ i=2 ⟨I ⟩ i θ i ∇ψ i . (12)
Voltage drops across the branches are linked to the nodal domains of the graph's eigenvectors.

A nodal domain D i related to mode i is defined by [START_REF] Dekel | Eigenvectors of random graphs: Nodal domains[END_REF]:

if k, l ∈ D i then ψ k,i ψ l,i ≥ 0. ( 13
)
In this equation, ψ k,i (respectively ψ l,i ) is the kth (resp. lth) entry of the eigenvector ψ i associated with mode i. Every eigenvector component is attached to a node. For instance, ψ k,i is attached to node k.

The definition [START_REF] Dorfler | Spectral analysis of synchronization in a lossless structure-preserving power network model[END_REF] means that two nodes k and l belong to the same domain D i if their related eigenvector components, respectively ψ k,i and ψ l,i are of the same sign. Some domains are therefore said to be positive if the signs of the eigenvectors components are positive. Negative domains are related to negative eigenvector entries. At the borders between two adjacent domains, the eigenvector components change of sign and the gradient of the eigenvector should be high.

According to the classical Courant's theorem, for every mode i, there is at most i nodal domains [START_REF] Dekel | Eigenvectors of random graphs: Nodal domains[END_REF]. A broad illustration is shown in Fig. 3 for an arbitrary 10-node graph. For the 2nd mode, there are only two domains, one positive and one negative. Indeed, the Courant's theorem states that there are at most two domains but the sum of the eigenvector components being equal to zero, some components must be positive where as some others must be negative. As a consequence, the exact number of domains for the second mode is exactly two. For the 10th mode, there are at most ten domains. The exact number of domains is not predictable by the theory. It could be less than ten. This strongly depends on the structure of the graph. Of course, if there are ten domains, every domain includes a single node and the sign of the eigenvector components change from node to node as it is broadly shown in Fig. 3.

The spectral graph theory provides therefore a multiscale reading of networks' connectivity. The second mode gives some indications about long-range connectivity of the system (i.e. connectivity between two sides of the graph, each side being covered by a nodal domain) whereas the higher modes are related to the local connectivity of nodes (i.e. connectivity between neighbours, each domain tending to be reduced to a few number of nodes).

Following Eq. ( 12), the critical voltage drops should be localized at the borders between nodal domains where the gradient is maximal due to the change of signs of the eigenvectors components. Because the voltage drops vary as the inverse of the eigenvalues, this effect should be magnified for the lower modes.

Spectral decomposition of DC resistance

Instead of looking at the voltage drops, the relation between modes and DC connectivity of the mesh grid can also be investigated through the spectral decomposition of the DC resistance of the path between two nodes. For that purpose, a constant nodal current is injected at node s and withdrawn at node l. All the other nodal currents are set to zero. The nodal currents vector I is therefore written as:

I = ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ 0 . . . I l = -I 0 . . . I S = I 0 . . . 0 ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ (14)
The components of the decomposition of this vector onto the Eigen basis are given by:

⟨I ⟩ i = ψ t i .I = ( ψ s,i -ψ l,i ) I 0 . (15)
The voltage drop between nodes s and l is given by the difference of the nodal voltages. Considering Eqs. ( 11) and ( 15), it is expressed by:

V s -V l = n ∑ i=2 ⟨I ⟩ i θ i ψ s,i - n ∑ i=2 ⟨I ⟩ i θ i ψ l,i = n ∑ i=2 1 θ i ( ψ s,i -ψ l,i ) 2 I 0 . (16)
The DC resistance is therefore given by:

R s,l = V s -V l I 0 = n ∑ i=2 ( ψ s,i -ψ l,i ) 2 θ i . (17)
As voltage drops, the DC resistance value is driven by the gradients of eigenvectors and by the inverse of eigenvalues.

Because the eigenvalues are ranked in ascending order such that 1

θ n ≤ • • • ≤ 1 θ 2
, DC resistance should be dominated by lower modes, especially by the algebraic connectivity θ 2 . Of course, it will depend on the nodes location too, meaning whether the two nodes are inside the same domain or not. This location will influence the values of eigenvectors gradients. For instance, if the nodes s and l are inside the same domain, it is expected that the DC resistance of the path is lower than if the nodes are in two distinct domains.

It results that DC connectivity of meshed electrical system is driven by its multiscale organization into nodal domains. An idea of its typical value is obtained by the average connectivity, expressed by the arithmetic mean of the DC resistance between two nodes. This mean resistance is defined by the sum of the individual path resistance divided by the number of pairs of distinct nodes s and l of the network.

