Dirk Slock
email: dirk.slock@eurecom.fr

Convergent Approximate Message Passing

Generalized Approximate Message Passing (GAMP) allows for Bayesian inference in linear models with non identically independently distributed (n.i.i.d.) priors and n.i.i.d. measurements of the linear mixture outputs. It represents an efficient technique for approximate inference which becomes accurate when both rows and columns of the measurement matrix can be treated as sets of independent vectors and both dimensions become large. It has been shown that the fixed points of GAMP correspond to extrema of a large system limit of the Bethe Free Energy (LSL-BFE), which represents a meaningful approximation optimization criterion regardless of whether the measurement matrix exhibits the independence properties. However, the convergence of (G)AMP can be notoriously problematic for certain measurement matrices and the only sure fix so far is damping (by a difficult to determine amount). In this paper we revisit the GAMP algorithm by rigorously applying an alternating constrained minimization strategy to an appropriately reparameterized LSL-BFE with matched variable and constraint partitioning. This guarantees convergence, at least to a local optimum.

I. INTRODUCTION

In the Gaussian case, the signal model for the recovery of a sparse signal vector x can be formulated as, y = Ax + v, where y are the observations or data, A is called the measurement or the sensing matrix which is known and is of dimension M ×N with typically M < N . In the sparse model case, x contains only K non-zero (or significant) entries, with K < M < N . In Bayesian inference, the Sparse Bayesian Learning (SBL) algorithm was first proposed by [START_REF] Tipping | Sparse Bayesian Learning and the Relevance Vector Machine[END_REF], [START_REF] Wipf | Sparse Bayesian Learning for Basis Selection[END_REF]. SBL is based on a two or three layer hierarchical prior on the sparse coefficients x. The priors for the hyperparameters (precision parameters) are chosen such that the marginal prior for x induces sparsity, allowing the majority of the coefficients to tend towards zero. It is worth mentioning that [START_REF] Giri | Type I and type II bayesian methods for sparse signal recovery using scale mixtures[END_REF] provides a detailed overview of the various sparse signal recovery algorithms which fall under l 1 or l 2 norm minimization approaches such as Basis Pursuit, LASSO etc and SBL methods. The authors justify the superior recovery performance of SBL compared to the above mentioned conventional methods. Nevertheless, the matrix inversion involved in the Linear Minimum Mean Squared Error (LMMSE) step in SBL at each iteration makes it computationally complex even for moderately large data sets. This complexity is the motivation behind approximate inference methods. Belief Propagation (BP) based SBL algorithms [START_REF] Tan | Computationally Efficient Sparse Bayesian Learning via Belief Propagation[END_REF] are computationally more efficient. A more detailed discussion on the various approximate inference methods for SBL appears in [START_REF] Thomas | Low Complexity Static and Dynamic Sparse Bayesian Learning Combining BP, VB and EP Message Passing[END_REF]. Various studies on the convergence analysis of Gaussian BP (GaBP) can be found in [START_REF] Du | Convergence Analysis of Distributed Inference with Vector-Valued Gaussian Belief Propagation[END_REF]- [START_REF] Cseke | Properties of Bethe Free Energies and Message Passing in Gaussian Models[END_REF]. Although BP achieves great empirical success [START_REF] Murphy | Loopy belief propagation for approximate inference: an empirical study[END_REF], not enough rigorous work exists to characterize the convergence behavior of BP in loopy networks. In [START_REF] Malioutov | Walk-Sums and Belief Propagation in Gaussian Graphical Models[END_REF] a convergence condition for GaBP is provided which requires the underlying distribution to be walk-summable. Their convergence analysis is based on the Gaussian Markov random field (GMRF) based decomposition, in which the underlying distribution is expressed in terms of the pairwise connections between the variables. The Approximate Message Passing (AMP) algorithm has been introduced to further reduce complexity of GaBP. In Generalized AMP (GAMP), the vector x can have non-Gaussian priors and the measurement process can be more general than linear with additive Gaussian noise. However, the convergence of (G)AMP can be problematic for certain measurement matrices A. Many variations have been introduced to help (G)AMP converge, such as adding Alternating Direction Method of Multipliers (ADMM), exploiting part of the singular value decomposition of the measurement matrix in Vector AMP (VAMP) (but which does not allow to handle n.i.i.d. priors conveniently), sequential updating in Swept AMP (SwAMP) which works almost always, and especially by introducing damping with the typically difficult to determine damping requirements. The AMP algorithm and its variations have many potential applications in (machine learning aided) wireless communications systems:

• multi-user detection [START_REF] Liu | Sparse Signal Processing for Grant-Free Massive Connectivity[END_REF],

• channel estimation [START_REF] Thomas | Variational Bayesian Learning for Channel Estimation and Transceiver Determination[END_REF],

• joint detection and channel estimation [START_REF] Ngo | Multi-user detection based on expectation propagation for the non-coherent simo multiple access channel[END_REF],

• compressive sensing [START_REF] Thomas | Low Complexity Static and Dynamic Sparse Bayesian Learning Combining BP, VB and EP Message Passing[END_REF],

• reduced complexity Linear Minimum Mean Squared Error (LMMSE) receiver or transmitter computation [START_REF] Thomas | Variational Bayesian Learning for Channel Estimation and Transceiver Determination[END_REF].

A. Contributions of this paper

• We propose a version of GAMP with guaranteed convergence, by rigorously applying an alternating constrained minimization strategy with matched variable and constraint partitioning. We apply this strategy to an appropriately augmented Lagrangian of the constrained Large System Limit of the Bethe Free Energy. • The new GAMP, dubbed AMBGAMP below, requires to solve for the mean constraint Lagrange multipliers s appearing in the posterior mean x(s) to make it satisfies this mean constraint. This can be done by bisection. We also propose a first-order approximation of x(s), which resembles the Method of Moments. We furthermore remark that this first-order approximation becomes exact in the Gaussian case, of which we work out the details also. • We also indicate that asymptotically, under an i.i.d. element model for A, the variance computations in AM-BGAMP are exact. This allows to analyze the steady-state MSE as a function of system dimensions, prior pdfs and measurement pdfs. In particular in the Gaussian case, this allows to analyze the performance for SBL.

II. GENERALIZED APPROXIMATE MESSAGE PASSING

The data model considered in GAMP is essentially a linear mixing model

z = A x , p x (x) , p y|z (y|z) (1)
with (possibly) non identically independently distributed (n.i.i.d.) prior

p x (x) = N i=1 p xi (x i ) and n.i.i.d. measure- ments p y|z (y|z) = M k=1 p y k |z k (y k |z k ).
In Bayesian estimation we are interested in the posterior, which is given by

p x,z|y (x, z|y) = 1 Z(y) e -N i=1 fx i (xi)-M k=1 fz k (z k ) 1 {z=Ax}
(2) where we have the negative loglikelihoods for prior and measurements

f xi (x i ) = -ln p xi (x i ) , f z k (z k ) = -ln p y k |z k (y k |z k ) (3)
where the equality in case of f z k (z k ) is up to constants that may depend on y (and which are absorbed in the normalization constant Z(y)). The problem in Bayesian estimation is the computation of this constant Z(y) and of the posterior means and variances. Belief propagation is a message passing technique that allows to compute the posterior marginals. However, due to loops in the factor graph, loopy belief propagation may have convergences issues and is furthermore still relatively complex. GAMP is an approximate belief propagation technique which is motivated by asymptotic considerations in which the rows and columns of the measurement matrix A are considered as random and independent, in which case GAMP can actually produce the correct posterior marginals. In any case, GAMP computes a separable approximate posterior of the form

q x,z (x, z) = q x (x) q z (z) = N i=1 q xi (x i ) M k=1 q z k (z k ) (4)
in which the dependence on y has been omitted. The GAMP algorithm [START_REF] Rangan | Generalized Approximate Message Passing for Estimation with Random Linear Mixing[END_REF], [START_REF] Rangan | Fixed Points of Generalized Approximate Message Passing with Arbitrary Matrices[END_REF] appears in the table for Algorithm 1. We only consider here Sum-Product GAMP (for MMSE estimation, as opposed to Max-Sum GAMP for MAP estimation).

III. AMBGAMP

AMB is short for ACM-LSL-BFE: Alternating Constrained Minimization of the Large System Limit of the Bethe Free Energy. As we shall see, AMBGAMP uses most of the same updates as GAMP, but GAMP does not rigorously follow

Algorithm 1 GAMP Require: y, A, S = A.A, f x (x), f z (z) 1: Initialize: t = 0, x t , τ t x , s t-1 = 0 2: repeat 3: [Output node update] 4: τ t p = S τ t x 5: p t = A x t -s t-
τ t r = 1./(S T τ t s )
12:

r t = x t + τ t r .A T s t 13:
x t+1 = E(x|r t , τ t r )

14:

τ t x = var(x|r t , τ t r ) 15: until Convergence the principle of alternating minimization (block coordinate descent) esp. in the presence of constraints. It has been shown that any fixed point of the GAMP algorithm is a critical point of the following constrained minimization of a Large System Limit (LSL) of the Bethe Free Energy (BFE) (see [START_REF] Rangan | Fixed Points of Generalized Approximate Message Passing with Arbitrary Matrices[END_REF] and references therein):

min qx,qz,τp J LSL-BF E (q x , q z , τ p ) s.t. E(z|q z ) = A E(x|q x ) τ p = S var(x|q x ) (5)
where the LSL BFE is given by

J LBF E (q x , q z , τ p ) = D(q x ||e -fx )+D(q z ||e -fz )+H G (q z , τ p ), with H G (q z , τ p ) = 1 2 M k=1 var(z k |q z k ) τ p k + ln(2π τ p k ) (6)
and where D(q||p) = E(ln( q p ) |q) is the Kullback-Leibler distance (KLD) and H G ( q z, τ p ) is a sum of a KLD and an entropy of Gaussians with identical means but different variances. The LSL BFE optimization problem (6) can be reformulated with the following augmented Lagrangian min qx,qz,τp,u max s,τs L(q x , q z , τ p , u, s, τ s ) with

