Luc Guillermo Hernandez Rodriguez

Guy Pibouleau

Catherine Azzaro Pantel

Serge Domenech

Optimization of Gas Transmission Networks under Energetic and Environmental Considerations *

Keywords: optimization, pipeline, fuel consumption, natural gas, carbon dioxide emissions

The transport of large quantities of natural gas (NG) is carried out by pipeline network systems across long distances. Pipeline network systems include one or several compressor stations which compensate for pressure drops. A typical network today might consist of thousands of pipes, dozens of stations, and many other devices, such as valves and regulators. Inside each station, there can be several groups of compressor units of various vintages that were installed as the capacity of the system expanded. The compressor stations typically consume about 3 to 5% of the transported gas. It is estimated that the global optimization of operations can save considerably the fuel consumed by the stations. Hence, the problem of minimizing fuel cost is of great importance. This study presents a mathematical formulation for NG transport through pipelines and compressors by considering the mass and energy balance equations on the basic elements of a didactic network from the literature. First, a deterministic optimization procedure is implemented. The objective of this formulation is the fuel minimization problem in the compressor stations for a fixed gas mass flow delivery. A second example is devoted to the simultaneous consideration of gas mass flow delivery maximization and fuel consumption minimization. In that case, two procedures are compared: a genetic algorithm coupled with a Newton-Raphson procedure and the scalarization method of ?-constraint. In both monobjective and biobjective cases, a study of carbon dioxide (CO 2 ) emissions is carried out. The Pareto front deduced from the biobjective optimization can be used either for identifying the minimum and maximum network capacity in terms of CO 2 emissions and mass flow delivery or *

Introduction

Natural Gas (NG) is an important source of energy for reducing pollution and maintaining a clean and healthy environment. In addition to being a domestically abundant and secure source of energy, the use of NG also offers a number of environmental benefits over other sources of energy, particularly other fossil fuels. The transport of large quantities of NG is carried out by pipeline network systems across long distances. As the gas flows through the network, pressure (and energy) is lost due to both friction between the gas and the pipe inner wall, and heat transfer between the gas and its environment. Typically, NG compressor stations are located at regular intervals along the pipeline to boost the pressure lost through the friction of the NG moving through the steel pipe. They consume a significant part of the transported gas (3 to 5%, [START_REF] Suming | Model relaxations for the fuel cost minimization of steady-state gas pipeline networks[END_REF], thus resulting in an important fuel consumption cost on the one hand, and in a significant contribution to CO 2 emissions, on the other hand. Nowadays, more than 50% of the total human-caused Greenhouse gas (GHG) emissions result from the production and use of energy. About 70% of GHG emissions from NG occur when it is burned to produce heat or energy. Pipelines emit CO 2 mainly due to energy used at compression stations. Therefore, pipeline companies reduce GHG emissions mainly by improving the use of energy by acquiring more efficient equipment and by adopting better operating practices [START_REF] Mora | Minimization of energy use in pipeline operations -an application to natural gas transmission systems[END_REF].

Thus, efficient operation of compressor stations is of major importance for enhancing the performance of the pipeline network. This paper is devoted to the presentation of (1) a gas transportation model taking into account the elements of the network under steady-state conditions that serve for operation condition optimization or design targets and, (2) two approaches for optimizing the performance of compression stations.

In the first part, gas pipeline network modelling provides the ability to represent the behaviour of the system under a variety of conditions, so that it can be used to support optimization procedures. In the second part, a classical deterministic optimization procedure based either on the nonlinear programming tool CONOPT3 of the GAMS (General Algebraic Modelling System) library or on the MATMAB toolbox has been implemented. The objective of this formulation is the fuel minimization problem in the compressor stations for fixed gas mass flow delivery. Then, a genetic algorithm (NSGA IIb, [START_REF] Gomez | Optimisation technico-économique multi-objectif de systèmes de conversion d'énergie : Cogénération électricité-hydrogène à partir d'un réacteur nucléaire de IVème génération[END_REF] coupled with a Newton-Raphson method and a scalarization procedure based on the -constraint technique are used to solve a biobjective problem, that is the simultaneous maximization of the gas mass flow delivery and the minimization of the fuel consumption in the compression stations. Both monobjective and biobjective optimization procedures are implemented on a didactic network coming from the literature [START_REF] Abbaspour | Nonisothermal compressor station optimization[END_REF], and a study of carbon dioxide emissions by the compression stations is carried out.

Transmission pipeline modelling

Previous works

Since 20 years, there has been an increased interest on the optimization of gas pipe distribution networks. [START_REF] Tian | Development of analytical design equation for gas pipelines[END_REF] have proposed a onedimensional compressible fluid flow equation. [START_REF] Lewandowski | Object-oriented modelling of the natural gas pipeline network[END_REF] has implemented an object-oriented methodology for modelling a natural gas transmission network, and [START_REF] Osiadacz | Dynamic optimization of high Pressure gas Networks using hierarchical systems theory[END_REF] has presented a dynamic optimization of high-pressure gas networks. [START_REF] Surry | A multi-objective approach to constrained optimization of gas supply networks[END_REF] have formulated the optimization problem based on a multiobjective genetic algorithm. [START_REF] Mohitpour | The importance of dynamic simulation on the design and optimization of pipeline transmission systems[END_REF] have used a dynamic simulation approach. [START_REF] Boyd | Evaluating the quality of pipeline optimization algorithms[END_REF] have studied steady-state gas pipeline networks by modelling the compressor stations. [START_REF]Optimization of pipeline networks with a hybrid MCST-CD networking model[END_REF] has based the modelling approach on a hybrid network. [START_REF] Sun | An integrated expert system/operations research approach for optimization of natural gas pipeline operations[END_REF] have used a software support system, called the Gas Pipeline Operation Advisor for minimizing the overall operating costs, subject to a set of constraints such as the horsepower requirement, availability of individual compressors, compressor types and cycles. A reduction technique for natural gas transmission network was implemented by [START_REF] Rios-Mercado | A Reduction technique for natural gas transmission network optimization problems[END_REF]. [START_REF] Martinez-Romero | Natural gas network optimization and sensibility analysis[END_REF] have used the software package "Gas Net". A MINLP model for the problem of minimizing the fuel consumption in a pipeline network was implemented by [START_REF] Cobos-Zaleta | A MINLP model for minimizing fuel consumption on natural gas pipeline networks[END_REF]. Costa et al. (2002) have developed a steady-state gas pipeline simulation. [START_REF] Mora | Minimization of energy use in pipeline operations -an application to natural gas transmission systems[END_REF] have determined the pipeline operation configurations requiring the minimum amount of energy. [START_REF] Chauvelier-Alario | Decision making software for Gaz de France distribution network operators[END_REF] have developed CARPATHE, a simulation package (French Company GdF Suez) for representing the behaviour of multi-pressure networks. Optimization methods for reinforcement planning on gas transportation networks and for minimizing the investment cost of an existing gas transmission network were used by [START_REF] André | Planning reinforcement on gas transportation networks with optimization methods[END_REF].

Modelling gas pipeline networks

The objective of this work is to propose a general framework able to embed formulations from design to operational purposes for steady-state problems. The goal is to implement, for a given pipeline network, an accurate numerical method able to be used within an optimization loop. The modelling of gas pipeline networks has already been presented by [START_REF] Tabkhi | Optimisation de réseaux de transport de gaz[END_REF]; only the main points are recalled here.

Gas pipeline equations

The governing equation giving the pressure at each point of a straight pipe can be derived as follows:

0 ) ( 2 2 2    v dx d v D f dx dP   (1)
where  is the gas density (kg/m 3 ), D the pipe diameter, v the average gas velocity (m/s).

The Darcy friction factor, f, is a dimensionless value that is a function of the Reynolds number, Re, and relative roughness of the pipeline, /D. The Darcy friction factor is numerically equal to four times of the Fanning friction factor that is preferred by some engineers. Since the regime of the gas passing through pipelines lies in the turbulent range, it is assumed that the wall roughness is the limiting factor compared with the Reynolds number to find out the value of the friction factor. The work of [START_REF] Romeo | Improved explicit equations for estimation of the friction factor in rough and smooth pipes[END_REF] is used to estimate the friction factor. The momentum balance in terms of pressure and throughput can be written in the following form:

0 ) ( 16 8 4 2 2 5 2 2    P ZT dx d MD m R P MD m fZRT dx dP     (2)
In this equation, Z is the compressibility factor, R the universal gas constant (8314 J/kmol-°K), T the temperature (°K), M the average molecular mass of the gas, and m  the pipe throughput (flow rate in kg/s).

