Z Belhachmi
email: zakaria.belhachmi@uha.fr

Z Mghazli

S Ouchtout
email: salih.ouchtout@uha.fr

Mathematical modelling and numerical appproximation of a leachate flow in the anaerobic biodegradation of waste in a landfill

Keywords: AMS subject classifications: 65M10, 78A48 Anaerobic biodegradation, Leachate flow, Biogas production, Coupled problem, Mixed finite element

We consider a coupled PDE-ODEs model that describes the bacterial dynamics of the anaerobic biodegradation of waste in a landfill taken as an unsatureted porous medium. We add a diffusion term in the ODEs part which gives a rise to a reactiondiffusion system coupled with porous media equations. This yields a new model of the biodegradation of the waste where the biological variables are varying in space and time. We perform the mathematical analysis of the resulting system and we consider a discretization based on BDF schemes and P 1 conforming finite elements for the reactiondiffusion subsystem and a mixed finite elements RT 0 /P 0 for the Darcy flow. We carry out the numerical analysis and we obtain the expected a priori estimates. We perform several numerical 2D and 3D simulations which are in agreement with the theoretical results.

Introduction

Waste management and renewable energy generation are two key issues in nowadays societies. A major research field arising in recent years focuses on combining the two aforementioned topics by developing new techniques to handle waste and to use it in the energy production. The anaerobic digestion process is a natural biological process of decomposition of organic mater by microorganisms (bacteria) that are activated under anaerobic conditions, that is to say without oxygen. It is characterized by a succession of complex reactions both in parallel and in series. In the long term, the organic mater is transformed into biogas, a mixture mainly composed of methane and carbon dioxide. The main stages of this process are hydrolysis, acidogenesis, acetogenesis and methanogenesis. Models such as the "ADM1" allow a detailed description of this process, cf. [START_REF] Hassam | A generic and systematic procedure to derive a simplified model from the anaerobic digestion model, No. 1 (ADM1)[END_REF]. However, such complex models are not well suited for mathematical analysis. A large number of simpler models have been investigated in the literature over theses last years [START_REF] Chenu | Modélisation des transferts réactifs de masse et de chaleur dans les installations de stockage de déchets ménagers: application aux installations de type bioréacteur[END_REF][START_REF] Dollé | Mathematical modeling and numerical simulation of a bioreactor landfill using Feel++[END_REF][START_REF] Fekih-Salem | Extensions of the chemostat model with flocculation[END_REF][START_REF] Rapaport | About biomass overyielding of mixed cultures in batch processes[END_REF][START_REF] Rapaport | About biomass overyielding of mixed cultures in batch processes[END_REF][START_REF] Rouez | Dégradation anaérobie de déchets solides: Caractérisation, facteurs d'influence et modélisations[END_REF]. When dealing with the digestion of wastewater, it is recognized that the limiting step is the methanoganesis. In such a case, modelling include one-, two-or three-step models. Of particular interest is the model by Bernard et al., 2001 ( [7]) which proposes to model the anaerobic digestion process as a two-step process involving both the acidogenesis (using a Monod kinetics) and the methanogenesis processes (using a Haldane kinetics). This model, for which the mathematical analysis has been conducted by Benyahia et al., 2012, cf. [6], is very popular, notably for control purposes, since it remains of moderate complexity while being quite easy to calibrate to predict process behaviour with satisfying performances, [START_REF] Arzate | Anaerobic digestion model (AM2) for the description of biogas processes at dynamic feedstock loading rates[END_REF][START_REF] Didi | A Luenberger-type observer for the AM2 model[END_REF] or still [START_REF] Hmissi | Evaluation of alkalinity spatial distribution in an up-flow fixed bed anaerobic digester[END_REF]. It has also been the basis for proposing a systematic way to link simple models to ADM1 predictions, cf. [START_REF] Hassam | A generic and systematic procedure to derive a simplified model from the anaerobic digestion model, No. 1 (ADM1)[END_REF].

The waste landfill is a multiphase medium consisting of solid, liquid and gas phases and is considered as a porous medium. The biodegradation process produces the biogas and leachate in the landfill [START_REF] Pinder | Essentials of multiphase flow and transport in porous media[END_REF] and thus the coupling of the biological activity and the fluid flow. We refer the interested reader to the review paper of Agostini, Sundberg, Navia [START_REF] Agostini | Is biodegradable waste a porous environment? A review[END_REF] on this subject. In this article, we present a new coupled model combining the biological and the mechanical aspects describing respectively the process of the biogas production and the leachate flow during the anaerobic biodegradation of the organic mater in a landfill. We give, as in [START_REF] Ouchtout | Analysis of an anaerobic digestion model in landfill with mortality term[END_REF], the standard two-step approach of the biogas production with two different specific growth rates. We modify the model by transforming the ordinary differential equations part into partial differential equations (PDEs) to obtain a system of diffusion-reaction. The added diffusion allows us to better represent the bacterial dynamic in the spatial domain and to take into account the nonhomogeneous distribution of methanogenese bacteria which is a more realistic assumption in a waste collection sites ( [START_REF] Shi | Coexistence of Competing Species for Intermediate Dispersal Rates in a Reaction Diffusion Chemostat Model[END_REF], [START_REF] Nguyen-Ngoc | Modeling microbial decomposition in real 3D soil structures using partial differential equations[END_REF], [START_REF] Vanrolleghem | Dynamical Modelling and Estimation in Wastewater Treatment Processes[END_REF]). Secondly, we give the full system governing the leachate flow and the biodegradation.

Besides the fact that describing a part of the bacterial dynamic as a diffusion-reaction system leads to more realistic and accurate modelling, it also acts as a regularization procedure for the initial ODEs system for which we have established in [START_REF] Ouchtout | Analysis of an anaerobic digestion model in landfill with mortality term[END_REF] the existence of several (stable and unstable) equilibria.

We consider discrete problems based on high order time discretization and conforming finite elements for the reaction-diffusion system and a mixed finite elements discretization (Raviart-Thomas/P 0 ) for the porous medium equations. We perform their numerical analysis and we obtain optimal a priori error estimates. We present some 2D and 3D numerical results that show the relevance of the new model. This study constitutes an important step towards the more suitable model of the dynamic of the waste in a landfill with multiple phases (gaz/liquid/solid).

The rest of this article is organized as follows: In section 2. we derive the mathematical model governing the chemical and physical phenomena taking place in the waste collection site. In section 3. we establish the existence and uniqueness results for the system and we perform the semi-discretization in time based on the second order BDF2 scheme and we derive the variational formulation for the full problem. Section 4. is devoted to the numerical analysis of the PDEs system. We use different discrete setting between the reaction diffusion subsystem, for which we use P 1 -conforming finite element approximation and the porous medium subsystem that we discretize as mixed variational formulation in the velocity-pressure variables, with Raviart-Thomas RT 0 for the velocity and P 0 finite elements for the pressure. We carry out the numerical analysis and we derive a priori error estimates in the spirits of [19] [5]. In section 5. we present some 2D and 3D numerical simulations and we detail the method for solving the proposed model.The implementation is based on the open source software FreeFem++ [26].

A coupled mathematical model for the biological dynamic and the flow

The bacterial dynamic

The anaerobic digestion system in two-step

The methane production process considered in this article is described as follows: a first step of hydrolysis / acidogenesis of the organic mater, represented by its concentration that we denote X , leads to the formation of carbon dioxide (CO 2 ) and simple soluble organic mater S. The latter is used as a substrate by methanogenic bacteria B which in turn produce carbon dioxide (CO 2 ) as well as methane (C H 4 ). At their death, the methanogenic bacteria in turn constitute a complex substrate to hydrolysis step (see FIGURE [START_REF] Agostini | Is biodegradable waste a porous environment? A review[END_REF]). The principle of mass conservation and the law of bacterial growth, show that the bacterial dynamic and the biogas production are solutions of the following system of ordinary differential equations

                                       d X d t = -K h X + αK d B d B d t = (µ(S) -K d ) B dS d t = f 1 K h X - 1 Y µ(S)B d[CO 2 ] d t = (1 -f 1 )K h X + (1 -f 2 ) 1 -Y Y µ(S)B d[C H 4 ] d t = f 2 1 -Y Y µ(S)B.
(2.1)

The parameters f 1 and f 2 are the stoichiometric coefficients which represent the parts of organic mater (OM) transformed into simple OM or methane during the "hydrolysis / acidogenesis" step or methanogenesis step respectively. Consequently (1 -f 1 ) and (1 -f 2 ) represent the parts of OM transformed into CO 2 during these two steps respectively. We have denoted by K h the rate of hydrolysis, Y the rate of use of the substrate, K d the bacterial mortality rate and α is a constant 0 < α ≤ 1 representing the fraction of the biomass mortality reused as a substrate in the methanogenesis. The dynamic variables of the system (2.1) are the degradable complex material X , the methanogenic biomass B, the soluble organic mater S, the carbon dioxide [CO 2 ] and the methane [C H 4 ]. This model was proposed by Rouez in [START_REF] Rouez | Dégradation anaérobie de déchets solides: Caractérisation, facteurs d'influence et modélisations[END_REF] with Monod law and α = 1. Several laws exist for the specific growth rate µ(S). The most used are the Monod law

µ(S) = µ m S K S + S (2.2)
and the Haldane law

µ(S) = µ m S K S + S + S 2 K I (2.3)
where µ m is the maximum growth rate, K S is the half-saturation constant and K I is the inhibition constant. The Monod law is related to saturation and limitation phenomena and the Haldane law is related to saturation and inhibition phenomena [START_REF] Bernard | Dynamical model development and parameter identification for an anaerobic wastewater treatment process[END_REF]. In this article we will compare the two laws. The system (2.1) allows us to compute the global rate of production of the biogas which constitutes the main part of the source/sink term in the PDEs model of the flow in the waste. The asymptotic behavior analysis of this ODEs system is performed in [START_REF] Ouchtout | Analysis of an anaerobic digestion model in landfill with mortality term[END_REF] and some coupled ODEs-PDEs system in this field are studied in [START_REF] Gallo | A fully coupled numerical model for two-phase flow with contaminant transport and biodegradation kinetics[END_REF] [13] [14] [START_REF] Ghogomu | Simulation du transport réactif dans un milieu poreux discrètement fracturé[END_REF]. In this article, we consider the realistic case of nonhomogenous landfill where we have to take into account the spatial dependency of the bacterial dynamic (represented by the variables of the ODEs system). This leads us to transform the ODEs system into a PDE system of diffusion-reaction form.

The reaction-diffusion PDEs system

Let Ω be an open bounded subset of d , d ≥ 2, representing the landfill. We suppose that its boundary Γ := ∂ Ω is Lipschitz-continuous and is the union of two parts Γ D and Γ N where Dirichlet and Neumann boundary conditions are respectively imposed, with Γ D ∩ Γ N = .