L = D(q x ||e -fx )+D(q z ||e -fz )+H G (q z , τ p ) +s T (E(z|q z ) -A E(x|q x )) -1 2 τ T s (τ p -S var(x|q x )) + 1 2 E(x|q x ) -u 2 τr + 1 2 E(z|q z ) -A u 2 τp ( 7
) where s, τ s are Lagrange multipliers, and τ r = 1./(S T τ s ) is just a short-hand notation for quantities that depend on τ s . We also use the notations: u 2 τ = i u 2 i /τ i , element-wise multiplication as in s.τ and element-wise division as in 1./τ . We interpret the constraints as follows: E(z|q z ) = A E(x|q x ) is interpreted as a constraint on E(z|q z ), and τ p = S var(x|q x ) (which is a vector of the individual variances) is interpreted as a constraint on τ p . To interpret constraints as constraints on a subset of the variables, such subset should be rich enough to allow to satisfy the constraints. Due to the updating order, the other variables will be fixed actually as can be seen further. So the alternating optimization of [START_REF] Du | Convergence Analysis of the Information Matrix in Gaussian Belief Propagation[END_REF], which corresponds to alternating minimization of the constrained problem [START_REF] Du | Convergence Analysis of Distributed Inference with Vector-Valued Gaussian Belief Propagation[END_REF], should be carried out in the following way. In the partitioning of the variables to be updated, the Lagrange multipliers for the constraints in which a given subset of variables is involved, should be optimized at the same time as that subset of variables. Such alternating optimization policy guarantees the cost function to decrease at each update, and hence to converge, to at least a local optimum. We propose to follow the following updating order

{q z , s} → {u} → {τ p , τ s } → {q x } . (8)
In other words, at iteration t we have the following sequence

{q t z , s t } = arg min qz max s L(q t-1 x , q z , τ t-1 p , u t-1 , s, τ t-1 s )( 9
)
{u t } = arg min u L(q t-1 x , q t z , τ t-1 p , u, s t , τ t-1 s ) ( 10
)
{τ t p , τ t s } = arg min τp max τs L(q t-1 x , q t z , τ p , u t , s t , τ s ) ( 11
)
{q t x } = arg min qx L(q x , q t z , τ t p , u t , s t , τ t s ) (12)
A. Update of {q z , s}

The most tricky part is the update of {q z , s}. To that end, consider

L(q t-1 x , q z , τ t-1 p , u t-1 , s, τ t-1 s ) = D(q z ||e -fz ) + 1 2 var(z|q z )./τ t-1 p +s T E(z|q z ) + 1 2 E(z|q z ) -A u t-1 2 τ t-1 p + const. = D(q z ||e -fz ) + 1 2 E(z T z|q z )./τ t-1 p -(E(z|q z )) T ((A u t-1 )./τ t-1 p -s) + const. = D(q z ||e -fz ) + 1 2 E( z -p t (s) 2 τ t-1 p ) + const. ( 13
)
where const. denotes constants w.r.t. z and

p t (s) = A u t-1 -s.τ t-1 p . (14)
This cost function is separable. We get per component

min qz k D(q z k ||g t z k /Z t z k ) ⇒ q t z k = g t z k /Z t z k Z t z k (s k ) = g t z k (z k ; s k ) dz k , -ln g t z k (z k ; s k ) = f z k (z k ) + 1 τ t-1 p k ( z 2 k 2 -z k A k,: u t-1 ) + z k s k (15)
where A k,: denotes row k of A. Note that the partition function Z t z k acts as cumulant generating function:

- ∂ ln Z t z k ∂s k = E(z k |q t z k ) = E(z k |p t k (s k ), τ t-1 p k ) = z t k (s k ) ∂ 2 ln Z t z k ∂s 2 k = var(z k |p t k (s k ), τ t-1 p k ) = τ t z k (s k ) - ∂ 3 ln Z t z k ∂s 3 k = E(z k -E z k ) 3 (16)
To satisfy the mean constraint in ( 5), we require s t k to satisfy

z t k = z t k (s t k ) = A k,: x t-1 , τ t z k = τ t z k (s t k ) . ( 17
)
The second derivative in [START_REF] Rangan | Generalized Approximate Message Passing for Estimation with Random Linear Mixing[END_REF] shows that z t k (s k ) is a monotonically increasing function, which means that solving [START_REF] Rangan | Fixed Points of Generalized Approximate Message Passing with Arbitrary Matrices[END_REF] for s t k can be done by bisection. Alternatively, we can approximate with a first-order Taylor series expansion (the thirdorder cumulant (kurtosis) in ( 16) will be zero for symmetric measurement pdfs, in which case the next non-zero term in the Taylor series expansion is a fourth-order term!)

z t k (s t k ) ≈ z t k (s t-1 k ) + τ z k (s t-1 k ) (s t k -s t-1 k ) = A k,: x t-1 (18)
from which we get

s t k = s t-1 k + 1 τ z k (s t-1 k ) (A k,: x t-1 -z t k (s t-1 k )) (19)
which is very similar to a Method of Moments (MM) update of the Lagrange multiplier. As a result, the algorithm can be expected to converge even with this (good) approximation (but one can also avoid this approximation). This MM update is actually exact in the Gaussian case.

B. Update of u

From ( 7), ( 10), we get

L(q t-1 x , q t z , τ t-1 p , u, s t , τ t-1 s ) = 1 2 x t-1 -u 2 τr + 1 2 A x t-1 -A u 2 τp + const. ( 20
)
where const. denotes constants w.r.t. u. The minimizer is obviously

u t = x t-1 . (21)
Note that due to the update of (only) {q z , s} just before, we have z t = A x t-1 which greatly simplifies this update of u. In contrast to [START_REF] Rangan | Inference for Generalized Linear Models via Alternating Directions and Bethe Free Energy Minimization[END_REF] where a complex update of u is required which is not compatible with the fast AMP style algorithms.

C. Update of {τ p , τ s } Due to the preceding update of u, the two quadratic terms shown in 20 are now zero. As a result, the dependence of those terms on τ p , τ s via the weights disappears for the update of {τ p , τ s } which comes next. Hence the terms of interest in [START_REF] Murphy | Loopy belief propagation for approximate inference: an empirical study[END_REF] for ( 8) are now

L(q t-1 x , q t z , τ p , u t , s t , τ s ) = H G (q t z , τ p ) -1 2 τ T s (τ p -S τ t-1 x ) + const. = const.+ 1 2 M k=1 τ t z k τ p k + ln(2π τ p k ) - 1 2 M k=1 τ s k (τ p k -S k,: τ t-1 x ) (22)
where const. denotes constants w.r.t. {τ p , τ s }. The optimization yields

∂L ∂τ s k = 0 ⇒ τ t p k = S k,: τ t-1 x (23) ∂L ∂τ p k = 1 2 (- τ t z k τ 2 p k + 1 τ p k -τ s k ) = 0 ⇒ τ t s k = 1 τ t p k (1 - τ t z k τ t p k ) (24)
(25)

D. Update of q x

For the update of q x in (12) finally, consider the relevant terms in the augmented Lagrangian (and remember that

τ t r = 1./(S T τ t s ) or 1./τ t r = S T τ t s ) (11) L(q x , q t z , τ t p , u t , s t , τ t s ) = D(q x ||e -fx ) -s t T A E(x|q x ) + 1 2 τ t T s S var(x|q x ) + 1 2 E(x|q x ) -u t 2 τ t r + const. = D(q x ||e -fx ) + 1 2 (1./τ t r ) T E(x.x|q x ) -s t T A E(x|q x ) -(u t ./τ t r )) T E(x|q x ) + const. = D(q x ||e -fx ) + 1 2 (1./τ t r ) T E(x.x|q x ) -(u t + τ t r .A T s t ) T (E(x|q x )./τ t r ) + const. = D(q x ||e -fx ) + 1 2 E( x -r t 2 τ t r |q x ) + const. (26)
where const. denotes constants w.r.t. x and

r t = u t + τ t r .A T s t . ( 27
)
This cost function is separable. We get per component

min qx k D(q x k ||g t x k /Z t x k ) ⇒ q t x k = g t x k /Z t x k Z t x k = g t x k (x k ) dx k , -ln g t x k (x k ) = f x k (x k ) + 1 2τ t r k [(x k -r k ) 2 -r 2 k ] . (28)
Again the partition function Z t x k acts as cumulant generating function:

τ t r k ∂ ln Z t x k ∂r k = E(x k |q t x k ) = E(x k |r t k , τ t r k ) = x t k (τ t r k ) 2 ∂ 2 ln Z t x k ∂r 2 k = var(x k |r t k , τ t r k ) = τ t x k . (29)
Again simplifications arise in the Gaussian case, and approximations for more general cases are possible, as suggested earlier for the moments of q z k .

IV. AMBGAMP LARGE SYSTEM ANALYSIS

In GAMP, as opposed to AMP, we may not have (simple) analytical updates for means and variances. As a result, the take on large system analysis (LSA) for GAMP is from a different angle. If both the rows or the columns of A are now modeled as independent, then given that also the priors on x and z are independent (factorized), the true posteriors for x and z will become factorized and will equal the approximate τ t r = 1./(S T τ t s )

Algorithm 2 AMBGAMP Require: y, A, S = A.A, f x (x), f z (z) 1: Initialize: t = 0, x t-1 , τ t-1 x , u t-1 , τ t-1 p , s t-1 = 0 2: repeat 3: [Output node update] 4: p t (s) = A u t-1 -s.τ t-1 p 5: z t (s) = E(z|p t (s), τ t-1 p ) 6: s t = arg{ z t (s) = A x t-1 }, p t = p t (
13:

[Input node update] 14:

r t = u t + τ t r .A T s t
15:

x t = E(x|r t , τ t r )

16:

τ t x = var(x|r t , τ t r ) 17: until Convergence posteriors q x (x), q z (z). So multiplication with A or A T acts like scrambling in CDMA communications, that renders the individual outputs independent. Furthermore, the marginal posteriors are the product of the respective prior and extrinsic distributions that correspond to information coming through A or A T , the random nature of which will lead to Gaussian extrinsic distributions by the central limit theorem. In other words, in the LSA, in which the dimensions of x and z (the two dimensions of A) tend to infinity at a constant ratio, the approximate posteriors handled in GAMP become asymptotically exact. As a result, the variance information propagated by GAMP corresponds asymptotically to the exact MSE of the (MMSE) estimates propagated by GAMP. The existing GAMP steady-state analysis results are valid, as they assume that the algorithm has converged to such steady-state. Such steady-state analysis appears in [START_REF] Rangan | Generalized Approximate Message Passing for Estimation with Random Linear Mixing[END_REF] (particularly in the extended arxiv version), or in [START_REF] Javanmard | State Evolution for General Approximate Message Passing Algorithms, with Applications to Spatial Coupling[END_REF].

