Nonlinear Energy-Maximizing Optimal Control of Wave Energy Systems: A Moment-Based Approach

Abstract

Linear dynamics are virtually always assumed when designing optimal controllers for wave energy converters (WECs), motivated by both their simplicity and computational convenience. Nevertheless, unlike traditional tracking control applications, the assumptions under which the linearisation of WEC models is performed are challenged by the energy-maximising controller itself, which intrinsically enhances device motion to maximise power extraction from incoming ocean waves. In this paper, we present a moment-based energy-maximising control strategy for WECs subject to nonlinear dynamics. We develop a framework under which the objective function (and system variables) can be mapped to a finite-dimensional tractable nonlinear program, which can be efficiently solved using state-of-the-art nonlinear programming solvers. Moreover, we show that the objective function belongs to a class of generalised convex functions when mapped to the moment-domain, guaranteeing the existence of a global energy-maximising solution, and giving explicit conditions for when a local solution is, effectively, a global maximiser. The performance of the strategy is demonstrated through a case study, where we consider (state and input-constrained) energymaximisation for a state-of-the-art CorPower-like WEC, subject to different hydrodynamic nonlinearities.

On Moment Matching for Stochastic Systems

“…Example 1: Consider system (6) and the generator (7) and assume χ t = X t ω t . In this example we show that equation ( 16) implies equation (12). First of all, equation ( 17) becomes…”

“…By factoring out ω t and noticing that the previous equation implies ∂ X t ∂ Wt = F X t − X t J + GL, yields (12). Note that there are simpler ways to achieve the same result, as pointed out in the next remark.…”

“…Note that d X t dω t = (F X t − X t J + GL) Jω t dt by ( 52), (53) and (12). Substituting the expressions of dx t , namely (6), of d X t , namely (12), and of dω t , namely (7), in (15) yields…”

“…Substituting these two expressions in (14) proves that X t , defined in (11), is the solution of the stochastic differential matrix equation (12). Now define the variable z t := x t −X t ω t .…”

“…Most of this field originally developed from the need of reducing the order of high-order linear ordinary differential equations obtained from the discretisation of partial differential equations for simulation purposes, see [8]. From there, the field expanded to much more general classes of systems, such as time-delay and nonlinear systems [9], and purposes, such as analysis and control, see for instance [10]- [12]. Many stochastic generalizations of deterministic methods have been proposed.…”