Felix Agner
email: felix.agner@control.lth.se

Jonas Hansson

Pauline Kergus

Anders Rantzer

Sophie Tarbouriech

Luca Zaccarian

Decentralized PI-control and Anti-windup in Resource Sharing Networks

Keywords: Energy Systems, Constrained Control, Decentralized Control

We consider control of multiple stable first-order systems which have a control coupling described by an Mmatrix. These agents are subject to incremental sector-bounded nonlinearities. We show that such plants can be globally asymptotically stabilized to a unique equilibrium using fully decentralized proportional integral anti-windup-equipped controllers subject to local tuning rules. In addition, we show that when the nonlinearities correspond to the saturation function, the closed-loop asymptotically minimizes a weighted 1-norm of the agents state mismatch. The control strategy is finally compared to other state-of-the-art controllers on a numerical district heating example.

I. INTRODUCTION

In this paper we consider the control of agents sharing a central distribution system with limited capacity as in [START_REF] Agner | Anti-windup coordination strategy around a fair equilibrium in resource sharing networks[END_REF]. We investigate systems where the positive action of one agent negatively impacts others. This type of competitive structure can arise in many domains, for instance internet congestion control [START_REF] Low | Internet congestion control[END_REF], [START_REF] Kelly | Fairness and stability of end-to-end congestion control[END_REF] and district heating systems [START_REF] Agner | Combating district heating bottlenecks using load control[END_REF]. In the district heating scenario, the structure arises because of the hydraulic constraints of the grid. If one agent (building) locally decides to increase their heat demand by opening their control valves, this will lead to higher flow rates and greater frictional pressure losses. These losses make it so that other agents now receive lower flow rates [START_REF] Agner | Combating district heating bottlenecks using load control[END_REF]. We consider a simple description of such systems as follows. Let x i be the state of each agent i. Then ẋi = -a i x i + n j=1 B i,j sat (u j ) + w i [START_REF] Agner | Anti-windup coordination strategy around a fair equilibrium in resource sharing networks[END_REF] where a i ∈ R and a i > 0, u i ∈ R is the control action of each agent i, and B ∈ R n×n having elements B i,j represents the network interconnection between the agents. w i ∈ R is a disturbance on agent i. The saturation function sat (•) represents the limitations of the network, but in the later analysis we consider a richer class of nonlinear functions. To capture the resource-sharing aspect of the system we consider the case where B is an M-matrix satisfying B i,i > 0 and B i,j ≤ 0 when i = j.

In this work, we analyze (1) under a fully decentralized PI (proportional-integral) control strategy. Semi-decentralized control strategies for multi-agent systems have been considered in the following works. In [START_REF] Col | Regional H∞ synchronization of identical linear multiagent systems under input saturation[END_REF], each networked agent is equipped with a local controller that receives the control input of its neighbors. In [START_REF] Ofodile | Decentralized approaches to antiwindup design with application to quadrotor unmanned aerial vehicles[END_REF], semi-decentralized anti-windup was considered for stable SISO plants that are decentralized in the linear domain, but become coupled during saturation. This is demonstrated on unmanned aerial vehicles. These and other works focus on stabilization when the disturbance w in plant ( 1) is energy bounded. In this work we focus instead on the asymptotic properties of plant [START_REF] Agner | Anti-windup coordination strategy around a fair equilibrium in resource sharing networks[END_REF], which become important when w is expected to vary slower than the system and can be assumed constant. We show that our strategy minimizes asymptotic costs of the form n i=1 c i |x i |. Previous works considering asymptotic optimality for plants of the form (1) are [START_REF] Bauso | The linear saturated decentralized strategy for constrained flow control is asymptotically optimal[END_REF] and [START_REF] Agner | Anti-windup coordination strategy around a fair equilibrium in resource sharing networks[END_REF]. In [START_REF] Bauso | The linear saturated decentralized strategy for constrained flow control is asymptotically optimal[END_REF], it was shown that the static controller u = -B ⊤ x asymptotically minimizes the cost

x ⊤ Ax + v ⊤ v where A = diag(a 1 , a 2 , • • • , a n ) and v = sat (u)
. This result also extends to the case when B is not an M-matrix. In [START_REF] Agner | Anti-windup coordination strategy around a fair equilibrium in resource sharing networks[END_REF], it was shown that, when B is an M-matrix and w fulfills certain criteria, decentralized PIcontrollers with a rank-one coordinating anti-windup scheme can minimize the cost max i |x i |. Both of these control strategies maintain certain scalability properties: With u = -B ⊤ x [START_REF] Bauso | The linear saturated decentralized strategy for constrained flow control is asymptotically optimal[END_REF], any sparsity structure in the B-matrix is maintained and the rank-one coordination scheme of [START_REF] Agner | Anti-windup coordination strategy around a fair equilibrium in resource sharing networks[END_REF] admits scalable implementations. However, the most scalable control solution is one that is fully decentralized. In general, it is nontrivial that decentralized PI-controllers are stabilizing, let alone fulfill any optimality criterion. In this paper we manage to not only show the aforementioned asymptotic optimality properties. We also show that the resulting equilibrium is globally asymptotically stable under decentralized controller tuning rules.

The paper is organized as follows. Section II presents the considered plant and control strategy. Section III presents the main results of the paper, namely equilibrium existence and uniqueness, global asymptotic stability, and equilibrium optimality for our considered closed-loop. A motivating numerical example consisting in the flow control of a simplified district-heating network is subsequently given in section IV. The proofs of the main results are presented in sections V, VI, and VII respectively. Conclusions and future work are covered in section VIII.

Notation: v i denotes element i of vector v ∈ R n , A i denotes row i of matrix A ∈ R n×m , and A i,j denotes its (i, j)-th element. A matrix A is strictly diagonally row- dominant if |A i,i | > j =i |A i,j | for all i.
A is strictly diagonally column-dominant if A ⊤ , denoting the transpose of A, is strictly diagonally row-dominant. Let the 2-norm of a vector x ∈ R n be given by x 2 = ( n i=1 x 2 i ) 1/2 . Let the 1-and-infinity-norms of a vector x ∈ R n be given by x 1 = n i=1 |x i | and x ∞ = max i |x i | respectively. Let the norm A 2 of a matrix A be the induced 2-norm. Let 1 ∈ R n be a vector of all ones, where n is taken in context. We say that a function f :

R → R is increasing (non-decreasing) if y > x implies that f (y) > f (x) (f (y) ≥ f (x)).

II. PROBLEM DATA AND PROPOSED CONTROLLER

We consider control of plants of the form

ẋ = -Ax + Bf (u) + w (2)
where vector x ∈ R n gathers the states x i of each agent, A ∈ R n×n is a diagonal positive definite matrix, and w ∈ R n is a constant disturbance acting on the plant. The input nonlinearity f : R n → R n satisfies Assumption 1 introduced below and B ∈ R n×n is an M-matrix, as characterized next.

If the matrix B has strictly positive diagonal elements and non-positive off-diagonal elements, then the following statements are all equivalent [START_REF] Horn | Topics in Matrix Analysis[END_REF]: B is an M-matrix. There is a diagonal positive definite matrix

Q such that QB + B ⊤ Q ≻ 0.
There is a diagonal positive definite matrix U such that U B and U BU -1 are strictly column-diagonally dominant. DB is an M-matrix for any positive definite diagonal D. For a more extensive list of equivalent statements, refer to [START_REF] Horn | Topics in Matrix Analysis[END_REF].

For (2), we focus on a certain class of functions f : R n → R n characterized by the following assumption.

Assumption 1:

f (x) = [f 1 (x 1 ), f 2 (x 2 ), . . . , f n (x n )]
⊤ has components f i satisfying f i (0) = 0 and incrementally sector-bounded in the sector [0, 1], namely satisfying 0 ≤

(f i (y) -f i (x)) / (y -x) ≤ 1 for all x ∈ R, y ∈ R, x = y.
Note that Assumption 1 implies that f is non-decreasing and Lipschitz with Lipschitz constant 1. Since f (0) = 0, f also enjoys a sector [0, 1] condition. Stability properties for feedback with incrementally sectorbounded nonlinearities has long been considered in the literature. As far back as [START_REF] Zames | On the input-output stability of time-varying nonlinear feedback systems part one: Conditions derived using concepts of loop gain, conicity, and positivity[END_REF] it was used for input-output stability analysis. Both [START_REF] Zhang | Robust synchronization of lur'e networks with incremental nonlinearities[END_REF] and [START_REF] Delellis | Adaptive pinning control of complex networks of lur'e systems[END_REF] consider the type of diagonally partitioned incrementally sector-bounded functions that we consider here, whereas [START_REF] Giaccagli | LMI conditions for contraction, integral action, and output feedback stabilization for a class of nonlinear systems[END_REF]- [START_REF] Giaccagli | Infinite gain margin, contraction and optimality: An LMI-based design[END_REF] consider a richer class of incremental sector-bound constraints of the form

(f (x) -f (y) -S 1 (x -y)) ⊤ (f (x) -f (y) -S 2 (x -y)) ≤ 0 for all x ∈ R n , y ∈ R n
where S 1 and S 2 are real symmetric matrices with 0 S 1 ≺ S 2 .

We will consider function pairs f (•), h(•) where f (x) + h(x) = x. These pairs fulfill the following property, the proof of which is in the appendix.

Lemma 1: Let f : R n → R n satisfy Assumption 1. Then h(u) = u -f (u) also satisfies Assumption 1.

The considered class of function pairs is well motivated by the common case f (x) = sat (x) where sat (x) = max (min (x, 1) , -11) and h(x) = dz (x) = xsat (x).

We propose controlling the plant (2) with fully decentralized PI controllers having decentralized anti-windup for each agent i = 1, . . . , n.

żi = x i + s i h i (u i ) (3) u i = -p i x i -r i z i (4)
where z i is the integral state, u i is the controller output, p i > 0 and r i > 0 are proportional and integral controller gains respectively, s i > 0 is an anti-windup gain, and h(u) = uf (u) is an anti-windup signal. Note that while the notation h is not needed (indeed we could equivalently replace h(u) with u -f (u)), we will use the pair f , h both to simplify the exposition and to highlight that f is the nonlinearity acting on the plant while h is the nonlinearity acting on the controller. We assume that the closed-loop system satisfies the following assumption.

Assumption 2: A is a diagonal positive definite matrix, B is an M-matrix, and w is a constant disturbance. The controller parameters p i , r i , and s i , for i = 1, . . . , n, are all positive.

III. MAIN RESULTS

In this section we will cover the main results of this paper. In particular, we will consider the proposed control law (3)-( 4) for the plant (2). We will show that this closed-loop system admits an equilibrium for any constant disturbance w. We will additionally show that this equilibrium is globally asymptotically stable and enjoys a notion of optimality. We will leave the proofs for Sections V to VII.

