From f49fbf04e270605743acdd59e2a86ce7538b2fe7 Mon Sep 17 00:00:00 2001 From: hivanov-nrel <53838851+hivanov-nrel@users.noreply.github.com> Date: Mon, 1 Jul 2024 14:49:48 -0600 Subject: [PATCH] updated notebooks --- examples/environmental_contours_example.html | 20 +- .../extreme_response_contour_example.html | 210 +++++++++ examples/extreme_response_contour_example.mlx | Bin 55367 -> 55391 bytes ...treme_response_full_sea_state_example.html | 443 ++++++++++++++++++ ...xtreme_response_full_sea_state_example.mlx | Bin 83163 -> 83613 bytes examples/short_term_extremes_example.html | 322 +++++++++++++ examples/short_term_extremes_example.mlx | Bin 249455 -> 249450 bytes 7 files changed, 985 insertions(+), 10 deletions(-) create mode 100644 examples/extreme_response_contour_example.html create mode 100644 examples/extreme_response_full_sea_state_example.html create mode 100644 examples/short_term_extremes_example.html diff --git a/examples/environmental_contours_example.html b/examples/environmental_contours_example.html index 22b542066..8460f27fe 100644 --- a/examples/environmental_contours_example.html +++ b/examples/environmental_contours_example.html @@ -6,9 +6,9 @@ .S4 { margin: 20px 10px 5px 4px; 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MHKiT Environmental Contours

Environmental contours of extreme sea states can be used as a part of reliability-based design for offshore structures, including wave energy converters (WECs). Environmental contours provide estimations of extreme sea states based on short-term data (e.g., 10 years used to estimate a 100-year event). These environmental contours describe extreme sea states by characterizing the resource, defining sea states for extreme condition analysis, and developing a framework for analyzing survivability of a design.
MHKiT includes functions adapted from the WDRT for creating environmental contours of extreme sea states using a principal component analysis (PCA) methodology, with additional improvements for characterizing the joint probability distribution of sea states. As a demonstration, this notebook will walk through the following steps to find a 100-year sea state for NDBC buoy 46022 using 16 years of spectral wave density data.
  1. Request Spectral Wave Density Data from NDBC
  2. Calculate Hm0 and Te using the requested data
  3. Find the data's 100-year contour
  4. Plot the data and the 100-year contour

1. Request Spectral Wave Density Data from NDBC

MHKiT can be used to request historical data from the National Data Buoy Center (NDBC). This process is split into the following steps:

Query available NDBC data

The NDBC_available_data function requires a parameter to be specified and optionally the user may provide a station ID as a string. We are interested in historical spectral wave density data 'swden' (from which we may calculate Hm0 and Te). Additionally, we will specify the buoy number as '46022' to only return data associated with this site.
% 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);
Data request URL: https://www.ndbc.noaa.gov/data/historical/swden
available_data
available_data = struct with fields:
Station_id: [25×1 string] - year: [25×1 double] - file: [25×1 string] -

Select years of interest

The NDBC_available_data function has returned a Table with columns 'Station_id', 'year', and 'file'. In this case, the years returned from NDBC_available_data span 1996 to the last complete year the buoy was operational (currently 2019 for 46022). For demonstration, we have decided we are interested in the data between the years 1996 and 2012 so we will create a new filenames_of_interest variable which only contains filenames of years less than 2013.
% Slice the available data to only include through year 2012
rows = (available_data.year < 2013) ;
filenames_of_interest = available_data.file(rows);
filenames_of_interest
filenames_of_interest = 17×1 string
"46022w1996.tx…
"46022w1997.tx…
"46022w1998.tx…
"46022w1999.tx…
"46022w2000.tx…
"46022w2001.tx…
"46022w2002.tx…
"46022w2003.tx…
"46022w2004.tx…
"46022w2005.tx…

Request Data from NDBC

To get the NDBC data we can use the NDBC_request_data function to iterate over each buoy id and year in passed filenames. This function will return the parameter data as a structure of structures which may be accessed by buoy id and then the year for multiple buoys or just the year for a single buoy. An additional data column called 'time' is created with time in datetime format.
ndbc_requested_data = NDBC_request_data(parameter, filenames_of_interest);

2. Calculate Hm0 and Te using the NDBC Data

A sea state may be characterized by significant wave height (Hm0) and energy period (Te). Using the historical spectral wave density data from NDBC, we can calculate these variables using MHKiT. Both Hm0 and Te return a single value for a given time (e.g., DateTime index).
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

