## 3 Simulation Model
### Hydrodynamic Interactions within the Array  
The hydrodynamical problem is solved within the framework of linear potential theory, specifically the Laplace equation, resulting in the interaction field illustrated in Figure 11. In this paper, the theoretical basis is only covered briefly. The work of [PERSON]'s _Marine Hydrodynamics_[19] is used as basis for this simulation work.  
**Figure 11.** Hydrodynamical interference between absorbers. (**a**) Illustration of the wave interaction with the array. The wave direction is from southwest and thus causes amplification on the southern side of the array and attenuation on the northern side; (**b**) Power correction factors for absorbers in array calculated from shadowing effects (\(Tz=6.5\ s\)).  
Since the velocity potential is linear, all contributions to forces and motions are linear. As a result, the principle of superposition applies. Therefore, it is convenient to split the complex problem into a set of simpler problems. The full solution is thus the sum of several simpler solutions. The potential arising from N absorbers placed in a string can thus be described as the sum of the following contributions.  
\[\phi=\phi_{0}+\phi_{D}+\phi_{R} \tag{3}\]  
The total velocity potential \(\phi\) due to the interaction of N absorbers on a string is the sum of the excitation potential due to incident waves \(\phi_{0}\), the diffraction potential due to the interaction of the incident potential with all absorbers at rest \(\phi_{D}\), and the radiation potential \(\phi_{R}\) due to the independent motion of every absorber in every mode of motion with no incident waves present.  
The diffraction problem and the radiation problem are solved independently. Thus, there are \(N+1\) independent problems to solve. Furthermore, the radiation potential from each absorber is separatedin \(6\) independent modes of motion. The total potential \(\phi_{N}^{i}\) acting on absorber \(N\) in mode \(i\) of motion is the sum of every other absorber's radiation and diffraction potential, in addition to the diffraction and radiation potential from absorber \(N\) acting on itself in mode \(i\) of motion. Combining the six modes of motions for each absorber, and allowing for all absorbers to interact, results in a total of \(N\times 6\) independent linear equations to be solved for each wave frequency. With a full description of the velocity potential, it is possible to integrate solutions in the frequency domain on specific wave climates and optimize the array energy output with respect to array layout angle and power take off damping coefficient.  
In order to represent the interactions within the array in the time domain model, a set of correction factors is applied to the power output from a time domain model of an array without interactions. Correction factors are calculated individually for each wave direction and wave period encountered. The correction factor for the individual WECs are plotted as a function of the array angle in Figure 11(b) for the design wave period (\(Tz=6.5\)\(s\)). Currently, only interactions within arrays are taken into consideration, and the hydrodynamical effect of the wave farm is not modeled. A detailed farm study will be required to produce an accurate figure for annual energy production, but based on experience with similar modeling, the interactions are not believed to have significant impact on the simulated power quality. The results presented here are therefore believed to be accurate and valid for evaluating power quality.