# Real-Time Wave Energy Converter Control Using Instantaneous Frequency
## 4 WEC Control Using Estimated Instantaneous Frequency
### WEC Modeling  
The response of a WEC is governed by its dynamic equations, which account for multiple interacting forces. Assuming the linear superposition of forces, the governing equation of motion is expressed as follows:  
\[m\mathbf{a}=\mathbf{F}_{\text{ex}}+\mathbf{F}_{\text{rad}}+\mathbf{F}_{\
u}+ \mathbf{F}_{\text{b}}+\mathbf{F}_{\text{pto}}, \tag{18}\]  
where \(\mathbf{F}_{\text{ex}}\) is the excitation force due to incident waves; \(\mathbf{F}_{\text{rad}}\) is the radiation force from the body's motion-induced waves; \(\mathbf{F}_{\
u}\) is the viscous damping force; \(\mathbf{F}_{\text{b}}\) is the buoyancy force; and \(\mathbf{F}_{\text{pto}}\) is the Power Take-Off (PTO) force.  
The hydrodynamic coefficients are used to estimate \(\mathbf{F}_{\text{ex}},\mathbf{F}_{\text{rad}}\), and \(\mathbf{F}_{\text{b}}\), which are derived using potential flow theory through BEM solvers such as WAMIT [12]. These coefficients are frequency-dependent and also influenced by the angle between the WEC body and incoming waves. However, for symmetrical geometries (e.g., cylindrical WECs), wave directionality effects can often be neglected.  
A significant challenge in modeling WEC responses arises from the time-varying nature of real sea conditions. While regular waves of a single frequency can be readily simulated, real ocean waves exhibit continuously varying frequencies. To address this, [PERSON] equation [1] uses an impulse response function to model radiational hydrodynamic coefficients through convolution. Although this method provides an accurate estimation, it imposes a high computational cost due to the need for continuous integration.  
An alternative approach, explored in [11], incorporates **instantaneous frequency estimation** to determine hydrodynamic coefficients in real sea conditions. This method effectively transforms the complex time-varying frequency problem into a regular wave problem  
Figure 9: Diagram illustrating the real-time PTO control strategy. The instantaneous frequency estimation feeds directly into the impedance matching calculations, dynamically adjusting PTO damping and stiffness to optimize energy extraction.  
by assigning hydrodynamic coefficients based on the momentary frequency at each time step. However, a limitation noted in [11] is that wave signals must be known in advance to compute the true instantaneous frequency, which is not feasible in real-time applications.  
This paper extends the previous sections by introducing the real-time estimation of instantaneous frequency using different methods. Based on these estimations, the PTO damping coefficient, or an additional stiffness coefficient for complex conjugate control [14], is dynamically adjusted in response to varying wave frequency. This adaptive approach enhances energy absorption efficiency.  
The next section presents the methodology and results, comparing the following:  
* **Fixed PTO damping:** Tuned to the dominant wave period for baseline comparison.
* **Variable PTO damping:** Adaptively adjusted in real-time based on the estimated instantaneous frequency.