The outward normal vector on Γ will be noted n. For T > 0, the time interval is [0, T ]. We write Q T = Ω×]0, T [ and Γ T = Γ ×]0, T [. The following system is obtained by adding the diffusion terms into the system (2.1) which describe the molecular agitation of the dynamic quantities ( [START_REF] Nguyen-Ngoc | Modeling microbial decomposition in real 3D soil structures using partial differential equations[END_REF], [START_REF] Shi | Coexistence of Competing Species for Intermediate Dispersal Rates in a Reaction Diffusion Chemostat Model[END_REF], [START_REF] Vanrolleghem | Dynamical Modelling and Estimation in Wastewater Treatment Processes[END_REF]).

                 d X d t = -K h X + αK d B + div(D 1 ∇X ) in Ω×]0, T [ d B d t = (µ(S) -K d ) B + div(D 2 ∇B) in Ω×]0, T [ dS d t = f 1 K h X -1 Y µ(S)B + div(D 3 ∇S) in Ω×]0, T [ (2.4)
where D 1 , D 2 and D 3 are, respectively, the diffusion coefficients of X , B and S. The equations on the concentrations CO 2 and C H 4 are not modified because their simple structure shows that they vary in space with the other variables. Therefore, to not increase the computational cost we do not add any diffusion to these equations a priori. Thus, the last equations in the system (2.4), remain unchanged. However, in the numerical section, we simulate and compare the different cases, including with added diffusion to the last variables, (CO 2 ) and (C H 4 ) (complete diffusion vs partial diffusion). Obviously, the analysis and the discretization remain essentially unchanged.We will denote by Dc the complete diffusion, that means the coefficients diffusion for (CO 2 ) and (C H 4 ) are added.

BOUNDARY AND INITIAL CONDITIONS OF THE SYSTEM (2.4)

We fix a time interval of biogas production [0, T ] where T corresponds to the moment when the leachate begins to stagnate at the bottom of the domain, or the moment when it is necessary to recover the biogas.We take the following no-flux boundary conditions associated with the system (2.4)

D 1 ∂ X ∂ n = D 2 ∂ B ∂ n = D 3 ∂ S ∂ n = 0 on Γ ×]0, T [. (2.5)
The initial conditions are given as follows

X (0, x) = X 0 (x), B(0, x) = B 0 (x), S(0, x) = S 0 (x) in Ω, (2.6) [CO 2 ](0, x) = [CO 2 ] 0 (x), [C H 4 ](0, x) = [C H 4 ] 0 (x) in Ω. (2.7)
We set

U = (u 1 , u 2 , u 3 ) t = (X , B, S) t , G = (u 4 , u 5 ) t = ([CO 2 ], [C H 4 ]) t (2.8)
and the diffusion coefficients matrix D is defined as follows

D =   D 1 0 0 0 D 2 0 0 0 D 3   .
The right-hand members of the equations in (2.1) will be denoted by the functions F i , i = 1, 2, .., 5 defined as follows

F 1 (U) = -K h u 1 + αK d u 2 , F 2 (U) = (µ(u 3 ) -K d ) u 2 , F 3 (U) = f 1 K h u 1 - 1 Y µ(u 3 )u 2 , F 5 (U) = f 2 1 -Y Y µ(u 3 )u 2 , F 4 (U) = (1 -f 1 )K h u 1 + (1 -f 2 ) 1 -Y Y µ(u 3 )u 2 ,
and we set the vector functions F 1 and F 2 such that

F 1 (U) = (F 1 (U), F 2 (U), F 3 (U)) T , F 2 (U) = F 4 (U), F 5 (U) T .
(2.9)

The system (2.4) can be written:

                         ∂ U ∂ t -div D∇U = F 1 (U) in Ω×]0, T [ dG d t = F 2 (U) in Ω×]0, T [ D ∂ U ∂ n = 0 on ∂ Ω×]0, T [ U(0, •) = U 0 (•), G(0, •) = G 0 (•) in Ω.
(2.10)

Remark 2.1. From the 2nd, the 4th and the 5th equation of the system (2.1), we remark that

dG d t = F 2 (U) =    (1 -f 1 )K h u 1 + (1 -f 2 ) 1 -Y Y du 2 d t + K d u 2 f 2 1-Y Y du 2 d t + K d u 2    . (2.11)
This remark will be used in the section (3.2) in order to linearize the system (11).

The flow system

The waste is considered as a porous medium consisting of a solid matrix, a liquid phase (leachate) and a gaseous phase formed by a binary mixture of methane and carbon dioxide (biogas). In this article, we neglect the biogas flow and consider only the leachate flow which is governed by the generalized Darcy's Law (low Reynolds number) and the mass conservation equation. The equations are given as follows [START_REF] Richards | Capillary conduction of liquids through porous mediums[END_REF]

∂ (ρθ (p)) ∂ t + div (ρu) = α l in Ω×]0, T [, (2.12
)

u = k i k r (p) µ ρ g(-∇p + e z ) in Ω×]0, T [ (2.13) or ∂ (ρθ (p)) ∂ t + div ρ k i k r (p) µ ρ g(-∇p + e z ) = α l in Ω×]0, T [ (2.14)
where u is the velocity of the leachate, k i its intrinsic permeability, k r the relative permeability of leachate, p its pressure head, µ the dynamic viscosity, ρ the density, g the modulus of the gravitational acceleration, z the vertical axis directed downwards, θ the water content of the leachate and α l is the source/sink term of leachate related to the rate of the biogas generation noted α b which will be described later. There are many empirical models to describe the variation of the water content θ (p) (called the retention curve) and the relative permeability k r (p) in terms of the pressure head p (see the Van Genuchten, Campbel and Brooks Corey model in [START_REF] Van Genuchten | A closed-form equation for predicting the hydraulic conductivity of unsaturated soils 1[END_REF], [START_REF] Campbell | A simple method for determining unsaturated conductivity from moisture retention data[END_REF], [START_REF] Brooks | Hydraulic Properties of Porous Media[END_REF]). We use here the Campbell model given by [START_REF] Campbell | A simple method for determining unsaturated conductivity from moisture retention data[END_REF]

θ (p) = θ s p a p 1/b if p < p a θ s if p ≥ p a (2.15)
where θ s is the saturation water content and p a is a scaling factor. (θ ≈ θ s ), and

k r (p) = p a p 2+3/b if p < p a K s if p ≥ p a
where K s is the saturation permeability, and b is a parameter describing the uniformity of the grain size (see [START_REF] Lanini | Analyse et modélisation des transferts de masse et de chaleur au sein des décharges d'ordures ménagères[END_REF], [START_REF] Aran | Modélisation des Ecoulements de Fluides et des Transferts de Chaleur au Sein des Déchets Ménagers[END_REF], [START_REF] Tinet | Contribution à l'étude des transferts de fluides dans les installations de stockage des déchets non dangereux[END_REF]). SOURCE/SINK TERM The production of the biogas needs water, so we have a sink term in equation (2.12) related to α b . We follow the approach of [3], [START_REF] Bernard | Dynamical model development and parameter identification for an anaerobic wastewater treatment process[END_REF], [START_REF] Agostini | Is biodegradable waste a porous environment? A review[END_REF], [START_REF] Kindlein | Numerical modelling of multiphase flow and transport processes in landfills[END_REF]:

α l = -γ.α b = -γg(ω)C T b λ cin , (2.16)
where

γ = M H 2 O /(3.4 * M b ) with M H 2 O
and M b are respectively the molar mass of H 2 O and of the biogas, ω is the humidity rate

θ = ρ 0 ρ ω,
where ρ 0 is the density of the dry medium. C T b is a positive constant, and λ cin is the rate of the biogas production (see [START_REF] Gallo | A fully coupled numerical model for two-phase flow with contaminant transport and biodegradation kinetics[END_REF] and [START_REF] Gholamifard | Modélisation des écoulements diphasiques bioactifs dans les installations de stockage de déchets[END_REF]) :

λ cin = d C biogaz d t = d([C H 4 ] + [CO 2 ]) d t .
(2.17)

g is an explicit function on the variable ω: the humidity rate. Experimental studies (see [START_REF] Aran | Modélisation des Ecoulements de Fluides et des Transferts de Chaleur au Sein des Déchets Ménagers[END_REF], [START_REF] Gil Diaz | Estimating material losses in sanitary landfills through biological degradation[END_REF]) show that below a minimal value ω min and above a maximum value ω max there is no biogas production. Between these two bounds, it increases, respectively decreases, linearly except on an interval where it is constant (that we normalize to 1), and where the medium is considered as saturated (see FIGURE [START_REF] Alt | Quasilinear elliptic-parabolic differential equations[END_REF]).

BOUNDARY AND INITIAL CONDITIONS

The boundary conditions associated with the system (2.12)-(2.13) are expressed in term of charges (Dirichlet conditions) on Γ D :

p = p D on Γ D ×]0, T [, (2.18)
and in term of flux (Neumann conditions) on Γ N :

u.n = 0 on Γ N ×]0, T [. (2.19)
The initial condition is given by

p(x, 0) = p 0 (x) in Ω. (2.20)
We introduce the capillary capacity (p) by rewriting the time derivative of the water content as follows

∂ θ ∂ t = ∂ θ ∂ p ∂ p ∂ t = (p) ∂ p ∂ t
and by reversing the relative permeability k r and the density ρ, we set

k(p) = µ l k i ρ g (k r (p)) -1 and f = ρ -1 α l .
The flow problem of leachate can be written finally as follows:

                           (p) ∂ p ∂ t + div u = f in Ω×]0, T [, k(p)u = -∇p + e z in Ω×]0, T [, p = p D on Γ D ×]0, T [, u.n = 0 on Γ N ×]0, T [, p(x, 0) = p 0 (x) in Ω.
(2.21)

Solving this problem under mixed formulation is more suited in our case than solving Richards equation where the velocity is eliminated. In fact, it allows us to have more accuracy for u and in following the leachate variation during the production of biogas. Therefore, the unknowns in this formulation are both the velocity u and the pressure p.

The final Model

The final mathematical model describing the degradation of the organic mater and the leachate flow during the anerobic process is given by:

(S1)                          ∂ U ∂ t -div D • ∇U = F 1 (U) in Ω×]0, T [, dG d t = F 2 (U) in Ω×]0, T [, D ∂ U ∂ n = 0 on ∂ Ω×]0, T [, U(0, •) = U 0 (•), G(0, •) = G 0 (•) in Ω, (2.22)
and

(S2)                            (p) ∂ p ∂ t + div u = f in Ω×]0, T [, k(p)u = -∇p + e z in Ω×]0, T [, p = p D on Γ D ×]0, T [, u.n = 0 on Γ N ×]0, T [, p(x, 0) = p 0 (x) in Ω.
(2.23)

Variational formulation and semi-discrete system

In what follows, We make some assumptions:

Hypothesis 3.1. Coefficients α, K d , Y , f 1 and f 2 fulfill the following conditions.

1. The proportion of nutrient recycling α cannot exceed 1

0 < α ≤ 1 (3.1)
2. The mortality rate K d is a positive parameter which is below the maximum growth rate 0

< K d < max s µ(s) (3.2)
3. The rate of use of the substrate is a strictly positive parameter such that

0 < Y < 1 (3.3)
4. The stoichiometric coefficients parameters f 1 and f 2 are strictly positive and satisfy

0 < f 1 < 1 and 0 < f 2 < 1. ( 3
< c θ ≤ θ (p) ≤ C θ , 0 < c p ≤ (p) ≤ C p , 0 < c k ≤ k(p) ≤ C k (3.5)
The final model obtained in (2.22)-(2.23) is weakly coupled, therefore, we can separate the analysis of the system (2.22) and (2.23).