3. Find the 100-year contour line

With the sea state data calculated, we can now use the modified I-FORM method to define reliability for a 100-year sea state based on the 17 years of spectral wave density data obtained from NDBC for buoy 46022. Reliability is the likelihood that a certain event will not occur in a given period. The period will define a line of constant probability in the joint probability of Hm0 and Te but individually each component different reliability (marginal distribution) which we can find by evaluating a normal cumulative distribution function (CDF). This CDF returns each component's quantiles along the iso-reliability line that finally allows us to calculate each sea state value (e.g., the 100-year contour values for Hm0 and Te).
For more detail on the environmental contour method used here please refer to: Eckert-Gallup et. al 2016
To apply the environmental contours function we will specify a 100-year sea state, the sea state data (Hm0, Te), and the time difference between measurements (dt in seconds).
% Return period (years) of interest
period = 100 ;
 
% Remove Hm0 Outliers and NaNs
filter = Hm0 < 20;
Hm0 = Hm0(filter);
Te = Te(filter);
[row, col] = 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);
 
% Get the contour values
contour = environmental_contour(Hm0, Te, dt, period);

4. Plot overlay of the data and contour

Lastly we can use the MHKiT graphics module to create a contour plot which shows the data and resultant contour line.
plot_environmental_contours(Te,Hm0,contour.contour2,contour.contour1,"x_label",...
'Energy Period (s)', "y_label",'Significant Wave Height (m)',"data_label",'NDBC 46022',...
"contour_label",'100 Year Contour');
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MHKiT Environmental Contours

Environmental contours of extreme sea states can be used as a part of reliability-based design for offshore structures, including wave energy converters (WECs). Environmental contours provide estimations of extreme sea states based on short-term data (e.g., 10 years used to estimate a 100-year event). These environmental contours describe extreme sea states by characterizing the resource, defining sea states for extreme condition analysis, and developing a framework for analyzing survivability of a design.
MHKiT includes functions adapted from the WDRT for creating environmental contours of extreme sea states using a principal component analysis (PCA) methodology, with additional improvements for characterizing the joint probability distribution of sea states. As a demonstration, this notebook will walk through the following steps to find a 100-year sea state for NDBC buoy 46022 using 16 years of spectral wave density data.
  1. Request Spectral Wave Density Data from NDBC
  2. Calculate Hm0 and Te using the requested data
  3. Find the data's 100-year contour
  4. Plot the data and the 100-year contour

1. Request Spectral Wave Density Data from NDBC

MHKiT can be used to request historical data from the National Data Buoy Center (NDBC). This process is split into the following steps:

Query available NDBC data

The NDBC_available_data function requires a parameter to be specified and optionally the user may provide a station ID as a string. We are interested in historical spectral wave density data 'swden' (from which we may calculate Hm0 and Te). Additionally, we will specify the buoy number as '46022' to only return data associated with this site.
% 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);
Data request URL: https://www.ndbc.noaa.gov/data/historical/swden
available_data
available_data = struct with fields:
Station_id: [28×1 string] + year: [28×1 double] + file: [28×1 string] +

Select years of interest

The NDBC_available_data function has returned a Table with columns 'Station_id', 'year', and 'file'. In this case, the years returned from NDBC_available_data span 1996 to the last complete year the buoy was operational (currently 2019 for 46022). For demonstration, we have decided we are interested in the data between the years 1996 and 2012 so we will create a new filenames_of_interest variable which only contains filenames of years less than 2013.
% Slice the available data to only include through year 2012
rows = (available_data.year < 2013) ;
filenames_of_interest = available_data.file(rows);
filenames_of_interest
filenames_of_interest = 17×1 string
"46022w1996.tx…
"46022w1997.tx…
"46022w1998.tx…
"46022w1999.tx…
"46022w2000.tx…
"46022w2001.tx…
"46022w2002.tx…
"46022w2003.tx…
"46022w2004.tx…
"46022w2005.tx…

Request Data from NDBC

To get the NDBC data we can use the NDBC_request_data function to iterate over each buoy id and year in passed filenames. This function will return the parameter data as a structure of structures which may be accessed by buoy id and then the year for multiple buoys or just the year for a single buoy. An additional data column called 'time' is created with time in datetime format.
ndbc_requested_data = NDBC_request_data(parameter, filenames_of_interest);

2. Calculate Hm0 and Te using the NDBC Data

A sea state may be characterized by significant wave height (Hm0) and energy period (Te). Using the historical spectral wave density data from NDBC, we can calculate these variables using MHKiT. Both Hm0 and Te return a single value for a given time (e.g., DateTime index).
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

3. Find the 100-year contour line

With the sea state data calculated, we can now use the modified I-FORM method to define reliability for a 100-year sea state based on the 17 years of spectral wave density data obtained from NDBC for buoy 46022. Reliability is the likelihood that a certain event will not occur in a given period. The period will define a line of constant probability in the joint probability of Hm0 and Te but individually each component different reliability (marginal distribution) which we can find by evaluating a normal cumulative distribution function (CDF). This CDF returns each component's quantiles along the iso-reliability line that finally allows us to calculate each sea state value (e.g., the 100-year contour values for Hm0 and Te).
For more detail on the environmental contour method used here please refer to: Eckert-Gallup et. al 2016
To apply the environmental contours function we will specify a 100-year sea state, the sea state data (Hm0, Te), and the time difference between measurements (dt in seconds).
% Return period (years) of interest
period = 100 ;
 