Let us first consider the existence of an equilibrium, which corresponds to well-posedness of the equations ( 2)-( 4) with ẋ = ż = 0.

Theorem 1: (Equilibrium Existence and Uniqueness) Let f satisfy Assumption 1 and let Assumption 2 hold. Then for each constant w ∈ R n , closed-loop ( 2)-( 4) has a unique equilibrium (x 0 , z 0 ), inducing input u 0 from (4), which satisfies ( 2)-( 4) with ẋ = ż = 0.

In addition to the existence of the unique equilibrium (x 0 , z 0 ), we can also show that it is globally asymptotically stable under the following assumption on the tuning of the control parameters.

Assumption 3: Assume that a i p i > r i and p i s i < 1 for all i, where a i are the diagonal elements of A in (2) and p i , r i , and s i are the controller gains in (3)-(4).

Theorem 2: (Global Asymptotic Stability) Let f satisfy Assumption 1 and let f (u) + h(u) = u. Let Assumptions 2 and 3 hold. Then there is a globally asymptotically stable equilibrium for the closed-loop (2)-( 4).

Remark 1: The tuning rules of Assumption 3 are fully decentralized. Each agent i can tune their own controller gains to satisfy r i < a i p i and s i < 1/p i . Let us now focus on the case where the function pair f (•) and h(•) are given by the pair sat (•) and dz (•) respectively, motivated by classical anti-windup for saturating controllers.

Let γ i be positive scalar weights, and consider the problem of minimizing the weighted sum of all state errors n i=1 γ i |x i |. We can define this problem through the optimization problem

minimize x, v n i=1 γ i |x i | = Γx 1 (5a) subject to -Ax + Bv + w = 0, (5b) -1 ≤ v ≤ 1. ( 5c
)
where Γ = diag{γ 1 , . . . , γ n }. The inequalities (5c) are considered componentwise. This problem can be motivated by a district heating example. Let w be the outdoor temperature, x i be the deviation from the comfort temperature for each agent i, and let Bv denote the heat provided to the agents, limited by (5c). Then if Γ = I, this corresponds to minimizing the total discomfort experienced by all agents.

One could consider γ i to be a cost describing the severity of agent i deviating from the comfort temperature, where γ i would be high for e.g. a hospital. Note that this cost does not capture the notion of fairness as considered in [START_REF] Agner | Anti-windup coordination strategy around a fair equilibrium in resource sharing networks[END_REF].

For instance, with Γ = I, x = [n, 0, . . . , 0] ⊤ , and y = [1, 1, . . . , 1] ⊤ we achieve the same costs Γx 1 = Γy 1 .

With the problem (5) defined, the following holds. Theorem 3: Let Assumption 2 hold and let ΓA -1 B be a strictly diagonally column-dominant M-matrix. Let f (u) = sat (u) and h(u) = dz (u) = usat (u). Let (x 0 , z 0 ), be an equilibrium for the closed-loop system in ( 2)-( 4), associated with input u 0 . Then x * = x 0 and v * = f (u 0 ) solves [START_REF] Col | Regional H∞ synchronization of identical linear multiagent systems under input saturation[END_REF].

Remark 2: Note that, since B is an M-matrix, A -1 B is also an M-matrix. In this scenario there always exists a diagonal, positive definite Γ such that ΓA -1 B is strictly diagonally column-dominant. This means that there is always a selection of weights γ i for which the control law (3)-( 4) stabilizes an optimal point.

IV. NUMERICAL EXAMPLE

This motivating example compares three different control strategies on a simplified, linear model of 10 buildings connected in a district heating grid. The compared strategies are the same as the ones considered in [START_REF] Agner | Anti-windup coordination strategy around a fair equilibrium in resource sharing networks[END_REF]. Each building i has identical thermodynamics on the form

ẋi = - a i C i (x c + x i -T ext (t)) + 1 C i Qi (u), (6)
where x i denotes agent i's indoor temperature deviation from the comfort temperature x c , C i is the heat capacity of each building and T ext is the outdoor temperature. Here the assumption of a constant disturbance w is replaced with the slowly time-varying disturbance T ext . Qi is the heat supplied to building i. This heat supply is given by

Q = Bsat (u) , (7)
where B represents the network interconnection. The simulation was conducted with

a i = 0.167 [kW/C • ], C i = 2.0 [kWh/C • ], p i = 2.5 [1/C • ], r i = 0.2 [1/C • h], and
s i = 2.0 [C • ] for all i.
The parameters a i , C i are chosen close to the values found in [START_REF] Bacher | Identifying suitable models for the heat dynamics of buildings[END_REF] which discusses parameter estimation for a single-family building. Matrix B is selected as 12.0 -0.75 -0.75 -0.75 -0.75 -0.75 -0.15 -0.3 -0.45 -0.6 -0.75 12.0 -0.9 -0.9 -0.9 -0.9 -0.15 -0.3 -0.45 -0.6 -0.75 -0.9

B =                 12
12.0 -1.05 -1.05 -1.05 -0.15 -0.3 -0.45 -0.6 -0.75 -0.9 -1.05 12.0 -1.2 -1.2 -0.15 -0.3 -0.45 -0.6 -0.75 -0.9 -1.05 -1.2

12.0 -1.35 -0.15 -0.3 -0.45 -0.6 -0.75 -0.9 -1.05 -1.2 -1.35 12.0

                (8) with elements B i,j in [kW].
Matrix B is constructed such that building 1 is closest to a production facility and thus has the best ability to extract heat, whereas building 10 is the farthest away. In addition, the entries of B are such that fully opened control valves at sat (u) = 1 gives Q representing a reasonable peak heat demand for small houses. We simulate the system using the DifferentialEquations toolbox in Julia [START_REF] Rackauckas | Differentialequations.jl-a performant and feature-rich ecosystem for solving differential equations in Julia[END_REF], for an outdoor temperature scenario given by data from the city of Gävle, Sweden in October 2022 during which the temperature periodically drops to almost -20 • C. The data is gathered from the Swedish Meteorological and Hydrological Institute (SMHI). We compare three different controllers and three different cost functions. The first controller is the fully decentralized PI-controller considered in this paper. Secondly the coordinating controller consists of the same PI-controllers as the decentralized case, but with the coordinating rank-1 anti-windup signal żi = x i + β1 ⊤ dz (u) considered in [START_REF] Agner | Anti-windup coordination strategy around a fair equilibrium in resource sharing networks[END_REF]. Finally, the static controller is given by u = -B ⊤ C -1 x as considered in [START_REF] Bauso | The linear saturated decentralized strategy for constrained flow control is asymptotically optimal[END_REF], where C is the diagonal matrix of all heat capacities C i .

Figure 1 shows the resulting deviations x during the simulations. At around hour 100, the outdoor temperature is critically low. At this time, the buildings do not receive sufficient heat, regardless of the control strategy. Figure 1a shows that with the decentralized strategy, the worst deviations become larger than with the coordinating strategy (Figure 1b). However, not all buildings experience temperature deviations, whereas with the coordinating strategy, all the buildings share the discomfort. Lastly, the static controller has large deviations experienced by many buildings. Even when the outdoor temperature is manageable, the static controller has a constant offset from the comfort temperature, highlighting the usefulness of the integral action.

We evaluate the performance through the three cost functions

J 1 = 1 T T 0 x(t) 1 dt, (9)
J ∞ = 1 T T 0 x(t) ∞ dt, (10)
J 2 = 1 T T 0 x(t) ⊤ Lx(t) + sat (u(t)) ⊤ sat (u(t)) dt. ( 11
)
where T is the simulation time and L is a diagonal matrix where each element is given by l i = qi Ci . The cost J 1 mimics the optimality notion considered in this paper, J ∞ mimics the optimality notion considered in [START_REF] Agner | Anti-windup coordination strategy around a fair equilibrium in resource sharing networks[END_REF], and J 2 mimics the optimality considered in [START_REF] Bauso | The linear saturated decentralized strategy for constrained flow control is asymptotically optimal[END_REF]. Figure 2 shows the resulting evaluations. Figure 2a shows that the coordinating controller gives minimal worst-case deviations J ∞ , but Figure 2b shows that the total discomfort J 1 is minimized in the decentralized strategy. This result, i.e. that the total discomfort is minimized by decentralized control but the worst-case discomfort is minimized by coordination, is found also in [START_REF] Agner | Combating district heating bottlenecks using load control[END_REF] where a nonlinear model of the grid hydraulics and a 2-state model of building dynamics is employed. On the weighted cost J 2 as seen in Figure 2c, all controllers provide similar performance. The static controller slightly outperforms the other two in this scenario, but it is outperformed in every other measure.

V. PROOF OF EQUILIBRIUM EXISTENCE AND UNIQUENESS

We will now prove Theorem 1 through the use of Banach's fixed-point theorem [START_REF] Agarwal | Fixed Point Theory in Metric Spaces: Recent Advances and Applications[END_REF]. This proof requires the following two lemmas, the proofs of which are found in the appendix.

Lemma 2: Let f : R n → R n and h : R n → R n where h(x) = x -f (x) satisfy Assumption 1. Then f : R n → R n and h : R n → R n given by f (x) = f (x + x 0 ) -f (x 0 ) and h(x) = h(x + x 0 ) -h(x 0 ) for some x 0 ∈ R n also satisfy Assumption 1 and h(x) + f (x) = x.

Lemma 3: Let f : R n → R n and h : R n → R n where h(x) = x -f (x) satisfy Assumption 1. Then f : R n → R n and h : R n → R n given by f (x) = Df (D -1 x) and h(x) = Dh(D -1 x) where D is a diagonal positive definite matrix also satisfy Assumption 1 and h(x) + f (x) = x.

