Hoai-Nam Nguyen

Paolino Tona

Continuously Adaptive PI Control of Wave Energy Converters under Irregular Sea-State Conditions

Abstract-The main objective in hydrodynamic control of wave energy converters (WECs) is the maximisation, for a broad range of sea states, of the energy captured from the waves. Latching control, model predictive control and "PID" control are examples of implementable strategies surveyed in the literature. "PID" control, while suboptimal, has the merit of being simple, requiring only straightforward calculations and can be considered a standard solution for WECs with a four-quadrant power takeoff system (PTO), in particular in its "PI" variant. Adaptive PI control has been already discussed in the literature, usually using a gain-scheduling approach, with optimal gains precomputed offline for a representative set of sea states and applied as a function of estimated sea state conditions. In most literature, only average online estimations of sea states have been proposed, with time windows of several minutes. Such intermittent (or switching) adaptive control laws are clearly suboptimal in terms of energy recovery, since the control gains are not continuously updated whereas the sea state is continuously time-varying. In this paper we present a continuously adaptive PI control strategy, which takes into account (non-ideal) PTO efficiency. The problem of optimising the gains of the PI controller is converted into a convex optimisation problem, thus guaranteeing that a unique and optimal solution can be found offline. The gains of the PI controller are continuously adapted online, on a wave-to-wave basis, based on a real-time estimate of the dominant wave frequency of the wave force, which is in turn estimated from (normally) available WEC measurements. Validation is performed on a case study based on a lab-scale point-absorber WEC. Simulation results show that the proposed adaptive control scheme can recover more energy than a fixed-gain PI controller even in a sea state with constant spectral characteristics. Furthermore, in a changing sea state, it outperforms switching gain-scheduled PI control where the gains are updated from time to time based on an evaluation of the current sea conditions. Though some preliminary wave basin tests have already given encouraging results, further experiments are needed to precisely quantify its energy harvesting potential. Moreover, further research is needed to include WEC motion or actuator constraints in the optimisation procedure.

Index Terms-Wave Energy Converters, PI Control, Conversion Efficiency, Adaptive Control, Unscented Kalman Filter, Dominant Frequency Estimation, Wave Force Estimation

I. INTRODUCTION

The main objective in hydrodynamic control of wave energy converters (WECs) is the maximisation, for a broad range of sea states, of the energy captured from the waves. In many WECs, active control can be performed acting on the force applied by the power take-off system (PTO) on the oscillating body responsible of primary energy conversion (f u in Fig. 1 which provides a simplified representation of a heaving-buoy WEC). Among the implementable strategies surveyed in the literature, we can recall for instance, latching control, model predictive control and "PID" control [START_REF] Ringwood | Control Optimisation and Parametric Design[END_REF]. While, in theory, latching control [START_REF] Saupe | Latching control strategies for a heaving buoy wave energy generator in a random sea[END_REF] and model predictive control [START_REF] Cretel | An application of model predictive control to a wave energy point absorber[END_REF], [START_REF] Hals | Constrained optimal control of a heaving buoy wave-energy converter[END_REF] allow to achieve high levels of energy capture, their practical implementation may be very challenging. This is due to the fact that these control strategies require a short-term prediction of wave excitation force, which can degrade the control performance, if the prediction is not perfect. Moreover, model predictive control, like other optimisation-based WEC control approaches, is a complex and computationally demanding algorithm.

"PID" control, on the other hand, has the merit of being simple, requiring only straightforward calculations [START_REF] Ringwood | Control Optimisation and Parametric Design[END_REF]. Because of that, it was one of the first control strategy to be implemented on WEC prototypes (the "Salter's duck" [START_REF] Salter | Development of the duck concept[END_REF], for instance). Even now, it can be considered a standard solution, when a four-quadrant PTO, capable of both harvesting and drawing power from the grid (respectively in generator and motor modes), is available. One embodiment of this approach is the PI (or BK) controller described in [START_REF] Sanchez | Control performance assessment and design of optimal control to harvest ocean energy[END_REF], which consists in computing the PTO force via a combined proportional feedback on the position and velocity of the oscillating element acting as primary converter (the float of Fig. 1) 1 . The damping controller (or P controller), where the PTO force is specified to be proportional to and oppositely directed to the velocity of the primary converter, is even more widely adopted, as it does not require the use of reactive power.

Assuming an ideal PTO (with 100 % conversion efficiency), well-established theoretical results are available to tune these kind of controllers for optimal energy recovery under the action of regular waves [START_REF] Falnes | Ocean waves and oscillating systems: linear interactions including wave-energy extraction[END_REF]. For practical applications, a dominant-frequency approximation is often used to extend these results to irregular sea-state conditions.

Whatever method is used to compute the P and I gains, they should not be kept constant, as the sea conditions change. Adaptive P or PI control has been already discussed in the literature [START_REF] Whittaker | The development of oystera shallow water surging wave energy converter[END_REF]- [START_REF] Hals | A comparison of selected strategies for adaptive control of wave energy converters[END_REF], at different level of details, in the form of a gain-scheduling approach. The main idea is to compute optimal gains for a representative set of sea states. The gains are calculated offline, analytically or numerically, using a gridding approach: for each sea state, the ones leading to the best average power are picked. Then the appropriate control action f u is found from a look-up table whose input is the current sea state, which can be identified, for instance, in terms of dominant wave period and significant wave height, or other characteristics of its spectrum. In most literature, only average online estimations of sea states have been proposed, with time windows of several minutes (10 minutes in [START_REF] Hansen | Modelling and control of the wavestar prototype[END_REF], 20-30 minutes in [START_REF] Hals | A comparison of selected strategies for adaptive control of wave energy converters[END_REF]). Such intermittent (or switching) adaptive control laws are clearly suboptimal in terms of energy recovery, since the control gains are not continuously updated whereas the sea state is continuously time-varying.

In this paper we present a (truly) adaptive PI control strategy, which takes into account (non-ideal) PTO efficiency. The problem of optimising the gains of the PI controller is converted into a convex optimisation problem, thus guaranteeing that a unique and optimal solution can be found offline. The gains of the PI controller are continuously adapted online, on a wave-to-wave basis, based on a real-time estimate of the dominant wave frequency of the wave force, which is in turn estimated from (normally) available WEC measurements.

Section II describes the modelling assumptions and defines the control problem, along with the energetic performance criterion to be optimised. Section III presents the procedure to design an optimal PI control under regular waves in the presence of a non-ideal PTO. Section IV explains how the results from the previous section can be used, following a dominant wave frequency approach to deal with realistic polychromatic sea states, to build an adaptive control scheme: notably, the variable-gains PI control law requires a real-time estimate of the dominant frequency of the wave excitation force acting on the WEC, which has to be estimated as well since it is not normally available while the WEC is running. Section V introduces a case study, based on a lab-scale pointabsorber WEC, and provides a preliminary assessment, in simulation, of the proposed control strategy. Finally, some concluding remarks end the paper.

II. WEC MODELLING AND CONTROL PROBLEM DEFINITION

A. WEC modelling

We consider here the type of WECs schematically represented in Fig. 1, that is, point-absorber WECs with an oscillating part that moves in one degree of freedom, the heaving direction for instance, with respect to a reference (fixed anchor or a submerged body). From the relative motion, useful energy can be extracted via the PTO.