Existence and uniqueness for problems (S1) -S(2)

We recall the definitions and notations used in this section. The usual Sobolev space

H 1 (Ω) = {v ∈ L 2 (Ω) / ∇v ∈ (L 2 (Ω)) d },
and the trace operator γ 0 : H 1 (Ω) -→ L 2 (Γ ). The trace space is

H 1/2 (Γ ) := γ 0 (H 1 (Ω))
and if Γ D is a part of the boundary of Ω we denote

H 1 Γ D (Ω) = {φ ∈ H 1 (Ω), φ = 0 on Γ D }.
We recall also that H -1 (Ω) is the dual of H 1 0 (Ω). The space L 2 (0, T ; H 1 (Ω)) is the set of measurable functions

u :]0, T [-→ H 1 (Ω) such that T 0 u(t) 2 H 1 (Ω) d t < +∞, equipped with the norm u L 2 (0,T ;H 1 (Ω)) =   T 0 u(t) 2 H 1 (Ω) d t   1/2 . L ∞ (Q T ) is the space of measurable functions u :]0, T [×Ω -→ such that |u(t, x)| is bounded almost everywhere in Q T , equipped with the norm u L ∞ (Q T ) = inf {C, |u(t, x)| < C in Q T } . Finally, C([0, T ], L 2 (Ω)) is the space of continuous functions from [0, T ] with values in L 2 (Ω) and L ∞ (Ω, 3 + ) is the space of positive measurable functions u : x ∈ Ω -→ u(x) ∈ 3
+ such that u i (x) is bounded almost everywhere in Ω. Due to the fact that the specific growth rate µ(•) is a nonnegative and bounded function in both Monod and Haldane cases, and under hypothesis (3.1)-(3.2), it is readily checked that assumptions of Theorem 1 in [START_REF] Bothe | Global wellposedness for a class of reaction-advection-anisotropic-diffusion systems[END_REF] are verified and we have the following proposition. Proposition 3.1. For any U 0 ∈ L ∞ Ω, 3 + , the system (2.10) has a unique global nonnegative weak solution U = (u 1 , u 2 , u 3 ) T in the following sense :

∀T > 0 ∀i ∈ {1, 2, 3}, u i ∈ C [0, T ]; L 2 (Ω) ∩ L ∞ (Q T ) ∩ L 2 0, T ; H 1 (Ω) ; ∀ψ ∈ C ∞ (Q T ) such that ψ(T ) = 0 - Ω u 0 i ψ(0) d x - Q T u i ∂ ψ ∂ t d x d t + Q T D i ∇u i • ∇ψ d x d t = Q T F i (U(t, x)). • ψ d x d t                    (3.6)
Moreover, for any T > 0, there exists C > 0 such that

U L ∞ (Q T ) 3 + U L 2 (0,T ;H 1 (Ω) 3 ) + ∂ t U L 2 (0,T ;H -1 (Ω) 3 ) ≤ C. (3.7)
Next, eliminating the velocity u in the Darcy equations, we obtain the well-known Richards equation on the pressure p ((2.14)). The existence and uniqueness for this parabolic PDE is obtained in [START_REF] Alt | Quasilinear elliptic-parabolic differential equations[END_REF]. More precisely, we have Proposition 3.2. For all p 0 ∈ L 2 (Ω) and p D ∈ H

Semi-discretization in time and variational formulation of the system (S1)

Let τ n = t n+1 -t n , n = 0, ..., N , where N ∈ * , be a partition of [0, T ] with t 0 = 0 and t N = T . In what follows we denote by f n the values of a function f at t n . The semi-discrete scheme for the system (S1) considered here is the second order backward differentiation formula BDF2 for the variables (X , S, B) and the implicit Euler scheme for the ODEs on ([CO 2 ], [C H 4 ]). The problem reads: find

U n+1 ∈ Z = H 1 (Ω) 3 such that Ω 3U n+1 -4U n + U n-1 2τ n • vd x + Ω D∇U n+1 • ∇vd x = Ω F 1 U n+1 • vd x ∀v ∈ Z (3.8)
where the initial values are computed as follow: find

U 1 ∈ Z such that Ω U 1 -U 0 τ 0 • vd x + Ω D • ∇U 1 • ∇vd x = Ω F 1 U 1 • vd x ∀v ∈ Z. ( 3.9)
We linearize the scheme by using the Taylor formula

F 1 U n+1 F 1 (U n ) + J F 1 (U n ) • (U n+1 -U n ) (3.10)
where J F 1 is the Jacobian matrix of the vector function F 1 . Consequently, the final BDF2 scheme reads : find

U 1 ∈ Z = H 1 (Ω) 3 such that Ω τ -1 0 I 3 -J F 1 U 0 • U 1 • vd x + Ω D • ∇U 1 • ∇vd x = Ω τ -1 0 U 0 + F 1 U 0 -J F 1 U 0 • U 0 • vd x ∀v ∈ Z, ( 3.11)
and for all n ≥ 1, find

U n+1 ∈ Z such that Ω 3 2τ n I 3 -J F 1 (U n ) • U n+1 • vd x + Ω D • ∇U n+1 • ∇vd x = Ω 4U n -U n-1 2τ n + F 1 (U n ) -J F 1 (U n ) • U n • vd x ∀v ∈ Z. (3.12)
On the other hand, by using the Remark 1, the approximation of G at t = t m+1 G m+1 = (u m+1 4 , u m+1

5

) T , for ,0 ≤ m ≤ n, is given by

u m+1 4 -u m 4 τ m = (1 -f 1 )K h u m+1 1 + (1 -f 2 ) 1 -Y Y u m+1 2 -u m 2 τ m + K d u m+1 2 , (3.13
)

u m+1 5 -u m 5 τ m = f 2 1 -Y Y u m+1 2 -u m 2 τ m + K d u m+1 2 .
(3.14)

Semi-discretization in time and variational formulation of the system (S2)

We use the semi-implicit scheme for (S2) :

k(p n )u n+1 = -∇p n+1 + e z in Ω, ( 3.15)
(p n ) p n+1 -p n τ n + div u n+1 = f n+1 in Ω, ( 3.16)
where the unknowns u n+1 and p n+1 refer to the approximation of u(t n+1 ) and p(t n+1 ) with

u n+1 • n = 0 on Γ N and p n+1 = p D on Γ D . (3.17)

MIXED VARIATIONAL FORMULATION

We introduce the following functional spaces

H(div, Ω) = v ∈ (L 2 (Ω)) d , div v ∈ L 2 (Ω) , V = H 0,N (Ω) = {v ∈ H(div, Ω), v.n = 0 on Γ N }, W = L 2 (Ω),
equipped with the norms

∀v ∈ V, v V = v H(div,Ω) = v 2 0,Ω + div v 2 0,Ω 1/2 (3.18) and ∀ϕ ∈ W, ϕ W = ϕ 0,Ω
where • 0,Ω is the L 2 -norm and we denote the dual spaces of V and W by V and W respectively. The weak formulation of (3.15)-(3.16) with the conditions (3.17) is now written as follows:

               find u n+1 ∈ V and p n+1 ∈ W such that Ω k(p n )u n+1 • ψd x - Ω div ψp n+1 = - Γ D p D ψ • nds + Ω e z • ψd x, ∀ψ ∈ V, Ω div u n+1 ϕd x + Ω (p n ) τ n p n+1 ϕd x = Ω (p n ) τ n p n ϕd x + Ω f n+1 ϕd x, ∀ϕ ∈ W,
which can be written in the following form

         find u n+1 , p n+1 ∈ V × W such that a u n+1 , ψ + b ψ, p n+1 = g 1 (ψ) , ∀ψ ∈ V, b u n+1 , ϕ -d p n+1 , ϕ = g 2 (ϕ) , ∀ϕ ∈ W, (3.19)
where a :

V × V -→ , b : V × W -→ and d : W × W -→ are the bilinear forms defined by a (u, ψ) = Ω k(p n )u • ψd x ; (u, ψ) ∈ V × V (3.20) b (u, ϕ) = - Ω div uϕ ; (u, ϕ) ∈ V × W (3.21) d (p, ϕ) = Ω (p n ) τ n pϕd x ; (p, ϕ) ∈ W × W (3.22)
and the second members g 1 : V -→ and g 2 : W -→ are the linear forms defined by

g 1 (ψ) = - Γ D p D ψ • nds + Ω e z • ψd x ; ψ ∈ V (3.23) g 2 (ϕ) = - Ω (p n ) τ n p n ϕd x - Ω f n+1 ϕd x ; ϕ ∈ W. (3.24)
We define the continuous operator B : u ∈ V -→ div u ∈ W and its transpose

B t : W -→ V such that < Bv, ϕ > W ×W =< v, B t ϕ > V ×V = b(v, ϕ) ∀v ∈ V, ∀ϕ ∈ W,
and then we introduce the subspace of V :

Ker B = {v ∈ V, ∀ϕ ∈ W, b(v, ϕ) = 0} .

Existence and uniqueness of the solution of (3.19)

Proposition 3.3. For all p D ∈ H 1/2 (Γ D ) and f ∈ L 2 (Ω) the system (3.19) has a unique solution, and we have the estimate

u n V ≤ C 1 g 1 V + C 2 g 2 W and p n W ≤ C 2 g 1 V + C 3 g 2 W (3.25)
with

C 1 = (1 + C 2 Ω ) -1 + 4τ -1 n-1 C(p n-1 )C k c k (1 + C 2 Ω ) -1 , C 2 = 2C 1/2 k c 1/2 k (1 + C 2 Ω ) -1/2 , C 3 = 4C k τ -1 n-1 C(p n-1 ) + 2(1 + C 2 Ω ) -1
, where C Ω is the Poincaré constant.

Proof. The bilinear form a(•, •) is symmetric, continuous and KerB-elliptic according to the hypothesis (3.2) and since in KerB, the norm (3.18) is reduced to the L 2 -norm. The bilinear form d(•, •) is symmetric, continuous and W -elliptic

|d (p, ϕ) | ≤ γ 1 p W ϕ W ∀(p, ϕ) ∈ W × W, d(ϕ, ϕ) ≥ γ 2 ϕ 2 W ∀ϕ ∈ W, with γ 1 = C p τ n and γ 2 = c p τ n . ( 3
   -∆v = ϕ in Ω, v = 0 on Γ D , ∇v.n = 0 on Γ N ,
we deduce in standard way that b(•, •) satisfies the in fsup condition (e.g. [START_REF] Boffi | Mixed finite element methods and applications[END_REF])

∃β = 1 1 + C 2 Ω > 0; ∀ϕ ∈ W, sup u∈V b(u, ϕ) u V ≥ β ϕ W .
Therefore, the existence follows from Brezzi-Babuska theorem.