R mean = 1 n (n -1) n ∑ s=1 n ∑ l=1 R s,l . (18)
Using the previous expression of R s,l [START_REF] Hien | Location and sizing of distributed generation units for loadabilty enhancement in primary feeder[END_REF]:

R mean = 1 n (n -1) n ∑ s=1 n ∑ l=1 n ∑ i=2 ( ψ s,i -ψ l,i ) 2 θ i . ( 19
)
Developing ( ψ s,i -ψ l,i
) 2 , the expression of the mean DC resistance becomes:

R mean = 1 n (n -1) n ∑ i=2 1 θ i [ n ∑ s=1 ( n ∑ l=1 ψ 2 s,i + n ∑ l=1 ψ 2 l,i -2ψ s,i n ∑ l=1 ψ l,i )] . (20)
Manipulating the summations over s and l, it is possible to write:

R mean = 1 n (n -1) n ∑ i=2 1 θ i [ 2 n ∑ s=1 n ∑ l=1 ψ 2 l,i - n ∑ s=1 ( 2ψ s,i n ∑ l=1 ψ l,i )] . ( 21
)
Due to the normalization of the eigenvectors, ∑ n l=1 ψ 2 l,i = 1. Hence:

R mean = 1 n (n -1) n ∑ i=2 1 θ i [ 2 n ∑ s=1 1 - n ∑ s=1 ( 2ψ s,i n ∑ l=1 ψ l,i )] . ( 22
)
For any i≥ 2, ∑ n l=1 ψ l,i = 0 because all the eigenvectors are orthogonal to the unitary eigenvector ψ 1 . Therefore:

R mean = 1 n (n -1) n ∑ i=2 2n θ i . ( 23
)
Since eigenvalues are sorted in an increasing order, it is obvious that, in average, the DC resistance of the network is limited by the inverse of the algebraic connectivity, i.e.:

R mean ≤ 2 θ 2 . ( 24
)
Consequently, and independently of the nodal domains distribution, DC impedance of the meshed grid should be lowered in average by a higher algebraic connectivity. This has motived the formulation of an optimization problem based on the maximization of the second eigenvalue.

Looking for an optimal network

In order to increase the connectivity, and therefore to decrease the voltage drops across meshed networks used for carrying direct current, the optimization process is intended to maximize the algebraic connectivity for a given network's mass. Indeed, increasing performances of on-board electrical systems shall not be paid by an extra-mass of the overall structure. The geometry of the network will not be changed by the optimization process, meaning that the nodes will not be moved of location. Only the branch sections will be changed. However, their variations will be kept bounded. In this section, two optimization formulations are proposed and tested.

Explicit maximization

Branch weights are given by branch admittances y branch . They are defined as follows:

y branch = S branch ρ.l branch (25)
ρ, l branch and S branch are respectively branch material resistivity, length and section. Because all nodes locations are chosen to be fixed, only branches sections can be changed and they will be defined as the optimization variables.

Moreover, the mass of a branch can be expressed as:

M branch = µ mat .l branch .S branch (26)
where µ mat is the material density. The total mass of the network is calculated as the sum of the mass of all branches. As it is not intended to increase the mass during the optimization, it is kept constant to its value M 0 before optimization. Finally, the optimization problem can be formulated as:

min 1/θ 2 subject to ∑ M branch = M 0 l b ≤ S branch ≤ u b (27)
In this formulation, θ 2 is maximized by minimizing 1/θ 2 . Branch sections are bounded between a lower value l b and an upper value u b , where the lower value has to be positive. This formulation raises two issues. It requires the computation of the algebraic connectivity at each iteration of the optimization process. It may be time consuming. Furthermore, this optimization problem is not guaranteed to be convex, meaning that the global minimum may be hard to be reached.

LMI-based formulation

To avoid the calculation of eigenvalues, numerous optimization problems are transformed into Linear Matrix Inequality problems (LMI) based on positive semi-definite matrix properties. It is worth mentioning the Generalized Eigenvalue Problem that consists in minimizing the largest eigenvalue [START_REF] Boyd | Linear Matrix Inequalities in System and Control Theory[END_REF]. Moreover, LMI problems can be solved numerically very efficiently using interior-point methods.