Under the assumption that the oscillations of the system are relatively small, the WEC motion can be expressed, in the frequency domain, as follows [START_REF] Falnes | Ocean waves and oscillating systems: linear interactions including wave-energy extraction[END_REF],

jωM + Z pa (jω) + K pa jω v(jω) = f ex (jω) -f u (jω) (1)
where

• v(jω) is the heaving velocity of the oscillating part;

• f ex (jω) and f u (jω) are the wave excitation force and the PTO force, respectively; • M is the mass of the float with its connected parts;

• Z pa (jω) is the radiation impedance;

• K pa is the stiffness coefficient. Equation ( 1) is based on the Cummins integro-differential equation [START_REF] Cummins | The impulse response function and ship motions[END_REF], whose hydrodynamic coefficients can be computed via boundary element method (BEM) software, such as WAMIT, Diodore, AQWA ou NEMOH. The radiation impedance Z pa (jω) is the result of an approximation of the radiation impulse response by an infinite impulse response filter and can be decomposed as

Z pa (jω) = B pa (ω) + jω(M pa (ω) + M ∞ ) = H pa (jω) + jωM ∞ (2)
where B pa (ω) is the radiation resistance, and M pa (ω) is the added mass after M ∞ , the asymptotic value of the added mass for ω → ∞ is removed,

H pa = B pa (ω) + jωM pa (ω).
As in most modeling studies on point absorbers, it is assumed that viscous, frictional forces are negligible compared to the other terms in the equation of motion.

From (1), the float velocity can be rewritten as

v(jω) = 1 Z i (jω) (f ex (jω) -f u (jω)) (3)
where the intrinsic impedance Z i (jω) is defined as

Z i (jω) = B pa (ω) + jω M + M ∞ + M pa (ω) - Kpa ω 2 = R i (ω) + jX i (ω) (4) where R i (ω) = B pa (ω), X i (ω) = ω M + M ∞ + M pa (ω) - Kpa ω 2 (5)
are, respectively, the intrinsic resistance and reactance.

B. Control structure

We define the control problem to be solved in the classic framework of impedance matching control for WECs [START_REF] Falnes | Ocean waves and oscillating systems: linear interactions including wave-energy extraction[END_REF].

In a monochromatic sea state, the wave excitation force is given as

f ex (t) = Asin(ωt + φ) (6)
where the phase φ can be set to be zero without loss of generality. Assuming that the control force is a linear feedback of the heaving velocity, see Fig. 2, its expression in the Laplace domain is

f u (s) = Z c (s)v(s) (7)
where s is a complex number, s = σ +jω. Z c can be seen as a load impedance, and designed following impedance-matching principles.

1/Z i (s) Z c (s) f u (s) f ex (s) v(s) - + Fig. 2. Block diagram with WEC internal impedance Z i (s) and control impedance Zc(s).
The following proportional-integral (PI) control structure is chosen for the control block

Z c f u (t) = K p v(t) + K i t 0 v(τ )dτ (8)
Rewriting [START_REF] Whittaker | The development of oystera shallow water surging wave energy converter[END_REF] in the frequency domain

f u (jω) = K p -j K i ω v(jω) (9)
one obtains

K p = R c , K i = -ωX c (10)
where R c and X c are, respectively, the control (or load) resistance and reactance, i.e. Z c (jω) = R c (ω) + jωX c (ω).

C. Control objective

The control objective is to maximise the average (electric) power produced by the WEC

P a = 1 T T t=0 ηf u (t)v(t)dt, (11)
where η is the efficiency coefficient.

In the literature, it is generally assumed that η = 1, i.e. the PTO system is perfect [START_REF] Falnes | A review of wave-energy extraction[END_REF], [START_REF] Hals | Constrained optimal control of a heaving buoy wave-energy converter[END_REF]. With this assumption, the mathematical procedure needed to calculate the optimal control action f * u (t) is greatly simplified. For the case of regular waves, there exists an elegant analytical expression for f * u (t) as a function of wave frequency. One of the well known results is that, in order to harvest the maximum amount of energy [START_REF] Falnes | Ocean waves and oscillating systems: linear interactions including wave-energy extraction[END_REF], the optimal velocity v * (t) should be in phase with the wave excitation force f ex (t). This result, however, does not hold if η = 1, as it will be shown later.

Unfortunately, the assumption of a perfect PTO with no conversion losses is unrealistic. The power that is withdrawn from the grid is always more expensive than the power that is delivered to the grid via the PTO. To take into account this fact, we can consider the efficiency coefficient η as a function of the ideal instantaneous power

f u v, η(f u v) = η p if f u v ≥ 0, η n if f u v < 0 (12)
where the coefficients 0 < η p ≤ 1 and η n > 1 depend on the PTO system, and may even be a function of f u v.

III. OPTIMAL PI CONTROL WITH NON-IDEAL PTO UNDER REGULAR WAVES

A. Average power calculation

For the given wave excitation force [START_REF] Sanchez | Control performance assessment and design of optimal control to harvest ocean energy[END_REF] and the control force [START_REF] Falnes | Ocean waves and oscillating systems: linear interactions including wave-energy extraction[END_REF], it can be proved that the average power [START_REF] Cummins | The impulse response function and ship motions[END_REF] is obtained as

P a = A 2 R c 2 ((X c + X i ) 2 + (R c + R i ) 2 ) C η (13)
where

C η = n p - η n -η p π X c R c -arctan X c R c ( 14
)
The proof is not detailed here for the sake of conciseness. Indeed, if we consider the limit case when the PTO is perfect, i.e. η p = η n = 1, then C η = 1 in (7) and the average power is calculated as

P a = A 2 R c 2 ((X c + X i ) 2 + (R c + R i ) 2 ) (15)
which is the well-known result [START_REF] Falnes | Ocean waves and oscillating systems: linear interactions including wave-energy extraction[END_REF], obtained when the nonlinear efficiency coefficient is not taken into account.

B. Optimal frequency response

Using [START_REF] Hals | Constrained optimal control of a heaving buoy wave-energy converter[END_REF], the next step is to compute the optimal R c and X c which maximise P a . We require that R c ≥ 0, since R c is used to convert the wave energy into useful mechanical or electrical energy. Notice that, because the factor A 2 2 has no influence on the optimal solution, it can be omitted in the optimisation problem. Thus, the problem to be solved is

min Rc,Xc - R c (X c + X i ) 2 + (R c + R i ) 2 C η (16)
With C η defined by ( 14), ( 16) is clearly a nonlinear optimisation problem. Hence it is impossible to obtain a closed analytical form for the optimal solution. Before proceeding further, define

d = X c R c Consider the function f (p) = η p - ηn-ηp π (d -arctan(d))
. By calculating the derivative of f (p), it is easy to see that this function is monotonically non-increasing. Hence the equation,

η p - η n -η p π (d -arctan(d)) = 0 (17)
has a unique solution d * for any positive η n and η p . Therefore, there exists a unique line

X c = d * R c (18)
that satisfies the equation

η p - η n -η p π X c R c -arctan( X c R c ) = 0 (19)
Figure 3 presents a numerical example of the average power P a as a function of R c , X c for a given and fixed ω, using the WEC intrinsic impedance and the efficiency coefficients defined in the case study of Section V. Figure 4 shows a contour plot of P a in the R c -X c plane. Using Figures 3 and4, two main remarks can be made 1) P a is not a concave function of R c and X c .

2) The line [START_REF] Julier | New extension of the kalman filter to nonlinear systems[END_REF] divides the plane R c -X c into two regions

• For all R c , X c such that

X c ≥ d * R c the function P a is convex. • For all R c , X c such that X c ≤ d * R c
the function P a is concave. Figure 5 shows the average power P a as a function of R c and X c in the concave region.

Figure 6 and Figure 7 show respectively, the optimal R c and X c (solid blue lines) as a function of ω. By solving [START_REF] Julier | A new approach for filtering nonlinear systems[END_REF], one obtains d * = 5.4403 for the given η p , η n . Figure 6 and Figure 7 also show R i and -X i (dashed red lines) as a function of ω. It can be noticed from Figure 7 that the optimal X c is very close but not equal to -X i . Hence the phase response ∠W (s) of the transfer function ( 20) of the WEC system in closed loop

v(s) = 1 Z i (s) + Z c (s) f ex (s) = W (s)f ex (s) (20)
is in general not equal to zero. Hence the optimal velocity is in general not in phase with the wave excitation force in the presence of the nonlinear efficiency coefficient. The result of Falnes [START_REF] Falnes | Ocean waves and oscillating systems: linear interactions including wave-energy extraction[END_REF], where it is stated that the optimal velocity is in phase with the wave excitation force, can be considered only as an ideal case, where the PTO system is perfect.

