Manon Doré

Yassine Ariba

Germain Garcia

Observer-based Switched Control of the Three Level Neutral Point Clamped Rectifier

In this paper, an observer-based switched control law is proposed for the three level neutral point clamped (NPC) converter operating as a rectifier. Modeling the converter as a switched affine system, the proposed control is based on the well known argmin control law to track a varying state reference trajectory. A full-order observer is introduced to compute the control law with only the measure of the input and the output voltages. The control aims at tracking a state reference defined from a power analysis and three objectives are addressed: to stabilize the output at a given DC voltage, to ensure a unit power factor by having the input current and voltage on phase and to have balanced capacitor voltages on the output. Based on a unified modeling methodology, the control and the observer are easily derived from LMI conditions. An outer loop is added to regulate the output when constant perturbations are considered. The results are illustrated by simulations on MATLAB/Simulink.

I. INTRODUCTION

Neutral Point Clamped (NPC) inverters or rectifiers is one of the most used topology of multilevel converter in high power electrical applications. Indeed, this converter has several advantages like a low total harmonic distortion (THD) of the output and a high efficiency [START_REF] Nabae | A New Neutral-Point-Clamped PWM Inverter[END_REF], which make it popular for AC/DC applications [START_REF] Carrasco | Power-electronic systems for the grid integration of renewable energy sources: A survey[END_REF], [START_REF] Franquelo | The age of multilevel converters arrives[END_REF]. The main objectives of the control are to achieve a unit power factor on the AC side while maintaining the DC side at a constant given voltage. Another objective is to ensure that the capacitor voltages are balanced to avoid the switches to undergo excessive voltage stress and preserve the quality of the output voltage [START_REF] Umbría | Voltages balance control in three phase three-level NPC rectifiers[END_REF].

From a control point of view, converter models belong to the class of switched affine systems. This nature comes from the fact that the dynamic of a converter depends on the state of electrical switches in the circuit. In the literature, most widely studied control solutions for power converters are based on averaged linearized models. More recently, an effort has been done to propose control methods that take into account the discontinuous behavior of such systems [START_REF] Deaecto | Switched affine systems control design with appliction to DC-DC converters[END_REF], [START_REF] Rosa | Practical considerations of nonlinear control techniques applied to static power converters: A survey and comparative study[END_REF]. However, control oriented papers on converters usually consider simple examples such as the Boost converter.

In the literature, some classical approaches can be found to control the NPC rectifier, like the direct power control methods based on an averaged model and the use of Pulse-Width Modulation (PWM) signal. In this case, linear or nonlinear control techniques are applied, involving several control loops to handle every control objectives [START_REF] Umbría | Voltages balance control in three phase three-level NPC rectifiers[END_REF], [START_REF] Yacoubi | Linear and nonlinear control techniques for a three-phase three-level npc boost rectifier[END_REF]. Advanced control strategies have also been considered, like M. Doré, Y. Ariba and G. Garcia are with LAAS-CNRS, Université de Toulouse, CNRS, INSA, Toulouse, France. Email: {mdore,yariba,garcia}@laas.fr space vector modulation [START_REF] Zhou | Vector shifted model predictive power control of three-level neutral-point-clamped rectifiers[END_REF] or predictive control methods [START_REF] Zhou | Vector shifted model predictive power control of three-level neutral-point-clamped rectifiers[END_REF], [START_REF] Barros | Optimal Predictive Control of Three-Phase NPC Multilevel Converter for Power Quality Applications[END_REF]. Regarding the switching nature of the converter, the number of papers dealing with it in the hybrid control framework is reduced [START_REF] Montero-Robina | Hybrid modeling and control of three-level npc rectifiers[END_REF].

In the same vein, the observation problem has been even less considered for this specific power converter, except in adaptive control schemes [START_REF] Umbría | Observerbased Direct Power Control for three-level NPC rectifiers[END_REF]- [START_REF] Yin | Observer-based adaptive sliding mode control of npc converters: An rbf neural network approach[END_REF], or based on approximated/averaged model [START_REF] Umbría | Voltage balancing in three-level neutral-point-clamped converters via Luenberger observer[END_REF]. Indeed, the three level NPC converter have a large number of switches and a dynamic reference which are the sources of difficulties when compared to numerous examples of converters usually considered [START_REF] Gonzalez-Fonseca | Observer-Based Controller for Switch-Mode DC-DC Converters[END_REF]- [START_REF] Beneux | Adaptive stabilization of switched affine systems with unknown equilibrium points: Application to power converters[END_REF]. Also, the use of an observer is relevant to reduce the number of sensors and maintenance costs.

This paper addresses the observer-based control problem of the Three Level Neutral Point Clamped Rectifier. Using a unified modeling methodology [START_REF] Garcia | A Unified Approach for the Control of Power Electronics Converters[END_REF] in Section II, a state space representation of the converter is establish from its electrical equations while considering its switching nature. In Section III-A, a full-order observer is designed through a set of Linear Matrix Inequalities (LMI) deduced from a Lyapunov analysis. Power analysis allows the derivation of a suitable state reference trajectory according to control objectives. The reconstructed state is then used in Section III-B to build an observer based control law. The proposed control is of an argmin state feedback control law type [START_REF] Deaecto | Switched affine systems control design with appliction to DC-DC converters[END_REF], [START_REF] Beneux | Adaptive stabilization of switched affine systems with unknown equilibrium points: Application to power converters[END_REF], [START_REF] Garcia | A Unified Approach for the Control of Power Electronics Converters[END_REF], and ensures the global asymptotic stability of the origin of the tracking error. To regulate the output in the presence of constant perturbations, Section III-C proposes to complete the control scheme adding an outer loop that adapts appropriately the state reference. Section IV presents simulation results carried out on MATLAB/Simulink to illustrate the effectiveness of the proposed approach.

II. MODELING

Fig. 1: Three Level Neutral Point Clamped Rectifier Circuit

The considered converter is a three level neutral clamped rectifier presented in Figure 1. The input supplies are three phase grid voltages e a , e b and e c , each connected to an inductor L including a parasitic resistance r L . On the DC side, the load is a resistor R, connected in parallel with two capacitors C with a parasitic resistance r C . For the AC side, the following Assumption 1 is taken into account.

Assumption 1: The circuit operates in balanced mode. Therefore • a +• b +• c = 0 with • = e, i, and e a = E cos(ωt) , e b = E cos(ωt -2π

3 ), and e c = E cos(ωt -4π 3 ). The circuit contains 12 switches and 6 diodes that aim at connecting each phase a,b and c to the point P , O or N . Therefore, we can model the switches with 9 boolean control variables u ij , where i = a, b, c and j = p, o, n. A u ij will be equal to one if the phase i is connected to the point j. Since a phase can be connected to only one point at a time, the following constraints exist:

u •p + u •o + u •n = 1 with • = a, b, c (1)
With this modeling, a phase of the circuit can be represented as in Figure 2. Applying Kirchhoff's laws:

e • = r L i • + L i• + v •o + v ok with • = a, b, c (2)
Using Assumption 1 and (2), the sum of the three input voltages is equal to 3v ok + v ao + v bo + v co . Therefore, we can deduce that

  v ak v bk v ck   = 1 3   2 -1 -1 -1 2 -1 -1 -1 2     v ao v bo v co   := T   v ao v bo v co   (3)
Since

v •o =    0 if u •o = 1 v 1 if u •p = 1 -v 2 if u •n = 1 with • = a, b, c Then   v ao v bo v co   =   u ap u bp u cp   v 1 -   u an u bn u cn   v 2 (4)
Substituting (3) and ( 4) in (2), we have the following model for the AC part:

L   ia ib ic   =   e a e b e c   -r L   i a i b i c   -T     u ap u bp u cp   v 1 -   u an u bn u cn   v 2  
Regarding the DC side, according to Figure 1, i 1 and i 2 are currents flowing through capacitors. Therefore, Kirchhoff's laws give:

i 1 = u ap i a + u bp i b + u cp i c -v1+v2 R -v1 r C i 2 = -(u an i a + u bn i b + u cn i c ) -v1+v2 R -v2 r C

Fig. 2: Schematic of phase a with control variables

Considering the objectives of stabilizing the output and balancing the capacitor voltages, it is interesting to express the model through the variables v

+ = v 1 + v 2 , v -= v 1 -v 2
and i a , i b and i c . Therefore, the model becomes:

           L   ia ib ic   =   e a e b e c   -r L   i a i b i c   - 1 2 T     u a- u b- u c-   v + +   u a+ u b+ u c+   v -   C v+ = i 1 + i 2 = u a-i a + u b-i b + u c-i c -v+ Re C v-= i 1 -i 2 = u a+ i a + u b+ i b + u c+ i c -v- r C (5)
where u i± = u ip ± u in for i = a, b, c and R e = Rr C R+2r C . In the context of balanced mode operation (see Assumption 1), it is possible to simplify the model using the Clarke-Concordia's transformation, which transforms an abc vector g a g b g c T into an αβγ vector g α g β g γ T according to:

  g α g β g γ   = 2 3    1 -1 2 -1 2 0 √ 3 2 - √ 3 2 1 √ 2 1 √ 2 1 √ 2      g a g b g c   := C   g a g b g c  
Multiplying (5) on the left by C and after some tedious calculations, the model can be written as:

             L iα iβ = e α e β -r L i α i β - √ 6 6 - √ 6 12 - √ 6 12 0 √ 2 4 - √ 2 4     u a- u b- u c-   v + +   u a+ u b+ u c+   v -   C v+ = √ 6 2ua--u b--uc- 6 i α + √ 2 u b--uc- 2 i β -v+ Re C v-= √ 6 2ua+-u b+ -uc+ 6 i α + √ 2 u b+ -uc+ 2 i β -v- r C
where e α = V αβ cos(ωt), e β = V αβ sin(ωt) and

V αβ = 3 2 E. If we choose x = i α i β v + v - T as state vector
and v + as the output, we have the following state space model:

