Frédéric Giraud
email: frederic.giraud@polytech-lille.fr

Michel Amberg

Romuald Vanbelleghem

Betty Lemaire-Semail

Power consumption reduction of a controlled friction Tactile Plate

Keywords: haptic interface, tactile interface, surface friction, power reduction

This paper describes design improvements of a friction reduction based tactile device, which yields to reduction of the supply power. We first evaluated the power consumption of four different plates. We found that a convenient design could cut the power losses down by 90%.

To explain these changes we propose a modelling of the dielectric losses in the piezoelectric actuators and of the vibration amplitude.

Introduction

One way to enhance interaction between man and the machine is to provide tactile stimulation to the user. By this way, information can be provided through the sense of touch. To achieve that, many tactile displays have been designed and can be divided into three groups, according to their working principle. Devices made up with a large number of pins arranged in an array [START_REF] Wagner | A tactile shape display using servomotors[END_REF] [START_REF] Valazquez | Tactile Rendering With Shape-Memory-Alloy Pin-Matrix[END_REF] reconstruct the geometry of a virtual surface. Other tactile stimulators don't create the exact geometry of a virtual surface, but the effect of this geometry on the finger pulp. For example, [START_REF] Summers | Results from a Tactile Array on the Fingertip[END_REF] uses an array of small pins that vibrate to excite each R.A. mechanoreceptor independently. In [START_REF] Levesque | Experimental Evidence of Lateral Skin Strain During Tactile Exploration[END_REF] [START_REF] Pasquero | STReSS: A Practical Tactile Display System with One Millimeter Spatial Resolution and 700 Hz Refresh Rate[END_REF], the skin is stretched to give the sensation of touching a real bump or a hole. Those two types of tactile stimulator create a local stimulation. But tactile stimulators using a global stimulation can be found as well. Devices based on friction reduction belong to this category. They use a vibrating plate at very high frequency (typically above 25kHz) on which the finger is free to move. The vibration creates an airgap which decreases friction between fingertip and the plate. By turning on and off the power supply of the device, the stimuli generated by this temporal friction variation create the illusion of a more or less finely textured surface. Previous designs of such a tactile plate have been made [START_REF] Biet | Implementation of tactile feedback by modifying the perceived friction[END_REF]. This plate, called the original plate suffered from the heat generated by the plate, which could hurt users. To solve this problem, we study the design of the tactile plate in the light of power consumption.

The paper is organized as follows. First we describe the original tactile plate and the squeeze film effect. Then we analyze the power consumption of three new designs of the tactile plate. A modelling of the power losses is then proposed, which explains the differences in power consumption between the new designs. Finally a discussion highlights the influence of the plate's dimension on power consumption.

Presentation of the original tactile plate

The friction reduction occurring between fingertip and a vibrating plate is often explained using squeeze film air bearing theory which is detailed in [START_REF] Wiesendanger | Squeeze Film air Bearings Using Piezo-electric Bending Elements[END_REF] and applied on a tactile device in [START_REF] Watanabe | A method for controlling tactile sensation of surface roughness using ultrasonic vibration[END_REF][START_REF] Winfield | T-PaD: Tactile Pattern Display through Variable Friction Reduction Proceeding of Worldhaptic[END_REF][START_REF] Biet | Betty Semail: Discrimination of Virtual Square Gratings by Dynamic Touch on Friction Based Tactile Displays[END_REF]. On a practical approach, an overpressure between the fingertip and the plate is created by the vibrations, resulting in a friction reduction. The plates described in this article use a standing flexural wave which propagates at the plate's surface. This design allows high vibration amplitude with small bulk size. To create the vibration, 28 piezo cells are bonded onto a copper substrate; when supplied by a sinusoidal voltage, they bend the substrate as shown in figure 1. One important thing to explain, is that when moving the fingertip on the plate, users don't feel the vibration by itself, but the effect of the vibration on the tribological behaviour of the contact between the finger and the plate. Moreover, the vibration is not homogenous, due to the vibration nodes and antinodes; however, the squeeze film effect is more or less filtered because touching area is in the same order of size as the vibration wavelength.

To be efficient, the plate should be designed so as to allow vibrations at 25kHz and above, otherwise the squeeze effect is not created. Moreover, a typical vibration amplitude of 1µm is necessary in order to attain a sufficiently low friction level. The original plate has been designed to offer those performances [START_REF] Biet | Implementation of tactile feedback by modifying the perceived friction[END_REF]. Its size was chosen in order to optimize the vibration amplitude for a given voltage amplitude V . However, in this design, plate thickness has been fixed to 2mm and the piezo cells'thickness to 1mm. Moreover, the optimization process paid no attention to the power requirements of the tactile plate, which actually needs a large amount of power to be supplied. Consequently, the device's temperature increases because of the power losses, leading to burning sensations. Reducing the losses is a key issue in these devices to allow a use over a long duration. The following section of this article presents several new designs which help to reduce the required power supplied to the device.

3 Loss reduction

Experimental approach

The tactile plate is built up with piezoceramics bonded on a copper-beryllium substrate. In order to find which design produces less power losses at a given vibration amplitude, we built up several tactile plates with different sizes:

plate 0: the original plate, the wavelength of which is λ = 20.6mm, plate 1: same as the plate 0 but the substrate's thickness is set to 0.7mm, plate 2: same as plate 1, but the ceramics' thickness is changed to 0.5mm, plate 3: a new plate with λ = 15.6mm, a substrate's thickness of 0.75mm and piezo ceramics' thickness of 0.5mm.

We then measured the required power as a function of W . Results are presented in figure 2. As it can be seen, the supply power required for the original tactile plate dramatically increases with the vibration amplitude. Moreover, it appears that plates with thicker piezoelectric cells exhibit larger vibration amplitude for the same amount of power consumption. We also measured the vibration of the plate as a function of the supply voltage. The results are presented in figure 3. It appears that for plate 0 and plate 1, the deflexion saturates to almost 1µm. When trying to vibrate more than 1µm, and thus providing more voltage to the plates, we supply with much more power but we don't obtain more deflection: the additional amount of power is lost into heat, without increasing the vibration amplitude. The other two designs plates 2 and 3, also saturate but at a higher level. These designs are more convenient to the use in a tactile stimulator. Comparing plate 0 and plate 1, we find that we can reduce power consumption using a thin substrate. In fact, both plates output the same deformation amplitude, but the thinnest plate consumes less. However, those plates saturate at a low level of deflection, and 1µm is a maximum value one may expect with them. A better design seems to be plates 2 and 3. For those plates, we used thin piezo cells: those plates don't saturate at 1µm, and the required power is very low compared to the other designs. As a conclusion to this experimental study, we can describe the effect of the plate's parameters as follows:

reducing substrate's thickness reduces power consumption and has no effect on the maximum vibration amplitude, reducing piezoelectric actuator's thickness reduces power consumption and also increases the maximum vibration amplitude.

The next section of this paper explains which role plays the thickness of the piezo layer, and also gives a modelling of the power losses and the vibration amplitude. This step is important to design an optimized plate.

Discussion

The reason why original plate requires so much power compared to the others is that the design was optimized to produce large vibration amplitude at a given voltage. This design was not optimized to reduce power losses. One mechanism leading to power losses is the presence of dielectric losses inside the piezoelectric cells. The piezo product manufacturer [11] gives the relationship between the voltage supplied to the piezo element V , its capacitance C 0 , the working frequency f and the power losses P by introducing the parameter δ:

P = (2πf )C 0 V 2 tan(δ) (1)
The capacitance C 0 is given by the cell's dimension:

C 0 = ε a × b e (2)
Where a × b is cells' surface, ε is the material's dielectric constant and e is the piezo-thickness. Introducing 2 into 1 leads to:

P = (2πf ) V 2 e ε(a × b) tan(δ) (3)
As it can be seen, the power losses depend on the resonance frequency of the plate. This is verified by experimental data, because the frequency of the three new plates is lower than the original one. In this study, the plates didn't fully fulfill requirements of section 2 because the resonance frequency was around 20kHz instead of the required 25kHz. However, conclusion to the work doesn't change.

Moreover, the power losses are proportional to the ratio V 2 e : for a given substrate and piezoelectric material, cells thickness should be as big as possible to reduce power losses. However, this fact is not consistent with conclusion of section 3.1. In fact, in this application, the device doesn't work at constant voltage, but at constant vibration amplitude. This is why, this modelling must be completed with the relationship between the vibration amplitude and the voltage.

This vibration modelling is extracted from [START_REF] Biet | Implementation of tactile feedback by modifying the perceived friction[END_REF] to deduce the vibration amplitude as a function of V , allowing the drawing of the curves in figure 3. In this paper, we do not fully describe this modelling, but we focus on the fact that the vibration amplitude is proportional to Q m d 31 V e 2 where d 31 is an elongation factor of the piezoelectric material, and Q m is a quality factor of the mechanical resonator.

Normally, Q m and d 31 are constant; however, to take into account non linearity of the piezo material, and reach to the behaviour shown in figure 3, these parameters should be varying. We decided to make d 31 vary with the electrical field inside a piezo cell, and we laid down:

d 31 = d 31n × E max E max + V e (4)
where d 31n is a value given by the material manufacturer and E max is a constant parameter depending on the material, not on the plate's size. This modelling is consistent with the experimental data as shown in figure 3, except for the plate 0 for which deflection is under estimated.

Finally, taking into account 4 into 3 yields to writing that P is proportional to

e 3 W 2 (Emax+ V e ) 2 E 2 max
: the thinner the piezo layer is, the smaller the power losses. This conclusion is consistent with what is found by experimental measurements.

Consequently, increasing the vibration amplitude can be achieved by choosing thin piezoceramics cells: to reach 1µm, the device needs lower voltage leading to a smaller amount of power. Then there exists a tradeoff between the level of voltage to reach 1µm and the power consumption leading to an optimized geometry of the tactile plate. By this way, we have shown that it is possible to reduce the power consumption to less than 0.5W at the rated vibration amplitude of the plate, while the original tactile plate needed 5W .

Conclusion

This paper presents a study of power losses in a controlled friction tactile stimulator. We found that optimal design should include power consumption in its cost function. Previous modelling used for the plate's design has been extended in order to take into account the power losses.

Finally, we found that a more convenient design could cut the power losses down by 90%. Consequently, the tactile plate doesn't get warm, allowing the user to play a long time with the device. Moreover, the rated power (0.5W) is sufficiently small to allow a power supply from a lightweight power source, or directly from an USB port for example.

Fig. 1 .

 1 Fig. 1. An ultrasonic tactile plate and the measured deformed shape.

Fig. 2 .

 2 Fig. 2. Required power as a function of the vibration amplitude, for several plate designs. line: modelling, dots: experimental measurements.

Fig. 3 .

 3 Fig. 3. Vibration amplitude as a function of the voltage supplied to the device, for several plate designs. line: modelling, dots: experimental measurements.