The previous optimization problem is transformed into an alternative problem that consists in imposing a LMI constraint to the set of eigenvalues of the Laplacian matrix [START_REF] Shafi | Designing node and edge weights of a graph to meet Laplacian eigenvalue constraints[END_REF]. The constraint is defined by introducing a variable λ and a matrix Q with dimensions n × (n -1). The columns of Q are the eigenvectors associated to the non-zero eigenvalues. The constraint is written as:

Q t (L + λE) Q ≽ 0. ( 28
)
Thanks to the orthogonal properties of matrix Q, the inequality (28) means that all the eigenvalues, except the first one θ 1 which is equal to zero, are higher than -λ. The minimization of λ therefore leads to a maximization of the algebraic connectivity.

The LMI-based optimization formulation is finally written as:

min λ subject to Q t (L + λE) Q ≽ 0 ∑ M branch = M 0 l b ≤ S branch ≤ u b . (29)
The values of branch sections S branch control the optimization. Laplacian matrix is defined by Eq. ( 1). According to [START_REF] Boyd | Linear Matrix Inequalities in System and Control Theory[END_REF], the LMI constraint is convex. And the two supplementary constraints on mass equality and sections are linear. Therefore, this formulation yields to a convex problem. This should make easier the solving because any local minimum will be a global minimum.

Application to a current return network

Presentation of the network

A reduced-scale mock-up of a current return network has been built by Safran Electrical & Power. It is called DEMLAB and is shown in Fig. 4.

The mock-up is 3.6 m long and 1m wide. The main conductive elements are the cabin floor, the frames, the raceways, the crossbeams and some copper strips. They are connected together to constitute a low impedance meshed network. It was previously used for validating some numerical solvers for low frequencies problems (grounding in normal conditions, lighting current carrying), reflectometry techniques and contact resistance measurements. One current source is located at one end of the network, and one load can be connected at 30 different nodes.

DEMLAB's graph is shown in Fig. 5. It has 244 nodes and 284 branches. All the branches are made of copper, and their sections are between 32 and 112 mm 2 .

Optimization results

The two formulations introduced in the previous section are now used for optimizing DEMLAB's network. Both of them have been implemented in Matlab. The first optimization is solved by the Fmincon function [START_REF] Byrd | A trust region method based on interior point techniques for nonlinear programming[END_REF]. The second formulation is performed using the Dual-Scaling Semidefinite Programming method (DSDP) from the OPTimization Interface (OPTI) Toolbox [START_REF] Benson | DSDP5: Software for semidefinite programming[END_REF]. In both cases, the solver uses the interior point method with gradient calculation performed by a finite differences approximation. In addition, the DSDP solver exploits the LMI properties in order to enhance the optimization performances.

During the optimization process, branch sections are allowed to vary between 2 and 200 mm 2 . The mass of the network is constant and fixed at its initial value of 28.8 kg. The initial value of the algebraic connectivity θ 2 is 71.7.

After optimization, both formulations provide a higher value of the algebraic connectivity. It has been increased by 9.8%, reaching the value of 78.7. The modifications of the branch sections are shown in Figs. 6 and7. The sections of the transverse elements of the cabin floor have been reduced whereas the sections of some top conductive elements and frames have been increased. It results in an improvement of the global connectivity while the total mass of the network is kept constant.

The optimization performances are summarized in Table 1. The solving of the LMI-based formulation is faster than the explicit maximization. Although both optimizations have nearly the same number of iterations, the explicit maximization is 2.5 times longer. The main reason is the calculation of θ 2 at each iteration. On the other hand, the LMI formulation exploits positive semi-definiteness of Laplacian matrix and calculations of eigenvalues are not needed. But the problem building is slower since it requires to calculate the eigenvectors of the Laplacian Matrix and to create the LMI constraint.

Verifying voltage drop criteria

As previously mentioned, the voltage drops at the loads connection nodes should not be higher than 0.5 V for a current return network operating in nominal conditions. As the proposed optimization approaches do not include voltage drop constraints, they have to be verified after processing the optimization.

For that purpose, a numerical circuit-oriented simulation of DEMLAB is performed. To simulate the nominal operating conditions, a 300 A DC current source is connected at node 212 (node circled in green in Fig. 5) and one single load is connected at any of the 30 possible load locations (nodes 1 to 30). Resulting voltage drops are presented in Fig. 8 according to load location.