     ẋ = A 0 + u ap A uap + u an A uan + u bp A u bp +u bn A u bn + u cp A ucp + u cn A ucn x + B 0 v in y = 0 0 1 0 x := C 0 x (6)
where :

A 0 =     -r L L 0 0 0 0 -r L L 0 0 0 0 -1 ReC 0 0 0 0 -1 Cr C     B 0 =     1 L 0 0 1 L 0 0 0 0     v in = e α e β A uap =      0 0 - √ 6 6L - √ 6 6L 0 0 0 0 √ 6 3C 0 0 0 √ 6 3C 0 0 0      A uan =      0 0 √ 6 6L - √ 6 6L 0 0 0 0 - √ 6 3C 0 0 0 √ 6 3C 0 0 0      A u bp =      0 0 √ 6 12L √ 6 12L 0 0 - √ 2 4L - √ 2 4L - √ 6 6C √ 2 2C 0 0 - √ 6 6C √ 2 2C 0 0      A u bn =      0 0 √ 6 12L - √ 6 12L 0 0 - √ 2 4L √ 2 4L √ 6 6C - √ 2 2C 0 0 - √ 6 6C √ 2 2C 0 0      A ucp =      0 0 √ 6 12L √ 6 12L 0 0 √ 2 4L √ 2 4L - √ 6 6C - √ 2 2C 0 0 - √ 6 6C - √ 2 2C 0 0      A ucn =      0 0 - √ 6 12L √ 6 12L 0 0 - √ 2 4L √ 2 4L √ 6 6C √ 2 2C 0 0 - √ 6 6C - √ 2 2C 0 0     
To apply the methodology proposed in [START_REF] Garcia | A Unified Approach for the Control of Power Electronics Converters[END_REF], modes should be ordered and a change of variable is required to obtain a polytopic model. A mode corresponds to a configuration of the circuit, thus constraints (1) must be considered in their definitions. Modes and new A λi matrices are summed up in Table I. After the change of variables, the model can be expressed as

ẋ = A 0 + 27 i=1 λ i A λi x + B 0 v in y = C 0 x (7)
This is a polytopic model where the λ i 's are the new control variables. Since only one mode can be active at a time, only one λ i can be equal to one at a time. Thus, λ i variables select a circuit configuration, and we have a switched affine system. Let us now gather the λ i variables into a vector λ ∈ Λ S where Λ S = λ ∈ {0, 1} 27 : 27 i=1 λ i = 1 . Finally, the model can be expressed in a compact form expressed as:

ẋ = A 0 x + B 0 (x)λ y = C 0 x (8)
where

B 0 (x) = A λ1 x + B 0 v in . . . A λ27 x + B 0 v in .

III. OBSERVATION AND CONTROL

A. Observer design

The state of a system is not always available to compute a control law. Moreover, the introduction of an observer can be useful to reduce the number of sensors. In this context, we want to design an observer to asymptotically reconstruct the state of system [START_REF] Zhou | Vector shifted model predictive power control of three-level neutral-point-clamped rectifiers[END_REF]. We assume that in addition of v in , v C1 and v C2 are measured. Thus, let us consider the measured output y m defined as:

y m = v C1 v C2 = 1 2 0 0 1 1 0 0 1 -1 x := C m x (9)
Theorem 1: For a given symmetric positive definite matrix

Q O ∈ R 4×4 , if it exists a symmetric positive definite matrix S ∈ R 4×4 and matrices W i ∈ R 4×2 such that the LMI A 0 +A λi T S +S A 0 +A λi -C T m W T i -W i C m +2Q O < 0 (10
) holds ∀i = 1, . . . , 27, then the observer

ẋ = A 0 x + B 0 (x)λ + 27 i=1 λ i L λi (y m -ŷm ) ŷm = C m x (11)
where L λi = S -1 W i , ensures the convergence of x to x. Proof: Let ε = x -x be the reconstruction error signal. Its dynamic is:

ε = A 0 + 27 i=1 λ i A λi -L λi C m ε
Consider the Lyapunov function V (ε) = ε T Sε where S = S T > 0. V (0) = 0 and V (ε) > 0 , ∀ε ̸ = 0. Regarding its time derivative, we have

V (ε) = ε T A 0 + 27 i=1 λ i A λi T S + S A 0 + 27 i=1 λ i A λi - 27 i=1 λ i SL λi C m -C T m L T λi S ε <-2ε T Q O ε < 0
The last inequality is obtained from LMI condition (10) by defining W i = SL λi . Since V (ε) < 0 , ∀ε ̸ = 0 and the dynamic of ε doesn't jump when the control change, we can conclude that the origin of ε is globally asymptotically stable (GAS).

B. Observer based control

The control aims at tracking a state reference computed according to three objectives: to stabilize the output y at a given DC voltage setpoint y e , to ensure a unit power factor by having the input currents and input voltages on phase, and to have balanced capacitor voltages on the output.

The first and third objectives directly set a part of the state reference since y = v + and the difference between the capacitor voltages is v -. The second objective suggests that the currents references should be of the form I αβ cos(ωt) for i α and I αβ sin(ωt) for i β . Therefore, the state reference can be expressed as:

x e (t) =     i αe (t) i βe (t) v +e v -e     =     I αβ cos(ωt) I αβ sin(ωt) y e 0     (12)
where I αβ has to be determined. To achieve this, the power balance of the system can be studied. Using [START_REF] Rosa | Practical considerations of nonlinear control techniques applied to static power converters: A survey and comparative study[END_REF], we can calculate instantaneous powers in capacitors and inductors, and recover the rest of the power balance expression:

Li α iα + Li β iβ + C 2 v + v+ + C 2 v -v- = -r L (i 2 α + i 2 β ) + e α i α + e β i β - v 2 + 2R e - v 2 - 2r C (13)
For x = x e given by ( 12), the power balance expression reduces to:

0 = -r l I 2 αβ + V αβ I αβ - y 2 e 2R e (14)
To simplify the problem, the objectives can be expressed in terms of instantaneous active and reactive power p and q instead of currents and voltages. They are expressed as [START_REF] Akagi | Instantaneous Reactive Power Compensators Comprising Switching Devices without Energy Storage Components[END_REF]:

p q = e α e β -e β e α i α i β (15)
With those definitions, the control objectives can now be summarized as follows: p = p * , q = 0, v + = y e and v -= 0, where p * has to be determined. The power balance is then expressed as follows:

0 = -r L p * 2 V 2 αβ + p * - y 2 e 2R e
Hence, for a fixed y e , p * is expressed as:

p * = V 2 αβ 2r L 1 ± 1 - 2r L V 2 αβ R e y 2 e with y e < V αβ R e 2r L (16)
To reduce the power in the system, the smallest solution should be selected. Finally, by definition [START_REF] Gonzalez-Fonseca | Observer-Based Controller for Switch-Mode DC-DC Converters[END_REF], I αβ = p * V αβ . Remark 1: Since the system is a switched system, the state reference is considered in the sense of Filipov [START_REF] Ingalls | An Infinite-Time Relaxation Theorem for Differential Inclusions[END_REF]. TABLE I: Modes and change of variables from the physical model ( 6) to the polytopic model ( 7)

Mode(i) uap uan uao u bp u bn u bo ucp ucn uco A λ i 1 0 0 1 0 0 1 0 0 1 0 2 0 0 1 0 0 1 0 1 0 Au cn 3 0 0 1 0 0 1 1 0 0 Au cp 4 0 0 1 0 1 0 0 0 1 Au bn 5 0 0 1 0 1 0 0 1 0 Au bn + Au cn 6 0 0 1 0 1 0 1 0 0 Au bn + Au cp 7 0 0 1 1 0 0 0 0 1 Au bp 8 0 0 1 1 0 0 0 1 0 Au bp + Au cn 9 0 0 1 1 0 0 1 0 0 Au bp + Au cp 10 0 1 0 0 0 1 0 0 1 Au an 11 0 1 0 0 0 1 0 1 0 Au an + Au cn 12 0 1 0 0 0 1 1 0 0 Au an + Au cp 13 0 1 0 0 1 0 0 0 1 Au an + Au bn 14 0 1 0 0 1 0 0 1 0 Au an + Au bn + Au cn 15 0 1 0 0 1 0 1 0 0 Au an + Au bn + Au cp 16 0 1 0 1 0 0 0 0 1 Au an + Au bp 17 0 1 0 1 0 0 0 1 0 Au an + Au bp + Au cn 18 0 1 0 1 0 0 1 0 0 Au an + Au bp + Au cp 19 1 0 0 0 0 1 0 0 1 Au ap 20 1 0 0 0 0 1 0 1 0 Au ap + Au cn 21 1 0 0 0 0 1 1 0 0 Au ap + Au cp 22 1 0 0 0 1 0 0 0 1 Au ap + Au bn 23 1 0 0 0 1 0 0 1 0 Au ap + Au bn + Au cn 24 1 0 0 0 1 0 1 0 0 Au ap + Au bn + Au cp 25 1 0 0 1 0 0 0 0 1 Au ap + Au bp 26 1 0 0 1 0 0 0 1 0 Au ap + Au bp + Au cn 27 1 0 0 1 0 0 1 0 0 Au ap + Au bp + Au cp
The addressed problem is to design a control law λ : R → Λ S which asymptotically stabilizes the state of the system (7) at the state reference x e defined by [START_REF] Portillo | Model based adaptive direct power control for three-level npc converters[END_REF], only measuring the output y m . The control is expressed based on the reconstructed state provided by the observer [START_REF] Umbría | Observerbased Direct Power Control for three-level NPC rectifiers[END_REF]. The observer and the control law can be designed separately, leading to a separation principle.

Theorem 2: Let ẽ := x-x e be the estimated tracking error signal, where x is the state estimated by the observer [START_REF] Umbría | Observerbased Direct Power Control for three-level NPC rectifiers[END_REF] and x e is defined by [START_REF] Portillo | Model based adaptive direct power control for three-level npc converters[END_REF]. For a given symmetric positive definite matrix Q C ∈ R 4×4 , if it exists a symmetric positive definite matrix P ∈ R 4×4 solution of

A T λi P + P A λi + 2Q C < 0 (17)
∀i = 1 . . . 27, then the control law

λ * = arg min d∈Λ S ẽT P A 0 x + B 0 (x)d (18)
= arg min i=1...27 ẽT P A λi x makes the origin of the tracking error e := x -x e globally asymptotically stable (GAS).