Remark: If the PTO system is constrained to be purely resistive, then X c = 0. In this case the optimisation problem (16) becomes

min Rc - n p R c X 2 i + (R c + R i ) 2 (21)
There exists an analytical closed form solution to ( 21)

R c = X 2 i + R 2 i (22)
Note that the efficiency coefficient does not play any role in the optimal R c in the resistive PTO system case. This is explained by the fact that there is no reactive power involved, so the energy harvested from the waves is always positive. Figure 8 presents the optimal R c as a function of frequency, for the purely resistive case.

IV. ADAPTIVE PI CONTROL

The approach described above allows to compute, for a specific WEC, the optimal control resistance R o c and reactance X o c , for each given regular wave excitation force. Using [START_REF] Hals | A comparison of selected strategies for adaptive control of wave energy converters[END_REF], the corresponding optimal PI gains can be computed off-line, frequency by frequency, as:

K o p = R o c (ω), K o i = -ωX o c (ω) (23)
The resulting curves can be implemented, as look-up tables for instance, in an adaptive PI control scheme where the scheduling variable is the dominant frequency of the wave force excitation 2 . A customary approach in control of real devices is indeed to choose the load so that impedance matching occurs at the dominant frequency in a wave spectrum [START_REF] Korde | Hydrodynamic Control of Wave Energy Devices[END_REF], though we should recall that the PI gains are truly optimal only in a harmonic excitation context. This approach is followed, for instance, for the design of the "simple-buteffective" controller in [START_REF] Fusco | A simple and effective real-time controller for wave energy converters[END_REF], a velocity tracking controller with adaptive parameters computed using online estimates of the dominant frequency and dominant amplitude of the wave excitation force.

The structure of the adaptive control scheme is shown in Fig. 9. Notice that the variable-gains PI control law requires an estimate of the dominant frequency of the wave excitation 2 The dominant frequency of the wave force is defined as the peak of its spectrum force as an input, and that the wave excitation force must be estimated as well, as it is a quantity that cannot be measured during normal WEC operation. To address the first problem, an estimation approach based on an extended Kalman filter (EKF), applied to a nonstationary, harmonic approximation of the wave excitation force, is proposed in [START_REF] Fusco | A simple and effective real-time controller for wave energy converters[END_REF]. However, it must be recalled that the EKF provides a solution to a nonlinear estimation problem via local linearisations of the underlying model. Thus, the variation of the wave force is large and/or the sampling time intervals are not sufficiently small, the linearisation may yield highly unstable filters, potentially leading to divergence phenomena [START_REF] Julier | A new approach for filtering nonlinear systems[END_REF], [START_REF] Julier | New extension of the kalman filter to nonlinear systems[END_REF]. A more robust solution consists in estimating the dominant frequency (and amplitude) estimation using an unscented Kalman filter (UKF), as described in [START_REF] Nguyen | Dominant wave frequency and amplitude estimation for adaptive control of wave energy converters[END_REF].

As to wave excitation force estimation, a few methods have been proposed recently and tested in real-time, in a model predictive control (MPC) framework (see for instance [START_REF] Kracht | Performance improvement of a point absorber wave energy converter by application of an observer-based control: Results from wave tank testing[END_REF], [START_REF] Ling | Real-time estimation and prediction of wave excitation forces for wave energy control applications[END_REF], [START_REF] Abdelkhalik | Estimation of excitation forces for wave energy converters control using pressure measurements[END_REF]), [START_REF] Nguyen | Experimental validation of a nonlinear MPC strategy for a wave energy converter prototype[END_REF]). The method assessed in [START_REF] Nguyen | Experimental validation of a nonlinear MPC strategy for a wave energy converter prototype[END_REF] (and fully described in [START_REF] Nguyen | Wave excitation force estimation for wave energy converters of the point absorber type[END_REF]), based on a linear Kalman filter and a random walk model for the variation of the wave excitation force, has the main advantages of using only standard measurements for reactive control (position, velocity, PTO force) and of yielding quite accurate estimates over a large range of sea states.

V. CASE STUDY

In this section, we consider a lab-scale 1-DoF WEC system (described in [START_REF] Møller Jakobsen | Practical performance of control strategies for point absorbers[END_REF]), whose intrinsic impedance can be modelled as the transfer function The idea is to compare the energy recovery performance of the proposed adaptive PI control using as a benchmark the (switching) gain-scheduled PI controller described in [START_REF] Hansen | Discrete displacement hydraulic power take-off system for the wavestar wave energy converter[END_REF] and [START_REF] Sánchez | Control performance assessment and design of optimal control to harvest ocean energy[END_REF]. Thus, optimal K p and K i gains are computed offline over a set of sea states using a gridding approach: for each given sea state and considered time interval, the nominal WEC model is simulated in closed loops for a grid of gains, and the combination leading to the best average power P a is picked. The control action is then to be determined from a lookup table with the current sea state (in terms of dominant wave period and significant height), being re-estimated at regular intervals, as an input. Notice, however, that the gainscheduling mechanism has not been detailed in [START_REF] Hansen | Discrete displacement hydraulic power take-off system for the wavestar wave energy converter[END_REF] nor in [START_REF] Sánchez | Control performance assessment and design of optimal control to harvest ocean energy[END_REF]: it has not been specified, for instance, if the gains are to be interpolated or chosen instead with a nearest-neighbour approach.

Z i (s) = s 6 + a 5 s 5 + a 4 s 4 + a 3 s 3 + a 2 s 2 + a 1 s b 7 s 7 + b 6 s 6 + b 5 s 5 + b 4 s 4 + b 3 s 3 + b 2 s 2 + b 1 s + b 0 (24)
The same set of irregular waves, generated using a JON-SWAP spectrum, which has been used to compute the optimal gains of the reference controller, is also used for validation. Of course, for a given irregular wave, the parameters of the switching gain-scheduled controller will not vary during the test, so the comparison is actually done with a fixed-parameters PI. The spectrum of the wave excitation force corresponding to one these irregular waves is shown in Fig. 11. Its dominant frequency is about 5 rad/s. Fig. 12 shows respectively, the control input of the adaptive PI control (solid blue) and of the reference PI control (dashed red) over a 8-seconds long time interval. Fig. 13 shows the velocity of the adaptive PI control (solid blue), of the PI control (dashed red) and of the wave excitation force (dash-dot green), over the same time interval. Adaptive PI Fig. 13. Velocity of the adaptive PI control law (solid blue), and of the PI control (dashed red), and the wave excitation force (dash-dot brown) for the WEC system [START_REF] Hansen | Discrete displacement hydraulic power take-off system for the wavestar wave energy converter[END_REF].

Finally, Fig. 14 presents the instantaneous power P a for the adaptive PI control (solid blue) and for the PI control (dashed red). Fig. 15 shows the total accumulated energy for the adaptive PI control (solid blue) and for the reference PI control (dashed red). Interestingly, even in a test where the sea state is considered constant, the proposed adaptive PI control performs better than the reference PI control. This is due to the continuous adaptation of PI gains to the dominant frequency estimate. Fig. 16 shows the variation of the dominant frequency estimate, obtained using the UKF approach mentioned in the previous section, over the last part of the test. Fig. 17 shows that the corresponding dominant magnitude estimated by the UKF (not used in the adaptive PI control law) follows well the wave force amplitude.

