Model Predictive and Integral Error Tracking Control of a Vertical Axis Pendulum Wave Energy Converter

By actively controlling the rotation of its pendulum, a vertical axis pendulum wave energy converter can generate significantly more net electricity from ocean waves than what would otherwise be possible through a passively swinging pendulum. This suggests that such converters can optimize their performance by incorporating active control schemes within their ever– changing ocean wave environment. The challenge of implementing such control, however, necessitates that an active controller be developed. Here, we address such challenges by: (i) deriving the equations of motion for a constrained generic vertical axis pendulum wave energy converter; (ii) modeling an irregular ocean wave environment; (iii) developing a model predictive and integral error tracking controller to enforce a desired control strategy; and (iv) simulating the converter within the modeled wave environment both with and without active control for the purpose of comparing their respective net power generation results. Here, we find that by actively controlling the rotation of the converter’s pendulum, both continuous–mean and peak power generation are increased significantly. Simulation results show that active control is capable of increasing continuous–mean net power generation by over 200 percent. These results give strong evidence and support for further vertical axis pendulum wave energy converter development and also for the continued investigation of applying such active controllers to real world scenarios and prototypes. ∗Corresponding Author: [email protected] NOMENCLATURE (X0,Y0,Z0) global coordinate system, origin at still water level (X1,Y1,Z1) . . . . . . . . . . . . . . . . . . . . . hull fixed coordinate system (X2,Y2,Z2) . . . . . . . . . . . . . . . . pendulum fixed coordinate system ~ H . . . . . . . . . . . . . . . . . . . . position vector of hull’s center of mass ~P . . . . . . . . . . . . . . . position vector of pendulum’s center of mass ~PH . . . . position vector of pendulum’s center of mass w.r.t. hull ω x1 . . . . . . . . . . . . . . . . . . . angular velocity of hull about X1 axis ω y1 . . . . . . . . . . . . . . . . . . . . angular velocity of hull about Y1 axis φ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . first Euler angle θ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . second Euler angle ψ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . third Euler angle α . . . .angular position of pendulum about Z1 axis w.r.t. X1 axis mh . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . mass of hull Ix1 . . . . . . . . . . . . . hull principal moment of inertia about X1 axis Iy1 . . . . . . . . . . . . . hull principal moment of inertia about Y1 axis mp . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . mass of pendulum L . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . length of pendulum arm g . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . acceleration due to gravity ~FE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . wave excitation force ~ ME . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .wave excitation moment ~FR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . hull radiation force ~ MR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . hull radiation moment