By integrating Eq. ( 2) between two points i and j, the following equation is obtained and will be used in the numerical formulations:

0 16 ln 32 5 2 2 4 2 2 2 2             MD L m fZRT P P MD m ZRT P P j i j i     (3)
where L (m) is the pipe length between points i and j. The compressibility factor, Z, is used to alter the ideal gas equation to account for the real gas behaviour. This factor can be expressed by the following relation:

c ij c p p T T Z ) 533 . 0 - 257 . 0 ( 1   (4)   i i c c y T T ( 5 )   i i c c y p p ( 6 )
The pseudo-critical temperature of natural gas, T c , and pseudo-critical pressure, p c , can be calculated using an adequate mixing rule starting from the critical properties of the NG components. In this work, average pseudo-critical properties of the gas are determined from the given mole fractions of its components by Kay's rule which is a simple linear mixing rule shown in Eq. ( 5) and ( 6). Then average pressure, p ij , can be calculated from two end pressures [START_REF] Mohring | Automated model reduction of complex gas pipeline networks[END_REF]):

) - ( 3 2 j i j i j i ij p p p p p p p    (7)

Maximum allowable operational pressure (MAOP)

The internal pressure in a pipe causes the pipe wall to be stressed, and if it is allowed to reach the yield strength of the pipe material, it could cause permanent deformation of the pipe and ultimate failure. In addition to the internal pressure due to gas flowing through the pipe, the pipe might also be subjected to external pressure which can result from the weight of the soil above the pipe in a buried pipeline, and also by the probable loads transmitted from vehicular traffic. The pressure at all points of the pipeline should be less than the maximum allowable operating pressure (MAOP) which is a design parameter in the pipeline engineering. This upper limit is calculated using Eq. ( 9), where t is the thickness of the pipe [START_REF] Menon | Gas pipeline hydraulics[END_REF]:

MAOP p  (8) T E F f f f t D t SMYS MAOP - 2  (9)
The yield stress used in Eq. ( 9), is called the specified minimum yield strength (SMYS) of pipe material. SMYS is a mechanical property of the construction material of the gas pipeline. The factor f F is the so-called design factor. This factor is usually 0.72 for cross-country or offshore gas pipelines, but can be as low as 0.4, depending on class location and type of construction. The class location, in turn, depends on the population density in the vicinity of the pipeline. The seam joint factor, f E , varies with the type of pipe material and joint type. Seam joint factors are between 1 and 0.6 for the most commonly used material types. The temperature factor, f T , is equal to 1 for the gas temperature below 393°K but it can reach 0.867 at 503°K. For each particular problem, these factors are well known by the NG practitioners.

Critical velocities

As flow rate increases due to the augmentation in pressure drop, so does the gas velocity. For a compressible flow, the increase in flow owing to the pressure drop increase is limited to the velocity of sound in the fluid, i.e., the critical velocity. Sonic or critical velocity is the maximum velocity which a compressible fluid can reach in a pipe. For trouble-free operation, velocities maintain under a half of sonic velocity. The sonic velocity in a gas, c is calculated using Eq. ( 11), where κ is the average isentropic exponent of the gas, and C p is the heat capacity at constant pressure in J/(kmol.K).

2 / c v  (10) M ZRT c   ( 1 1 ) R y C y C i i p i i p     ) ( ) (  (12)
Increasing gas velocity in a pipeline can have a particular effect on the level of vibration and increase the noises too. The upper limit of the velocity range should be such that erosion-corrosion cavitations or impingement attack will be minimal. The upper limit of the gas velocity for the design purposes is usually computed empirically with the following equation [START_REF] Menon | Gas pipeline hydraulics[END_REF]. The erosional velocity must always be lower than the speed of sound in the gas.

Compressor characteristics

As shown on Figure 1, a centrifugal gas compressor is characterized by delivered flow rate and pressure ratio, the ratio between suction side pressure of the compressor and its discharge pressure. The compression process in a centrifugal compressor can be well formulated using isentropic process for calculating horsepower for a compressor station. The pressure ratio of a centrifugal compressor is usually linked with a specific term named "head" carried over from pump design nomenclature and expressed in kJ/kg for compressors. The "head" developed by the compressor is defined as the amount of energy supplied to the gas per unit mass of gas.

Figure 1. A typical centrifugal compressor map

The equation for power calculation can be expressed as follows:

is i ac h m Pw    ( 15
)
where ac m  (kg/s) is the mass flow rate of compressed gas, h i (kJ/kg) the compressor isentropic head, η is the compressor isentropic efficiency.

For adiabatic compressor the adiabatic efficiency is defined by:

w ideal w is P P ,   ( 1 6 )
As shown in the following equation, head is an index of the pressure ratio across the compressor. In this equation, p d is the discharge pressure of the compressor and p s , the suction pressure. The compressibility factor and the temperature here are considered at suction side of the compressor, where M is the average molecular mass of the gas [START_REF] Smith | Introduction to chemical engineering thermodynamics (4 th edition)[END_REF].

Q s (m 3 /s) h i (kJ/kg) ω=cte η is =cte Q surge Surge limit          1 ) ( 1 - 1     s d s s i p p M RT Z h (17)
Centrifugal compressors in the station are assumed to be driven by turbines whose supply energy is provided from a line of the gas derived from the pipeline passed through the station in order to be compressed as shown in Figure 2. The flow rate of the consumed gas as fuel for the compression process in each compressor is obtained by dividing required power for compression, P w , by the mechanical efficiency, η m , driver efficiency, η d and LHV (low heating value):

LHV n n n h m m d m is i ac f   3 10  ( 18
)
According to [START_REF] Abbaspour | Nonisothermal compressor station optimization[END_REF], we have chosen a value of m  (respectively d  ) equal to 0.90 (respectively 0.35). LHV represents the quantity of energy released by mass unity of the gas during complete combustion. It is considered at 298°K and 1 bar in (kJ/kg) and is calculated from the mass lower heating values, LHV i of the molecules composing the gas (M i is the molecular mass of specie i):

   i i i i i M y LHV M y LHV (19) Figure 2.

Representation of the centrifugal compressor and its incorporated turbine

The adiabatic efficiency is  is defined by Eq. ( 21). Applying standard polynomial curve-fitting procedures for each compressor, the normalized head h i /ω 2 can thus be obtained under the form of the following equation [START_REF] Abbaspour | Nonisothermal compressor station optimization[END_REF].

2 3 2 1 2 ) (    s s i Q b Q b b h    (20)
where Q s is the volumetric flow rate (m 3 /s).

In the same way, contours of constant isentropic efficiency could be fitted in the polynomial form of second degree shown in Eq. ( 21):

2 6 5 4 ) (    s s is Q b Q b b    (21)
The rotational speed  (defined from Eq. 17 and 20) of all compressors is comprised between lower and upper bound as represented below. To prevent from surge phenomenon, by considering a surge margin, λ surge , the following constraint is introduced [START_REF] Odom | Tutorials on modelling of gas turbine driven centrifugal compressors[END_REF]:

u l      (22) s surge s surge Q Q Q -   (23)
There is a surge flow rate, Q surge , corresponding to each compressor rotational speed [START_REF] Pugnet | Pompage des compresseurs[END_REF]:

2 / 1 2 1 2 2 7 ) ) ( ) ) ( 1 (( M p RT Z M p RT Z h Mp RT Z b Q s s s s s s surge s s s surge          (24)
In this equation, h surge is the surge head at a specified compressor speed and can be calculated using following equation:

2 3 2 1 2 ) (    surge surge surge Q b Q b b h    (25)
Considering a fixed value for a given surge efficiency, the surge efficiency will be introduced as a parameter during the optimization procedure. To avoid chocking occurrence at inlet, the following inequality should be considered.

) 1 ( 2 1 1 2              c A Q s s (26)
A s is the cross sectional area and c is the gas sonic velocity at the compressor inlet. Another inequality is introduced corresponding to the protection of a compressor against chocking phenomenon in impeller passages as shown in Inequality ( 27). In this expression, the impeller radius, r in (m) and A, the flow rate area in (m 2 ), are considered at the section of rotating passages as well as Q, Z, T, p. The index 1 indicates the impeller inlet state.

) 1 ( 2 1 2 1 2 1 1 1 / ) )( 1 ( 2                   c r A c pM ZRT Q (27)
To prevent from diffuser choking, another inequality similar to that of the compressor inlet is considered, but as shown below, in this relation the gas properties are in the conditions of the diffuser and index 2 is used for diffuser inlet.

) 1 ( 2 1 02 02 1 2               f f f f f A c M p RT Z Q (28)

Representing network topology by using incidence matrices

The different links between the elementary sections of a network can be defined using incidence matrices. Each pipe, each compressor and each fuel stream are represented by an arc. Consider a network with N n nodes, N p pipe arcs and N c compressor arcs. Therefore, there will be N c fuel streams since for each compressor unit there is a stream that carries fuel to it. Because in a compressor, compression process is carried out, a compressor unit can be named an active arc.

In this way, a pipe segment, in where the pressure decreases, may be called a passive arc. Let us note that the fuel streams have been considered as inert arcs regarding pressure change through them. A flow direction is assigned preliminarily to each pipe that can or not coincide with the real flow direction of the gas that running through the arc. N v valves can be introduced into the network to break the pressure between some pairs of arcs in order to balance the network.