In order to validate the proposed adaptive control against a continuous change in sea state, corresponding to a transition from the wave of Fig. 11 to another wave, whose excitation force is shown in Fig. 18, a second test is performed, under the excitation given by a wave force generated with a time-varying dominant frequency, which varies linearly from 5 rad/s to 2 rad/s. For comparison purposes, a switching PI controller is implemented, using a table of optimal pre-computed gains for the two waves and for a set of intermediate sea states. The switching is performed on the basis of an average estimation of the dominant wave frequency. The first gain switch takes

VI. CONCLUSIONS

This paper focuses on a novel approach to design a continuously adaptive PI control strategy for wave energy converters.

A procedure, based on the solution of a convex optimisation problem, is presented, which allows to compute the optimal PI gains for a given regular wave excitation force, while taking into account (non-ideal) PTO efficiency. It is shown that the optimal velocity is in general not in phase with the wave excitation force when realistic PTO efficiencies are considered. Optimal gain curves, as a function of frequency, can be obtained from this procedure, and implemented in an adaptive control scheme based on a dominant frequency approach. Combining two robust and accurate methods for realtime estimation of the wave excitation force from normally available WEC measurements and of the dominant frequency of the (estimated) wave excitation force, the gains of the PI controller can be continuously adapted online, on a wave-towave basis.

A case study, based on a lab-scale point-absorber WEC, has been used for validation. Simulation results show that the proposed adaptive control scheme can recover more energy than a fixed-gain PI controller even in a sea state with constant spectral characteristics. Furthermore, in a changing sea state, it outperforms switching gain-scheduled PI control where the gains are updated from time to time based on an evaluation of the current sea conditions.

In the future, a more extensive assessment is planned, both in simulation and through experimentation. Though the proposed adaptive PI control system has already been tested in a wave basin on the lab-scale prototype used as a case study, with encouraging results, further experiments are needed to precisely quantify its energy harvesting potential.

Finally, further research is needed to take into account WEC motion or actuator constraints (PTO force limits, stroke limits), which are not dealt with in the current optimisation procedure, as well as more complex WEC dynamics.

Fig. 1 .

 1 Fig. 1. Schematic diagram of a generic (heaving-buoy) wave energy converter

Fig. 3 .

 3 Fig. 3. Average power Pa as a function of Rc and Xc.

Fig. 4 .

 4 Fig. 4. Contour plot Pa in the Rc -Xc plane.

Fig. 5 .

 5 Fig. 5. Average power Pa as a function of Rc and Xc in a concave region.

Fig. 6 .

 6 Fig. 6. Optimal Rc and R i as a function of ω.

Fig. 7 .

 7 Fig. 7. Optimal Xc and -X i as a function of ω.

Fig. 8 .

 8 Fig. 8. Optimal Rc as a function of ω in the resistive PTO system case.

Fig. 9 .

 9 Fig. 9. Adaptive PI control approach.

  whose coefficients are given in TABLE I Numerator Denominator b 7 = 1.44 b 6 = 300.4 a 5 = 208.6 b 5 = 1.237 × 10 5 a 4 = 8.583 × 10 4 b 4 = 1.284 × 10 7 a 3 = 8.899 × 10 6 b 3 = 1.652 × 10 8 a 2 = 1.074 × 10 8 b 2 = 2.106 × 10 9 a 1 = 7.031 × 10 8 b 1 = 9.988 × 10 9 b 0 = 6.539 × 10 10

Figure 10

 10 Figure10presents the corresponding Bode plot, typical of WECs of the point-absorber type, with a resonance frequency of f r = 7.84 rad/s.

Fig. 10 .

 10 Fig. 10. Bode plot of the considered WEC system

Fig. 11 .

 11 Fig. 11. First wave force spectrum (dominant frequency of ∼5 rad/s)

Fig. 14 .Fig. 15 .

 1415 Fig. 14. Instantaneous power Pa of the adaptive PI control law (solid blue), and of the PI control (dashed red) for the WEC system.

Fig. 16 .

 16 Fig. 16. Dominant wave force frequency estimation.

Fig. 17 .Fig. 18 .Fig. 19 .

 171819 Fig. 17. Dominant wave magnitude estimation.

TABLE I MODEL COEFFICIENTS

 ICOEFFICIENTS

This corresponds to a proportional-integral action on the velocity of the primary converter, which is the reason for it being (confusingly) referred to as a "PI" controller.


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PRIOR ART

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[0006] Only the A lamin and the B lamins undergo matu ration after the synthesis of a precursor (for an overvieW, see Gruenbaum et al., 2000 (20)). Lamin C is synthesiZed directly in the mature form thereof.

[0007] The precursor of laminA and lamins B terminates in a characteristic CaaX motif (C is a cysteine, a an amino acid With an uncharged aliphatic chain and X is any amino acid, Which, in this case is methionine (for an overvieW, see Levy & Cau 2003 (29)).

[0008] The C-terminal CaaX motif enables the attachment of a fatty acid (generally a fatty acid at C15, farnesyl), oWing to a famesyl transferase. This prenylation (the famesyl unit derived from a basic aliphatic unit at C5 called isoprene) enables the prelamins to be inserted into the membrane of the endoplasmic reticulum after the synthesis of same in the cytosol. They undergo the effects therein of an endoprotease, Which is itself inserted into the envelope membrane of the reticulum and the active site of Which is cytosolic. The spe ci?c endoprotease of prelamin A is Facel (or ZMPSTE24, Zinc Metallo-Protease STE24 homologue of yeast), While Jul. 7, 2011 Face2 (or Rce1, ras-converting enZyme) is speci?c to the prelamins B. These enZymes catalyZe the hydrolyse of the peptide bond betWeen the cysteine and the folloWing (ali phatic) amino acid, shortening the prelamins of 3 amino acids. The carboxyl end of the farnesylated cysteine is next recogniZed by an isoprenylcysteine-carboxymethyl trans ferase (ICMT), Which attaches a methyl group thereto via esteri?cation.

[0009] The maturation of the prelamin A continues only With a second endoproteolytic cleavage via Facel, Which releases a famesyl-peptide of 15 amino acids and the mature lamin A. This lamin A, Which no longer contains the fatty acid, becomes soluble, is imported into the nucleus oWing to the nuclear localiZation signal thereof, and is localiZed in the nuclear lamina itself as Well as in the rest of the nuclear compartment, thereby forming a true nuclear skeleton (Gruenbaum et al., 2005 (19)). On the other hand, at the C-terminal end, the cysteine of the mature lamin B is alWays farnesylated and methyl-esteri?ed. It therefore remains inserted into the envelope membrane of the reticulum, and then into the nucleoplasmic face of the nuclear envelope, hence its exclusive localiZation at the nuclear lamina, beneath the inner membrane of the nuclear envelope Wherein it is anchored.

[0010] The term prenylation is understood to mean the attachment to the thiol group of a cysteine of either a famesyl chain of 15 carbon atoms, Which is then referred to as fame sylation, or of a geranyl-geranyl chain of 20 carbon atoms,

Which is then referred to as geranyl-geranylation (Reid et al., 2004 (39)), or else of any other isoprene derivative.

[0011] The famesylation step, Which is catalyZed by the famesyltransferase (FTase), Which recogniZes the C-terminal consensus sequence (CaaX), preferably attaches a famesyl group onto the cysteine residue of the unit.

[0012] Geranylgeranylation is the attachment by the gera nylgeranyl-transferase (GGTase) of a geranylgeranyl group onto the cysteine residue of the unit.

[0013] The fatty acids are derived from the biosynthesis pathWay, Which is used by the cells to manufacture choles terol, steroids, the heme in haemoglobin and ubiquinones (Hampton et al., 1996 (20)).