Surprisingly, even if the network algebraic connectivity has been increased by the optimization, the network quality has been downgraded. Indeed, most of the voltage drops have been increased except for loads locations 1, 18, 29 and 30. To understand why such a surprising result has been obtained, a further look at path spectrum is required.

Path spectrum analysis

Firstly, the path between the source and node 18 is analysed. The spectrum decomposition of its DC resistance is expressed by [START_REF] Hien | Location and sizing of distributed generation units for loadabilty enhancement in primary feeder[END_REF]. The values of the first 22 modal components of the decomposition are shown in Fig. 9. The other components are negligible.

It can be seen that, for this path, the most dominant mode is the second one. The optimization objective being the maximization of θ 2 , the DC connectivity has been increased by the optimization and therefore the voltage drop has been decreased as shown in Fig. 8. even still not below 0.5 V.

On the other side, the spectrum of the path between the source and the node 12 is shown in Fig. 10. The five most dominant modes are the 3rd, 13th, 5th, 8th and 4th ones. Furthermore, the second mode is not influential at all.

Even if the optimization process has increased the algebraic connectivity, it has decreased at the same time the modes 5, 8 and 13 as shown in Table 2. As a result, the path resistance has been increased and therefore its voltage drop too. This is why most of the voltage drops have been deteriorated for almost all the loads location in spite of the higher algebraic connectivity.

To conclude, enforcing θ 2 may improve the connectivity of certain paths of the network. However, the voltage drops depend on all the eigenvalues, and increasing θ 2 is not always sufficient to improve the overall voltage profile. For paths not depending dominantly on θ 2 , this optimization does not guarantee a better network performance. Our results show that it may even downgrade the voltage drops. A further optimization should be carried on to maximize all the dominant eigenvalues of the paths between electrical sources and loads. Another option is to optimize the DC resistance itself as a function of Laplacian eigenvalues [START_REF] Boyd | Convex optimization of graph Laplacian eigenvalues[END_REF].

Conclusion

In this paper, the DC connectivity of aircraft current return networks was analysed by using spectral graph analysis. Thanks to the multi-scale reading offered by the graph's spectrum, we show that a better connected network is achieved by increasing the second eigenvalue. It was also expected that a network with higher θ 2 has lower voltage drops.

An LMI-based algorithm was presented to perform the algebraic connectivity optimization of a meshed graph by changing sections of all branches while keeping the total mass constant. The algorithm was tested with success for a representative aircraft return current network. On the other hand, it has been shown that not all the voltage drops were improved by optimization because they are not all ruled by the algebraic connectivity.

Despite of these limitations, the developed approach is promising because it offers to scientists and engineers an in-depth dissection of complexity of meshed electrical pathways used for ensuring current return network functions in more composite aircraft. In addition, it provides a pre-design tool based on spectral optimization. Some theoretical works have also proven that it could certainly be used with success to perform in the future dynamic analysis of complex electrical networks or assess their robustness to the removal of some branches [START_REF] Mahoney | Lecture Notes on Spectral Graph Methods[END_REF].

  Lower bound of the branch sections (m) l branch Branch length (m) m Number of branches of a graph n Numbers of nodes of a graph u b

Fig. 1 .

 1 Fig. 1. ESN structure of A350 (picture taken from [1]).

Fig. 2 .

 2 Fig. 2. Two subgraphs connected by a set of branches k (m ≥ p).

Fig. 3 .

 3 Fig. 3. Broad illustration of the nodal domains of a 10-node graph. Positive values of ψ k,i are shown by yellow ascending arrows. Negative values are shown by green descending arrows.

Fig. 4 .

 4 Fig. 4. DEMLAB mock-up at Safran Electrical & Power's laboratory.

Fig. 5 .

 5 Fig. 5. DEMLAB's graph with source location (green circle) and possible load locations (red squares).

Fig. 6 .

 6 Fig. 6. Branch sections of DEMLAB before optimization.

Fig. 7 .

 7 Fig. 7. Branch sections of DEMLAB after optimization.

Fig. 9 .

 9 Fig. 9. Spectrum of path source-node 18.

Fig. 10 .