Let A be a matrix of dimension ) (

V c p n N N N N   
, where each of its elements, a ij , is given the following attribution:

       otherwise 0 i node into goes j arc if 1 i node from out comes j arc if 1 ij a ( 29
)
A is called the node-arc incidence matrix. Similarly, let B be another matrix of dimension

c p N N 
whose elements, b ij , are defined below and it is named the pipe-compressor incidence matrix:

       otherwise 0 j compressor of node suction to connected is i pipe if 1 j compressor of node discharge to connected is i pipe if 1 ij b (30)
The last matrix is the node-fuel incidence matrix which describes the existing fuel stream derivations from a node, and it is called the compressor-fuel matrix. The dimension of this matrix is

c n N N 
and elements are defined below:

    otherwise 0 j node from derived be i stream fuel if 1 ij c (31)
This matrix indicates which fuel stream belongs to which compressor. These three incidence matrices are used to write the material balances around each node i, the flow rate of the consumed gas as fuel for the compression process in each compressor, and the equation of movement. For example, the material balance around the node i is expressed as Eq. ( 32). In this equation S i represents the gas delivery or supply relative to this node. It is negative if the node is a delivery one and positive for a supply node where the gas is injected to the node.

i s compressor j j f j i arcs j j j i S m b m a       , , (32)

Optimization procedures

Previous works

Monobjective optimization

A great diversity of optimization methods was implemented to meet the industrial stakes and provide competitive results. But if they prove to be well fitted to the particular case they consider, the numerical performances cannot be constant whatever the treated problem is. Actually, the efficiency of a given method for a particular example is hardly predictable, and the only certainty we have is expressed by the No Free Lunch Theory [START_REF] Wolpert | No free lunch theorems for optimization[END_REF]: there is no method that outdoes all the other ones for any considered problem. Among the diversity of optimization techniques, two important classes have to be distinguished: deterministic methods and stochastic ones. Complete reviews are proposed in literature for the two classes [START_REF] Hao | Métaheuristiques pour l'optimisation combinatoire et l'affectation sous contrainte[END_REF]Grossmann, 2002, Biegler and[START_REF] Biegler | Retrospective on optimization[END_REF]. A thorough analysis of both classes was previously studied by [START_REF] Ponsich | Stratégies d'optimisation mixte en Génie des Procédés -Application à la conception d'ateliers discontinus[END_REF] with the support of batch plant design problems.

The deterministic methods assume the verification of mathematical properties of the objective function and constraints, such as continuity, differentiability and convexity. In practice, these assumptions (particularly convexity) do not always hold, and the convergence towards a global optimum is no longer guaranteed. This working mode enables only to ensure to get a local optimum, which is for all that a great advantage versus stochastic methods. Among the deterministic class, particularly for NLP and MINLP problems, the following procedures can be mentioned: the Outer Approximation algorithm [START_REF] Duran | An outer-approximation algorithm for a class of mixed-integer nonlinear programs[END_REF], the Branch & Bound methods for scanning trees [START_REF] Gupta | Branch and bound experiments in convex nonlinear integer programming[END_REF][START_REF] Ryoo | Global optimization of nonconvex NLPs and MINLPs with applications in process design[END_REF][START_REF] Smith | A symbolic reformulation/spatial branch-andbound algorithm for the global optimization of nonconvex MINLPs[END_REF], the Generalized Benders Decomposition [START_REF] Geoffrion | Generalized benders decomposition[END_REF], the Extended Cutting Plane method for problems with a moderate degree of nonlinearity [START_REF] Westerlünd | An extended cutting plane method for solving convex MINLP problems[END_REF], disjunctive programming for quasi-convex problems [START_REF] Raman | Modelling and computational techniques for logic based integer programming[END_REF]. Even though most of the abovementioned methods are only academic tools, some computational codes are available: the SBB, BARON, DICOPT++ and LOGMIP solvers within the GAMS modelling environment [START_REF] Brooke | GAMS: a user's guide[END_REF], MINLP_BB [START_REF] Leyffer | User manual for MINLP_BB[END_REF] and ECP [START_REF] Westerlünd | Alpha-ECP, version 5.04, An interactive MINLP-solver based on the extended cutting plane method[END_REF]. Concerning the global optimization of non convex problems, the interval analysis method (Kearfott, 2001[START_REF] Csendes | Generalized subinterval violation criteria for interval global optimization[END_REF]) is a promising tool, but restricted at the present time to small problems, due to very high computational times.

The second class, namely stochastic methods, is based on the evaluation of the objective function at different points of the search space. These points are chosen through a set of heuristics, combined with generations of random numbers. Thus, stochastic procedures cannot guarantee to obtain an optimum.

However by allowing occasional objective function increases (for minimization problems) they may go out of local optimum gaps. They are divided into neighbourhood techniques such as Simulated Annealing [START_REF] Kirkpatrick | Optimization by simulated annealing[END_REF], Tabu Search [START_REF] Teh | Tabu search for global optimization of continuous functions with application to phase equilibrium calculations[END_REF], and evolutionary algorithms comprising genetic algorithms [START_REF] Holland | Adaptation in natural and artificial systems[END_REF], evolutionary strategies [START_REF] Beyer | Evolution strategies, a comprehensive introduction[END_REF] and evolutionary programming [START_REF] Yang | An evolutionary programming algorithm for continuous global optimization[END_REF]. Even if stochastic methods do not require any mathematical property for the objective function and constraints, they are difficult to implement for problems involving a significant number of equality constraints.

Multiobjective optimization

In many real world cases, the problem often involves several competing measures of performance, or objectives [START_REF] Collette | Optimisation multiobjectif[END_REF]. Using the formulation of multiobjective constrained problems of Fonseca and Fleming, (1998), a general multiobjective problem is made up of a set of n criteria f k , k = 1, …, n to be minimized or maximized. Each f k may be nonlinear with respect to the decision variable vector x in an m-dimensional universe U.

        x f x f x f n , ,  1  (33)
This kind of problem has not a unique solution in general, but presents a set of non-dominated solutions, named Pareto-optimal set or Pareto-optimal front. The Pareto-domination concept lies on two basic rules:

In the universe U a given vector u = (u 1 , …, u n ) dominates another vector v = (v 1 , …, v an ), if and only if,

    i i i i v u n i v u n i        : , , 1 : , , 1   ( 34
)
For a concrete mathematical problem, Eq. 34 gives the following definition of the Pareto front: for a set of n criteria, a solution f(x), related to a decision variable vector x = (x 1 , …, x m ), dominates another solution f(y), related to y = (y 1 ,…, y m ) when the following condition is checked (for a minimization problem),

            y f x f n i y f x f n i i i i i        : , , 1 : , , 1   ( 35
)
The last definition concerns the Pareto optimality: a solution x u  U is called Pareto-optimal if and only if there is no

x v  U for which v = f(x v ) = (v 1 , … , v n ) dominates u = f(x u ) = (u 1 , … , u n ).
These Pareto-optimal non-dominated individuals represent the solutions of the multiobjective problem. In practice, the decision maker has to select a single solution by searching among the whole Pareto front, and it may be difficult to pick one "best" solution out of a large set of alternatives. [START_REF] Branke | Finding knees in multiobjective optimization, Parallel problems solving from nature[END_REF], and [START_REF] Taboada | Data mining techniques to facilitate the analysis of the Pareto-optimal set for multiple objective problems[END_REF] suggest picking the knees in the Pareto front, that is to say, solutions where a small improvement in one objective function would lead to a large deterioration in at least one other objective. Other MCDM methods [START_REF] Miettinen | Nonlinear Multiobjective Optimization[END_REF] like Topsis [START_REF] Olson | Comparison of weights in TOPSIS model[END_REF], can be used.

As in the monobjective case, for solving a multiobjective optimization problem, two classes of numerical methods are available. In the deterministic approach, the MOOP (MultiObjective Optimization Problem) is solved by means of NLP or MINLP procedures by optimizing a unique objective resulting from the aggregation of the various conflicting objectives to be optimized [START_REF] Sawaragi | Theory of multiobjective optimization[END_REF]. Classical toolboxes like GAMS or MATLAB can be used. The Pareto front is obtained by scanning the search space corresponding to the possible values of weighting factors. Another approach, called -constraint method consists in optimizing a single objective function, the other objective being considered as parameterized constraints [START_REF] Mavrotas | Generation of efficient solutions in multiobjective mathematical programming problems using GAMS[END_REF][START_REF] Bérubé | An exact e-constraint method for biobjective combinatorial optimization problems[END_REF]. The major inconvenience of these methods is that they are highly dependent on the choice either of weighting factors or of the RHS of the constraints defined by the objectives.

An alternative way consists in implementing stochastic methods. All the algorithms cited above can be adapted to the multiobjective case, and as it can be observed in the list of references recently proposed by [START_REF] Coello | [END_REF]. Since 1990, the number of published papers per year is very important (more than 3700). Starting from a quasi null value in 1990, this number continuously increases, reaching 100 in 1995, 200 in 2000, 400 in 2005 and 500 the last year. The analysis of the list shows that genetic algorithms (GA) are cited in almost 40% of cases, and far behind them they are simulated annealing and particle swarms, followed by tabu search and differential evolution algorithms, then come the ant colonies and artificial neural networks, followed by constraint propagation methods, honeybee colonies, artificial immune systems and Monte-Carlo procedures. The two most popular methods in the chemical engineering field are MOGA (MultiObjective Genetic Algorithm, see [START_REF] Konac | Multi-objective optimization using genetic algorithms: A tutorial[END_REF], and MOSA (MultiObjective Simulated Annealing, see [START_REF] Shu | A novel Multi-objective orthogonal simulated annealing algorithm for solving multi-objective optimization problems with a large number of parameters[END_REF][START_REF] Smith | Dominance measure for multiobjective simulated annealing[END_REF][START_REF] Bandyopadhyay | A simulated annealing-based multiobjective optimization algorithm: AMOSA[END_REF]. None of these two methods is perfect and selecting one depends on the requirements of the particular design situation considered. From the literature survey [START_REF] Van Veldhuizen | Multiobjective evolutionary algorithms: analyzing the sate-of-the-art[END_REF][START_REF] Branke | Finding knees in multiobjective optimization, Parallel problems solving from nature[END_REF][START_REF] Turinsky | Evolution of nuclear fuel management and reactor operational aid tools[END_REF][START_REF] Mansouri | Bicriteria two-machine flowshop scheduling using metaheuristics[END_REF] it appears that MOGA is generally preferred to MOSA.