Discrete problem and analysis

We assume in this section that Ω is a polygonal domain of d . Let h be a partition of Ω into triangles T (in 2 ) or tetrahedra (in 3 ) and we denote by P k (O) the space of polynomial functions defined in a subset O of d of total degree at most k. In order to define the full discritization of the problem (2.22)-(2.23), we introduce the following finite-dimensional space

Z h = v h ∈ C( Ω) 3 ; ∀T ∈ h v h | T ∈ (P 1 (T )) 3 . (4.1)
We recall the definition of the local Raviart-Thomas space of lowest order (see [START_REF] Brezzi | Mixed and hybrid finite element methods[END_REF]), ∀T ∈

h RT 0 (T ) := v h ∈ H(div, T ) : v h ∈ P d 0 (T ) ⊕ xP 0 (T ), x ∈ T , (4.2)
and the global Raviart-Thomas space of lowest order is given by

RT 0 ( h ) := {v h ∈ H(div, Ω) : v h | T ∈ RT 0 (T ), ∀T ∈ h } . ( 4.3)
We introduce the discrete spaces

V h = v h ∈ RT 0 ( h ) v h • n |Γ N = 0 . (4.4)
and

W h := q h ∈ L 2 (Ω) : q h | T ∈ P 0 (T ), ∀T ∈ h . (4.5)

Full discretization of the system (S1)

The discrete problem associated to (3.11)-(3.14) reads : find

U 1 h ∈ Z h such that Ω τ -1 0 I 3 -J F 1 U 0 • U 1 h • vd x + Ω D • ∇U 1 h • ∇vd = Ω τ -1 0 U 0 + F 1 U 0 -J F 1 U 0 • U 0 • vd x ∀v ∈ Z h , (4.6)
and for all n ≥ 1, find

U n+1 h ∈ Z h such that Ω 3 2τ n I 3 -J F 1 U n h • U n+1 h • vd x + Ω D • ∇U n+1 h • ∇vd x = Ω 4U n -U n-1 2τ n + F 1 (U n ) -J F 1 (U n ) • U n • vd x ∀v ∈ Z h . (4.7)
On the other hand, we seek

G m+1 h = (u m+1 4h , u m+1 5h ) T , ∀m, 0 ≤ m ≤ n, such that u m+1 4h -u m 4h τ m = (1 -f 1 )K h u n+1 1h + (1 -f 2 ) 1 -Y Y u m+1 2h -u m 2h τ m + K d u m+1 2h , (4.8
)

u m+1 5h -u m 5h τ m = f 2 1 -Y Y u m+1 2h -u m 2h τ m + K d u m+1 2h . (4.9)
Remark 4.1. In practice we choose m = n in our numerical simulation to solve in one step the discrete problem. Nevertheless, we write under this general form because it may be suitable for some applications two have to time grid for the ODEs part and the reactiondiffusion term.

Full discretization of the system (S2)

The mixed finite element method formulation of (3.19) reads Proof. We denote B h :

         find u n+1 h , p n+1 h ∈ V h × W h such that a u n+1 h , ψ h + b ψ h , p n+1 h = g 1 (ψ h ) ∀ψ h ∈ V h b u n+1 h , ϕ h -d p n+1 h , ϕ h g 2 (ϕ h ) ∀ϕ h ∈ W h
V h -→ W h the operator b(v h , ϕ h ) =< B h v h , ϕ h > W h ×W h ∀v ∈ V h , ∀ϕ h ∈ W h
and we introduce the kernel of B h :

Ker B h = {v h ∈ V h , ∀ϕ h ∈ W h , b(v h , ϕ h ) = 0} .
Let P h be the orthogonal projection operator from W to W h , then for all v h ∈ V h and q ∈ W Thus, Ker B h ⊂ Ker B, and a(•, •) is uniformly (Ker B h )-elliptic: there exists a constant α 0 independent of h such

a(u h , u h ) ≥ α 0 u h 2 V ∀u h ∈ Ker B h . ( 4.11)
Moreover, the discrete inf-sup condition follows from the continuous one. There exists a constant β > 0 independent of h such that inf

v h ∈V h sup w h ∈W h b(v h , w h ) v h V w h W ≥ β
We deduce the existence and uniqueness of the solution of (3. Proof. Combining the ellipticity of the bilinear form a(., .) and the uniform discrete inf-sup condition on b(., .) it is standard to derive the abstract estimate

u -u h V + p -p h W ≤ C inf v h ∈V h u -v h V + inf q h ∈W h p -q h W .
Using the interpolation operators Raviart-Thomas operator Π h and the the orthogonal L 2projection operator P h , we get ( [START_REF] Girault | Finite element methods for Navier-Stokes equations: theory and algorithms[END_REF], [START_REF] Brezzi | Mixed and hybrid finite element methods[END_REF])

u -u h V + p -p h W ≤ Ch |u| 1,Ω + |div u| 1,Ω + |p| 1,Ω .

Numerical results

In this section we present some numerical results for the system (S 1 )-(S 2 ). In all the experiments, we take

Ω =]0, 1[×]0, 1[×]0, 1[⊂ 3 , ∂ Ω = Γ D ∪ Γ N and Γ D = {(x, y) ∈ ]0, 1[×]0, 1[
and z = 1} for 3D simulations and Ω = {(x, y) ∈]0, 1[×]0, 1[ and z = 0}, with the corresponding boundaries for 2D simulations. The various constants used are fixed as in [START_REF] Rouez | Dégradation anaérobie de déchets solides: Caractérisation, facteurs d'influence et modélisations[END_REF] and [START_REF] Aran | Modélisation des Ecoulements de Fluides et des Transferts de Chaleur au Sein des Déchets Ménagers[END_REF] and given in TABLE [START_REF] Agostini | Is biodegradable waste a porous environment? A review[END_REF], and from the same references, we take the initial conditions for the biodegradation system (in mgC/L) as given in TABLE [START_REF] Alt | Quasilinear elliptic-parabolic differential equations[END_REF]. The boundary conditions for the biodegradation system are all homogeneous Neumann conditions (see (2.5)). The boundary and initial conditions for the flow system are given in TABLE [START_REF] Aran | Modélisation des Ecoulements de Fluides et des Transferts de Chaleur au Sein des Déchets Ménagers[END_REF].

We recall that, at each time step, the system (S 1 ) is approximated by (4.6)-(4.9), and its solution gives the source term in the flow system (S 2 ), approximated by (4.10). the examples, except (5.4), we consider only the system (S1) for comparisons with existing results in the literature. This is not a restriction due to the weak coupling of (S1) and (S2) in the present model. We first compare the numerical results obtained with the reaction-diffusion system (S 1 ) and the ones obtained in [START_REF] Rouez | Dégradation anaérobie de déchets solides: Caractérisation, facteurs d'influence et modélisations[END_REF] and in [START_REF] Ouchtout | Analysis of an anaerobic digestion model in landfill with mortality term[END_REF]. In example (5.2), we use different values of the diffusion coefficients in order to show the effect on the final solution. Example (5.3) consists of solving (S 1 ) with the partial diffusion and with the complete diffusion (on all the variables of (S1)). The last example consists of solving the full system (S1) -(S2).

Model verification

In this example, we denote Dval = diag (D X , D S , D B ), with D X = 0.1, D S = 0.3 and D B = 0.5, the diffusion matrix. We set B 0 = 2 and we use Monod law, and we plot in FIGURE (3) the evolution in time of all the variables. We notice that the results are identical to similar numerical experiments in the literature for (S1) without diffusion (e.g. [START_REF] Rouez | Dégradation anaérobie de déchets solides: Caractérisation, facteurs d'influence et modélisations[END_REF]). Next, we use Haladane law and we represent in FIGURE (4) the same curves but with different values of X 0 , which also confirms the previous results. Moreover, in FIGURE [START_REF] Belhachmi | Finite element approximation of the Neumann eigenvalue problem in domains with multiple cracks[END_REF] we plot the curve of the biogas [C H4]+[CO2] as a function of the initial value X 0 and we obtain a discontinuity, already noticed in [START_REF] Ouchtout | Analysis of an anaerobic digestion model in landfill with mortality term[END_REF]. In fact, if X 0 < 475 the curve is monotonically increasing otherwise, it is decreasing. The explanation of this behaviour of the dynamical system is thouroughly detailed in [START_REF] Ouchtout | Analysis of an anaerobic digestion model in landfill with mortality term[END_REF] and the added diffusion do not change qualitatively the expected dynamic. In particular, Haladane law, usually, considered as more realistic (and the Monod function can be seen as a particular case for large values of the parameter K I , see FIGURE [START_REF] Benyahia | Bifurcation and stability analysis of a two step model for monitoring anaerobic digestion processes[END_REF]) yields more accurate predictions on the biogas production. Figures [START_REF] Bernard | Dynamical model development and parameter identification for an anaerobic wastewater treatment process[END_REF] and [START_REF] Boffi | Mixed finite element methods and applications[END_REF] show that the lack of methanogenic bacteria in a given area (i.e. B 0 = 0 in 1 ), induces nul biogas production in Σ 1 and sharp transition between Σ i , i = 1, 2, 3 in the case of no diffusion, whereas with the additional diffusion, the transition become more diffuse and biogas production expand to the entire domain.

Comparison of the partial diffusion D and the complete diffusion Dc .

In this example, we compare the partial diffusion with the complete diffusion case where D CO 2 = 0 and D C H 4 = 0. We set D5 = {D X = 0.005, D S = 0.007, D B = 0.009} In FIGURES ( 9) and ( 10), we notice that the concentration of C H 4 is more smooth (regularizing effect) but its average value do not increases. Besides, the CPU time is smaller with the partial diffusion, (7.605s in 2D and 71s in 3D), than with the complete one (8.7s in 2D, and 103s in 3D). Therefore the partial diffusion seems more suitable choice in 3D.

Solution of the full problem (S1)-(S2)

We turn now to the full system (S1)-(S2) and we take Haldane law and D3 as a diffusion term. We set B 0 B 0 (x, y) =    1 in {0 ≤ x ≤ 0.5 , 0.5 ≤ y ≤ 1 and 0.5 ≤ z ≤ 1}, 2 in {0.5 ≤ x ≤ 1 , 0.5 ≤ y ≤ 1 and 0.5 ≤ z ≤ 1}, 3 in {0 ≤ x ≤ 1 n 0 < y < 0.5 and 0 ≤ z < 0.5}. Figures [START_REF] Campbell | A simple method for determining unsaturated conductivity from moisture retention data[END_REF], ( 13) and [START_REF] Celia | Contaminant transport and biodegradation: 1. A numerical model for reactive transport in porous media[END_REF] show respectively, the velocity field of the Darcy flow, the pressure head, and the water content (via the function θ (p)). We notice that after 2 days there is a stagnation zone of the leachate (i.e the liquid phase) in the bottom indicating that the medium is saturated therefore there is no biogas production there (we recall that the humidity rate ω lies between ω 1 and ω 2 ). It follows that injecting water in the landfill will improve the biogas production which should be addressed in a more precise way with an optimal control approach. This also suggests to investigate more deeply the role of the humidity rate ω (see Figure

Conclusion

In this article we have considered a coupled system of ODE-PDEs representing the bacterial activity in a landfill taken as an unsaturated and inhomogeneous porous medium. We have introduced new diffusion terms in the ODEs system in order to take into account the spatial evolution of the bacterial dynamic. We analyzed the new model of reactiondiffusion coupled with Darcy flow system and we carried out the analysis of the discrete systems with finite element method. We obtained well-posedness and a priori error estimates for the discrete problems. Finally, we have performed several numerical 2D and 3D simulations which are in agreement with the theory and the literature. In particular, we obtain that the introduction of the additional diffusion in the bacterial dynamic improves the existing models and enforces the accuracy of the coupling with the porous medium flow. The discrete problems with a BDF2 scheme and a mixed finite element formulation with RT 0 /P 0 constitute a highly efficient numerical method to solve such a complex problem. In a forthcoming paper we extend this approach to a two phase flow where the gaseous phase is included in the model. Moreover, for more realistic models in waste management and renewable energy production, the transport of the biogas should be considered and the approach followed in this article seems to be the right way to do so.

Figure 1 :

 1 Figure 1: Scheme used for modelling anaerobic degradation of organic mater.

Figure 2 :

 2 Figure 2: The empirical function g(ω).

. 4 ) 3 . 2 .