Proof: Consider the Lyapunov function V (ẽ, ε) = ẽT P ẽ + αε T Sε where P = P T > 0 is solution of ( 17), S = S T > 0 is solution of (10) and α > 0 is a scalar. V (0, 0) = 0 and V (ẽ, ε) > 0 , ∀ẽ, ε ̸ = 0. The time derivative is expressed as

V (ẽ, ε) = ẽT P A 0 x + B 0 (x)λ + 27 i=1 λ i L λi C m ε -ẋe + αε T S 27 i=1 λ i A 0 + A λi -L λi C m ε
For the proposed switching control λ * (18), by construction and convexity 17) and [START_REF] Montero-Robina | Hybrid modeling and control of three-level npc rectifiers[END_REF] imply

ẽT P A 0 x + B 0 (x)λ * ≤ ẽT P A 0 x + B 0 (x)λ e Therefore V (ẽ, ε)≤ ẽT P A 0 ẽ + B 0 (x)-B 0 (x e ) λ e + 27 i=1 λ i L λi C m ε + αε T S 27 i=1 λ i A 0 + A λi -L λi C m ε = ẽT P A 0 + 27 i=1 λ ei A λi ẽ + 27 i=1 λ i L λi C m ε + αε T S 27 i=1 λ i A 0 + A λi -L λi C m ε Inequalities (
V (ẽ, ε) ≤ -ẽ T Q C ẽ + ẽT P 27 i=1 λ i L λi C m ε -αε T Q O ε = - ẽ ε T Q C γ γ T αQ O ẽ ε := - ẽ ε T Q ẽ ε where γ = -1 2 P 27 i=1 λ i L λi C m . Consequently, V (ẽ, ε) < 0 if Q > 0. By Schur complement, this condition holds for αQ O -1 4 27 i=1 λ i C T m L T λi P Q -1 C P L λi C m > 0 Hence, a sufficient condition is α > max i=1...27 λ max 1 4 Q -1 2 O C T m L T λi P Q -1 C P L λi C m Q -1 2 O
Finally, for a sufficiently large α and since the state doesn't jump when the control change, we can conclude that the origins of ε and ẽ are GAS. Noticing that e = ẽ + ε, it proves that the origin of e is GAS too. In this section, the objective is to regulate the output v + when constant perturbations are considered. Such perturbations may stem from, for example, some variations of the resistive load R or some variations of the amplitude of the input source voltages V αβ . The idea consists in modifying the current reference amplitude I αβ adding an extra term ∆I αβ . This extra term is obtained from an outer loop on v + with an integral action whose output ∆I αβ is expressed as

∆I αβ (t) = K I t 0 (y e -v + (u))du
where K I is a constant gain. The control scheme is depicted in Figure 3 where the inner loop corresponds to the closed-loop system designed in the previous paragraph: the converter controlled by the observer based control [START_REF] Garcia | A Unified Approach for the Control of Power Electronics Converters[END_REF]. To design gain K I , it is assumed that the dynamic of the inner loop is much faster than the one of the introduced outer loop. In that context, the transfer function between ∆I αβ and ∆v + = y e -v + can be approximated from the power balance equation. Introducing i α = (I αβ + ∆I αβ ) cos(ωt), i β = (I αβ + ∆I αβ ) sin(ωt), v + = y e -∆v + and v -= 0 -∆v -, the power balance equation ( 13 The transfer function is given by

G v+ (s) = ∆v + ∆I αβ = (V αβ -2r L I αβ )Re ye 1 + ReC 2 s := K 1 + T s
Considering the control scheme in Figure 3, the outer openloop transfer function is given by

KK I s(1 + T s)
If the desired phase margin is π/3 radians, a simple calculation gives

K I = 2 3KT = 4y e 3R 2 e C(V αβ -2r L I αβ )
.

IV. SIMULATION

To illustrate the performance of the proposed control, simulations were performed on MATLAB/Simulink. Components parameters are summed up in Table II. With those values, the equilibrium power is p * = 782.02 W. Using MATLAB and a SDP solver to minimize the trace of P and S while satisfying [START_REF] Beneux | Adaptive stabilization of switched affine systems with unknown equilibrium points: Application to power converters[END_REF] for Q C = I 4 , (10) and S > 10 -4 I 4 for Q O = 10 -2 I 4 , we obtain: P = diag(600, 600, 23, 30) and S = diag(4, 4, 1, 1) × 10 -4 and 27 L λi gains.

During simulation, the outer loop is turned on after 0.2s to prevent overshoots at start-up and to observe the system behavior without it. For a simulation step of T = 5 × 10 -5 s and with the initial condition x 0 = 0 0 15 5

T , Figure 4 presents the evolution of the currents at start-up. The observer states reaches system states as expected, and currents follows their references. Perturbations that affect the output and the amplitude of the input voltage are reported in Figure 5, along with the evolution of the amplitude of the reference currents I αβ +∆I αβ due to the outer loop. As expected, the reference is adapted when a perturbation affects the system, which ensures that the control objectives are still met.

The evolution of instantaneous powers p and q, currents stabilization and reconstruction errors, as well as system and observer voltages with references are presented in Figure 6. It can be noticed that the output y = v + and p present a steady state error before the outer loop is activated, of 2% and 3.2% respectively. Once the outer loop is turned on, steady state errors are cancelled and the output is regulated to its reference. p, q and system states are affected by perturbations, but are regulated too. The reactive power q converge to 0 after each perturbation, the unit power factor is therefore maintained. As expected, all reconstruction errors converge to 0 despite the perturbations, which ensures that the observer-based control law works properly.

V. CONCLUSION

In this paper, an observer-based switched control law for the three-level neutral point clamped rectifier has been designed. The modeling process was based on the modeling methodology proposed in [START_REF] Garcia | A Unified Approach for the Control of Power Electronics Converters[END_REF] and adapted to take into account constraints between control variables specific to this converter. Despite the complexity and the large number of configurations, a full order observer is easily designed based on LMI conditions that can be solved with numerical tools. It allows to reconstruct the state of the system with the measurement of the input and both capacitors voltages. The stability of the system with the proposed control law is assessed with a Lyapunov analysis, and illustrated by simulations on MATLAB/Simulink. To complete the control scheme, an outer loop is proposed to regulate the output for constant perturbations. The future work could consist in validating those results on an experimental setup.

Fig. 3 :

 3 Fig. 3: Block diagram of the output regulation loop

Fig. 4 :Fig. 5 :Fig. 6 :

 456 Fig. 4: System and observer currents with references

  -∆v + )∆ v+ = -r L (I αβ + ∆I αβ ) 2 -2r L I αβ ∆I αβ + V αβ ∆I αβ -

				) becomes
	C 2 (y e +V αβ (I αβ + ∆I αβ ) -	(y e -∆v + ) 2 2R e	-	(0 -∆v -) 2 2r C
	Taking into account (14) and retaining the first order terms,
	we obtain
	Cy e 2	∆ v+ = y e R e	∆v +

TABLE II :

 II Parameters of simulations

	Parameters	Values	Parameters	Values
	L	15 mH	r L	0.4 Ω
	C	1500 µ F	r C	20 kΩ
	E	72 V	R	30 Ω
	ye	150 V	ω	2π×50 rad/s


Y Cocheril
email: cocheril@inrets.fr

M Berbineau
email: berbineau@inrets.fr

P Combeau
email: combeau@sic.sp2mi.univ-poitiers.fr

Y Pousset
email: pousset@sic.sp2mi.univ-poitiers.fr

On the importance of the MIMO channel correlation in underground railway tunnels

Keywords: MIMO channels, correlation, underground tunnels, modal theory, models, Kronecker, Weichselberger

This paper deals with MIMO channel modeling according to the correlation level in underground railway tunnels for various antenna configurations for the transmitting and receiving arrays. MIMO channel matrices have been computed with a 3D ray-tracing based software at 2.4 GHz and 5.8 GHz in two different tunnel environments: 1) a 1-track empty tunnel with a square cross section, 2) a 1-track tunnel with a square cross section in which a train is parked between the transmitter and the receiver. In this paper, two different strategies are investigated to model the MIMO channel using the Kronecker and the Weichselberger correlation based channel models. The first one is to model the MIMO channel using a single model over the total tunnel length. The second one takes into account the correlation at the receiving side according to the transmitter-receiver distance. In the latter solution, it is possible to isolate specific areas in the tunnel with specific correlation properties and model them in an independent way to take them into account in a system simulation. In this paper, these two modeling strategies are compared in terms of channel capacity.

I. INTRODUCTION

New technologies of communication and information are today key components for mass transit systems operation with applications, such as control and command, embedded surveillance, maintenance reporting or video on demand [START_REF] Berbineau | Existing or emerging telecommunication systems and their use in guided transports[END_REF]. These wireless systems are often deployed using radiating cables, wave guides, or antennas, using in this case free propagation in tunnels. The wireless systems must be able to maximize data rate (several applications on the same radio medium), or robustness (decreasing the number of radio access points, or increasing the QoS) while avoiding the increase in transmitting power and/or transmission bandwidth.