Choice of optimization methods

The compressor adjustment problem presented in the next section is a NLP one, first solved for minimizing the total fuel consumption in compressors at the compression stations, and then solved in the multiobjective case by considering besides the fuel consumption, the throughput (delivery gas mass flow) of the system. As mentioned above, in the monobjective case, the deterministic methods are the most efficient. A solver (CONOPT3) of the GAMS environment was chosen, since this optimization package is widely used, and even stands as a reference for the solution of problems coming from the Process Systems Engineering field. The MATLAB optimization toolbox (namely FMINCON) was also used for comparison purposes.

In the multiobjective case, two solution methods are implemented and compared. A genetic algorithm called NSGA-IIb belonging to the genetic algorithm library (MULTIGEN) recently developed in the research group by [START_REF] Gomez | Optimisation technico-économique multi-objectif de systèmes de conversion d'énergie : Cogénération électricité-hydrogène à partir d'un réacteur nucléaire de IVème génération[END_REF] has been first retained. Compared with the well-known NSGA II of [START_REF] Pratap | A Fast and Elitist Multiobjective Genetic Algorithm: NSGA-II[END_REF], new genetics operators are introduced for clones creation limiting. The classical crossover operator SBX has been modified to produce children different from parents. The objective is to prevent unnecessary calculations for clones of existing solutions: all solutions generated by the crossover and mutation procedures are statistically different [START_REF] Gomez | Optimisation technico-économique multi-objectif de systèmes de conversion d'énergie : Cogénération électricité-hydrogène à partir d'un réacteur nucléaire de IVème génération[END_REF]. The set of constraints is made up of mass and momentum balances on the one hand and of compressor equations on the other hand. The numerical solution of this set of equations must be done carefully, making sure that the equality system of equations captures all the relevant aspects of the associated network problem. This square set of nonlinear equations is parameterized by the optimization variables (rotational speeds of compressors), and so the MOOP is solved by means of a two-stage strategy. At the upper level the GA returns values of the rotational speeds, which are reported into the set of constraints, written under the format of MATLAB R2008a and solved by a Newton-Raphson procedure of the MATLAB toolbox. To efficiently solve this set of nonlinear equations, suitable variable bounds and initial values have to be applied at each node of the network. These values are taken from Tabkhi's PhD Thesis (2007).

The second method implemented is the -constraint procedure. In this method one of the objective functions is minimized while all the other objective functions are upper bounded by introducing supplementary constraints. So the multiobjective problem is transformed into the following monobjective one:

U x x f k  ) ( Min with the additional constraints: (36) , to 1 , ) ( k i n i x f i i    
By parametrical variation in the RHS of the constrained objective functions the efficient solutions of the problem can be obtained. In order to properly apply the ε-constraint method, the range of the n-1 objective functions that are used as constraints, must be known. The calculation of the range of the objective functions over the efficient set is not a trivial task. While the best value is easily attainable as the optimum of the individual optimization, the worst value over the efficient set (Nadir value) is not. The most common approach is to calculate these ranges from the payoff table (the table with the results from the individual optimization of the n objective functions). From [START_REF] Ehrgott | Multiobjective Programming, Multiple Criteria Decision Analysis. State of the Art Surveys[END_REF], the optimal solution of problem ( 36) is guaranteed to be an efficient solution only if all the (n-1) objective function constraints are binding. To overcome this difficulty, [START_REF] Mavrotas | Generation of efficient solutions in multiobjective mathematical programming problems using GAMS[END_REF] proposes the transformation of the objective function constraints to equalities by explicitly incorporating the appropriate slack or surplus variables. In the same time, the sum of these slack or surplus variables is used as a second term (with lower priority) in the objective function forcing the program to produce only efficient solutions. A quite similar approach based on slack variables is presented by [START_REF] Ehrgott | Improved -constraint method for multiobjective programming[END_REF].

Numerical example: compressor adjustment problem

Network characteristics

The example is inspired from the work of [START_REF] Abbaspour | Nonisothermal compressor station optimization[END_REF] (see Figure 3). The network consists of three long pipelines of 100 kilometres. There are two compressor stations that operate to compensate for pressure drop in the pipelines. Each compressor station includes three parallel centrifugal compressors. In each station, there are six short pipe segments of about a hundred meters linked to the entrances and outlets of the compressors. Although the length and the diameter of these pipes is lower than those of the three major pipelines, their role in the pressure change through the network may not be negligible and may even sometimes become bottleneck of the system. Therefore, these pipelines are also considered in the model. The technical features of the pipeline system corresponding to Figure 3, considered as fixed parameters for the optimization problem, are proposed in Table 1.

Figure 3. Schema of the considered pipeline network Table 1.Technical features of the pipelines of the system shown in Figure 3 Pipeline tag [START_REF] Menon | Gas pipeline hydraulics[END_REF]. The network includes 18 nodes, 15 pipes arcs and 6 compressor arcs. As for each compressor unit, there is a stream that carries fuel to it; there are 6 fuel streams which have not been shown in Figure 3 to avoid complexity. For each compressor, this stream originates from suction node. A flow direction is assigned to each pipe so the gas flows from 0 to 17. For example, the node-arc incidence matrix A (see Table 3) is a matrix of dimension 18 × (15+6) (the network does not involve any valve). The material balance around the nodes can be stated in a very concise way by using this matrix.

G-1 G-2 G-3 G-4 G-
Roughness of inner surface of the pipes is considered to be equal to 46×10 -6 (traditional value reported for stainless steel). The temperature is assumed to be isothermal and equal to 330°K all over the system. The adiabatic efficiency is  is defined by Eq. ( 21), and mechanical efficiency and driver efficiency for the The pressure is considered to be equal to 60 bar with a margin of ±2% at the entrance point of the network, node 0, as well as the delivery pressure, at node 17 (in other words the lower bound is 58.8 bar and the upper one is 61.2 bar). The gas flows from node 0 towards node 17, and there is no input or output in the other nodes. The typical composition of NG considered in the numerical runs is presented in Table 2 with also the thermodynamic properties of gas components. Table 3. Node-arc incidence matrix corresponding to Figure 3 (+ represents +1, -represents -1, blank represents 0)

G 1 G 2 G 3 G 4 G 5 G 6 G 7 G 8 G 9 G 10 G 11 G 12 G 13 G 14 G 15 C 1 C 2 C 3 C 4 C 5 C 6 Node + 0 _ + + + 1 _ + 2 _ + 3 _ + 4 + _ 5 + - 6 + _ 7 _ + 8 _ + 9 _ + 1 0 + _ 1 1 + _ 1 2 + _ 1 3 _ _ _ + 14 + + + _ 1 5 + _ _ _ 1 6 _ 1 7
Monobjective case: total fuel consumption optimization

NLP formulation

The continuous variables of this problem are: 18 pressure variables governing the nodes and 21 flow rate variables (including fuel streams) corresponding to pipes and compressors. The rotational speeds of the compressors have not been explicitly considered as variables, since the flow rates of the fuel streams have already been considered as variables for each compressor. As shown in Eq. ( 18) to (25), the rotational speeds are directly dependent on the fuel stream flow rates.

The coefficients b i of Eq. ( 20) and ( 21) are reported in Table 4.

m 2 m -1 m -4 - m -3 m -6
The equality constraints consist of 18 mass balances around nodes, 15 equations of motion for the pipe arcs, 6 isentropic head equations for compressors as shown in Eq. ( 17), 6 relationships between rotational speed, suction volumetric flow rate and head of each compressor (Eq. 20), 6 equations to calculate isentropic efficiency according to Eq. ( 21), 6 equations to determine fuel consumption at each compressor unit. The set of inequality constraints is constituted by a lower bound for delivery flow rate (flow rate in arc G2) equal to 150 kg/s, an upper bound as well as a lower bound for the pressures of the nodes (MAOP as an upper bound and atmosphere pressure as a lower bound, the following values were chosen for computing the MAOP: f F = 0.72, f E = 1, f T = 1), sonic velocity and erosional velocity in the role of upper bounds of the velocities through pipes, lower and upper bounds on the rotation speed of all compressors (166.7 and 250 rps respectively), a lower bound on compressor throughput in order to avoid pumping phenomenon, an upper bound on compressor throughput to prevent from chocking phenomenon. In total, there are 57 equality constraints and 76 inequality constraints.

The total sum of the fuel consumption in compressors is the objective function, as expressed in Eq. ( 37). For each compressor, fuel consumption flow rate, i f m  is obtained by using Eq. [START_REF] Raman | Modelling and computational techniques for logic based integer programming[END_REF].

   s compressor i i f obj m f  (37)

Problem solution

As indicated above, in this monobjective case, the solver CONOPT3 of the GAMS package has been retained for solving the problem. The initialization of the variables is performed directly through the software CONOPT3 under the condition that the problem is well-scaled and that bounds are assigned adequately. For bounded variables, CONOPT3 takes the initial values in the middle of the bounds. Several other initial points were randomly selected (inside the bounds, for bounded variables), the same solution was obtained. Strictly considering the nonconvexity feature, the example is not so strongly nonconvex.