 10 Fig. 10. Spectrum of path source-node 12.

Table 1

 1 Optimization performances.Fig. 8. Source-load voltage drops according to load location.

		Explicit maximization LMI-based formulation
	Number of iterations	73	68
	Problem building time [s] 1,3	9,3
	Resolution time [s]	668,4	227,4
	Total time [s]	669,7	236,7

Table 2

 2 Variation of some eigenvalues of path source-node12 after optimization.

	Eigenvalue	Variation [%]
	θ 2	9.96
	θ 3	2.45
	θ 13	-8.64
	θ 5	-8.86
	θ 8	-17.85
	θ 4	1.87


Patrice Droz

Jean-François Seignol
email: jean-francois.seignol@ifsttar.com

Raphaël Leroy

Liviu Boldea

STRATEGY FOR THE REHABILITATION OF SALANFE DAM

Keywords: AAR-treatment, monitoring, gravity dam, numerical modelling

This paper presents the ongoing study of a gravity dam affected by alkali-silica reaction (ASR) and the strategy planned for its rehabilitation. The aim is to lower its stress-state by slot-cutting. The position and number of these slot cuttings will be determined by use of numerical modelling. The first steps of the study are described: analysis of in situ monitoring of the dam, laboratory tests on concrete samples, numerical model representing the consequences of ASR on the structure, introduction of special elements to take into account the slot-cutting and preliminary computations to elaborate a global strategy.

INTRODUCTION

Dam managers have sometimes to face internal swelling reaction in their structure (Charlwood, 2011). The main problem induced by these pathologies lies in the change occurring in the material stress-state, with potential consequences on structural reliability. Hence, evaluation of the modified stress-state is the main goal of structure re-assessment based on numerical modelling. But the numerical tool may also make it possible to design and calculate treatment solutions such as reducing internal stresses by slot-cutting.

These questions are illustrated in this paper by the case-study of Salanfe dam, a gravity dam in which Alkali-Silica Reaction (ASR) was proven and the displacement of which have been monitored for several years.

Material investigations and structure monitoring are first described, focusing on the ASR-induced displacements. The strategy for the dam rehabilitation is then introduced. Last, the numerical tool aimed at dimensioning the slot-cutting procedure is introduced, as well as first results obtained on preliminary numerical models.

THE DAM AND ITS PATHOLOGY 2.1 Description of the dam

The Salanfe Dam was built in 1952. It is located near the city of Martigny (Valais, Switzerland) at an altitude of 1925 m above sea level on the Salanfe Plateau in Swiss Alps. It is used to produce hydroelectricity from the waters of the Salanfe and Saufla Rivers. The dam holds a reservoir of 40 mio. m 3 of capacity.

Salanfe is a concrete dam classified in the gravity-dam category, according to its profile. The upstream face is vertical and the downstream face has a slope of 1:0.742 to the 1915.60 m level, then 1:0.2 to the crest. The crest length can be divided into 4 straight sections: a central part which is 260.65 m long, the right wing composed of 2 sections of respectively 74 m and 76 m, and the left wing measuring 206 m. The total length at the crest is 616.65 m; the maximum height above the foundations is 52 m. The foundation of the dam is located in sound gneissic rocks and of excellent geomechanical resistance. See Figures 1 and2.

The dam consists of 42 blocks which are for the most part 14 m long and separated by construction joints. Each joint contains a vertical shaft with a diameter of 100 cm. Some of these shafts are equipped with a pendulum. Inspection galleries measuring 2 m wide by 2.5 m high are located at various levels between levels 1882.00 and 1908.00 from block 6 to 33. They are accessed from the downstream face via seven transversal galleries as well as via vertical shafts positioned on the construction joints. The dam is equipped with an overflow spillway and the intake is located in the dam body (block 18).

Monitoring

The dam is equipped with several monitoring systems such as pendulums and extensometers and, since 1993, with a geodetic and levelling network to follow the evolution of its deformations. The results of this equipment allow disorders to be detected so that the engineers can intervene to correct the problem in due time. The dam has constantly been deformed by the hydrostatic load and the effects of seasonal thermal variations.

Blocks 13 and 23 are equipped with direct pendulums whose measures are interpreted since the initial setting in November 1953. New pendulums were added in blocks 18, 28 and 29 since 2005.

Pendulums in blocks 13 and 23 allow the long-term monitoring of horizontal displacements. As shown on figure 3, the drift started soon after the dam was built and then accelerated in the 1970s. Since the dam was commissioned, the total irreversible downstream-upstream displacement is approximately 35 mm.

A geodetic network was set up in 1993 in order to monitor deformations of the dam and its supporting soil. As for the downstream-upstream movements, the vertical displacement of the crest has continued over time. In 17 years, the elevation has reached 35 mm at points 125 and 127. It has also doubled in the last ten years compared to the reference state. Figure 4 shows the downstream-upstream movements of the crest.