Acknowledgement

This work has been carried within the framework of the INRIA Mint project and is supported by the IRCICA (Institut de Recherche sur les Composants logiciels et matériels pour l´Information et la Communication Avancée)


Peter Sergeant
email: peter.sergeant@ugent.be

Frédéric Giraud
email: frederic.giraud@polytech-lille.fr

Betty Lemaire-Semail

Geometrical optimization of an ultrasonic tactile plate for surface texture rendering

Keywords: Piezo-electricity, Optimization, Tactile Plate PACS: 77.65.Fs, 02.60.Pn, 02.70.Dh

The Tactile plate consists of piezo-ceramics glued on a copper-beryllium resonator. Its purpose is to create programmable tactile sensations, which give the illusion of finely textured surfaces. The illusion originates from the variable friction between a finger and the vibrating resonator, caused by the squeeze film effect. In order to obtain a maximal deflection of the plate for a minimal supply voltage, an optimization is carried out of the length, thickness, and width of both the resonator and the ceramics. Constraints are realistic geometrical dimensions, a resonance frequency of at least 25 kHz, and a low supply voltage. The plate is modelled by both an analytical and a numerical model. The maximal dynamical deflection per volt was achieved with thin piezo ceramics (0.5 mm) at the minimal frequency of 25 kHz. A high deflection can be obtained in a wide range of the resonator length. With increasing length, the optimal resonator thickness increases too. The plate width seems to have little influence. Experiments are carried out on two plates with different geometry.

Introduction

A tactile stimulator provides information to a user through the sense of touch. It can be used as a texture or shape simulator into an immersive virtual environment, or as an input-output tactile pad as those found on any new electronic hand-held device. To achieve that, some stimulators try to create the illusion of touching a virtual shape by using pins array. These devices suffer from their large bulk size which increases with the resolution with which the virtual shape is simulated [START_REF] Wall | Sensory substitution using tactile pin arrays: Human factors, technology and applications[END_REF]: the touching area is thus limited, and the user's fingertip can't move with a free motion. To tackle this problem, an other proposal is deluding a user by simulating the effect of the shape on the fingertip. Pin array is then used to recreate lateral stretches of the skin appearing when a user is touching a bump or a hole [START_REF] Hayward | Tactile display device using distributed lateral sjin stretch[END_REF].

The tactile stimulator studied in this paper belongs to another family of tactile devices. It is named "the tactile plate" as it uses a vibrating plate to produce tactile stimuli. However, the vibration is not used to directly create stimuli, but to modify the friction between fingertip and the plate. The working principle of the tactile plate is based on the squeeze film effect: if the thin layer of air between the plate and the finger is compressed and expanded very rapidly by the vibration, then an overpressure is generated that tries to lift the finger. The squeeze film effect reduces the friction between the plate and the finger during the vibration of the plate. Several experimental studies have shown that it is possible to create tactile feedback with this operating principle. For example [START_REF] Watanabe | A method for controlling tactile sensation of surface roughness using ultrasonic vibration[END_REF] modifies roughness perception of samples of sand papers, while [START_REF] Biet | Discrimination of Virtual Square Gratings by Dynamic Touch on Friction Based Tactile Displays[END_REF] has shown that it is possible to give the illusion of touching finely textured surfaces with a programmable spatial period. The main advantage of such devices is that they allow a free motion of the fingertip on the virtual surface. By this way, user's movement is not constrained: interaction between the user and the device is thus natural.

There exist several designs which produce tactile feedback based on friction reduction: [START_REF] Watanabe | A method for controlling tactile sensation of surface roughness using ultrasonic vibration[END_REF] uses two langevin transducers to make a thin iron plate vibrate; [START_REF] Winfield | Tactile pattern display through variable friction reduction in WorldHaptic[END_REF] uses a disc shape ring resonator to reduce the size of the device. In [START_REF] Biet | Discrimination of Virtual Square Gratings by Dynamic Touch on Friction Based Tactile Displays[END_REF], a rectangular copper plate with thin piezo-electric elements bonded on it was designed. This solution uses a standing wave with multiple wavelength propagating on top of the device. This design allows a large exploration area, with a small thickness and thus results in size reduction. However, in each case, squeeze film effect is effectual if the vibration amplitude is in the order of several micrometers at 25kHz or above [START_REF] Biet | Squeeze film effect for the design of an ultrasonic tactile plate[END_REF].

To complete surface texture rendering, the friction coefficient has to be a function of fingertip's position, and then we need its precise and accurate control. For example, gratings can be simulated by switching between a high friction level, to a smaller one. Thus, the plate's capability to produce high vibration amplitude is a key feature for tactile simulation since smaller friction level is achieved with larger vibration amplitude. Previous work in [START_REF] Biet | Squeeze film effect for the design of an ultrasonic tactile plate[END_REF] has shown that friction reduction depends on plate's original roughness, but it's also found that a value af 300nm can be considered as a minimum deflection below which friction reduction is too small. Optimization of the tactile plate is required in order to obtain a large vibration amplitude for a low supply voltage, without violating the several constraints. These constraints are 1) realistic dimensions of the plate 2) resonance frequency above 25 kHz to have the squeeze film effect 3) low supply voltage for safety reasons and for low power consumption to avoid heating of the plate. The optimization uses a 3D numerical model in combination with an analytical model to accelerate the computation.

After explaining these models in section 2, we analyze the influence of several geometrical parameters in section 3 in order to find some general design rules, similar to [START_REF] Doelle | Geometry optimization for planar piezoresistive stress sensors based on the pseudo-Hall effect[END_REF][START_REF] Monturet | Optimal dimensioning of a piezoelectric bimorph actuator[END_REF][START_REF] Chung | Development of piezoelectric fans for flapping wing application[END_REF]. Next to geometrical parameters, also the issue of damping is discussed. Experiments are carried out for two geometries (section 4). Finally, the conclusion summarizes which parameters are important to develop an efficient tactile plate.

Efficient optimization requires two models of the tactile plate

Analytical model

The first model is an analytical model based on the one in [START_REF] Biet | Squeeze film effect for the design of an ultrasonic tactile plate[END_REF]. It is recalled in the following paragraphs. Firstly, the static deflection is studied on a simply-supported beam (see Fig. 1) whose length is half of the wavelength λ. Secondly, the dynamic deflection is calculated. Thirdly, the resonance frequency is determined, and finally, the cost calculation is explained.

Static deflection

The modelled beam (Fig. 1) consists of a resonator with thickness h i in copper beryllium -the mechanical resonator -glued to a layer of piezoelectric ceramics with thickness h p . It is assumed that the thickness of the glue is zero and that the strains are continuous on the resonatorpiezoelectric ceramic interface. The beams length is λ/2 and its width b. Firstly, the displacement vector in the cartesian frame is described using the Bernoulli-Euler theory [START_REF] Landau | Theory of elasticity, course of theoretical physics[END_REF]

U =   u v w   =   (z 0 -z) δw δx (z 0 -z) δw δy w(x, y, t)   (1)
where w(x, y, t) represents the displacement along the z axis, and z 0 the neutral plan ordinate, which distinguishes the compressed and the stretched zones in the plate. The radius of curvature, ρ, due to bending moments caused by contraction of the piezoceramic, can be determined from the resulting stresses and strains in the composite beam (Fig. 1). The plate does not vibrate according to a planar movement but bends. In fact, it is easier to realize bending deformation than planar deformation.

The resonator is considered to be isotropic and the stress T i can be expressed by the reduced form as follows:

T i = E i S xx (2)
where E i is the Young modulus of the Copper Beryllium and where S xx (z) is the strain component along the xaxis acting on the plane normal to the x-axis. As for the piezoelectric part of the plate, the constitutive relationships between the stress, strain and applied fields can be expressed in terms of the piezoelectric stress relations

T p = c E S -e t E D = eS + ǫ S E ( 3
)
where E is the electric field intensity vector and D is the electric flux density vector. c E , e and ǫ S are respectively the elastic constants short circuit matrix, the voltage coefficients and the dielectric constants matrixes.

Finally, the stress distributions on the x-faces due to an applied electric field in the z-direction can be written as follows:

T xx (z) = E i S xx (z) h p < z < h p + h i c E 11p S xx -d 31 c E 11p E z 0 < z < h p (4)
where

e t = d 31 c E 11p
Considering that no external force acts on this structure, we can deduce that curve c can be expressed as follows [START_REF] Biet | Squeeze film effect for the design of an ultrasonic tactile plate[END_REF]:

c = 1 ρ = d 2 w dx 2 = 3 2 d 31 E z h p a (5)
where

1 a = 1 -2f0 1 -3f0 + 3f 2 0 + α(3β + 3β 2 + β 3 -6βf0 -3β 2 f0 + 3βf 2 0 )
and

α = E i c E 11p , β = h i h p , f 0 = zo h p (6)
By taking into account the boundary conditions of a simply-supported beam and by integrating equation ( 5) twice, we obtain the displacement profile:

w(x) = 3 4 d 31 E z h p a x 2 - λ 2 x (7)
and the static deflection w max = w(λ/4) at λ 4 is given by:

w max = -3 16
d 31 E z h p (λ/2) 2 a = - 3 16
d 31 V z h 2 p (λ/2) 2 a ( 8
)
where V z is the voltage applied between the piezo-ceramic electrodes.

Determination of the dynamic deflection

In order to increase the deflection, we work at resonance frequency. This explains why, in this subsection, we must determine the dynamic deflection, which is the static deflection times the dynamic amplification factor.

The dynamic amplification factor Q = Q m U mono /U piezo is the mechanical quality factor Q m of the piezoceramics balanced by the ratio of the strain energy of the entire monomorph U mono to the strain energy of the piezoelectric layer U piezo [START_REF] Nashif | Vibration damping[END_REF].

The stored elastic energy in the piezoelectric layer is

U piezo = λb 2 1 2 z S xx T spiezo dz (9)
and the stored elastic energy in the entire monomorph is given by

U mono = λb 2 1 2 z S xx (T spiezo + T sreson ) dz (10)
From [START_REF] Biet | Squeeze film effect for the design of an ultrasonic tactile plate[END_REF], we can deduce the stored energy as a function of the material properties and the geometrical dimension of the structure:

U piezo = λb 4 hp 0 c E 11p S 2 xx dz (11) = 3 16 λbh p d 2 31 E 2 z c E 11p 1 -3f 0 + 3f 2 0 a 2 (12)
and

U mono = λb 4 hp 0 d 31 c E 11p E z z -z 0 ρ dz (13)
The neutral line ordinate z 0 is found by expressing that the sum of the stresses at both sides of the line equals zero: 0 = z S xx (T spiezo + T sreson ) dz, resulting in

z 0 = h p (1 + αβ 2 + 2αβ) 2(1 + αβ) (14)
Here, we assumed that the materials are purely elastic, i.e. with a null electric field. Then, by inserting ( 14) into (13), we obtain

U mono = 3 16 λbh p d 2 31 E 2 z c E 11p 1 -2f 0 a Finally, Q = Q m a(1-2f0) 1-3f0+3f 2 0
and w dyn = w λ 4 Q:

w dyn = Q m -3 16
d 31 V z h 2 p λ 2 4 1 -2f 0 1 -3f 0 + 3f 2 0 (15)
Notice that the deflection in the analytical model is independent of the beam width b.