[0014] The family of prenylated proteins comprises approximately 300 members in the human genome, the majority of Which is identi?able by the CaaX C-terminal motif (Reid et al., 2004 (39)). The proteins of the Ras, Rho, Rab families (Leung et al., 2006 (28)), some proteins provid ing in import function toWards the mitochondria (HDJ2), and some proteins mitotic proteins (CENPE, CENPF) are in par ticular prenylated (Winter-Vann & Casey 2005 (51)). In gen eral if, in the CaaX motif, X is a serine, a methionine, a cysteine, an alanine or a glutamate, the preferably grafted isoprenoid is farnesyl. If X is a leucine, recognition of the CaaL motif Will preferably occur via the GGTase, Which Will catalyZe the transfer of a geranylgeranyl group (Basso et al., 2006 (1)). It is probable that other isoprene derivative groups can also be attached to this cysteine, even though that is not described in literature.

[0015] Three lamin genes exist in man. The LMNA gene situated at 1q21. 2-q21.3 (Wydner et al., 1996 (52)), yields [0016] In fact, the gene codes differently for 4 spliced prod ucts, of Which the lamins C and the prelamin A are the 2 primary ones (Lin & Worrnan 1993 (31)). The differential production of lamins A and C is carried out by the use of an alternative splicing site of exon 10 of the pre-messenger, Whereby lamin C is coded by exons 1 to 10 and lamin A is coded by exons 1 to 9, the ?rst 90 base pairs of exon 10, and exons 11 and 12 (lamin A-speci?c).

[0017] Consequently, the prelamin A and lamin C peptides are identical in the ?rst 566 amino acids, the C-terminal ends of lamins C and of prelaminA then contain 6 and 98 speci?c amino acids, respectively.

[0018] The type B lamins include three different proteins (Shelton et al., 1981 (43)): lamins B1, B2 (the tWo isoforms most represented) and B3. The LMNBl gene is situated at 5q23.3-q31.1 and comprises 11 exons coding lamin B1 (Lin & Worrnan 1995 ( 30)). The LMNB2 gene is located at 19p13.3 and codes for lamins B1 and B3 via an alternative splicing mechanism (Biamonti et al., 1992 (2)).

[0019] Type B lamins are expressed consecutively in all cells from the ?rst stages of development, Whereas type A lamins are generally absent in embryonic stem cells (SteWart et al., 1987 (45)) and are expressed in all differentiated somatic cells. Their expression is subject to regulations, depending on the tissue and over the course of life (Duque et al., 2006 (9)). It appears that their expression is not necessary, since mice in Which the expression of lamin A Was speci? cally blocked, but Which nonetheless express lamin C and the other lamins, do not have any apparent phenotype (Fong et al., 2006 (14)).

[0020] Lamins interact With a very high number of protein partners having very diverse functions; they are therefore involved in a large number of nuclear processes, including DNA replication and repair, transcription and splicing con trol, organisation of the chromatin structure (for an overvieW, see Shumaker et al., 2003(44), ZastroW et al., 2004(54), Hutchinson et al., 2004(26), Gruenbaum et al., 2005 (19)).

The alterations of the lamina structure are the source of numerous hereditary human diseases. They are due to muta tions of the genes coding the lamins, or of other proteins of the lamina. These diseases have been grouped together under the generic term of laminopathies (Broer et al., 2006(5), Mattout et al., 2006 (33)). Mutations in the genes of the enZymes responsible for the maturation of the lamins (Facel in par ticular) have recently been identi?ed, Which give rise to dis eases likeWise belonging to the group of laminopathies (N a varro et al., 2004 (36) and 2005 (35)).

[0021] To date, the only disease in man Which is associated gene. Furthermore, it has recently been shoWn that the B lamins required for the formation of the spindle during mito sis, Which tends to prove that the role thereof is dynamic and multiple over the cell cycle, and not solely restricted to main taining the architecture of the nucleus (Tsai et al., 2006 (48)).

As concerns this latter role, a recent article demonstrates the structural function of the lamins B: cells arti?cially deprived of lamins B1 have a "?oating" nucleus in the cell, Which rotates about itself (Liu et al., 2007 (45)). The functional redundancy existing betWeen the tWo lamins B1 and B2 is undoubtedly also a direct re?ection of the criticality thereof, Which exerts heavy selection pressure and masks the effect of possible mutations in the sequence of the corresponding genes.

[0022] The functional alterations of the lamins A/C, Which are due to mutations in the LMNA gene, are at the source of at least 15 disorders encompassing very diverse diseases Within a clinical spectrum ranging from less serious forms affecting a single tissue in an isolated manner, to lethal sys temic forms during the perinatal period.

[0023] A number of mutations of the LMNA gene signi? cantly modify the assembly of proteins in the nuclear enve lope and disrupt the operation thereof. The morphology of the nuclei is altered in the cells of various tissues: they often have hernias Which extrude from the genetic material in the cyto plasm (Goldman et al., 2004 (18)).

[0024] The proteins commonly associated With the nuclear envelope, the lamins B, certain proteins of the nuclear pores and the LAP2 proteins are absent from the periphery of these hernias.

[0025] These morphological anomalies are folloWed by functional alterations, Which result in cell death. Amongst all of the diseases grouped together under the name of lamino pathies, only those associated With the abnormal accumula tion of a prenylated form of protein are of concern in this invention.

[0026] These primarily involve the Hutchinson-Gilford or Progeria syndrome (De Sandre-Giovannoli et al., 2003 (7), Eriksson et al., 2003 (11)), and restrictive dermopathy (Na varro et al., 2004 (36)). In these tWo syndromes, the physio pathological cause is an accumulation and persistence of immature farnesylated prelamin in the cells of patients.

[0027] Restrictive dermopathy, Which is lethal around the natal period, is characterised by clinical signs Which are nearly all the result of a de?ciency in the skin Which restricts in utero movements. This disease is very rare. The skin is rigid and tight, it sags in places, thereby causing tears, for example, in the vicinity of the arm pits and neck. The eyelashes, eye broWs and hair on the skin are absent or very sparse. Hydram nios is often present, and the decrease in foetal movement is reported from the 6th month of pregnancy. In terms of the skeletal system, X-rays reveal contractures in all of the joints, "ice-pick" feet, thin, dysplastic and bipartite clavicles, ribbon ribs, long tubular arm bones and demineralisation of the skull. Transmission of the lethal restrictive dermopathy is autoso mal recessive.

[0028] Mutations of LMNA and ZMPSTE24/Face1 Were have been reported for this disease (Navarro et al., 2004 (36)).

In both cases, the physiopathological mechanism is the same: the prelamin A cannot mature (no mutation of Facel or dis appearance of the cleavage site by mutation of the prelamin A), and remains farnesylated, and therefore inserted into nuclear membrane. The accumulation and persistence of these abnormal precursors in the cells, Which probably pre vents the normal interaction of the lamins B and C With their partners, results in the death of the cells and, in the short run, of the patient. It Was clearly shoWn that it is indeed the persistence of the famesyl group, and not the absence of the mature lamin A, as might ?rst have been thought, Which is responsible for cell toxicity (Fong et al., 2004 (16)).

[ [0030] The responsible mutation, situated in exon 11 of the LMNA gene, activates a cryptic splicing site of the pre RNAm, resulting in an RNAm With 150 nucleotides deleted (De Sandre-Giovannoli et al., 2003 (7), Eriksson et al., 2003 (11)). This deleted RNAm is converted into an abnormal prelamin A, progerin, Which cannot be matured into normal et al., 2004 (18)). Here again, it is the persis tence of the famesyl group, Which is also necessary for anchoring the envelope membrane of the reticulum in Which some of the enZymes responsible for maturation are located (cleavages, methylation), Which is responsible for the cell toxicity of the progerin (Fong et al., 2004 (16)).

