INTERNAL MODEL CONTROL APPLIED TO THE ARCHIMEDES WAVE SWING. Duarte Valério 1,3

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1 INTERNAL MODEL CONTROL APPLIED TO THE ARCHIMEDES WAVE SWING José Sá da Costa 1, Pedro Beirão,3, Duarte Valério 1,3 1 IDMEC / IST, TULisbon Av. Rovisco Pais 1, Lisboa, Portugal, {sadacosta,dvalerio}@dem.ist.utl.pt Instituto Superior de Engenharia de Coimbra, Department of Mechanical Engineering, Rua Pedro Nunes, Coimbra, Portugal, pbeirao@isec.pt Abstract: This paper addresses the control of a wave energy converter (WEC) called Archimedes Wave Swing (AWS). The internal model control (IMC) approach is used and the results compared with those achieved with the signal-based control currently implemented in this device. The AWS is a WEC of which a MW prototype has already been tested in Portugal. IMC is, to the best of the authors knowledge, being applied for the first time to a WEC. It is found out that wave energy absorption increases thereby, from 179 % up to 531 %. Such efficiency increases are important for this type of WEC to achieve break-even point. Keywords: Internal Model Control, Renewable energy, Electricity production, Wave energy converter, Archimedes Wave Swing 1. INTRODUCTION Environmental concerns urge humankind to find renewable sources of energy alternative to the current options, heavily dependent on fossil fuels. The inexorable rise of oil prices is another reason why renewable energies are receiving increased attention. Production of electricity from sea waves, 3 Pedro Beirão was partially supported by the Programa do FSE-UE, PRODEP III, acção 5.3, no âmbito do III Quadro Comunitário de Apoio. Duarte Valério was partially supported by Fundação para a Ciência e a Tecnologia, grant SFRH/BPD/636/4, funded by POCI 1, POS C, FSE and MCTES. Research for this paper was partially supported by grant PTDC/EME-CRO/7341/6 of FCT, funded by POCI 1, POS C, FSE and MCTES. that carry enormous amounts of energy, promises to do so as soon as sufficiently efficient wave energy converters (WECs) are developed. Control engineering plays an important role towards that objective. This paper is a step towards that objective, presenting preliminary results of the control of one such device, the Archimedes Wave Swing (AWS). Internal model control (IMC) is employed, which is (to the best of the authors knowledge) a novelty in what WECs are concerned. The AWS is a WEC of which a MW prototype (Fig. 1) has already been built and tested at the Portuguese northern coast during 4. Since this prototype has been decommissioned, results presented were obtained with an accurate, non-linear Simulink

2 Fig. 1. The MW AWS prototype model thereof. Several parameters and significant values of the model, however, have been altered, due to industrial protection reasons. Nevertheless, these results are expected to be of use during the development of a new, improved secondgeneration AWS prototype. The paper is organised as follows: section addresses the reasons why sea waves can become an important source of renewable energy; section 3 briefly presents the AWS, originally controlled with a PID controller as described in section 4; section 5 introduces IMC, implemented in section 6; the results are given in section 7; conclusions are drawn in section 8.. SEA WAVES AS A SOURCE OF ENERGY On the issues addressed by this section, see (Falcão, 6; Clément et al., ; Boud, ; Pontes and Falcão, 1) for more complete and detailed overviews. Ocean energy resources can be roughly divided into several different forms, depending on the different origins. The most developed are: Tidal energy (gravitational fields of the moon and the sun); Ocean thermal energy (solar radiation); Marine current energy (thermal and salinity differences added to tidal effects); Ocean wave energy (interaction of winds with the ocean surface). One of the most significant forms is ocean wave energy. Ocean waves and therefore wave energy can be generally considered as a concentrated form of solar energy. The majority of ocean waves are wind generated. The uneven heating of the Earth s atmosphere causes density variations, which influenced by gravity and the spinning globe, generate winds. When these winds blow across large expanses of water surface, friction (shear stress) develops between wind and water surface and little ripples arise at still water surface. At the same time, part of energy from wind flow is being transferred