 432 HypothesisThe functions p -→ θ (p), p -→ (p) = ∂ θ ∂ p and p -→ k(p) are uniformly bounded, i.e. there exist nonnegative constants c θ , C θ , c θ , C θ , c k and C k such that 0

1 2 (

 2 Γ D ), There exists a unique weak solution pp D ∈ L 2 (0, T ; H 1 Γ D (Ω)) of the Richards equation (2.14).

. 26 )

 26 Moreover, the bilinear form b(•, •) is continue on V×W , and applying Lax-Milgram theorem to the auxiliary problem

(4. 10 )

 10 EXISTENCE AND UNIQUENESS FOR THE DISCRETE PROBLEMSProposition 4.1. Problem (4.10) admits a unique solution.

p

  0 in Ω p D on Γ D u • n on Γ N examples, less sensitive to the dimension, are 2D simulations for comparison purposes. The two last examples are the preliminary 3D simulations. Notice that in all

Figure 3 : 1 TimeFigure 6 :

 316 Figure 3: Evolution of the averages variables of (S1) using the Monod law and the diffusion Dval and B 0 = 2, X 0 = 1751, α = 1

Dc 5 =

 5 D X = 0.005, D S = 0.007, D B = 0.009, D CO 2 = 0.009, D C H 4 = 0.009

  (a)-D = D0 (no diffusion term) (b)-D = D2

Figure 7 :

 7 Figure 7: The production of methane after 2 days using the Haldane law without the diffusion in (a) and with the diffusion D2 in (b).

Figure 8 :

 8 Figure 8: The production of methane after 2 days using Haldane law with the diffusion D3 in (c) and D4 in (d).

D5 Dc 5 Figure 9 : 5 Figure 10 :

 59510 Figure 9: The production of methane after 2 days using Haldane law with D5 and Dc 5 in 2D.

  (

  2)) in the modelling. (a) 1 day (b) 2 days

Figure 11 :

 11 Figure 11: Evolution of biogas production with D3 and Haldane law.

Figure 12 :

 12 Figure 12: The velocity of leachate (liquid phase) with D3 and Haldane law.

Figure 13 :

 13 Figure 13: Evolution of the pressure head p with D3 and Haldane law.

Figure 14 :

 14 Figure 14: The water content evolution with D3 and Haldane law.

Table 1 :

 1 Parameters.

	K H	µ m	f 1	f 2	K S	K d	Y	α	K I	µ l
	0.176 0.3 0.7 0.76 160 0.04 0.05 0.9 500 4.61027 × 10 -4
	θ r	θ s	k i	p a	b	M b (g/mol) M H 2 O (g/mol) A m C T b (m 3 /K g)
	0.27 0.9715 10 -4 -0.0323 2.5	30			18.01	0.8	0.178

Table 2 :

 2 Initial conditions for the biodegradation system.

	X 0 (x, y) S 0 (x, y) [CO 2 ] 0 (x, y) [C H 4 ] 0 (x, y)
	1751	0	0	0

Table 3 :

 3 Initial and boundary conditions for the flow system.

Table 4 :

 4 Four days evolution of the average quantity of methane using different values of diffusion with the Haldane law.

	Diffusion Di	0.5 day	1 days	1.5 days	2 days
	D0 (no diffusion term) 1.60651 4.48873	7.89388	11.5530
	D2	1.60671 4.48927	7.89465	11.5537
	D3	1.60737 4.49098	7.89703	11.5560
	D4	1.60949 4.49647	7.90468	11.5633

Acknowledgments

The authors are gratefully for the French-Moroccan exchange program "PHC TOUBKAL 17/47".

Effect of the diffusion parameters on the biogas production

In this example, we use and compare several choices of the diffusion coefficients on the solution of the reaction-diffusion system. We set In TABLE (4), we observe that the values of the diffusion coefficients have a negligible effect on the average values of the biogas production.


domain: spi phys phys.meca

text:
Y Medghaghet

O Bennis

F Benchabane
email: fateh_benchabane@yahoo.fr

A Titaouine

A Guettaf

Characteristics of Speed Sensorless Vector Controlled Induction Motor with High Efficiency Taking Core Loss into Account

Keywords: Induction motor, Core loss, Direct field oriented control, Estimation, Extended Kalman filter, High efficiency

This paper presents the characteristics of a speed sensorless vector control method of an induction motor operating at high efficiency and high response, in which core loss is taken into account. An improved method of speed estimation that operates on the principle of speed adaptive flux and current observer has been proposed. An observer is basically an estimator that uses a plant model and a feedback loop with measured stator voltage and current. Simulation results show that the proposed direct field oriented control with the proposed observer provides good performance characteristics. The IM is fed by an indirect power electronics converter. This indirect converter is controlled by a pulse width modulation PWM technique that enables minimization of harmonics introduced by the line converter, as well as the control of the power factor and DC-link voltage. We study the robustness of the overall system-using simulation of different modes to operating modes and varied parameters.

INTRODUCTION

The conventional field-oriented method of controlling an induction motor ignores core loss. However, in practice, core loss causes the rotor current and rotor flux to interfere with each other [START_REF] Matsuse | High-Response Flux Control of Direct-Field-Oriented Induction Motor with High Efficiency Taking Core Loss into Account[END_REF]. This makes the output torque different from the reference torque. To solve this problem, we have also proposed a method of compensating for core loss to ensure that the rotor flux and rotor current do not affect each other.

Research interest in induction motor (IM) sensorless drives has grown significantly over the past few years due to some of their advantages, such as mechanical robustness, simple construction, and maintenance. Present efforts are devoted to improve the sensorless operation, especially for low speed and to develop robust control strategies.

Direct and indirect vector control methods for the speed and torque control of IM have found intensive application through the last two decades. For the indirect control of IM, in addition to the rotor speed, accurate knowledge of the slip frequency (calculated as a function of the IM parameters) is required. On the other hand, direct control of IM necessitates accurate information on the rotor speed, as well as rotor flux as referred to the stator stationary frame [START_REF] Barut | An Extended Kalman Filter Based Sensorless Direct Vector Control of Induction Motors[END_REF][START_REF] Sturtzer | Modélisation et commande des moteurs triphasés, commande vectorielle des moteurs synchrones, commande numérique par contrôleurs DSP[END_REF][START_REF] Ouhrouche | Application of an Extended Kalman Filter to Rotor Speed and Resistance Estimation in Induction Motor Vector Control[END_REF].

However, the possibility of using the EKF structure but making a direct parameterization of the observer gain matrix. Variants of this approach has been suggested by e.g. by Kubota et al [START_REF] Kubota | DSP-based speed adaptive flux observer of induction motor[END_REF], Nilsen and Kaimierkowski [START_REF] Nilsen | Reduced-order observer with parameter zdaption for fast rotor flux estimation in induction machines[END_REF], Peterson [START_REF] Peterson | Induction machine speed estimation: observations on observers[END_REF], and Harnefors and Nee [START_REF] De Doncker | The universal field oriented controller[END_REF].

This paper present a method of estimation of speed and flux of induction motors with taking core loss into account based on EKF. The paper is organized as follows. Section 2 describes the model of induction motor model with taking core loss into account. In Section 3 an estimator using EKF for the speed and flux is designed. Simulations of the scheme are carried out in Section 4 and some conclusions are given in Section 5.

INDUCTION MOTOR MODEL

For the purpose of using EKF for the estimation of the rotor flux of an induction machine; it is possible to use various machine models. For example, it is possible to use the equations expressed in the rotor flux-oriented reference frame, or in stator flux-oriented reference frame. In order to avoid extra calculations and non-linear transformations, stationary reference frame is preferred. The main advantages of using the model in stationary reference frame are reduced computation time, smaller sampling time, higher accuracy, more stable behaviour [START_REF] Matsuse | High-Response Flux Control of Direct-Field-Oriented Induction Motor with High Efficiency Taking Core Loss into Account[END_REF]. Thus, we have chosen stationary reference frame in our study. So, the proposed model for induction motor with high efficiency taking core loss into account is shown in fig 1 .  Assuming that core loss occurs due to the eddy current, the induction motor model can be as shown in Fig. 1. In this figure, eddy current is assumed to flow in d, q winding. The formula for the voltage of an induction motor in which stator core loss is taken into consideration is given by (l), in the rotating reference frame [START_REF] Matsuse | High-Response Flux Control of Direct-Field-Oriented Induction Motor with High Efficiency Taking Core Loss into Account[END_REF][START_REF] Kubota | DSP-based speed adaptive flux observer of induction motor[END_REF]

.i .i ) 2L = φ -φ (2)
The core loss resistance is function of frequency and flux level. But the change of R f , value depends on the variation of frequency more than that of rotor flux. Therefore, R f is approximately a function of frequency and gains A, B.

2 f R A.f B.f = + (3)
Where:

2 r s M 1 L L δ = - , r r r L T R = , s s s L T R =
The model of the induction motor takes core loss into account. Expressed in the Park reference frame is given in the suitable state form [START_REF] Barut | An Extended Kalman Filter Based Sensorless Direct Vector Control of Induction Motors[END_REF][START_REF] Kubota | DSP-based speed adaptive flux observer of induction motor[END_REF]

L d i R d M V (R R )i L L d t L L d t d L d i R M V (R R )i L L d t L L d t L d R R M 0 ( R )i L T L d t d L R R M 0 ( R )i L T L d t δ δ δ δ φ ⎧ = + +δ + φ + ⎪ ⎪ ⎪ φ = + +δ + φ + ⎪ ⎪ ⎨ φ + ⎪ = - + φ + +ωφ ⎪ ⎪ φ + ⎪ = - + φ + +ωφ ⎪ ⎩ (4)

DESIGN OF EKF OBSERVER

Accurate and robust estimation of motor variables which are not measured is crucial for high performance sensorless drives. A multitude of observers have been proposed, but only a few are able to sustain persistent and accurate wide speed range sensorless operation. At very low speed, theirs performances are poor. One of the reasons is the high sensitivity of the observers to unmodeled nonlinearities, disturbance and model parameters detuning.

The Kalman filter provides a solution that directly cares for the effects of disturbance noises including system and measurement noises. The errors in parameters will also normally be handled as noise [START_REF] Harnefors | fill-order observers for flux and parameter estimation of induction motorb[END_REF]. The dynamic state model for non non-linear stochastic machine is as follows where all symbols in the formulations denote matrices or vectors [START_REF] Grabowski | A simple direct-torque neuro-fuzzy control of PWM-inverter-fedinduction motor drive[END_REF][START_REF] Benchabnae | Sensorless Control Strategy For Permanent Magnet Synchronous Motor Fed By AC/DC/AC Converter[END_REF] x(t) f (x(t), u(t), t) w(t) y(t) h(x(t), t) v(t)

Nomenclature

• ⎧ ⎪ = + ⎨ ⎪ = + ⎩
(5) w(t) : System noise vector. v(t) : Measurement noise vector w, v : are unrelated and zero mean stochastic processes. A recursive algorithm is presented for the discrete time case. For the given sampling time s T , both the optimal estimate sequence k / k x and its covariance matrix k / k P generated by the filter go through a two step loop. The first step (prediction) performs a prediction of both quantities based on the previous estimates k 1/ k 1

x --and the mean voltage vector actually applied to the system in the period from k 1

Tto k T .F is the system gradient matrix (Jacobean matrix).