% Remove Hm0 Outliers and NaNs
filter = Hm0 < 20;
Hm0 = Hm0(filter);
Te = Te(filter);
[row, col] = 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);
 
% Get the contour values
contour = environmental_contours(Hm0, Te, dt, period, "PCA");

4. Plot overlay of the data and contour

Lastly we can use the MHKiT graphics module to create a contour plot which shows the data and resultant contour line.
plot_environmental_contours(Te,Hm0,contour.contour2,contour.contour1,"x_label",...
'Energy Period (s)', "y_label",'Significant Wave Height (m)',"data_label",'NDBC 46022',...
"contour_label",'100 Year Contour');

+
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Extreme Conditions Modeling - Full Sea State Approach

The full sea state approach consists of the following steps:
  1. Take N samples to represent the sea state. Each sample represents a small area of the sea state and consists of a representative (H, T) pair and a weight W associated with the probability of that sea state area.
  2. 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).
  3. Integrate over the entire sea state to obtain the long-term extreme. This is a sum of the products of the weight of each sea state times the short-term extreme.
See more details and equations in
[1] Coe, Ryan G., Carlos A. Michelén Ströfer, Aubrey Eckert-Gallup, and Cédric Sallaberry. 2018. “Full Long-Term Design Response Analysis of a Wave Energy Converter.” Renewable Energy 116: 356–66.
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);
Data request URL: https://www.ndbc.noaa.gov/data/historical/swden
 
% 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 sampling the sea state to get samples (H, T) and associtated weights. For this we will use the waves.contours.samples_full_seastate function. We will sample 20 points between each return level, for 10 levels ranging from 0.001—100 years return periods. For more details on the sampling approach see
[1] Coe, Ryan G., Carlos A. Michelén Ströfer, Aubrey Eckert-Gallup, and Cédric Sallaberry. 2018. “Full Long-Term Design Response Analysis of a Wave Energy Converter.” Renewable Energy 116: 356–66.
[2] Eckert-Gallup, Aubrey C., Cédric J. Sallaberry, Ann R. Dallman, and Vincent S. Neary. 2016. “Application of Principal Component Analysis (PCA) and Improved Joint Probability Distributions to the Inverse First-Order Reliability Method (I-FORM) for Predicting Extreme Sea States.” Ocean Engineering 112 (January): 307–19.
% return levels
levels = [0.001; 0.01; 0.05; 0.1; 0.5; 1; 5; 10; 50; 100];
 
% points per return level interval
npoints = 20;
 
% create samples
[sample_hs, sample_te, sample_weights] = samples_full_seastate( ...
Hm0, Te, npoints, levels, dt, "PCA", 250);
We will now plot the samples alongside the contours. First we will create the different contours using contours.environmental_contours. See environmental_contours_example.ipynb for more details on using this function. There are 20 samples, randomly distributed, between each set of return levels.
Hm0_contours = zeros(length(levels),1000);
Te_contours = zeros(length(levels),1000);
for p = 1:length(levels)
contour = environmental_contours(Hm0, Te, dt, levels(p), "PCA", "return_fit",true);
Hm0_contours(p,:) = contour.contour1;
Te_contours(p,:) = contour.contour2;
end
 
plot_environmental_contours(sample_te,sample_hs,Te_contours,Hm0_contours,"x_label",...
'Energy Period (s)', "y_label",'Significant Wave Height (m)',"data_label",'NDBC 46022',...
"contour_label",string(levels'));

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 200 (H, T) sampled sea states.
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 Weibull-tail-fitting method to estimate the 3-hour short-term extreme distributions for each of the 200 samples.
For more details on short-term extreme distributions 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.
NOTE: For calculating the short_term_extreme of each state, we set the optional argument output_py to true in order to return the native python output that we can then feed into our long term extreme function. Consequently, we can put arbitrary inputs to the x and method inputs of the short_term_extreme function.
% 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;
 