Determination of the resonance frequency

The resonance frequency f r is [START_REF] Rose | Ultrasonic waves in solid media[END_REF][START_REF] Graf | Wave motion in elastic solids[END_REF]:

f r = 2π λ 2 G b M b (16)
with

G b = c E 11p hp 0 (z -z 0 ) 2 dz +E i hp+hi hp (z -z 0 ) 2 dz the flexional rigidity of the beam in [N.m] and M b = ρ b h b + ρ i h i the total mass per length in [kg/m 2 ].

Analytical cost function

For use in optimization routines, a cost function is created that contains the analytical model.

The objective is to have a maximal deflection per volt supply voltage. This means that the cost is given by ( 15) with V z equal to a constant voltage. Since users directly touch the tactile plate. We chose V z = 15V because it is a safe voltage level.

The six parameters to optimize are the length, thickness and width of the resonator and the ceramics. Nevertheless, the cost function has only three inputs: the wavelength λ, the resonator thickness h i , and the piezo-ceramics thickness h p . As the number of piezo-ceramics n = 7 and the space between adjacent ceramics δ are constant, the wavelength also determines two additional parameters: the length l p of the ceramics and the length L of the resonator as shown in Table 1. The two remaining parameters -the width of the ceramics and the resonator -are equal and denoted by b. However, the width b cannot be optimized in the analytical model, because the cost [START_REF] Sergeant | Space mapping method for the design of passive shields[END_REF] does not depend on b.

Numerical model 2.2.1. Finite element model

The second model is a 3D linear finite element model (FEM) in the software package Comsol. The number of degrees of freedom varies between 10 4 and 10 5 , depending on the geometry. The modelled geometry and boundary conditions are shown in Fig. 2: the FEM geometry is only a quarter of the tactile plate because symmetry is exploited both along the x-and the y-direction.

The model in Comsol was validated by comparing it with the Ansys model in [START_REF] Biet | Squeeze film effect for the design of an ultrasonic tactile plate[END_REF] for the same plate geometry. The geometry studied in the following sections is however slightly different from the one in the cited paper: the length of the resonator is increased by λ/8 at both ends in comparison with the plate in [START_REF] Biet | Squeeze film effect for the design of an ultrasonic tactile plate[END_REF]. The aim of the modification is to slightly modify the wavelength at resonance so that λ = 2(l p + δ). This means that the nodes of the The dots represent the points used to determine the average deflection (antinodes). Dimensions and material properties can be found in Table 1.

standing wave are exactly between adjacent ceramics (at x = k(l p + δ)/2, k = -7, -5, ..., 7) and the anti-nodes are exactly in the middle of the ceramics (k = -6, -4, ..., 6). This increases the efficiency of the piezo-ceramics to generate a strong vibration.

In contrast with the analytical model, the numerical model has also transverse resonance modes for some geometries. Fig. 3 shows that a small modification of the plate width b can make the difference between a pure longitudinal resonance mode, and a mixed longitudinal/transverse resonance mode.

Numerical cost function

The cost function containing the parametrized numerical model should be able to determine w dyn for each possible geometry of the tactile plate. The resonance frequencies and the corresponding resonance modes (along the xdirection and/or along the transverse y-direction) are not known a priori. To tackle this problem, the cost function is organized as follows. After creating the geometry and the mesh, the cost function searches for eigenfrequencies. The analytical model is very useful to limit the calculation time of the FEM model, because it estimates the resonance frequency f r . Without a priori information about f r , all resonance modes in the considered frequency range should be evaluated in a time-harmonic FEM evaluation. By using the predicted f r from the analytical model for the same inputs, the numerical cost function has to evaluate the 3D FEM only for a few resonance frequencies close to the predicted f r . In practice, evaluating the six "nearest" eigenfrequencies seemed to be sufficient for the whole range of h i and λ in Table 1. On a 2.4 GHz CPU, the evaluation time of the numerical cost function considering six eigenfrequencies is between 50 s and 650 s, depending on the number of degrees of freedom.

The cost of the numerical cost function is the same as for the analytical model: the dynamical deflection for V z = 15 V. The function has four inputs: the same three inputs λ, h i , and h p as the analytical model, and the plate width b as an additional input.

Optimization

Both the analytical and the numerical cost functions are optimized, in order to see the difference between both functions. The optimization routines try to maximize the deflection for a given voltage by iteratively evaluating the analytical or the numerical cost function. The optimization routines consider two real input parameters λ and h i , and one discrete parameter h p : not all thicknesses are commercially available. In order to cope with the mixed discrete/real optimization problem, the optimization for the two real parameters only is repeated for several values of h p . The parameter b is not included because it has no or few influence on the cost. The influence of b in the numerical model is shown in section 3.3.

From the three constraints mentioned in the introduction, the first one -realistic dimensions -is achieved by setting the lower and upper boundaries of the parameters as given in Table 1; the second one -f r > 25 kHz -is programmed as a constraint in the optimization routine; the third one is taken into account by choosing a constant, safe V z of 15 V.

The analytical cost function is optimized by the Matlab function fmincon, which is a gradient based technique.

For the numerical cost function, two optimization routines are compared. The first one is also fmincon, wherefor the starting value is the optimum of the analytical cost function. The second routine is the Space Mapping technique [START_REF] Bandler | Space mapping: The state of the art[END_REF][START_REF] Sergeant | Space mapping method for the design of passive shields[END_REF][START_REF] Encica | Lomonova Electromagnetic Actuators for Gravity Compensation and Vibration Isolation[END_REF] which is an efficient optimization technique if a fast but inaccurate model and a slow but accurate model are available. This technique combines both models to take advantage of the speed of the fast model without loosing the accuracy of the slow model. The analytical model is useful as fast inaccurate model and the numerical model as slow accurate model. The analytical model speeds up the optimization in two ways: on the one hand, by reducing the number of numerical cost function evaluations in the space mapping algorithm, and on the other hand by limiting the number of resonance frequencies to investigate in the numerical cost function.

Effects of geometrical parameters and damping,

simulated by both models

Resonator thickness and length of the piezo ceramics

For all simulations and experiments, the resonator is made from a copper-beryllium alloy, and the ceramics are lead titanate zirconate ceramics type PI-91, manufactured by Saint-Gobain Quartz. The characteristics of the materials are given in Table 1. Additional properties of the ceramics can be found in [START_REF] Biet | Squeeze film effect for the design of an ultrasonic tactile plate[END_REF]. As already mentioned, the voltage supply of the piezo-ceramics is set to 15 V in order to avoid dangerous voltages. For this section, the quality factor Q is chosen constant and equal to 54. This value is obtained from experiments on the plate from [START_REF] Biet | Squeeze film effect for the design of an ultrasonic tactile plate[END_REF] with λ = 24 mm, h i = 2.0 mm, h p = 1.0 mm and b = 49 mm. In the following sections, this plate is called the "original tactile plate". For the chosen Q-value, the simulated dynamical deflection of the original plate equals the experimentally observed one. The black circle shows the optimum (λ * = 20.54 mm, h * i = 0.712 mm, w dyn = 1.4908 µm, fr = 25.00 kHz), and the cross shows the original plate (λ = 24.00 mm, h i = 2.00 mm, w dyn = 1.070 µm, fr = 32.13 kHz). The dotted line indicates the minimal frequency of 25 kHz and the dashed line indicates the minimal resonator thickness of 0.5 mm. An additional contour line is added at w dyn = 1.4908 µm to indicate the points with equal deflection as the optimum.

For a piezo-ceramics thickness of 1.0 mm, the deflection of the beam and its resonance frequency calculated by the analytical cost function are shown as a function of the wavelength λ and the resonator thickness h i in Fig. 4. It is recalled that the results of the analytical model are independent of the plate width b, and that the wavelength is related to the length L = (2n + 1)λ/4 of the plate, and to the length of the piezo-ceramics: l p = λ/2 -δ with δ the separation between adjacent ceramics. When moving to the right in the figure, the total plate length L increases; when moving upward, the total thickness increases. High deflections can be found in the lower right corner, but here the resonance frequency is below 25 kHz. Because of the constraint that the frequency should be above 25 kHz to produce squeeze film effect, the plate dimensions should be chosen in the region left of the dotted line. Furthermore, the minimal resonator thickness should be 0.5 mm for practical reasons, so that the optimum is not allowed to be below the dashed line. The minimization results in the point indicated by a circle in Fig. 4. This is the point where the curve with frequency equal to 25 kHz (dotted line) is tangential to the curve with deflection equal to 1.49 µm (thick gray line).

The results of the numerical model are shown in Fig. 5 for b = 49 mm. The contour lines of equal resonance frequencies correspond well with the ones of the analytical model; the contour lines of equal deflections are less smooth. This is caused by transverse modes that influence the determination of the average deflection. These modes are not taken into account by the analytical model. The optima of the analytical cost function (circle in Fig. 4) and of the numerical cost function (square in Fig. 5) are very close to each other. In spite of the good correspondence, it is not a good idea to design a tactile plate using the analytical cost function only. The numerical model is useful in order to avoid possible undesired transverse modes that are not predicted by the analytical model. For the analytical cost function, the optimum was found quickly by the Matlab routine fmincon. The numerical cost function was also optimized by fmincon, using the optimal solution of the analytical model as a starting value. Although the "landscape" of the numerical cost function is not very smooth, the gradient based algorithm was able to find the optimum in 48 evaluations. When using space mapping, the optimization required only four evaluations of the numerical cost function. In addition, four optimizations were needed of the analytical cost function, but the time to optimize the analytical cost function is negligible compared to the time to evaluate the numerical model. Although the total CPU time is ten times less than for the conventional gradient approach, the optimum found was almost the same (difference less than 0.1 mm).

Thickness of the piezo-ceramics

The influence of the thickness can be seen by comparing the deflection for h p = 1.0 mm in Fig. 4 and Fig. 5, with the deflection for h p = 2.0 mm in Fig. 6 and for h p = 0.5 mm in Fig. 7 and Fig. 8. The supplied voltage and the Q-factor are unchanged.

The deflection decreases for thicker piezo-ceramics. The contour lines of equal deflection have a bend that occurs at a thickness that is roughly half of the ceramics thickness. If the wavelength increases, the corresponding h i that generates the highest deflection increases too. The optimum h i increases if h p increases: for h p = 0.5 mm it is on the minimal allowed thickness. For h p = 1 mm, it is at 0.7 mm, and for h p = 2 mm, it is on 1.5 mm. Also the optimal wavelength increases for increasing h p . To summarize: if the piezo thickness increases, then all other dimensions of the plate should increase also. Nevertheless, the amplitude of deflection reduces with increasing h p .