MIMO (Multiple-Input Multiple-Output) systems appear to answer the needs for robust and high data rate communications, without an additional power or bandwidth consumption [START_REF] Foschini | On limits of wireless communications in a fading environment when using multiple antennas[END_REF]. In an environment full of multipaths, the use of multiple antenna arrays at both the transmitting and receiving sides leads to the identification of several independent propagation channels which are linked to the rank of the channel matrix H [START_REF] Gallager | Elements of information theory[END_REF], [START_REF] Berezansky | Functional analysis[END_REF]. The capacity of the MIMO channel depends on this rank. With spatial correlation or key hole effect in the channel, the H matrix will be degenerated [START_REF] Shiu | Fading correlation and its effect on the capacity of multielement antenna systems[END_REF], [START_REF] Chizhik | Capacity of multi element transmit and received antennas: Correlation and keyholes[END_REF]. Previous works have shown the interest and efficiency of such a system in transport environments [START_REF] Moniak | Robust and high data rate transmissions for security between a bus and a control center[END_REF]. Nevertheless, in a tunnel environment when there is no train, the number of scatters is generally low as well as the spread of the angle of arrival of the rays due to the guided effect. In this context the use of MIMO systems and their efficiency is not obvious [START_REF] Cocheril | MIMO channel propagation characteristics in tunnels[END_REF].

The modal theory [START_REF] Laakman | Waveguides: characteristic modes of hollow rectangular dielectric waveguides[END_REF] shows that in infinite rectangular cross section tunnels, free propagation is possible when the transverse dimensions are large compared to the wavelength. In this specific case, the tunnel can be compared to an oversized lossy waveguide. In this condition, the modal theory shows that only the hybrid modes denoted EH mn are able to propagate, where m and n stand for the mode order. The higher order modes are very numerous near the transmitter, and fade rapidly with the increasing of the distance between the transmitter and the receiver [START_REF] Lienard | Wave propagation in tunnels in a MIMO contexta theoretical and experimental study[END_REF]. Far from the transmitter, it remains only two main modes which interfere together. From [START_REF] Lienard | Wave propagation in tunnels in a MIMO contexta theoretical and experimental study[END_REF], three areas can be clearly identified: 1) approximately from 0 to 150 m, 2) from 150 to 400 m, 3) above 400 m. According to the decrease of the number of active modes, the full rank of the channel matrix H can not be guaranteed over all the tunnel length. Subsequently, the ergodic channel capacity C decreases significantly. [START_REF] Cocheril | MIMO channel propagation characteristics in tunnels[END_REF] has shown that confined environments like tunnels lead to narrowband channels at high frequencies. Thus, assuming a time-invariant and flat fading channel, the ergodic channel capacity formulation is [START_REF] Foschini | On limits of wireless communications in a fading environment when using multiple antennas[END_REF] :

C = log 2 det I NRx + SNR N Tx HH H ,N Tx ≥ N Rx (1)
where N Tx and N Rx are the number of elements at the transmitter and receiver arrays, respectively, I is the identity matrix, SNR stands for the Signal-to-Noise Ratio, and HH H traduces the correlation at the receiving side. Assuming this channel capacity formulation, we have to study the influence of the correlation at the receiver according to the transmitter-receiver distance onto the channel modeling, and thus onto channel capacity.

In this paper, we investigate 4×4 MIMO channel correlation properties for several configurations of the transmitting and receiving arrays, in two underground railway tunnels. A full 3D ray-tracing based wave propagation simulator [START_REF] Chartois | A siso and mimo radio channel characterization with 3d ray tracing propagation model in urban environment[END_REF] is used to compute the deterministic channel matrix H according to the transmitter-receiver distance at 2.4 GHz and 5.8 GHz. In this paper, two channel modeling are proposed. First, the channel is modeled in a global way over all the tunnel length, using the Kronecker [START_REF] Kermoal | A stochastic MIMO radio channel model with experimental validation[END_REF] and Weichselberger [START_REF] Weichselberger | A stochastic MIMO channel model with joint correlation of both link ends[END_REF] correlation based MIMO models. Then, the tunnel is separated in multiple areas to ensure the stationarity of the correlation properties. In each area, the channel is modeled using the two models. Finally, we present a comparative study between these specific channel models and the simulation results in terms of channel capacity.

This paper is structured as follows. We first introduce in Section II the configurations simulated thanks to a 3D ray-tracing based software. In addition, the Kronecker and Weichselberger models are also described. In Section III, the influence of the geometric configuration of the antennas, in the tunnel and on the trains, on the correlation at the receiver is analyzed. In Section IV, the influence of the channel modeling considering 1) a unique model, or 2) two specific models over two different areas according to the correlation level, is discussed. We will then conclude and give the perspectives to this work.

II. MIMO CHANNEL SIMULATIONS

Firstly, we present the two underground railway tunnels, varying the antenna configurations in the tunnel and on the train. Each 4×4 MIMO channel matrix H, obtained for specific antenna configurations in an environment, has been simulated thanks to a 3D ray-tracing based software [START_REF] Chartois | A siso and mimo radio channel characterization with 3d ray tracing propagation model in urban environment[END_REF]. This software computes all the possible paths followed by an electromagnetic wave between a receiver and a transmitter, assuming the electromagnetic parameters (relative permittivity, conductivity) of the objects in the scene and a specific number of interactions (reflection and/or diffraction). A previous study has shown that the channel matrix H can be studied in narrowband [START_REF] Cocheril | MIMO channel propagation characteristics in tunnels[END_REF] for the considered configurations. In this paper, this MIMO channel matrix H is used to model the channel using existing MIMO channel models such as the Kronecker and the Weichselberger models. These channel models are detailed in a second step.

Rx

A. Description of the tested configurations

Several realistic configurations are simulated, varying the position of the receiver in a tunnel and the orientation of the transmitter and the receiver antennas. The 1-track tunnel (4.5×4.5×500 m) can be 1) empty, or 2) with a parked train (3×4×120 m) between the transmitter and the receiver. Figure 1 illustrates these two configurations. Table I gives the relative permittivity and the conductivity of the materials used to model the environment in the 3D ray-tracing based software. Notice that in the empty tunnel case, this software is configured to compute all the paths which exist between the transmitter and the receiver assuming 10 reflections. On the contrary, the number of reflections is reduced to 9 in the second environment, and the number of diffractions is set to 1. Indeed, many paths can be diffracted by an edge of the parked train. Both transmitter and receiver are 4-elements antennas, disposed at various places as indicated in Figure 2. The four elements of the transmitter are 1 m spaced and fixed on the tunnel ceiling. Three configurations are tested (TP1, TP2, TP3): the angle between the orientation of the transmitter axis and the main direction of the tunnel varies from 0 to 90 • passing through 45 • . The elements of the receiver are closer (0.65 m spaced) due to the lack of space available on the roof of the train. Three configurations are also studied (RP1, RP2, RP3). The two first ones are located on the roof of the train, oriented in the same direction as the longitudinal axis of the tunnel and perpendicular to it, respectively. The third one is located onto the back windshield of the train in a rectangular shape. The coordinates of the elements of each antenna configurations are given in Table II.

Channel matrices H are computed for all the configurations of the transmitting and receiving arrays 1) over all the tunnel length (X = [0; 500] m) in an empty tunnel, 2) after the end of the parked train (X = [310; 500] m) in the second environment. Indeed, it is not realistic to have the receiver above the parked train in a 1-track tunnel. The sampling rate is equal to 0.5 m.

B. Channel modeling

The development and the evaluation of new digital wireless transmission systems need faithful channel models. A lot of models exist in the literature, and they can be classified in two main categories [START_REF] Almers | Survey of channel and radio propagation models for wireless mimo systems[END_REF].

Physical models are often based on an accurate geometrical description of the propagation environment. They can be deterministic, when they use channel parameters deduced from measurement campaigns or simulation tools (with a 3D ray-tracing based simulator for example) [START_REF] Schumacher | Recent advances in propagation characterisation and multiple antenna processing in the 3GPP framework[END_REF]. In this case, the accuracy is high, as the cost in materials, human or computing resources. To eliminate previous drawbacks, researchers have developed many statistical physical models, based on the characterization of the scatters present in the propagation environment [START_REF] Yu | Models for MIMO propagation channels: a review[END_REF]. The most used for example are the one-ring and two-ring models [START_REF] Shiu | Fading correlation and its effect on the capacity of multielement antenna systems[END_REF] and the distributed scatters model [START_REF] Gesbert | Outdoor MIMO wireless channels: models and performance prediction[END_REF]. The major drawback of these models remains the determination of the statistical distributions of the scatters in the environment. Analytical models or stochastic models are independent of the geometric environment description. They are based on the statistical properties of the channel such as the correlation computed from measurements or simulations. There are different possibilities to take into account the correlation between the arrays elements at the receiver and at the transmitter but also in the channel. Several stochastic models were compared in [START_REF] Pardonche | Présentation de quelques modèles stochastiques de canal mimo et comparaison expérimentale[END_REF]. One of the simplest is the Kronecker model [START_REF] Kermoal | A stochastic MIMO radio channel model with experimental validation[END_REF] assuming perfect independence of the correlation between transmission and reception sides, while the coupling between the transmitting and receiving arrays can be taken into account using the Weichselberger model [START_REF] Weichselberger | A stochastic MIMO channel model with joint correlation of both link ends[END_REF].

The two models are based on the following decomposition of the channel matrix H, filled with complex coefficients, which can be written as follows:

vec {H} = R 1/2 H g (2)
R H is the correlation/covariance matrix of the channel, g is an i.i.d. random fading vector with unit variance and the operator vec {•} stacks a matrix into a vector, columnwise.

1) Kronecker model:

The Kronecker model assumes a correlation at the transmitter independent from the correlation at the receiver. So the total correlation of the channel R H can be expressed as the Kronecker product (⊗) of the correlation matrices at the transmitter R Tx and at the receiver R Rx :

R H = R Tx ⊗ R Rx (3)
Thus, using (3) in (2), we obtain the following relation for the Kronecker model:

H = R 1/2 Rx G R 1/2 Tx T (4)
Notice that the covariance matrices can be used instead of the correlation matrices in (4). This formulation is very simple and easily usable once we have obtained the correlation/covariance matrices.

2) Weichselberger model: Contrary to the Kronecker model, the Weichselberger model takes into account the correlation in the channel between the transmitter and the receiver.

