Method of controlling a wave energy conversion system maximizing the power output

Abstract

The invention is an improved wave energy conversion system (1, 2) including a model predictive control method for an energy conversion machine (1) that maximizes the power output by accounting for the energy conversion efficiency and prediction of a wave (3).

Claims

1. A method of controlling a wave energy conversion system that converts energy of waves into electrical or hydraulic energy comprising a mobile system cooperating with an electric machine or a hydraulic machine, the mobile system having an oscillating motion with respect to the electric machine or the hydraulic machine comprising: a) constructing a dynamic model of the wave energy conversion system relating velocity of the mobile system to a force exerted by the waves on the mobile system and to force exerted by the electric machine or the hydraulic machine on the mobile system; b) constructing a wave energy model of the wave energy conversion system relating average power generated by the electric machine or the hydraulic machine to the force exerted by the electric machine or the hydraulic machine on the mobile system to the velocity of the mobile system and to energy conversion efficiency of the wave energy conversion system, the wave energy model accounts for conversion efficiency from mechanical energy into electrical or hydraulic energy with the wave energy model being expressed as a formula $P_m^c=-\frac{1}{T}{}_{t=0}^T\mathrm{uvdt},$ with P.sub.m.sup.c being average power output, t being time, T being a predetermined duration, being energy conversion efficiency, u being force exerted by the electric machine or the hydraulic machine on the mobile system and v being velocity of the mobile system in relation to the electric machine or the hydraulic machine; c) predicting a force exerted by waves on the mobile system for a predetermined time period; d) determining a control value of a force exerted by the electric machine or the hydraulic machine on the mobile system which maximizes the average power generated by the electric machine or the hydraulic machine by use of the predicted force exerted by the waves on the mobile system of the dynamic model of the wave energy conversion system and on the wave energy model of the wave energy conversion system; and e) controlling the electric machine or the hydraulic machine with the control value.

2. A method as claimed in claim 1, wherein the force exerted by the waves on the mobile system is predicted by at least one of a measurement and an estimation of the force exerted by the waves on the mobile system using a set of pressure detectors associated with the mobile system associated with force sensors between the mobile system and the electric machine or the hydraulic machine.

3. A method as claimed in claim 1, wherein the force exerted by the waves on the mobile system is predicted by measurement of waves upstream from the wave energy conversion system.

4. A method as claimed in claim 1, wherein the dynamic model of the wave energy conversion system is constructed using a model of dynamics of the electric machine or the hydraulic machine and of a model of a mechanical part and a hydrodynamic part of the wave energy conversion system.

5. A method as claimed in claim 2, wherein the dynamic model of the wave energy conversion system is constructed using a model of dynamics of the electric machine or the hydraulic machine and of a model of a mechanical part and a hydrodynamic part of the wave energy conversion system.

6. A method as claimed in claim 3, wherein the dynamic model of the wave energy conversion system is constructed using a model of dynamics of the electric machine or the hydraulic machine and of a model of a mechanical part and a hydrodynamic part of the wave energy conversion system.

7. A method as claimed in claim 4, wherein the dynamic model of the electric machine or the hydraulic machine is expressed with equations: x.sub.a=A.sub.a.sup.cx.sub.a+B.sub.a.sup.cu.sub.c and u=C.sub.a.sup.cx.sub.a, and the model of the mechanical part and the hydrodynamic part as: x.sub.s=A.sub.s.sup.cx.sub.s+B.sub.s.sup.c(wu) and v=C.sub.s.sup.cx.sub.s, with x.sub.a being a state vector of the electric machine or the hydraulic machine, x.sub.s being a state vector of the mechanical part and the hydrodynamic part, A.sub.a.sup.c, B.sub.a.sup.c, C.sub.a.sup.c, A.sub.s.sup.c, B.sub.s.sup.c and C.sub.s.sup.c being dynamic matrices, inputs, outputs of a dynamic model of the electric machine or the hydraulic machine and of a mechanical part and a hydrodynamic part, u.sub.c being the control of the force exerted by the electric machine or the hydraulic machine on the mobile system, w being control of the force exerted by the waves on the mobile system, u being exerted by the electric machine or the hydraulic machine on the mobile system and v being velocity of the mobile system in relation to the electric machine or the hydraulic machine.

