On the control design of wave energy converters with wave prediction
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Abstract
This paper presents a shapebased approach to compute, in an optimal sense, the control of a single degreeoffreedom point absorber wave energy converter. In this study, it is assumed that a prediction for the wave is available. The control is computed so as to maximize the energy extraction over a future time horizon. In the shapebase approach, one of the system states is represented by a series expansion. The optimization variables are selected to be the coefficients in the series expansion instead of the history of the control variable. A gradientbased optimizer is used to optimize the series coefficients. The available wave prediction is used to compute an initial guess for the unknown series coefficients. This concept is tested on two different dynamic models: a simplified reducedorder model and a dynamic model with radiation dynamic states necessary to compute the radiation force. Several test cases are presented that cover a range of different sea states. The results show that the shapebased approach finds efficient solutions in terms of the extracted energy. The results also show that the obtained solutions are suitable for realtime implementation in terms of the smoothness of the obtained control and the speed of computations. Comparisons between the results of the shapebased approach and other techniques are presented and discussed.
Keywords
Wave energy Wave energy converter Wave energy converter optimal control Shapebased control1 Introduction
Reference Ringwood et al. (2014) presents a comprehensive study for the methods of control for wave energy conversion; a brief is here presented. Using a linear dynamic model, the frequency domain analysis leads to the wellknown complexconjugate conditions that provide a means to compute the optimal float velocity that guarantees maximum energy extraction, regardless of the spectral distribution of the excitation force (Bacelli 2014; Falnes 2002). This complexconjugate control, however, requires having the complex bidirectional energy flow mechanisms in the system. There are also practical constraints on the amplitude of maximum heave excursion and float velocity that limit the maximum extracted energy to bounds lower than the ideal complexconjugate bound (Evans 1981; Falnes 2007a). Moreover, the complexconjugate solution is not causal, which means a prediction for the wave elevation or the excitation force is needed for realtime implementation. Several control concepts are suggested in the literature. A feedforward control can be implemented assuming the availability of the excitation force (wave) model to compute the control force (Naito and Nakamura 1986). Also a feedback approach can be implemented through computing the control force using both the measurements and the wave prediction data (Korde et al. 2001; Korde 1999). Another way to implement the complexconjugate control is to use a velocitytracking control where the estimates of the excitation force are used to compute the optimal float velocity (through the feedforward loop) which is imposed on the WEC through a feedback loop (Maisondieu and Clement 1993).
This paper presents a shapebased (SB) approach to solve the WEC optimal control problem. The SB approach has its roots in the space trajectory optimal control problem. When sending a spacecraft on an interplanetary mission, the problem of controlling the spacecraft thrust magnitude and direction, in an optimal sense, is a complex optimization problem. One category of methods to solve this problem in a computationally efficient way is the SB methods, in which the shape of the space trajectory is assumed to be known. References Petropoulos and Longuski (2004), Petropoulos (2001), Paulino (2008), for instance, assume that the trajectory of the spacecraft takes a spiral shape and use that to limit the search space to spiral trajectories. In a broader view, reference De Pascale and Vasile (2006) presents an SB method that approximates the states using two shaping functions suitable for solar and nuclear electric propulsion systems. References Wall and Conway (2009) and Wall (2008) developed a sevenparameter inverse polynomial shape for lowthrust rendezvous trajectories. Recently, reference Taheri and Abdelkhalik (2012) presented an SB method that does not assume a specific shape for the states; rather it assumes a series representation of some/all of the states and use it to satisfy any problem specific constraint and boundary conditions. This SB approach is suitable for the WEC optimal control problem as detailed in Sect. 4. The SB approach assumes a shape for the float velocity history over a finite future horizon using a finite Fourier series. The initial guess for coefficients in the Fourier series is obtained using the available estimates about the wave in the future horizon. These Fourier coefficients are then tuned during the search for the optimal solution. The main difference between this SB approach and other pseudospectral methods is the significant savings in terms of computational cost which makes the SB approach suitable for realtime implementation. The savings in computational cost is due to the fact that pseudospectral methods approximate all the states and the control as functions of basis functions, while the SB method approximates only one state in terms of a finite Fourier series.
Several references present the dynamic models for different types of wave energy converters in detail (Fusco 2012). The proposed SB approach is tested on two different models, both are single degreeoffreedom; yet one of them is a simplified reducedorder model that does not have dynamic states representing the radiation force. This paper is organized as follows. Sections 2 and 3 describe the two dynamic models used in this paper. Section 4 describes the SB approach and its implementation in the WEC optimal control problem. The results of SB approach are presented in Sect. 6, along with comparisons between the SB approach and each of the dynamic programming, pseudospectral, resistive loading, and model predictive control approaches.