T x(t) x(t) f (x(t), u(t), t) F(x(t), t) x (t) = ∂ = ∂ % % ( 6
) k/k 1 k 1/k 1 s k 1/k 1 k 1 x x T. f(x ,u ) - - - - - - = + (7)
T k/k 1 k/k 1 k 1/k 1 k 1/k 1 s P P (F P P F ). T Q - - - - - - = + + + (8)
The second step corrects the predicted state estimate and its covariance matrix trough a feedback correction scheme that makes use of the actual measured quantities; this is realized by the following recursive relations

k / k k / k 1 k k k / k 1 x x K (Y Hx ) - - = + - ( 9
) k / k k / k 1 k k / k 1 P P K HP - - = - (10)
Where the filter gain matrix is defined by

T T 1 k k / k 1 k / k 1 k P H (HP H R) - - - = + (11) H is transformation matrix x(t) x(t) h H(x(t), t) x = ∂ = ∂ % % (12)
The proposed EKF observer (fig 2) is designed in rotor reference frame ( , α β frame).

where k/k denotes a prediction at time k based on data up to time k. similarly, (k+1)/k denotes a prediction at time k+1 based on data up to time k.

The discrete model of the IM can be given as follows

d d d x(k 1) f (x(k), u(k)) w(k) A x(k) B U(k) W(k) y(k) h(x(k)) V(k) C x(k) V(k) + = + = + + = + = + (13)
With: w(k) is the measurement noise and v(k): is the process noise, A d , B d and C d matrix of discrete system. The state vector is chosen to be:

I: identity matrix of system depending on the size of the state vector.

t s s r r x i i α β α β ⎡ ⎤ = ψ ψ Ω ⎣ ⎦ ; t s s u u u α β ⎡ ⎤ = ⎣ ⎦ ; t s s y i i α β ⎡ ⎤ = ⎣ ⎦
The critical step in the EKF is the search for the best covariance matrices Q and R have to be set-up based on the stochastic properties of the corresponding noise. The noise covariance R accounts for the measurement noise introduced by the current sensors and quantization errors of the A/D converters [START_REF] He | A new variable structure controller for direct torque controlled interior permanent magnet synchronous motor drive[END_REF][START_REF] Yahia | Indirect vector control of induction motor with on line rotor resistance identification[END_REF][START_REF] Benchabane | Sensorless fuzzy sliding mode control for permanent magnet synchronous motor fed by AC/DC/AC converter[END_REF]. Increasing R indicates stronger disturbance of the current. The noise is weighted less by the filter, causing also a slower transient performance of system. The noise covariance Q reflects the system model inaccuracy, the errors of the parameters and the noise introduced by the voltage estimation [START_REF] Benchabane | Systematic fuzzy sliding mode approach combined with extended Kalman filter for permanent magnet synchronous motor control[END_REF][START_REF] Guohan | Estimation of rotor resistance of induction motor based on extended kalman filter[END_REF][START_REF] Taibi | Stability analysis of the extended Kalman filter for Permanent Magnet Synchronous Motor[END_REF]. Q has to be increased at stronger noise driving the system, entailing a more heavily weighting of the measured current and a faster transient performance.

An initial matrix 0 P represents the matrix of the covariance in knowledge of the initial condition. Varying 0 P affects neither the transient performance nor the steady state condition of the system.

In this study, the value of these elements is tuned "manually", by running several simulations. This is maybe one of the major drawbacks of the Kalman filter.

SIMULATION RESULT

To verify the effectiveness of the proposed scheme, simulations were conducted in Simulink. The extended Kalman speed and flux estimation is applied to an induction motor direct field oriented controlled as shown in fig. 3.

In fig. 3, Assumed the reference is constant with the value of 1 Wb, and the reference speed and load is 157rad/s and 10N.m respectively.

The outputs of a PWM voltage source inverter are used as the control inputs for the EKF. These signals contain components at high frequencies, which are used as the required noise by the Kalman filter. Thus, no additional external signals are then needed. From fig. 5, we can notice that the proposed EKF works in very low speed region, where many speed estimators or observers have poor performances. Finally, all these results confirm that EKF's performance is quite good under no load, full load and in speed estimation for all quadrants without causing instability.

MEASUREMENT NOISE

In many applications, the measurement noise caused by either hardware or environment affects the system significantly. The estimated states or sensed variables are the inputs into the nonlinear control. These states or variables are offen multiplied with other states or constant coefficients and amplified which degrades the outputs of the system. In this case, the filtering characteristic of the observer becomes more critical. Since the Kalman filter has the measurement noise uncertainty modeling property, one can eliminate measurement noise up to a certain limit. The estimation accuracy of EKF is tested in this paper under noisy current measurement.

In Fig 6, the injected noise to the d and q stator currents in the range of [0-2 A] is shown. The noise is zero mean, white Gaussian. The aim of the current injection is to observe the low pass filter characteristics of EKF. Fig 7 (overleaf) shows the responses of the speed, the currents and the flux between the actual and estimated states for step reference with 75% of the rated load at t=0.5 s. At t=1.2 s the speed is reversed from +100rad/s to -100rad/s and at t=2.2 s, the reference speed becomes 10 rad/s. As shown in Figure . 7, the estimated speed, currents, and flux are not affected too much from the injected noise. The state estimation accuracy may be increased by increasing the measurement noise covariance under noisy conditions thus the system model will have more importance.

CONCLUSION

The extended Kalman filter has been used for the speed sensorless direct vector control of an induction motor, the knowledge of the rotor space vector position is essential to implement the direct field-oriented control strategy to induction motor drives.

In this paper, a simple estimation algorithm based on EKF in direct vector control with taking core loss into account has been presented. Good results are obtained through the simulations performed under loads variation, references variation and low speed. The effectiveness and the validity of the proposed EKF estimator are verified.

Fig. 1 .

 1 Fig. 1. Equivalent circuit for the IM with taking core loss into account.

Fig. 2 .

 2 Fig. 2 . S-Function representation of the EKF.

Fig. 4 .

 4 shows the step response of speed with 0.6 a time rating load (5Nm). The real speed and the estimated speed are shown in fig.4.a. when the references of 100 rd/s with the load torque, are given. We can notice the good estimation of the motor speed with a few rd/s error in steady state and when load applied which illustrated by fig 4.b. Fig.4.c shows the real and estimated flux module and the error between real and estimated flux is dressed in fig.4.d.

Fig. 3 .

 3 Fig.3. Speed control of an induction machine using direct field oriented control method and an EKF for rotor flux and speed estimation.

Fig. 4 .Fig. 5 .

 45 Fig.4.Step response with application of torque at the 0.6 s.

Fig. 6 .Fig. 7 .

 67 Fig.6. Injected noise to the currents of IM.

Appendix

Parameters of the system used in simulation: Type: three-phase, 0.75 Kw. 220/380V, squirrel-cage induction motor.


Valery F Timkov
email: valerytimkov@gmail.com

Serg V Timkov

Vladimir A Zhukov

Konstantin E Afanasiev

ANALYTICAL EVALUATION OF THE NUMERICAL VALUES OF THE HUBBLE CONSTANT AND MAIN SPATIAL-ENERGY CHARACTERISTICS OF THE OBSERVABLE UNIVERSE

Keywords: The Hubble constant, Universe, fractals, proportions and constants of Planck, black hole, the Hubble sphere Планка, SDSS-III Baryon Oscillation Spectroscopic

Since the baryonic matter of the observable Universe consists mainly of protons and neutrons, then the numerical value of its mass can be represented and calculated on the basis of an additivemultiplicative golden algebraic fractal, based on golden algebraic fractals of the masse of proton, neutron, and muon. Based on an analytical estimate of the mass of the observable Universe, using the law "Planck's Universal Proportions", an analytical estimate of the Hubble constant and the main spatial-energy characteristics of the observed Universe is obtained. An analytical estimate of the Hubble constant is consistent with the experimental data of Planck's mission, SDSS-III Baryon Oscillation Spectroscopic Survey, DES Collaboration. The objectivity of the experimental estimation of the Hubble constant from the H0LiCOW, Riess et al, Hubble Space Telescope collaborations does not raise any doubts. This means that the Hubble constant describes two similar, but different physical processes and has at least two values. The value of the Hubble constant from the collaborations Planck's mission, SDSS-III Baryon Oscillation Spectroscopic Survey, DES Collaboration describes the process of rotation of the space of the observed Universe, and the value of the Hubble constant from the collaborations H0LiCOW, Riess et al, Hubble Space Telescope describes the process of rotation of substance in the space of the observed Universe. New estimates of the values of Dirac's large energy numbers are presented. It is confirmed that all coefficients in Planck's proportions, which are calculated on the basis of Planck's constantslength, mass, time, are equal to similar coefficients, which are calculated on the basis of the main characteristics of the observed Universe. Estimation of the numerical values of the Schwarzschild and Hubble radii for the observable Universe suggests that the Hubble sphere is inside the Schwarzschild sphere, therefore the observable Universe is a black hole. The presence of a redshift from distant objects of the observable Universe suggests that this is a rotating black hole. It has been shown that the vacuum pressure cannot be a source of dark energy. It is shown that the force of imaginary dark energy is equal in magnitude to the force of gravitational compression of the observed Universe and is equal to the Planck force. It is shown that after the Big Bang, the space of the observable Universe made one incomplete revolution of at 345 degrees, and the substance in it made one complete revolution of approximately 379 degrees. Estimates are given for the energy of dark matter, as the kinetic energy of rotation of the space of the observable Universe, as well as the equivalent mass of dark matter. New estimates are given: of the gravitational constant, of the Planck energy, of the Planck acceleration, of the Planck force, of the gravity factor of the observable Universe, of the Planck temperature, of the angular velocity of rotation of the space of the observable Universe. Estimates of temperature and wavelength of thermal radiation of the observable Universe, as the Hubble sphere, are given.

Introduction. Formulation of the problem.

It is known that fractals are self-similar algebraic and geometric constructions, in which each part of them is similar to the whole structure, and the entire structure is similar to each its part. In [1,2,3] it was shown that physical constants can be represented in the form of gold algebraic fractals (GAF). The basis of these fractals is the number g f , which is the coefficient of similarity of fractals: 5 0.5 2sin(18 ) 2 cos(72 ) 0.6180339887498948482045868343656...

2 g f        (1)
Then any physical constant  in the form of a golden algebraic fractal has the form:

,

y gg Mf   (2)
where g M is the mantissa of the GAF constant  , y is the structural level of the GAF. The number g M and integer y satisfy the conditions: 1 1 1.6180339887498948482045868343656...; .

g g My f         (3)
Mantissas g M and gi M physical constants  and 1  satisfy the condition:

1 .

g gi g MM f  (4)
The mantissa gi M will be called the inversion of the mantissa g M . The numerical value of the mass [4] of some elementary particles: electron, muon, tau, proton, neutron and Planck's hypothetical particle will be presented in the form of GAF (2), and the data will be entered in Table 1. We take into the account that the muon and the hypothetical Planck particle are the same GAF [1], so the characteristics of a hypothetical Planck particle are determined on the basis of the characteristics of the muon: Table 1. The mantissa, the inversion of mantissa, the structural level of the GAF: electron, muon, tau, proton, neutron, hypothetical Planck particle. Based on experimental data estimates of the Hubble constant, which were obtained from various sources: the Planck Mission [5,6,7,8], the Wilkinson Microwave Anisotropy Probe (WMAP 9 years, combined with other measurements) [9], SDSS-III Baryon Oscillation Spectroscopic Survey [10], DES Collaboration [11], using the law "Planck's Universal Proportions" [12], calculate the estimate of the mass of the observable Universe. The obtained data, including the value of the estimated mass of the observable Universe in the form of a GAF, are presented in Table 2. The GAF can be divided into: the simple fractals (e.g., such as in Table 1); multiplicativethose in which the mantissa is formed by multiplying the mantissas of the simple fractals with their subsequent normalization to the form (3); additive ones are those whose mantissa can be represented as a sum of an infinite series of power functions n g f [2]; mixed type -additive- multiplicative or multiplicative-additive.