Its formulation is based on the well known Singular Value Decomposition (SVD) of the correlation matrices R Tx and R Rx , as:

R Tx = U Tx Λ Tx U H Tx (5) R Rx = U Rx Λ Rx U H Rx (6)
where:

-U Tx and U Rx are unitary matrices; their columns contain the eigen vectors of R Tx and R Rx respectively, -Λ Tx and Λ Rx are diagonal matrices filled with the eigen values of R Tx and R Rx respectively. Using ( 5) and ( 6) in (4), the following relation can be obtained:

H = U Tx (Ω G) U T Rx (7)
where is the Schur-Hadamard product, and Ω traduces the coupling between the transmitter and the receiver. Its coefficients w mn > 0 are equal to:

w mn = E H U H Rx,m HU * Tx,n 2 (8)
In ( 8), E H {•} denotes expectation with respect to H.

III. CHANNEL CORRELATION In this section, the importance of the orientation of the transmitting and receiving antennas is studied, computing the correlation level at the receiver. The correlation level at the receiver is directly linked with the theoretical channel capacity (1) through the product HH H . Smaller the correlation level is, greater the theoretical channel capacity is. So, the aim of this section is to identify the best antenna configurations which maximize the channel capacity, and thus offer the maximal diversity which can be traduced into robustness or high data rate capabilities in a wireless communication.

A. Empty tunnel

For each position of the train (from 0 to 500 m), the correlation ρ at the receiver is computed. Figure 3 presents these correlation coefficients according to the distance for all the 9 transmitter-receiver combinations at 2.4 GHz and 5.8 GHz. This figure highlights the increase of the correlation coefficient with the transmitter-receiver distance when the transmitter and/or receiver arrays (TPj × RP1, TP1 × RPk, j, k ∈{1, 2, 3}) are oriented in the same direction as the main tunnel axis x. In these specific cases, two areas can be distinguished. The first one, called A1, starts from 0 to 150 m and shows fluctuations of the correlation level. The second one, named A2, starts from 150 to 500 m and presents high correlation values roughly equal to 1.

On the contrary, the correlation coefficient at the receiver is quite constant over all the tunnel length for specific antenna configurations. The main orientation of these antenna arrays is transverse to the longitudinal axis of the tunnel (TP2 × RP2 for example). These configurations lead to a smaller coefficient correlation due to the increase of the spatial diversity. The solution, which consists in positioning the antenna arrays onto the back windshield of the train (TP3 × RPk, k ∈{2, 3}), shows similar performance.

Table III summarizes the correlation coefficients averaged over both areas A1 and A2 identified previously (called ρ), at 2.4 GHz and 5.8 GHz, respectively. When the angle between the array and the tunnel axis is reduced to 0 • (TPj × RP1, TP1 × RPk, j, k ∈{1, 2, 3}), the mean correlation values are different in each area. In the area A1 (first 150 m), the mean correlation values are between 0.76 and 0.93, while they reached 0.99 in the area A2 (from 150 to 500 m). [START_REF] Molina-Garcia-Pardo | Interpretation of mimo channel characteristics in rectangular tunnels from modal theory[END_REF] has explained that spatial diversity can be compared in the tunnel with the concept of modal diversity. Consequently, the correlation between the received modes can increase rapidly with the transmitter-receiver distance, specifically when the transmitting and receiving arrays are oriented along the x axis.

To maximize the modal diversity, the element arrays have to be oriented in a plane coplanar with the tunnel cross section. Thus, it appears that the correlation level in the TPj × RPk, j, k ∈{2, 3} configurations decreases significantly (0.49 < ρ< 0.64 at 2.4 GHz and 0.50 < ρ< 0.60 at 5.8 GHz) and is independent from the transmitter-receiver distance. This small values are possible thanks to the modal diversity which can be maintained in these specific antenna configurations even far from the transmitter [START_REF] Molina-Garcia-Pardo | Interpretation of mimo channel characteristics in rectangular tunnels from modal theory[END_REF].

The main observations are the same at 2.4 GHz and 5.8 GHz: 1) two areas A1 and A2 for the configurations TPj × RP1 and TP1 × RPk, j, k ∈{1, 2, 3}, 2) a unique global behavior for the configurations TPj × RPk, j, k ∈ {2, 3}. The correlation level is smaller at 5.8 GHz than at 2.4 GHz for the configurations TPj × RP1 and TP1 × RPk, j, k ∈{1, 2, 3} in the area A2. At this frequency, the modal diversity increases due to a higher number of active hybrid modes that propagate [START_REF] Molina-Garcia-Pardo | Interpretation of mimo channel characteristics in rectangular tunnels from modal theory[END_REF].

B. Tunnel with a parked masking train between the transmitting and receiving sides

In the presence of a masking train parked in the middle of the tunnel, the correlation level at the receiver can be computed only between 310 and 500 m. Figure 3 summarizes these results for all the antenna configurations. Only one behavior can be observed: the correlation level varies around a constant mean value from 310 to 500 m. In comparison with the empty tunnel, the correlation level is significantly smaller for the configurations TPj × RPk, j, k ∈{1, 2, 3} at 2.4 GHz (Table IV). In these specific cases, the parked train added some spatial diversity at the receiver.

C. Conclusion

In this section, two different behaviors of the correlation level have been observed. When the transmitting and receiving antenna arrays are oriented in a plane coplanar with the tunnel cross section, the modal diversity leads to an uncorrelated channel in the overall tunnel length (from 0 to 500 m). On the contrary, two different behaviors can be observed if the antenna arrays at the transmitting and/or receiving sides are oriented along the direction of the longitudinal tunnel axis: close from the transmitter the channel is highly correlated (about 0.90), far from the transmitter it is totally correlated (about 0.99).

IV. CHANNEL MODELING

The previous section has shown that the correlation at the receiver can evolve significantly with the transmitter-receiver distance. This behavior has been only observed in the empty tunnel environment, and particularly when the transmitter and/or receiver are oriented in the longitudinal tunnel direction. This section focuses on the importance to match with real channel behavior in the system performance analysis, computing a channel model over the area A1, and another one over the area A2. The analysis has been performed for the empty tunnel environment. First, we compute a single model over the total tunnel length, using the Kronecker and the Weichselberger models. Then, we compute these two different models over the areas A1 and A2, respectively. Finally, the channel capacity (1) is computed using the two channel models for each areas (A1, A2, and over the total tunnel length) and we compare these results with those obtained directly from the channel matrices computed with the 3D ray-tracing based software in identical area conditions.

Figures 4 and5 show the channel capacity results using the channel matrices H obtained respectively from the simulations and the Kronecker model and the Weichselberger model in the TP2 × RP2 and TP1 × RP1 configurations, respectively. Notice that these results are averaged over the areas where they are computed (all the tunnel, area A1 or area A2). In the configuration TP2 × RP2, the antennas are oriented in a plane coplanar with the tunnel cross section. In this case, the previous section has shown that the correlation at the receiver is independent from the transmitter-receiver distance; so the strategy which consists in modeling the channel in two different areas does not seem to be really interesting. Indeed, the channel capacity values are quite similar whatever the strategy we consider. and for those (TPj × RPk, j, k ∈{2, 3}) which have highlighted a constant correlation level whatever the transmitter-receiver distance is. On the contrary the choice of a strategy before modeling the channel is very important in the TP1 × RP1 configuration. The different values of the correlation level according to the distance lead to a great error onto the mean channel capacity over the total tunnel length. The estimated channel capacity for a SNR = 30 dB is equal to 18.3 bit.s -1 .Hz -1 . This estimation is far from the mean values obtained in the two areas A1 and A2: 23.1 bit.s -1 .Hz -1 and 14.6 bit.s -1 .Hz -1 considering the Kronecker model (see Table V), respectively. So, modeling the channel in two different areas appears as a better choice to well model the channel behavior. This observation can be generalized in the other antenna configurations (TPj × RP1, TP1 × RPk, j, k ∈{1, 2, 3}) for which a high correlation degree far from the transmitter has been obtained when the transmitter and/or receiver arrays are oriented according to the longitudinal axis of the tunnel.

Moreover, the Kronecker model gives better channel modeling results in terms of channel capacity compared to those obtained directly from the simulations, unlike the Weichselberger one. This observation can be performed in the areas A1 and A2, and also over the total tunnel length. Nevertheless, the Weichselberger model is sometimes better, in some specific configurations, but the differences with the Kronecker model results are too small. So, due to its implementation facility and its great performance, the Kronecker model can be chosen to model the channel in the underground tunnels that we have tested.

Finally, this section has shown that it is useful to identify the different areas in the tunnel where the correlation level varies significantly. With such considerations, an accurate description of the channel capacity in the different areas of the tunnel can be obtained.

V. CONCLUSION This paper has presented an interesting study which has focused on the importance to have an accurate description of the correlation variations in an underground tunnel environment, in order to well model the MIMO channel, using Kronecker and Weichselberger correlation based channel models. This study has been performed in two different underground railway environments: 1) a 1-track empty tunnel, and 2) a 1-track tunnel with a masking train parked between the transmitting and receiving sides. In such environment, the 4×4 MIMO channel matrix has been computed thanks to a 3D ray-tracing based software for various antenna configurations at 2.4 GHz and 5.8 GHz.

The study of the correlation level at the receiver has highlighted two different behaviors according to the antenna configurations. When the antennas are oriented in the same direction as the longitudinal axis of the tunnel at the transmitting or receiving sides, the correlation level increases with the transmitter-receiver distance. Thus, two areas have been identified: the first one called A1 starts from 0 to 150 m; the second one named A2 follows the first one until the end of the tunnel (from 150 to 500 m).

On the contrary, when the transmitting and receiving antennas are located in a plane coplanar with the tunnel cross section, the correlation is constant over the total tunnel length, due to the diversity of the hybrid modes which propagate. In terms of channel modeling, it is really interesting to subdivide the tunnel in areas where the correlation properties are significantly different, in order to have a better modeling of the channel behavior whatever the transmitter-receiver distance is. When the correlation level is quite constant over the total tunnel length (antennas perpendicular to the tunnel axis), it is sufficient to consider only a single model, valid over the whole tunnel length.