2 Performance model
The order of system, n, is usually within \(3 \le n \le 8\). The matrices \(\mathbf {A_r}\), \(\mathbf {B_r}\) and \(\mathbf {C_r}\) must be chosen to produce a vector \(\overline{\mu }\) to mimic \(\mu \).
3 Simplified dynamic model
4 Shapebased approach for simplifiedmodel WEC optimal control
 1.
\(\tau _u \le \tau _{\mathrm{umax}}\),
 2.
\(D \le D_{\mathrm{max}}\),
 3.
the equations of motion defined in Eq. (7).
5 Shapebased approach for higherorder model WEC optimal control

\(N_H\) an integer that represents the horizon length in units of wave period,

\(N_{\mathrm{cw}}\) an integer that determines the number of control updates in one wave period,

\(N_{\mathrm{FFT}}\) the number of fourier terms.
6 Numerical results
Two categories of results are presented in this paper. First, Sect. 6.1 presents the SB results when the simplified dynamic model described in Sect. 2 is used. The results obtained using SB are compared to those obtained using a Pseudospectral optimal control approach and a dynamic programming control optimization approach. Section 6.2 presents the SB results obtained when using the performance model described in Sect. 3. Comparison between the SB, a model predictive control approach, and a resistive loading approach are presented.
It is assumed in this study that a wave prediction over a future time horizon at all times is available. The length of the foreknowledge horizon ranges from 1 to 3 wave peak periods. For all the simulations presented in this paper, an Intel Sandy Bridge processor is used at 2.60 GHz.
6.1 Numerical results using simplified dynamic model
The simplified dynamic model presented in Sect. 2 is used. In this study, the simulation time is selected to be 45 seconds. Reference Li et al. (2012) presents a dynamic programming approach to solve the WEC optimal control problem. For comparison purposes, the same case study presented in reference Li et al. (2012) is investigated in this paper. In this case study, the float has a diameter of 9 m, a mass of \(10^{4}\) kg, and the added mass is \(7 \times 10^{4}\) kg. The height of the float is 2.4 m. The damping coefficient is \(D = 2 \times 10^4 ~\mathrm{N\,s}/\mathrm{m}\). The stiffness is \(K = 6.39 \times 10^{5}\) N/m. The friction damping ratio is \(D_f = 2 \times 10^{4}\) N s/m. The nonlinearity coefficient is \(k = 4\). The maximum control input is \(\tau _{\mathrm{umax}} = 3 \times 10^{5}~N\). Let \(\phi \) be the difference between the sea level and the float hight. The constraint on the float hight is \(\phi \le 1.2\) m. The seawater density is \(1025~\mathrm{kg}/\mathrm{m}^{3}\). The gravity constant is 9.8 N/kg.
6.1.1 Comparisons and discussions
This section presents the comparisons between the results of implementing the SB approach and the results of implementing each of the dynamic programming (DP) and the pseudospectral (PS) approaches, in controlling a point absorber WEC. Comparison is made based on the extracted energy, the smoothness of the obtained control, and the computational time.
Reference Li et al. (2012) presents a thorough explanation for the DP approach and its implementation to the WEC control problem. The DP approach in Li et al. (2012) is reproduced and the results are compared to the SB results. A fundamental step in implementing the DP approach is the discretization, in which the ranges of each of the states and the time are discretized into nodes. This way, the infinite continuous domain is converted into a finite discrete domain. The number of nodes impacts the obtained numerical results. As the number of nodes increases, the obtained solution is more accurate. However, the computational cost is higher. It is possible to show that in the limit, the DP approach finds the optimal solution. The computational cost limitation, however, is significant since, in a realtime implementation, all the computations should be carried out in less than the control time step which is \(1/u_{rate}\) s. If a coarse space grid is used, the DP algorithm might generate suboptimal solutions, and hence the extracted energy could be less than that of the optimal solution of the problem.
To compare the DP to the SB approach, it is important to note that the SB approach is suboptimal, since it is based on representing one of the states by a finite number of Fourier series. DP on the other hand searches for the optimal solution. The implementation, however, may alter this advantageous situation of the DP. As discussed above, the computational time of the DP increases as we increase the number of nodes. Also, increasing the number of states in higherorder models significantly increases the computational costs. In the SB approach, on the other hand, the dynamics of the problem can be captured reasonably with as few as \(11 \times 2 + 1 = 23\) Fourier terms.