The options used for implementing CONOPT3 are following: optimality tolerance = 10 -8 , maximum feasibility tolerance = 10 -5 , number of stalled iterations = 100. The resolution takes about 0.5s CPU on a PC (processor Intel Core 2 Duo, 2.99 GHz, RAM 1.96 Go). With same tolerances MATLAB gives identical results, but the CPU time is higher (a few seconds). Table 5 presents the results relative to pressure values at each node. Observe that at node 0 (i.e., supply node), the algorithm has taken the maximum possible pressure (61.2 bar) whereas the minimum possible value (58.8 bar) was obtained at node 17 (i.e., delivery node). The value of objective function that is the total fuel consumption in the compressor stations is equal to 0.749 kg/s (sum of individual compressor consumptions, see Table 6, bold line) leads to a significant reduction of 13% from the initial solution (0.863 kg/s for initial values between their bounds). Other results are listed in Table 6. The optimum percentage of the input gas that is consumed in the stations can thus be calculated and is found equal to 0.499 %. For each compressor, consumption ratio is defined as the fuel consumption divided by the input mass flow rate. Let us mention in this example that the compressors involved in the second station work at their minimum rotational speeds, whereas the compressors of the first station work close to their maximum speeds. Finally, the transmitted power of the pipeline, that is the product of the pipeline delivery throughput (150 kg/s) and the lower heating value (LHV) of the NG (48830 kJ/kg) is found to be equal to 7324 MW at this optimal point. The Lagrange multipliers obtained at the solver convergence can be used to carry out a sensitivity analysis. All these parameters are null or quasi-null except for the supply pressure at node 0 (value=-0.047) and the delivery pressure at node 17 (value 0.017). This means for example that if the supply pressure is increased of one bar, the total fuel consumption will be decreased of 0.047 kg/s. In the same way, if the delivery pressure is decreased of one bar, the total fuel consumption will be decreased of 0.017 kg/s.

Carbon dioxide emissions

The total fuel consumption in the compressor stations is found equal to 0.749 kg/s, that is to say 23 640 ton/year. The combustion reaction of one molecule of methane (molar mass = 16 g) produces one molecule of CO 2 (molar mass = 44 g). One molecule of ethane (molar mass = 30 g) gives two molecules of CO 2 , and for one molecule of propane (molar mass = 44 g), three molecules of CO 2 are obtained.

The results are summarized in Table 7. The carbon dioxide emissions are 66 332 ton/year. Let us recall that the NG delivery is 150 kg/s that is to say 4 730 400 ton/year. The carbon dioxide emissions represent only 1.4% of the delivery gas, which is very acceptable.

Multiobjective case: total fuel consumption minimization and NG delivery maximization

Problem formulation

In the previous section the fuel consumption in the compressor stations was minimized for a given gas mass flow delivery. However for a natural gas delivery Company, the demand may vary according to climatic conditions or industrial requirements. So the problem which arises is to determine, for a given supply at the network entrance nodes, the minimal and maximal network capacities in terms of NG mass flow delivery and fuel consumption in compressor stations. This problem can be formulated as a biobjective optimization problem.

In fact, it's not about a problem a decision making strictly speaking, as far as the practical problem formulates as follows. For a NG delivery Company, the total mass flow delivery is imposed on a given period, and the problem is to operate the compressor stations so as to minimize the fuel consumption in the stations. When performing the biobjective optimization, the Pareto front (see Figure 4) provides an easy way for: 1 -identifying the minimum and maximum network capacities in terms of mass flow delivery and fuel consumption; 2 -for a given mass flow delivery between the above bounds, the minimal fuel consumption and thus the minimal carbon dioxide emission can be deduced.

Concerning the optimization variables and constraints, the problem is identical to the previous one, but here the NG mass flow delivery is not fixed at 150 kg/s. The goal is to simultaneously minimize the total fuel consumption (this objective is noted f 1 ) in the compressor stations, while maximizing the NG delivery mass flow (objective noted f 2 ).

Problem solution

As mentioned above, the solver NSGA-IIb of the MULTIGEN library, coupled with a Newton-Raphson procedure was implemented for solving the multiobjective problem. To simplify the scanning of the search space, it was assumed that the rotational speeds can only take integer values. The options used for implementing NSGA-IIb are: population size = 100, maximum number of generations = 300. The GA was run 10 times with different initial values for the rotational speeds (randomly generated); the Pareto front (symbols crosses) reported on Figure 4 has been obtained five times. The resolution of one GA takes an average time of 9 hours CPU on the same PC as above. were found again. Conversely, the NG mass flow delivery was computed with compressor rotational speeds given at the two extreme points, the mass flow delivery of 133 kg/s and of 157 kg/s were obtained again.

It can be observed on the Pareto front, that for NG mass flow delivery of 150 kg/s, the same value (0.749 kg/s) of the total fuel consumption as in the monobjective case is found again. The values of pressures, discharge flow rates, rotational speeds, fuel consumptions, isentropic head and isentropic efficiency for the compressors are the same as the ones listed in Tables 5 and6. The carbon dioxide emissions are given in Table 7.

A supplementary verification was carried out by implementing the constraint method. The obtained front is given by symbols circles for 40 generated solutions and triangles for 100 generated solutions on Figure 4. The two fronts obtained from genetic algorithm and -constraint are superimposed but the front of -constraint is much more restricted than the one of genetic algorithm. If the problem was a MCDM one, the -constraint would be a more efficient tool than the genetic algorithm due to its restricted front.

From these checks, we can assume that the Pareto front given by the genetic algorithm is correct and brings more information because more extended than the one of -constraint.

Carbon dioxide emissions

For the pair (f 1 = 0.749 kg/s, f 2 = 150 kg/s), the results are reported in Table 7. Two new studies for the extreme solutions (case 1: f 1 = 0.540 kg/s, f 2 = 133 kg/s) and (case 2: f 1 = 0.980 kg/s, f 2 = 157 kg/s) are carried out, the results are indicated in Tables 8 and9.

Case 1

The carbon dioxide emissions are 47 823 ton/year (see Table 8). The NG delivery is 133 kg/s that is to say 4 194 288 ton/year. The carbon dioxide emissions represent 1.1% of the delivery gas. The carbon dioxide emissions are 86 794 ton/year (see Table 9). The NG delivery is 157 kg/s that is to say 4 951 152 ton/year. The carbon dioxide emissions represent 1.8% of the delivery gas.

Discussion

Along the Pareto front, the carbon dioxide emissions vary from 1.1% to 1.8% of the NG mass flow delivery. These values are very lower than those usually admitted; indeed as mentioned in the Introduction section it is estimated [START_REF] Suming | Model relaxations for the fuel cost minimization of steady-state gas pipeline networks[END_REF] that the compressor stations typically consume about 3 to 5% of the transported gas. So the optimization of compression operations yields significant savings for the fuel consumed in the stations.

Conclusion

Efficient operation of compressor stations is of major importance for enhancing the performances of pipeline networks. In this paper, a pipeline network system including two compressor stations is optimized by implementing two strategies.

In the monobjective case a deterministic optimization procedure is used. In the multiobjective case a genetic algorithm and a -constraint method are implemented and compared. A didactic example coming from the literature illustrates the approach.

In the monobjective study, the objective function is the total fuel consumption in the compressor stations to be minimized for a fixed gas delivery mass flow, since reduction of the energy used in pipeline operations will have a significant economical impact. Typical results are analyzed and the characteristic values of some key parameters like isentropic head and isentropic efficiency are computed. The numerical results show that numerical optimization is an effective tool for optimizing compressor rotational speeds, and can yield significant reductions in fuel consumption. The carbon dioxide emissions evaluated at the optimal solution represent only 1.4% of the delivery gas, which is very acceptable.

For the multiobjective study, the goal consists in simultaneously minimizing the total fuel consumption while maximizing the gas mass flow delivery. The problem is solved by means of a genetic algorithm and -constraint procedure. The two methods give superimposed Pareto fronts, but the one from genetic algorithm is much larger than the one from -constraint.

Along the Pareto front provided by the genetic algorithm, the carbon dioxide emissions vary from 1.1% to 1.8% of the NG mass flow delivery. It is estimated that the compressor stations typically consume about 3 to 5% of the transported gas. So the optimization of compression operations yields big savings for the fuel consumed in the stations, and thus has a real environmental impact. Contrary to classical decision making problems, where the best points have to be found on the Pareto front by searching "knees" on the front or by using MCDM method like Topsis, the Pareto front supplies two significant information. First, bounds on the network capacity in terms of mass flow delivery and CO 2 emissions can directly be obtained from the curve. Second, for an imposed mass flow delivery that corresponds to practical case for a NG delivery Company, the minimal fuel consumption directly linked to CO 2 emissions can be obtained by tuning compressor stations (particularly rotational speeds of compressors) at values provided by the optimizer.

  to be 0.90 and 0.35 respectively according to values proposed in the dedicated literature

Figure 4 .

 4 Figure 4. Pareto frontRelevant information lies in the two extreme points of the front, insofar as they represent the minimum and maximum capacities of the network in terms of NG delivery and fuel consumption. These solutions were verified by performing monobjective optimizations of the fuel consumption for a NG mass flow delivery of 133 kg/s and then of 157 kg/s; the same solutions for the fuel consumption

Table 2 .

 2 Thermodynamic properties of the components of gas flowing in the pipelines

	Component	Methane	Ethane	Propane
	Mole percent (%)	70	25	5
	Molecular weight (kg/mol)	16	30	44
	Critical temperature (°K)	190.6	305.4	369.8
	Critical pressure (bar)	46 0	48.8	42.5
	Lower Heating Value (kJ/kg)	50009	47794	46357
	Heat capacity at constant pressure,	35.663	52.848	74.916
	kJ/(kmol-°K)

Table 4 .