Proof: (of Theorem 1) Denoting by S a diagonal positive definite matrix gathering the anti-windup gains s i , i = 1, . . . n, we can rearrange (2)-( 4) by imposing ẋ = ż = 0, which yields

0 = h(u 0 ) + S -1 A -1 Bf (u 0 ) + S -1 A -1 w. ( 12
)
If there is a unique u 0 solving (12) then x 0 = A -1 Bf (u 0 ) + w and z 0 = R -1 (-P x 0 -u 0 ) are uniquely determined by ( 2) and ( 4) respectively, where R = diag{r 1 , . . . , r n } is invertible by Assumption 3. Hence we need only show that there is a unique u 0 solving [START_REF] Giaccagli | LMI conditions for contraction, integral action, and output feedback stabilization for a class of nonlinear systems[END_REF] for the proof to be complete. Let D be a diagonal positive definite matrix such that DS -1 A -1 BD -1 is strictly diagonally column-dominant. Note that such a D always exists because A and S are diagonal positive definite and B is an Mmatrix. Left-multiply ( 12) by D and insert multiplication by

I = D -1 D before f (u 0 ) to obtain 0 = Dh(u 0 ) + DS -1 A -1 BD -1 Df (u 0 ) + DS -1 A -1 w. (13)
Introduce the change of variables B = DS -1 A -1 BD -1 , ζ = Du 0 , and ŵ = DS -1 A -1 w. Then (13) yields

0 = Dh(D -1 ζ) + BDf (D -1 ζ) + ŵ. (14)
Here we can use Lemma 3 to replace f (•), h(•) with f (•), ĥ(•), which satisfy Assumption 1 and

f (ζ) + ĥ(ζ) = ζ.
Introduce a scalar k satisfying k > max(1, 2max i Bi,i ). Divide ( 14) by -k, add ζ to the left-hand side, and ζ = f (ζ) + ĥ(ζ) to the right-hand side of ( 14) to obtain

ζ = - 1 k (1 -k) ĥ(ζ) + ( B -kI) f (ζ) + ŵ . (15)
We define the right-hand side of this expression as T w (ζ), defined for a specific w. By showing that T w is a contractive mapping for any ŵ, we can use Banach's fixed point theorem [START_REF] Agarwal | Fixed Point Theory in Metric Spaces: Recent Advances and Applications[END_REF] to show that there is a unique solution ζ = T w (ζ) (and thus a unique u 0 = D -1 ζ) for any ŵ (and thus any w

= ASD -1 ŵ). Consider any α ∈ R n , β ∈ R n . Then T w (α) -T w (β) = -1+k
k ĥ(α) -ĥ(β)

+ -B+kI k f (α) -f (β) . (16)
Here we use Lemma 2 to introduce h(α -β) = ĥ(α) -ĥ(β) and f (α -β) = f (α) -f (β). Denote ∆ = α -β and ∆ + = T w (α) -T w (β). Then

|∆ + i | ≤ k -1 k | hi (∆ i )|+ k -Bi,i k | fi (∆ i )|+ j =i | Bi,j | k | fj (∆ j )|. (17) Therefore ∆
+ 1 = n i=1 |∆ + i | ≤ n i=1 k -1 k | hi (∆ i )| + k -Bi,i k | fi (∆ i )| + j =i | Bj,i | k | fi (∆ i )| . ( 18
)
Due to the diagonal column-dominance of B and the def-

inition of k, it holds that k > Bi,i > j =i | Bj,i |. Thus, selecting λ = k-1 k < 1, µ i = k-( Bi,i-j =i | Bj,i|) k < 1, γ i = max(λ, µ i ) < 1, and γ = max i γ i < 1, we obtain ∆ + 1 ≤ n i=1 λ| hi (∆ i )| + µ i | fi (∆ i )| ≤ n i=1 γ i | hi (∆ i )| + | fi (∆ i )| ≤ n i=1 γ|∆ i | = γ ∆ 1 . ( 19
) Note that | hi (∆ i )|+| fi (∆ i )| = |∆ i | since fi (∆ i ) and hi (∆ i )
always have the same sign by Assumption 1, and sum to ∆ i . This proves that T w is a contraction mapping with respect to the metric • 1 . Thus, by Banach's fixed point theorem, for each w and the ensuing ŵ = DS -1 A -1 w there is a unique ζ such that (15) holds, and thus a u 0 = D -1 ζ such that [START_REF] Giaccagli | LMI conditions for contraction, integral action, and output feedback stabilization for a class of nonlinear systems[END_REF] holds, which completes the proof.

VI. PROOF OF GLOBAL ASYMPTOTIC STABILITY

Given the existence of an equilibrium (x 0 , z 0 ) and the associated input u 0 , consider the change of variables z = -R(z -z 0 ), ũ = u -u 0 , f (ũ) = f (u 0 + ũ) -f (u 0 ), and

h(ũ) = h(u 0 + ũ) -h(u 0 ). Due to Lemma 2, f (•), h(•)
satisfy Assumption 1, and f (ũ) + h(ũ) = ũ. This allows rewriting the ( 2)-( 4) as

ż u = -RP -1 RP -1 A -RP -1 -A + RP -1 z ũ - 0 P B f (ũ) - RS RS h(ũ)
(20) where P , R, and S are diagonal matrices gathering the controller parameters p i , r i , and s i . Stabilizing this system to z = ũ = 0 is equivalent to stabilizing the original system system to the equilibrium x = x 0 , z = z 0 , and u = u 0 . We will therefore now prove Theorem 2 with a Lyapunov-based argument considering system (20).

Proof: (of Theorem 2) Consider the Lyapunov function candidate

V (z, ũ) = n i=1 zi 0 q i (a i p i r i -1) fi (ζ) + ǫζ) dζ + n i=1 ũi 0 q i fi (ζ) + ǫζ dζ (21)
where scalars q i > 0 and ǫ > 0 are parameters to be fixed later. For any such choice of parameters, V is positive definite and radially unbounded because fi (ζ) + ǫζ is increasing in ζ and zero at zero. Also a i pi ri -1 > 0 due to Assumption 3. The time derivative of V along the trajectories of system (20) is given by

V (z, ũ) = -f (z) + ǫz -f (ũ) -ǫũ ⊤ D(z -ũ) (22a) -f (z) + ǫz ⊤ DP S h(ũ) (22b) -f (ũ) + ǫũ ⊤ QRS h(ũ) (22c) -f (ũ) + ǫũ ⊤ QP B f (ũ) ( 22d
)
where D is a diagonal positive definite matrix gathering the elements q i (a i -r i /p i ) and Q is a diagonal positive definite matrix gathering the elements q i . To simplify this expression, we split it into

V (z, ũ) = V1 (z, ũ) + V2 (z, ũ) (23
i ) + ǫz i -fi (ũ i ) -ǫũ i Di,i (z i -ũi ) = | fi (z i ) + ǫz i | + | fi (ũ i ) -ǫũ i | Di,i (|z i | + |ũ i |).
f (ũ) + ǫũ ⊤ QRS h(ũ) = f (ũ) + ǫ f (ũ) + ǫ h(ũ) ⊤ QRS h(ũ) = (1 + ǫ) f (ũ) ⊤ QRS h + ǫ h(ũ) ⊤ QRS h(ũ) ≥ ǫβ h(ũ) 2 2 ( 24
)
where β is the minimum diagonal element of QRS. Note also that

f (ũ) + ǫũ ⊤ QP B f (ũ) = (1 + ǫ) f (ũ) ⊤ QP B f (ũ) + ǫ h(ũ) ⊤ QP B f (ũ). (25)
Fix now the weights q i in such a way that QP B + B ⊤ P Q is positive definite. This is possible because B is an M-matrix according to Assumption 2. Therefore ∃α > 0 such that QP B + B ⊤ P Q ≻ 2αI. Thus the first term of (25) satisfies

(1 + ǫ) f (ũ) ⊤ QP B f (ũ) ≥ (1 + ǫ)α f (ũ) 2 2 . (26)
We also note that the second term in (25) satisfies

ǫ h(ũ) ⊤ QP B f (ũ) ≥ -ǫγ f(ũ) 2 h(ũ) 2 (27)
where γ = QP B 2 . Thus, combining the bounds in (24), ( 26) and ( 27) within (22c)-(22d), we obtain

V2 (z, ũ) ≤ -(1 + ǫ)α f (ũ) 2 2 -ǫβ h(ũ) 2 2 + ǫγ f(ũ) 2 h(ũ) 2 = f (ũ) 2 h(ũ) 2 ⊤ -(1 + ǫ)α 1 2 ǫγ 1 2 ǫγ -ǫβ f (ũ) 2 h(ũ) 2 . (28)
We may now select the Lyapunov function parameter ǫ sufficiently small such that α + ǫα -ǫγ 2 4β > 0. This makes the quadratic form (28) negative definite. Thus V2 (z, ũ) = 0 if and only if f (ũ) = h(ũ) = 0, i.e. if and only if ũ = 0. In this case, V1 (z, ũ) is clearly negative definite in z. Thus V (z, ũ) is negative definite, which implies that the origin is globally asymptotically stable for system (20). Equivalently, the equilibrium (x 0 , z 0 ), with input u 0 , is therefore globally asymptotically stable for the original system (2)-(4).

VII. PROOF OF EQUILIBRIUM OPTIMALITY

Here we prove Theorem 3.

Proof: Firstly, it is clear that v * = sat u 0 and x * i = x 0 i = -s i dz u 0 i for all i satisfies (5b) due to x 0 , z 0 being an equilibrium, and satisfies (5c) because sat (•) is bounded in the range [-1, 1]. Consider, for establishing a contradiction, that there exists µ = 0 such that v † = v * + µ and x † = A -1 Bv † + A -1 w = x * + A -1 Bµ is the optimal solution to (5) with a smaller cost (5a) than the one obtained by x * , v * . Then µ solves the optimization problem

minimize µ n i=1 |γ i x * i + Bi µ| (29a) subject to -1 ≤ v * + µ ≤ 1. ( 29b
)
where Bi is row i of the matrix B = ΓA -1 B. The equilibrium of (3) implies x * i = -s i dz u 0 i . Therefore we can leverage (29b) to see that x * i > 0 =⇒ u 0 i < -1 =⇒ v i = -1 =⇒ µ i ≥ 0 and conversely x * i < 0 =⇒ u 0 i > 1 =⇒ v i = 1 =⇒ µ i ≤ 0. Combining this with Γ and A both being diagonal, positive definite and the fact that B is an M-matrix which implies that Bi,i > 0, we obtain |γ i x i + Bi,i µ i | = |γ i x i | + | Bi,i µ i | for all i. Thus (29a) can be expanded as follows

VIII. CONCLUSIONS

In this paper we considered fully decentralized PI-control for a class of interconnected systems subject to incrementally sector-bounded nonlinearities. We showed that for systems where the input matrix is an M-matrix, fully decentralized PI-controllers globally asymptotically stabilize a specific equilibrium. Furthermore, this equilibrium is optimal in that it minimizes costs of the form n i=1 γ i |x i |. The proposed control strategy was employed in a numerical example of a simplified district heating system model. The example showed that, with our decentralized strategy, the total discomfort in the system is minimized, at the cost of higher worst-case discomforts when compared with a alternative coordinated control strategies. We have thus demonstrated that a fully decentralized and easily tuned control law constitutes a relevant design for a large class of systems.

Open questions include analysis of the transient response, and finding controller tuning rules accordingly. Furthermore, to better capture the district heating application, a richer class of systems should be considered: Multi-state models for each individual building, as well as more complex, nonlinear models of the interconnection B should be considered.