6.2 Numerical results using the performance model
Tested sea states characteristics and simulation times
ID  Tp (s)  Hs (m)  Duration (s) 

RS06  2.5  0.194  180 
RS07  3  0.278  180 
RS08  3.5  0.37  180 
RS09  4  0.464  180 
RS10  4.5  0.556  300 
RS11  5  0.646  300 
RS12  6  0.8222  300 
RS13  7  0.992  360 
RS14  8  1.158  360 
Selected SB parameters for some of the tested sea states
\(N_H\)  \(N_{\mathrm{cw}}\)  \(N_{\mathrm{FFT}}\)  Ctrl Integ  

RS06  4  10  7  6 
RS07  5  15  6  3 
RS08  5  15  7  2 
RS09  3  15  7  2 
RS10  5  30  8  10 
RS11  5  30  7  10 
RS12  7  40  7  64 
RS13  7  80  7  49 
Figure 15 shows the aggregated results for all the tested sea states where the average power is shown for each sea state. The average power is computed over the simulation duration of each sea state. A comparison is made of the resulting average power from the SB control versus a resistive loading control and a model predictive control (Cretel et al. 2011). Plotted also in Fig. 15 is the trend line of the SB solutions. As can be seen from Fig. 15, the SB solution is close to that obtained by the model predictive control and is much better than the resistive loading control, in terms of the extracted energy.
7 Conclusion
This paper presents a shapebased approach that searches for the optimal control of a one degreeoffreedom wave energy point absorber. Two linear dynamic models are used representing two levels of complexity. Wave prediction is assumed to be available. The shapebased approach assumes a series expansion for one of the states and conducts a search for the optimal shape of the state so that the energy extracted is optimized. A key point in this approach is the use of the wave prediction data to initialize the search for the optimal shape of the state. As a result, the speed of convergence is fast, enabling a realtime implementation for the shapebased approach. Another feature about the shapebased approach is the use of all the available wave data to initialize the optimization of the states (compared to the dynamic programming that uses only a shorter time horizon of data for computational cost reasons). This feature results in a smoother control; i.e., a lower switching frequency between the two control extremes. The obtained solution using shapebased approach is very comparable to that obtained using the pseudspectral approach, dynamic programming, or model predictive control. The shapebased approach is faster than the pseudospectral and dynamic programming methods when higherorder models are used. The shapebased method can be applied to a nonlinear dynamic model witch is the future plan of work.
Notes
Acknowledgments
This study was supported by Sandia National Laboratories. Sandia National Laboratories is a multiprogram laboratory operated by Sandia Corporation, a Lockheed Martin Company, for the U.S. Department of Energys National Nuclear Security Administration under contract DEAC0494AL85000. The resilience nonlinear control project at Sandia National Labs is supported by The Department of Energy (DOE), Office of Energy Efficiency and Renewable Energy (EERE), Wind and Water Power Technologies Office (WWPTO). Superior, a highperformance computing cluster at Michigan Technological University, was used in obtaining some of the results in this presentation.
References
 Bacelli G (2014) Optimal control of wave energy converters. National University of Ireland, Maynooth, Maynooth, Ireland, PhDGoogle Scholar
 Bacelli G, Ringwood J (2015) Numerical optimal control of wave energy converters. IEEE Trans Sustain Energy 6(2):294–302. doi: 10.1109/TSTE.2014.2371536 CrossRefGoogle Scholar
 Bacelli G, Ringwood J, Gilloteaux JC (2011) A control system for a selfreacting point absorber wave energy converter subject to constraints. In: Proceedings of 18th IFAC world congress, Milan, Italy, http://eprints.nuim.ie/3555/
 Coe RG, Bull DL (2014) Nonlinear timedomain performance model for a wave energy converter in three dimensions. In: OCEANS2014, IEEE, St. John’s, CanadaGoogle Scholar
 Coe RG, Bull DL (2015) Sensitivity of a wave energy converter dynamics model to nonlinear hydrostatic models. In: Proceedings of the ASME 2015 34th international conference on ocean, offshore and arctic engineering (OMAE2015), ASME, St. John’s, NewfoundlandGoogle Scholar
 Cretel JAM, Lightbody G, Thomas GP, Lewis AW (2011) Maximisation of energy capture by a waveenergy point absorber using model predictive control. In: IFAC World Congress, August 28September 2, Milano, ItalyGoogle Scholar
 Cummins WE (1962) The impulse response function and ship motions. Tech. Rep. DTNSDRC 1661, Department of the Navy, David Taylor Model Basin, Bethesda, MD, http://dome.mit.edu/handle/1721.3/49049