[0031] These systemic diseases have the particular charac teristic of being associated With the premature appearance of signs commonly associated With ageing. Their common physiopathological characteristic is that of generating a pre nylated lamin, along With the described consequences.

Jul. 7, 2011

This pro gerin is localised in the lamina, at the periphery of the cell nucleus. The nucleus of cells of "normal" elderly patients may have hernias characteristic of a laminopathy caused by accidental splicing events, Which result in anomalies in the cell functions and are probably at least partially responsible for the ageing thereof.

[0034] In the case of in vivo skin, progerin is also synthe siZed by a sub-population of dermal ?broblasts and kerati nocytes, cells in Which it accumulates With age. Progerin could therefore be a marker for skin ageing (McClintock et al., 2007 (34)).

[0035] It appears that identical molecular mechanisms are, on the one hand, responsible for the signs of premature ageing in individuals afflicted with Progeria and, on the other hand, to a much less extent, involved in the physiological ageing of individuals not carrying mutations.

[0036] There are tWo therapeutic approaches described in the prior art for improving the cell phenotype caused by the pathological production of progerin. The ?rst of these solu tions is quite simply to prevent the use of this cryptic splicing site in exon 11 via the spliceosome, While (47), Fong et al., 2006 (15)).

[0037] HoWever, blocking the farnesylation can induce a compensatory geranylgeranylation (Bishop et al., 2003(3), Varela et al., 2008 (55)).

[0038] On the other hand, it Was recently reported that FTIs cause a stoppage of the cell cycle by blocking the proteasome (Demyanets et al., 2006(8), Efuet & Keymarsi 2006 (10)).

Thus, the treatment undoubtedly causes an accumulation of progerin in the nucleoplasm, Which is probably ubiquitiny lated and undegraded by the proteasome.

[0039] Additionally, recent studies report that the decrease in the farnesylation rate of the progerin in vivo is very loW, of the order of 5% (Young et al., 2006 (53)), Which is not su?i cient to explain the restoration of the nuclear morphology observed in vitro.

[0040] Finally, the FTIs are speci?c to a single one of the protein prenylation pathWays, and cannot be anticipated as global inhibitors of post-translational prenylation.

[0041] In addition, it is reported that the total absence of one of the enZymes of this pathWay, mevalonate kinese, is lethal during childhood (loss-of-function homozygous muta tion of the gene coding for this enZyme, a syndrome reported by Hoffmann et al., 2003 (24)). and an inhibitor of farnesyl pyrophosphate synthase has a synergistic effect in the restoration of the normal phenotype in ?broblasts of patients afflicted with Progeria. The effect of the association is markedly greater than the effect of either of the inhibitors used individually.

[0043] Use of the association on cells of patients af?icted With Progeria and in a mouse model reproducing restrictive dermopathy results in an inhibition of protein prenylation, and therefore in the appearance of non-farnesylated prelamin A and in the improvement of the nuclear symptoms (Varela et al., 2008 (24)).

[0044] This invention relates therefore to the use of at least one inhibitor of hydroxymethylglutaryl-coenZyme A (HMG CoA) reductase and at least one inhibitor of farnesyl pyro phosphate synthase, or of one of the physiologically accept able salts thereof, in the preparation of a cosmetic and/ dermatological composition. This composition, for example, can be intended for the treatment of pathological or non pathological situations associated With the accumulation and/ or persistence of prenylated proteins in cells.

[0045] In particular, this invention relates to a cosmetic and/ or dermatological composition including:

[0046] a. at least one inhibitor of hydroxymethylglu taryl-coenZyme A (HMG-CoA) reductase,

[0047] b. at least one inhibitor of farnesyl pyrophosphate synthase, and

[0048] c. at least one cosmetic and/or dermatological product.

[0049] According to another aspect, this invention likeWise Jul. 7, 2011

[0054] Thus, according to this aspect, the aforesaid com pounds can be used in a mixture for simultaneous adminis tration or separately for concomitant or successive adminis tration.

[0055] For example, the at least one inhibitor of hydroxym ethylglutaryl-coenZyme A (HMG-CoA) reductase can be administered ?rst, then the at least one inhibitor of farnesyl pyrophosphate synthase can be administered secondly. In another example, the at least one inhibitor of farnesyl pyro phosphate synthase can be administered ?rst, then the at least one inhibitor of hydroxymethylglutaryl-coenZyme A (HMG CoA) reductase can be administered secondly.

[0056] According to a third example, the at least one inhibi tor of farnesyl pyropho sphate synthase, on the one hand, and the at least one inhibitor of hydroxymethyl glutaryl-coenZyme A (HMG-CoA) reductase, on the other hand, are adminis tered at the same time but, for example via different routes of administration, e.g., chosen from amongst oral administra tion, topical administration or by injection.

[0057] Irrespective of the embodiment of this invention, in particular in the aforesaid aspects, [0058] For example, the composition used in this invention can be prepared extratemporaneously at the moment of administration or immediately prior thereto. The preparation can be made, for example, by means of a syringe or a pump system Which enables the tWo aforesaid compounds to be mixed together at the moment of administration or immedi ately prior thereto. The administration can be carried out, for example, by means of a dual syringe and/ or cosmetic pump, e.g., by means of a pump marketed by the Kemai Company (trademark) .

[0059] In other Words, even though in this description ref erence is made to a composition, it is Well understood that each of the compounds of the composition can be adminis tered jointly or concomitantly With the other compounds (e.g., in a single composition or in tWo compositions or in three compositions, each of these compositions including one or more of the aforesaid components, the method of admin istering each of the compounds or composition being option ally identical or different), or independently of one another, e.g., successively, e. g., separate administration of at least one inhibitor of hydroxymethylglutaryl-coenZyme A (HMG CoA) reductase, separate administration of at least one inhibi tor of farnesyl pyrophosphate synthase, these administrations being carried out With respect to a single patient, jointly, concomitantly or successively or alternately, in the aforesaid order or in another order. These various administrations can be carried out independently of one another or in an interre lated manner (composition or co-administration), by an iden tical or different method of administration (injection, inges tion, topical application, etc.), one or more times per day, for one or more successive or non-successive days.

[0060] This composition, for example, can be intended for the treatment of pathological or non-pathological situations associated With the accumulation and/or persistence of pre nylated proteins in cells.

[0061] It is also Within the scope of the invention to use compounds Which are both inhibitors of hydroxymethylglu taryl-coenZyme A (HMG-CoA) reductase and inhibitors of farnesyl pyrophosphate synthase.

[0062] Very particularly, the cosmetic and/or dermatologi cal composition according to the invention can be intended for the treatment of pathological or non-pathological situa tions associated With the accumulation and/or persistence of pro gerin in cells; even more particularly, for the treatment of situations associated With the accumulation and/or persis tence of farnesylated prelamin A in cells, Whether truncated or non-truncated or modi?ed.

[0063] In particular, since it is admitted that physiological ageing is in particular a consequence of the presence of pro gerin in cells over the course of life. The progerin is concen trated in particular in the mesenchymal cells, the composition according to the invention can be intended to prevent the effects of cell ageing, particularly in the skin and/or vascular endothelium.

[0064] For example, said cosmetic composition can be used [0065] Progeria is accompanied by very pronounced alope cia. A?licted children lose their hair starting at 2 years old, Which hair becomes sparse, White and silky. The eyelashes and eyebroWs also fall out, and body hair is rare or completely absent (see the detailed description of clinical symptoms compiled by R. Hennekam (2006). This alopecia is likeWise observed in the murine model of progeria (Zmpste24-de? cient mice). In a completely unexpected Way, the treatment With an association of pravastatin and Zoledronate according to this invention at least partially restores the fur coat of these mice.