  .0 -0.15 -0.15 -0.15 -0.15 -0.15 -0.15 -0.15 -0.15 -0.15 -0.15 12.0 -0.3 -0.3 -0.3 -0.3 -0.3 -0.3 -0.3 -0.3 -0.15 -0.3 12.0 -0.45 -0.45 -0.45 -0.45 -0.45 -0.45 -0.45 -0.15 -0.3 -0.45 12.0 -0.6 -0.6 -0.6 -0.6 -0.6 -0.6 -0.15 -0.3 -0.45 -0.6

Fig. 1 : 1 ( 2 (

 112 Fig. 1: Simulation results showing temperature deviations x for each control strategy. The blue lines (with values on the left axis) are the temperature deviations x. The dotted, black line (with values on the right axis) is the outdoor temperature. At about hour 100, the temperature becomes critically low. This induces saturation in the controllers, leading to a drop of the indoor temperatures.

Fig. 2 :

 2 Fig. 2: Cost functions evaluated for each control strategy.

  n i=1 |γ i x i + Bi µ| ≥ i =j   |γ i x i + Bi,i µ i | -| i =j Bi,j µ j |   ≥ n i=1 |γ i x i | + | Bi,i ||µ i | -n i=1 j =i | Bi,j ||µ j | = n i=1 |γ i x i | + n k=1   | Bk,k |j =k | Bj,k |   |µ k |.

  B is diagonally column-dominant, then | Bk,k |j =k | Bj,k | isstrictly positive for all k. Thus this expression is minimized by µ = 0, which completes the proof.

  and DP S are diagonal, V1 can be analyzed for each i individually. fi(ζ i ) + ǫζ i is increasing in ζ i , therefore sign fi (z i ) + ǫz i -fi (ũ i ) -ǫũ i = sign (z i -ũi )and thus (22a) is negative semi-definite. If zi and ũi have the same sign, (22b) contributes negatively to V1 . If they have opposite signs the contribution is positive, but in this case (22a) only comprises negative terms because fi (z

	)
	where V1 (z, ũ) corresponds to the terms (22a)-(22b) and V2 (z, ũ) corresponds to the terms (22c)-(22d). Since D

  Indeed, since p i s i < 1 from Assumption 3 and | hi (ũ i )| ≤ |ũ i | from Assumption 1, then (22a) as developed above dominates (22b) which is upper bounded by | fi (z i ) + ǫz i | Di,i | hi (ũ i )|,and thus V1 is negative semidefinite. We now turn our attention to V2 . Note that ũ, f (ũ), and h(ũ) elementwise have the same sign and QRS is diagonal, positive definite. Thus

† This work is funded by the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation program under grant agreement No 834142 (ScalableControl). * This work was partially funded by Wallenberg AI, Autonomous Systems and Software Program (WASP) funded by the Knut and Alice Wallenberg Foundation.

APPENDIX

We prove here suitable properties of the function class characterized by Assumption 1, as stated in Lemmas 1, 2 and 3. To simplify the exposition, we drop the index i.

Proof: (of Lemma 1)

Clearly, h(0

which clearly shows that if x = y then 0 ≤ (h(y) -h(x))/(y -x) ≤ 1, thus concluding the proof.

Proof:

which clearly shows that if x = y then 0 ≤ ( f (y) -

which clearly shows that if x = y then 0 ≤ ( f (y)f (x))/(y -x) ≤ 1. Finally f (x) + h(x) = Df (D -1 x) + Dh(D -1 x) = D f (D -1 x) + h(D -1 x) = DD -1 x = x, thus concluding the proof.


Vera Bikbaeva

Nikolay Nesterenko
email: nikolay.nesterenko@sulzer.com

Stanislav Konnov

Thanh-Son Nguyen

Jean-Pierre Gilson

Valentin Valtchev

Affiliations

A Low Carbon Route to Ethylene: Ethane Oxidative Dehydrogenation with CO 2 on Embryonic Zeolite Supported Mo-Carbide Catalyst

Keywords: CO 2 utilization, Oxidative dehydrogenation, Embryonic zeolite, Mo modification, CO 2 negative ethylene production

 Embryonic zeolite supported Mo 2 C catalyst is prepared and comprehensively characterized  The catalyst is employed for oxidative dehydrogenation of ethane with CO 2 (ODH-CO 2 )  The proposed catalyst allows designing a potentially CO 2 negative technology  109% of CO 2 reduction in comparison to state-of-the-art ethane cracker is achieved  A techno-economical analysis of the proposed ODH-CO 2 process is performed

Introduction

Ethylene is the largest-volume petrochemical building block, produced primarily by steam cracking of oil fractions [START_REF] Foley | Ethylene production by steam cracking of normal paraffins[END_REF]. In 2020, the worldwide ethylene production capacity was 201 million tons, with an expected increase to 263 million tons in 2030 [START_REF]Ethylene production capacity globally 2025, Statista, Oxidative Dehydrogenation of Ethane Technology[END_REF]3]. Ethylene is used to produce a wide spectrum of products like plastics, fibers, and other chemicals in the packaging, transportation, and construction industries. Polyethylene is the single-largest outlet, accounting for about 60% of global ethylene consumption in 2020. Other major applications include the production of ethylene oxide, ethylene dichloride, and ethylbenzene.

Steam cracking of oil fractions is the major process currently used to produce ethylene; a smaller contribution comes from refinery processes such as fluid catalytic crackers and cokers. It is a mature and well-established technology, which operation is considered near-optimal considering constant incremental improvements of the technology over 80 + years [START_REF] Zimmermann | Ullmanns Encycl. Ind. Chem[END_REF][START_REF] Ren | Olefins from conventional and heavy feedstocks: Energy use in steam cracking and alternative processes[END_REF], leading to a thermal efficiency of close to 95% [START_REF] Yao | Understanding Variability To Reduce the Energy and GHG Footprints of U.S. Ethylene Production[END_REF]. However, cracking is a very endothermic reaction taking place at high temperatures. As a result, ethylene production via steam cracking consumes about 30 % of the energy used in the chemical industry. In particular, as ethylene production by steam cracking consumes roughly between 17-25 GT/ton of ethylene while releasing about 1.2 tons of CO 2 per ton of olefin produced [START_REF] Ren | Olefins from conventional and heavy feedstocks: Energy use in steam cracking and alternative processes[END_REF]; it is with about 260 million tons of CO 2 emissions worldwide per year [START_REF] Yao | Understanding Variability To Reduce the Energy and GHG Footprints of U.S. Ethylene Production[END_REF], one of the largest contributors to greenhouse gas emissions in the manufacturing sector. Alternative routes and feedstocks need to be explored and developed to foster sustainable development of olefins production. In the last decade, the ongoing shift to lighter hydrocarbons as steam cracking feedstock has prompted research and development of alternative on-purpose production routes to ethylene.

One of the options for sustainable ethylene production is in the utilization of bio feedstock: either bioethanol or bio hydrocarbon fractions [START_REF] Mohsenzadeh | Bioethylene Production from Ethanol: A Review and Techno-economical Evaluation[END_REF][START_REF] Johnson | A carbon footprint of HVO biopropane, Biofuels Bioprod[END_REF][START_REF] Vermeiren | Procédé de production de produits de craquage par vapocraquage de bio-naphtha et bio-propane obtenus à partir de mélanges complexes d'huiles et de graisses existant à l'état naturel[END_REF]. Bioethanol dehydration shows advantages in energy consumption and in capital investments relative to conventional steam cracking due to simpler product slate and lower reaction temperature. Until the mid-1940s, a significant portion of the ethylene produced industrially was produced from bioethanol.

Afterward, steam cracking of hydrocarbons, because of their superior economies of scale and lower feedstock costs, became the main source of ethylene. Currently, with the significantly increased cost of natural gas and crude oil together with the need to reduce the carbon intensity of the petrochemical industry, the downstream industry is renewed interest in ethylene-fromethanol plants [START_REF] Jamil | Greener and sustainable production of bioethylene from bioethanol: current status, opportunities and perspectives[END_REF][START_REF] Zhang | Dehydration of Ethanol to Ethylene[END_REF]. So, this route will certainly regain importance but will not fulfill the total demand for ethylene due to the competition for the feedstock with biofuel production. The availability of bio-naphtha and bio-propane as a feedstock for olefins production is still limited [START_REF] Johnson | A carbon footprint of HVO biopropane, Biofuels Bioprod[END_REF]. The direct conversion of the cellulosic biomass or bio-oil remains an attractive route but mainly provides aromatic fractions [START_REF] Corma | Processing biomass-derived oxygenates in the oil refinery: Catalytic cracking (FCC) reaction pathways and role of catalyst[END_REF][13][START_REF] Yang | Prospective contributions of biomass pyrolysis to China's 2050 carbon reduction and renewable energy goals[END_REF].

So, a reduction of the carbon intensity of ethylene production routes remains a very important task for the petrochemical industry. The oxidative dehydrogenation of ethane (ODH) shows good prospects to reduce greenhouse gases emissions in comparison to a conventional ethane cracker due to a number of advantages, including lower energy consumption, simpler production slate, and potential elimination of the equilibrium limitations [START_REF] Heracleous | Investigation of Ni-based aluminasupported catalysts for the oxidative dehydrogenation of ethane to ethylene: structural characterization and reactivity studies[END_REF][START_REF] Chen | Isotopic Tracer and Kinetic Studies of Oxidative Dehydrogenation Pathways on Vanadium Oxide Catalysts[END_REF]. One of the main limitations remains the development of an efficient and stable high-temperature catalyst.

Already in the early 80 ' s, Phillips Petroleum proposed the first industrial Li-SnCl 2 -MgO catalyst for oxidative dehydrogenation of ethane with oxygen [START_REF] Kolts | Composition of matter and method of oxidative conversion of organic compounds therewith, US4672145A[END_REF]. This process did not require acetylene hydrogenation and had the potential to reduce energy consumption in an ethylene plant, allowing operations at 700 °C instead of 850 °C, and providing a higher ultimate yield of 90.5% ethylene [START_REF] Kolts | Composition of matter and method of oxidative conversion of organic compounds therewith, US4672145A[END_REF] compared to the thermal process. However, as CO 2 was an important by-product, the economics were not better than state-of-the-art steam crackers. Recently, the topic regain attention the two semi-commercial oxidative dehydrogenation (ODH) of ethane processes were proposed [START_REF]Oxidative Dehydrogenation of Ethane Technology[END_REF][START_REF] Gärtner | Oxidative Dehydrogenation of Ethane: Common Principles and Mechanistic Aspects[END_REF][START_REF] Cavani | Oxidative dehydrogenation of ethane and propane: How far from commercial implementation?[END_REF]. Oxygen as an oxidant could be used in gaseous form or from catalysts' metal oxides lattice. One such promising on-purpose Oxidative Dehydrogenation of Ethane (ODH-E) technology is proposed by the Linde/Clariant partnership; a high selectivity to valuable products (ethylene and acetic acid) compared to steam cracking combined with a significant reduction potential of the CO 2 footprint is promised. Moreover, the reaction takes place under mild conditions, for instance, at atmospheric pressure and a maximum temperature of 400 °C [START_REF] Arnold | Process for producing ethylene via oxidative dehydrogenation (ODH) of ethane[END_REF].