In future works, we will focus on the improvement of the channel modeling results, and on their understanding using the modal theory. More complicated scenes will be also studied (2-track tunnels, stations, etc.), and other antenna configurations will be investigated. Measurements are planned to validate this approach.

Figure 1 .

 1 Figure 1. Shapes and dimensions of the underground railway tunnels

Figure 2 .

 2 Figure 2. Transmitter (TPx) and Receiver (RPx) antenna configurations

Figure 3 .

 3 Figure 3. Correlation coefficient ρ at the receiver for all the antenna configurations

Figure 4 .Figure 5 .

 45 Figure 4. Average Channel capacities for the simulated channel matrices H using a ray-tracing based software and both Kronecker and Weichselberger models -TP2 × RP2 (2.4 GHz)

TABLE I

 I

	.
	ELECTRICAL PROPERTIES OF MATERIALS
		Tunnel Train
	Relative permittivity ε r	10	2
	Conductivity σ (S.m -1 )	0.01	10 8

TABLE II

 II

			.
		4-ELEMENTS POSITIONS (x, y, z) FOR EACH ANTENNAS
		CONFIGURATION (i ∈{0, 1, 2, 3})
		x	y	z
	TP1	i	0.20	4.30
	TP2	0	{0.2; 1.2; 2.2; 3.2}	4.30
	TP3	0.707 i	0.2 + 0.707 i	4.30
	RP1	X +0.65 i	2.25	4.10
	RP2	X	1.275 + 0.65 i	4.10
	RP3	X	{1.6; 1.6; 2.9; 2.9}	{3.8; 3.15; 3.8; 3.15}

TABLE III .

 III AVERAGE CORRELATION COEFFICIENT ρ IN AREAS A1, A2 -EMPTY TUNNEL AT 2.4 GHz AND 5.8 GHz

			f =2 .4 GHz
		TP1		TP2	TP3
		A1	A2	A1	A2	A1	A2
	RP1	0.83 0.99	0.89	0.98	0.93	0.99
	RP2	0.76 0.96	0.50	0.49	0.63	0.59
	RP3	0.81 0.97	0.57	0.56	0.59	0.64
			f =5 .8 GHz
	RP1	0.86 0.96	0.79	0.95	0.80	0.94
	RP2	0.84 0.88	0.50	0.52	0.51	0.58
	RP3	0.79 0.91	0.60	0.54	0.60	0.56

TABLE IV .

 IV AVERAGE CORRELATION COEFFICIENT ρ -TUNNEL WITH A PARKED MASKING TRAIN AT 2.4 GHz AND 5.8 GHz

		TP1	TP2	TP3
	f (GHz)	2.4	5.8	2.4	5.8	2.4	5.8
	RP1	0.91	0.97	0.99	0.97	0.99	0.96
	RP2	0.87	0.89	0.52	0.53	0.57	0.53
	RP3	0.80	0.93	0.48	0.55	0.61	0.55

  Table V confirms these observations for this specific configuration

TABLE V .

 V AVERAGE CHANNEL CAPACITY C FOR SNR = 30 dB OVER AREAS A1, A2, AND THE TOTAL TUNNEL LENGTH (A1 + A2) -RESULTS

	WITH KRONECKER MODEL -EMPTY TUNNEL AT 2.4 GHz AND
				5.8 GHz
			f =2 .4 GHz
		TP1		TP2		TP3
		A1	A2	A1	A2	A1	A2
		A1 + A2	A1 + A2	A1 + A2
	RP1	23.1 14.6 18.3	24.4 20.2 16.9	25.1 21.3 16.8
	RP2	28.6 20.5 25.3	33.8 34.6 34.6	32.7 34.3 34.5
	RP3	28.9 19.8 25.8	34.6 34.4 34.0	34.6 34.4 34.0
			f =5 .8 GHz
	RP1	24.8 18.0 20.7	28.7 23.5 20.5	29.0 24.2 21.3
	RP2	30.0 26.1 27.9	34.8 34.0 33.6	34.5 33.7 33.4
	RP3	30.0 23.9 26.7	33.7 34.7 34.4	33.6 34.6 34.5

ACKNOWLEDGMENT

This work was performed in the framework of the PREDIT research project MOCAMIMODYN (MOdèles de CAnaux MIMO DYNamiques en tunnels pour des applications transport public) in the framework of the I-Trans cluster and the regional CISIT (Campus International Sécurité et Intermodalité des Transports) program. The authors would like to thank the french National Research Agency (ANR), the North Region and the ERDF (European Regional Development Fund) for financial support.

He is currently a postdoctoral researcher at the french national institute for transport and security research (INRETS). His current research interests include the SISO and MIMO channel modeling in underground tunnels and the optimization of the physical layer in such confined environments, for robust and high data rate transmissions in railway applications.

Marion Berbineau was born in


E Masson

P Combeau
email: combeau@sic.sp2mi.univ-poitiers.fr

M Berbineau
email: marion.berbineau@inrets.fr

R Vauzelle

Y Pousset
email: pousset@sic.sp2mi.univ-poitiers.fr

Radio Wave Propagation in Arched Cross Section Tunnels -Simulations and Measurements

Keywords: Radio wave propagation, Arched shape tunnels, Ray tracing, tessellation, E-field measurements. I

For several years, wireless communication systems have been developed for train to infrastructure communication needs related to railway or mass transit applications. The systems should be able to operate in specific environments, such as tunnels. In this context, specific radio planning tools have to be developed to optimize system deployment. Realistic tunnels geometries are generally of rectangular cross section or arched shape. Furthermore, they are mostly curved. In order to calculate electromagnetic wave propagation in such tunnels, specific models have to be developed. Several works have dealt with retransmission of GSM or UMTS [1], [2]. Few theoretical or experimental works have focused on 2.4 GHz or 5.8 GHz bands [3]. In this paper, we propose an approach to model radio wave propagation in these frequency bands in arched shape cross section straight tunnels using tessellation in multi-facets. The model is based on a Ray-Tracing tool using the image method. The work reported in this paper shows the propagation loss variations according to the shape of tunnels. A parametric study on the facets size to model the cross section is conducted. The influence of tunnel dimensions and signal frequency is examined. Finally, some measurement results in an arched cross section straight tunnel are presented and analyzed in terms of slow fadings and fast fadings.

INTRODUCTION

The need for wireless communication systems is increasing for train to ground or train to train communications in the railway or mass transit domains. These systems are developed to satisfy operational needs, such as traffic management, maintenance, information applications and security of passengers and staff. Among these applications we can mention control-command systems and also high data-rate transmissions for multimedia or other operational applications. Several systems are already deployed in the world using free propagation in tunnels in the 2.4 GHz or 5.8 GHz bands (Urbalis system for ALSTOM, Airlink® system used by SIEMENS). The prediction of radio coverage levels is required to optimize deployment phases and ensure availability and robustness of the links. Generally, minimal field levels are required to guarantee key performance indicators related to safety constraints or QoS requirements for video transmissions. In tunnels, the usual laws of free space propagation are no longer valid and the propagation phenomenon has to be specifically analyzed to develop statistical models which are easy to use to predict radio coverage [START_REF]Document Cost231 TD 031 -Thomas Klemenschits, Ernts Bonek[END_REF]. Thus, up to now, the case of tunnel areas is generally treated via intensive measurement campaigns. The development of specific models is then very relevant.

The paper focuses on results obtained at 2.4 GHz and 5.8 GHz bands in arched shape cross section straight tunnels. The model is based on a Ray-Tracing tool using the image method. A method of tessellation in multifacets of the cross section is used, and obtained results are compared to measurements. The first part of the paper presents different modeling techniques that can be used to describe radio wave propagation in tunnels. The second part details the considered method of tessellation. Particularly, obtained results in arched shape cross section straight tunnels at 1 GHz are presented and compared to existing results. The influence of shape of tunnels is highlighted and a parametric study on the number of facets to model the cross section is realized. The third part of the paper is devoted to the study of the influence of tunnel dimensions and signal frequency, focusing on 2.4 GHz and 5.8 GHz bands. The last part of the paper is dedicated to measurement results obtained in an arched shape cross section tunnel. Comparisons between simulations and measurements are realized in terms of slow fadings and fast fadings. Finally, conclusions and perspectives of the work are given.

II. SEVERAL METHODS OF RADIO WAVE PROPAGATION

MODELING IN TUNNELS

In order to describe the radio wave propagation in tunnels, several modeling approaches can be set up. The numerical resolution of Maxwell's equations would be an ideal solution. However, this kind of technique is not feasible due to the enormous computational burden. A more conventional way to solve the problem is given by modal theory. The tunnel is here treated as a hollow waveguide with dielectric boundaries. Unfortunately, analytical expressions of the different constants, such as cut-off frequency and wave impedance, only exist for few canonical types of configuration, e.g. rectangular or circular cross section waveguides [START_REF] Balanis | Advanced Engineering Electromagnetics[END_REF]. The prediction of radio wave propagation with an adequate accuracy in finite time is given by ray optics solutions. These solutions can be adopted in tunnels because dimensions in tunnels are generally large compared to the considered wavelength (frequency above 1 GHz). Several methods based on the ray-optical modeling approach have been proposed. They use ray launching [START_REF] Zhang | Rayoptical prediction of radio-wave propagation characteristics in tunnel environments part 1: Theory[END_REF], [START_REF] Wang | Simulations and measurements of wave propagations in curved road tunnels for signals from GSM base stations[END_REF], the image method [START_REF] Torres | CINDOOR: An engineering tool for planning and design of wireless systems in enclosed spaces[END_REF] or combination of each [START_REF] Cheng | SBR image approach for radio wave propagation in tunnels with and without traffic[END_REF]. They all consider only the case of straight and rectangular tunnels. The classical approach of Ray-Tracing techniques cannot be transposed to the case of curved surfaces. Indeed, the image method is not applicable because of the infinite line of images of one source compared to a single point for plane surface. For the ray launching, the concept of reception sphere is no longer valid for curved surfaces because of the non-conservation of reflection angles [START_REF] Didascalou | Ray-optical Wave Propagation Modelling in Arbitrarily Shaped Tunnels[END_REF]. The first conceivable solution to model the propagation in environments where curved surfaces are present is the development of a completely novel model based on ray optics, as in [START_REF] Didascalou | Ray-optical Wave Propagation Modelling in Arbitrarily Shaped Tunnels[END_REF]. The second option for non-rectangular cross section straight tunnels is to consider the equivalent rectangular cross section tunnel with an equal area. The last solution is to tessellate geometries into multiple planar facets, as proposed in [START_REF] Cheng | SBR image approach for radio wave propagation in tunnels with and without traffic[END_REF], [START_REF] Torres | CINDOOR: An engineering tool for planning and design of wireless systems in enclosed spaces[END_REF], [START_REF] Baranowski | Optimisation des règles d'ingénierie radio-modélisation de la propagation d'ondes radioélectriques en tunnels courbes[END_REF].