 4 Coefficients of the isentropic head equation and coefficients of the isentropic efficiency equation of the compressors

	Coefficient:	b 1	b 2	b 3	b 4	b 5	b 6
	Value:	3.8113 4 10  	3.849 ×10 -6	-6.3985 ×10 -9	17.269 0.3237	-4.1789 ×10 -4
	Unit:

Table 5 .

 5 Pressure of NG at all of the nodes of the pipeline network

	Node Pressure (bar)	Node Pressure (bar)
	0	61.2	9	58.3
	1	47.4	10	58.4
	2	47.0	11	65.2
	3	47.1	12	65.5
	4	47.2	13	65.2
	5	67.0	14	66.8
	6	66.9	15	58.4
	7	67.0	16	65.0
	8	58.3	17	58.8

Table 6 .

 6 Optimal values of discharge flow rates, rotational speeds, fuel consumptions, isentropic head and isentropic efficiency for the compressor units of the network

	Compressor		C1	C2	C3	C4	C5	C6
	Discharge flow rate	49.186 50.450 50.559 50.200 49.521 50.279
	(kg/s)
	Rotational	speed	244.3	246.5	246.6	166.7	166.7	166.7
	(rpm)
	Fuel consumption	0.182	0.186	0.187	0.064	0.066	0.064
	(kg/s)
	Consumption ratio	0.369	0.367	0.369	0.127	0.133	0.127
	(%)
	Isentropic	head	42.592 42.188 42.201 12.664 13.367 12.607
	(kJ/kg)
	Isentropic efficiency	74.917 74.215 74.207 64.195 65.331 64.101
	(%)

Table 7 .

 7 CO 2 emissions

	Total consumption	fuel	23 620	emission Total CO 2	66 332
	(ton/year)			(ton/year)
	Methane (70%)	16 534	CO 2	4 5 4 6 8
	Ethane (25%)	5 905	CO 2	1 7 3 2 1
	Propane (5%)	1 181	CO 2	3 5 4 3

Table 8 .

 8 CO 2 emissions (case 1)

	Total consumption	fuel	17 029	emission Total CO 2	47 823
	(ton/year)			(ton/year)
	Methane (70%)	11 920	CO 2	3 2 7 8 0
	Ethane (25%)	4 257	CO 2	1 2 4 8 7
	Propane (5%)	852	CO 2	2 5 5 6

Table 9 .

 9 CO 2 emissions (case 2)

	Total consumption	fuel	30 905	emission Total CO 2	86 794
	(ton/year)			(ton/year)
	Methane (70%)	21 635	CO 2	5 9 4 9 6
	Ethane (25%)	7 726	CO 2	2 2 6 6 3
	Propane (5%)	1 545	CO 2	4 6 3 5

The authors graciously acknowledge the funding of this work to the Secretaria de Educacio Publica (SEC) and the Consejo Nacional de Ciencia y Tecnologia (CONACYT) from the Mexican Government.


Agathe Legrand
email: agathe.legrand@etu.parisdescartes.fr

Karine Doré Mazars

Julie Lazzareschi

Christelle Lemoine

Isabelle Olivier

Julien Barra

Maria Pia Bucci

K D Mazars

Differing effects of prosaccades and antisaccades

Keywords: Posture, Eye movements, Antisaccades, Prosaccades, Dual-task

à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

Postural control has long been considered as an automatic system, but recent studies have identified a process of postural regulation in cases of both simple and more complicated tasks, especially when the latter involve attentional processes [START_REF] Blanchard | The influence of concurrent cognitive tasks on postural sway in children[END_REF][START_REF] Palluel | Postural control and attentional demand during adolescence[END_REF]. The mobilization of such attentional resources depends on many factors, such as availability of sensory information [START_REF] Shumway-Cook | Attentional demands and postural control: the effect of sensory context[END_REF], task complexity and age [START_REF] Woollacott | Attention and the control of posture and gait: a review of an emerging area of research[END_REF], or postural skills of the participants [START_REF] Lajoie | Attentional demands for static and dynamic equilibrium[END_REF].

In 1985, Kerr et al. were the first to show, using the dualtask paradigm, that the postural control of young adults is attention dependent and that postural stability is not affected by all the cognitive tasks in the same way. In 1993, [START_REF] Lajoie | Attentional demands for static and dynamic equilibrium[END_REF] and more recently [START_REF] Barra | Increasing cognitive load with increasing balance challenge: recipe for catastrophe[END_REF] described the correlation between the complexity of postural conditions and the associated attentional costs.

In other words, visual inputs which have a central role in the control of posture are known to require attention, and attention is also involved in the execution of saccadic eye movements [START_REF] Deubel | Saccade target selection and object recognition: evidence for a common attentional mechanism[END_REF][START_REF] Belopolsky | When are attention and saccade preparation dissociated?[END_REF]. Thus, saccades have an impact on balance but also on the adjustment of the vestibuloocular function [START_REF] Leigh | The neurology of eye movement, 4th edn[END_REF]. Several structures of the central nervous system in the cerebral cortex (frontal, parietal, occipital) and in the brainstem (paramediane pontine reticular formation and superior colliculus) play an important role in postural control as well as in the programming and execution of saccadic eye movements [START_REF] Leigh | The neurology of eye movement, 4th edn[END_REF]; consequently, one could expect interferences between oculomotor and postural control.

Indeed, the effect of saccadic eye movements on postural control has long been investigated. For instance, some authors [START_REF] Uchida | Effects of periodic saccades on the body sway in human subjects[END_REF][START_REF] Stoffregen | Postural sway and the frequency of horizontal eye movements[END_REF][START_REF] Rougier | Performing saccadic eye movements or blinking improves postural control[END_REF] showed an improvement in postural stability while saccadic eye movements are performed. In contrast, other researchers [START_REF] White | Saccadic eye movements and body sway[END_REF][START_REF] Straube | Visual ataxia induced by eye movements: posturographic measurements in normal and patients with ocular motor disorders[END_REF][START_REF] Glasauer | How the eyes move the body[END_REF] found that eye movements did not modify or decrease postural stability. Note, however, that the controversial results could be due to the varying experimental conditions used in the different studies, such as the type of postural test (bipodal or unipodal position) or characteristics of saccades (saccade amplitudes varying from 4° to 40°, horizontally or vertically directed saccades).

In normal young participants, the surface of center of pressure (CoP), length of CoP, and mean speed of CoP are good spatio-temporal variables currently used to study the variation of postural control [START_REF] Stoffregen | Postural sway and the frequency of horizontal eye movements[END_REF][START_REF] Rougier | Performing saccadic eye movements or blinking improves postural control[END_REF]. Other postural parameters can be used when a visual isolated event is manipulated. For example, the time to onset of a modification of amplitude of CoP after neck muscle or galvanic stimulation in the opposite direction of gaze is observed [START_REF] Bove | Interaction between vision and neck proprioception in the control of stance[END_REF][START_REF] Grasso | Time course of gaze influences on postural responses to neck proprioceptive and galvanic vestibular stimulation in humans[END_REF]. Nevertheless, the examination of those parameters is not possible when a series of saccades is manipulated.

The goal of the present study was to examine the effect of saccadic eye movements on postural control in young adults. In order to gain a better insight into how saccades influence postural control, we examined posture while different types of saccades were performed: prosaccades (saccades made toward the target, in a reactive and voluntary mode) as well as antisaccades (saccades made in the opposite direction of the target) and two types of fixation: without and with distractors. Furthermore, two postural conditions were used to change postural stability, the Standard Romberg (ST), and the Tandem Romberg (TR) positions, and the latter being more challenging still than the former. Our driving hypothesis was that a high-level cognitive task such as that involved in antisaccades could decrease postural stability, in particular in the more complex posture (TR).

Material

Platform posturography

A platform (principle of strain gauge) consisting of two dynamometric clogs (Standards by Association Française de Posturologie, produced by TechnoConcept©, Céreste, France) was used to measure postural stability. Excursions of CoP were recorded for 25.6 s. The platform posturography is equipped with an analog-digital converter (16 bit), and the acquisition frequency is 40 Hz.

Participants

Ten young adults (25 ± 3 years) participated in the study. All participants had good corrected visual acuity (≥20/25), tested with the Monoyer scales, at a distance of 5 m; they had normal binocular vision (<60 s of arc) as established by the TNO random dot test (Netherlands Organisation of Applied Scientific Research Test of Stereoacuity).

The investigation adhered to the principles of the Declaration of Helsinki and was approved by our Institutional Human Experimentation Committee. We obtained the participants consent after we explained the experimental procedure.

Methods

Visual tasks

Five visual tasks were designed and tested in separate sessions: two fixation tasks (without and with distractors) and three saccadic tasks (prosaccades: reactive and voluntary saccades and antisaccades). Stimuli were presented on the PC screen (19 inches, resolution: 800 × 600 pixels, luminance of background: 7 cd/m 2 ) adjusted at the eye level in front of the participant. It should be noted that eye movements are not recorded in this study. Most participants were accustomed to oculomotor experiments, and each participant was trained to the oculomotor tasks used here before postural recording.

Stimuli were white on gray background (4.8 cd/m 2 ). The fixation cross (0.36°) was displayed at the center of the screen. For all the visual tasks (except Fixation without distractor), a peripheral stimulus (saccade target or distractor) was a white disk (1.5° diameter, 38 cd/m 2 ) with a small gray disk in the center (0.2°), randomly presented at 4° leftward or rightward from the central fixation cross along the horizontal axis. The experimental paradigms are illustrated in the Fig. 1.

Fixation tasks

Fixation without distractor

Participants had to fixate the white cross that appeared at the center of the screen for 25.6 s (Fig. 1a).