However, as oxygen is applied as an oxidant, CO 2 is still present in products.

Another ODH option is based on chemical looping (CL-ODH). In CL-ODH, ethane is oxidatively dehydrogenated to ethylene and water following a typical Mars-van Krevelen mechanism with the participation of lattice oxygen from the catalyst (Catalyst reduction, Reaction 1a) and its exothermic air regeneration (Reaction 1b), further providing heat to the endothermic ethane dehydrogenation to ethylene [START_REF] Neal | Intensified Ethylene Production via Chemical Looping through an Exergetically Efficient Redox Scheme[END_REF].

CL-ODH (Reduction)

C 2 H 6 + Me y O x → C 2 H 4 + H 2 O + Me y O x-1 (Reaction 1a) CL-ODH (Regeneration) Me y O x-1 + 1/2O 2 →Me y O x (Reaction 1b)
Other alternative approaches, currently at the laboratory scale, also using CO 2 as an oxidant to produce value-added ethylene from ethane, are proposed [START_REF]Oxidative Dehydrogenation of Ethane Technology[END_REF][START_REF] Gärtner | Oxidative Dehydrogenation of Ethane: Common Principles and Mechanistic Aspects[END_REF]. Such processes are attractive because they may lead to a CO 2 -negative route in ethylene production. An additional benefit is a significant decrease in energy consumption downstream as hydrogen will not need purification: it will be consumed in situ to convert CO 2 to CO, and the latter could further be used to produce methanol, carbonylate, or hydroformylate other feedstocks. This approach is particularly interesting in view of recent trends to "electrify" steam crackers [START_REF] Basf | join forces to realize the world's first electrically heated steam cracker furnace[END_REF][START_REF]Petrochemical companies form Cracker of the Future Consortium and sign R&D agreement -Borealis, Borealisgroup[END_REF][START_REF]Dow and Shell demonstrate progress in joint technology development for lower CO2 emission crackers[END_REF].

It is of note that the enthalpies of reactions 2 and 3 (Δ r Hᵒ 298 ) to produce ethylene from ethane by steam cracking and ODH-CO 2 are very close: The ODH-CO 2 reaction was first reported by Lunsford et al. [START_REF] Wang | Effect of carbon dioxide on the selectivities obtained during the partial oxidation of methane and ethane over Li+/MgO catalysts[END_REF] using Li/MgO catalysts.

C 2 H 6 (g) + CO 2 (g) → C 2 H 4 (g) + CO(g) + H 2 O (l), Δ
Krylov et al. [START_REF] Krylov | The regularities in the interaction of alkanes with CO2 on oxide catalysts[END_REF] investigated the ODH-CO 2 using several oxide catalysts (Fe 2 O 3 , Cr 2 O 3 , MnO 2 ) and multi-component systems. In the ODH-CO 2 at 800 °C, MnO 2 -based catalysts exhibited high activities but low ethylene selectivity [START_REF] Krylov | Oxidation of Hydrocarbons and Alcohols by Carbon Dioxide on Oxide Catalysts[END_REF]. Nakagawa et al. [START_REF] Nakagawa | Dehydrogenation of ethane over gallium oxide in the presence of carbon dioxide[END_REF] studied the dehydrogenation of C 2 H 6 by CO 2 on several oxides and reported that gallium oxide (Ga 2 O 3 ) is an effective catalyst for this reaction, giving a C 2 H 4 yield of 18.6% with a selectivity of 94.5% at 650 °C.

Several reactions may occur in parallel, including oxidative dehydrogenation (ODH), reverse water-gas-shift (RWGS), or dry reforming of ethane (DRE) [START_REF] Yao | Combining CO 2 Reduction with Ethane Oxidative Dehydrogenation by Oxygen-Modification of Molybdenum Carbide[END_REF]. For the ODH-CO 2 reaction, precious metals, oxide [START_REF] Talati | Oxidative dehydrogenation of ethane to ethylene by carbon dioxide over Cr/TiO2-ZrO2 nanocatalyst: Effect of active phase and support composition on catalytic properties and performance[END_REF], and carbide [START_REF] Yao | Combining CO 2 Reduction with Ethane Oxidative Dehydrogenation by Oxygen-Modification of Molybdenum Carbide[END_REF]-based catalysts are reported. Most of the catalytic systems are based on Pt [START_REF] Myint | Reforming and oxidative dehydrogenation of ethane with CO2 as a soft oxidant over bimetallic catalysts[END_REF], Ni [START_REF] Qiao | Oxidative dehydrogenation of ethane to ethylene over Ni-Nb-M-O catalysts: Effect of promoter metal and CO2-admixture on the performance[END_REF], Co [START_REF] Koirala | Effect of cobalt loading on structure and catalytic behavior of CoOx /SiO 2 in CO 2 -assisted dehydrogenation of ethane[END_REF], Cr [START_REF] Bai | Oxidative dehydrogenation of ethane with carbon dioxide over Cr 2 O 3 /SBA-15 catalysts: the influence of sulfate modification of the support[END_REF][START_REF] Bugrova | Oxidative dehydrogenation of ethane with CO 2 over CrO x catalysts supported on Al 2 O 3 , ZrO 2 , CeO 2 and Ce x Zr 1-x O 2[END_REF], Fe [START_REF] Theofanidis | CO 2 -oxidative ethane dehydrogenation over highly efficient carbon-resistant Fe-catalysts[END_REF] oxides deposited on zeolite or oxide carriers.

Only a few Mo-based materials were reported for the ODH-CO 2 of ethane [START_REF] Bakhtyari | Simultaneous production of ethylene and hydrogen through carbon-dioxide-assisted conversion of ethane over cobalt-molybdenum catalysts[END_REF][START_REF] Watson | Mo Loading Effects over Mo/Si : Ti Catalysts in the Oxidative Dehydrogenation of Ethane[END_REF][START_REF] Heracleous | Surface Properties and Reactivity of Al 2 O 3 -Supported MoO 3 Catalysts in Ethane Oxidative Dehydrogenation[END_REF], but molybdenum carbide is well known as an alternative catalyst to noble metals in the highly selective reduction of CO 2 to CO [START_REF] Liu | Effective and Highly Selective CO Generation from CO 2 Using a Polycrystalline α-Mo 2 C Catalyst[END_REF][START_REF] Porosoff | Molybdenum Carbide as Alternative Catalysts to Precious Metals for Highly Selective Reduction of CO 2 to CO[END_REF]. Very high conversions of ethane in co-processing with CO 2 to produce hydrogen, ethylene, and CO were reported on the catalysts made up of the oxides of cobalt and molybdenum [START_REF] Bakhtyari | Simultaneous production of ethylene and hydrogen through carbon-dioxide-assisted conversion of ethane over cobalt-molybdenum catalysts[END_REF][START_REF] Zafarnak | Conversion of ethane to ethylene and hydrogen by utilizing carbon dioxide: Screening catalysts[END_REF]. 58% conversion of ethane, 10.3% of CO 2 , and 21.9% ethylene yield were achieved over these systems at 700 °C [START_REF] Zafarnak | Conversion of ethane to ethylene and hydrogen by utilizing carbon dioxide: Screening catalysts[END_REF]. The conversion of ethane in these studies was higher relative to the thermodynamic equilibrium values of ethane conversion alone due to a significant contribution of secondary reactions [START_REF] Bikbaeva | Ethane oxidative dehydrogenation with CO 2 on thiogallates[END_REF]. Relatively low selectivity to ethylene showed a significant contribution to the direct ethane dehydrogenation resulting in high hydrogen production [START_REF] Bakhtyari | Simultaneous production of ethylene and hydrogen through carbon-dioxide-assisted conversion of ethane over cobalt-molybdenum catalysts[END_REF][START_REF] Zafarnak | Conversion of ethane to ethylene and hydrogen by utilizing carbon dioxide: Screening catalysts[END_REF].

Mo 2 C supported on SiO 2 was found to be an effective catalyst for the dehydrogenation of ethane to ethylene in the presence of CO 2 . The selectivity to ethylene at 600-650 °C was 90-95% at an ethane conversion of 8-30%. At higher temperatures, dry reforming of ethane penalized ethylene yield. It is assumed that Mo oxycarbides formed in the reaction between CO 2 and Mo 2 C play an important role in ethane activation [START_REF] Solymosi | The oxidative dehydrogenation of ethane with CO 2 over Mo 2 C/SiO 2 catalyst[END_REF]. Recently, a Mo-containing catalyst was reported for ODH-CO 2 but again only at 600 °C with relatively modest performance [START_REF] Yao | Combining CO 2 Reduction with Ethane Oxidative Dehydrogenation by Oxygen-Modification of Molybdenum Carbide[END_REF][START_REF] Marquart | Conversion of CO 2 and small alkanes to platform chemicals over Mo 2 C-based catalysts[END_REF]. Oxygen from CO 2 appears crucial in cleaving the ethane C-C bond, increasing ethylene production. The addition of a Fe promoter accelerated the formation of surface oxygen species and stabilized them from reduction by ethane, leading to a shorter induction period, higher ethylene yield, and improved stability [START_REF] Yao | Combining CO 2 Reduction with Ethane Oxidative Dehydrogenation by Oxygen-Modification of Molybdenum Carbide[END_REF]. However, the main problems with Mo 2 C catalysts operating at high temperatures remain the coking, sintering of the active Mo-species from cycle to cycle, and a significant contribution of the dry reforming reaction.

In order to design a stable at high-temperature Mo 2 C-supported catalyst, we propose a different approach consisting of the utilization of MoS 2 precursors in place of oxides and embryonic zeolites (EZ) [START_REF] Akouche | Synthesis of Embryonic Zeolites with Controlled Physicochemical Properties[END_REF] as support in place of silica or alumina. An advantage of embryonic zeolites is the high surface area with a significant micropore volume resulting in the good dispersion and accessibility of all types of active sites [START_REF] Haw | Supported Embryonic Zeolites and their Use to Process Bulky Molecules[END_REF]. In addition, EZ undergoes changes in surface area after the first heating cycle above 600 °C [START_REF] Haw | Supported Embryonic Zeolites and their Use to Process Bulky Molecules[END_REF], which stabilizes the metal against sintering while keeping the active sites still accessible for the small molecules. After the first surface reduction, substantially no changes occur with these materials upon consecutive thermal cycles. In this context, the embryonic zeolites as support could also stabilize the formed molybdenum carbide phase at high temperatures. However, the aging of EZ zeolite is a long process, what may lead to the agglomeration of MoO x -oxides species before carburization. In order to limit the agglomeration of the MoO x species during the first heating cycle, the idea was to use a MoS 2 -precursor supported on EZ. Supported sulfides, especially MoS 2 , are also known for CO 2 hydrogenation at high-pressure [START_REF] Hu | Sulfur vacancy-rich MoS 2 as a catalyst for the hydrogenation of CO2 to methanol[END_REF], but using MoS 2 for oxidative or non-oxidative ethane dehydrogenation was never reported to the best of our knowledge. This work reports the preparation of a novel thermally stable Mo-based catalyst supported on EZ carriers from a MoS 2precursor (Mo 2 C@EZ), its comprehensive characterization, and catalytic performances in ODH-CO 2 of ethane.

