Extreme Conditions Modeling - Contour Approach
Extreme conditions modeling consists of identifying the expected extreme (e.g. 100-year) response of some quantity of interest, such as WEC motions or mooring loads. Three different methods of estimating extreme conditions were adapted from WDRT: full sea state approach, contour approach, and MLER design wave. This noteboook presents the contour approach.
The contour approach consists of the following steps:
- Sample the environmental contour for the return period of interest at N points near the maximum H.
- For each sample (H, T) calculate the short-term (e.g. 3-hours) extreme for the quantity of interest (e.g. WEC motions or mooring tension).
- Select the sea state with the highest short-term expected value for the quantity of interest as the design sea state, and some metric (e.g. 95th percentile) of the distribution as the extreme/design response.
NOTE: Prior to running this example it is recommended to become familiar with environmental_contours_example.ipynb and short_term_extremes_example.ipynb since some code blocks are adapted from those examples and used here without the additional description.
Obtain and Process NDBC Buouy Data
The first step will be obtaining the environmental data and creating the contours. See environmental_contours_example.ipynb for more details and explanations of how this is being done in the following code block.
% Specify the parameter as spectral wave density and the buoy number to be 46022
parameter = 'swden';
buoy_number = '46022';
available_data= NDBC_available_data(parameter,"buoy_number", buoy_number);
% Slice the available data to only include through year 2012
rows = (available_data.year < 2013) ;
filenames_of_interest = available_data.file(rows);
ndbc_requested_data = NDBC_request_data(parameter, filenames_of_interest);
Hm0 = [];
Te = [];
for field = fieldnames(ndbc_requested_data)'
Hm0 = [Hm0 ; significant_wave_height(ndbc_requested_data.(field{1}))];
Te = [Te ; energy_period(ndbc_requested_data.(field{1}))];
end
% Remove Hm0 Outliers and NaNs
filter = Hm0 < 20;
Hm0 = Hm0(filter);
Te = Te(filter);
[row, ~] = find(~isnan(Te));
Hm0 = Hm0(row);
Te = Te(row);
[row, col] = find(~isnan(Hm0));
Hm0 = Hm0(row);
Te = Te(row);
% Delta time of sea-states in seconds
dt = ndbc_requested_data.year_1996.time(2)- ndbc_requested_data.year_1996.time(1);
dt = seconds(dt);
1. Sampling
The first step is to create the environmental contour for the return period of interest, which is 100-years in this case, using the wave.contours.environmental_contours. See environmental_contours_example.ipynb for more details on using this function.
Next we choose 5 sample sea states along this contour. We choose 5 equally spaced energy period values, between 15-22 seconds, to sample the contour. We then obtain the significant wave height for each sample using the wave.contours.samples_contour function.
% Get the contour values
period = 100; % 100-year contour
contour = environmental_contours(Hm0, Te, dt, period, "PCA");
hs_contour = contour.contour1;
te_contour = contour.contour2;
% 5 samples
te_samples = linspace(15, 22, 5);
hs_samples = samples_contour(te_samples, te_contour, hs_contour);
We now plot the contour and samples.
plot_environmental_contours(Te,Hm0,te_contour,hs_contour,"x_label",...
'Energy Period (s)', "y_label",'Significant Wave Height (m)',"data_label",'NDBC 46022',...
"contour_label",'100 Year Contour');
hold on;
scatter(te_samples, hs_samples,"red","filled","o"); legend("data","100-year contour","samples")
hold off;
2. Short-Term Extreme Distributions
Many different methods for short-term extremes were adapted from WDRT, and a summary and examples can be found in short_term_extremes_example.ipynb. The response quantity of interest is typically related to the WEC itself, e.g. maximum heave displacement, PTO extension, or load on the mooring lines. This requires running a simulation (e.g. WEC-Sim) for each of the 5 sampled sea states (H, T). For the sake of example we will consider the wave elevation as the quantity of interest (can be thought as a proxy for heave motion in this example). Wave elevation time-series for a specific sea state can be created quickly without running any external software.
NOTE: The majority of the for-loop below is simply creating the synthetic data (wave elevation time series). In a realistic case the variables time and data describing each time series would be obtained externally, e.g. through simulation software such as WEC-Sim or CFD. For this reason the details of creating the synthetic data are not presented here, instead assume for each sea state there is time-series data available.
The last lines of the for-loop create the short-term extreme distribution from the time-series using the loads.extreme.short_term_extreme function. The short-term period will be 3-hours and we will use 1-hour "simulations" and the Weibul-tail-fitting method to estimate the 3-hour short-term extreme distributions for each of the 5 samples.
For more details see short_term_extremes_example.ipynb and
[3] Michelén Ströfer, Carlos A., and Ryan Coe. 2015. “Comparison of Methods for Estimating Short-Term Extreme Response of Wave Energy Converters.” In OCEANS 2015 - MTS/IEEE Washington, 1–6. IEEE.
% create the short-term extreme distribution for each sample sea state
t_st = 3.0 * 60.0 * 60.0;
gamma = 3.3;
t_sim = 1.0 * 60.0 * 60.0;
n = length(hs_samples);
for i=1:length(hs_samples)
tp = te_samples(i) / (0.8255 + 0.03852 * gamma - 0.005537 * gamma^2 + 0.0003154 * gamma^3);
% time & frequency arrays
df = 1.0 / t_sim;
T_min = tp / 10.0; % s
f_max = 1.0 / T_min;
Nf = int32(f_max / df) + 1;
time = linspace(0, t_sim, 2 * Nf + 1);
f = linspace(0.0, f_max, Nf);
fprintf("Sea state %g/%g. (Hs, Te) = (%.2f m, %.2f s). Tp = %.2f s\n",i,n,hs_samples(i),te_samples(i),tp);
% spectrum
S = jonswap_spectrum(f, tp, hs_samples(i), gamma);
% 1-hour elevation time-series
data = surface_elevation(S, time);
% 3-hour extreme distribution
ste_all(i) = short_term_extreme(time, data.elevation, t_st, "peaks_weibull_tail_fit", 1, "expect");
ste_95(i) = short_term_extreme(time, data.elevation, t_st, "peaks_weibull_tail_fit", 0.95, "ppf");
end
3. Select Design Sea State and Design Response
Finally, we will choose the sea state with the largest expected value as the design sea state. In the code snippet above, we stored the expected value for each sea state in ste_all. We will then use the 95th percentile of the short-term extreme distribution as the extreme design response.
First we find the sampled sea state with the largest expected value.
% find max expected wave height
[~,max_ind] = max(ste_all);
hs_design = hs_samples(max_ind);
te_design = te_samples(max_ind);
fprintf("Design sea state (Hs, Te): (%.2f m, %.2f s)", hs_design, te_design)
Next, we choose the 95th percentile of the short-term extreme distribution at this sea state as the extreme design response.
response_design = ste_95(max_ind);
fprintf("Design (extreme) response: %.2f meters", response_design)