to water in the form of kinetic (movement of water molecules) and potential energy (mass of water present in waves above the mean sea level). This energy exchange causes air pressure differences in the Earth s atmosphere. As a result the ripples initially formed become wavelets and finally are converted into waves. As this process continues, waves become higher and distance between its crests (wavelength) longer. Because waves are induced from storms far out in the ocean, storm waves are known as wind waves. Where wind waves arise (the generation area) they exhibit an irregular pattern. Once created, wind waves can travel long distances with a minimum of energy losses, far out from the storm areas, even after the wind turns or dies down. For instance, waves created on the American side of the Atlantic will travel to the western coast of Europe, supported by prevailing west winds, until breaking on some distant shore. During this lengthy voyage (thousands of kilometres) and if the fetch (or length of water over which wind has blown) is long enough, wind waves become progressively more regular and smooth, the socalled swell waves. Wavelength of swell waves is in the range of hundreds of metres and their effect extends to a greater water depth in comparison to wind waves. Another characteristic of swell waves is their relatively much larger presence in the world. During each stage of this conversion process, power is being progressively concentrated. The original solar power levels of usually 1 W/m can be converted into waves with power levels of typically 1 kw/m of wave crest length (wcl) to 5 kw/m wcl up to a power level of over 1 kw/m wcl. The amount of energy transferred from wind to water and thus the size of the resulting waves depends on the wind velocity, the duration of time that wind blows over the waves and the distance over which it interacts with the sea, also known as fetch. Although wave power produced from swell waves is steadier and predictable, it suffers great variations in several time scales starting from few seconds, with the passage of each wave (wave to wave), sea-state variations (hourly or daily basis) and ending in seasonal variations (month to month). Hence this natural seasonal variation is a vital factor to estimate the different locations of great wave power potential, being in general favourable in temperate zones. Usually the strongest wave energy potential coincides with

3 Table 1. Characteristics of several irregular waves according to ONDATLAS Fig.. AWS working principle winter months and greatest energy demand in the northern hemisphere. Some estimates show that, if properly harnessed, total worldwide wave power could contribute with nearly 1 % of the current world total electricity consumption. That would be around TWh/year, comparable to the actual production of worldwide large scale hydroelectric plants. Like most forms of other renewable energies available worldwide, the amount of energy contained in the waves is unequally distributed over the globe. Increased wave activity and therefore highest wave power concentration, with annual average levels between kw/m wcl and 7 kw/m wcl with peaks to 1 kw/m wcl, is found in the temperate zones between latitudes 3 o to 6 o on both hemispheres, induced by the occurrence of strong storms. Nevertheless, reasonable wave climates still exists within ±3 o of latitude due to regular trade winds (western winds) blowing in these zones. The lower wave power level is compensated by a less significant variability. 3. THE AWS The AWS is an off-shore, fully-submerged (43 m deep underwater), point absorber (that is to say, of neglectable size compared to the wavelength) WEC (Valério et al., 7b), consisting mainly in a bottom-fixed air-filled cylindrical chamber (the silo) and a movable upper cylinder (the floater) which heaves due to the changes in wave pressure (Fig. ): under a wave top the floater moves down compressing the air inside the AWS; under a wave trough pressure decreases and consequently the air expands and the floater moves up. An electric linear generator (ELG) converts the floater s heave motion into electricity. The AWS can hence be expected to behave much like a mass spring damper system, though with relevant non-linearities. In the dynamic behaviour of the AWS two motions can be modelled separately: the low frequency motion due to changes in tide, atmospheric pressure and/or temperature inside the AWS