Experimental section

Preparation of EZ carriers

An all-silica (EZ Si ) and an aluminosilicate (EZ Al-Si ) embryonic zeolites were used as carriers in all experiments.

The initial gel composition for the EZ Si synthesis was 9 TPAOH:0.25 Al 2 O 3 :25 SiO 2 :430 H 2 O:100 EtOH, and for EZ Al-Si , 9 TPAOH:25 SiO 2 :430 H 2 O:100 EtOH where TPAOH stands for tetrapropylammonium hydroxide [START_REF] Akouche | Synthesis of Embryonic Zeolites with Controlled Physicochemical Properties[END_REF]. A typical preparation procedure includes mixing the TPAOH solution (20% in water, Alfa Aesar) with distilled water. In the case of synthesis of the embryonic zeolite Al 2 (SO 4 )•18 H 2 O (Sigma-Aldrich,98%) was subsequently added and the system homogenized by stirring. Tetraethyl orthosilicate (TEOS Sigma-Aldrich, 98%) was then added, and the mixture stirred for 6 h leading to the formation of water clear homogeneous sol.

The solid product was recovered by freeze-drying (-94 °C under vacuum). The recovered solids were ion-exchanged with 0.5 mol ammonium nitrate solution, washed with distilled water, and dried under air. The resulting samples are referred to as EZ Al-Si and EZ Si . A ZSM-5 (Si/Al=47.6) provided by Tosoh was used as a reference catalyst.

1.2.Metal loading and sulfidation

Mo oxide-decorated catalysts were prepared by incipient wetness impregnation on the embryonic zeolite carriers. The samples were loaded with 5 wt. % of Mo and denoted as EZ Al-Si -5%Mo, EZ Si -5%Mo, and ZSM-5-5%Mo. (NH 4 ) 6 Mo 7 O 24 •4H 2 O (Alfa Aesar, 99%) was used as a source of Mo. After drying at room temperature, all samples were calcined at 550 °C for 5h in static air.

Mo sulfidation was performed under an H 2 S/H 2 gas flow (30 ml/min, 10 vol%H 2 S). 300 mg of the pelletized sample (35/45 mesh) was loaded in a Pyrex reactor, first heated under an N 2 flow at 140 °C for 2 h, then at 350 °C (2,8 °C•min -1 ramp rate) for 2 h under an H 2 S/H 2 flow, cooled down to 180 °C and finally under a flow of dry N 2 to room temperature and kept there for 3 h. The samples were denoted EZ Al-Si -5%MoS 2 , EZ Si -5%MoS 2 , and the reference ZSM-5-5%MoS 2 .

Physicochemical characterization

The powder X-ray diffraction (PXRD) patterns were recorded on a PANalytical X'Pert Pro diffractometer using the Cu Kα radiation (λ = 1.5418 Å). The scanning electron micrographs were collected on a MIRA-LMH (Tescan) SEM equipped with a field emission gun.

Nitrogen physisorption was performed on a Micromeritics ASAP 2020 surface area analyzer. The calcined samples were analyzed after degassing at 300 °C. The microporous volume (V mic , cm 3 •g -1 ) and the external surface area (S ext , m 2 •g -1 ) were obtained from the t-plot method based on the Harkins-Jura equation.

All 27 Al NMR measurements were done with 4 mm (o.d.) zirconia rotors spun at 14 kHz on a Bruker Avance 400 spectrometer operating at 100.6 MHz. 29 Si MAS NMR was recorded on a Bruker Avance 500 spectrometer operating at 130.3 MHz. TMS was the reference for 29 Si, while a 1M Al(NO 3 ) 3 solution was used for 27 Al. The quantification of the spectra was done by integrating the peaks using the Dmfit program.

The formation of the Molybdenum oxide and sulfide phases was monitored by Raman spectroscopy. The Raman spectra were acquired with a green laser at 532 nm (Jobin Yvon Labram 300 confocal Raman spectrometer coupled to a microscope and CCD detector). Both accumulation time and laser power were adjusted for the samples (embryonic zeolites after metal loading: 40-60 s accumulation 3 times at 50% laser power; after sulfidation: 15-25 s accumulation 3 times at 1% laser power).

Pyridine adsorption monitored by IR spectroscopy was selected to evaluate the acidity.

The spectra were collected on a Nicolet Magna 550 FTIR spectrometer equipped with a DTGS detector at a 4 cm -1 optical resolution with one level of zero filling for the Fourier transform.

Prior to measurement, each sample was ground and pressed into a self-supporting disc (diameter: 2 cm, approximately 5 mg•cm -2 ) and activated in a vacuum (ca. 10 -6 hPa) at 450 °C for 2 h at 2 °C•min -1 . After cooling to room temperature, a spectrum was recorded as a reference. A pressure of 1.33 hPa of pyridine was established in the cell at room temperature till saturation was reached. The wafer was then heated to 100 °C for 15 min to facilitate pyridine diffusion in the whole disc. Successive evacuations were performed at 100, 150, 200, 250, 300, and 350 °C at 15 min intervals, and spectra were recorded at each step. All spectra were normalized to a 2.5 mg• cm -2 wafer. The number of acidic sites was determined using the areas of the bands at 1450 cm -1 (Lewis) and 1545 cm -1 (Brønsted). The molar extinction coefficients (ε) used for quantification were ε1545 (B-pyridine) = 1.8 cm•μmol -1 , ε1455 (L-pyridine) = 1.5 cm•μmol -1 [START_REF] Akouche | Synthesis of Embryonic Zeolites with Controlled Physicochemical Properties[END_REF]. The OMNIC software program was used for data processing. Thermogravimetric analysis (TGA) was carried out with a SETSYS SETARAM analyser.

The temperature was increased from 30 to 800 °C at 5 °C/min in reconstituted (80 % N 2 + 20 % O 2 ) dry air (40 mL/min).

Catalytic test

300 mg of the 100-400 µm fraction was used for all the CO 2 oxidative ethane dehydrogenation tests. Before the reaction, all the catalysts were heated at 300 °C under reactants flow for 2 hours, as the accuracy of flow rates was verified. The catalysts were brought to the reaction temperature under a flow of reactants at a 10 °C/min ramp rate. After the reaction, the reactor cooled down to room temperature under a dry N 2 flow.

The effluents were analyzed online with a gas chromatograph (Interscience Compact GC) equipped with two thermal conductivity detectors (TCD), and a one flame ionization detector (FID). A Molsieve 5A, Rt-QBond, Rtx-1 were used to separate light gases (H 2 , N 2 , CO, CO 2 ), light hydrocarbons (CH 4 , C 2 H 4 , C 2 H 6 ), and aromatics (from benzene till naphthalene), respectively. In all the experiments, N 2 was used as an internal standard to measure conversion and establish mass balance (Table S1). The mass balance in all the reported experiments was at least 99% (Table S1).

The tests were performed with a C 2 H 6 /CO 2 /N 2 = 49.5/36.5/14 (vol.%) mixture in the temperature range 700-800 °C, at atmospheric pressure with a weight hourly space velocity WHSV= 5.9 h -1 (g C2H6 /g catalyst × h).

In order to reveal the catalytic impact of the catalyst on the reaction yield, a blank test without any catalyst was performed under the same operating conditions (same linear velocity, same temperature, same total and partial pressure of the reagents). In the blank test C2 (ethane), N 2 , and CO 2 were co-fed in the same ratio as in the catalytic reaction (C 2 H 6 /CO 2 /N 2 =49.5/36.5/14 (vol.%)). The blank test without a catalyst is referred to in the text as a thermal process ("thermal").

Ethane conversion (C 2 conversion), ethylene selectivity (C 2 = selectivity), and CO selectivity (CO selectivity) were obtained as follows; the sum of the selectivities of the C3-C6 fraction and aromatics was labeled heavies:

A techno-economic analysis was done using an Aspen Plus simulator to compare our results with those typical of ethane steam cracking from the literature. The simulations were for a typical industrial-scale capacity of ethylene production of 1000 kt/year. The physical property databanks and methods as well as ASPEN Plus® models used are summarized in Table S2.

Results and discussion

Characterization of the catalysts

X-ray diffraction (XRD) indicates that on all samples, the embryonic carrier and the supported metal phase are X-ray amorphous (Fig. 1a), and EDX mapping indicates the metal is uniformly distributed on the EZ (Fig. 1b). The 77-K nitrogen sorption-desorption isotherms of the EZ carriers (Fig. 2a) and the corresponding Mo-and MoS 2derivatives (Fig. 2b) show the Type IV isotherms. The mesoporosity is attributed to the intercrystallite spaces between the embryonic units and is observed in all the cases. After impregnation with the metal precursor, micropore volume drops from about 90 cm 3 /g to 70 cm 3 /g for the Al-containing embryonic sample (EZ Al-Si ) and to 45 cm 3 /g for all-silica embryos (EZ Si ). Mesopore volume decrease is observed only to a very small extent after each treatment for the catalysts. The results show that the metal precursors are mainly located in micropores of the supports. Raman spectroscopy was used to identify the nature of the metal phase in the EZsupported catalysts (Fig. 3). The spectrum of EZ Si -5%Mo shows a wide band with two overlapping peaks at 990 cm -1 and 950 cm -1 and two better-resolved bands at 854 cm -1 and 821 cm -1 . These bands can be assigned to polymolybdates (950 cm -1 , 854 cm -1 ) [START_REF] Garcia | Εffect οf the suppοrt οn the activity and mοrphοlοgy οf hydrοdesulfurizatiοn catalysts[END_REF] and ɑ-MoO 3 (990 cm -1 , 821 cm -1 ) [START_REF] Lee | Raman spectroscopic studies of electrochromic a-MoO 3 thin films[END_REF] phases. In contrast, the EZ Al-Si -5%Mo spectrum displays only an intense band at 990 cm -1 with a shoulder at 854 cm -1 . We assume that both phases are also present on the latter catalyst, the polymolybdate phase being predominant.