for i=1:length(sample_hs)
tp = sample_te(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,200,sample_hs(i),sample_te(i),tp);
% spectrum
S = jonswap_spectrum(f, tp, sample_hs(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", 100, "cdf", "output_py",true);
end
Sea state 1/200. (Hs, Te) = (2.87 m, 10.18 s). Tp = 11.26 s +Sea state 2/200. (Hs, Te) = (3.23 m, 10.46 s). Tp = 11.57 s +Sea state 3/200. (Hs, Te) = (3.78 m, 10.59 s). Tp = 11.72 s +Sea state 4/200. (Hs, Te) = (3.49 m, 9.51 s). Tp = 10.53 s +Sea state 5/200. (Hs, Te) = (3.12 m, 9.07 s). Tp = 10.04 s +Sea state 6/200. (Hs, Te) = (2.53 m, 9.01 s). Tp = 9.97 s +Sea state 7/200. (Hs, Te) = (2.78 m, 8.23 s). Tp = 9.11 s +Sea state 8/200. (Hs, Te) = (2.41 m, 8.94 s). Tp = 9.89 s +Sea state 9/200. (Hs, Te) = (2.32 m, 7.81 s). Tp = 8.64 s +Sea state 10/200. (Hs, Te) = (2.04 m, 8.03 s). Tp = 8.89 s +Sea state 11/200. (Hs, Te) = (1.66 m, 7.45 s). Tp = 8.25 s +Sea state 12/200. (Hs, Te) = (1.48 m, 7.66 s). Tp = 8.48 s +Sea state 13/200. (Hs, Te) = (2.35 m, 9.11 s). Tp = 10.08 s +Sea state 14/200. (Hs, Te) = (2.35 m, 9.12 s). Tp = 10.09 s +Sea state 15/200. (Hs, Te) = (2.11 m, 9.11 s). Tp = 10.08 s +Sea state 16/200. (Hs, Te) = (2.21 m, 9.32 s). Tp = 10.32 s +Sea state 17/200. (Hs, Te) = (2.25 m, 9.37 s). Tp = 10.37 s +Sea state 18/200. (Hs, Te) = (2.37 m, 9.16 s). Tp = 10.14 s +Sea state 19/200. (Hs, Te) = (2.38 m, 9.19 s). Tp = 10.17 s +Sea state 20/200. (Hs, Te) = (2.39 m, 9.20 s). Tp = 10.18 s +Sea state 21/200. (Hs, Te) = (4.44 m, 13.23 s). Tp = 14.64 s +Sea state 22/200. (Hs, Te) = (4.62 m, 12.60 s). Tp = 13.95 s +Sea state 23/200. (Hs, Te) = (3.92 m, 10.88 s). Tp = 12.04 s +Sea state 24/200. (Hs, Te) = (4.38 m, 10.45 s). Tp = 11.56 s +Sea state 25/200. (Hs, Te) = (4.60 m, 9.34 s). Tp = 10.34 s +Sea state 26/200. (Hs, Te) = (3.24 m, 8.11 s). Tp = 8.98 s +Sea state 27/200. (Hs, Te) = (3.20 m, 7.16 s). Tp = 7.92 s +Sea state 28/200. (Hs, Te) = (2.37 m, 6.24 s). Tp = 6.91 s +Sea state 29/200. (Hs, Te) = (2.15 m, 6.98 s). Tp = 7.72 s +Sea state 30/200. (Hs, Te) = (1.55 m, 5.77 s). Tp = 6.39 s +Sea state 31/200. (Hs, Te) = (1.14 m, 5.86 s). Tp = 6.48 s +Sea state 32/200. (Hs, Te) = (1.04 m, 6.56 s). Tp = 7.26 s +Sea state 33/200. (Hs, Te) = (0.80 m, 7.16 s). Tp = 7.92 s +Sea state 34/200. (Hs, Te) = (0.52 m, 8.36 s). Tp = 9.26 s +Sea state 35/200. (Hs, Te) = (1.00 m, 8.74 s). Tp = 9.67 s +Sea state 36/200. (Hs, Te) = (1.22 m, 10.30 s). Tp = 11.40 s +Sea state 37/200. (Hs, Te) = (1.50 m, 10.77 s). Tp = 11.92 s +Sea state 38/200. (Hs, Te) = (1.84 m, 13.76 s). Tp = 15.23 s +Sea state 39/200. (Hs, Te) = (2.14 m, 14.16 s). Tp = 15.67 s +Sea state 40/200. (Hs, Te) = (3.64 m, 13.32 s). Tp = 14.74 s +Sea state 41/200. (Hs, Te) = (0.40 m, 11.12 s). Tp = 12.31 s +Sea state 42/200. (Hs, Te) = (0.73 m, 11.89 s). Tp = 13.16 s +Sea state 43/200. (Hs, Te) = (1.23 m, 13.83 s). Tp = 15.31 s +Sea state 44/200. (Hs, Te) = (2.74 m, 15.95 s). Tp = 17.65 s +Sea state 45/200. (Hs, Te) = (3.90 m, 15.72 s). Tp = 17.40 s +Sea state 46/200. (Hs, Te) = (4.60 m, 15.40 s). Tp = 17.05 s +Sea state 47/200. (Hs, Te) = (5.08 m, 14.47 s). Tp = 16.01 s +Sea state 