slow Jan Feb Mar Apr May Jun H s / m T e,min / s T e,max / s Jul Aug Sep Oct Nov Dec H s / m T e,min / s T e,max / s dynamics, and the high frequency motion due to sea waves fast dynamics. In this paper simulations are brief enough to allow neglecting the effects of slow dynamics. The control of the fast dynamics is provided to ensure that the amplitude of the floater heave motion is as large as possible to extract the maximum wave energy, but maintained within certain operational preset limits (±3.5 m). This is done varying the damping of the AWS, part of the damping being provided by the ELG and part by hydraulic damping devices, called water dampers. These are necessary when the damping force from the ELG is not enough. A model of the AWS in the time domain is based on Newton s law applied to the floater s vertical acceleration ξ. The equation of motion of the floater is: f tot = m ξ f pi f pe w f f n f v f m f wd f lg = = (m f + m wt ) ξ (1) The total mass comprises the mass of the floater m f and the mass of the water trapped inside the floater m wt. The total force acting on the floater is the sum of the following forces: the internal air pressure f pi, the external water pressure f pe, the weight of the floater w f, the force exerted by a nitrogen cylinder extant inside the AWS f n, the hydrodynamic viscous drag f v, the mechanical friction f m, the force exerted by the water dampers f wd, and the force exerted by the ELG f lg. Notice that the convention of signs assumes that positive values are given to the most natural direction hence among all forces the only one pointing upwards is f pi. Also notice that in this paper capital letters are used for variables in the frequency domain, while the corresponding lowercase letters are employed for the corresponding variables in the time domain. A detailed description and complete explicit expressions of all these terms cannot be given here for lack of space. They may be found for instance in (Pinto, 4; Sá da Costa et al., 3; Sá da Costa et al., 5). A non-linear simulator of the AWS, the AWS Time Domain Model (TDM), was already developed in Simulink implementing these expressions (see the references above).

4 4 3. m e F G* G d G' ξ / m m Fig. 4. Block diagram for Internal Model Control e C G d Fig. 3. Output of the AWS TDM for regular waves with 1 s of period and amplitudes of.5 m, 1. m, 1.5 m and. m Yet, before thinking about the extensive use of a non-linear model of the AWS (such as the AWS TDM) for control purposes, a linear model approximation of that same WEC should be identified in the first place (Beirão et al., 7b). This is possible because, even though the AWS is a nonlinear system, a sinusoidal input causes a fairly sinusoidal output, with an amplitude practically proportional to that of the input, for all wave periods and amplitudes expected to occur. Fig. 3 shows this for several regular waves of different amplitudes. So a second-order linear approximate model, relating the floater s vertical position Ξ to the wave excitation force F exc (this is the force that would act on the floater if it were fixed), has been identified from AWS TDM simulation data (Beirão et al., 7b): Ξ(s) F exc (s) = s s + 1 () This simplified linear model () will not be used in simulations throughout this paper, but will be used to design the controller. 4. ORIGINAL PID CONTROL The original version of the AWS TDM was implemented together with a simple controller that may easily be replaced by another one to test any desired control strategy. This controller provides a control force f u given by f u = ξ k ξ ξ stp (3) ( ) ξ π ξ stp = if ξ < 3.5 m 3.5 (4) if ξ 3.5 m In (3), k is the adjustable gain of a proportional controller. Constant 3.5 m shows up in (4) because Fig. 5. Block diagram equivalent to that of Fig. 4 it is the maximum possible amplitude for the floater s vertical oscillations, while constant 1 s shows up because it is a reasonable value for the expected period of an incoming wave (Pinto, 4). Actually the proportional controller is in reality a PID (proportional integrative derivative) controller, with the integral and derivative control actions set to zero. This is because no values of these control parameters were found leading to results better than those with proportional control only. 5. INTERNAL MODEL CONTROL The internal model control methodology (Hägglund and Åström, 1996) makes use of the control scheme of Fig. 4. In that control