8. A method as claimed in claim 3, wherein the dynamic model of the electric machine or the hydraulic machine is expressed with equations: x.sub.a=A.sub.a.sup.cx.sub.a+B.sub.a.sup.cu.sub.c and u=C.sub.a.sup.cx.sub.a, and the model of a mechanical part and a hydrodynamic part as: x.sub.s=A.sub.s.sup.cx.sub.s+B.sub.s.sup.c(wu) and v=C.sub.s.sup.cx.sub.s, with x.sub.a being a state vector of the electric machine or the hydraulic machine, x.sub.s being a state vector of the mechanical part and the hydrodynamic part, A.sub.a.sup.c, B.sub.a.sup.c, C.sub.a.sup.c, A.sub.s.sup.c, B.sub.s.sup.c and C.sub.s.sup.c being dynamic matrices, inputs, outputs of a dynamic model of the electric machine or the hydraulic machine and of a mechanical part and a hydrodynamic part, u.sub.c being the control of the force exerted by the electric machine or the hydraulic machine on the mobile system, w being control of the force exerted by the waves on the mobile system, u being exerted by the electric machine or the hydraulic machine on the mobile system and v being velocity of the mobile system in relation to the electric machine or the hydraulic machine.

9. A method as claimed in claim 2, wherein the dynamic model of the electric machine or the hydraulic machine is expressed with equations: x.sub.a=A.sub.a.sup.cx.sub.a+B.sub.a.sup.cu.sub.c and u=C.sub.a.sup.cx.sub.a, and the model of a mechanical part and a hydrodynamic part as: x.sub.s=A.sub.s.sup.cx.sub.s+B.sub.s.sup.c(wu) and v=C.sub.s.sup.cx.sub.s, with x.sub.a being a state vector of the electric machine or the hydraulic machine, x.sub.s being a state vector of the mechanical part and the hydrodynamic part, A.sub.a.sup.c, B.sub.a.sup.c, C.sub.a.sup.c, A.sub.s.sup.c, B.sub.s.sup.c and C.sub.s.sup.c being the dynamic matrices, inputs, outputs of a dynamic model of the electric machine or the hydraulic machine and of a mechanical part and a hydrodynamic part, u.sub.c being the control of the force exerted by the electric machine or the hydraulic machine on the mobile system, w being control of the force exerted by the waves on the mobile system, u being exerted by the electric machine or the hydraulic machine on the mobile system and v being velocity of the mobile system in relation to the electric machine or the hydraulic machine.

10. A method as claimed in claim 4, wherein a Kalman filter is synthesized from two linear models used by a state observer for observing a state of the mobile system.

11. A method as claimed in claim 7, wherein a Kalman filter is synthesized from two linear models used by a state observer for observing a state of the mobile system.

12. A method as claimed in claim 1, wherein efficiency is a function of force u exerted by the electric machine or the hydraulic machine on the mobile system and of velocity v of the mobile system in relation to the electric machine or the hydraulic machine.

13. A method as claimed in claim 1, wherein the energy conversion efficiency is calculated with a formula: $\left(\mathrm{uv}\right)=\{\begin{array}{cc}{}_0& \mathrm{if}\mathrm{uv}0\\ \frac{1}{{}_0}& \mathrm{if}\mathrm{uv}<0\end{array}$ with .sub.0 being motor and generator efficiency of the electric machine or the hydraulic machine, with 0.sub.01.

14. A method as claimed in claim 12, wherein energy conversion efficiency is calculated with a formula: $\left(\mathrm{uv}\right)=-\frac{\frac{1}{{}_0}-{}_0}{}\arctan \left(\frac{r_a\mathrm{uv}}{2}\right)+\frac{\frac{1}{{}_0}-{}_0}{2}+{}_0$ with .sub.0 being motor and generator energy conversion efficiency of the electric machine or the hydraulic machine, with 0.sub.01, and r.sub.a being a smoothing parameter of the function.

15. A method as claimed in claim 13, wherein efficiency is calculated with a formula: $\left(\mathrm{uv}\right)=-\frac{\frac{1}{{}_0}-{}_0}{}\arctan \left(\frac{r_a\mathrm{uv}}{2}\right)+\frac{\frac{1}{{}_0}-{}_0}{2}+{}_0$ with .sub.0 being motor and generator energy conversion efficiency of the electric machine or the hydraulic machine, with 0.sub.01, and r.sub.a being a smoothing parameter of the function.