[0066] The cosmetic and/or dermatological composition of this invention can be intended for the treatment of any living being, man or animal, particularly for preventing the effects of cell ageing. The composition therefore ?nds application in both human medicine and veterinary medicine, in cosmetics and/or in dermatology. For example, it makes it possible to repair and/ or prevent physiological or non-physiological skin disorders, including ageing of the skin and the alteration or disappearance of the hair.

[0067] According to the invention, any inhibitor of famesyl pyrophosphate synthase, or of one of the physiologically acceptable salts thereof, can be used in the preparation of the composition according to the invention.

[0068] The physiologically acceptable salts, for example, can be salts formed With hydrochloric, hydrobromic, nitric, sulphuric, phosphoric acids, carboxylic acids, e.g., such as acetic, formic, propionic, benZoic, maleic, fumaric, succinic, tartaric, citric, oxalic, glyoxylic and aspartic acids, sulfonic alkanes such as sulfonic methane or ethane acids, arylsulfonic acids such as benZene or para toluene sulfonic acids.

[0069] In particular, the inhibitor of famesyl pyrophos phate synthase can be one of the members of the family of polyphosphonates, particularly aminobiphosphonates (NBPs), or one of the physiologically acceptable salts thereof.

[0070] Polyphosphonates are synthetic molecules Widely used in the treatment of osteoporosis and for bone regenera tion.

Jul. 7, 2011

[0071] The term phosphonate applies to molecules very similar to phosphate:

Phosphonate Phosphate

[0072] The core structure of biphosphonates (BPs) is the equivalent of a P4OiP bond in ATP, for example, but Where the oxygen is replaced by a carbon. This endoWs these mol ecules With a completely distinctive degree of stability.

[0073] A simple biphosphonate Would be equivalent to ADP, the tWo phosphate groups (O3Pi) being replaced by the biphosphonate group.

oR2o

[0074] Unlike the oxygen of the phosphates, the central carbon can also be involved in 2 bonds, and it is the nature of the groups grafted onto this carbon Which determines the speci?city of the biphosphonates.

[0075] When the "side" chains (R1 and R2) comprise an amine function (NH), or more generally one or more nitrogen atoms, reference is made to an aminobipho sphonate or NBP.

[0076] Of course, other substituents can be attached to the oxygen atoms.

[0077] Pyrophosphoric acid or pyrophosphate in solution [0078] is used in numerous metabolic reactions as a sub strate transporter, and it is restored at the end of the reaction.

One of the metabolic pathWays using molecules coupled With pyrophosphate is that of protein prenylation.

[0079] Grafting of an isopentenyl-PP (base unit at C5) onto a geranyl-PP (C10) in order to yield a famesyl-PP, Which is a reaction catalyZed by the famesyl pyrophosphate synthase (FPS) enZyme, releases a PPi.

[0080] It is this step Which is speci?cally inhibited by the NBPs.

[0081] In this regard, and as an example, the aminobiphos phonate (inhibitor of the famesyl pyrophosphate synthase) can be chosen from amongst:

[0082] alendronic acid or the ionic form thereof, alendr onate;

[0083] clodronic acid or the ionic form thereof, clodr onate;

[0084] etidronic acid or the ionic form thereof, etidr onate;

US 2011/0165092 A1 [0093] 4-N,N-dimethylaminomethane diphosphonic acid or the ionic form thereof, dimethylaminomethane diphosphonate; ibandronic acid or the ionic form thereof, ibandr medronic acid or the ionic form thereof, medr neridronic acid or the ionic form thereof, neridr risedronic acid or the ionic form thereof, risedr tiludronic acid or the ionic form thereof, tiludr Zoledronic acid or the ionic form thereof, Zoledr

[0094] ot-amino-(4-hydroxybenZylidene)diphospho nate.

[0095] Preferentially, according to the invention, it is pre ferred to use Zoledronic acid (also called Zolendronic acid) or the ionic form thereof, Zoledronate (also called Zolendr onate).

[0096] According to the invention, any HMG-CoA reduc tase inhibitor, or one of the physiologically acceptable salts thereof, can be used in preparing the compositions.

[0097] In particular, the HMG-CoA reductase inhibitor can be a molecule of the family of statins, Whether liposoluble or Water-soluble, or one of the physiologically acceptable salts thereof.

[0098] Statins have been shoWn to exist in fungi. They have an inhibiting activity With respect to the HMG-CoA reduc tase, a key enZyme in the biosynthesis of cholesterol and steroids, Which catalyZes the reduction of the hydroxymeth ylglutarate coupled With the CoenZymeA into mevalonic acid (mevalonate in solution). This inhibition is ensured by their structural analogy With the skeleton of hydroxymethylglut arate. The metabolic pathWay involved is admittedly that of cholesterol biosynthesis, but is also that of the synthesis of phenyl groups, Which are 5-carbon isoprene base unit poly mers used to modify approximately 300 proteins in cells and to attach a lipophilic tail thereto, enabling, in particular, the anchoring of same to the membranes.

[0099] The principle polyprenes, all of Which are derived from pyruvate and HMG-CoA, are geranyl (C10), famesyl (C15) and geranylgeranyl (C20).

[0100] All of the statins are globally hepatoselective, but all do not have the same mode of entry into cells. As a matter of fact, pravastatin and rosuvastatin are both hydrophilic, and therefore Water-soluble, on the other hand, all of the others are lipophilic and therefore can freely diffuse through the plasmic membranes (double layers of lipid molecules), Which undoubtedly explains the higher degree of toxicity thereof.

Water-soluble statins need a speci?c transporter in order to enter the cell, Organic Anion Transporter 3, or OAT3, or SLC22A8 (Takedaa et al., 2004 (46)).

[0101] They are Widely used for treating hypercholester olemia, and the side effects thereof, Which are rare, are Well characterized. This is the case in particular of rhabdomyolysis (1 to 7% of the cases depending on the molecule used, Evans Jul. 7, 2011Jul. 7, et al., 2002 (12) (12)), the early signs of Which are muscle pain in the patient being treated and result in the immediate cessation of treatment.

[0102] In this regard, and as an example, a statin can be chosen from amongst atorvastatin, simvastatin, pravastatin, rivastatin, mevastatin (or compactin), ?uindostatin, velosta tin, ?uvastatin, dalvastatin, cerivastatin, pentostatin, rosuvas tatin, lovastatin, pitavastatin, or one of the physiologically acceptable salts thereof.

[0103] Lovastatin, pravastatin and simvastatin are mol ecules derived from fungal metabolites, Whereas the others (atorvastatin, cerivastatin, ?uvastatin, pitavastatin and rosuv astatin) are entirely synthetic. According to the invention, pravastatin is preferably used, Which is a Water-soluble semi natural statin.

[0104] According to the invention, it is of course possible to use one, tWo or more inhibitor(s) of famesylpyrophosphate synthase associated With one, tWo or more inhibitor(s) of HMG-CoA reductase.

[0105] According to one particular form of the invention, the composition can be intended for the treatment of patho logical or non-pathological situations requiring protein pre nylation to be inhibited. These diseases may or may not be labelled, e.g., Noonan's syndrome, cardio-fascio-cutaneous syndrome, or diseases associated With abnormal prenylation or persistence of Ras and signal transduction proteins.

[0106] According to another particular form of the inven tion, the composition can be intended for the treatment of pathological or non-pathological situations exhibiting signs of ageing, Whether natural, premature or accelerated. In par ticular, in cases of signs of deterioration of the vascular endot helium (protection of the vascular endothelium), ageing of the skin, hair loss of body hair and/or head hair, and bone lysis.

[0107] The composition according to the invention is pref erably a dermatological and/or cosmetic composition intended for the treatment of Progeria (HGPS, Hutchinson Gilford Progeria Syndrome) and restrictive dermopathy (DR or RD).

[0108] According to the invention, the inhibitor of fame sylpyrophosphate synthase and the inhibitor of HMG-CoA reductase are advantageously present in the composition at physiologically effective doses.