V. SBL-AMBGAMP LARGE SYSTEM ANALYSIS

In this Gaussian case, MMSE estimation becomes LMMSE, for which we have investigated LSA in [START_REF] Thomas | Posterior Variance Predictions in Sparse Bayesian Learning under Approximate Inference Techniques[END_REF] using large random matrix theory. It can be checked that the LSA of the general GAMP case above reduces to these same results in the Gaussian case.

VI. CONCLUDING REMARKS

We have drawn attention to an approach to perform alternating constrained optimization, which is also called block coordinate descent for optimization problems with constraints. The approach consists of not only partitioning the variables appearing in the cost function, but also to partition the constraints according to this variable partitioning and to identify for each constraint subset the variable subset that can be used to satisfy these constraints. In the alternating optimization of the cost function w.r.t. each variable subset, the possible corresponding constraint subset should be involved in the constrained optimization sub-problem. To arrive at the convergent AMBGAMP algoritrhm, a particular formulation of the Bethe Free Energy criterion is considered with a judiciously chosen Method of Moments extension to incorporate constraints quadratically. The Lagrangian of the resulting augmented BFE is then introduced. The alternating optimization of the resulting cost function only leads to the desired algorithm of low complexity when alternating optimization is done in one particular order. Other updating orders would also lead to convergent algorithms but not to low complexity updates.

  s t )

	7: 8:	τ t z = var(z|p t , τ t-1 p u t = x t-1	)
	9:	[Variance matching]
	10: 11:	τ t p = S τ t-1 x τ t s = (1 -τ t z ./τ t p )./τ t p
	12:

ACKNOWLEDGEMENTS

EURECOM's research is partially supported by its industrial members: ORANGE, BMW, SAP, iABG, Norton LifeLock, and by the French ANR project CellFree6G.


AVERTISSEMENT

Ce document est le fruit d'un long travail approuvé par le jury de soutenance.

La propriété intellectuelle du document reste entièrement celle du ou des auteurs. Les utilisateurs doivent respecter le droit d'auteur selon la législation en vigueur, et sont soumis aux règles habituelles du bon usage, comme pour les publications sur papier : respect des travaux originaux, citation, interdiction du pillage intellectuel, etc.

Il est mis à disposition de toute personne intéressée par l'intermédiaire de l'archive ouverte DUMAS (Dépôt Universitaire de Mémoires Après Soutenance).

Si vous désirez contacter son ou ses auteurs, nous vous invitons à consulter en ligne les annuaires de l'ordre des médecins, des pharmaciens et des sagesfemmes.

Contact à la Bibliothèque universitaire de Médecine Pharmacie de Grenoble : bump-theses@univ-grenoble-alpes.fr

L'UFR de Médecine de Grenoble n'entend donner aucune approbation ni improbation aux opinions émises dans les thèses ; ces opinions sont considérées comme propres à leurs auteurs.

[Données à caractère personnel]

REMERCIEMENTS

Je souhaite tout d'abord remercier chaleureusement les membres du jury :

Le président du jury, le Professeur Jean François PAYEN, de me faire l'honneur de présider ce jury.

Ma directrice de thèse, le Professeur Anne Claire TOFFART de m'avoir confié un sujet si passionnant. Tu as toujours fait preuve d'une grande disponibilité et d'une grande implication dans ma formation. Je te remercie sincèrement de ton aide, de ta confiance et de tes précieux conseils même quand je te sollicite à des heures tardives.

Le Professeur Charlotte COSTENTIN, un chaleureux merci pour votre présence aujourd'hui.

Le Docteur Louis Marie GALERNEAU, pour avoir accepté ́ de faire partie de ce jury, merci pour ce semestre en réanimation que j'ai adoré malgré la première vague COVID et ces nombreuses gardes en USC ensemble. Un grand merci d'avoir pris le temps de me faire ce fameux cours qui m'a permis de beaucoup mieux comprendre les stat' et d'avoir été disponible pour toutes nos questions de dernière minute.

Le Docteur Alexandre GODON, merci d'avoir participé à ce projet et d'avoir accepté de faire partie du jury.

Je remercie également notre statisticien, François Bettega, merci pour ta patience face à nos nombreuses demandes et explications sur les OR, multivariée et autres choses bien abstraites pour nous.

Je remercie tout particulièrement ma binôme Sophie JERUSALEM, avec qui j'ai pris beaucoup de plaisir à travailler. Nous avons formé une belle équipe vraiment complémentaire, sans toi rien n'aurait été possible. A mes moments de doute où tu as su me rassurer, à nos moments de solitude téléphonique à répéter constamment les mêmes questions de ce SF36. Cette galère est enfin derrière nous, nous pouvons être fières.

Je remercie également l'ensemble des médecins et des équipes paramédicales avec qui j'ai eu le plaisir de travailler :

Mes premières assistantes, le Docteur Johanna CLAUSTRE et le Docteur Anne Laure MOUNAYAR : quelle chance d'avoir eu des personnes aussi calées que vous en 1 er semestre, vous avez accompagné mes premiers pas d'internes, merci de votre aide et de votre confiance. Anne Laure, merci pour tous ces avis infectieux, pour ta spontanéité et pour toutes tes petites histoires que j'adore écouter.

Aux autres médecins du service de pneumologie, Pr PISON, Pr DEGANO, Pr MORO-SIBILOT, Pr TAMISIER, Pr BAYAT, mention spéciale pour Sébastien qui a toujours été disponible durant tout mon internat, merci de ton implication pour l'organisation du poste de Dr Junior, Christel, Rebecca, Amandine, Marie, Rita, François merci pour tous vos enseignements dans cette ambiance conviviale et familiale. Vous m'avez tant appris.

Léonie, on a eu l'occasion de travailler ensemble seulement 3mois, mais tu es devenue une amie, je te remercie pour ta gentillesse et ta bienveillance. J'espère pouvoir un jour te rejoindre dans le cabinet pour continuer à bosser ensemble. En attendant, on continue nos repas avec nos rhumatologues préférés. Julian (alias K-maro) et Hélène merci pour votre bonne humeur à toute épreuve. Victor, merci pour ce semestre à Annecy, le meilleur semestre de mon internat ! Mes cointernes, devenu Docteur : Thomas, j'espère que tu ne ronfleras pas au prochain CPLF, Mickaël, j'attends toujours l'apéro sur ta terrasse à Gières, on a passé des bons moments au cours du GOLF, CPLF, et autres congrès. Justin, le plus gentil des râleurs, j'espère avoir mis beaucoup de paillettes dans ta vie pendant ce semestre en HDJ (rappelle-toi les fiches greffes, les embrouilles à J2 dans le couloir d'HDJ, le secrétariat du bip d'avis). Hubert pour ta bonne humeur et ton rire que l'on entend à des km, Loïc, pour être toujours disponible pour répondre à mes questions et surtout pour manger tous les oeufs de la salade à l'internat. Arnaud le roi des endoscopistes, Margaux, Salomé, Héloise.

A mes cointernes : Romain, le Réunionnais par excellence, merci d'avoir supporté nos discussions de filles avec Giulia pendant ces longs trajets vers Clermont Ferrand pour nos cours de DES. Geoffroy, tes nombreuses chemises vont nous manquer. Merci aux autres internes, Lucie devenu anesthésiste mais qu'on aime quand même, Amy, Marie, Perrine, Murielle, Estelle et Camille.

Le service de pneumologie (la team qui m'a trouvé mon nouveau surnom Wassi'), d'oncologie thoracique (mention spéciale aux gazelles du 6 ème ) et EFR du CHU de Grenoble, infirmières, aides-soignantes, ASH, cadre, secrétaires, kinés. J'ai eu plaisir et j'ai toujours plaisir à travailler avec vous dans une très bonne ambiance. L'ensemble des équipes médicales et paramédicales des services d'infectiologie et de réanimation médicale. Merci de votre accueil, du partage d'expériences et de m'avoir rappelé l'importance du travail pluri-disciplinaire.

À mes amis

Ma Dreamteam : à mes copines de toujours, mes soeurs de coeurs, à celles qui ont sont là depuis des années, on a tout connu ensemble. Merci pour cette amitié très précieuse. Laura, si je devais te définir en 1 mot ce serait la positivité (non, pas le retard haha), merci d'être cette personne qui ne se prend jamais la tête, d'être cette amie débordante d'énergie et toujours partante pour tout, je te souhaite toute la réussite du monde dans tes nouveaux projets professionnels. Magali, notre maman à tous, merci pour ces longues heures de papouille, tous les cupcakes et autres muffins, merci d'avoir pris soin de nous à chaque voyage, hâte de rencontrer la petite merveille que tu portes. Mélissa, tout a commencé à St Martin de Crau dans un camping en tente (nous dans une tente, qui l'eut cru..) tu es un petit clown au quotidien, une amie vreuuuument, sur qui je pourrais toujours compter sauf peut -être le matin très tôt (c'est à dire avant 12h..). Nawelle, ma plus vieille amie, de la marelle, à la danse orientale les jours de cantine, au lit cassé à Passins, aux sorties ski où je me cassais la gueule avec Philippe qui me laissait descendre avec un seul ski, 21ans d'amitié et ce n'est pas fini. Roxanne, mon extraordinaire amie (oui tu as bien lu), ma tulipe, je cherche encore pourquoi je t'aime autant mais je dirais probablement pour le fameux micro-onde, je ne vois pas d'autre chose.

Holfa, tu as quand même failli nous abandonner pour partir vivre en Australie, heureusement que tu es vite revenue au bout de quelques mois. A tous nos week-ends Genevois, à ta joie de vivre et ton grain de folie qu'on aime tant.

Ma team Paillettes : Giulia ma première cointerne, je me rappelle encore de ce 1 er jour au 4eme D, nos dictées à n'en plus finir, nos confidences dans les vestiaires, nos beignets Nutella qui égayaient notre après-midi, surtout les jeudis sans fin (visite-staff greffe-colloque..), on n'a pas réussi à tenir notre promesse de faire tout notre internat ensemble mais ce qui est sûr c'est qu'on restera amies. Juliette et Bébé Domi enfin Zoé, la femme enceinte la + active que je n'ai jamais connu, c'est bon grâce à Zozo tu vas enfin devenir la reine de l'organisation. Cessou et son doux accent de Perpignaaan, rappelle-toi la première fois qu'on s'est vu dans ce petit hôtel parisien pour le CPLF, les temps ont bien changé depuis ce jour. Orlane, notre pneumologue de coeur, merci pour le soutien pendant notre semestre à la Mut' et surtout merci d'être toi, la zinzin de notre team. A nos repas où on ne peut pas s'empêcher de toute parler en même temps, à nos soirées potins, à nos week-ends en Provence et à tous nos prochains week-ends où on évitera de prendre un logement avec 5 chambres pour finalement toutes dormir dans la même (on attend toujours le Doodle d'ailleurs….).

