I spent some time exploring the use of the Richardson Extrapolation to achieve an accurate solution with lower resolution. Specifically, I defined a WecOptTool optimization with the Wavebot, a regular wave, and a nonlinear constraint (max PTO force) and increased the number of frequencies used.
I started with 2 frequencies with the fundamental frequency (f1) being equal to the wave frequency, then incrementally doubled the number of frequencies. The results for 2 frequencies and 16 frequencies are shown below for reference with the larger frequency array allowing for more nonlinearities to be captured.
Testing 2 frequencies through 256 frequencies leads to a series of 8 sets of frequencies and thus 8 different optimization solutions. The average electrical power for those solutions is shown below:
It is clear that the average power converges to a solution. Using the Richardson Extrapolation (RE) method, we can take any 3 consecutive points and calculate a theoretical exact solution using the following formula
With 8 solutions, there are 6 sets of 3 which RE can be applied to ([2,4,8], [4,8,16]...). The results from these 6 sets are shown below.
For the RE method to apply, the order has to be constant, which is not the case for most of the results, although it does potentially start to converge to a constant result at the end. For the different sets of results, RE improves some results (3,4) while making others (1, 2, 3) less accurate. It is possible that for the last few results it is reaching the asymptotic convergence region, but the errors are so small at this point that the improvements are miniscule. Furthermore, the optimization results change slightly if run again and the errors can become slightly different.
I spent some time exploring the use of the Richardson Extrapolation to achieve an accurate solution with lower resolution. Specifically, I defined a WecOptTool optimization with the Wavebot, a regular wave, and a nonlinear constraint (max PTO force) and increased the number of frequencies used.
I started with 2 frequencies with the fundamental frequency (f1) being equal to the wave frequency, then incrementally doubled the number of frequencies. The results for 2 frequencies and 16 frequencies are shown below for reference with the larger frequency array allowing for more nonlinearities to be captured.
Testing 2 frequencies through 256 frequencies leads to a series of 8 sets of frequencies and thus 8 different optimization solutions. The average electrical power for those solutions is shown below:
It is clear that the average power converges to a solution. Using the Richardson Extrapolation (RE) method, we can take any 3 consecutive points and calculate a theoretical exact solution using the following formula
With 8 solutions, there are 6 sets of 3 which RE can be applied to ([2,4,8], [4,8,16]...). The results from these 6 sets are shown below.
For the RE method to apply, the order has to be constant, which is not the case for most of the results, although it does potentially start to converge to a constant result at the end. For the different sets of results, RE improves some results (3,4) while making others (1, 2, 3) less accurate. It is possible that for the last few results it is reaching the asymptotic convergence region, but the errors are so small at this point that the improvements are miniscule. Furthermore, the optimization results change slightly if run again and the errors can become slightly different.