The band at 480 cm -1 [START_REF] Stojkovic | Investigation of the NaA zeolite crystallization mechanism by i.r. spectroscopy[END_REF] corresponds to Si-O stretching in the SiO 2 tetrahedra and is attributed to 4-membered rings often reported at 472 cm -1 [START_REF] Dutta | Correlation of framework Raman bands of zeolites with structure[END_REF]. Both, EZ Si -IEx and EZ Al-Si -IEx display low-intensity bands around 970 and 1035 cm -1 . The 970 cm -1 band is assigned to Si-O stretching vibrations in terminal Si-OH groups [START_REF] Martinelli | Local anisotropy in single crystals of zeotypes with the MFI framework structure evidenced by polarised Raman spectroscopy[END_REF], while the 1035 cm -1 band is characteristic for the stretching of alkyl chains of the protonated nitrogen atom of the organic template. The conversion of the oxide to the sulfide phase can also be monitored by Raman spectroscopy. Bands at 370 and 403 cm -1 are attributed to 2H-MoS 2 phase [START_REF] Yao | MoS2 Coexisting in 1T and 2H Phases Synthesized by Common Hydrothermal Method for Hydrogen Evolution Reaction[END_REF][START_REF] Attanayake | Effect of Intercalated Metals on the Electrocatalytic Activity of 1T-MoS 2 for the Hydrogen Evolution Reaction[END_REF]. The wideband with a maximum of about 230 cm -1 is characteristic of silica tetrahedra in a bulk SiO 2 phase and remains unchanged after full oxide conversion.

The high-temperature stability of the catalyst carrier is studied by solid-state 27 Al and 29 Si NMR. The nature of the silicon environment in the parent all-silica EZ and Mo loaded after one calcination at 550 °C and subsequent calcination at 550 °C and at 700 °C is presented in Fig. 4a.

All spectra display large peaks centered around -110 ppm, and the CP MAS spectrum exhibited a low resolved peak around -100 ppm. Only the parent contains a relatively large amount of Q 3 species. Small amounts of Q 2 species (-97 ppm) can be observed in EZ Si -IEx and EZ Si -5%Mo-550. All spectral deconvolutions are summarized in Table 1. They indicate that metal impregnation increases the number of Q 4 units along with a full disappearance of the Q 3 and Q 2 's. G) 1600 cm -1 on the sample after the reaction. They are typically attributed to the graphitized carbon (Fig. 5) [START_REF] Kakihana | Raman Spectroscopy as a Characterization Tool for Carbon Materials[END_REF]. The intensity ratio of G (1580 cm -1 ) to D (1350 cm -1 ) band reflects the graphitization degree of catalysts [START_REF] Mo | Effect of carburization protocols on molybdenum carbide synthesis and study on its performance in CO hydrogenation[END_REF].These data suggest that the excess of methane led to the formation of graphitic carbon with the characteristic bands at 1350 and at 1600 cm -1 , which are also typically observed when the Mo 2 C phase is formed [START_REF] Mo | Effect of carburization protocols on molybdenum carbide synthesis and study on its performance in CO hydrogenation[END_REF]. The same covering of Mo-C bands was demonstrated under ethane, while XRD confirmed the formation of the claimed phase [START_REF] Mo | Effect of carburization protocols on molybdenum carbide synthesis and study on its performance in CO hydrogenation[END_REF].

Impact of Mo-precursor:MoO x vs MoS 2

Thermal, i.e., the non-catalytic transformation of the mixture of ethane and CO 2 at 700 °C, results in ethane to ethylene conversion level, which is close to the thermodynamic equilibrium, with the ethylene selectivity above 95%. However, only a very limited CO 2 conversion is observed in this case. In contrast, the presence of the Mo 2 C-supported on embryonic zeolite catalyst, formed from MoS 2 or MoO x -precursors (EZ Si -5%(Mo 2 C) S or EZ Si -5%(Mo 2 C) O ), leads to a significantly higher CO 2 conversion. One can also see that the EZ Si -5%(Mo 2 C) S from MoS 2 -precursor performs CO 2 transformation substantially without decreasing the ethane conversion in comparison with the one in the thermal process (Fig. 6a). Moreover, the selectivity to ethylene is higher (Fig. 6b). In contrast, the MoO x -derived catalyst displays not only lower CO 2 conversion in comparison with the MoS 2 -derived catalyst but also significant activity in ethylene hydrogenation back to ethane and very high methane selectivity (Fig. 6a, Fig. 6b). The results summarized in Table S3 show that only a few by-products are produced in all the cases. The MoS 2 -derived catalyst also demonstrates higher stability than the MoO x -derived one (Fig. S1) and the highest selectivities in all cases. Therefore, the MoS 2 -derived catalyst was selected for further investigation.

Impact of the catalyst carrier

The performances in the ODH-CO 2 reaction of a series of MoS 2 -derived catalysts supported on EZ of different acidity (EZ Si -5%(Mo 2 C) S, EZ Al-Si -5%(Mo 2 C) S, ZSM-5-5%(Mo 2 C) S ) was evaluated. An embryonic zeolite and a conventional ZSM-5 with a crystal size of about 1 µm with a similar Si/Al atomic ratio were used for the study. The Bronsted acidity (BAS) and the Lewis acidity (LAS) of the supports before metal impregnation was assessed by FTIR adsorption of pyridine (Table 2). While the siliceous EZ Si did not display any acidity, the Alcontaining embryonic zeolite (EZ Al-Si ) showed some acid properties (Table 2). As expected, the highest acidity was measured on the zeolite carrier (ZSM-5). The initial CO 2 and C 2 H 6 conversions on the catalysts prepared on acidic support (ZSM-5-5%(Mo 2 C) S and EZ Al-Si -5%(Mo 2 C) S ) were significantly higher than on siliceous one (Fig. S2).

In the case of ZSM-5-5%(Mo 2 C) S , ethane and CO 2 conversions were 37 % and 30 % , respectively. However, the main products were coke and aromatics, leading to rapid deactivation: down to 18% C 2 H 6 and 13 % CO 2 after 140 min-on-stream, respectively. The result showed that the strong acidity of the support, which is typical for the zeolites is not desirable and leads to many side reactions.

The embryonic zeolite with a bulk Si/Al ratio close to the ZSM-5 one showed lower acid site concentration and, as a consequence, slower deactivation in comparison to the one on ZSM-5 support. It should be mentioned that the selectivity patterns on the Al-containing EZ Al-Si -5%(Mo 2 C) S catalyst and on Al-free EZ Si -5%(Mo 2 C) S catalysts looked more similar. One of the potential advantages of the Al-containing embryonic zeolite is the higher activity in CO 2 hydrogenation. However, higher coke content, in addition to higher methane selectivity, makes the acidity a disadvantage in this reaction (Fig. S3).

Performance of Al-free EZ

Si -5%(Mo 2 C) S in ODH-CO 2 of ethane

The selected purely silicious EZ Si material showed to be advantageous in the dispersion and stabilization of Mo 2 C while preventing deactivation due to coke formation (Fig. 7a, Fig. S3).

Indeed, MoS 2 -precursor supported on EZ extends the operating temperature range of Mo 2 C catalysts compared to previously published results [START_REF] Bugrova | Oxidative dehydrogenation of ethane with CO 2 over CrO x catalysts supported on Al 2 O 3 , ZrO 2 , CeO 2 and Ce x Zr 1-x O 2[END_REF][START_REF] Theofanidis | CO 2 -oxidative ethane dehydrogenation over highly efficient carbon-resistant Fe-catalysts[END_REF][START_REF] Solymosi | The oxidative dehydrogenation of ethane with CO 2 over Mo 2 C/SiO 2 catalyst[END_REF], in addition to suppressing the unwanted side reactions of methanation and dry-reforming. Remarkably, the selectivity to ethylene is even higher at the same process temperature than in the thermal process (Fig. 7b). The difference in ethane's catalytic and thermal conversion is due to some ethylene hydrogenation. Although the activity in ethylene hydrogenation is significantly lower on the catalyst than the one in CO 2 conversion to CO, it is worth comparing the selectivity at different temperatures at the same level of conversion.

The selected catalyst (EZ Si -5%(Mo 2 C) S ) was tested at different temperatures, and its performance was compared with the thermal (without catalyst) transformation of ethane in the presence of CO 2 (Fig. 8). Due to some partial hydrogenation, the apparent ethane conversion level at the same temperature was slightly lower than in the thermal process (Fig. S4). In order to compare the results, the data were linearly extrapolated to the same level of conversion. Three conversion levels of ethane were selected for the comparison: low conversion (15%), mid conversion (40%), and high conversion (66%). In contrast to literature data on Mo 2 C type of materials [START_REF] Solymosi | The oxidative dehydrogenation of ethane with CO 2 over Mo 2 C/SiO 2 catalyst[END_REF], one can see that at about 50% of CO 2 conversion to CO (reverse water-gas-shift reaction), the methanation and the dry reforming remained insignificant in the studied system (Fig. 8). In all cases, the ethylene selectivities in -CO 2 of ethane were found very close between catalytic and non-catalytic processes. The impact on ethylene selectivity did not exceed 1-3 % for all the conversion levels of ethane, which could be considered insignificant. One can see that the presence of catalyst allows achieving a 50% conversion level of CO 2 without decreasing the selectivity to ethylene in comparison with the thermal process at the conversion of ethane up to 66% (Fig. 8). All the results were obtained in the temperature range 700-800 °C what is the industrially relevant temperature range for the ethane cracking. A detailed comparison with the ethane cracking of the obtained catalyst performance is given below. ethane at the same conversion level of ethane (coke-free basis, C 2 H 6 /CO 2 =1/0.74, WHSV = 5.9 h -1 ).

Techno-economical analysis of the ODH-CO 2 process based on an EZ

Si -5%(Mo 2 C) S catalyst
As mentioned in the introduction section, the emerging technological solutions aim to produce ethylene from ethane with lower CO 2 emissions in comparison with an ethane cracker.

The endothermic cracking of ethane (its C-C and C-H bonds require dissociation energies of 345 and 413 kJ/mol, respectively) to ethylene yield is limited. It requires high temperature and leads to the formation of side products such as aromatics and light gaseous. The catalytic conversion of ethane to ethylene in the presence of an oxidant is a potential solution to the problems outlined above. The ODH-E technology recently developed by Linde/Clariant reports CO 2 emissions reductions in the 23-32% range compared to conventional steam cracking [START_REF]Oxidative Dehydrogenation of Ethane Technology[END_REF].

The EcoCatalytic's CL-ODH technology also claims a 64% substantial reduction in CO 2 emissions compared to a state-of-the-art ethane steam cracker of the same capacity [START_REF] Neal | Intensified Ethylene Production via Chemical Looping through an Exergetically Efficient Redox Scheme[END_REF]. In the case of the state-of-the-art ethane steam cracker, hydrogen is the by-product. Such hydrogen could alternatively be used as a fuel for heat generation, leading to overall CO 2 emission reduction.

