Discussion on the mode mixing in wave energy control systems using the Hilbert-Huang transform Paula B. Garcia-Rosa, Marta Molinas, and Olav B. Fosso Abstract—A great improvement in the absorption of energy of a wave energy converter (WEC) is obtained with a time-varying power take-off (PTO) damping over a constant damping. In a passive control scheme based on the Hilbert-Huang transform (HHT), the PTO damping is time-varying and tuned to the instantaneous frequency of the wave excitation force. The HHT method relies on the use of the empirical mode decomposition (EMD) method to decompose the wave signal into a number of compo- nents (IMFs) from the highest to the lowest frequency component. However, the decomposition process is not always perfect and may result in mode mixing, where an IMF will consist of signals of widely disparate frequency scales, or different IMFs will consist of signals with similar frequency scales. Mode mixing can be caused by intermittent/noisy signals, and by specific amplitude and frequency relations of the original modes in the signal. The aim of this paper is to extend the studies on the use of the HHT for WEC tuning purposes by revealing how the EMD mode mixing problem affects the WEC performance. A comprehensive study using firstly synthetic two-tone waves (i.e., superposition of two sinusoidal waves) is performed. Then, the observations from the two-tone studies are used to further improve the energy absorbed by WECs using the HHT in real ocean wave scenarios resembling the analytic scenario. Index Terms—wave energy, Hilbert-Huang transform, mode mixing. I. I NTRODUCTION R ECENT studies have shown that tuning the power take-off (PTO) damping of a wave energy con- verter (WEC) to time-frequency estimations obtained from the Hilbert-Huang transform (HHT) results in greater energy absorption than tuning the PTO to a constant frequency of the wave spectrum [1], or to time-frequency estimations from the extended Kalman filter (EKF), and frequency-locked loop (FLL) method [2]. Both the EKF and FLL methods provide single dominant frequency estimates, whereas the HHT pro- vides the instantaneous wave-to-wave frequency of the oscillation modes present in a wave profile. In addition, by adopting other methods to, e.g., estimate the on- line dominant wave frequency [3], or determine the optimal time-varying PTO damping [4], other studies have also shown that continuously tuning the PTO ID 1275 track GPC. P. B. Garcia-Rosa and O. B. Fosso are with the Department of Electric Power Engineering, Norwegian University of Science Technology, Trondheim, Norway (e-mails: p.b.garcia-rosa@ieee.org, olav.fosso@ntnu.no). M. Molinas is with the Department of Engineering Cybernetics, Norwegian University of Science Technology, Trondheim, Norway (e-mail: marta.molinas@ntnu.no). result in greater energy absorption than tuning it to a constant frequency of the wave spectrum. The HHT method [5] relies on the use of the em- pirical mode decomposition (EMD) to decompose the wave signal into a number of components, named intrinsic mode functions (IMFs), from the highest to the lowest frequency component. However, the decom- position process may result in mode mixing, and an IMF will consist of signals of widely disparate fre- quency scales, or different IMFs will consist of signals with similar frequency scales [6]. Mode mixing can be caused by an intermittent/noisy signal, by spe- cific amplitude and frequency relations of the original modes in the signal, and by a combination of both cases. Additionally, the mode mixing effects can be attenuated by applying, e.g., masking signals prior to the EMD procedure [7] [8], or by using the ensemble EMD (EEMD), a white noise-assisted EMD method [9]. Focusing on the case when mode mixing is caused by specific amplitude and frequency relations of the original modes in a signal, [10] presents a rigorous mathematical analysis that shows how the EMD sepa- rates the original modes in signals with two frequency components. Three different domains are identified by the authors depending on the frequency and amplitude ratios of the modes. After the EMD procedure, the components can be separated and correctly identified (domain 1), considered as a single wave-form (domain 2), or the EMD does something else (domain 3) [10]. In this paper, the focus is also on mode mixing caused by specific amplitude/frequency relations of original modes in a wave signal. In this framework, the aim is to extend the studies on the use of the HHT for WEC tuning purposes with a passive control (PC) strategy, by revealing how the EMD mode mixing problem affects the WEC performance. A comprehen- sive study using initially synthetic two-tone waves (superposition of two sinusoidal waves) is performed. Then, the mode mixing conditions observed for the two-tone wave studies are used to further improve the energy absorbed by WECs using the HHT in real ocean wave scenarios resembling the synthetic two- tone wave scenario. II. PASSIVE CONTROL USING THE HHT Here, we assume linear hydrodynamic theory, and consider a single oscillating-body represented as a truncated vertical cylinder constrained to move in heave. By neglecting friction and viscous forces, the