Mathilde, alias Schach, mon hématologue préférée. Merci à la réanimation et au COVID de m'avoir permis de te rencontrer et d'avoir noué une vraie amitié. Merci pour ta bonne humeur, notre concert de Jennifer (cf video), notre défilé de mode (surblouse orange fluo, nouvelle co'), notre plus belle interprétation du Titanic sur les guéridons, à nos transports de galère à l'IRM, sans parler de mes tentatives de pose de KT sur toi, mais surtout à nos pauses quotidiennes potins-chocolat dans le vestiaire.

Sana, tu as fait de mon semestre à Annecy le meilleur de tout mon internat. Merci pour ta générosité, ta gentillesse et ton sens de la fête. Grosse dédicace à « Jean-Michel chimiothérapie alias Jean-Mi» qui a adoré écouter les L5 toute la soirée.

Chawki, on se connait depuis maintenant plus de 20ans, première rencontre à l'école primaire des Ruires, depuis ce jour on a toujours été là l'un pour l'autre. Tu as rencontré et épousé Naomi qui te comble de bonheur. Même si on se voit beaucoup moins depuis votre départ à Lyon, je ne vous oublie pas.

Sabrine, merci pour ta gentillesse, ta sérénité et ta joie de vivre. Je suis sûre que tu vas devenir un très bon médecin.

Luca, mon frate, le plus rital des italiens (sicilien pardon). Merci à nos 2 noms de familles qui se suivent et qui nous ont permis d'être dans les mêmes groupes durant tout l'externat. Enfin je ne sais pas si tu as été externe ou brancardier d'ailleurs. Merci d'avoir été toujours présent. Tu es aujourd'hui à plus de 15000km tranquille sur la plage, mais je sais que tu penses à moi.

À ma famille

Mes parents, je vous dois absolument tout. Je vous remercie d'avoir fait de moi la personne que je suis devenue, merci pour votre éducation et votre confiance. Merci pour votre dévouement de chaque instant durant toutes ces années d'études où j'ai pu être chiante. Si je suis là aujourd'hui, c'est grâce à vous. Je vous aime Mes frères et soeurs : Adel, mon grand frère tu as été comme un 2 ème père durant mon enfance, merci pour tous ces diabolos et tous ces bonbons que tu as pu m'acheter (c'est le dentiste qui a dû être content..), tu n'es pas présent aujourd'hui mais promis on t'enverra pleins de vidéos, j'espère te revoir rapidement dès que le Covid sera derrière nous. Somaya, merci de m'avoir considéré comme ta fille, merci pour toutes les fois où tu m'as trimbalé en vacances (la pause repas sur l'aire de 2h qui nous a mis en retard au camping, les merguez dans le coffre direction Coulommiers et j'en passe). Sabrina, comment faire dans la vie sans une soeur qui gère tes impôts, ta paperasse, tes prises de RDV, ton organisation du pot de thèse et même tes retours de bouée licorne trouée quand tu n'oses pas le faire, merci pour toutes ces petites choses du quotidien qui m'ont bien facilité la vie. Brahim, merci d'avoir été le témoin de mon mariage, merci pour tes petites blagues (non, mettre sa cigarette entièrement dans la bouche pour faire rire la galerie ce n'est pas bien, dixit la nouvelle pneumologue), et merci pour tes énigmes que tu adores, entre autres l'énigme de l'euro manquant que j'ai dû entendre des millions de fois. Ilhem, ma plus fidèle patiente, merci pour ton franc-parler, merci pour toutes ces aventures à Marseille (les galères en bus, métro et bateau), à Paris (c'est vraiment un travail de m…), Lyon (il a di ci yeux …) etc. Toto, merci d'avoir été la meilleure wedding-planner que l'on puisse avoir, merci d'être toujours disponible pour des conseils de tout genre. Merci de m'avoir nourrie toutes les fois où j'étais seule, merci pour tous ces bonbons que j'ai pu manger dans tes placards (en cachette parfois, pour ne pas rendre jalouse princesse Miraculouuus).

Kheira, ma presque soeur, nous avons grandi ensemble. A nos jeux de rôle où on finissait toujours par appeler le vigile, à nos sauts au-dessus du Verderet comme Tarzan, à nos fameux repas « viens on mange comme des riches » où on s'essuyait délicatement le coin de la bouche, à nos délires de s'inventer une nouvelle langue dans la rue, à nos rires de « folle ». Cette année 2021 est la fin de nos études à toutes les 2, nous pouvons être fières.

Mes beaux-frères et belles-soeurs : Rima, Ben, Krimo, Dalila, Achour, Mohamed. Merci de vous être occupé de moi comme de votre soeur. Merci pour tous ces week-end, anniversaires et autres repas de famille.

Mes neveux, je ne vais pas tous vous citer un par un sinon ça risque d'être plus long que ma thèse en elle-même haha. Même si je vous ai engueulé de nombreuses fois quand je devais réviser et que vous préfériez jouer et faire du bruit, même si je vous exploite parfois pour me masser, vous occupez une place importante dans mon coeur.

Ma grand-mère qui doit être fière de là-haut.

Mon oncle, parti il y a 1an, toi qui attendais tant ce jour pour que je devienne enfin ta pneumologue. Results: Overall, 23 surviving ICU cancer patients were matched with 46 non-ICU cancer patients. All patients answered the survey except for four pertaining to the ICU group. MCS was significantly higher in ICU patients than non-ICU patients, with a median of 54 (interquartile range [IQR]: 42-57) versus (vs.) 47 (IQR: 37-52), respectively (p = 0.01). PCS did not differ between both groups, with a median of 35 (IQR: 31-47) vs. 42 (IQR: 34-47) (p = 0.24).

Ma belle famille

The multivariate analysis revealed a non-significant association between ICU/non-ICU patient status and patient quality-of-life scores (MCS or PCS) at 3 months post-discharge. The therapeutic strategy based on anticancer treatment at 3 months post-discharge was similar between both groups. A good performance status (PS) (coef. 1.46, confidence interval

[CI]95%: 1.11-1.90, p = 0.01) and non-metastatic cancer status (coef. 1.21, CI95%: 1.05-1.40, p = 0.01) were independently associated with a higher PCS.

INTRODUCTION

Over the last decades, improvements in care of cancer patients (1-3) has resulted in their increased admission into intensive care units (ICUs). Indeed, cancer patients occupy 15 to 20% of ICU beds. [START_REF] Soares | Effects of Organizational Characteristics on Outcomes and Resource Use in Patients With Cancer Admitted to Intensive Care Units[END_REF][START_REF] Darmon | Changes in critically ill cancer patients' short-term outcome over the last decades: results of systematic review with meta-analysis on individual data[END_REF] The reasons for their ICU admission are mostly cancer-related complications, treatment-associated undesirable effects, or organ failures unrelated to cancer.

If these patients require intensive care, they chiefly display the same illness severity than noncancer patients, with comparable outcomes [START_REF] Taccone | Characteristics and outcomes of cancer patients in European ICUs[END_REF]. The ICU and hospital mortality rates of cancer patients are around 20-50% and 30-60%, respectively, according to studies. (7-13) Known risk factors associated with a higher mortality include poor performance status (PS), recurrent or progressive malignancy, and higher Sequential Organ Failure Assessment (SOFA) scores at admission. Therefore, new policies of ICU admission have been proposed to cancer patients, including early admission, non-invasive diagnostic and therapeutic strategies, and timelimited treatment trial. [START_REF] Thiéry | Outcome of cancer patients considered for intensive care unit admission: a hospital-wide prospective study[END_REF][START_REF] Lecuyer | The ICU trial: a new admission policy for cancer patients requiring mechanical ventilation[END_REF][START_REF] Azoulay | Diagnostic strategy for hematology and oncology patients with acute respiratory failure: randomized controlled trial[END_REF][START_REF] Azoulay | Intensive care of the cancer patient: recent achievements and remaining challenges[END_REF][START_REF] Shrime | Time-Limited Trials of Intensive Care for Critically Ill Patients With Cancer: How Long Is Long Enough?[END_REF][START_REF] Azoulay | The Intensive Care Medicine research agenda on critically ill oncology and hematology patients[END_REF] Short-and long-term health related quality of life (QoL) of critically ill patients is lower than that of an age-and gender-matched population [START_REF] Oeyen | Quality of life after intensive care: a systematic review of the literature[END_REF][START_REF] Langerud | Health-related quality of life in intensive care survivors: Associations with social support, comorbidity, and pain interference[END_REF][START_REF] Vogel | Health-related quality of life after general surgical intensive care[END_REF][START_REF] Gerth | Changes in health-related quality of life after discharge from an intensive care unit: a systematic review[END_REF][START_REF] Hofhuis | Health-related quality of life in ICU survivors-10 years later[END_REF]. Major reductions in long-term QoL are observed in cases of severe acute respiratory distress syndrome, prolonged mechanical ventilation, and severe sepsis, which represent complications that likely affect cancer and noncancer patients [START_REF] Oeyen | Quality of life after intensive care: a systematic review of the literature[END_REF]. Only few data are available on QoL of cancer patients following ICU discharge. In a cohort of 483 cancer patients, a significantly reduced QoL was reported one year following ICU discharge compared to QoL recorded prior to admission [START_REF] Oeyen | Long-term outcomes and quality of life in critically ill patients with hematological or solid malignancies: a single center study[END_REF]. No study has so far compared QoL of cancer patients requiring ICU admission with that of those not requiring such an admission but whose QoL may be impaired by the malignancy itself.

The aim of this study was to assess the impact of an ICU stay on QoL of cancer patients at 3 months post-ICU discharge, in comparison with that of those cancer patients not requiring intensive care.