16. A method as claimed in claim 1, wherein c), d) and e) are repeated for a model predictive control with a moving horizon.

Description

BRIEF DESCRIPTION OF THE DRAWINGS

(1) Other features and advantages of the method according to the invention will be clear from reading the description hereafter, with reference to the accompanying figures wherein:

(2) FIG. 1 illustrates stages of the method according to the invention;

(3) FIG. 2 illustrates two curves for two examples of equations of efficiency as a function of an instantaneous mechanical power (product uv);

(4) FIG. 3 illustrates curves of an instantaneous mechanical power (product uv) as a function of time for various efficiency values;

(5) FIG. 4 illustrates a model of predictive control with a moving horizon according to the invention;

(6) FIG. 5 illustrates an example of a wave energy conversion system;

(7) FIG. 6 illustrates an amplitude as a function of frequency of various wave motion types which are simulated;

(8) FIG. 7 shows curves relative to displacement, velocity, control and instantaneous mechanical power (product uv) for the wave energy conversion system of FIG. 5 and a wave motion of HO 1 type of FIG. 6 according to a method of the prior art and according to the control method of the invention;

(9) FIG. 8 corresponds to FIG. 7 for a wave motion of HO 5 type in FIG. 6; and

(10) FIG. 9 is a flow chart the method of controlling wave energy conversion system that converts energy of waves into electrical or hydraulic energy comprising a mobile system cooperating with the wave energy conversion machine the mobile system having an oscillating motion with respect to the wave energy system.

DETAILED DESCRIPTION OF THE INVENTION

(11) The invention is a method of controlling a wave energy conversion system that comprises at least one mobile system cooperating with at least one wave energy conversion machine also referred to as Power Take-Off (PTO). The mobile system has an oscillating motion with respect to the wave energy conversion machine, under the action of the waves (or wave motion) and of the wave energy conversion machine. The wave energy conversion machine converts mechanical energy of motion of the mobile system into electrical energy. The wave energy conversion machine can therefore be an electric or a hydraulic machine.

(12) Notations

(13) The following notations are used in the description below: u is force exerted by the converter machine on the mobile means, and u.sub.c is control value of force exerted by the wave energy conversion machine on the mobile system, w is force exerted by waves on the mobile system, v is velocity of the mobile system in relation to the converter machine, x.sub.a is a state vector of the wave energy conversion machine of the wave energy conversion system; x.sub.s is a state vector of a mechanical and a hydrodynamic part of the wave energy conversion system; A.sub.a.sup.c, B.sub.a.sup.c, C.sub.a.sup.c, A.sub.s.sup.c, B.sub.s.sup.c and C.sub.s.sup.c are dynamic matrices, inputs, outputs of dynamic models of the wave energy conversion machine and of the mechanical and the hydrodynamic part. The model can be calculated by a balance of forces or an experimental identification procedure. If the model is linear, it can be represented by matrices (it is a formalism), with: P.sub.m.sup.c, being average power output; t being time; T.sub.f being a predetermined duration; being an energy conversion efficiency; with .sub.0 being motor and generator efficiency of the wave energy conversion machine; these are manufacturer's data or experimentally determined data; and r.sub.a is a predetermined smoothing parameter of an efficiency function.

(14) In the description below and in the claims, the terms waves, sea waves and wave motion are considered to be equivalent.

(15) The invention relates to a method of controlling a wave energy conversion system. FIG. 1 shows the various stages of the method according to the invention:

(16) 1. Construction of a dynamic model (MOD DYN)

(17) 2. Construction of an energy model (MOD ENE)

(18) 3. Prediction of the force exerted by the waves (PRED)

(19) 4. Estimation of the state of the system (ETAT)

(20) 5. Determination of the control value (VAL)

(21) 6. Control of the converter machine (COM).

(22) Stages 1 and 2 are stages that can be carried out beforehand and are part of a calibration procedure when the machine is installed. Stages 3 to 6 are carried out in real time, in a real-time loop (BTR).

(23) Stage 1Construction of a Dynamic Model (MOD DYN)

(24) In this stage, a dynamic model of the wave energy conversion system is constructed. The dynamic model represents the dynamic behavior reflecting the motion of the elements making up the wave energy conversion system under action of waves and under action of the wave energy conversion machine. The dynamic model is a model that relates velocity of the mobile system to a force exerted by the on the mobile system and to force exerted by the wave energy conversion machine on the mobile system.