Obviously, two or more algebraic fractals are the same fractal if their mantissas are equal.

The Hubble constant has a dual nature. It describes with the same success two different physical processes in the observable Universe: expansion and rotation of its space. That is, the redshift from distant objects of the observable Universe can be described with equal success on the one hand by the removal (scattering) of these objects under the action of the force of so-called dark energy, and on the other hand by the rotation of the space of the observed Universe. Both hypotheses have flaws. In the first case, a scientifically based source of dark energy is required, and in the second case, under certain conditions, the basic principle of cosmology, the principle of isotropy of space, is violated. Currently, of the many hypotheses about the nature of the source of dark energy, the most widely used hypothesis is the cosmological constant or, as say, the lambda constant.

According to this hypothesis, the lambda constant in Einstein's equations gets a physical meaningthe pressure of the vacuum. The expansion of the space of the observable Universe occurs under the action of the pressure of the vacuum. However, space can expand only from the area of high pressure to the area of low pressure, that is, only under the action of pressure drop. The lambda calibration coefficient is constant for all points of the Universe, that is, the vacuum pressure at all points of the Universe is the same and there is no pressure differential in it. This means that the hypothesis of the cosmological constant, as a source of dark energy, is not consistent. The solution of the Einstein equations for the rotating space of the Universe was proposed by Gödel in 1949 [13].

Further development of the hypothesis of the rotation of the space of the observable Universe was obtained in [14,15,16,17,18]. Professor Michael Longo (University of Michigan in Ann Arbor), having studied [14,15] in the framework of Sloan Digital Sky Survey (SDSS) database project SDSS DR5, which contains about 40,000 galaxies ( of them -more than 15,000 spiral galaxies ), for where the value of redshift z <0.04, came to the conclusion that left twisted spiral galaxies is much larger than spiral galaxies swirling right. Conclusions of professor Michael Longo confirmed a group of scientists led by Professor Lior Shamir (Lawrence Technological University) [16]. Was investigated about 250 thousand spiral galaxies, for which the value of redshift z <0.3. Professor

Lior Shamir also found that galaxies left more than right. Symmetry breaking between the rightand left -twisted spiral galaxies are about seven percent, but the probability that is a cosmic accident is very low -claims Professor Michael Longo. The results of research professors Michael

Longo and Lior Shamir contradict the notion that the Universe is homogeneous and symmetric.

Scientists believe that the asymmetry of the Universe emerged in the Big Bang at the expense of the initial rotation of the Universe space counterclockwise.

This means, firstly, that the space of the Universe could have existed before the Big Bang, and secondly, as claims, Professor Michael Longo universe revolves now. In [17] it is shown that the rotation of the space of the observable Universe can be a source of dark energy and dark matter.

Analytical estimate of the numerical value of the Hubble constant.

It is known [19,20] that the mass of the baryonic matter of the observable Universe consists of more than 75% protons.

Then, taking into account that the electron mass is much less than the proton and neutron masses, and also that the neutron and proton masses are close in value, we can conditionally assume that the mass of the observable Universe is 75% composed of protons and 25% of neutrons.

Analysis of the data in Tables 1 and2 shows that the mantissas of the masse of the observable Universe are very close to the mantissas of the masses of the proton and neutron, in all cases, except for the data from WMAP.

Let the mass of the observable universe is formed only from protons. Then, assuming that the mass of the observable Universe and the mass of the proton are one and the same golden algebraic fractal, but on different structural levels of matter, we choose the proton mantissa as the standard, and on its basis, we estimate the mass of the observed Universe :

p U M  254 53 1.8435885153510... 10 . 1.52336288580... p U g M f kg      (5)
Let the mass of the observable Universe and the mass of the neutron -is one and the same golden algebraic fractal, but on different structural levels of matter, then choose of the mantissa of the neutron as standard, and on its basis to estimate of the mass of the observable Universe

p n p n U U U M M M kg         (7)
Since the hypothetical Planck particle and the muon are the same golden algebraic fractal, then the

           (9)
Then the Hubble constant estimate p H  for the observable Universe, which consists only of protons, is equal to:

p p p U pt U p U T M s H s T                (10)
n n n U pt U n U T M s H s T            (12)
The value of the evaluation of the Hubble constant in the traditional form:

0 67.4879634 ( / ) . n H km s Mpc  (13)
The Hubble constant estimate

0 pn H  
for the observable Universe, which consists of protons and neutrons, is equal to:

0 0 0 0 1 17 18
p n p n p n U pt U pn U T M s H s T                    (14)
The value of the evaluation of the Hubble constant in the traditional form: 0 67.55770 ( / ) / .

pn km s Mp H c    (15)
Then the analytical evaluation of the Hubble constant can be represented as:

      (18)
The impulse of a unit rarefaction of the mass of Planck 1 pt   is represented by formula (9), that is:

1 1 36 1 1
2.4766349727317... 10 .

p pt p p p t t kg s m                 (19)
The quantum of Planck's mass

The linear rarefaction of Planck's mass 1 pl   :

1 28 1 1

7.424764860... 10 .

p pl p l kg m m            (22)
The quantum of Planck's length, or the speed of light in vacuum c : 11 299792458 .

p p l c m s t       (23)
In electromagnetism [22], electrical resistance has the dimension of velocity, then by analogy, we call the quantity, which is inverse to velocity, the conductivity of light in a vacuum 1 c  :

1 9 1 1
3.33564095198... 10 .

p p t c m s l           (24)
In addition to the coefficients that are presented in formulas (19)(20)(21)(22)(23)(24), we calculate the coefficients that characterize the density of mass (and its rarefaction) of Planck and the product of the Planck constants: of mass, of length and of time.

The surface density of Planck's mass

The surface rarefaction of Planck's mass

           (26)
The density of Planck's mass

p pv p m kg m l          (27)
The rarefaction of Planck's mass .

U U U M kg m R      ( 47
)
The estimate of the rarefaction of the mass of the observable Universe

F m a h c h kg m s

          (52) 3 3 4 1 2 1 2 1 2 11 3 1 2 2 2 () () 6.6730459... 10 . () () p U pl Ul U U U p pp l R c G c c H m kg s M T F mt                         ( 53
)
Clarify the value of the estimates of the total energy of the observable Universe U E  , taking into account the rotation of its space [3,17]: 70 0.268 2.1017172440... 10 .

U U U E E E J       ( 54
)
The estimate of the energy of dark matter, as the kinetic energy of rotation of the space of the

   ( 60
)
where k is the Boltzmann constant [4]:

      ( 62
)
where  is the constant of Stephen -Boltzmann [4]:

W c m k K           h ( 63
)
where h is Planck's constant over 2pi.

Obviously (62) can also be represented as:

P S W m l         ( 64
)
By analogy with (64), the estimation of the total of energy luminosity U S of the observable Universe, as a Hubble sphere, without taking into account the rotation of its space:

U U U P S W m R      (65)
Note. With accuracy 4 10  , formula (65) can be represented as:

1 . U S W m    (66)
The estimation of the total of energy luminosity of the observable Universe, as the Hubble sphere, taking into account the rotation its space:

2 12 2 . 0.403696039 . 6 0 .. 0 ( ) U U U P S W m R        (67)
The estimation of the temperature of the observable Universe, as the Hubble sphere, without taking into account the rotation of its space U T  , and also taking into account the rotation of its space

U U S TK      (69)
We will determine, according to Wien's law, the estimation of the maximum wavelengths max U  and max U   , which correspond temperature by formulas (68) and (69):

5 max 0.0028999 [ ] 5.95733476... 10 , U U mK m T        (70)
U U mK m T          (71)
The wavelength range (56-60) mkm is a far infrared region that is far beyond the range of visible light.

Taking into account the new value of the Hubble constant according to the formula (17)

FR E M c M R T N E m c F l m l t                      (72) 2 121
=7.178843892933870... 10 .

p U U U U U U De U p p p p p p S M R M T R T N S m l m t l t                     (73)
Estimation of the angular velocity U  of rotation of the space of the observable Universe:

In [18] it was shown that if the angular velocity of rotation of the space of observable Universe is less than 9 1 1 10 , rad year  then the principle of isotropy of its space is not violated. The value of the angular velocity (75) confirms the principle of isotropy of the space of the observable Universe.

Since the Big Bang (for the age of the observable Universe -13.88 billion years [25]), the observable Universe, as the Hubble sphere, has made one incomplete turn at of 345 degrees, and the substance in it made one complete revolution of approximately 379 degrees [21]. The difference between the rotation angles of the space of the observable Universe and the substance in it is a value that is close to the angle value: 1) 12 33.69 Q  o the neutrino mixing matrix [2], then, if we take the analytical estimate of the Hubble constant as a basis: 2) .

Since, according to the basic principle of cosmology, the space of the observable Universe is isotropic, then we can assume that the Hubble sphere is an axisymmetric sphere. Then we clarify the value of the estimation of the Schwarzschild radius US R for the observable Universe:

U US GM Rm c     (76)
From formulas (69) and (32,33) it follows that: 2.

US U RR  (77)

Since the radius of the observed Universe is two times smaller than the Schwarzschild radius, that is, two times smaller than the gravitational radius or the radius of the event horizon, the observed Universe, as the Hubble sphere, is a black hole. Additional indirect signs that the observable Universe is a black hole are: 1) the wavelength range of its thermal radiation is in the far infrared region radiation; 2) the presence of gravitational-electromagnetic resonance, which is confirmed experimentally [12,23,24]. At present, there are no cases of expansion with the acceleration of the black hole space, but at the same time, there is a red shift from distant objects of the observable Universe, which means that the black hole, like the Hubble sphere, rotates. The rotation of the space of the observable Universe is also confirmed by the fact that the acceleration of the force of imaginary dark energy is equal in absolute value to the gravity factor of the observable Universe [17].

Let us calculate the estimate of the magnitude of the modulus of the force of imaginary dark energy

                 (81)
Based on the formula (80), it can be argued that at present, there is no expansion of the space of the observed Universe after the Big Bang.

Results. Estimates of the main spatial and energy characteristics of the observable

Universe are presented in Table 3.

Abbreviations in the table:

(s.) -static;

(d.) -dynamic. Universe and the absence of dark energy can only be explained by the rotation of the space of the observable Universe [17]. Improving the accuracy of the analytical evaluation of the Hubble constant will be possible when the percentage composition of protons and neutrons in the mass of the observable Universe is more accurately estimated. More accurate than in formula (17), the experimental definition of the Hubble constant will make it possible to specify the percentage ratio of protons and neutrons in the mass of the observed Universe. Analytical estimations: of the Hubble constant, of the gravity factor of the observable Universe and of the acceleration of the force of imaginary dark energy, of the force of imaginary dark energy, of the Schwarzschild radius of the observable Universe, allow us to state that of dark energy not exist. The observable Universe, as of electromagnetic fractal of the Universe [26], is a slowly rotating black hole in the space of a gravitational fractal of the Universe. The magnitude of the angular velocity of rotation of the space of the observable Universe is such that the cosmological principle of isotropy is not violated.