METHODS

Patients

Patients were eligible for enrollment into the ICU patient group if they exhibited lung, colorectal, or head and neck cancer, in the event of unscheduled ICU admission, hospitalization for at least 24 hours, and discharge alive. We matched these patients to non-ICU cancer patients with respect to three criteria including primitive cancer type, treatment goal (curative/palliative), and treatment line. Two non-ICU cancer patients were matched to one ICU cancer patient. Exclusion criteria were age <18 years, cancer in remission for more than 2 years, patients unable to answer a survey by phone, patients' scheduled admission into ICU, patients with guardianship, and patient opposition to the study. At first step, an unmatched pool of non-ICU patients was established comprising two patients of each cancer type (lung, colorectal or head and neck). Each time one ICU patient was identified, two options were available, including either two non-ICU patients were identified at medical oncological consultation within the next two weeks or from the unmatched control pool; if no non-ICU patient was available, the ICU patient was entered into the pool of cases.

(Figure 1) Patients in the control pool could be matched with a case patient at any time. ( 26) If a non-ICU patient was secondarily admitted into the ICU, this patient then became an ICU patient at discharge, whereas he/she could no longer be a non-ICU patient. In case of death of a case or control patient before the 3-month evaluation, the associated case or control was returned in the respective pool. ICU and non-ICU patients could stay for one year within these pools and be matched at any time. Non-ICU patients were regularly searched for in order to be matched with ICU patients from the pool of cases. When there were no more patients in the pool of controls, other non-ICU patients were recruited from medical oncological consultations and included in the control pool.

Data collection

Demographic data (age, gender, most recent PS, and Charlson comorbidity Index) [START_REF] Quan | Updating and validating the Charlson comorbidity index and score for risk adjustment in hospital discharge abstracts using data from 6 countries[END_REF] and cancer-related characteristics (metastatic status and anticancer treatment) were collected from medical records. Intensive care data (severity scores like SOFA at 24 hours and the worst score, (28) IGS2 ( 29), organ failures during ICU stay, and length of stay) were similarly recorded. At 3 months, quality of life and anticancer treatment data were collected, as were PS, independence, anxiety and depression scores evaluated using validated surveys (see below). These data were derived using standardized phone calls conducted by the same evaluators or from medical records.

Endpoints

The primary outcome was the QoL evaluated using the 36-Item Short Form survey instrument (SF-36) through a telephone call 3 months after inclusion. The SF-36 questionnaire contains 36 items measuring eight domains including physical and social functioning, role limitations due to either physical or emotional problems, mental health, vitality, bodily pain, and general perception of health. Two summary scores, a Physical Component Summary (PCS) and Mental Health Component Summary (MCS), were calculated based on these items and expressed as percentages, higher scores indicating better perceived health [START_REF] Ware | The MOS 36-item short-form health survey (SF-36). I. Conceptual framework and item selection[END_REF][START_REF] Mchorney | The MOS 36-Item Short-Form Health Survey (SF-36): II. Psychometric and clinical tests of validity in measuring physical and mental health constructs[END_REF][START_REF] Mchorney | The MOS 36-item Short-Form Health Survey (SF-36): III. Tests of data quality, scaling assumptions, and reliability across diverse patient groups[END_REF][START_REF] Perneger | Validation of a French-language version of the MOS 36-Item Short Form Health Survey (SF-36) in young healthy adults[END_REF]. The validity and reliability of the SF-36 were previously confirmed in the critically ill population [START_REF] Chrispin | Short Form 36 in the intensive care unit: assessment of acceptability, reliability and validity of the questionnaire[END_REF]. General health, which was evaluated using the PS ranging from 0 (good general status) to 4 (bad general status), independence, which was evaluated using the activities of daily living (ADL) questionnaire ranging from 0 (dependent) to 6 (independent), anxiety and depression, which was evaluated using the hospital anxiety and depression scale (HADS) including a score for anxiety (HADS-A) and another for depression (HADS-D), each ranging from 0 to 21 (no symptomatology if < 7), in addition to the oncological care project (continuation or modification of anticancer treatment according to the disease progression), which were all secondary outcomes evaluated at 3 months.

Statistical analysis

We hypothesized that the mean SF-36 at 3 months would be 35/100 in ICU cancer patients versus 45/100 in non-ICU cancer patients, based on scientific literature [START_REF] Langerud | Health-related quality of life in intensive care survivors: Associations with social support, comorbidity, and pain interference[END_REF][START_REF] Oeyen | Long-term outcomes and quality of life in critically ill patients with hematological or solid malignancies: a single center study[END_REF][START_REF] Van Vliet | Longterm health related quality of life following intensive care during treatment for haematological malignancies[END_REF][START_REF] Ehooman | Long-term health-related quality of life of critically ill patients with haematological malignancies: a prospective observational multicenter study[END_REF]. With a statistical power of 90%, an alpha risk of 5%, and 20% of patients lost to follow-up, we calculated that we needed to include 20 cases and 40 controls.

Population characteristics were expressed as number and percentages for qualitative variables, and median and interquartile range (25 to 75 th quartiles) for quantitative variables.

Comparisons between the two groups were performed using a Chi-squared test or Fisher's exact test for qualitative variables, and a nonparametric Mann-Whitney U test for quantitative variables. We performed all tests in a two-sided manner. The alpha risk was set at 0.05. The primary outcome analysis was performed after excluding patients who did not answer the QoL questionnaire at 3 months, in addition to their matched cases or controls. The multivariate analysis was carried out using a generalized linear model with a gamma distribution on factors not used for matching (age and gender) and clinically pertinent variables (PS and metastatic cancer status). Multivariate analysis assumptions were supported by doing so. R version 4.0.3 was employed for all statistical analyses.

RESULTS

Patients' characteristics at inclusion

A prospective individual matching (Figure 1) was applied [START_REF] Charpentier | An algorithm for prospective individual matching in a non-randomized clinical trial[END_REF]. During the study period, 123 cancer patients (lung, colorectal, and head and neck cancers) were discharged alive from the ICU, 31 of whom were recruited, with 23 still alive at 3 months. Overall, 46 non-ICU cancer patients were identified, thereby matching with these last ICU patients (Figure 2).

Most patients (n=39, 57%) displayed lung cancer, 18 (26%) colorectal cancer, and 12 (17%) a head and neck cancer (Table 1). Demographic characteristics were similar between ICU and non-ICU patients, except for PS at inclusion, which was better in non-ICU patients (0 or 1 for all of them) than in ICU patients (PS 0 or 1 for 78%; PS 2 or 3 for 22%). Half of the patients displayed metastatic disease. Regarding anticancer treatment, 43% patients underwent treatment in a curative goal, and only 13% were in second-or further-line treatment (Table 1).

Reasons of ICU admission were mostly cancer-related complications (n=15, 65%). Most patients experienced respiratory failure (n=13, 57%) or cardiovascular failure (n=9, 39%) (Table 2). Median number of organ failure was 1 (IQR, 1-2). Mechanical ventilation was employed in nine (39%) patients, and vasopressors in nine (39%), as well.

Quality of life at 3 months

Only 19/23 ICU patients answered the QoL questionnaire at 3 months, whereas all the 46 non-ICU patients did so. We therefore performed the analysis on the 38 matched patients, as well as on the 19 patients who answered the 3-month questionnaire. The median MCS of SF-36 significantly differed between ICU and non-ICU cancer patients, being 54 (IQR: 42-57) and 47 (IQR: 37-52), respectively (p=0.01). The median PCS of SF-36 was not significantly different, being 35 (IQR 31-47) vs. 42 (IQR 34-47), respectively (p=0.24) (Table 3).

The multivariate analysis revealed no significant association between ICU/non-ICU patient status and quality-of-life scores (MCS or PCS) 3 months after discharge (Table 4). The nonmetastatic cancer status (coef. 1.21, CI95%: 1.05-1.40, p=0.01) and PS 0-1 (coef. 1.46, CI95%:

1.11-1.90, p=0.01) were independently associated with an increase in PCS scores.

General conditions and anticancer treatment at 3 months

At 3 months, PS was still significantly better in non-ICU than ICU cancer patients (for PS 0-1, 18 cases (78%) vs. 45 controls (98%), p=0.01). The HADS-A was significantly better in ICU than non-ICU patients (5 (IQR 4-6) vs 8 (IQR 5-10), p=0.01). The HADS-D was not significantly different. ADLs were similar between ICU and non-ICU cancer patients (Table 3). Seven (30%) ICU patients underwent anticancer treatment modifications (change of treatment line because of disease progression or curative treatment discontinuation for exclusive palliative care), whereas only six of 46 non-ICU patients (13%) underwent such modifications. All of the 12 (100%) non-ICU patients who were controlled without any treatment at baseline had still an exclusive follow-up without any cancer treatment at 3 months, whereas two out of six (33%) ICU patients had to start a treatment beforehand.

Concerning the question of a potentially further ICU admission, 19 (83%) ICU and 43 (94%) non-ICU cancer patients agreed with such a proposal.

DISCUSSION

We herein report the results of a 1-year case-control prospective study comparing cancer patients either admitted (n=19) or not admitted (n=38) into ICU, the study's design being to compare QoL at 3 months between the two groups. MCS was significantly higher in ICU patient vs. non-ICU patient groups. However, the ICU stay was not independently associated with variation in mental or physical components of SF36 at 3 months in a multivariate analysis.

It must be stressed that this study followed an originally prospective design, whereas most case-control studies are retrospective in nature. Various clinical data were collected at inclusion and at 3 months post-discharge like general condition, characteristics of ICU stay, anticancer treatment, and oncologic care project. QoL was assessed using the SF36, a commonly used survey questionnaire that had previously been validated in critically ill population [START_REF] Chrispin | Short Form 36 in the intensive care unit: assessment of acceptability, reliability and validity of the questionnaire[END_REF]. Among the screening patients admitted into ICU, the ICU and 3-month (35% and 44% respectively) mortality rates were similar to those published in the scientific literature [START_REF] Vincent | In-hospital and day-120 survival of critically ill solid cancer patients after discharge of the intensive care units: results of a retrospective multicenter study-A Groupe de recherche respiratoire en réanimation en Onco-Hématologie (Grrr-OH) study[END_REF][START_REF] Oeyen | Long-term outcomes and quality of life in critically ill patients with hematological or solid malignancies: a single center study[END_REF][START_REF] Gheerbrant | Factors associated with survival of patients with solid Cancer alive after intensive care unit discharge between 2005 and 2013[END_REF]. Nevertheless, our study was a single-center study with a small number of patients, especially given that four patients (17%) refused to answer the questionnaires at 3 months. Moreover, most of our ICU patients displayed low SOFA scores, which could restrict the possibility of extrapolating our results to the more severely-ill patients that are usually admitted into ICUs. Owing to these issues, we must interpret our results with caution.