III. MODELING CURVED WITH FACETS -INFLUENCE OF NUMBER OF FACETS

A Ray-Tracing method [START_REF] Aveneau | Efficient computation of radio coverage zone using a spatial partionment approach[END_REF] combined to a tessellation of the curvature of the tunnel will be used. First, the simulation tool is presented as well as the tessellation principle. Then, simulation configurations are detailed. Finally, a parametric study on the facets size is realized to model the arched section.

A. Method of tessellation

The Ray-Tracing method consists of a direct research of geometric paths followed by the waves. It allows us to determine exactly the set of paths from a transmitter to a receiver. This technique is based on the image theory based on Snell-Descartes formulas. From these paths, the Electric field is computed from Geometrical Optic (GO) laws and from Fresnel coefficients. In the case of a curved surface, one source generates an infinite number of images. To solve this problem, we choose to approximate the curved surface by facets. Several problems appear, such as the position of the facets and the optimal size of facets. In [START_REF] Baranowski | Optimisation des règles d'ingénierie radio-modélisation de la propagation d'ondes radioélectriques en tunnels courbes[END_REF], a first approach of tessellation of a curved surface is presented in the case of a 2D curvature. A method developed in [START_REF] Cheng | SBR image approach for radio wave propagation in tunnels with and without traffic[END_REF] uses triangular facets to model the curved surface. A hybrid method based on a Shooting and Bouncing Ray (SBR) combined with the image theory is tested in some particular configurations of tunnels. Finally in [START_REF] Torres | CINDOOR: An engineering tool for planning and design of wireless systems in enclosed spaces[END_REF], a tool developed for planning and design of wireless systems is presented. It is based on Ray-Tracing techniques. Similar results to Cheng & Jeng [START_REF] Cheng | SBR image approach for radio wave propagation in tunnels with and without traffic[END_REF] are obtained. The use of the tessellation raises the problem of the size of facets to be used to represent a given curvature. A compromise has to be found on the number of facets retained. It has to be sufficient to represent the geometry of the arched section but not too large to stay in the limit of validity of the physical model based on a high frequency approach. None of the mentioned publications gives criteria to determine the optimal number of facets.

B. Configurations of simulations at 1 GHz

Three configurations of simulations of straight tunnels with equivalent cross section area from Cheng & Jeng [START_REF] Cheng | SBR image approach for radio wave propagation in tunnels with and without traffic[END_REF] are considered, illustrated on Figure 3.1: tunnel A is a rectangular cross section tunnel, tunnel B represents the intermediate tunnel, an arched cross section tunnel modeled with 3 facets, and tunnel C is an arched cross section tunnel modeled with n facets. A 1 GHz frequency is considered. The transmitting and receiving antennas are dipoles vertically polarized and placed respectively in (4 , 0 , 4.5) m and (2.1 , y , 1.5) m, where y represents the longitudinal direction of the tunnel and varies from 10 m to 150 m with a step of 1 m. Results for tunnels A and B are rapidly obtained. For tunnel C, a study on the number of facets used to approximate the arched section has to be realized. To guarantee a valid physical model, the high frequency approach requires a size of facets d >> , where is the wavelength. We assume that these conditions are respected for d > 2 . Given the tunnel dimensions, the condition on the number of facets n is as follows:

4 n 20

(1)

C. Simulation Results

All the results presented are given in terms of signal attenuation (in dB) depending on the distance between transmitter and receiver. Only the reflection phenomenon is considered in the simulations. It represents the dominant effect in an empty tunnel: there is no transmission because of the tunnel walls properties and no diffraction because of the lack of edge.

All the results presented in the paper are displayed with the same scale in order to facilitate the comparisons between the different curves. An important signal level variation can be observed from one configuration to another. Results obtained in simulations are compared to results presented in [START_REF] Cheng | SBR image approach for radio wave propagation in tunnels with and without traffic[END_REF] considering an arched section approximated by facets with a size of 3, which gives the higher signal level. It is important to note that the higher level does not inevitably correspond to the real signal level. We will see later the comparison with measurement results in order to validate, or not, this observation. Figure 3.3 shows the results obtained for tunnels A, B and C and highlights a concordance with results presented in [START_REF] Cheng | SBR image approach for radio wave propagation in tunnels with and without traffic[END_REF]. A « focusing » effect in terms of energy can be observed for the arched section compared to the rectangular section. This phenomenon is characterized by smaller depth fadings and a higher global signal level. The signal level is also compared to the free space case (dotted line), in Line Of Sight (LOS). This result highlights the guided effect in tunnels, compared to free space.

IV. INFLUENCE OF FREQUENCY AND TUNNEL DIMENSIONS

A. Frequency influence

In this part, the frequency varies for similar tunnel geometry. High frequency conditions have to be respected. At 2.4 GHz, equation ( 1 The « focusing » effect observed at 1 GHz is greatly attenuated at 2.4 GHz and 5.8 GHz. The number of facets and the size of the facets have an important effect on the signal level. It has to be noticed that these conclusions concern tunnels of large dimensions (8 m x 5 m), such as TGV tunnels where GSM-R is deployed, or large motorway tunnels. We will then consider tunnels of smaller dimensions corresponding to mass transit tunnels. Similar conclusions to the previous ones can be made in the case of small tunnels. An important variation is observed on the signal level for different facets sizes. A « focusing » effect for the arched section is also highlighted compared to the rectangular one.

V. MEASUREMENT RESULTS

A. Trial conditions

Measurements were conducted in the Tunnel of Roux, which is a two ways arched cross section straight road tunnel located in Ardèche region in France. This tunnel is perfectly straight and has a length of 3,336 m (Figure 5.1). The transverse section of the tunnel is semicircular and has a diameter of 8.3 m. The maximum height is 5.8 m at the centre of the tunnel. The transmitting part is static. It is composed of a large bandwidth horn antenna (9.2 dBi gain at 2.4 GHz, 10.1 dBi gain at 5.8 GHz), vertically polarized, and connected by a low loss cable (63 dB/100 m at 2.4 GHz, 100 dB/100 m at 5.8 GHz) to a transmitter delivering a sinusoidal signal at the required frequency. The mobile reception system is composed of the same horn antenna, also vertically polarized, and connected by a low loss cable to a THALES VUH-TRC8025 receiver used in an analyzer mode and allowing an acquisition rate of 6 points/sec, which corresponds to one measurement every /2 at 2.4 GHz with a speed of 1.4 km/h. The system is installed on a go-kart allowing a very small and regular velocity.

Two different configurations of transmitter-receiver locations have been studied, as is shown in Figure 5.2. The EP1-RP1 configuration corresponds to a transmitter located in the center of the cross section, at (3.2 , 0 , 4.8) m and a receiver in the middle of one of the two ways, at (2.4 , y , 4.1) m, where y represents the longitudinal direction of the tunnel and varies from 1 m to 500 m with a step of 0.0125 m. For the EP2-RP1 configuration the transmitter is moved near the tunnel wall at (1.9 , 0 , 4.8) m.

The measurements were conducted at 2.4 GHz and 5.8 GHz.

B. Measurement results

All the results presented are normalized by the maximum of the received power along the tunnel in order to eliminate the effects of antenna gain and cable loss. Figures 5.3 and 5.4 illustrate measurement results at 2.4 GHz and 5.8 GHz for the EP1-RP1 and EP2-RP1 configurations respectively. For the first configuration (cf. figure 5.3), we can observe a very weak influence of the frequency on the received power. In the second configuration, the attenuation is lower at 5.8 GHz compared to 2.4 GHz. These results are similar to the results obtained by modal theory for the case of rectangular tunnel [START_REF] Collin | Field Theory of Guided Waves[END_REF]. These two configurations have been tested to evaluate the influence of transmitter location in the cross section. In our case, no major difference is observed from one location to another.

C. Comparison of simulation and measurement results

The configurations of measurements were reproduced in simulations using our tessellation approach of the cross section.

Once again, to guarantee the validity of the high frequency hypothesis, some conditions on the number of facets n have to be satisfied according to the given tunnel: at 2.4 GHz: 4 n 66 [START_REF] Torres | CINDOOR: An engineering tool for planning and design of wireless systems in enclosed spaces[END_REF] at 5.8 GHz:

4 n 160 (8)

Figures 5.5 and 5.6 illustrate the simulated received power for different facets sizes compared to the measured power, for the EP1-RP1 configuration at 2.4 GHz. Globally, we can note a good concordance between measurements and simulations, whatever the facets size. To determine the best facets size, we proposed the following method.

To be free of fast fading variations, we smooth the received signal by using a running mean. The window's length is 40, on the first 50 meters, and 100 for the rest [START_REF] Lienard | Propagation in Wide Tunnels at 2 GHz: A statistical analysis[END_REF]. Then for each simulation, i.e. for each tessellation, the standard deviation is calculated between the measured and simulated mean powers. Different zones were considered to evaluate the behavior of configurations in relation to the transmitter-receiver distance. Table I presents the results obtained for the EP1-RP1 configuration at 2.4 GHz. The bold values present the lowest standard deviations which potentially correspond to the best tessellation. A similar study was conducted at 5.8 GHz. The simulation results are illustrated in Figures 5.7 and 5.8. Table II presents the quantitative analysis. The results led to different conclusions. The two tables, Tables I andII, show that, despite a good matching between simulations and experimentations, we cannot conclude on one preferable size of facets or another. In the following paragraph, we present a statistical analysis of the fast fadings on the measured and simulated data.