In such a broader context, oxidative dehydrogenation of ethane in the presence of CO 2 (ODH-CO 2 ) is a promising economical and environmental alternative. This process directly utilizes hydrogen to transform CO 2 into CO, which could further be used as the precursor for producing valuable chemicals, mitigating CO2 emissions while simultaneously achieving ethane dehydrogenation. This process can not only reduce CO 2 emission but also simplifies production slate processing by reducing CH 4 and heavies production.

Steam cracking proceeds through free-radical mechanisms [START_REF] Froment | Thermal Cracking of Ethane and Ethane-Propane Mixtures[END_REF], which are inherently characterized by a vast number of species and reactions leading to wide product distribution.

Moreover, in steam crackers, acetylene is a by-product. Typically, the acceptable acetylene contaminant level in polymer grade ethylene products is 1 ppmv or lower. Since acetylene will fractionate with ethylene, it needs to be removed before the C2 fractionation step. An acetylene hydrogenation unit is often required to convert it to ethylene. Consequently, the downstream processing of a steam cracker requires multiple separation units, as shown in Fig. S5a.

In a state-of-the-art ethane steam cracker, the traces of CO are hydrogenated to CH 4, and the unconverted H 2 is of low grade with a significant amount of methane. The ODH-CO 2 process results in a significant amount of CO, significantly lower methane production, and some remaining hydrogen. So, the light end from an ODH-CO 2 could be considered as syngas, which after an adjustment of hydrogen content, could be directly suitable for MeOH synthesis.

ODH-CO 2 results in a lower amount of heavies, with almost no C 3 fraction and no acetylene, detected in the effluent of the reactor, as seen in Table 3. That is very interesting for separation and, eventually, for the energy balance. On the other hand, the C 2 /CO 2 fraction should not be necessarily separated and could be recycled back to the reaction session as a feedstock (Fig S5b).

Conclusions

Embryonic zeolites show potential as supports for Mo 2 C active species in ODH-CO 2 .

Supported metal sulfides proved to be efficient precursors of the Mo 2 C active phase, which allows the designing of novel thermally stable and highly active catalysts for CO 2 hydrogenation in co-processing with alkenes. This "sulfide" route shows a clear advantage over the "oxide" one and requires only about 5 wt.% of Mo.

Although a detailed formation mechanism from MoS 2 to Mo 2 C is still under investigation, it seems that MoS 2 supported on embryonic zeolites prevents sintering of the Mo-species during the carbide formation and results in a more active and stable catalyst for ODH-CO 2 of ethane.

The latter allows operating at higher temperatures, avoiding dry reforming and methanation side reactions while producing more selectively ethylene.

The supported Mo 2 C displays a good activity in CO 2 hydrogenation without negatively impacting the thermal C 2 conversion to ethylene at a conversion up to 66%. The resulting ODH-CO 2 process could lead to a potentially promising CO 2 -negative technology with a muchreduced separation section due to a lower amount of heavies, no acetylene production, and better carbon efficiency. ODH-CO 2 thus brings significant advantages in terms of investment (CAPEX) and operating (OPEX) costs of a greener process.

Fig. 1 .

 1 Fig. 1. a) XRD patterns of EZ Si -IEx, EZ Si -5%Mo, EZ Si -5%MoS 2 ; b) EDX elemental mapping of Mo in EZ Si -5%Mo.

Fig. 2 .

 2 Fig. 2. N 2 adsorption-desorption isotherms: a)EZ Al-Si -based; b)EZ Si -based.

Fig. 3 .

 3 Fig. 3. Raman spectrum of EZ decorated with molybdenum oxide and sulfide phases.

Fig. 4 .

 4 Fig. 4. Solid-state NMR analysis of the parent materials after ion exchange and Mo-loading: a) 29 Si NMR (1pulse) and CP MAS spectra; b) 27 Al NMR spectra.

hypothesis on Mo 2 C

 2 formation is based on the typical profile of the TG curve[START_REF] Choi | Structural Evolution of Molybdenum Carbides in Hot Aqueous Environments and Impact on Low-Temperature Hydroprocessing of Acetic Acid[END_REF] and on the observation of the characteristic bands for the graphitized carbon in Raman spectra synthesized with non-methane carburization agent[START_REF] Xiao | Preparation of Molybdenum Carbides Using Butane and Their Catalytic Performance[END_REF][START_REF] Mo | Effect of carburization protocols on molybdenum carbide synthesis and study on its performance in CO hydrogenation[END_REF]. The TGA curves of the spent Mo 2 C-containing samples demonstrated a slight weight increase up to 470 °C due to the oxidation reaction of Mo 2 C to MoO 3 (Fig.S3). However, the analysis of Mo 2 C on the spent catalysts by Raman spectroscopy is a difficult task. The sulfidic form on the precursor is confirmed by the two bands (a) (b) around 370 and 403 cm -1 (Fig.5), both bands are disapered on the spent Mo 2 C-containing samples. That shows the transformation of the MoS 2 -species to a different species. In addition, two new intense bands are detected in the region of C-C vibrations -(band D) at 1350 and (band

Fig. 5 .

 5 Fig. 5. Raman spectrum of embryonic zeolite decorated with sulfide phase before and after the catalytic test at 700 °C over EZ Si -5%(Mo 2 C) S .

Fig. 6 .

 6 Fig. 6. ODH-CO 2 of ethane on EZ Si -5%(Mo 2 C) O , EZ Si -5%(Mo 2 C) S , and without catalyst: a) conversion of CO 2 and ethane; b) selectivities to the products (coke-free basis, 700 °C, C 2 H 6 /CO 2 =1/0.74, WHSV = 5.9 h -1 , average level for 2-5 hours-on-stream).

Fig. 7 .

 7 Fig. 7. ODH-CO 2 of ethane (C2) on EZ Si -5%(Mo 2 C) S and without catalyst: a) conversion of CO 2and ethane as a function on time-on-stream; b) selectivities to the products (coke-free basis, T =800 °C, P = 1 atm, C 2 H 6 /CO 2 = 1/0.74, WHSV =5.9 h -1 ).

Fig. 8 .

 8 Fig.8. Comparison of catalytic (EZ Si -5%(Mo 2 C) S ) and thermal performances in ODH-CO 2 of

Figure 9

 9 Figure9highlights the unit operations of a conventional SC plant and an ODH-CO 2 plant. The ODH-CO 2 requires 50% less unit operations compared to an ethane cracker. One can see that ODH-CO 2 may bring significant advantages in terms of investment costs and separation duties and therefore reduce CAPEX and OPEX.Regarding energy consumption, ethylene production via steam cracking of ethane is very energy consuming: producing 1 ton of ethylene typically requires 21 GJ of thermal energy (process energy consumed in pyrolysis and separation) and results in approximately 1.2 tons of CO 2 emissions[START_REF] Ren | Olefins from conventional and heavy feedstocks: Energy use in steam cracking and alternative processes[END_REF]. In our case, the simulation of the ODH-CO 2 process indicates an energy consumption of 17.4 GJ of thermal energy to produce 1 ton of ethylene and consumes 1,09 tons of CO 2 . Therefore, if ODH-CO 2 is implemented based on the current catalytic results, the reduction of CO 2 would be up 109% in comparison with the state-of-the-art ethane cracker making the technology CO 2 negative.

Fig 9 .

 9 Fig 9. Comparison in term of unit operations between SC and OHD-CO 2 ([55])

Table 1 .

 1 Relative percentage of Q 2 , Q 3 , and Q 4 silicon environments in EZ Si -parents and calcined derivatives.It is known that above 650 °C, MoO x and MoS 2 species react with hydrocarbons to produce Mo 2 C[START_REF] Li | In Situ Synthesis of Carbon Nanotube Hybrids with Alternate MoC and MoS 2 to Enhance the Electrochemical Activities of MoS 2[END_REF][START_REF] Zhao | Vertically Aligned MoS2/Mo2C hybrid Nanosheets Grown on Carbon Paper for Efficient Electrocatalytic Hydrogen Evolution[END_REF]. The mechanism of Mo 2 C phase formation from MoS 2 in the presence of CH 4 and H 2 is described[START_REF] Jeon | Epitaxial Synthesis of Molybdenum Carbide and Formation of a Mo 2 C/MoS 2 Hybrid Structure via Chemical Conversion of Molybdenum Disulfide[END_REF]. The carbide formation from MoO 3 under C 2 H 6 and C 4 H 10 is also reported[START_REF] Xiao | Effect of carburising agent on the structure of molybdenum carbides[END_REF]. In this contribution, the active Mo 2 C phase is obtained in situ by subjecting the Mo-containing precursor in a contact with the mixture of C 2 H 6 /CO 2 =1/0.74 at 700 °C. The

	Sample	Q 2	Q 3	Q 4
	EZ Si	-IEx	7	48	45
	EZ Si	-5%Mo-550	2	22	76
	EZ Si	-5%Mo-550-700	-	27	73
		During the high-temperature treatment of silica-alumina embryos (550 °C or consecutive
	calcination at 550 and 700 °C) after Mo loading, Al is extracted from tetrahedral positions,
	Fig.4b while the parent, ion-exchanged with NH 4 NO 3 , does not contain any octahedral Al. The
	calcined samples loaded with 5 wt.% Mo, three types of aluminum coexist (Al IV , Al VI , and
	Al 2 (MoO 4 ) 3 ), witnessing the catalysts alteration at high temperatures.
	3.2. Catalytic performances

Table 2 .

 2 Acid site concentration on ZSM-5 and Al-containing embryonic zeolite measured by FTIR with pyridine probe molecule.

	Sample	LAS, µmol/g	BAS, µmol/g
	ZSM-5 (Si/Al=47)	235	1094
	EZ Al-Si (Si/Al=43)	57	185

Table 3 .

 3 Comparison of steam cracking (SC) of ethane and ODH-CO 2 .

		SC of ethane [66],	ODH-CO 2
		845 °C-outlet	[this work]
	Ethane Conversion (%)	65.0	66.1
	Carbon Yield (dry mol%)
	CO	0.00	27.34
	CO2	0.00	20.71
	CH 4	3.40	4.81
	C 2 H 2	0.50	0.00
	C 2 H 4	56.10	27.29
	C 2 H 6	35.00	19.46
	C 3 H 6	1.20	0.00
	C 3 H 8	0.10	0.00
	C 4 H 6	2.10	0.00
	C 4 H 8	0.20	0.00
	C 4 H 10	0.20	0.30
	C 5 's	0.60	0.10
	Aromatics (benzene, toluene)	0.60	0.08

Acknowledgments

This research was supported by TotalEnergies, the Industrial Chair ANR-TOTAL "NanoClean Energy" (ANR-17-CHIN-0005-01), and project NanoCleanEnergy + (Region Normandy).

CRediT authorship contribution statement

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.