48/200. (Hs, Te) = (5.74 m, 12.87 s). Tp = 14.24 s +Sea state 49/200. (Hs, Te) = (6.28 m, 12.58 s). Tp = 13.92 s +Sea state 50/200. (Hs, Te) = (5.22 m, 9.61 s). Tp = 10.64 s +Sea state 51/200. (Hs, Te) = (4.59 m, 8.55 s). Tp = 9.46 s +Sea state 52/200. (Hs, Te) = (3.85 m, 7.33 s). Tp = 8.11 s +Sea state 53/200. (Hs, Te) = (2.66 m, 5.95 s). Tp = 6.59 s +Sea state 54/200. (Hs, Te) = (2.13 m, 5.38 s). Tp = 5.95 s +Sea state 55/200. (Hs, Te) = (1.81 m, 5.17 s). Tp = 5.72 s +Sea state 56/200. (Hs, Te) = (1.33 m, 5.11 s). Tp = 5.65 s +Sea state 57/200. (Hs, Te) = (0.91 m, 5.41 s). Tp = 5.99 s +Sea state 58/200. (Hs, Te) = (0.71 m, 6.13 s). Tp = 6.78 s +Sea state 59/200. (Hs, Te) = (0.37 m, 6.53 s). Tp = 7.23 s +Sea state 60/200. (Hs, Te) = (0.42 m, 7.52 s). Tp = 8.32 s +Sea state 61/200. (Hs, Te) = (0.41 m, 13.51 s). Tp = 14.95 s +Sea state 62/200. (Hs, Te) = (0.75 m, 14.83 s). Tp = 16.41 s +Sea state 63/200. (Hs, Te) = (2.15 m, 16.55 s). Tp = 18.32 s +Sea state 64/200. (Hs, Te) = (3.37 m, 17.22 s). Tp = 19.05 s +Sea state 65/200. (Hs, Te) = (4.01 m, 17.20 s). Tp = 19.03 s +Sea state 66/200. (Hs, Te) = (6.27 m, 16.59 s). Tp = 18.35 s +Sea state 67/200. (Hs, Te) = (6.33 m, 15.75 s). Tp = 17.43 s +Sea state 68/200. (Hs, Te) = (6.78 m, 14.08 s). Tp = 15.58 s +Sea state 69/200. (Hs, Te) = (6.28 m, 11.10 s). Tp = 12.29 s +Sea state 70/200. (Hs, Te) = (5.70 m, 9.76 s). Tp = 10.80 s +Sea state 71/200. (Hs, Te) = (5.07 m, 8.50 s). Tp = 9.41 s +Sea state 72/200. (Hs, Te) = (4.14 m, 7.24 s). Tp = 8.02 s +Sea state 73/200. (Hs, Te) = (3.38 m, 6.33 s). Tp = 7.00 s +Sea state 74/200. (Hs, Te) = (2.79 m, 5.68 s). Tp = 6.28 s +Sea state 75/200. (Hs, Te) = (1.65 m, 4.95 s). Tp = 5.47 s +Sea state 76/200. (Hs, Te) = (1.61 m, 4.99 s). Tp = 5.53 s +Sea state 77/200. (Hs, Te) = (1.15 m, 4.90 s). Tp = 5.42 s +Sea state 78/200. (Hs, Te) = (0.60 m, 5.13 s). Tp = 5.68 s +Sea state 79/200. (Hs, Te) = (0.25 m, 5.88 s). Tp = 6.50 s +Sea state 80/200. (Hs, Te) = (0.17 m, 6.27 s). Tp = 6.94 s +Sea state 81/200. (Hs, Te) = (0.78 m, 16.17 s). Tp = 17.89 s +Sea state 82/200. (Hs, Te) = (1.97 m, 17.29 s). Tp = 19.14 s +Sea state 83/200. (Hs, Te) = (3.66 m, 18.06 s). Tp = 19.98 s +Sea state 84/200. (Hs, Te) = (4.52 m, 19.50 s). Tp = 21.57 s +Sea state 85/200. (Hs, Te) = (6.21 m, 18.91 s). Tp = 20.92 s +Sea state 86/200. (Hs, Te) = (7.49 m, 16.74 s). Tp = 18.52 s +Sea state 87/200. (Hs, Te) = (8.16 m, 15.00 s). Tp = 16.60 s +Sea state 88/200. (Hs, Te) = (7.65 m, 13.41 s). Tp = 14.84 s +Sea state 89/200. (Hs, Te) = (7.20 m, 11.74 s). Tp = 12.99 s +Sea state 90/200. (Hs, Te) = (6.61 m, 10.39 s). Tp = 11.50 s +Sea state 91/200. (Hs, Te) = (5.05 m, 8.41 s). Tp = 9.31 s +Sea state 92/200. (Hs, Te) = (4.24 m, 7.09 s). Tp = 7.85 s +Sea state 93/200. (Hs, Te) = (3.56 m, 6.19 s). Tp = 6.85 s +Sea state 94/200. (Hs, Te) = (2.61 m, 5.36 s). Tp = 5.93 s +Sea state 95/200. (Hs, Te) = (1.94 m, 4.82 s). Tp = 5.34 s +Sea state 96/200. (Hs, Te) = (1.40 m, 4.45 s). Tp = 4.93 s +Sea state 97/200. (Hs, Te) = (1.36 m, 4.71 s). Tp = 5.22 s +Sea state 98/200. (Hs, Te) = (0.72 m, 4.61 s). Tp = 5.10 s +Sea state 99/200. (Hs, Te) = (0.67 m, 4.84 s). Tp = 5.36 s +Sea state 100/200. (Hs, Te) = (0.07 m, 5.34 s). Tp = 5.91 s +Sea state 101/200. (Hs, Te) = (0.23 m, 17.16 s). Tp = 18.99 s +Sea state 102/200. (Hs, Te) = (2.04 m, 19.19 s). Tp = 21.24 s +Sea state 103/200. (Hs, Te) = (2.49 m, 19.57 s). Tp = 21.65 s +Sea state 104/200. (Hs, Te) = (6.08 m, 19.65 s). Tp = 21.74 s +Sea state 105/200. (Hs, Te) = (6.77 m, 19.04 s). Tp = 21.07 s +Sea state 106/200. (Hs, Te) = (8.08 m, 18.35 s). Tp = 20.30 s +Sea state 107/200. (Hs, Te) = (8.66 m, 15.55 s). Tp = 17.21 s +Sea state 108/200. (Hs, Te) = (8.18 m, 12.97 s). Tp = 14.36 s +Sea state 109/200. (Hs, Te) = (8.06 m, 12.36 s). Tp = 13.68 s +Sea state 110/200. (Hs, Te) = (6.49 m, 9.89 s). Tp = 10.95 s +Sea state 111/200. (Hs, Te) = (5.13 m, 7.71 s). Tp = 8.53 s +Sea state 112/200. (Hs, Te) = (4.07 m, 6.58 s). Tp = 7.28 s +Sea state 113/200. (Hs, Te) = (3.18 m, 5.54 s). Tp = 6.13 s +Sea state 114/200. (Hs, Te) = (3.04 m, 5.36 s). Tp = 5.93 s +Sea state 115/200. (Hs, Te) = (1.95 m, 4.52 s). Tp = 5.01 s +Sea state 116/200. (Hs, Te) = (1.40 m, 4.24 s). Tp = 4.69 s +Sea state 117/200. (Hs, Te) = (0.97 m, 4.20 s). Tp = 4.65 s +Sea state 118/200. (Hs, Te) = (0.75 m, 4.31 s). Tp = 4.77 s +Sea state 119/200. (Hs, Te) = (0.51 m, 4.55 s). Tp = 5.03 s +Sea state 120/200. (Hs, Te) = (0.06 m, 5.15 s). Tp = 5.70 s +Sea state 121/200. (Hs, Te) = (1.41 m, 19.66 s). Tp = 21.76 s +Sea state 122/200. (Hs, Te) = (2.54 m, 20.29 s). Tp = 22.46 s +Sea state 123/200. (Hs, Te) = (3.47 m, 21.14 s). Tp = 23.39 s +Sea state 124/200. (Hs, Te) = (7.10 m, 20.16 s). Tp = 22.31 s +Sea state 125/200. (Hs, Te) = (8.64 m, 19.45 s). Tp = 21.53 s +Sea state 126/200. (Hs, Te) = (9.72 m, 17.61 s). Tp = 19.48 s +Sea state 127/200. (Hs, Te) = (9.35 m, 16.58 s). Tp = 18.34 s +Sea state 128/200. (Hs, Te) = (8.45 m, 13.05 s). Tp = 14.44 s +Sea state 129/200. (Hs, Te) = (7.97 m, 11.42 s). Tp = 12.64 s +Sea state 130/200. (Hs, Te) = (6.37 m, 9.31 s). Tp = 10.30 s +Sea state 131/200. (Hs, Te) = (6.28 m, 8.86 s). Tp = 9.80 s +Sea state 132/200. (Hs, Te) = (4.57 m, 6.78 s). Tp = 7.50 s +Sea state 133/200. (Hs, Te) = (3.53 m, 5.70 s). Tp = 6.30 s +Sea state 134/200. (Hs, Te) = (2.70 m, 4.83 s). Tp = 5.34 s +Sea state 135/200. (Hs, Te) = (2.55 m, 4.62 s). Tp = 5.11 s +Sea state 136/200. (Hs, Te) = (1.61 m, 4.09 s). Tp = 4.53 s +Sea state 137/200. (Hs, Te) = (1.14 m, 4.12 s). Tp = 4.56 s +Sea state 138/200. (Hs, Te) = (0.97 m, 3.90 s). Tp = 4.32 s +Sea state 139/200. (Hs, Te) = (0.42 m, 4.09 s). Tp = 4.52 s +Sea state 140/200. (Hs, Te) = (0.08 m, 4.49 s). Tp = 4.97 s +Sea state 141/200. (Hs, Te) = (0.25 m, 19.72 s). Tp = 21.82 s +Sea state 142/200. (Hs, Te) = (2.18 m, 21.97 s). Tp = 24.31 s +Sea state 143/200. (Hs, Te) = (3.88 m, 22.22 s). Tp = 24.59 s +Sea state 144/200. (Hs, Te) = (7.30 m, 22.13 s). Tp = 24.49 s +Sea state 145/200. (Hs, Te) = (9.22 m, 20.77 s). Tp = 22.98 s +Sea state 146/200. (Hs, Te) = (9.69 m, 19.12 s). Tp = 21.16 s +Sea state 147/200. (Hs, Te) = (10.60 m, 17.39 s). Tp = 19.24 s +Sea state 148/200. (Hs, Te) = (10.20 m, 14.75 s). Tp = 16.32 s +Sea state 149/200. (Hs, Te) = (8.50 m, 11.62 s). Tp = 12.86 s +Sea state 