Implementation

As described in the Theory & Practice page, WecOptTool operates by solving a constrained optimization problem, in which an equality constraint ($r(x)=0$) is used to enforce the dynamics of the system.

(1)$$\begin{array}{r}\underset{x}{min}J(x)\\ \text{s.t.}& \\ r(x)& =0\\ c_{ineq}(x)& \le 0\\ c_{eq}(x)& =0\end{array}$$

For the linear case, which is the default formulation in WecOptTool, the WEC dynamics can be expressed, in residual form, as

(2)$$r=-(M+m_a)\ddot{x}-kx(t)-B\dot{x}(t)-C\underset{-\mathrm{\infty }}{\overset{t}{\int }}\dot{x}(t)+f_{ext}+f_{exc}=0.$$

The decision variable, $x$, includes states of the body motion as well as other external states, e.g., related to controllers, PTO systems, mooring systems, etc.

(3)$$x=[x_{pos},x_{ext}{]}^T$$

Here, $x_{pos}\in {\mathbb{R}}^{m(2n+1)}$, where $m$ is the number of body modes of motion and $n$ is the number of frequency components. Similarly, $x_{ext}\in {\mathbb{R}}^{p(2n+1)}$, where $p$ is the number of external states. Body modes could include multiple degrees-of-freedom for a single body (e.g., heave, surge, and pitch) and/or degrees-of-freedom from multiple bodies (e.g., heave modes from each body in a large array of bodies). The frequency vector $\omega \in {\mathbb{R}}^n$ should span the relevant frequencies of the device operation, and also super harmonics to capture nonlinearities.

(4)$$x_{pos}=[x_{pos,1},x_{pos,2},\dots ,x_{pos,m}{]}^T$$

For each mode, $x_{pos}$ has $2n+1$ elements ($x_{pos,k}\in {\mathbb{R}}^{2n+1}$). There are $2n$ elements because each frequency is described by two components: one each corresponding to the cosine and sine amplitudes. The additional element in the first position is used to capture a mean (“DC”) displacement. Thus, the portion of $x_{pos}$ describing the $k$-th mode would be.

(5)$$x_{pos,k}=[{\overline{x}}_{pos,k},a_{k,1},b_{k,1},\dots ,a_{k,n},b_{k,n}{]}^T$$

The external state vector ($x_{ext}$) can have an analogous composition. The mean components in $x_{ext}$ can be used to capture, for example, pretension in a tether.

The corresponding time vector for the chosen frequency array

$$f=[0,f_1,2f_2,\dots ,n{f}_1{]}^T$$

is given by

$$t=[0,\mathrm{\Delta }t,2\mathrm{\Delta }t,\dots ,(2n-1)\mathrm{\Delta }t{]}^T,$$

where $\mathrm{\Delta }t=t_f/(2n)$ and the repeat period is $t_f=2n\mathrm{\Delta }t$. Since the pseudo-spectral method results in continuous in time solutions, the resulting dynamic time-series can be evaluated with additional substeps in the time vector.

The code deals with the same dynamic system in different representations and there is a need to convert between them. In general, a state $x$ is represented as the mean (DC) component followed by the real and imaginary components of the Fourier coefficients as

$$x=[X_0,\mathrm{\Re }(X1),\mathrm{\Im }(X1),\dots ,\mathrm{\Re }(Xn),\mathrm{\Im }(Xn){]}^T$$

In the time-domain, $x(t)$, this is given as $Mx$, where $M$ is the time matrix. Similarly, the state of its derivative is given as $Dx$, where $D$ is the derivative matrix. Alternatively, we sometimes need to represent the state as a complex vector $x=[X_0,X_1,\dots ,X_n{]}^T$. Finally, sometimes we need to relate the state $x$ to another state $y$ (e.g. a force component) via an impedance matrix $Z$ as $y=Zx$. The matrices $M$, $D$, and different matrices $Z$, and relevant functions are constructed internally in wecopttool.