This study focused on a highly selected population, with patients discharged alive from ICU and still alive at 3 months. Our patients displayed a particularly good PS and low SOFA scores with mostly only one organ failure, which are factors known to be independently associated with survival [START_REF] Taccone | Characteristics and outcomes of cancer patients in European ICUs[END_REF][START_REF] Azoulay | Predictors of short-term mortality in critically ill patients with solid malignancies[END_REF][START_REF] Soares | Short-and long-term outcomes of critically ill patients with cancer and prolonged ICU length of stay[END_REF][START_REF] Soares | Characteristics and outcomes of patients with cancer requiring admission to intensive care units: a prospective multicenter study[END_REF][START_REF] Azevedo | Outcomes for patients with cancer admitted to the ICU requiring ventilatory support: results from a prospective multicenter study[END_REF][START_REF] Vincent | In-hospital and day-120 survival of critically ill solid cancer patients after discharge of the intensive care units: results of a retrospective multicenter study-A Groupe de recherche respiratoire en réanimation en Onco-Hématologie (Grrr-OH) study[END_REF][START_REF] Gheerbrant | Factors associated with survival of patients with solid Cancer alive after intensive care unit discharge between 2005 and 2013[END_REF][START_REF] Soares | Intensive care in patients with lung cancer: a multinational study[END_REF][START_REF] Mcgrath | ICU and 6-month outcome of oncology patients in the intensive care unit[END_REF]. In a retrospective cancer patient cohort with suspected infection admitted to ICU, the PS impairment influenced the in-hospital mortality of those with lower SOFA scores (≤6), which concerned 32% of patients with PS 0-1 versus 52% of those with . This could explain why most of our patients (69%) continued their anticancer treatments without any modifications following their ICU stay, whereas Soares et al. reported that only 50% of survivors at 6 months could still benefit from any anticancer treatment [START_REF] Soares | Intensive care in patients with lung cancer: a multinational study[END_REF]. At 3 months, the MCS of SF36 was significantly higher in cancer patients after their ICU stay. This result could be accounted for by the probably more relevant psychological care these patients received during their ICU stay. Regarding the SF36 values, the MCS was slightly higher than the PCS one, as recorded in other studies [START_REF] Oeyen | Long-term outcomes and quality of life in critically ill patients with hematological or solid malignancies: a single center study[END_REF][START_REF] Ehooman | Long-term health-related quality of life of critically ill patients with haematological malignancies: a prospective observational multicenter study[END_REF]. This difference probably reflects a gradual process in which patients progressively adapt and accept their physical limitations. This interpretation is further supported by the fact that the majority of our patients would accept a potential admission or re-admission into ICU as necessary. According to Oeyen et al., the SF36-assessed QoL was shown to decline during the 3 months after an ICU stay (from 45/100 to 38/100 for PCS and 50/100 to 44/100 for MCS) in solid cancer patients, while improving again at 1 year (to 41/100 for PCS and to 50/100 for MCS), yet without returning to baseline values [START_REF] Oeyen | Long-term outcomes and quality of life in critically ill patients with hematological or solid malignancies: a single center study[END_REF]. In multivariate analysis, the ICU stay was unlikely to impact the QOL scores in patients who were still alive 3 months following ICU discharge. A good PS and the non-metastatic cancer status were independently associated with a higher PCS of SF36, reflecting a higher health-related QoL. This observation was consistent with the results of studies in which PS and recurrent or progressive malignancy were proven associated with mortality in critically ill cancer patients [START_REF] Azoulay | Predictors of short-term mortality in critically ill patients with solid malignancies[END_REF][START_REF] Soares | Short-and long-term outcomes of critically ill patients with cancer and prolonged ICU length of stay[END_REF][START_REF] Soares | Characteristics and outcomes of patients with cancer requiring admission to intensive care units: a prospective multicenter study[END_REF][START_REF] Azevedo | Outcomes for patients with cancer admitted to the ICU requiring ventilatory support: results from a prospective multicenter study[END_REF][START_REF] Vincent | In-hospital and day-120 survival of critically ill solid cancer patients after discharge of the intensive care units: results of a retrospective multicenter study-A Groupe de recherche respiratoire en réanimation en Onco-Hématologie (Grrr-OH) study[END_REF][START_REF] Gheerbrant | Factors associated with survival of patients with solid Cancer alive after intensive care unit discharge between 2005 and 2013[END_REF][START_REF] Mcgrath | ICU and 6-month outcome of oncology patients in the intensive care unit[END_REF][START_REF] Soares | Intensive care in patients with lung cancer: a multinational study[END_REF]. These factors, in addition to mortality and short-term health-related QoL, could be used by intensivists to help them decide if specific cancer patients should be admitted into ICUs.

Finally, our results highlight the absence of a negative impact of an ICU stay on QoL and anticancer treatments in cancer patients still alive at 3 months. This, again, stresses the relevance of identifying those patients who would benefit from an ICU stay and be able to handle their oncological project. The good prognostic factors previously reported in literature remain the most robust ones, including a good PS and non-metastatic controlled disease state with a curative therapeutic project and a limited number of organ failures. Multidisciplinary discussion along intensivists and oncologists are still a relevant part of patient care, as shown by a multicenter study where daily meetings among them for care planning purposes was associated with lower mortality rates and more efficient resource use. ( 4)

CONCLUSION

At three months post-discharge, cancer patients admitted into ICUs exhibited comparable QoL with matched patients not admitted into ICUs. Thus, the challenging issue is to identify those patients with good prognostic factors who would benefit from an ICU stay in terms of QoL and continue their anticancer treatments. Nevertheless, these results still require confirmation by multicenter studies with higher patient numbers.

FIGURES AND TABLES

  DES PATIENTS ATTEINTS DE CANCER À 3 MOIS D'UN SÉJOUR EN RÉANIMATION : UNE ÉTUDE CAS TÉMOINS PROSPECTIVE THÈSE PRÉSENTÉE POUR L'OBTENTION DU TITRE DE DOCTEUR EN MÉDECINE SPÉCIALITÉ : PNEUMOLOGIE Par Wassila M'SALLAOUI THÈSE SOUTENUE PUBLIQUEMENT À LA FACULTÉ DE MÉDECINE DE GRENOBLE Le 10/09/2021 DEVANT LE JURY COMPOSÉ DE : Président du jury : M. le Professeur Jean François PAYEN Membres : Mme le Professeur Charlotte COSTENTIN M. le Docteur Louis Marie GALERNEAU M. le Docteur Alexandre GODON Mme le Professeur Anne-Claire TOFFART (directrice de thèse)

  : Sophie, Antoine, Thierry, Nico & Marine, Papy Jacques, Papy Pierre et Mamie Michelle. Merci de m'avoir accueilli dans votre famille, merci pour votre gentillesse. J'espère qu'on continuera à passer de bons moments sur la chaise « Lafuma » à jouer aux jeux des mimes, à manger des raclettes près de la cheminée à Corrençon en écoutant Radio Oxygene, et à bronzer autour de la piscine à Aix. Mon mari, Guillaume, je ne sais pas par où commencer tellement il y a des choses à dire. Merci d'être un rayon de soleil au quotidien (bon parfois une VP aussi), merci de m'apporter un amour inconditionnel, merci d'être si attentionné, merci d'avoir signé pour me supporter chaque jour de ta vie pour le meilleur et pour le pire. Personne ne me connait aussi bien que toi, merci pour tous ces bons moments ensemble. Merci pour tout. Je t'aime TABLES DES MATIERES ABRÉVIATIONS ...................................................................................................................................................................................... ARTICLE .................................................................................................................................................................................................... ABSTRACT .......................................................................................................................................................................................... INTRODUCTION ............................................................................................................................................................................... METHODS ............................................................................................................................................................................................ RESULTS .............................................................................................................................................................................................. DISCUSSION ....................................................................................................................................................................................... CONCLUSION ..................................................................................................................................................................................... REFERENCES ..................................................................................................................................................................................... FIGURES AND TABLES ................................................................................................................................................................... CONCLUSION .......................................................................................................................................................................................... SERMENT D'HIPPOCRATE ................................................................................................................................................................ ABRÉVIATIONS ADL : Activities of Daily Living CI : Confidence Interval ECOG-PS : Eastern Cooperative Oncology Group-Performance Status HADS : Hospital Anxiety and Depression Scale ICU : Intensive Care Unit IGS2 : Indice de Gravité Simplifié IQR : Interquartile Range MCS : Mental Component Score OR : Odds Ratios PCS : Physical Component Score QOL : Quality of life SF-36 : Short Form 36 Quality of life of cancer patients at 3 months after ICU stay : a prospective case-control study AC Toffart, S Jerusalem, W M'Sallaoui, A Godon, F Bettega, G Roth, J Pavillet, E Girard, LM Galerneau, C Schwebel, and JF Payen ABSTRACT Background: Although short-and long-term survival in critically ill cancer patients was extensively studied, only limited data exist concerning their quality of life (QoL) following an intensive care unit (ICU) stay. We thus sought to determine the impact of an ICU stay on QoL at 3 months in solid cancer patients. Method: A prospective observational case-control study was conducted between February 2020 and February 2021 involving three French ICUs. Surviving ICU patients with lung, colorectal, or head and neck cancers at 3 months after ICU discharge were matched with non-ICU cancer patients in terms of their cancer type, anticancer treatment goal (curative/palliative), and treatment line. QoL was assessed using the Short Form 36 (SF-36) health survey (ranging 0-100, higher values indicating better QoL) including mental (MCS) and physical (PCS) components. Anticancer treatment at 3 months was additionally assessed.

  This prospective study occurred between February 2020 and February 2021. Three adult ICUs of the University Hospital of Grenoble, France, participated to the study, including a surgical ICU (19 beds), cardio-surgery ICU (20 beds), and medical ICU (28 beds). The ethical committee for the Clinical Investigation Centers of Rhône-Alpes-Auvergne (IRB 5891) approved the study design on February 20, 2020. Given the study's observational nature, the requirement for written informed consent from patients or relatives was waived (clinical trial ID NCT04310033). Each patient or his/her relatives received oral and written information about the research, and they could refuse at any time to participate to the trial according to the French law (Toulouse E et al, ACCPM 2020).