(25) According to an embodiment of the invention, the dynamic model can comprise a linear model of dynamics of the wave energy conversion machine. This linear model can be written in form as follows: x.sub.a=A.sub.a.sup.cx.sub.a+B.sub.a.sup.cu.sub.c and u=C.sub.a.sup.cx.sub.a. The dynamic model can also comprise a linear model of a mechanical part and a hydrodynamic part of the wave energy conversion system. This linear model can be written in the form as follows: x.sub.s=A.sub.s.sup.cx.sub.s+B.sub.s.sup.c(wu) and v=C.sub.s.sup.cx.sub.s.

(26) Stage 2Construction of an Energy Model (MOD ENE)

(27) In this stage, an energy model of the wave energy conversion system is constructed. The energy model represents an energy balance between the energy generated by the wave energy conversion machine (that is the energy supplied to the grid) and the wave energy. According to the invention, this model accounts for an imperfect efficiency of conversion of, mechanical energy into electrical or hydraulic energy, and of the imperfect efficiency of conversion of electrical or hydraulic energy into mechanical energy. The energy model relates the average power generated by the wave energy conversion machine to a force exerted by the wave energy conversion machine on the mobile system, to velocity of the mobile system and to efficiency of energy converters.

(28) According to an embodiment of the invention, the energy model of the wave energy conversion system can be determined from the average power that is extracted for a duration T, which can be calculated with a formula of the type:

(29) $P_m^c=-\frac{1}{T}{}_{t=0}^T\mathrm{uvdt}.$
The definition of the above average power output is such that the average power has a negative sign if the energy is extracted from the system and for example supplied to the power grid. A maximization of the average power output therefore corresponds to a minimization of this power.

(30) According to the invention, function , which is the wave energy conversion efficiency, is used to model an imperfect efficiency of the energy conversion chain. In this case, the amount of energy generated in motor mode is decreased and a cost of energy supplied to the system (to bring it into resonance with the waves in motor mode) increases. A simple model using the hypothesis that efficiency .sub.0 is the same in motor and generator mode can be written with a first equation (Eq 1):

(31) $\left(\mathrm{uv}\right)=\{\begin{array}{cc}{}_0& \mathrm{if}\mathrm{uv}0\\ \frac{1}{{}_0}& \mathrm{if}\mathrm{uv}<0\end{array}.$
Another possibility for modelling efficiency avoiding the use of a discontinuity can be written with a second equation (Eq 2):

(32) $\left(\mathrm{uv}\right)=-\frac{\frac{1}{{}_0}-{}_0}{}\arctan \left(\frac{r_a\mathrm{uv}}{2}\right)+\frac{\frac{1}{{}_0}-{}_0}{2}+{}_0.$
The two options (Eq 1 and Eq 2) are illustrated in FIG. 2 as a function of product uv. The discontinuity of the first equation (Eq 1), unlike the second equation (Eq 2), can be observed in this figure. Function can also model other energy losses.

(33) Stage 3Prediction of the Force Exerted by the Waves (PRED)

(34) In this stage, force exerted by the waves on the mobile system is predicted in real time for a future period of predetermined duration T.sub.f. This predetermined duration T.sub.f can be short, ranging for example from 5 to 10 seconds. A prediction method is then selected and applied to the time being considered.

(35) According to an embodiment of the invention, one option is estimating or measuring in real time force exerted on the mobile system by the wave motion, with for example a set of pressure detectors arranged in a vicinity of the mobile system or force sensors arranged between the mobile system and the wave energy conversion machine, or wave elevation sensors. For the prediction, the force exerted on the mobile means by the wave motion can be extrapolated using, for example, an autoregressive model identified online.

(36) According to an alternative, the force exerted by waves on the mobile system is predicted using a set of detectors arranged upstream from the device. These detectors can notably measure the amplitude and the frequency of the waves.

(37) Stage 4Estimation of the State of the System (ETAT)

(38) The current state of the wave energy conversion system is determined in real time. For this stage, the current state can be estimated by use of a system state observer. This state observer can be achieved by synthesis of a Kalman filter from a dynamic model of the wave energy conversion system. For example, the observer is constructed from the linear models described in stage 1.

(39) Furthermore, the observer can take current control of the converter machine into account to determine a current state of the wave energy conversion system, for example by use of control at times preceding the time being considered.