The frequency decreases only a bit with increasing h p . For all three considered thicknesses, the optimum has a resonance frequency of 25 kHz. Higher frequencies always have lower deflections.

Plate width

In Fig. 9, the dashed line shows that in the analytical model, the plate width b has no influence on the deflection, neither on the frequency. The numerical model shows that the deflection and the resonance frequency slightly decrease for low b < 40 mm. The analytical model does not predict this decrease. Indeed, for small plate widths, the analytical model is not valid as it assumes that the plate thickness h i +h p is much smaller than its length and width. The deflection should in theory be constant for b > 40 mm. The deflection obtained by the numerical model as a function of b is not smooth at all. This is caused by transverse modes that disturb the efficient deformation of the platesee also Fig. 3. In the hypothetic case without these transverse modes, the deflection would be given by a curve that connects all the tops of the curve. This curve would predict a deflection that corresponds well with the analytical one. The resonance frequency is not much affected by the transverse modes, as can be observed in Fig. 9, taking into account the detailed scale.

Damping

The Q-factor for the damping was experimentally determined based on the deflection of the original tactile plate. In the previous sections, the same Q-factor of 54 was chosen also for other geometries. For the optimized plate with h p = 0.5 mm (black square in Fig. 8), this results in the deflection w dyn shown in Fig. 10 as a function of time and in Fig. 11 as a function of the frequency. The maximal deflection is about 4 µm and the time constant is approximately 1 ms.

However, according to literature, the Q-factor decreases with the vibration velocity of the piezo-ceramics. The figure 3.8 in [START_REF] Uchino | Piezoelectric actuators and ultrasonic motors[END_REF] shows for a piezo-ceramic without resonator i and fr equal to the constraints of 0.5 mm and 25 kHz resp. (dashed lines). If the constraint on h i is removed, the optimum is at λ * = 14.52 mm, h * i = 0.356 mm, w dyn = 2.9814 µm, fr = 25.00 kHz.

a Q-factor that strongly decreases as a function of the displacement speed v: Q is about 2000 for v < 0.1 m/s, and decreases rapidly if v increases: Q is about 200 at v = 0.25 m/s. To obtain a more realistic map of the deflection w dyn (λ, h i ), the Q-factor was experimentally measured as a function of v for the original setup. The result -displayed in Fig. 12 -is similar to [START_REF] Uchino | Piezoelectric actuators and ultrasonic motors[END_REF] for v >0.2 m/s, and much lower for smaller v. The latter is due to the additional damping by the glue that is not present in the cited figure.

Modifying the map with a variable Q-factor gives rise to Fig. 13. The resonance frequency is not shown because it remains the same as in Fig. 5. The deflection remains the same for the original plate (indicated by a cross). The deflection becomes much lower for most other geometries, but the position of the optimum doesn't change.

In some applications, a prescribed deflection should be obtained with the lowest possible voltage. Fig. 13 shows the voltage needed to obtain a deflection of 1.15 µm. For the original plate, evidently, the required voltage is 15 V.

Changing the thickness of the piezo-ceramics results in Fig. 14. The deflection is still larger than for h p = 1 mm, but the difference is much smaller than with constant Q.

The optimal wavelength and resonator thickness for h p = 1 mm and 0.5 mm changes only slightly in comparison with the ones for Q constant, but the deflection at the optimum is strongly reduced. This is an important conclusion, because it means that the changes in the damping type has almost no influence on the optimal dimensions of the plate.

Experiments

Tactile plates are built up with piezo cells which are bonded on the copper-beryllium substrate. First, a thin layer of Epoxy is spread onto the substrate. Then the cells are precisely positioned. The assembly is pressed, and put into an oven for polymerization of the glue. Figure 15 presents a tactile plate after the polymerization process. Two tactile plates were manufactured according to optimization's results. However, we could not be supplied with h i = 0.673mm, we have chosen h i = 0.75mm instead. We name these two plates P1 and P2, and they have the following characteristics respectively:

• P1: h p = 1.0mm, h i = 0.75mm and λ * = 20.56mm,

• P2: h p = 0.5mm, h i = 0.5mm and λ * = 15.66mm. Dynamical deflection, resonance frequency , and the voltage necessary to reach 1.15µm were measured. Results are presented in the following paragraphs.

Experimental characterization of the optimized plates

The resonant frequency was measured on each tactile plate. We found: • for P1: f r = 23.160kHz,

• for P2: f r = 23.950kHz. These figures don't exactly match the target value of the optimization process. However, the small shift observed is compatible with squeeze film effect: sensation may be softened but is still perceptible to human finger.

We also measured the dynamical deflection for the two plates, using a laser interferometer. We present in figure 16 the contour and a 3-d view of the same measurements for the plate P1. We also depicted the same plots for the plate P2 in figure 17. In order to avoid resonant frequency drift due to temperature rise during the run, measurements were done at low voltage level, and thus low vibration amplitude.

These figures show that the resonance mode found is the same for both plates, and the wavelength matches the value found for the optimization process. However, the two plates do not behave exactly the same. In fact, dynamical deflection of the big plate (h i = 0.75mm,h p = 1.0mm) is more disrupted than the thin one: tactile stimulation is not homogenous all over the plate, more particularly for the big plate. This problem is due to a transverse mode which resonance frequency is too close to the working frequency. Compared to the original plate, the transverse mode of the optimized plates has a more harmful effect, as shown in figure 18.

We finally measured the dynamical deflection of each plate as a function of the voltage supplied to the resonator. The results, presented in figure 19, show that the voltage which is necessary to attain the nominal deflection is lower for the thin plate than for the big one or the original tactile plate.

Thus, the optimized tactile plate needs less voltage to produce the same dynamical deflection, and requires less energy during operation. Best results are obtained with the thin plate because the required supply voltage is 6[V ] only to produce 1.15µm. This confirms the analytical conclusion of chapter 3.2, because for the same voltage amplitude, dynamical deflection reduces by increasing h p .

Conclusion

This paper presents the optimization process of a tactile plate. Optimized plates require higher deflection for the same voltage level and a resonance frequency higher than 25kHz to allow squeeze film effect. We obtained two optimized plates which were manufactured and characterized. The new designs present more transverse mode than on the original plate configuration, and a resonance frequency slightly lower than the optimization condition. However, more deflexion at lower voltage level is obtained in return. The optimized plate with the thinner layer of piezo-ceramic offers better performances than the bigger one. Deflection is more homogenous over the tactile plate, and a supply voltage of 6[V ] only is required to work at nominal deflection. This new design has an important property: with such a low voltage, the tactile plate can be used in a computer environment more easily.

Figure 1 :

 1 Figure 1: Half-wavelength beam which bends by contracting the piezoceramics.

Table 1 : 2 -Figure 2 :

 122 Figure2: Geometry of the tactile plate in the 3D FEM, which is only one quarter of the entire plate. The dots represent the points used to determine the average deflection (antinodes). Dimensions and material properties can be found in Table1.

Figure 3 :

 3 Figure 3: Deformation in µm of (one quarter of) the tactile plate with λ = 24 mm, h i = 2.0 mm, hp = 1.0 mm. In (a), the width b is 55 mm giving rise to a pure resonance mode in x-direction. In (b), the width b is 48 mm, resulting in an additional transverse resonance mode

Figure 4 :

 4 Figure 4: Contour lines of equal dynamical deflection in (a) and equal resonance frequency in (b) found by the analytical model for hp = 1.0 mm and Q = 54.The black circle shows the optimum (λ * = 20.54 mm, h * i = 0.712 mm, w dyn = 1.4908 µm, fr = 25.00 kHz), and the cross shows the original plate (λ = 24.00 mm, h i = 2.00 mm, w dyn = 1.070 µm, fr = 32.13 kHz). The dotted line indicates the minimal frequency of 25 kHz and the dashed line indicates the minimal resonator thickness of 0.5 mm. An additional contour line is added at w dyn = 1.4908 µm to indicate the points with equal deflection as the optimum.

Figure 5 :

 5 Figure 5: Contour lines of equal dynamical deflection in (a) and equal resonance frequency in (b) found by the numerical model for hp = 1.0 mm and Q = 54. The square shows the optimum (λ * = 20.60 mm, h * i = 0.683 mm, w dyn = 1.8417 µm, fr = 25.04 kHz). The original plate indicated by the cross has λ = 24.00 mm, h i = 2.00 mm, w dyn = 1.151 µm, fr = 31.17 kHz

Figure 6 :

 6 Figure 6: Contour lines of equal dynamical deflection in (a) and equal resonance frequency in (b) found by the analytical model for hp = 2.0 mm. The circle shows the position of the optimum (λ * = 29.04 mm, h * i = 1.425 mm, w dyn = 0.7454 µm, fr = 25.00 kHz)

Figure 7 :

 7 Figure 7: Contour lines of equal dynamical deflection in (a) and equal resonance frequency in (b) found by the analytical model for hp = 1.0 mm and Q = 54. The circle shows the position of the optimum (λ * = 15.76 mm, h * i = 0.500 mm, w dyn = 2.9308 mm, fr = 25.00 kHz), which has h *i and fr equal to the constraints of 0.5 mm and 25 kHz resp. (dashed lines). If the constraint on h i is removed, the optimum is at λ * = 14.52 mm, h * i = 0.356 mm, w dyn = 2.9814 µm, fr = 25.00 kHz.

Figure 8 :

 8 Figure 8: Contour lines of equal dynamical deflection in (a) and equal resonance frequency in (b) found by the numerical model for hp = 0.5 mm and Q = 54. The black square shows the position of the optimum (λ * = 16.10 mm, h * i = 0.500 mm, w dyn = 3.9743 µm, fr = 25.07 kHz).

Figure 9 :

 9 Figure 9: Deflection as a function of the plate width shows the effect of transverse modes for the original configuration. The deformation for b = 48 mm and for b = 55 mm is shown in Fig. 3.

Figure 10 :

 10 Figure10: Time simulation at 25070 Hz for the optimized tactile plate with hp = 0.5 mm, λ * = 16.10 mm, h * i = 0.500 mm. The gray line shows the average deflection that has at the resonance frequency a phase lag of 90 • compared to the applied voltage over the piezoceramics (black line). The time constant is approximately 1 ms.

Figure 11 :Figure 12 :

 1112 Figure 11: Time harmonic simulation (a) around 25070 Hz for the tactile plate with hp = 0.5 mm, λ * = 16.10 mm, h * i = 0.500 mm and (b) around 31170 Hz for the tactile cell with hp = 1.0 mm, λ * = 24.00 mm, h * i = 2.0 mm. The peak values correspond with Fig. 8and Fig. 5 respectively.