Fixation with distractors

Participants had to fixate the cross for 25.6 s while a peripheral distractor randomly appeared leftward or rightward from the central cross (Fig. 1b).

Saccadic tasks

Prosaccade tasks

Horizontal, visually guided saccades were elicited by using two different temporal paradigms, namely simultaneous (gap = 0 ms) and overlap paradigms. In the simultaneous paradigm, the target appears simultaneously with the offset of the central fixation point, while in the overlap paradigm, the central fixation point stays on and, at the same time, a target appeared to the right or to the left side. The simultaneous paradigm (gap = 0 ms) is well known to induce reactive saccades, whereas the overlap paradigm stimulates voluntary saccades needing disengagement of attention in order to perform the saccade [START_REF] Fischer | The preparation of visually guided saccades[END_REF].

For more details, before each saccade target, a central fixation cross was switched on for a random period of 400, 600, or 800 ms.

For reactive saccade, after a variable delay (400, 600 or 800 ms), the central fixation cross was switched off, and simultaneously, a peripheral target stayed on for 300 ms. The screen was blank for the following 500 ms (Fig. 1c).

For voluntary saccades, the central fixation cross was illuminated for 400, 600, or 800 ms. The saccade target appeared while the fixation cross stayed on, both were visible for 600 ms, that is, the overlap period. Then, the central fixation cross was switched off, and the peripheral target remained for 300 ms, followed by the blank screen of 500 ms (Fig. 1d).

In both prosaccade tasks (simultaneous and overlap paradigms), participants were instructed to make a saccade to the peripheral target (around 18 reactive and 12 voluntary saccades for the postural recording duration).

Antisaccade task

Antisaccades have also been stimulated (Fig. 1e). This task was originally introduced by [START_REF] Hallet | Primary and secondary saccades to goals defined by instructions[END_REF]: It consisted in presenting a visual stimulus on one side and asking participants to make a saccade to the opposite side. The central cross was illuminated for a random period of 400, 600, or 800 ms. Simultaneously to the cross extinction, a peripheral stimulus appeared and stayed on for 300 ms. The screen was blank for the following 500 ms. The sequence of visual events was similar to those used in the simultaneous (gap = 0 ms) paradigm, but in this case, participants were invited to make an antisaccade to the opposite side of the stimulus as soon as possible.

Postural recording procedure

Participants stood on the platform, in front of the screen located 114 cm away from them. Postural measurements were performed in two different postural conditions: Standard Romberg and Tandem Romberg. In the Standard Romberg condition, the heels were placed 4 cm apart and feet positioned symmetrically with respect to the participant's sagittal axis at a 30° angle. In the Tandem Romberg position, the feet were placed slightly apart (4 cm) in a semi-tandem position with the dominant foot in front of the non-dominant one.

The duration time of each postural recording was 25.6 s. For each condition (5 visual tasks × 2 postural positions), four trials were run. Thus, a total of 40 postural measures were recorded for each participant. The order of the visual tasks and of the postural conditions varied randomly across participants. Participants were asked to stay as stable as possible, with the arms along the body.

Data processing

To quantitate the effect of visual tasks and the postural performance from data obtained from the platform, we analyzed the surface area, the length, and the mean speed of the CoP. The surface area and the length allow to efficiently measure CoP spatial variability [START_REF] Chiari | Stabilometric parameters are affected by anthropometry and foot placement[END_REF]). The surface of CoP corresponds to an ellipse with 90 % of CoP excursions. The length of CoP is the path of the center of pressure. These two postural parameters are uncorrelated; indeed, the inner surface of the same length may be different [START_REF] Gagey | Posturology; Regulation and disruption of standing[END_REF][START_REF] Vuillerme | Can a plantar pressure-based tongue-placed electrotactile biofeedback improve postural control under altered vestibular and neck proprioceptive conditions?[END_REF]. The mean of speed represents a good index of the amount of neuromuscular activity required to regulate postural control [START_REF] Maki | Aging and postural control: a comparison of spontaneous-and induced-way balance tests[END_REF][START_REF] Geurts | Intrasubject variability of selected force platform parameters in the quantification of postural control[END_REF]. Figure 2 shows stabilogramms of a representative participant in two different postural conditions recorded during the five visual tasks.

Statistical analysis

Analyses of variance (using the ANOVA test) were performed to compare data in the different visual tasks with the different type of postural conditions: two-way ANOVA: 5 Visual tasks (fixation without and with distractors, reactive and voluntary prosaccades, and antisaccades) × 2 postural conditions (Standard and Tandem Romberg). Post hoc comparisons were made with the Fischer's least significant differences (LSD) test used to explore further and compare the mean of one oculomotor task or postural positions with the mean of another. The effect of a factor was considered as significant when the p value was below 0.05.

Results

Analysis of the surface of CoP (Fig. 3) showed a significant effect of the postural condition (F (1,9) = 40.10, p < 0.001) that was greater in Tandem than in Standard Romberg condition. There was also a significant effect of the visual task (F (4,36) = 11.71, p < 0.001). Interestingly, there was a significant interaction between the visual task and the postural condition (F (4,36) = 2.66, p < 0.05). The effect of the visual task did not reach significance in standard Romberg condition (F (4,36) = 2.0, p = 0.1), in contrast to the tandem Romberg condition (F (4,36) = 7.70, p < 0.01). While the antisaccade task did not differ from the prosaccade task (compared to reactive saccades, p = 0.5 and to voluntary ones p = 0.2), the surface of CoP was greater in both fixation tasks compared to that of saccadic tasks (F (1,9) = 19.09, p < 0.01).

Analysis of the length of CoP (Fig. 4) showed a significant effect of postural condition, with smaller values in the Standard than in the Tandem Romberg condition (F (1,9) = 162.27, p < 0.001). There was also a significant global effect of the visual task (F (4,36) = 6.60, p < 0.001), but this effect did not interact with the postural position (F < 1). Nevertheless, although the length of CoP was significantly greater in both fixation tasks (without and with distractors) than those measured in prosaccadic tasks (reactive as well as voluntary saccades) for both postural position, the length of CoP was significantly greater in the antisaccade task than in both prosaccade tasks (F (1,9) = 7.26, p < 0.05 and F (1,9) = 7.25, p < 0.02, respectively, for reactive and voluntary prosaccades), only in the Standard Romberg position.

Analysis of the mean speed of CoP (Fig. 5) showed a significant effect of postural condition, with smaller values in the Standard than in the Tandem Romberg condition (F (1,9) = 164.79, p < 0.001). There was also a significant global effect of the visual task on the mean speed of CoP (F (4,36) = 6.43, p < 0.001), but this effect did not interact with the postural position (F < 1). Nevertheless, similar to the length of CoP, values of the mean speed of CoP in both fixation tasks (without and with distractors) were significantly greater than those measured in prosaccadic tasks (reactive as well as voluntary prosaccades) for both postural conditions. While in the antisaccade task, the mean speed of CoP was significantly greater than the value reported in both prosaccade tasks (F (1,9) = 7.26, p < 0.05 and F (1,9) = 7.25, p < 0.02 for the reactive and voluntary saccades, respectively) only in the Standard Romberg position.

Discussion

The goal of the study was to compare the effect of different visual tasks (fixation without and with distractors, prosaccades-reactive and voluntary-, antisaccades) on the surface, length, and mean speed of CoP when using two types of postural conditions (Standard Romberg and Tandem Romberg). Main findings are the important effect of postural positions, with more body sways in Tandem Romberg compare to Standard Romberg. A global effect of visual tasks showed less body sways when participant made saccades compared to fixation conditions. A difference between saccades was found in Standard Romberg position, in which antisaccades increased body sways compared to reactive saccades (for length and mean speed of CoP).

Different effects of saccades and fixations on postural control

As suggested in the Introduction, the effect of visual tasks in postural control is still controversial. Our results showed that postural stability seems to be improved when the participant is making eye movements rather than staring at the fixation point. In saccadic tasks, the surface of CoP decreased with respect to the fixation tasks (fixation with or without distractors). These results are in line with those obtained by [START_REF] Uchida | Effects of periodic saccades on the body sway in human subjects[END_REF] and [START_REF] Rougier | Performing saccadic eye movements or blinking improves postural control[END_REF] who reported an improvement of postural stability during the execution of saccades, and they contrast with the results showed by [START_REF] Glasauer | How the eyes move the body[END_REF] who found a decrease of postural stability during the execution of eye movements.

Further explanations on the decrease of body sways while performing eye movements are exposed below. [START_REF] Uchida | Effects of periodic saccades on the body sway in human subjects[END_REF] attributed this effect to signals related to the initiation or execution of saccades, which are integrated by the postural system via vestibulo-spinal and reticulo-spinal signals in the muscle tone of the lower limb. Furthermore, [START_REF] Rougier | Performing saccadic eye movements modifies postural control organization[END_REF] underlined the importance of attentional demands in postural control. Executing a secondary task (motor or cognitive) during a postural task allows attention not to be focused on postural control [START_REF] Vuillerme | How attentional focus on body sway affects postural control during quiet standing?[END_REF], which leads to a better postural stability (automatic attentional system). Such improvement might be due to the fact that postural control becomes more automatic [START_REF] Geurts | Dual-task assessment of reorganization of postural control in persons with lower limb amputation[END_REF][START_REF] Melzer | Age-related changes of postural control: effect of cognitive tasks[END_REF][START_REF] Hunter | Postural control: visual and cognitive manipulations[END_REF]. [START_REF] Beilock | When paying attention becomes counter productive: impact of divided versus skill-focused attention on novice and experienced performance of sensorimotor skills[END_REF] compared performance of novice and experienced golf putting groups during skill-focused and dual-tasks, and they showed that two groups increased performance in dual-task when the attention was not focused on sensorimotor skills. More generally, the focus of attention on body movements can be detrimental to the performance of well-learned skills [START_REF] Mcnevin | Carlson C effects of attentional focus, self-control, and dyad training on motor learning: implications for physical rehabilitation[END_REF][START_REF] Wulf | Directing attention to movement effects enhances learning: a review[END_REF].