 De Pascale P, Vasile M (2006) Preliminary design of lowthrust multiple gravity assist trajectories. J Spacecr Rockets 43:1065–1076. doi: 10.2514/1.19646 CrossRefGoogle Scholar
 Evans D (1981) Maximum wavepower absorption under motion constraints. Appl Ocean Res 3(4):200–203CrossRefGoogle Scholar
 Falcao A (2010) Wave energy utilization: a review of the technologies. Renew Sustain Energy Rev 14(3):899–918, doi: 10.1016/j.rser.2009.11.003, http://www.sciencedirect.com/science/article/pii/S1364032109002652
 Falnes J (2002) Ocean waves and oscillating systems—linear interactions including waveenergy extraction. Cambridge University Press, CambridgeCrossRefGoogle Scholar
 Falnes J (2007) A review of waveenergy extraction. Mar Struct 20(4):185–201. doi: 10.1016/j.marstruc.2007.09.001 CrossRefGoogle Scholar
 Falnes J (2007b) A review of waveenergy extraction. Mar Struct 20(4):185–201, doi: 10.1016/j.marstruc.2007.09.001, http://www.sciencedirect.com/science/article/pii/S0951833907000482
 Forsgren A, Gill PE, Wright MH (2002) Interior methods for nonlinear optimization. SIAM Rev 44(4):525–597. doi: 10.1137/S0036144502414942 MathSciNetCrossRefzbMATHGoogle Scholar
 Fusco F (2012) Realtime forecasting and control for oscillating wave energy devices. PhD thesis, NUI MAYNOOTH, Faculty of Science and Engineering, Electronic Engineering DepartmentGoogle Scholar
 Hals J, Falnes J, Moan T (2011) Constrained optimal control of a heaving buoy waveenergy converter. J Offshore Mech Arct Eng 133(1):1–15. doi: 10.1115/1.4001431 CrossRefGoogle Scholar
 Korde U (1999) Efficient primary energy conversion in irregular waves. Ocean Eng 26(7):625–651, doi: 10.1016/S00298018(98)000171, http://www.sciencedirect.com/science/article/pii/S0029801898000171
 Korde UA, Schoen MP, Lin F (2001) Time domain control of a single mode wave energy device. In: Proceedings of the Eleventh international offshore and polar engineering conference. Stavanger, Norway, pp 555–560Google Scholar
 Li G (2015) Nonlinear model predictive control of a wave energy converter based on differential flatness parameterisation. Int J Control 89:1–10: doi: 10.1080/00207179.2015.1088173
 Li G, Belmont MR (2014) Model predictive control of sea wave energy converters part I: a convex approach for the case of a single device. Renew Energy 69(0):453–463, doi: 10.1016/j.renene.214.03.070, http://www.sciencedirect.com/science/article/pii/S0960148114002456
 Li G, Weiss G, Mueller M, Townley S, Belmont MR (2012) Wave energy converter control by wave prediction and dynamic programming. Renew Energy 48(0):392–403, doi: 10.1016/j.renene.2012.05.003, http://www.sciencedirect.com/science/article/pii/S0960148112003163
 Maisondieu C, Clement A (1993) A realizable force feedbackfeedforward control loop for a piston wave absorber. 8th International Workshop on Water Waves and Floating Bodies, St John’s, Newfoundland, Canada, pp 79–82Google Scholar
 Naito S, Nakamura S (1986) Wave energy absorption in irregular waves by feedforward control system. In: Evans D, de Falco A (eds) Hydrodynamics of ocean waveenergy utilization, international union of theoretical and applied mechanics. Springer, Berlin, pp 269–280CrossRefGoogle Scholar
 Paulino T (2008) Analytical representation of lowthrust trajectories. Master’s thesis, Delft University of TechnologyGoogle Scholar
 Petropoulos AE (2001) A shapebased approach to automated, lowthrust, gravityassist trajectory design. PhD thesis, Purdue Univ., Purdue, INGoogle Scholar
 Petropoulos AE, Longuski JM (2004) Shapebased algorithm for automated design of lowthrust, gravityassist trajectories. J Spacecr Rockets 41:787–796CrossRefGoogle Scholar
 Rao AV, Benson DA, Darby C, Patterson MA, Francolin C, Sanders I, Huntington GT (2010) Algorithm 902: GPOPS, a matlab software for solving multiplephase optimal control problems using the gauss pseudospectral method. ACM Trans Math Softw 37(2):22:1–22:39Google Scholar
 Ringwood J, Bacelli G, Fusco F (2014) Energymaximizing control of waveenergy converters: the development of control system technology to optimize their operation. Control Syst IEEE 34(5):30–55. doi: 10.1109/MCS.2014.2333253 CrossRefGoogle Scholar
 Taheri E, Abdelkhalik O (2012) Shape based approximation of constrained lowthrust space trajectories using Fourier series. J Spacecr Rockets 49(3):535–545Google Scholar
 Wall B, Conway BA (2009) Shapebased approach to lowthrust rendezvous trajectory design. J Guid Control Dyn 32:95–101. doi: 10.2514/1.36848 CrossRefGoogle Scholar
 Wall BJ (2008) Shapebased approximation method for lowthrust trajectory optimization. In: AIAA/AAS astrodynamics specialist conference and exhibit, Honolulu, HI, AIAA Paper 2008–6616Google Scholar