Figure 13 :

 13 Figure 13: Contour lines of equal dynamical deflection in (a) and required voltage to obtain a deflection of 1.15 µm in (b) found by the numerical model for hp = 1 mm and Q-factor given in Fig. 12. The optimum λ * = 20.56 mm, h * i = 6.73 mm has the same position as in Fig. 5, but a lower w dyn = 1.2731 µm. The dashed line shows the configurations with fr = 25 kHz.

Figure 14 :

 14 Figure 14: Contour lines of equal dynamical deflection in (a) and required voltage to obtain a deflection of 1.15 µm in (b) found by the numerical model for hp = 0.5 mm and Q-factor given in Fig. 12. The optimum λ * = 15.66 mm, h * i = 0.500 mm has almost the same position as in Fig. 8, but a lower w dyn = 1.215 µm. The dashed line shows the configurations with fr = 25 kHz.

Figure 15 :

 15 Figure 15: A manufactured tactile plate made up with 28 piezo cells glued onto the copper beryllium plate.

Figure 16 :

 16 Figure 16: Measured dynamical deflection for the plate P1(hp = 1.0mm, h i = 0.75mm and λ * = 20.56mm).

Figure 17 :

 17 Figure 17: Measured dynamical deflection for the plate P2 (hp = 0.5mm, h i = 0.5mm and λ * = 15.66mm).

Figure 18 :

 18 Figure 18: Measured dynamical deflection for the original plate configuration

Acknowledgment

This work was carried out within the framework of the INRIA Alcove project and is supported by the IR-CICA (Institut de Recherche sur les Composants logiciels et matériels pour l´Information et la Communication Avancée) and The European Commission (FEDER). Furthermore, this work was supported by the FWO project G.0082.06, by the GOA project BOF 07/GOA/006 and by the Interuniversity Attraction Poles (IAP) project P6/21. The first author is a postdoctoral researcher of the FWO.

Betty Semail received her Ph. D degree in 1990 from University of Paris XI, Orsay and habilitation degree in 1997 from University of Science and Technology of Lille. Since 1990 she has been assistant professor in Ecole Centrale of Lille, till 1998 and she is now professor in University of Lille 1. She is a member of the electrical engineering and power electronics laboratory of Lille (L2EP) and responsible of the research axis upon control of electrical systems. She has been studying electromagnetic motors and now, her main field of interest deals with the modelling and control of piezo-electric actuators, for positioning and force feedback applications.


Eléazar Zerbo
email: eleazar.zerbo@univ-nantes.fr

CO2 emissions, growth, energy consumption and foreign trade in Sub-Sahara African countries

Keywords: CO 2 emissions, Economic growth, Energy consumption, Foreign trade, Sub-Sahara Africa. JEL classification: Q43, Q56, O13, C32

CO2 emissions, growth, energy consumption and foreign trade in Sub-Sahara African countries

Eléazar

Introduction

In economic literature, there are several studies that link economic activity and environmental degradation, including empirical studies that have used either panel data or time series. Most of these studies have focused on the Environmental Kuznets Curve (EKC).1 EKC is a hypothesis which postulates that the relationship between economic growth and pollution can be represented in an inverted-U curve. During the first stage of economic growth, the evolution of the pollution would have an upward trend, but only up to a certain income level and then this trend declines. This hypothesis has been the subject of several empirical studies and there is many of them critical concerning the logic and methodology used to test this hypothesis. The use of new and more appropriate econometric methods on time series has given a significant advance in the estimation of the relationship between economic growth and pollution.

However, few studies have focused on developing countries, especially Sub-Sahara African (SSA) countries; most of them have been concerned with the developed and emerging countries. This can be explained by the low level of pollution in SSA in absolute terms and its level of development compared to the rest of the world. Indeed in 2010, the SSA countries recorded 0.71 billion tons of carbon dioxide compared to 5.43 billion tons for the United States of America (USA), 3.71 for the European Union and 8.29 billion tons for China (WDI, 2013).

Studies on the countries of the SSA countries are more focused on the causality between energy consumption and economic growth [START_REF] Akinlo | Energy consumption and economic growth: Evidence from 11 Sub-Sahara African countries[END_REF][START_REF] Wolde-Rufael | Energy consumption and economic growth: The experience of African countries revisited[END_REF] in order to study the dependence or not of different countries on energy. Those which have studied the effect of economic growth and other economic factors on pollution relate mainly to South Africa, which is considered as an emerging country [START_REF] Kohler | CO2 emissions, energy consumption, income and foreign trade: A South African perspective[END_REF][START_REF] Menyah | Energy consumption, pollutant emissions and economic growth in South Africa[END_REF][START_REF] Shahbaz | The effects of financial development, economic growth, coal consumption and trade openness on CO2 emissions in South Africa[END_REF]. This paper examines the dynamic early in the development process, because the SSA has an economy highly dependent on natural and environmental resources and is the region which suffers most from natural disasters and the phenomenon of climate change due to its financial difficulties in adapting to these changes. In addition, in order to control the current poverty level, it is necessary to improve the dynamics of economic growth. In fact, during the last decade SSA has experienced dynamic economic growth at around 5.6% (AfDB, 2013), which was mainly driven by the exploitation of natural and environmental resources. This also raises the issue of the sustainability of the growth path. Despite the good economic performance recorded over the period, poverty has not declined significantly due to an average decrease in poverty of 0.84 % per annum (AfDB, 2013). Economic growth is one of the main way to improve the well-being of the population in developing countries [START_REF] Fagnart | Environment and economic development : An introduction[END_REF][START_REF] Hugon | Can we talk of a crisis or a renewal of development economics in the last decade?[END_REF]. And yet production involves polluting nature. Moreover, [START_REF] Liousse | Explosive growth in African combustion emissions from 2005 to 2030[END_REF] forecast the level of organic emissions in Africa by 2030 compared to the 2005 level and find that it could be 50% of the global level. This then summarizes some reasons to study the factors affecting pollution and economic growth in SSA.

This study aims to fill this gap by studying the effect of economic growth, energy consumption and international trade on carbon dioxide in different SSA countries, namely Botswana, Cameroon, Gabon, Ivory Coast, Kenya, Senegal, South Africa and Togo. The results will be particularly important in policy decisions in order to act on the level of emissions while pursuing economic growth objectives.

The rest of the paper will focus on the following points: Section 2 presents briefly the literature review on the inter-relationship between environment pollution, output, energy consumption, trade openness and financial development. Section 3 describes the econometric methodology used in this study. Section 4 presents the results. Section 5 discusses empirical results and concludes.

Literature review

In the economic literature, there are a variety of empirical studies that focus on the link between environmental degradation and economic activity. The main study in this area concerns the EKC that connects economic growth and pollution. The pioneers of this study were [START_REF] Grossman | Environmental Impacts of a North American Free Trade Agreement[END_REF], [START_REF] Shafik | Economic Growth and Environmental Quality: Timeseries and Cross-country Evidence[END_REF] and [START_REF] Panayotou | Empirical tests and policy analysis of environmental degradation at different stages of economic development[END_REF]. The EKC is a hypothesis which postulates that the relationship between the growth of per capita income and the level of pollution could be represented as an inverted-U shape.

Pollution increases in the early stages of economic growth, and after a certain level of development, it will present a downward trend.

Few studies on EKC have focused on SSA countries. Studies which consider the Africa continent have used mainly deforestation as an environmental pressure measure. For example, [START_REF] Bhattarai | Institutions and the Environmental Kuznets Curve for Deforestation: A Crosscountry Analysis for Latin America, Africa and Asia[END_REF] examined the relationship between income and deforestation in 66 countries of Latin America, Asia and Africa. In the case of African countries, their results support the EKC hypothesis. In the same way, [START_REF] Culas | Deforestation and the environmental Kuznets curve: An institutional perspective[END_REF] studied the dynamic between income and deforestation in a panel of 15 countries including four SSA countries, namely the Congo Democratic Republic, Ivory Coast, Kenya and Nigeria. He failed to support the EKC hypothesis in these countries. Recently, [START_REF] Onafowora | Bounds testing approach to analysis of the environment Kuznets curve hypothesis[END_REF] use carbon dioxide to check for EKC in Brazil, China, Egypt, Japan, Korea, Mexico, Nigeria and South Africa.

They employ the ARDL bound testing approach to cointegration and find that the long-run relationship between CO2 emissions and economic growth follows an N-shaped trajectory in Nigeria and South Africa. This mean that carbon dioxide rises with economic growth, declines from a first turning point to increase again from a second turning point. However, the estimated turning points are outside the sample.

While the EKC hypothesis has been the subject of several empirical studies, it has not been verified for all indicators of pollution, such as carbon dioxide (CO 2 ). It has mainly been verified for sulfur dioxide (SO 2 ) though (Nourry, 2007). 2 Moreover, some studies focus on the causality between energy consumption and economic growth. The objective of these studies is to verify whether the relationship between energy consumption and income growth may be unidirectional, bidirectional, or neutral. A unidirectional relationship of energy consumption to income growth (or vice versa) will involve an expansion economic policy (or energy conservation policy). This allows economic governors to pursue an effective policy in reducing emissions which does not have a significant impact on income growth, assuming that the consumption of energy, especially fossil fuel, has an important share in pollutant emissions. The seminal article in this field was published by [START_REF] Kraft | On the relationship between energy and GNP[END_REF], who studied causality between gross national product (GNP) and energy consumption in the United States and found that the relation runs from GNP to energy consumption. Some authors have focused on SSA countries by investigating the hypothesis that prevailed in these countries. [START_REF] Ebohon | Energy, economic growth and causality in developing countries: A case study of Tanzania and Nigeria[END_REF] examined the causal relationship between economic growth and energy consumption in Tanzania and Nigeria. His results suggest a complementarity between energy consumption and income growth. This result requires massive investments in the energy sector to stimulate economic activity. In the same way, other authors have examined the nature of this relationship in various SSA countries [START_REF] Akinlo | Energy consumption and economic growth: Evidence from 11 Sub-Sahara African countries[END_REF][START_REF] Fondja Wandji | Energy consumption and economic growth: Evidence from Cameroon[END_REF][START_REF] Odhiambo | Energy consumption, prices and economic growth in three SSA countries: A comparative study[END_REF][START_REF] Ouédraogo | Electricity consumption and economic growth in Burkina Faso: A cointegration analysis[END_REF][START_REF] Tamba | Causality analysis of diesel consumption and economic growth in Cameroon[END_REF][START_REF] Wolde-Rufael | Energy demand and economic growth: The African experience[END_REF], and the results of these studies reveal that the direction of causality vary from one country to another according to its specificity.