(40) Stage 5Determination of the Control Value (VAL)

(41) In this stage, a Control value of the force exerted by the wave energy conversion machine on the mobile system is determined in real time. The control value maximizes the average power generated by the converter machine. Determination is therefore performed using prediction of force exerted by the waves (Stage 3), the dynamic model (Stage 1) and an energy model (Stage 2). Furthermore, this determination can be achieved by taking a state of the system into account (Stage 4).

(42) Using prediction of the force exerted by the waves gives the predictive characteristic of the control method according to the invention. Using an energy model accounting for energy conversion efficiency involves consideration energy losses, which enables an optimum control that maximizes the average power generated by the wave energy conversion machine.

(43) Indeed, if efficiency is different from 1, the product between control u and optimum velocity v changes significantly due to cost of the energy supplied to the machine, notably related to the energy losses, as shown in FIG. 3. In this figure, the curves show, for one example, product uv as a function of time for different values of efficiency .sub.0, which is why it is important to take efficiency into account in the calculation of the control.

(44) With the formulations of the dynamic and energy models, the search for the optimum control with constraints on control u and on the state of the system x can be formulated in a general manner: min.sub.u.sub.c P.sub.m.sup.c, as a function of the models and the following constraints: u.sub.min<u<u.sub.max and x.sub.min<x<x.sub.max. For minimization of the average power output, variable can be parameterized. One option is to select a variable u.sub.c that is piecewise constant.

(45) According to an embodiment of the invention, maximization of the average power output P.sub.m.sup.c is performed by use of an optimization algorithm.

(46) According to a variant embodiment of the invention, in order to smooth the control and to avoid unwanted oscillations, a penalty for the variations of u.sub.c can be added to the target function.

(47) According to an embodiment of the invention, a model predictive control (MPC) with a moving horizon is applied for real-time calculation of a control:

(48) 1. At a current step, step i which is the state of the system is estimated (stage 4) and the wave force is predicted (stage 3). The optimum control on a horizon limited to the predetermined period T (around 5 sec) is calculated from these values. This gives a series of optimum controls u.sub.c,i of length n.

(49) 2. The first element of the series of optimum controls u.sub.c,i is applied to the system as the target value for the converter machine (PTO). The value is maintained constant during a time step.

(50) 3. At a next step, step i+1, a state of the system is estimated (stage 4) and the wave force is predicted (stage 3). The optimum control on a horizon limited to the predetermined period T is calculated from these values. The initial values for this optimization are selected from results of a previous step (from the second input of u.sub.c,i to the last value that is repeated once). Optimization provides a new series of optimum controls u.sub.c,i+1 of length n.

(51) 4. The first element of the new series of optimum controls u.sub.c,i+1 is applied.

(52) These stages 1 to 4 are repeated for each time step.

(53) This model predictive control (MPC) with moving horizon is illustrated in FIG. 4. This figure shows the velocity v and control u.sub.c curves for two consecutive time steps: i and i+1. PA indicates the past of the considered time, AV is the future of the considered time, PRED i indicates the velocity prediction at time i and PRED i+1 is the velocity prediction at time i+1. It can be noted in this figure that the prediction has been modified between the two-time steps, even though the general form is similar. It can also be observed that the control has been slightly modified between the two-time steps.

(54) The algorithms that solve optimization problems are iterative algorithms. Since the time allowed for executing them is limited in real time, it is important for all the steps to give solutions that satisfy the constraints, in cases where it would be necessary to end the algorithm before convergence. An algorithm providing upon each iteration values that satisfy the constraints, such as for example an algorithm of interior point type, can be used to solve the optimization problem that gives the optimum control.

(55) Stage 6Control of the Converter Machine (COM)

(56) In this stage, the converter machine is controlled as a function of value determined in the previous stage. The wave conversion energy machine (electrical or hydraulic machine) is therefore actuated to reproduce a new value of force u.sub.c as determined in stage 5.

(57) For example, a new expression of force u.sub.c exerted by the wave energy conversion machine on the mobile system is applied to the electrical machine. Controlling the electrical machine so that it applies force u.sub.c to the mobile system is achieved by modifying the electrical current applied to the electrical machine. More precisely, to provide a torque or a force that drives the mobile system, a current is applied by supplying an electrical power.