The upstream drift of the crest is amplified by the bends giving an "arched" form to the plan. This situation is typical of arch dams where swelling of the concrete leads to an increase in the length of the arch, which is only possible outwards, namely upstream.

The swelling in the main part of the dam between blocks 11 and 27 pushes the extremities of this part in opposite directions. Detailed inspections exhibited cracks in the galleries and on the downstream face. Crack meters have been set in transversal gallery in block 29 and in the lower inspection gallery in block 28. Their monitoring exhibits irreversible displacements both as crack opening and in the direction perpendicular to the wall facing.

ASR detection General

The different symptoms observed on the dam are signs of an expansive reaction occurring in the concrete. To ascertain its nature, material samples are analyzed. The aim of these tests is the identification of the concrete components, the characterisation of the microstructure and the presence of swelling reaction traces.

The aggregate is siliceous limestone. Main mineral phases identified by XRD are quartz (silica) and calcite. Petrographic examination reveals the presence of a large quantity of aggregates showing signs of potential ASR (Alkali-silica reaction), particularly gneiss and mica schists exhibiting rough and wavy-edged quartz. Observation on a Philips XL 30 SEM (Scanning Electron Microscope) was also performed.

Results

Cement grains are relatively well hydrated and the remains of non-hydrated clinker grains are rare. Next to the clinker, neither slag nor flying ashes are visible in the binder. Chemical analyses have shown the presence of alkali in the cement paste that should not be neglected.

The concrete shows a high and heterogeneously distributed apparent porosity (macropores) in its microstructure. A relatively high porosity in the transition areas between the cement paste and the aggregates is also observed, which tends to weaken the concrete cohesion.

In every sample the presence of siliceous-calcareous-alkali gel was observed (see Figure 5), demonstrating the presence of alkali reaction. This gel is cracked or showing hillocks and is principally situated at the interface between the cement paste and the aggregates.

Small quantities of ettringite are also observed in every sample. Its habitus is frequently in aggregates of thin needles, but can also be massive in some rare cases. It corresponds to secondary ettringite, which is commonly visible in concretes showing alkali reaction; in such cases, the available alkali are used by the silicagel formation. Hence, in the case of the Salanfe concrete, the presence of ettringite is not linked to an internal sulfatic reaction.

These observations and measurements conducted on the seven concrete samples have led to the conclusion that an alkali reaction has developed in the Salanfe dam concrete.

Material tests

Once the ASR ascertained, different laboratory tests have been performed on a large set of sample cores. Some specimens are submitted to residual expansion tests [START_REF] Lcpc | Alcali-réaction du béton -Essai d'expansion résiduelle sur béton durci[END_REF] in order to evaluate the residual swelling potential in the concrete. These tests are still ongoing. Other are used to assess physical and mechanical properties. Testing was performed in compliance with SIA standards. They have been performed by Holcim Technical Center in Eclépens. Sample cores have been drilled out from blocks 13, 15 and 29, in areas considered as representative with regards to observed concrete degradation.

Compressive strength :

The obtained compressive strength varies between 9.8 and 50.2 MPa with a mean value around 27 MPa. Whereas this mean value is normal for this kind and this age of concrete, the large scattering (even amongst samples from the same block) means a strongly heterogeneous material. 60% of the samples gave strength lower than the mean value. It can be due to drilling in areas with initial weakness (such as honeycombing or bad compaction) or in areas where material was damaged by ASR.

Young's modulus :

Once again a large scattering is observed, from 7,000 to 36,500 MPa. One third of tested samples were under the mean value of 22,100 MPa, for the same reasons as compressive strength tests.

Tensile strength

The tensile strength varies from 0.2 to 2.7 MPa with a mean value of 1.2 MPa, which is 1 MPa lower than expected value when considering material compressive strength.

Water permeability

These measurements are consistent with the strong heterogeneity observed with regards to mechanical properties, being scattered from 5.7 to 38.6 g/m 2 /h.

REPAIR TECHNIQUE

Managing and repairing ASR-affected structures is a difficult matter. Up to now, there is no efficient technique to prevent the reaction from developing. There only exists methods affecting ASR-effects and trying to limit the induced degradation [START_REF] Godart | The Rilem guidance on appraisal and management of structures damaged by AAR[END_REF].

Purpose of repairing

The rehabilitation project for Salanfe dam aims at returning it to acceptable stress and strain states in order to guarantee dam safety for every situation.