D. Statistical analysis of fast fading

The two previous windows are considered to perform a running mean that is then subtracted from the signal in order to extract its fast variations. A comparison between measured and simulated data is realized. For the simulations, we consider the configuration that gives the nearest results compared to experimentations. For the EP1-RP1 configuration at 2.4 GHz, this corresponds to a tessellation size of 32.

The Cumulative Density Functions of the measured and simulated data are calculated. They are both compared to the Rayleigh, Weibull and Nakagami distributions. We used the Kolmogorov-Smirnov test in order to decide which distribution best fits the results. Figures 5.9 and 5.10 show the Cumulative Density Functions of the measured and simulated data respectively compared to the fitted models for the EP1-RP1 configuration at 2.4 GHz. The parameters of the theoretical distributions are determined by Maximum Likelihood estimators. The same analysis was also realized for the other configurations and frequencies, and similar results were obtained. The analysis shows a similar behavior for both measured and simulated data. The curves show that the Weibull distribution fits both the measurement and the simulation statistics quite well [START_REF] Yacoub | The alpha-mu General Fading Distribution[END_REF], [START_REF] Sagias | Gaussian class multivariate Weibull distributions: Theory and applications in fading channels[END_REF]. This paper illustrates a method to model the radio wave propagation in tunnels. After the presentation of several possible techniques found in the literature, we detailed the results obtained using a Ray-Tracing process combined with a tessellation of an arched cross section straight tunnel. The influences of signal frequency and tunnel dimensions were analyzed. Finally, some measurements were performed in real conditions in a road tunnel.

We realized the study on three configurations of straight tunnels. It allows us to observe a « focusing » effect on the received signal for the arched section compared to the rectangular section, even if this phenomenon is less marked at higher frequencies. The study was conducted at 1 GHz, 2.4 GHz and 5.8 GHz. Two dimensions of tunnels were considered: a large tunnel (TGV type) and a small tunnel (mass transit type). An important variation in signal level can be observed for different facets length. A compromise has to be found on the number of facets. It has to be sufficient to represent the geometry of the arched section but not too large to respect the limits of the physical model based on a high frequency approach.

The simulations with the tessellation approach were then compared to measurements performed in an arched cross section straight road tunnel. An analysis on slow fadings and fast fadings is made on the results. The comparisons in terms of slow fading were realized by the calculation of standard deviations. Cumulative Distribution functions were computed on the fast fadings and fitted with classical statistical distributions.

We observed that the results obtained with the tessellation approach depend on the facets size. An optimal facets length cannot be defined. However, a good concordance with measurement results is observed in terms of average signal level and fast fading statistics.

The tessellation of the cross section into multiple planar facets was a first approach to model the wave propagation in non-rectangular tunnels. The results presented for different configurations and different frequencies do not allow us to set up any conclusion in terms of optimal number of facets and optimal length. Consequently, we envisaged as the continuation of this work, to consider another solution, based on ray launching techniques associated to real curved surfaces. At the same time, we envisage to compare the presented simulation results with those obtained with exact methods, such as FDTD on small sections, for computation time reasons. Dr Berbineau field of expertise are EM propagation, channel characterization and modeling for transport environments, signal processing for wireless communication systems, MIMO systems. She is involved in several national and European projects. She is author and co-author of several publications and patents. Dr Berbineau is an IEEE member, affiliated to the VTS society.

Emilie Masson was born in

Figure 3 . 1 :

 31 Figure 3.1: Geometry of tunnels

Figure 3 . 2 :Figure 3 . 3 :

 3233 Figure 3.2: Results for Tunnel C vs. facets size 1 GHz

Figures 4 .

 4 Figures 4.1and 4.3 illustrate simulation results for tunnel C at the two frequencies. Some significant variations can be observed from one result to another. Figures 4.2and 4.4 show, respectively at 2.4 GHz and 5.8 GHz, the results for the 3 tunnels A, B and C (considering a facets size providing the higher signal level).The « focusing » effect observed at 1 GHz is greatly attenuated at 2.4 GHz and 5.8 GHz. The number of facets and the size of the facets have an important effect on the signal level. It has to be noticed that these conclusions concern tunnels of large dimensions (8 m x 5 m), such as TGV tunnels where GSM-R is deployed, or large motorway tunnels. We will then consider tunnels of smaller dimensions corresponding to mass transit tunnels.

  Figures 4.1and 4.3 illustrate simulation results for tunnel C at the two frequencies. Some significant variations can be observed from one result to another. Figures 4.2and 4.4 show, respectively at 2.4 GHz and 5.8 GHz, the results for the 3 tunnels A, B and C (considering a facets size providing the higher signal level).The « focusing » effect observed at 1 GHz is greatly attenuated at 2.4 GHz and 5.8 GHz. The number of facets and the size of the facets have an important effect on the signal level. It has to be noticed that these conclusions concern tunnels of large dimensions (8 m x 5 m), such as TGV tunnels where GSM-R is deployed, or large motorway tunnels. We will then consider tunnels of smaller dimensions corresponding to mass transit tunnels.

Figure 4 . 1 :Figure 4 . 3 :Figure 4 . 4 :

 414344 Figure 4.1: Results for Tunnel C vs. facets size 2.4 GHz

  Figures 4.5, 4.7 and 4.9 illustrate results obtained for tunnel C at the 3 frequencies depending on the facets size. Figures 4.6, 4.8 and 4.10 show the results for tunnels A, B and C (considering a facets size providing the higher signal level) to illustrate the effect of section geometry.Similar conclusions to the previous ones can be made in the case of small tunnels. An important variation is observed on the signal level for different facets sizes. A

  Figures 4.5, 4.7 and 4.9 illustrate results obtained for tunnel C at the 3 frequencies depending on the facets size. Figures 4.6, 4.8 and 4.10 show the results for tunnels A, B and C (considering a facets size providing the higher signal level) to illustrate the effect of section geometry.Similar conclusions to the previous ones can be made in the case of small tunnels. An important variation is observed on the signal level for different facets sizes. A

Figure 4 . 5 :Figure 4 . 6 :

 4546 Figure 4.5: Results for Tunnel C vs. facets size 1 GHz

Figure 4 . 7 : 4 GHzFigure 4 . 8 :

 47448 Figure 4.7: Results for Tunnel C vs. facets size 2.4 GHz

Figure 4 . 9 : 8 GHzFigure 4 . 10 :

 498410 Figure 4.9: Results for Tunnel C vs. facets size 5.8 GHz

Figure 5 Figure 5 . 2 :

 552 Figure 5.1: Tunnel of Roux

Figure 5

 5 Figure 5.3: Measurement results EP1-RP1 configuration

Figure 5 . 5 :Figure 5 . 6 :Figure 5 . 7 :Figure 5 . 8 :

 55565758 Figure 5.5: Simulation results vs. facets size EP1-RP1 configuration -2.4 GHz (1)

Figure 5 . 9 :Figure 5 . 10 :

 59510 Figure 5.9: Measured data and fitted distributions

  Armentières, France, on October 21, 1982. She received the Engineer degree from the Institut Supérieur de l'Electronique et du Numérique, Lille, in 2005. Currently, she works as a PhD student in ALSTOM-Transport (Saint-Ouen) and she is tutored by the LEOST (Transport Electronics and Signal Processing laboratory) of INRETS (French National Institute for research on Transport and Safety) and the Xlim UMR CNRS 6172 SIC Laboratory of the University of Poitiers. She is working in the field of electromagnetic wave propagation in non-rectangular and curved tunnels for railway and mass transit applications. Pierre Combeau was born in Angoulême, France, on June 12, 1978. He received the M.S. degree in mobile radiocommunication and image processing and the Ph.D. degree in signal processing and telecommunications from the University of Poitiers, respectively in 2001 and 2004. Since 2005, he is professor assistant at the University of Poitiers in the department SIC (Signal Image Communications) of the Xlim Laboratory. His fields of expertise include the study of the electromagnetic waves propagation for the SISO and MIMO wireless communication systems. Marion Berbineau was born in Toulouse, France, on September 18, 1962. She received the Engineer degree in electronics, automatic and metrology from Polytech'Lille (France) and the PhD in electronics from the University of Lille respectively in 1986 and 1989. She joined INRETS as a full time researcher in telecommunications in 1989. She is currently Research Director and Director of the LEOST laboratory.

TABLE I .

 I STANDARD DEVIATION (IN DB) OF SIMULATIONS COMPARED TO MEASUREMENTS (EP1-RP1 2.4 GHZ)

	Zone	0m-	0m-	50m-	150m-	350m-
	Facet	500m	50m	150m	350m	500m
	size
	32	5.24	5.48	3.59	5.76 11.33
	22	5.48	3.64	4.85	6.23 10.92
	10	6.50	3.70	3.57	8.97 11.07
	5	6.43	3.73	4.31	8.81 11.19
	3	7.90	3.94	5.83	10.31 12.49
	2	7.08	3.97	5.17	9.29 12.10

TABLE II .

 II STANDARD DEVIATION (IN DB) OF SIMULATIONS COMPARED TO MEASUREMENTS (EP1-RP1 5.8 GHZ)

	Zone	0m-	0m-	50m-	150m-	350m-
	Facet	500m	50m	150m	350m	500m
	size
	77	5.33	5.15	6.09	3.89	6.46
	32	5.07	5.52	3.50	4.84	5.70
	16	6.06	4.42	3.36	7.19	6.10
	8	6.60	4.87	4.25	7.98	6.13
	4	5.81	6.01	3.13	6.54	5.93
	3	5.57	6.42	3.06	6.12	5.88

ACKNOWLEDGMENT

The authors would like to thank the ALSTOM-TIS (Transport Information Solution) who supported this work.