(58) On the other hand, to produce a torque or a force withstanding the motion of the mobile system, a current is applied by recovering electrical power.

(59) FIG. 9 illustrates an example 900 of a method relates to a method 902 of which is controlling a wave energy conversion system that converts energy of waves into electrical or hydraulic energy comprising a mobile system cooperating with a wave energy conversion machine with the mobile system having an oscillating motion with respect to the wave energy conversion machine. The steps of the method start with step 904 which constructs a dynamic model of the wave energy conversion system relating velocity of the mobile system to a force exerted by the waves on the mobile system and to a force exerted by the wave energy conversion machine on the mobile system. The method then proceeds to step 906 which constructs a wave energy model of the wave energy conversion system relating average power generated by the wave conversion machine to the force exerted by the wave energy machine on the mobile system to the velocity of the mobile system and to energy conversion efficiency of the wave energy conversion system which accounts for conversion efficiency from mechanical energy into electrical or hydraulic energy. The method they proceed to step 908 which predicts a force exerted by the waves on the mobile system for a predetermined time period. Then the method proceeds to step 910 which determines a control value of a force exerted by the wave energy conversion machine on the mobile system which maximizes the average power generated by the wave energy conversion machine by use of the predicted force exerted by the waves on the mobile system of the dynamic model and on the wave energy model. Finally, at step 912 controlling the wave energy of conversion machine with the control value occurs.

Application Example

(60) A non-limitative example of a wave energy conversion system is an oscillating buoy as shown in FIG. 5. This wave energy conversion system comprises a buoy 2 as the mobile means of mass m, a converter machine 1 of damping d and elasticity k that is stationary. The buoy is subjected to an oscillating motion through waves 3 and to hydraulic forces (ordinary differential equation ODE of order 5).

(61) In this example, a model predictive control MPC is compared with a moving horizon according to the invention with a PI (Proportional Integral) control according to the prior art. Five different sea states are considered, whose (smoothed) spectra are shown in FIG. 6. This figure shows the curves, for five sea states (wave motions HO 1, HO 2, HO 3, HO 4 and HO 5), of the wave elevation h in meters as a function of frequency fin Hertz.

(62) For the control according to the invention, the dynamic model takes into account the dynamics of the mechanical part and hydrodynamic part with a fifth-order linear system and the actuator dynamics with a second-order linear system. The wave energy conversion machine stress is limited and the non-linear efficiency, corresponding to

(63) $\left(\mathrm{uv}\right)=\{\begin{array}{cc}{}_0& \mathrm{if}\mathrm{uv}0\\ \frac{1}{{}_0}& \mathrm{if}\mathrm{uv}<0\end{array},$
with .sub.0=0.7, is modelled and a saturation of the control can be achieved.

(64) A conventional control according to the prior art wave energy conversion system couples the PTO control on the velocity of the mobile system via a PI control:
u.sub.c=k.sub.p.sub.t.sub.0.sup.tdk.sub..

(65) For the comparison with the model predictive control according to the invention, has gains .sup.kp and .sup.k which are optimally calibrated for all the sea states being considered.

(66) The results of the comparison between the MPG strategy according to the invention (INV) and the PI strategy according to the prior art (AA) are summed up in Table 1. The energy generation gain ranges between 16% and 50%.

(67) TABLE-US-00001 TABLE 1 Average power of the model predictive control and of a conventional PI control Max. pos. Max. vel. P.sub.m PI P.sub.m MPC Gain ratio ratio (AA)/kW (INV)/kW P.sub.m (MPC/PI) (MPC/PI) Wave 2.83 4.26 50.43% 1.31 1.48 motion 1 Wave 8.64 11.89 37.54% 0.89 0.88 motion 2 Wave 16.42 20.76 26.44% 0.81 0.91 motion 3 Wave 24.44 29.26 19.75% 0.88 0.91 motion 4 Wave 32.14 37.14 15.55% 0.90 0.97 motion 5

(68) The trajectory of the system d is in rad, velocity v is in rad/s, control u is in 10.sup.6 Nm and product uv is in kW, controlled by MPC (INV) and PI (AA), as shown in FIGS. 7 and 8 respectively for a wave motion seq states HO 1 and HO 5 (see FIG. 6). It can be seen that the maximum position and the maximum velocity are reduced (except in the case of wave motion HO 1, which is very weak and does not excite high oscillations).