Recent econometric methods on time series have allowed several authors to analyze the relationship between pollution, economic growth and energy consumption in the same multidimensional framework. For emerging and developing countries, Saboori and Sulaiman (2013a) and Saboori and Sulaiman (2013b) considered CO 2 emissions, energy consumption and economic growth in Southeast Asia countries and Malaysia in the same framework by using the cointegration approach. This approach allows one to check the effects of energy consumption and economic growth on the level of CO 2 emissions. In the same way, [START_REF] Halicioglu | An econometric study of CO2 emissions, energy consumption, income and foreign trade in Turkey[END_REF] adds international trade to these variables in order to estimate the effect of trade openness on pollution. For the author, the sign of international trade depends on the level of economic development of the country. In the case of Turkey, he finds that this effect has certainly less impact but it increases pollution. Contrary to this result, [START_REF] Kohler | CO2 emissions, energy consumption, income and foreign trade: A South African perspective[END_REF] found that international trade contributes to reducing CO 2 emissions in South Africa.

It is clear that previous studies did not concern the effect of these variables (economic growth, energy consumption and international trade) on the variation of CO 2 emissions in SSA countries. The objective of this study is to fill this gap, by proposing to study the impact of economic growth, energy consumption and international trade on the variation of CO 2 emission in some SSA countries.

Methodology

Model

Following the empirical literature, it is plausible to form the relationship between CO 2 emissions, economic growth, energy consumption and foreign trade, with a view forward testing the effect of those variables on CO 2 emissions as follows:

where C t is CO 2 emissions per capita, E t is energy use per capita, Y t is per capita real income, T t is sum of importation and exportation per capita used as a proxy for foreign trade; and Ɛ t is the regression error term assumed to be normally distributed with zero mean and constant variance.

As for the expected signs, one expects to be positive; it indicates that an increase in economic growth is linked with high CO 2 emissions. The sign of α 2 is expected to be positive because a higher level of energy consumption should result in greater economic activity and stimulate CO 2 emissions. The expected sign of α 3 is mixed in economic literature. For [START_REF] Kohler | CO2 emissions, energy consumption, income and foreign trade: A South African perspective[END_REF], it depends on the level of the economic development stage of a country. In the case of developed countries, it is expected to be negative as countries develop; they cease to produce certain pollution intensive goods and begin to import these from other countries with less restrictive environmental protection laws. In the case of developing countries, this sign expectation is reversed as they tend to have dirty industries with a high share of pollutants.

Because in SSA there are developing countries, this sign is expected to be positive.

Data

Based on the specification detailed above, annual time series on carbon dioxide (CO 2 ) emissions, income, energy consumption and trade openness have been collected for eight SSA high level of energy consumption relatively to the population and a high degree of participation in international trade along with Botswana. Gabon shows a great variation in all the variables with respect to the standard deviation. In general the degree of asymmetric distribution falls within an interval of -1 and 1, but some exceptions exist. In the case of Togo for example, the values of skewness (2.45) and Kurtosis (13.10) suggest that CO 2 emission distribution doesn't match with Gaussian distribution.

The ARDL bound testing cointegration approach

We apply the ARDL bounds testing cointegration approach developed by [START_REF] Pesaran | Bounds testing approaches to the analysis of level relationships[END_REF] to investigate whether a long run dynamic relationship exists between economic growth, energy consumption, trade openness and carbon emissions. Various approaches have been applied to test the presence of cointegration between variables in numerous studies. Two of these approaches are taken from [START_REF] Engle | Co-integration and error correction: representation, estimation, and testing[END_REF] in the bivariate case and [START_REF] Johansen | Maximum likelihood estimation and inference on cointegration-with applications to the demand for money[END_REF] when multivariate, and require that all the series should be integrated at the order of integration I(1). The ARDL bounds testing approach is more appropriate compared to those traditional cointegration approaches. The approach avoids endogeneity problems and the inability to test long run relationships of variable associates with the traditional Engel-Granger method. Both short run and long run parameters are calculated simultaneously and the ARDL approach can be used regardless of whether the data are integrated of order I(0) or I(1). [START_REF] Narayan | The saving and investment nexus for China: evidence from cointegration tests[END_REF] argues that the ARDL approach is superior in small samples to other single and multivariate cointegration methods. As set out in [START_REF] Halicioglu | An econometric study of CO2 emissions, energy consumption, income and foreign trade in Turkey[END_REF] and [START_REF] Kohler | CO2 emissions, energy consumption, income and foreign trade: A South African perspective[END_REF], the ARDL representation is formulated as follows:

The bounds test is based on the Wald's statistic, where the null hypothesis joint significance test implies no relationship (i.e.

), and the alternative implies cointegration (i.e.

). This approach does not use a normal 2) is formulated as follows:

where λ is the speed of the adjustment parameter, and is the residuals that are obtained from the estimated cointegration model of eq. (2).

Toda and Yamamoto causality test

To complement our study, we have also carried out the Granger non-causality test using the [START_REF] Toda | Statistical inference in vector autoregressions with possibly integrated processes[END_REF] procedure. The issue of causality between variables in the literature has been undertaken via the Engle and Granger causality test. Nevertheless, this approach has been criticized for its low power test, restrictive nature and small sample properties [START_REF] Kofi Adom | Carbon dioxide emissions, economic growth, industrial structure, and technical efficiency: Empirical evidence from Ghana, Senegal, and Morocco on the causal dynamics[END_REF]. Consequently, we use the Toda and Yamamoto (T-Y) procedure to test the causal link between CO 2 , income, energy consumption and international trade. The advantage of this method is that it is valid regardless of whether a series is I (0), I

(1) or I (2), cointegrated or not-cointegrated. The T-Y Granger causality test augments a VAR model in level with extra lags (the maximum order of integration d of the series). This ensures that the Wald statistics possess the necessary power properties. Then the augmented VAR model with a total of (k+d) lags is then estimated on levels of the selected variables (not on their first difference) and restricting the first k-lags to zero. Thus using this approach, we estimate the following VAR model to determine the causal dynamics between CO 2 emission, income, energy consumption and international trade. The augmented VAR model is represented as follows:

After estimating the above systems of eqs. (4-7), the T-Y approach is based on the Wald statistic test for the significance of the first k lags by restricting the coefficients to zero, under a null hypothesis of no causality.

Empirical results

Unit roots tests

Before performing the ARDL bound test, it may be useful to test for unit roots. Here, the augmented [START_REF] Dickey | Likelihood ratio statistics for autoregressive time series with a unit root[END_REF] test, the [START_REF] Phillips | Testing for a unit root in time series regression[END_REF] test and the modified Dickey and Fuller test proposed by [START_REF] Elliot | Efficient tests of the unit root hypothesis[END_REF] are used to identify the order of integration of our variables.

ARDL test result

The cointegration test consists firstly of determining the optimum lag length m to be considered in the estimation of equation (2). To do this, we used the AIC, SBIC and HQIC criteria within an unrestrictive autoregressive (VAR) model. This allowed us to obtain appropriate lag levels for each country. Then the F-test is applied to the equation (2) to determine whether or not a long-term relationship exists between the dependent and independent variables. The results are summarized in Table 3. Compared with the critical values of [START_REF] Narayan | The saving and investment nexus for China: evidence from cointegration tests[END_REF], the ARDL test suggests the existence of a long-term relationship between our variables in South Africa and Togo. According to these results, we will therefore estimate a short-term relationship in Botswana, Cameroon, Ivory Coast, Gabon, Kenya and Senegal where the cointegration relationship has not been proven and a long-term relationship for South Africa and Togo. The estimated short-run relationship is presented in Table 4. We observe that the energy consumption has a significant effect on the variation of CO 2 in Botswana (1.61), Kenya (2.79), South Africa (0.81) and Togo (3.62) in the short term. This means that the energy consumption increases CO 2 emissions for these countries. However, the proportion is more or less high depending on the country's development stage. Indeed, it should be noted that this proportion is high for the poorest countries. From these four countries, Kenya and Togo, classified by the World Bank as "Low income" compared to Botswana and South Africa ("middle income"), have larger effects of energy consumption. In addition, the effect of international trade is statistically significant in Kenya and South Africa. Trade openness has a positive effect in reducing the level of CO 2 emissions in South Africa contrary to that of Kenya. One can also assume that this is consistent with the hypothesis of [START_REF] Halicioglu | An econometric study of CO2 emissions, energy consumption, income and foreign trade in Turkey[END_REF], which postulated that the effect of trade on pollution depends on the country level of development. In comparison to others studies, our results for South Africa are consistent with those of [START_REF] Kohler | CO2 emissions, energy consumption, income and foreign trade: A South African perspective[END_REF] and [START_REF] Shahbaz | The effects of financial development, economic growth, coal consumption and trade openness on CO2 emissions in South Africa[END_REF], who found that energy consumption contributes to the deterioration of the environment while trade openness improves it. Moreover [START_REF] Kohler | CO2 emissions, energy consumption, income and foreign trade: A South African perspective[END_REF] found a non-significant effect of income growth in rising CO 2 emissions. However, they contradict those of [START_REF] Onafowora | Bounds testing approach to analysis of the environment Kuznets curve hypothesis[END_REF] who found that trade increases CO 2 emissions level in South Africa. 3 It is only in Togo that income has a significant effect during the period with a negative coefficient. This might suggest that income growth would support a reduction in the level of CO 2 . However in Togo, while the level of CO 2 was increasing, the product per capita fell. The risk here is that even with resumption of economic growth, this will not be accompanied by an improvement in weather conditions. The results also show that in other countries (Cameroon, Ivory Coast, Gabon and Senegal), none of our variables significantly influence the variation in the level of pollution. This could be explained by the level fluctuation of the variables over the period as shown by the standard deviation (Table 1) and the stability tests of [START_REF] Brown | Techniques for Testing the Constancy of Regression Relationships over Time[END_REF], known as the cumulative sum (CUSUM) and cumulative sum of squares (CUSUMSQ) test, based on recursive regression residuals. The test-statistics are given in the Appendix. When these statistics fall inside the critical bounds of 5% significance, one could assume that the coefficients of the given regression are stable. Thus on the graphs, we can observe that in almost all countries, the CUSUM and the CUSUMSQ statistics don't fall outside the critical bounds during all the period covered. The estimation of an error correction model has allowed the observation of the effect in the long term of our different variables on emissions in South Africa and Togo (Table 5). The error correction term (EC t-1 ) is statistically significant and has the correct sign (negative) in both countries, which confirms the existence of a long term relationship between CO 2 emissions, income, energy consumption, and foreign trade in South Africa and Togo. This term shows the speed of an adjustment process following a short run shock. The high level of the error correction term coefficient in both countries in absolute value suggests a high speed of adjustment to the long run equilibrium following an external shock in the short term. In other words, when per capita CO 2 emission moves away from its equilibrium level, it adjusts by almost 66.2% within one year in South Africa and 92.1% in Togo. Furthermore, income that affects the variation of CO 2 in the short term is not significantly involved in the decrease of the pollution level in Togo.