[0109] Generally speaking, the quantities to be adminis tered can be adapted based on the patient, the disease, the method of administration, etc. It is understood that repeated applications can be carried out, possibly in combination With other active ingredients or any vehicle.

[0110] In general, the daily dose of the inhibitors Will be the maximum dose for obtaining the desired dermatological and/ or cosmetic effect.

[0111] According to the invention, the inhibitor of hydroxymethylglutaryl-coenZyme A (HMG-CoA) reductase and the inhibitor of farnesylpyrophosphate synthase can be used in the composition, mixed With one or more inert excipi ents or vehicles, i.e., physiologically inactive and nontoxic. Reference can be made, for example, to the ingredients com monly used in cosmetic or dermatological preparations.

[0112] The cosmetic and/or dermatological composition of this invention includes at least one cosmetic and/ or dermato logical product. A "cosmetic and/ or dermatological product" is understood to mean any product Which makes it possible to form a composition, together With the aforesaid inhibitors, enabling a cosmetic and/or dermatological application, i.e., a

  Figure 15

  lamins A and C by Way of alternative splicing. The LMNA gene consists of 12 exons. The start of exon 1 codes the N-terminal globular end common to lamins A and C; the end of exon 1 and up to the start of exon 7 codes the central US 2011/0165092 A1 helicoidal portion; ?nally, the other exons code the C-termi nal globular end (Levy & Cau 2003 (29)).

  With mutations of the LMNBl or 2 genes is a leukodystrophy caused by a complete duplication of the LMNBl gene(Padi ath et al., 2006 (37)). Doubt remains as to the potential involvement of variations of sequences found in LMNB2 in patients afflicted With Barraquer-Simons syndrome(Hegele et al., 2006 (22)). HoWever, in vitro experiments With RNAi (RNA-interference)(Harborth et al., 2001 (21)), as Well as in With the murine model(Vergnes et al., 2004 (50), have shoWn that type B lamins are essential to cell development and integrity. As a matter of fact, B1 lamin de?ciency in mice results in perinatal lethality. In addition, the nuclei of the embryonic ?broblasts of the same LMNBl-de?cient mice shoW remarkable alterations in the nuclear morphology, close to those observed in patients carrying mutations of the LMNAJul. 7, 2011

  laminA: the absence of 50 amino acids of exon 11 comprising the recognition site of the protease blocks the 2"" cleavage of the pro gerin, the C-terminal end of Which retains the famesyl group thereof. It therefore remains inserted into the nucleo plasmic face of the nuclear envelope, Which has characteristic alterations, hernias of the nucleoplasm in the cytosol and anomalies in the distribution of the peripheral heterochroma tin (Goldman

[ 0032 ]

 0032 TWo recent studies have shoWn that a reduction in intranuclear accumulation of the truncated or non-truncated famesylated prelamin effectively prevents appearance of the cell phenotype. The ?rst one Was conducted on the Facel protease-de?cient progeroid murine model(Pendas et al., 2002 (38)). When they are crossed With mice expressing less than half LaminA (Lmna+/-mice), the effects of the absence of Facel are lessened(Varela et al., 2005 (49)). The second study shoWs that the treatment of cells of HGPS patients With a morpholino (antisense oligonucleotides) targeting the cryp tic splicing site abolishes the mutant phenotype(Scaf?di & Mistelli 2005 (43)).[0033] Several recent studies (see Scaf?di & Mistelli 2006 (42)) shoW the involvement of lamin A in the physiological ageing process. In particular, it has been demonstrated that, during physiological ageing, progerin is synthesiZed by the cells in the absence of any mutation of the LMNA gene, due to the loW-noise use of the cryptic splicing site of exon 11.

  "masking" it by treating With an antisense oligonucleotide (Scaf?di & Misteli 2005 (41)), or With a retrovirus producing an siRNA (Huanget al., 2005 (25)). The results are promising in vitro, but, in this case, it is a matter of "gene" therapy, and the development of a drug based on this approach is inevitably long and com plicated, With all of the draWbacks related to vectoriZation of the ASOs in order to obtain an in vivo effect. The second solution consists of inhibiting the farnesyltransferase, the enZyme Which catalyZes the transfer of the famesyl group to the prelamins using famesyl pyrophosphate. When such inhibitors (FTI) are used, a "normal" nuclear envelope is only partially restored on HGPS (Progeria) cells in culture, and the survival of RD mice (KO ZMPSTE24) is improved(Glynn & Glover 2005 (17),Capell et al., 2005 (6),Toth et al., 2005

  DISCLOSURE OF THE INVENTION[0042] After lengthy research, the inventors have shoWn that the association of an inhibitor of hydroxymethylglutaryl coenZyme A reductase (family of statins), an inhibitor of famesyl pyrophosphate synthase (aminobiphosphonate fam US 2011/0165092 A1 ily, NBPs), or of one of the physiologically acceptable salts thereof, is an effective treatment for the pathological or non pathological situations associated With the accumulation and/ or persistence of prenylated proteins in cells, insofar as it acts on the entire protein prenylation pathWay, both at C15 and at C20 or in the non-characterised forms. The inventors have further ascertained that the association of an inhibitor of hydroxymethylglutaryl-coenZyme A (HMG-CoA) reductase

  Joint or simultaneous use, i.e., the at least tWo folloWing products (the at least one inhibitor of hydroxymethylglutaryl-coenZyme A reductase, on the one hand, and the at least one inhibitor of farnesyl pyro phosphate synthase) are administered in a single com position containing same, [0051] Concomitant use, i.e., separately, each of the at least tWo folloWing products (the at least one inhibitor of hydroxymethylglutaryl-coenZyme A reductase, on the one hand, and the at least one inhibitor of farnesyl pyro phosphate synthase) being administered independently via identical or different routes of administration, the administration of these tWo products being carried out concomitantly or [0052] Successive use, that is separately, each of the at least tWo folloWing products (the at least one inhibitor of hydroxymethylglutaryl-coenZyme A reductase, on the one hand, and the at least one inhibitor of farnesyl pyro phosphate synthase) being administered independently via identical or different routes of administration, the administration of these tWo products being carried out successively, one product after the other, [0053] the use of at least one inhibitor of hydroxymethyl glutaryl-coenZyme A (HMG-CoA) reductase, and the at least one inhibitor of farnesyl pyrophosphate synthase in the treat ment of skin and/or hair disorders.

  0029] InApril 2003, based on an overlapping of the symp toms common to acromandibular dysplasia and to certain diseases resulting in premature ageing, the inventors demon strated that Progeria, Which is the most typical and serious form of premature ageing, results from a mutation of the LMNA gene(De Sandre-Giovannoli et al., 2003 (7)). Chil dren af?icted With this disease, Which is also called Hutchin son-Gilford syndrome, suffer from accelerated ageing, up to ten times more rapid than that of a normal individual, and have a life expectancy Which does not exceed 13 years. In Europe, one child in approximately six million is affected. The symptoms are ageing of the skin, baldness, reduction in the siZe of the jaW and problems associated With old age, e.g., stiffness in the joints and cardiovascular disorders. The latter,

such as myocardial infarction or atherosclerosis, are often the cause of death.

  in order to prevent and/or treat skin disorders. These skin disorders can be disorders of natural or pathological origin or associated With a therapeutic treatment. The disorders of natural origin are those associated With natural ageing, e.g.,

	skin ageing. The pathological disorders, for example, are
	those associated With hormonal skin ageing, photo-induced
	ageing, premature skin ageing, myolipocutaneous ageing,
	lipodystrophy and restrictive dermopathy. The disorders
	associated With a treatment, for example, are those associated
	With chemotherapeutic treatments, X-ray irradiation by
	X-rays, gamma rays, ultraviolet rays or those associated With
	side effects of the treatments.