loop, G is the plant to control, G is an inverse of G (or at least a plant as close as possible to the inverse of G), G is a model of G and F is some judiciously chosen filter. If G were exact, the error e would be equal to disturbance d. If, additionally, G were the exact inverse of G and F were unity, control would be perfect. Since no models are perfect, the error will not be exactly the disturbance. That is also exactly why F exists and is usually a low-pass filter: to reduce the influence of highfrequency modelling errors. It also helps ensuring that product F G is realisable. The interconnections of Fig. 4 are equivalent to those of Fig. 5 if the controller C is given by F G C = 1 F G G (5) 6. IMPLEMENTATION It may be shown (Falnes, ) that, in order to maximise the power absorbed from the waves, the velocity of the floater should be in phase with the wave excitation force acting thereupon. (See the

5 Table. Power in kw obtained under several regular waves (figurative data) % increase from % increase from Wave period / s Wave amplitude / m No control PID no control IMC no control Appendix for an outline of the proof.) This, however, requires non-causal control actions. Hence all implementations of optimum control are in reality sub-optimum, since several approximations have to be indulged in so as to render control physically possible. Because of this, it seems reasonable to try to employ IMC to control the AWS in order to have the velocity of the floater in phase with the wave excitation force. For that purpose, a Simulink block was used, implementing IMC with the configuration of Fig. 4, to control the AWS. Within that block, G was given by () multiplied by s (this additional zero at the origin serving to have the floater s vertical velocity and not its position as the output): Ξ(s) F exc (s) = s.634s s + 1 (6) The inverse model was G = 1 G. Since G is not causal, filter F had to have more poles than zeros. It was found by trial and error that a second-order filter without zeros was the best option. The position of the poles was adjusted so as to maximise the absorbed wave energy for the simulation that uses an irregular wave with parameters corresponding to the month of March (deemed to be a significant month). The values found were F = 6 (s + 3)(s + ) (7) This is reasonable since it corresponds to a lowpass filter that preserves the frequencies where waves are expected to appear, while cutting off higher ones. Because of this, the product F G has an integral action. Since the signal it acts upon (labelled e in Fig. 4) has a residual non-null average, this lead to an ever-increasing (or ever-decreasing) control action, something that was not intended. To prevent this, the control action had to be corrected, by subtracting its average, computed from the beginning of the simulation and actualised on-line. The AWS was submerged 5 km offshore Leixões, Portugal. Data for wave climate in several locations in Portugal may be found in the ONDATLAS software (Pontes et al., 5). The nearest location available is the Leixões buoy location (41 o 1. N, 9 o 5.3 W). The corresponding data on significant wave height H s (from trough to crest) and on maximum and minimum values of the wave energy period T e is found in Table 1. Throughout this paper, simulations are performed using regular and irregular waves. The first are sinusoidal; values for the amplitude and the period of the sinusoids within the ranges found in Table 1 are used. For the latter, twelve waves (one for each month of the year) satisfying the Pierson- Moskowitz s spectrum, that accurately models the behaviour of real sea waves (Falnes, ), were used. This spectrum is given by S(ω) = A ( ω 5 exp B ) ω 4 (8) where S is the wave energy spectrum (a function such that + S(ω)dω is the mean-square value of the wave elevation). The numerical values A =.78 (SI) and B = 3.11/Hs were used. Values for H s and for T e (from which the limits of the range of frequencies ω were then found) were provided by Table 1. Simulations with a duration of 6 s (1 min) were carried out, employing several incident waves, as mentioned above. In all cases, the absorbed wave energy is given by 6 W u = f lg (t) ξ(t)dt (9)