Different effects of prosaccades and antisaccades on postural control

Our study shows that antisaccades, in contrast to prosaccades, lead to more body sway. Indeed, the length and mean speed of CoP increased during antisaccades in comparison with the prosaccade conditions, but just only in the Standard Romberg position. In the Tandem Romberg position, these differences between saccadic tasks are not present, perhaps in line with a ceiling effect in this more complex postural condition [START_REF] Olivier | Dual-task study of cognitive and postural interference in 7-yearolds and adults[END_REF]). Surprisingly, this result had not been found either for the surface of CoP, perhaps because this parameter is not sensitive enough to be modified by the effect of different types of saccades. This result suggests the importance of measuring different spatial and temporal postural parameters for an accurate study of postural control [START_REF] Winter | Human balance and posture control during standing and walking[END_REF][START_REF] Gagey | Posturology; Regulation and disruption of standing[END_REF][START_REF] Vuillerme | Can a plantar pressure-based tongue-placed electrotactile biofeedback improve postural control under altered vestibular and neck proprioceptive conditions?[END_REF]. Recall that, the mean speed of COP has been shown to reflect neuromuscular energy used by the body for self stabilization [START_REF] Maki | Aging and postural control: a comparison of spontaneous-and induced-way balance tests[END_REF][START_REF] Geurts | Intrasubject variability of selected force platform parameters in the quantification of postural control[END_REF]. Consequently, our results regarding the increase of the mean speed during the antisaccade task could reflect a difficulty to perform this task together with a postural test.

For a neuropsychological approach, cortical circuits implied in the triggering and executing of saccades in the antisaccade task are broader than those of prosaccades, because more cognitive processes are necessary to realize antisaccade task [START_REF] Leigh | The neurology of eye movement, 4th edn[END_REF]. Indeed, in order to correctly perform an antisaccade, the reactive saccade toward the peripheral stimulus has to be inhibited and a voluntary saccade in the opposite direction (antisaccade) has to be programmed. Neuropsychological studies have shown an important role of the frontal cortices when antisaccades are performed. For instance, [START_REF] Everling | Neuronal correlates for preparatory set associated with pro-saccades and anti-saccades in the primate frontal eye field[END_REF] revealed that several frontal structures (frontal eye field, dorsolateral prefrontal cortex, and supplementary eye field) are more activated during antisaccades than during prosaccades. Furthermore, [START_REF] Matsuda | Functional MRI mapping of brain activation during visually guided saccades and anti-saccades: cortical and subcortical networks[END_REF] reported increased activity in the inferior parietal cortex during antisaccades compared to prosaccades. Interestingly, [START_REF] Ettinger | Decomposing the neural correlates of anti-saccade eye movements using event-related FMRI[END_REF] showed that the antisaccade tasks activated an extensive fronto-parieto-subcortical network, and other studies found out that the parietal cortex (some regions in the intraparietal sulcus) is responsible for the vector inversion required to generate an antisaccade to the correct location [START_REF] Clementz | When does the brain inform the eyes whether and where to move? An EEG study in humans[END_REF].

Different structures of central neural control are involved in the control of posture, namely the spinal cord and brainstem, basal ganglia, cerebellum, cerebral cortex: somatosensory, premotor, and motor cortices [START_REF] Horak | Balance orientation and equilibrium[END_REF]. Some of these structures are implicated in postural regulation and in inhibition process, which may interfere with one another, leading to a decrease in postural performance during dual-task.

Finally, an interesting point of our study is the absence of significant differences between fixation and antisaccade tasks for the length and mean speed of CoP. This could suggest that the fixation task, frequently used in postural studies as baseline or control task, requires an increase in attentional resources allocated to control posture, which diminishes the stability of participants [START_REF] Olivier | Effects of attentional focus on postural sway in children and adults[END_REF]. Moreover, the inhibition of saccades in the fixation with distractor condition is necessary to complete this fixation task successfully. This could explain similar results on postural parameters during fixation and antisaccade tasks.

Effect of dual-tasks on postural control

The dual-task paradigm is frequently used to assess attentional resources required to control our movements [START_REF] Schaefer | Cognitive performance is improved while walking: differences in cognitive-sensorimotor couplings between children and young adults[END_REF]. Their resources vary with the complexity of the cognitive and/or sensorimotor tasks added to the motor task. It is now well established that even highly practiced activities, such as walking and postural control, are attention demanding [START_REF] Huang | Dual-task methodology: applications in studies of cognitive and motor performance in adults and children[END_REF]. The first study of [START_REF] Kerr | Cognitive spatial processing and the regulation of posture[END_REF] described the attentional demand of postural control in young adults and suggested that cognitive tasks do not influence postural stability in the same way. So, the secondary task could either increase or decrease postural stability depending on the type of secondary task and on the attentional cost of such a task. Because the interference effects of our visual concurrent tasks on the posture performing are different, we can conclude that our results are in line with this proposition. An interesting model, called the U-shaped nonlinear interaction model, described by [START_REF] Lacour | Posture control, aging, and attention resources: models and posture-analysis methods[END_REF] explained why the effect of a secondary task during postural task varies. The U-shaped nonlinear interaction model suggests that the performance of an easy cognitive task can shift the focus of attention away from postural control and can lead to a better postural control relative to a single-task baseline. However, increasing the difficulty of the cognitive task can result in a degradation of postural sway. In accordance with this U-shaped model, [START_REF] Huxhold | Dualtasking postural control: aging and the effects of cognitive demand in conjunction with focus of attention[END_REF] showed an improvement of the postural sway under conditions in which participants were focusing their attention on an easy cognitive task (automatization of the postural control), as compared to focusing their attention exclusively on the execution of the sensorimotor task of balancing. When the attentional demand for the cognitive task increased (i.e., complex cognitive task), a degradation of the postural sway was observed. Our results are also in line with this U-shaped model. Indeed, an easy cognitive task (prosaccades) shifting the attentional focus away from postural control leads to a better automatic postural performance, while the increase in cognitive task complexity decreases postural performance (antisaccades). In a visual-postural dual-task when the visual condition is easy, the attentional resources are sufficient and the focusing on the realization of a reactive saccade, for example, automates the postural control, improving body stabilization. Moreover, when the complexity of the visual condition increases and more again when the complexity of the postural task increases also, the attentional resources are insufficient and the addition of an antisaccade task, for example, exceeds the attentional resources implying a decrease of postural stability because of too many attentional resources mobilized to control visual task. So, the visual and postural tasks study in our dual-task study seems to use the same attentional resources (i.e., unique attentional resources model [START_REF] Wickens | Temporal limits of human information processing: a developmental study[END_REF]. To summarize, the present study shows that an easily secondary task, as prosaccades, shifts the attentional focus away from postural control leading to a better automatic postural performance; in contrast, an increased complexity of the secondary task, as antisaccades, leads progressively to an overtaking of attentional capacities and a decrease in postural performance.

Conclusion

In conclusion, this study provides evidence about an effect on postural stability when performing saccadic eye movements. However, we showed the existence of interference between visual inhibition processes and postural executive ones at the level of the attentional resources which are pooled and shared between these two functional processes (oculomotor and postural control). Our results highlight that the complexity of the oculomotor task leads to a U-shaped postural performance. The secondary task depending on its attentional cost affects in a different way postural stability. Prosaccades direct participants' attention to postural control thereby improving postural stability; in contrast, a more complex task (i.e., antisaccades) could be responsible for shifting the attention away from postural control, decreasing postural performance.

Further studies, in which different types of eye movements and postural performance are simultaneously recorded, are needed to explore such an issue.

Fig. 1

 1 Fig. 1 Examples of visual events displayed in the five visual tasks: fixation without distractor (a), fixation with distractors (b), reactive saccades elicited with the simultaneous paradigm (c), voluntary saccades elicited with the overlap paradigm (d), and antisaccades (e). The arrows represent the saccadic eye movements. See text for details

Fig. 2

 2 Fig. 2 Stabilogramms representing CoP displacements for one participant as a function of five visual tasks (fixation without and with distractors, reactive and voluntary prosaccades, and antisaccades) and difficulty of postural task (SR Standard Romberg and TR Tandem Romberg). The vertical and horizontal axes represent antero-posterior (AP) and medio-lateral (ML) CoP displacements, respectively

Fig. 3 Fig. 4

 34 Fig. 3 Mean and standard deviation of surface of CoP in SR and TR conditions in the five visual tasks (fixation without and with distractors, reactive, and voluntary prosaccades and antisaccades)

Fig. 5

 5 Fig. 5 Mean and standard deviation of mean speed of CoP in SR and TR conditions in the five visual tasks (fixation without and with distractors, reactive and voluntary prosaccades, and antisaccades)

Acknowledgments Agathe Legrand was supported by the prize of "Association Posture et Equilibre," Grenoble 2010.

Conflict of interest

Authors have no financial and personal relationships with other people or organizations that could inappropriately influence or bias their work in this study.