150/200. (Hs, Te) = (7.08 m, 9.44 s). Tp = 10.45 s +Sea state 151/200. (Hs, Te) = (6.06 m, 8.16 s). Tp = 9.03 s +Sea state 152/200. (Hs, Te) = (5.13 m, 7.16 s). Tp = 7.92 s +Sea state 153/200. (Hs, Te) = (3.76 m, 5.57 s). Tp = 6.16 s +Sea state 154/200. (Hs, Te) = (3.00 m, 4.84 s). Tp = 5.35 s +Sea state 155/200. (Hs, Te) = (2.24 m, 4.31 s). Tp = 4.77 s +Sea state 156/200. (Hs, Te) = (1.55 m, 3.85 s). Tp = 4.27 s +Sea state 157/200. (Hs, Te) = (1.48 m, 3.89 s). Tp = 4.31 s +Sea state 158/200. (Hs, Te) = (0.76 m, 3.83 s). Tp = 4.24 s +Sea state 159/200. (Hs, Te) = (0.40 m, 4.06 s). Tp = 4.49 s +Sea state 160/200. (Hs, Te) = (0.22 m, 4.18 s). Tp = 4.62 s +Sea state 161/200. (Hs, Te) = (0.65 m, 22.88 s). Tp = 25.32 s +Sea state 162/200. (Hs, Te) = (3.04 m, 23.41 s). Tp = 25.90 s +Sea state 163/200. (Hs, Te) = (4.59 m, 23.43 s). Tp = 25.92 s +Sea state 164/200. (Hs, Te) = (8.08 m, 22.66 s). Tp = 25.08 s +Sea state 165/200. (Hs, Te) = (10.23 m, 20.21 s). Tp = 22.37 s +Sea state 166/200. (Hs, Te) = (10.97 m, 19.67 s). Tp = 21.76 s +Sea state 167/200. (Hs, Te) = (10.73 m, 16.24 s). Tp = 17.97 s +Sea state 168/200. (Hs, Te) = (10.20 m, 13.75 s). Tp = 15.22 s +Sea state 169/200. (Hs, Te) = (10.02 m, 12.77 s). Tp = 14.13 s +Sea state 170/200. (Hs, Te) = (7.91 m, 10.02 s). Tp = 11.09 s +Sea state 171/200. (Hs, Te) = (6.50 m, 8.46 s). Tp = 9.36 s +Sea state 172/200. (Hs, Te) = (5.40 m, 7.21 s). Tp = 7.98 s +Sea state 173/200. (Hs, Te) = (3.84 m, 5.51 s). Tp = 6.10 s +Sea state 174/200. (Hs, Te) = (3.11 m, 4.86 s). Tp = 5.38 s +Sea state 175/200. (Hs, Te) = (2.28 m, 4.15 s). Tp = 4.59 s +Sea state 176/200. (Hs, Te) = (1.58 m, 3.72 s). Tp = 4.12 s +Sea state 177/200. (Hs, Te) = (1.27 m, 3.64 s). Tp = 4.03 s +Sea state 178/200. (Hs, Te) = (0.96 m, 3.60 s). Tp = 3.99 s +Sea state 179/200. (Hs, Te) = (0.73 m, 3.72 s). Tp = 4.12 s +Sea state 180/200. (Hs, Te) = (0.20 m, 4.00 s). Tp = 4.43 s +Sea state 181/200. (Hs, Te) = (0.76 m, 23.17 s). Tp = 25.64 s +Sea state 182/200. (Hs, Te) = (3.91 m, 24.51 s). Tp = 27.12 s +Sea state 183/200. (Hs, Te) = (7.32 m, 24.46 s). Tp = 27.07 s +Sea state 184/200. (Hs, Te) = (9.48 m, 23.50 s). Tp = 26.01 s +Sea state 185/200. (Hs, Te) = (11.81 m, 21.25 s). Tp = 23.52 s +Sea state 186/200. (Hs, Te) = (12.09 m, 19.36 s). Tp = 21.43 s +Sea state 187/200. (Hs, Te) = (12.01 m, 16.87 s). Tp = 18.67 s +Sea state 188/200. (Hs, Te) = (10.65 m, 13.32 s). Tp = 14.74 s +Sea state 189/200. (Hs, Te) = (10.36 m, 13.08 s). Tp = 14.47 s +Sea state 190/200. (Hs, Te) = (8.78 m, 10.69 s). Tp = 11.83 s +Sea state 191/200. (Hs, Te) = (6.12 m, 7.75 s). Tp = 8.57 s +Sea state 192/200. (Hs, Te) = (4.99 m, 6.40 s). Tp = 7.09 s +Sea state 193/200. (Hs, Te) = (4.18 m, 5.61 s). Tp = 6.21 s +Sea state 194/200. (Hs, Te) = (2.87 m, 4.41 s). Tp = 4.87 s +Sea state 195/200. (Hs, Te) = (2.43 m, 4.09 s). Tp = 4.53 s +Sea state 196/200. (Hs, Te) = (1.67 m, 3.61 s). Tp = 3.99 s +Sea state 197/200. (Hs, Te) = (1.48 m, 3.53 s). Tp = 3.91 s +Sea state 198/200. (Hs, Te) = (0.92 m, 3.44 s). Tp = 3.81 s +Sea state 199/200. (Hs, Te) = (0.45 m, 3.53 s). Tp = 3.90 s +Sea state 200/200. (Hs, Te) = (0.30 m, 3.67 s). Tp = 4.06 s