6 absorbed energy / J absorbed energy / J absorbed energy / J 1 x Regular wave, amplitude.5 m, period 14 s IMC PID no control x Irregular wave for March IMC PID no control x Irregular wave for June IMC PID no control Fig. 6. Evolution of absorbed wave energy with time 7. RESULTS From these simulations, values for the absorbed power (time-averaged in the case of irregular waves) are given in Table and in Table 3. The corresponding evolution of wave energy absorption with time for one regular wave and two irregular waves (deemed significative of the rest) is given in Fig. 6. For comparison purposes, absorbed wave energy when the AWS has no control at all is also given 4. It is worthy of notice that wave energy absorption increases are important when PID control is used, but become much more significant when IMC is used. Since nowadays the major problem of WECs 4 Values given in this paper for absorbed power when the AWS has no control follow (Valério et al., 7a; Valério et al., 7c; Beirão et al., 7a) and are higher than those given in (Beirão et al., 6). This is because a residual force exerted by the ELG that had been neglected was now taken into account, which seems to be more correct. velocity / m s 1 ; force / MN velocity / m s 1 ; force / MN velocity / m s 1 ; force / MN Regular wave, amplitude.5 m, period 14 s Irregular wave for March excitation force 1.5 velocity (no control) velocity 5 (IMC) 55 6 velocity (PID) Irregular wave for June Fig. 7. Evolution of floater s vertical velocity and wave excitation force with time is their low efficiency, these are very satisfactory and promising results. This happens because IMC succeeds to a very great extent in putting the floater s vertical velocity in phase with the wave excitation force (something that does not happen when no control action is applied). This can be seen in Fig. 7, that also shows that PID control is as efficient as IMC in that endeavour, but with PID control oscillations are more limited, and hence less wave energy is absorbed. The major factor affecting the different performance along the year is the significant height H s of the waves. This is shown in Fig. 8, where the absorbed power with IMC is shown as a function thereof. It is possible to find a relation between the two variables; though a linear relation suffices, a clearly better fit is found with the following quadratic: power in kw = 1.8H s 15.9H s (1)

7 absorbed power / kw Jul Ago Jun May Sep Oct Apr Mar Nov Feb Dec Jan APPENDIX This Appendix closely follows (Falnes, ). As said above in section 3, the AWS is not unlike a mass-spring-damper system. This corresponds to a dynamic behaviour given by (Beirão et al., 7b) m ξ(t) + R ξ(t) + Sξ(t) = f exc (t) (11) H / m s Fig. 8. Absorbed power per month with IMC as function of H s 8. CONCLUSIONS From the last section it is seen that wave energy absorption increases when IMC is used (compared with the situation when there is no control applied) are very significant. IMC is also an improvement over the original PID control; corresponding wave energy absorption increases are often important, especially for the simulations of irregular waves, the ones deemed to be the most significant. This shows the importance of carefully chosen control strategies for the economic viability of WECs. Further refinements in control algorithms may be possible. Algorithms for estimating the incident wave and the consequent wave excitation force from data collected by buoys placed around the WEC or by some other means have to be studied and improved. But the main future task will certainly be the application of control strategies (these and others) to the second generation of AWS prototypes first in simulation, then in hardware, to help making them economically viable alternative sources of electrical power. Table 3. Power in kw obtained under several irregular waves (figurative data) Controller Jan Feb Mar Apr None PID % increase from no control IMC % increase from no control Controller May Jun Jul Aug None PID % increase from no control IMC % increase from no control Controller Sep Oct Nov Dec None PID % increase from no control IMC % increase from no control Parameters R (resistance) and S (stiffness) are both positive. This linear model can provide an approximate description of the non-linear AWS behaviour. Defining complex-valued phasors ˆf exc and ˆ ξ for fexc and ξ, respectively, f exc (t) = ˆf exc eiωt + ˆf exc e iωt (1) ˆ ξ ˆ ξ ξ(t) = eiωt + e iωt (13) equation (11) becomes ( e iωt ˆf exc R + iωm + S ) ˆ ξ iω ( + e iωt ˆf exc R iωm S ) ˆ ξ = (14) iω Defining an impedance Z = R + i ( ) ωm S ω, expression (14) can be rewritten as ( ) ( ) e iωt ˆfexc Zˆ ξ + e iωt ˆf exc Z ˆ ξ = (15) For (15) to be satisfied for all values of time t, condition ˆ ξ = ˆfexc Z ˆ ξ = ˆf exc Z must be verified. The impedance can be rewritten as Z = R + ix. The real part R = ReZ is called resistance and the imaginary part X = ImZ = ωm S ω is called reactance. Suppose now that a control force f u is applied to the AWS WEC. This will be needed to ensure that the conditions leading to maximum wave energy absorption (or at least conditions as close as possible to those) are met. Then Z ξ = f exc + f u Z(ω) Ξ(ω) = F exc (ω) + F u (ω) (16) The absorbed wave energy W u can be given by W u = + f u (t) ξ(t)dt (17) (Notice that no decomposition of f u is assumed, unlike what was done in (9). There, it is taken into account that the wave energy absorbed by the water dampers is wasted and not transformed

8 into electricity.) Considering that f u and ξ are real functions, i.e., Fu (ω) = F u ( ω) and Ξ (ω) = Ξ( ω), by applying Parseval s theorem, W u can be given by W u = 1 π + Knowing that W u is real, F u (ω) Ξ (ω) dω (18) F u (ω) Ξ (ω) = Re F u (ω) Ξ (ω) = = 1 F u (ω) Ξ (ω) + Fu (ω) Ξ(ω) (19) expression (18) can be rewritten as W u = 1 π + F u (ω) Ξ (ω) Fu (ω) Ξ(ω) dω() It will be convenient to add and subtract the term Fexc(ω)F exc (ω) R to the integrand of (), and finally W u is now given by (omitting the frequency argument) W u = 1 π + F exc R α dω (1) R In (1) α(ω) is the so-called optimum condition coefficient, given by α(ω) = F exc (ω)fexc(ω) + R F u (ω) Ξ (ω) + Fu (ω) Ξ(ω) () After some tedious manipulations, it turns out that α = F exc (ω) R Ξ(ω). So, it is when α(ω) = that W u is maximal, and an optimum condition can be written as F exc (ω) = R Ξ(ω). This means that Ξ must be in phase with F exc. REFERENCES Beirão, P., D. Valério and J. Sá da Costa (6). Phase control by latching applied to the Archimedes Wave Swing. In: Proceedings of the 7th Portuguese Conference on Automatic Control. Lisbon. Beirão, P., D. Valério and J. Sá da Costa (7a). Comparison of control strategies applied to the Archimedes Wave Swing. In: European Control Conference. Kos. Beirão, P., D. Valério and J. Sá da Costa (7b). Linear model identification of the Archimedes Wave Swing. In: IEEE International Conference on Power Engineering, Energy and Electrical Drives. Setúbal. Boud, R. (). Status and research and development priorities 3: Wave and marine current energy. Technical Report DTI No. FES- R-13, AEAT No. AEAT/ENV/154. UK Department of Trade and Industry. Clément, A., P. McCullen, A. Falcão, A. Fiorentino, F. Gardner, K. Hammarlund, G. Lemonis et al. (). Wave energy in Europe: current status and perspectives. pp Vol. 6 of Renewable and Sustainable Energy Reviews. Elsevier Science. Falcão, A. (6). The history of and progress in wave energy conversion devices. In: 9th World Renewable Energy Conference. Firenze. Falnes, J. (). Ocean waves and oscillating systems. Cambridge University Press. Cambridge. Hägglund, T. and K. Åström (1996). Automatic tuning of PID controllers. In: The control handbook (W. S. Levine, Ed.). pp CRC Press. Boca Raton. Pinto, P. (4). Time domain simulation of the AWS. Master s thesis. Technical University of Lisbon, IST. Lisbon. Pontes, M. and A. Falcão (1). Ocean energies: Resources and utilisation. In: 18th World Energy Conference. Buenos Aires. Pontes, M. T., R. Aguiar and H. Oliveira Pires (5). A nearshore wave energy atlas for Portugal. Journal of Offshore Mechanics and Arctic Engineering 17, Sá da Costa, J., A. Sarmento, F. Gardner, P. Beirão and A. Brito-Melo (5). Time domain model of the Archimedes Wave Swing wave energy converter. In: Proceedings of the 6th European Wave and Tidal Energy Conference. Glasgow. pp Sá da Costa, J., P. Pinto, A. Sarmento and F. Gardner (3). Modelling and simulation of AWS: a wave energy extractor. In: Proceedings of the 4th IMACS Symposium on Mathematical Modelling. Agersin-Verlag. Vienna. pp Valério, D., P. Beirão and J. Sá da Costa (7a). Feedback linearisation control applied to the Archimedes Wave Swing. In: 15th IEEE Mediterranean Conference on Control and Automation. Athens. Valério, D., P. Beirão and J. Sá da Costa (7b). Optimisation of wave energy extraction with the archimedes wave swing. Ocean Engineering. Submitted. Valério, D., P. Beirão and J. Sá da Costa (7c). Reactive control and phase and amplitude control applied to the Archimedes Wave Swing. In: 17th International Offshore (Ocean) and Polar Engineering Conference & Exhibition. Lisbon.