Different techniques

Different techniques have been tested on various ASR-affected concrete structures. They can be divided into four types.

Protection against water ingress

Various waterproof membranes or coatings can be applied to reduce water ingress and, hence, slow down ASR-kinetics [START_REF] Fasseu | L'expérience française sur le traitement des ouvrages[END_REF], but their efficiency on structures as large as dams is not assured [START_REF] Charlwood | Managing AAR affected dams[END_REF]

Reinforcement with anchoring, prestressing or rebars

This set of techniques can be applied to reduce concrete strain by confining it and to improve structural safety [START_REF] Fasseu | L'expérience française sur le traitement des ouvrages[END_REF][START_REF] Charlwood | Rehabilitation Techniques for AAR Affected Dams and Hydro-Electric Projects[END_REF]. However, its efficiency often decreases when applied to large and massive structures.

Injection of chemical or gas (lithium)

This technique consists in having lithium ions infiltrating the concrete, since this chemical species can inhibit ASR. The main problem lies in the ion-diffusion process, which limits this technique to very thin structures (such as concrete roads, for instance).

Stress-release by slot-cutting

The fourth possibility consists in lowering the compressive stresses induced by prescribed concrete swelling by sawing a part of the structure, as detaile infra.

Slot-cutting

The rehabilitation project developed for Salanfe dam is based on sawing the upper part of the dam in order to release compressive stress and ergo lower the arching effect due to the elbows in the dam. Since Salanfe is a gravity-dam, it does not need any arching effect to correctly resist the water pressure and this is why the rehabilitation aim could be to totally restore the initial stress state. Nevertheless, aiming to have a small remaining compressive stress level after slot-cutting was considered as a better option in order to use the arching-effect to keep the joints closed.

Slot-cutting with diamond-wire has already been applied to other dams, for instance Mactaquac in Canada [START_REF] Curtis | Analysis and structure response to recent slot cutting at Mactaquac Generating Station[END_REF], Chambon in France, Pian Tessio in Italy, and Center Hill, TN in the USA [START_REF] Hull | Center-Hill Dam. Mitigating Effects of Alkali-Aggregate Reaction with Vertical Slots[END_REF].

The sawing device is composed of a diamond-wire the diameter of which is about 10 to 15 mm, a main wheel anchored in the downstream face and pulleys. Wire diameter may vary according to the slot height to compensate for strain induced by sawing.

The rehabilitation project does not take into account other techniques in order to restrain or stop future ASR-development, but some crack injections, mainly in the galleries, may also be scheduled. Complementary measures as well as repeating slot-cutting can of course be studied in further phases of rehabilitation, depending on the monitoring of the dam evolution.

In this initial step, the aim is to define a first slot-cutting scenario. A scenario consists in giving the following directives:

 position of the (various) slot-cutting;  their height (from crest down to altitude z);

 width of the slot-cutting.

In the case of future supplementary slot-cutting, optimization of schedule should also been studied, in particular in the case of Salanfe dam, the accessibility of which is complex.

USE OF A NUMERICAL TOOL

Numerical modelling is used for preliminary testing of different slot-cutting scenarios. The following section presents the preliminary computations used to determine the strategy of the global study.

Purpose of the modelling (vérifier tout l'orthographe avec un dictionnaire anglais US

The main purpose of the study is to provide the dam manager with different slot-cutting scenarios, and for each one its advantages with regards to the dam behaviour. These scenarios will then be compared, taking into account their cost (varying from one scenario to another) and practical feasibility. Advantages brought by a given scenario will be assessed according to:

 First, the short-term re-closure of each slot-cutting in order to avoid water ingress (even if it is possible to inject the slot-cutting afterwards); from a numerical point of view, it means the existence of compressive stresses in the two opposite faces of the slot-cutting.  Lowering of stress in the dam, which is compared to initial stress state without rehabilitation; the new stress state will be compared to dam safety requirement (concerning both stresses in the dam concrete and in its supporting soil). The resulting stress state must be assessed near the slotcuttings, but also at a significant distance in order to estimate a sort of "influence zone" around each slot-cutting.  Lowering of dam drift (both in the upstream-downstream and vertical directions), or even a displacement back to the initial deformation of the sound dam. Note that decreasing these abnormal displacements is not the real aim of the rehabilitation project, but they represent the easiest means to monitor symptoms of ASR.