Causality test result

Since we focused on the effect of income, energy consumption and international trade on the variation of CO 2 level in some SSA countries, it may be useful to conduct causality tests between CO 2 , income, energy consumption and trade openness in order to determine which variable causes each other in the sense of Granger. The results of the causality test based on T-Y approach are summarized in Table 6. The results show that income and CO 2 emissions are mutually dependent in Gabon, and that previous values of emissions are useful in predicting the income level of Kenya and Togo. What needs to be noted is that, in none of these countries, does income lead to CO 2 emissions. This fact is consistent with our estimation where we don't find any significant effect of income in explaining the variation of carbon dioxide emissions level. Furthermore, energy consumption is found to lead to higher pollution levels in Gabon, South Africa and Togo. This confirms the hypothesis postulated that energy is an important factor in explaining the variation of CO 2 levels. The result in Gabon concerning emission-trade causality also confirms the role that trade openness can play in dealing with the pollution level. However, the low degree of integration of SSA countries in international trade explains why in the rest of sample, both variables have an independent distribution.

Concerning the economic structure variables (income, energy consumption and international trade), results suggest a bi-directional relationship between energy consumption and income in Botswana and Gabon, and a unidirectional relationship running from energy to income in Kenya. This means that energy consumption is an important factor in the production process of these countries. Consequently, energy consumption measures may have an adverse effect on economic growth. Yet the previous results prove that energy consumption contributes mostly to a deteriorating environment. Thus an investment in efficient energy measures and renewable energy must be conducted in order to control for the pollution level and to promote economic growth. Energy conservation policy would not have an adverse effect on economic growth in Ivory Coast, where it is income which leads to energy consumption. On the other hand, in Cameroon, Senegal, South Africa and Togo the neutrality hypothesis between energy consumption and economic growth is supported. Our results are consistent with [START_REF] Akinlo | Energy consumption and economic growth: Evidence from 11 Sub-Sahara African countries[END_REF] for Cameroon and Togo and with [START_REF] Wolde-Rufael | Energy consumption and economic growth: The experience of African countries revisited[END_REF] for Ivory Coast and Gabon but contradict the results of both authors in the case of Kenya and Senegal.4

For the trade-income relationship, results suggest that international trade and income are mutually dependent in Botswana and that income causes trade in Gabon and Senegal in the sense of Granger. In the rest of the sample, we find that there are neutral relationships between both variables. What needs to be pointed out is that, in none of these cases, does international trade lead to income. This could mean that international trade is dominated by importation, or that these countries have a low degree of participation in world trade. We demonstrate that trade is an important factor that can be involved in reducing emissions level.

The important part of necessity goods in these countries prevents trade from contributing to reducing pollution. This fact can also be seen in the energy-trade relationship. Indeed, most of the countries are dependent on the importation of oil for their energy consumption, so that energy consumption is independent of international trade if we consider oil as a necessity commodity. The exception is Gabon, where both variables have dependent evolutions. This can be explained by the fact that the main export product in Gabon is oil.

Discussion and conclusion

The results of this study us helped to learn more about the impact of various economic factors on changes in the level of CO 2 in some SSA countries. Thus we observed that countries where energy consumption strongly influenced CO 2 emissions include Botswana, Kenya, South Africa and Togo. The extent of this effect depends on the level of development. Indeed, the more the country has a high income per capita, the less is the impact of its energy consumption on CO 2 compared to countries with low per capita income. These countries were not able to develop innovative technology, and thus their energy consumption is less efficient and therefore more polluting. As was pointed out, the effect of international trade was also dependent on the development of the country and its integration rate in the world trade level.

In poorer countries such as Kenya, it has an amplifying effect on CO 2 , unlike the case of South Africa. Data from the CIA World Factbook show that Kenya exports mainly natural resources and imports manufactured goods. Fluctuations in commodity prices also compromise the possibility of purchasing efficient technologies with priority given to basic necessities. South Africa has a relatively developed industrial sector, allowing it to have a diversified commercial sector, so that we have a positive effect of trade on improving the level of emissions.

From these results, what emerges is that SSA countries should pay particular attention to energy and trade sectors to reduce CO 2 emissions and to stimulate economic activity. In order to propose an appropriate economic policy for each country, it was necessary to study the nature of the relationship between these factors. The T-Y causality test result is consistent with our estimation and reveals that each country has its own specificity. For example, in Kenya, energy consumption stimulates economic activities, whereas it is economic growth which promotes energy consumption in Ivory Coast. Furthermore, rather than international trade stimulating growth, it is the reverse in Gabon and Senegal while the two variables have an independent trend in the rest of the countries. This could highlight the importance of imports compared with exports in the international trade of SSA countries or the fact that trade is dominated by imports of vital commodities so that their volume is not affected by the economic situation.

Thus the specificity of each country (low or middle income country, dependent or not on energy, the importance of international trade), must be considered in economic policy decisions that use energy consumption and international trade as instruments.

What also needs to be emphasized is the non-significant effect of income on the variation of CO 2 levels. This could mean that an increase in income doesn't affect the variation of pollution emissions considering the low level of CO 2 in absolute terms and the per capita income in the sub-Saharan region. This result is supported by the T-Y causality test suggesting that income doesn't lead to CO 2 emission, except in Gabon where both variables are mutually dependent. These econometric results contribute to learning more about the environmental policy to consider in SSA while pursuing a goal of increasing per capita income. However, they cannot be generalized to all countries in SSA because as we see with the results, each country has its own specificities in terms of its level of development. In addition they must be considered with particular attention, owing to the weak stability of the coefficients suggested by the [START_REF] Brown | Techniques for Testing the Constancy of Regression Relationships over Time[END_REF] test. According to these authors, the fact that significance is not achieved by the cusum squares of the recursive residuals plot may be due to shifts in values of regression coefficients. This could underline the fact that the SSA countries concerned by this study have experienced exogenous shocks during the period covered. For future research, the present study may be reinvestigated by taking into account the possible presence of structural breaks in the data series and the threshold effect in the dynamic relationship between CO 2 emissions, income, energy consumption and foreign trade.

  countries, namely Botswana (BWA), Cameroon (CMR), Ivory Coast (CIV), Gabon (GAB), Kenya (KEN), Senegal (SEN), South Africa (ZAF) and Togo (TGO). The choice of these countries included in the study was based on the availability of data on the variables incorporated in the work. The period covered by the data varies across countries: 1981-2010forBotswana; 1971-2010 for Cameroon, Kenya, South Africa, Senegal and Togo; 1971-2008 for Ivory Coast; and 1971-2007 for Gabon. The per capita CO 2 emissions (measured in metric tons) are used as a proxy for the environmental quality. Gross domestic product per capita at constant price of 2005 expressed in US dollar serves as a proxy for real income per capita.Energy consumption is proxied by total energy use per capita (measured in kgs of oil equivalent per capita). Data on trade openness is proxied by the ratio of total trade (sum of importation and exportation at constant 2005 US dollar) to population.

  Table 1 shows the summary statistics of the variables for each selected country. All the data are collected from the World Development Indicators (WDI, 2013) provided by the World Bank, and are converted to natural logarithms. results of Table 1 show that Gabon (US $ 7,270.90) and South Africa (US $ 5,006.92) have the highest mean of income per capita while Togo has the lowest (US $ 425.42). The high value of CO 2 emission per capita is displayed in South Africa (8.86 metric tons per capita) and the lowest in Togo (0.22 metric tons per capita). Gabon and South Africa have a

		Table 1: descriptive statistics for the selected countries
			mean	median	maximum	minimum standard deviation skewness	kurtosis
	BWA	C	1.897	2.238	2.657	0.912	0.367	-0.512	1.624
		Y	3917.805	3558.471	6295.876	1708.1	1405.662	0.145	1.864
		E	941.334	965.352	1149.242	730.229	122.983	-0.420	1.987
		T	3856.063	3993.204	5640.222	2138.245	830.289	-0.161	2.801
	CMR	C	0.283	0.217	0.674	0.089	0.166	1.299	3.531
		Y	942.752	915.991	1356.138	689.188	171.402	0.806	2.919
		E	394.434	396.442	429.407	326.368	24.939	-1.173	4.209
		T	400.327	428.834	583.459	227.974	100.451	-0.277	2.061
	CIV	C	0.526	0.517	0.809	0.323	0.121	0.519	2.499
		Y	1199.604	1104.489	1759.053	932.792	252.806	0.720	2.239
		E	423.072	403.626	570.109	355.112	61.652	1.069	3.233
		T	752.848	746.128	985.624	545.104	102.857	0.252	2.489
	GAB	C	4.903	4.612	10.918	0.859	3.162	0.246	1.606
		Y	7270.897	7124.092	12451.72	4961.261	1358.628	1.778	7.750
		E	1584.896	1378.122	2474.94	1176.041	385.199	0.633	2.078
		T	8300.617	8873.008	11743.05	4683.855	1899.172	-0.351	2.056
	KEN	C	0.283	0.271	0.386	0.192	0.056	0.189	1.907
		Y	516.544	512.007	575.060	424.036	29.178	-0.464	4.110
		E	450.297	450.296	482.024	430.172	10.733	0.653	3.700
		T	280.727	279.677	414.455	178.476	63.591	0.307	2.439
	SEN	C	0.440	0.435	0.602	0.311	0.069	0.300	2.561
		Y	723.375	719.728	813.792	634.353	47.674	0.000	2.072
		E	250.471	250.702	285.546	207.759	23.853	-0.059	1.868
		T	538.374	514.862	715.572	435.730	64.207	0.797	3.219
	ZAF	C	8.855	9.001	10.357	7.306	0.947	-0.107	1.880
		Y	5006.923	4967.908	5848.042	4472.486	363.991	0.651	2.769
		E	2547.905	2574.123	3007.979	1997.498	260.066	-0.525	2.438
		T	2272.221	2268.399	3460.257	1647.61	466.423	0.854	3.345
	TGO	C	0.223	0.221	0.523	0.129	0.064	2.447	13.099
		Y	425.420	425.521	533.540	319.663	44.738	0.285	2.976
		E	362.426	332.655	446.161	303.179	48.798	0.497	1.503
		T	526.72	487.519	1029.723	275.133	167.035	0.820	3.486

C indicates per capita carbon dioxide emissions in metric tons, Y indicates per capita real GDP in constant 2005 $ US, E indicates per capita energy consumption in kgs oil equivalent and T indicates sum of importation and exportation in constant 2005 $ US per capita.