Figure 1 .Figure 2 .

 12 Figure 1. Method of prospective individual matching[START_REF] Charpentier | An algorithm for prospective individual matching in a non-randomized clinical trial[END_REF]

Table 1 .

 1 Case and control characteristics at baseline Do not meet test assumption. Quantitative variables have been expressed as median (first quartile-third quartile) and qualitative as number (percentage) ICU: intensive care unit; Non-ICU: non-intensive care unit

		ICU patients,	Non-ICU patients,	p-value
		n=23	n=46
	Demographic data
	Age (in year)	70 [60;75]	66.5 [60;75]	0.91
	Male gender	18 [78]	27 [59]	0.18
	Weight at inclusion (kg)	62 [55;69]	68 [58;79]	0.26
	Charlson comorbidity index	1 [0;3]	2 [0;3]	0.32
	Performance status
	-0-1	18 (78)	16 (100)	<0.01
	-2-3	5 (22)		<0.01
	History of cancer
	Primitive cancer			*
	-Lung	13 (57)	26 (57)
	-Colorectal	6 (26)	12 (26)
	-Head and neck	4 (17)	8 (17)
	Non -metastatic disease	12 (52)	23 (50)	1
	Previous anticancer treatments
	-Chemotherapy	15 (65)	30 (65)	1
	-Immunotherapy	6 (26)	13 (29)	1
	-Curative radiotherapy	5 (22)	12 (26)	0.92
	-Curative surgery	10 (44)	20 (44)	1
	-Targeted therapy	3 (13)	5 (11)	1
	Actual anticancer treatment
	Treatment goal			*
	-Palliative	13 (57)	26 (57)
	-Curative	10 (43)	20 (43)
	Treatment line			*
	-Only surgery or radiotherapy	6 (26)	12 (26)
	-Line 1 of systemic treatment	14 (61)	28 (61)
	-Line 2-3 of systemic treatment	3 (13)	6 (13)
	Cancer status			**
	-Treatment not-reassessed	10 (44)	19 (41)
	-Controlled without treatments	7 (30)	11 (24)
	-Controlled with treatments	5 (22)	16 (35)
	-Progression	1 (4)	0

*Not performed because of matching variables. **

Conclusions: At 3 months, cancer patients admitted into ICU displayed comparable QoL with matched patients not admitted into ICU. Thus, the challenging issue is to identify patients with good prognostic factors who would benefit from an ICU stay in terms of QoL and continue their anticancer treatments. These results require confirmation by a multicenter study. KEY WORDS: ICU, quality of life, long-term outcomes, solid cancer, anticancer treatments


Heitor Farias De Barros
email: heitor.farias-de-barros@g2elab.grenoble-inp.fr

Bertrand Raison

Marie-Cécile Alvarez Herault

Quoc Tuan Tran

A COMPARATIVE STUDY OF OPTIMAL PLANNING OF DISTRIBUTION SYSTEMS: AC/DC ARCHITECTURE VS. CONVENTIONAL STRATEGIES

As the integration of DC distributed energy resources increases in recent years, hybrid AC/DC topology appears as a promising alternative in the planning of distribution systems. This paper aims to compare conventional planning strategies with conversion of AC lines into DC through the formulation of optimal planning models. Both operational and reinforcement solutions are considered in the conventional planning. The results indicate that line reinforcement is more cost-effective from a DSO point-ofview for the case study considered.

INTRODUCTION

Distribution systems are facing an increasing integration of highly variable distributed energy resources (DERs) in recent years. This tends to raise the risks of security constraints violations and accelerates the need to upgrade these systems. Historically, Distribution System Operators (DSOs) rely on conventional network development solutions such as conductor reinforcement and operational/ non-wire alternatives (e.g. DER reactive power support and setting of transformers tap) to deal with aforementioned risks.

Nevertheless, because the majority of DERs is DC native, converting AC systems into hybrid AC/DC emerges as a solution to alleviate overloaded networks and keep voltage into safe limits while facilitating the connection of DC DERs. The benefits of such a conversion are an increase of maximum transfer capacities as well as reduction of losses and voltage drop in the feeders [START_REF] Zhang | Converting AC Distribution Lines to DC to Increase Transfer Capacities and DG Penetration[END_REF]. Indeed, [START_REF] Yu | Initial designs for the ANGLE DC project; converting existing AC cable and overhead line into DC operation[END_REF] and [START_REF] Liu | The Upgrading Renovation of an Existing XLPE Cable Circuit by Conversion of AC Line to DC Operation[END_REF] present concrete projects where converting existing AC cables into DC provided a cost-effective solution.

This solution can be especially worthwhile in Medium Voltage (MV) systems with a significant increase of highpower DC DERs integration. In these cases, a local MVDC network can avoid network developments that would be otherwise required if the network remained operated full AC.

Although AC/DC expansion planning strategies have been developed in the literature [START_REF] Ahmed | A Planning Approach for the Network Configuration of AC-DC Hybrid Distribution Systems[END_REF], a comparison between this strategy and conventional solutions is rarely found. Therefore, this paper aims to compare current solutions with AC/DC topology in order to identify most suitable strategies and propose a systematic optimization tool to aid in DSO planning decision-making process.

PROBLEM DESCRIPTION

To deal with DERs connections over long-term horizon, DSOs typically carry out planning studies to determine best solutions to host these resources without jeopardizing system security [START_REF] Dutrieux | Assessing the Impacts of Distribution Grid Planning Rules on the Integration of Renewable Energy Resources[END_REF]. In this work, optimal planning models are developed to study DERs integration into MV distribution systems ensuring grid code compliance. Different solutions developed to respect voltage and current limits are presented in the next paragraphs.

First, an optimal operational planning considering nonwire alternatives is developed and tested. This strategy considers reactive power support from DERs according to an optimal set of power factor following the French DSO grid code [START_REF]Principe et conditions de mise en oeuvre d'une régulation locale de puissance réactive pour les Installations de Production raccordées au Réseau Public de Distribution HTA[END_REF]. Moreover, optimal system configuration (choice of normally open points) and on-load tap changer (OLTC) voltage control are also considered.

For weak grids with high integration of DERs, non-wire alternatives are sometimes unable to ensure a secure operation. In this case, two strategies are further tested, namely AC/DC and reinforcement planning.

The first strategy studies the optimal conversion of a part of the system into DC to create a link between AC feeders, improving feeder balance and avoiding constraints violation. At the same time, DC resources can be connected to the MVDC systems, providing a less costly and more efficient connection of DC resources to the network from a DERs holder point-of-view.

The second strategy models the conventional solution used by DSOs to propose replacement of a conductor by another, i.e. line reinforcement. In this way, network developments are determined to minimize the costs while keeping current and voltage within the limits by changing line capacity and impedance.

Both AC/DC and reinforcement planning include also operational solutions applied for operational planning (Volt-VAR control, new configuration and OLTC). All models are formulated as Mixed Integer Quadratic Programming (MIQP) optimization problems, guaranteeing optimality within a reasonable computation time with off-the-shelf solvers. Figure 1 summarizes flowchart of methodology used to compare developed planning models for a given case study.

Figure 1 -Flowchart of planning model

OPTIMAL PLANNING MODELLING

The Distribution System Planning (DSP) strategies defined in the next subsections aim to optimize economic expenditures from a DSO point-of-view over a set of illustrating relevant operating points for a given planning horizon. The costs for the DERs holders are not modelled here.

Representative Scenarios

A Wasserstein distance minimization-based scenario reduction technique [START_REF] Wang | Optimal siting and sizing of soft open points in active electrical distribution networks[END_REF] is used to determine representative scenarios for stochastic parameters, i.e. level of load demand, electric vehicle charging station demand, wind turbine power and photovoltaic power. Hence, each scenario is defined by a modulation index that multiplies nominal power and by a weight coefficient (𝑤 𝑠 ) representing the probability of occurrence of the scenario.

Objective function

The 2)-( 4). Operational Expenditures (𝑂𝑃𝐸𝑋) consider the losses cost of the converters (𝐿𝐶 𝑣𝑠𝑐 ) and lines (𝐿𝐶 𝑙𝑖𝑛𝑒𝑠 ) in ( 5)-( 7) taking into account losses unit cost (𝑙𝑐) and a weight coefficient (𝑤 𝑠 ).

The line losses computation is done by an approximation of current for nominal voltage operation in ( 8)-( 9) to obtain a quadratic objective function. Note also that indices s, i, ij and k are used to refer respectively to the variables for each scenario, node, line and conductor type.

Operational constraints

Power flow constraints are formulated through a LinDistFlow model [START_REF] Wu | AC/DC Hybrid Distribution System Expansion Planning Under Long-Term Uncertainty Considering Flexible Investment[END_REF], which solves the power flow for squared nodal voltage (𝑣 𝑖 ) and branch power (𝑝 𝑖𝑗 and 𝑞 𝑖𝑗 ), neglecting line losses in power balance nodal constraints ( 10)-( 11) as well as in voltage drop equation [START_REF] Garry | Probabilistic modelling of renewable generation to account for uncertainties in interconnection studies[END_REF]. Line security limit is modelled through apparent power limitation presented by a piecewise linearization of (13) [START_REF] Wu | Hybrid AC/DC Microgrid Planning with Optimal Placement of DC Feeders[END_REF]. Voltage security constraint is defined in (14

These conversions define base values for the models and allow the coupling of per unit power constraints of AC and DC systems. The values considered (𝑘 𝑝 > 1 and 𝑘 𝑟 < 1) result therefore in an increase of maximum transfer capacities and in the reduction of losses and voltage drop in feeders converted into DC.

VSC constraints

The VSC constraints ensure balance between AC and DC active power and losses in ( 22)-( 23). Losses are assumed to be proportional to VSC rated and operating apparent power in (24), producing 5% losses at nominal power.

Operating apparent power of the VSC is computed by a piecewise linearization of (25), which is a relaxation of exact constraint for each scenario. Finally, VSC operating apparent power is limited by its rated power in (26).

Reinforcement constraints

The reinforcement planning determines the replacement or not of a conductor by selecting between the conductor sections typically used by a French DSO (Table 2). This planning strategy is defined through same constraints of operational planning, but considering different cross sections options for each network feeder in power flow constraints. Thus, constraints ( 10) -( 12) are written as ( 27) -(29) when a line is reinforced.