Numerical model for the dam

The finite element model used to assess the dam state is composed of a 760 m by 325 m by 98 m block representing the soil and the dam itself (see figure 6). Two models have been developed for this study. The first one, used for the following preliminary tests, is composed of 30,000 quadratic elements (14,000 for the sole dam) representing about 130,000 nodes. A second one, more refined, will be used to better represent large stress gradients occurring around the slot cutting as well as the galleries.

ASR-affected concrete is represented by LCPC model [START_REF] Li | Concrete ASR degradation: from material modelling to structure assessment[END_REF] completed by the relation binding stressstate and ASR-expansion anisotropy proposed by Multon and Toutlemonde [START_REF] Multon | Effect of applied stress on alkali-silica reaction-induced expansion[END_REF]. Since residual expansion tests are still ongoing at the beginning of the numerical modelling step, mode parameters are adjusted with the help of monitored displacements [START_REF] Leroy | Re-assessment and treatment-design of an ASR-affected gravity dam[END_REF].

Special elements for slot-cutting

Special elements have been developed [START_REF] Omikrine-Metalsi | Integration of contact elements and creep in RGIB-modulus of the FE-program Cesar-LCPC. Application to structures affected by internal swelling reactions[END_REF] in order to represent the mechanical effect of slot-cutting. They are based on classical contact elements, the resolution of which is obtained through stiffness-penalty technique. Their main characteristic is to represent the slot-cutting width w by authorizing a limited interpenetration (of value w) of initially superposed nodes to represent the sudden disappearance of concrete. This makes it possible to represent each slot-cutting by elements of zero thickness, otherwise a very strong refinement of the FE mesh would be compulsory, which could dramatically increase the size of the problem to model.

In addition to the interpenetration test, which provides information on whether compressive stress can occur again through slot-cutting, these elements take into account friction (based on Coulomb's law) which checks whether the two parts will stay together in the future or if some kind of "shear-locking" device would be necessary.

Preliminary study: position of slot cutting

In these preliminary computations, two slot cuttings have been defined, as shown on figure 7. Their influence on displacements and stresses in the left elbow is discussed. Slot-cutting SC1 is located on the right of the elbow, at 170 m; slot-cutting SC2 is on the left at 40 m. Both are 20 mm wide and represent one-third of the total height of the dam at the same position. Four modelling with an ASR free-expansion of 0.25% are compared: no slot-cutting (SC0), only SC1, only SC2, and SC1 and SC2 together (SCall). Figure 8 shows the left-right displacements when only SC1 is active, which exhibits the strong discontinuity along the slotcutting.

When considering stresses and displacements in the left elbow, very similar results are obtained with models SC0 and SC1 on one hand, with SC2 and SCall on the other. This exhibits the fact that slot cutting has a limited range of effect since sawing the dam at 170 m away from the elbow does not modify its behaviour, whereas slot-cutting at 40 m does. For instance, compressive stress in the elbow 5 m below the crest remains unchanged with model SC1, whereas it decreases by 37% with models SC2 and SCall.

This preliminary study advocates making a large number of slot-cuttings, regularly spaced, in order to obtain the best influence on the global dam behaviour.

CONCLUSIONS

This paper presents an ongoing study of a gravity dam affected by ASR. The purpose of this study is to define and validate a rehabilitation strategy aimed at restoring structure safety by lowering its stress-state. Up to now, the following points have been achieved:

 exploiting the various monitoring devices to exhibit the abnormal behaviour of the dam due to ongoing expansive reaction;  ascertaining the occurrence of ASR by laboratory tests on concrete samples;  proposing a remedial treatment based on slot-cuttings;  developing a numerical model of the dam taking into account the effect of diamond-wire sawing of the concrete;  preliminary tests to elaborate a global strategy. Although numerical modelling of the dam is still in progress, first conclusions can be drawn from preliminary models:

 performing slot-cutting do have an influence on dam stress-state and displacements;

 this influence has a limited range, which advocates for slot-cutting in several places along the dam. Aknowledgements: The authors would like to express their greatest thanks to Patricia Roure (Itech) for her help.

FIGURE 1 :FIGURE 4 :FIGURE 5 :

 145 FIGURE 1: upstream developped elevation of the dam

FIGURE 6 :

 6 FIGURE 6: general view of the mesh of the dam and its soil

FIGURE 8 :

 8 FIGURE 8: discontinuity of longitudinal displacement around slot-cutting SC1.