The

  distribution, thus[START_REF] Pesaran | Bounds testing approaches to the analysis of level relationships[END_REF] have computed critical values for given significance levels. However, because of the small sample size used in this study, critical value ranges of the F-statistic as set out in[START_REF] Narayan | The saving and investment nexus for China: evidence from cointegration tests[END_REF] are used as they are more appropriate. If the computed F-statistic exceeds the upper bound, H 0 is rejected, if it falls below the lower bound, H 0 cannot be rejected, and if lies between both bounds, the test is inconclusive. Therefore, there is need to choose the appropriate number of lags in the model.The lag orders of the variables can be the selected Schwarz's Bayesian information criterion (SBIC), the Hannan and Quinn information criterion (HQIC), and the Akaike's information criterion (AIC). The SBIC selects the smallest possible lag length while AIC is employed to select maximum relevant lag length. The long run relationship among variables can be estimated after the selection of the ARDL model by AIC or SBIC criterion. Once a long-run relationship has been established, a general error correction model of eq. (

Table 2 :

 2 The null hypothesis in those tests is that the series have unit roots contrary to the alternative of stationarity. The results of the Augmented Dickey Fuller (ADF), Phillip-Perron (PP) and Elliot-Rothenberg-Stock (ERS) unit-root tests on the natural logarithm of the level and the first differences of the variables are summarized in Table2. The result suggests that all series are stationary in first differences, indicating that none of the variables is I(2) or beyond. This validates the use of ARDL bounds testing for cointegration. Test for integration

	country				level			first difference
			lag	ADF	PP	ERS	lag	ADF	PP	ERS
	Botswana	C	0	0.694	0.694	-0.579	0	-5.648*** -5.648*** -6.136***
		Y	0	-2.318	-2.318	0.263	0	-3.931*** -3.931*** -3.711***
		E	0	-2.581	-2.581	-2.672	0	-5.683*** -5.683*** -6.174***
		T	4	-2.838	-2.561	-1.540	3	-3.324**	-4.974*** -2.717**
	Cameroon	C	0	-3.169**	-3.169**	-2.607**	1	-6.575*** -7.188*** -6.340***
		Y	4	-3.352**	-2.014	-2.448**	4	-1.781*	-4.117***	-1.812
		E	1	-0.727	-0.865	-0.668	0	-4.394*** -4.394*** -4.475***
		T	0	1.005	1.005	-0.842	0	-5.032*** -5.032*** -5.118***
	Ivory Coast	C	0	-3.772**	-3.772**	-3.526**	0	-7.914*** -7.914*** -7.742***
		Y	1	-1.069	-1.459	-0.775	3	-2.639**	-3.775*** -3.202***
		E	0	0.511	0.511	-1.021	0	-6.999*** -6.999*** -7.059***
		T	1	-1.731	-1.731	-1.470	0	-5.791*** -5.791*** -5.809***
	Gabon	C	0	-0.985	-0.985	-1.005	0	-6.008*** -6.008*** -6.058***
		Y	1	-4.609**	-3.515**	-2.639	0	-4.317*** -4.317*** -4.046***
		E	2	-1.016	-2.798	-1.313	4	-1.283	-5.911***	-1.510
		T	1	-3.712**	-2.924	-1.881	0	-4.338*** -4.338*** -4.190***
	Kenya	C	3	-3.053**	-2.236	-2.081*	0	-3.673*** -6.245*** -6.235***
		Y	1	-1.314	-3.337**	-0.415	4	-2.009 *** -5.672***	-0.583
		E	1	0.465	0.592	-1.806	0	-5.069*** -5.069*** -4.856***
		T	4	-1.684	-1.958	-1.403	4	-2.627*** -6.478***	-1.123
	Senegal	C	0	-3.491*** -3.491***	-2.308*	0	-8.370*** -8.370*** -8.201***
		Y	0	0.144	0.144	-1.454	0	-7.423*** -7.423*** -6.504***
		E	3	-0.133	-0.373	-1.437	2	-2.044*** -6.270***	-1.884
		T	0	-2.837*	-2.837*	-2.876	2	-5.350*** -8.435*** -4.945***
	South
	Africa	C	4	-2.450	-1.871	-1.518	0	-5.821*** -5.821*** -5.586***
		Y	1	0.788	0.936	-0.99	0	-3.820*** -3.820*** -3.868***
		E	0	-2.025	-2.025	-0.693	0	-5.788*** -5.788*** -5.890***
		T	0	-1.864	-1.864	-1.383	0	-4.940*** -4.940*** -4.441***
									-
	Togo	C	0	-4.988*** -4.988*** -3.383***	0	-10.049***	10.049*** -8.954***
		Y	0	-2.850	-2.850	-2.692	3	-4.128*** -6.307*** -3.081***
		E	0	-2.134	-2.134	-1.788	0	-6.293*** -6.293*** -6.446***
		T	1	-3.188	-2.449	-2.983*	0	-4.734*** -4.734*** -4.670***
	*, ** and *** indicate significance level at 10%, 5% and 1%; C indicates per capita carbon dioxide emissions in
	metric tons, Y indicates per capita real GDP in constant 2005 $ US, E indicates per capita energy consumption in
	kgs of oil equivalent and T indicates sum of importation and exportation in constant 2005 $ US per capita.

Table 3 :

 3 Result for the F-test for cointegration

	result of the F-test of cointegration Fc(C/Y, E, T)
	country	AIC, SBIC optimal lag	F-statistic	result
	Botswana	1		2.21		No cointegration
	Cameroon	1		2.86		No cointegration
	Ivory Coast	1		1.98		No cointegration
	Gabon	2		3.29		No cointegration
	Kenya	1		0.23		No ointegration
	Senegal	1		3.07		No cointegration
	South Africa	1		4.34*	cointegration
	Togo	1		7.50***	cointegration
		n=30		n=35		n=40
	Critical value	Lower I(0)	Upper I(1)	Lower I(0)	Upper I(1) Lower I(0) Upper I(1)
	1%	5.333	7.063	5.198	6.845	5.018	6.610
	5%	3.710	5.018	3.615	4.913	3.548	4.803
	10%	3.008	4.150	2.958	4.100	2.933	4.020
	*, **, *** denote significance level at 10%, 5% and 1% respectively.

Table 4 :

 4 Short run results based on ARDL model * and *** indicate significance level at 10%, 5% and 1%; standards errors are in parentheses; and dof denotes degree of freedom; C indicates per capita carbon dioxide emissions in metric tons, Y indicates per capita real GDP in constant 2005 $ US, E indicates per capita energy consumption in kgs of oil equivalent and T indicates sum of importation and exportation in constant 2005 $ US per capita; BWA stands for Botswana, CMR for Cameroon, CIV for Ivory Coast, GAB for Gabon, KEN for Kenya, SEN for Senegal, ZAF for South Africa, and TGO for Togo.

		BWA	CMR	CIV	GAB	KEN	SEN	ZAF	TGO
	∆Y	-0.408	0.105	0.167	-0.326	0.564	-0.336	0.206	-1.486*
		(0.68)	(1.58)	(0.77)	(0.78)	(0.67)	(0.72)	(0.35)	(0.80)
	∆E	1.617*** 2.547	0.587	0.901	2.798**	0.410	0.815*** 3.625**
		(0.38)	(3.52)	(0.46)	(0.61)	(1.33)	(0.74)	(0.17)	(1.20)
	∆T	-0.095	0.227	0.188	0.530	0.361*	-0.040	-0.235*	-0.149
		(0.29)	(0.90)	(0.44)	(0.63)	(0.20)	(0.25)	(0.12)	(0.32)
	constant	0.031	0.033	-0.010	-0.022	-0.011	0.016	-0.002	-0.016
		(0.03)	(0.08)	(0.03)	(0.05)	(0.02)	(0.02)	(0.01)	(0.04)
	R 2	0.455	0.020	0.072	0.093	0.278	0.015	0.477	0.251
	dof	25	35	33	32	35	35	35	35
	*, *

Table 5 :

 5 ECM long run result based on ARDL model * and *** indicate significance level at 10%, 5% and 1%; standards errors are in parentheses; and dof denotes degree of freedom; C indicates per capita carbon dioxide emissions in metric tons, Y indicates per capita real GDP in constant 2005 $ US, E indicates per capita energy consumption in kgs of oil equivalent and T indicates sum of importation and exportation in constant 2005 $ US per capita; ZAF stands for South Africa, and TGO for Togo.

		ZAF	TGO
	∆Y	-0.066	-0.869
		(0.29)	(0.54)
	∆E	0.850***	2.755***
		(0.14)	(0.82)
	∆T	-0.219**	-0.262
		(0.09)	(0.21)
		-0.662***	-0.921***
		(0.15)	(0.14)
	constant	-0.001	-0.010
		(0.00)	(0.03)
	R 2	0.669	0.674
	dof	34	34
	*, *

Table 6 :

 6 Toda and Yamamoto causality test *, *** denote significance level at 10%, 5% and 1% respectively, C indicates per capita carbon dioxide emissions in metric tons, Y indicates per capita real GDP in constant 2005 $ US, E indicates per capita energy consumption in kgs of oil equivalent and T indicates sum of importation and exportation in constant 2005 $ US per capita; BWA stands for Botswana, CMR for Cameroon, CIV for Ivory Coast, GAB for Gabon, KEN for Kenya, SEN for Senegal, ZAF for South Africa, and TGO for Togo.

	country	BWA	CMR	CIV	GAB	KEN	SEN	ZAF	TGO
	Y→C	0.071	0.270	2.404	8.201*	0.080	2.582	0.255	0.276
	C→Y	0.642	1.398	0.266	9.040*	5.092**	1.489	0.269	6.334**
	E→C	0.134	0.010	0.454	7.854*	0.879	0.480	2.720*	4.152**
	C→E	2.156	0.015	1.979	3.069	5.573**	1.078	0.485	0.054
	T→C	0.010	0.335	0.080	8.149*	0.368	0.448	2.406	1.123
	C→T	1.040	3.146*	0.022	6.169	1.337	0.010	0.033	1.980
	E→Y	6.548**	0.205	0.238	40.866*** 14.355*** 0.918	0.006	0.857
	Y→E	2.770*	1.232	4.100**	29.699*** 0.811	0.605	0.055	2.057
	T→Y	3.523*	1.072	1.475	2.111	1.181	0.010	0.009	0.785
	Y→T	5.611**	0.029	0.377	21.19*** 2.693	3.284*	0.123	0.944
	T→E	0.613	1.297	0.495	33.958*** 0.600	0.106	1.223	0.757
	E→T	5.127**	0.111	0.163	33.671*** 1.265	0.079	0.227	1.527
	*, *

See Grossman and Krueger (1991) for a survey.

For a fairly detailed review on EKC, refer to[START_REF] Dinda | Environmental Kuznets Curve Hypothesis: A Survey[END_REF].

[START_REF] Onafowora | Bounds testing approach to analysis of the environment Kuznets curve hypothesis[END_REF] estimated an ordinary least squares model on levels of the variables (not on their first difference) to check for the long run dynamic between CO 2 emissions, income, energy consumption, population density and foreign trade.

The difference might be due to the sample, the approach adopted or the additional variables used. Akinlo (2008) uses the Granger causality test and considers price and government expenditure to study income and energy causality. His sample is from World development indicators of 2005. Wolde-Rufael (2009) uses T-Y Granger causality test but adds stock of capital and labour to income and energy consumption. His sample is from World Development Indicators of 2004 and 2007.