∑ ( ∑ 𝑝

Topological modelling

Aforementioned constraints actually depend on the topology of the network, i.e. they are activated as a function of the choice between AC or DC, open or closed and conductor type, for instance. This selection between constraints and radiality constraints are modelled by using binary variables in this work.

RESULTS

The goal of this planning is to determine the most economical topology of a distribution system from a DSO point-of-view considering the integration of DC distributed resources. To evaluate planning models, a case study based on part of a French rural MV distribution The proposed formulations are implemented in the YALMIP optimization toolbox [START_REF] Lofberg | YALMIP: a toolbox for modeling and optimization in MATLAB[END_REF] with MATLAB R2020b, solved by Gurobi [START_REF] Optimization | Gurobi Optimizer Reference Manual[END_REF], on a laptop equipped with a 1.70GHz Intel® CoreTM i5-10310U CPU and 16 GB RAM.

Case Study

The case study considered has 18 nodes and 18 lines (2 Normally Open Points) and includes 14 large MV loads. The integration of DC DERs is considered by adding to the grid a photovoltaic power plant (PV), a wind turbine power plant (WT), a DC load and 2 electric vehicle charging stations according to Table 3. EV charging station 0.75 3 scenarios are considered to represent the probability distribution function of each stochastic resource. Then, under the hypothesis that resources availability and load demand are independent [START_REF] Garry | Probabilistic modelling of renewable generation to account for uncertainties in interconnection studies[END_REF], the scenarios are combined to generate 3 4 = 81 representative scenarios.

A 30 years planning is performed considering a 55% load growth, equivalent approximately to a 1.5%/year rate. This value is chosen to emulate a high load growth scenario.

The converters cost took into consideration the replacement of the VSCs at the end of their service life. Other planning parameters are provided in Table 4.

Planning

The operational planning considering non-wire alternatives did not converge, i.e. it is an infeasible optimization problem, because security constraints cannot be ensured. The AC/DC and reinforcement planning results optimal topology obtained are shown respectively in Table 5 and Figure 2. The reinforcement requires a changing in configuration and replacement of line 1 by a Cu 240 mm 2 cross section conductor. The reinforced line is quite short (less than 50m), which represents a very low capital cost to ensure system operation under security constraints. In contrast, optimal AC/DC topology converts lines 5, 6 and 7 into a DC, creating a MVDC link between MVAC feeders improving balance between them by active power transfer. However, power converter high installation costs and losses expenditures required to implement this topology greatly increase both capital and operational expenditures. Accordingly, reinforcement solution significantly outperforms economically AC/DC planning.

This result applies for the given case study, where reinforcement needs are very limited, but it shows that AC/DC topology tends to be less cost-effective from the DSO point-of-view at the current conditions, i.e. VSC cost and performance. Notwithstanding, AC/DC topology

  𝑑) 𝑡 (𝐶𝐴𝑃𝐸𝑋 𝑡 + 𝑂𝑃𝐸𝑋 𝑡 ) 𝑦 𝑡=0 (1) 𝐶𝐴𝑃𝐸𝑋 𝑡 = 𝐶𝐶 𝑣𝑠𝑐 𝑜𝑟 𝐶𝐶 𝑟𝑒𝑖𝑛𝑓 (2) 𝐶𝐶 𝑣𝑠𝑐 = 𝑐𝑐 𝑣𝑠𝑐 ∑ 𝑆 𝑣𝑠𝑐,𝑖 𝑟𝑎𝑡𝑒𝑑 𝑛𝑜𝑑𝑒𝑠 (3) 𝐶𝐶 𝑟𝑒𝑖𝑛𝑓 = ∑ (𝑐𝑐 𝑟𝑒𝑖𝑛𝑓,𝑘 ∑ 𝑙𝑒𝑛𝑔𝑡ℎ 𝑖𝑗 𝑙𝑖𝑛𝑒𝑠 )

2 (

 2 𝑟 𝑎𝑐,𝑖𝑗,𝑘 𝑝 𝑎𝑐,𝑖𝑗,𝑘 𝑠 + 𝑥 𝑎𝑐,𝑖𝑗,𝑘 𝑞 𝑎𝑐,𝑖𝑗,𝑘 𝑠 ) (29)

Figure 2 -

 2 Figure 2 -Optimal topology planning of case study: (a) AC/DC and (b) Line reinforcement

  ).

							Rome, 12-15 June 2023
							Paper n° 10570
	AC/DC Architecture
	DC operational constraints
	Analogously to AC power flow constraints, (15)-(18)
	represent the DC power flow constraints valid whenever a
	line is converted into DC.
	∑ 𝑝 𝑑𝑐,𝑖𝑗 𝑠	+ 𝑃 𝑑𝑐,𝑖 𝑠 + 𝑃 𝑣𝑠𝑐,𝑑𝑐,𝑖 𝑠	= 0	(15)
	𝑙𝑖𝑛𝑒𝑠 𝑣 𝑑𝑐,𝑖 𝑠	-𝑣 𝑑𝑐,𝑗 𝑠 𝑝 𝑑𝑐,𝑖𝑗 𝑠 𝑣 𝑑𝑐 ≤ 𝑣 𝑑𝑐,𝑖 = 2𝑟 𝑑𝑐,𝑖𝑗 𝑝 𝑑𝑐,𝑖𝑗 𝑠 ≤ 𝑝 𝑑𝑐,𝑖𝑗 𝑠 ≤ 𝑣 𝑑𝑐	(16) (17) (18)
	Double circuit feeders were considered in system.
	Since isolation of three-phase AC conductors is roughly
	designed to withstand peak phase-to-phase voltage, it is
	assumed here a conversion factor of 𝑘 𝑣 = 2√2/√3
	between phase-to-phase AC voltage and pole-to-pole DC
	voltage. This results in a conversion factor of 𝑘 𝑟 = 0.5 for
	resistance and 𝑘 𝑈 𝑑𝑐 ≅ 𝑘 𝑣 𝑈 𝑎𝑐 𝑟 𝑑𝑐,𝑖𝑗 = 𝑘 𝑟 𝑟 𝑎𝑐,𝑖𝑗		(19) (20)
		𝑝 𝑑𝑐,𝑖𝑗 = 𝑘 𝑠 𝑎𝑐,𝑖𝑗
						∑ 𝑝 𝑎𝑐,𝑖𝑗 𝑠	+ 𝑃 𝑎𝑐,𝑖 𝑠 + 𝑃 𝑣𝑠𝑐,𝑎𝑐,𝑖 𝑠	= 0	(10)
						𝑙𝑖𝑛𝑒𝑠
						∑ 𝑞 𝑎𝑐,𝑖𝑗 𝑠	+ 𝑄 𝑎𝑐,𝑖 𝑠	+ 𝑄 𝑣𝑠𝑐,𝑎𝑐,𝑖 𝑠	= 0	(11)
					𝑣 𝑎𝑐,𝑖 𝑠	𝑙𝑖𝑛𝑒𝑠 -𝑣 𝑎𝑐,𝑗 𝑠	= 2(𝑟 𝑎𝑐,𝑖𝑗 𝑝 𝑎𝑐,𝑖𝑗 𝑠	+ 𝑥 𝑎𝑐,𝑖𝑗 𝑞 𝑎𝑐,𝑖𝑗 𝑠	)	(12)
						√(𝑝 𝑎𝑐,𝑖𝑗 𝑠	) 2 + (𝑞 𝑎𝑐,𝑖𝑗 𝑠	)	2 ≤ 𝑠 𝑎𝑐,𝑖𝑗	(13)
							𝑣 𝑎𝑐 ≤ 𝑣 𝑎𝑐,𝑖 𝑠	≤ 𝑣 𝑎𝑐	(14)
					These approximations yield to a linear formulation of the
					problem that is piecewise convex, reducing computational
					burden with good accuracy, as later discussed in the
					Results section.
	CIRED 2023						3/5

𝑝 = √2 between AC and DC lines power capacity in per unit

[START_REF] Zhang | Converting AC Distribution Lines to DC to Increase Transfer Capacities and DG Penetration[END_REF]

. This conversion is further illustrated by Table

1

and (

19

)-(

21

).

Table 1 -

 1 Double circuit feeder conversion of AC 3-phase into DC symmetrical without neutral wire (based on [1]) Configuration 𝑷 𝒅𝒄 or 𝑺 𝒂𝒄

	AC	or	√3𝑈 𝑎𝑐 × 2𝐼
	DC		𝑈 𝑑𝑐 × 3𝐼
		or

Table 2 -Reinforcement parameters Conductor type R (ohm/km) X (ohm/km) Cost (k$/km)

 2

	Al 95 mm 2	0.320	0.100	109
	Al 150 mm 2	0.206	0.100	110
	Al 240 mm 2	0.125	0.100	112
	Cu 240 mm 2	0.075	0.100	150
	2x Al 240 mm 2	0.062	0.050	224

Table 3

 3

		-DER integration data
	Node	Type	Nominal Power (MW)
	5	PV power plant	2.00
	6	EV charging station	1.00
	11	MV load	0.50
	12	WT power plant	3.00
	16

Table 4 -

 4 Planning parameters

	Parameter	Value
	Base values (𝑺, 𝑼 𝒂𝒄 and	10MVA, 20kV and
	𝑼 𝒅𝒄 )	32kV
	Voltage magnitudes limits	±5%
	Discount rate	7.5%/year
	VSC service life	15 years
	Line service life	30 years
	VSC unit capital cost	170$/kVA
	Losses cost	0.08$/kWh

Table 5 -

 5 Planning results

	model	NPC	CAPEX	OPEX
			(M$)	(M$)
	Operational		Infeasible
	AC/DC	1.307	0.380	0.927
	Reinforcement	0.497	0.008	0.489

CONCLUSION

Proposed models represent an advance towards systematic integration of DC into long-term distribution system planning studies. Although preliminary results show an advantage to conventional planning strategies from a point-of-view of a DSO, cost modelling did not consider cost reduction from DERs holder side related to the connection to a DC system instead of an AC one. Other case studies with larger and more overloaded distribution systems where more reinforcement is required might also highlight the interest of an AC/DC distribution planning.

Therefore, costs sharing between DSO and DER holders as well as scalability issues could be the subject of further investigation to find case studies where DC conversion is a cost-effective solution. In addition, future research should also focus on protection and reliability analysis of hybrid AC/DC systems to better evaluate costeffectiveness of considered planning strategies.