The dark energy hypothesis allows only an unambiguous value of the Hubble constant. The hypothesis of the rotation of the space of the observed Universe admits a multi-valued value of the Hubble constant.

6 )

 6 Let us estimate the mass of the observable Universe 0 pn U M   , provided that it consists of 75% protons and 25% neutrons:

H

  time, or the delay in the propagation of the light signal at a distance that is equal to the radius of the observed Universe. The value of the evaluation of the Hubble constant in the traditional form: for the observable Universe, which consists only of neutrons, is equal to:

  the Hubble constant made it possible to clarify the main spatial and energy characteristics of the observable Universe.5. Conclusions. The mass of the observableUniverse is an additive golden algebraic fractal of a proton and of a neutron. The Hubble constant is an inverse additive-multiplicative fractal of the temporal characteristics of the proton, of neutron and of the muon. The analytical estimate of the Hubble constant is within the tolerances of the experimental Hubble constant values, which are obtained by Planck Mission, DES Collaboration, SDSS-III BOSS. Based on the proposed method, it is possible to clarify the mass of distant objects of the observable Universe. Analytical evaluation of the Hubble constant only on the basis of the characteristics of elementary particles suggests that the Hubble constant is an internal immanent characteristic of the observed Universe, which excludes the presence of dark energy in it. The presence of a red shift from distant objects of the observable

Table 2

 2

	Name of the data	Citation	Hubble const.,	Mass of the	GAF form of a
	source		(km/s)/Mpc	Universe, kg	mass of the
					Universe, kg
	Planck Mission Planck Mission	[5] [6,7]	67.66±0.42 67.74±0.46	1.8414357•10 53 1.839260•10 53	1.52158398•f g 1.51978700•f g	-254 -254
	Planck Mission WMAP SDSS-III BOSS DES Collaborat.	[8] [9] [10] [11]	67.80±0.77 69.32±0.80 67.6(+0.7)(-0.6) 67.77±1.30	1.837633•10 53 1.797339•10 53 1.8430700•10 53 1.838447•10 53	-254 -254 1.485146740•f g 1.5184420•f g -254 1.522934498•f g 1.51911424•f g -254

. The Estimation of the mass of the observable Universe based on the Hubble constant, which is obtained from the various experimental sources of the information.

Evaluation of the main spatial and energetic characteristics of the observable Universe

  The analytical estimate of the Hubble constant is within the tolerances of the experimental Hubble constant values, which are obtained by Planck Mission, DES Collaboration, SDSS-III BOSS.There are two main ways to experimentally determine the Hubble constant: by studying the relict radiation and by measuring the distances to distant objects of the Universe, for example, quasars, Cepheids and red giant stars. Relic radiation is distributed throughout the space of the observable Universe, therefore, an experimental estimate of the Hubble constant based on it reflects the physical process that is inherent in the entire space of the observable Universe. An experimental estimate of the Hubble constant based on measuring changes in the distances to quasars, Cepheids, and red giant stars reflects the physical process associated with the movement of substance in the observable Universe. This means that the Hubble constant objectively has at least two values. In this case, the value of the Hubble constant about the value: 1)

	H		67.55770	 	0.02329 0.06974	( / ) / km s Mpc .	(16)
	Or taking into account the percentage ratio of protons and neutrons:
	H		67.55770		0.02329	( / ) / km s Mpc .	(17)
								H	67.55770	0.02329	( / ) / km s Mpc ,
			H		74.03		1.42( / ) / km s Mpc	[21], displays the process
	Planck's mass 1   :
	p
	p	1	1 m 	7 4.594239449... 10	kg	1	.
			p



 displays the process of rotation of the space of the observed Universe, and the value of the Hubble constant, for example, about the value: 2) of rotation of the substance in the space of the observed Universe. It is possible that an estimate of the Hubble constant based on the motion of a substance whose mass density is significantly lower than the mass density of quasars, Cepheids, and red giant stars will differ from the known values 1) and 2).

3. . Based on the values of Planck's constants -of the mass, of length and of time, which are represented in formula

(8)

, we calculate the coefficients of proportionality for the law "Planck's Universal Proportions". To do this, we first introduce an auxiliary coefficient: the unit rarefaction of

  , we can estimate two other characteristics. A similar statement is true for any, having a mass, body of the observable Universe. Let us estimate the radius of the observable

	1.17336937272... 10  The estimate of the surface density of the mass of the observable Universe Us . pm p p m t kgs         :	(30)
	The impulse of Planck's length Us  : pl M U 2 U R  	9.836055336170...	12 kg m 	.	(44)
	Note. With accuracy	79  10  , formula (44) can be represented as: 8.7119916869... 10 pl p p l t        2	. ms	(31)
	Formulas (7) and (14) represent the mass  UU pn MM    2 1 kg m 	0  2 . of the observable Universe as well as the (45)
												Us
	Hubble time The estimate of the surface rarefaction of the mass of the observable Universe 1 0 pn TT    . Based on the law "Planck's Universal Proportions" and formulas (19-24),   :
		UU										Us
	using one of the three characteristics of the observed Universe: mass 2 1 1 2 0.101666772484... . U R kg m   	U M , Hubble time	U T , or	(46)
				Us
	Hubble radius								M	U
	The estimate of the density of the mass of the observable Universe
	Universe	R , or Hubble radius :
		U
				R				1  		M		26 1.36929282469... 10 , m 	(32)
					U						pl	U
						R			c T   	26 1.36929282469... 10 . m 	(33)
								UU
	Also true:
			M			R  	53 1.84422382448890... 10 	kg	,	(34)
				U						pl	U
			M		h T  	53 1.84422382448890... 10 	kg	,	(35)
				U						pm	U
				T				1  		M		17 69221 .. 10 4.5674 27 	,. s	(36)
					U					pt	U
				T					1 c R  		17 922127. . 10 4.56746 	.. s	(37)
					UU
	It follows from the law of "Planck's Universal Proportions" that formulas (19-24) are also valid
	when instead of the Planck constants: of mass, of length and of time are substituted for the
	corresponding characteristics of the observed Universe. Then:
			1   	1 			U T		2.4766349727317... 10 	36 	1 1 kg s 	,	(38)
			Ut			pt
											M
												U
			h			h			M	U		35 4.0377367315... 10 	1 1 kg s 	,	(39)
			Um					pm
												T
												U
			Ul  	pl	1   : R M 	U		1.34684399950... 10 	27	1 kg m	1 	,	(40)
											pv
												U
			3  () p l     pl 1   1 pv    Ul 1  	1.9391922174000... 10 M   7.424764860... 10 U R  	97  28 	1 3 kg m  1 1 , . kg m 	(28) (41)
												U
							m
										p
	The moment of Planck's mass		j : c	U R 	299792458	11 m s 	,	(42)
							pm
												T
												U
			j	pm c	 1 	43  1 1 kgm . . m s 3.51767288389... 10  9 3.33564095198... 10   p m l p   U T     	(29) (43)
	The impulse of Planck's mass	U R  :
							pm

U R

  As can be seen from the formulas (57) and (58), the Planck power and the power of the observable Universe, without taking into account the rotation, are equal to each other.

	The estimate of the equivalent of the mass of dark matter Udm M :
					M	Udm		2 Udm E c		52 4.94251984960... 10 	kg	.	(56)
	The estimation of the Planck power	P :
										p
			p P		p t   E		3.62893679784... 10 	52 W kg m s 1 2 3 ,   	.	(57)
						p
	Estimation of the power of the observable Universe	P without taking into account the rotation of
										U
	its space, and also taking into account the rotation of its space	U P  :
		U P		U E 	3.62893679784... 10 	5 2	13 2 g m s W k ,   	,	(58)
						T
						U
		U P		U E  	4.60149185966... 10 	5	2	12 k m W gs  3 ,  	.	(59)
						T
						U
										E
										p
							TK
									p
										k
	observable Universe	E :
		Udm
						E				0.268		E		69 4.442115310... 10 . J 	(55)
						Udm			U

Let us specify the value of the Planck temperature estimate p T : 32 1.41691777191... 10 ,

  , let us specify the values of estimations of large energy numbers of Dirac[1]

										N and 2 N :
										De	De
				2
	De	U	U	2	pU	U	U	U	60 8.472805847494600... 10 ,
		p
			p		p	p	p	p	p

  This force is applied to each point of the boundary of the Hubble sphere and it is directed outward from the Hubble sphere. Let us calculate the estimate of the magnitude of the force of gravitational compression of the observable Universe g F . This force acts on every point of the boundary of the Hubble sphere and it is directed into the interior of the Hubble sphere:

		F		M c H   	44 1.21048301950... 10 	1 1 2 kg m s 	.	(79)
		gU
	Obviously, that the force	F , the force	F and the Planck force	F are equal to each other:
		ide				g	p
								F	F 	F	.	(80)
								ide	g	p
	Obviously, that:
	1 Us Gg 1 Us 2 U R	=	1 Us	с H	6.673045869... 10	11 3 m kg s 1 2	.
		T
		U
		F		M c H   	1.21048301950... 10 	kg m s 	.	(78)
		ide			U

Table 3 .

 3 Estimates of the main spatial and energy characteristics of the observable Universe.

	The name of characteristics	The characteristic	The
		value	dimension
	the Hubble constant	2.1893962533…•10 -18	s -1
	the Hubble constant (traditional form)	67.55770 +0.02329	(km/s)/Mpc
	the Hubble constant (for the substance)	74.15 +0.02329	(km/s)/Mpc
	the Hubble time	4.5674692213…•10 17	s
	the mass of the observable Universe	1.8442238245…•10 53	kg
	the radius of the observable Universe	1.3692928247…•10 26	m
	the quantum of the mass of the observable Universe	4.0377367315…•10 35	kg 1 s -1
	the linear density of the mass of the observable Universe 1.3468439995…•10 27	kg 1 m -1
	the linear rarefaction of the mass of the observable	7.424764860…•10 -28	kg -1 m 1
	Universe
	the surface density of the mass of the observable	9.8360553362…	kg 1 m -2
	Universe
	the surface rarefaction of the mass of the observable	0.1016667725…	kg -1 m 2
	Universe
	the density of the mass of the observable Universe	7.183310362…•10 -26	kg 1 m -3
	the rarefaction of the mass of the observable Universe	1.392115820…•10 25	kg -1 m 3
	the energy of the observable Universe (s.)	1.6575057130…•10 70	J
	the total energy of the observable Universe (d.)	2.1017172440…•10 70	J
	the energy of dark matter	4.442115310…•10 69	J
	the equivalent of the mass of dark matter	4.9425198496…•10 52	kg
	the gravity factor of the observable Universe	6.5636448430…•10 -10	m 1 s -2

integral an angle between the direction Coriolis force and the axis of rotation

[17]

Acknowledgments

Regarding this manuscript, the authors do not have a conflict of interest in any area of human activity: neither legal, nor financial, nor ethical ... and any other, both in relations with each other and in relations with other citizens and organizations. The work is done exclusively on the personal funds of the co-authors.