3. Long-Term Extreme Distribution

Finally we integrate the weighted short-term extreme distributions over the entire sea state space to obtain the extreme distribution, assuming a 3-hour sea state coherence. For this we use the full_seastate_long_term_extreme function. The integral reduces to a sum over the 200 bins of the weighted short-term extreme distributions.
Similar to the short-term extreme functions, the output of long-term extreme function is a probability distribution. Here, we will look at the survival function and the 100-year return level.The value of the survival function at a given return level (e.g. 100-years) may be calculated using the return_year_value function.
t_st_hr = t_st / (60.0 * 60.0);
t_return_yr = 100.0;
probability_of_exceedance = 1 / (t_return_yr * 365.25 * 24 / t_st_hr);
x_t = full_seastate_long_term_extreme(ste_all, sample_weights, 1-probability_of_exceedance, "ppf");
fprintf("100-year elevation: %.3f meters", x_t)
100-year elevation: 10.299 meters
% plot survival function
x = linspace(0, 20, 1000);
lte_sf = full_seastate_long_term_extreme(ste_all, sample_weights, x, "sf");
figure;
semilogy(x, lte_sf);
xlabel('elevation [m]'); ylabel('survival function (1-cdf)'); ylim([1e-10, 1]); xlim([0, x(end)]);
% add 100-year return level to plot
hold on
st_sf = full_seastate_long_term_extreme(ste_all, sample_weights, x_t, "sf");
xline(x_t, '--')
yline(st_sf, '--')
+
+ +
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zcoa>>b7@Pw4Z(6kyxf8oK8=Kl;ih+5g!ptVf3ZToi53W>bV)P>(TE_(L73Rdh>yxf zPD=akx>3+b0v%{D!ILb_$TzE1*juR000{R000000003100000>^%XO%0B@dm)%7H5SOAw0U82RKbO8m0Z9XI pKmnIeLjf9>XhH!EmoY{GDg$sq0hdoh0V$V +MHKiT Short-term Extreme Loads

MHKiT Short-term Extreme Loads

Short-term extreme loads are typically required as part of a long-term extreme load estimate. A short-term extreme distribution is a probability distribution for the extreme value of a short-term (e.g. 3-hours) time-series.
MHKiT includes functions adapted from the WDRT for estimating the short-term extreme distribution. Several methods are included, which can be divided into two main approaches:
  1. Block maxima: approximate the short-term extreme distribution directly by fitting a probability distribution to the maxima from N distinct short-term blocks.
  2. Peaks distribution: approximate the short-term extreme distribution from the distribution of peaks from a time-series of arbitrary length.
The two sets of methods will be demonstrated.
NOTE: This notebook will walk through the two different approaches, and each method in them. The short-term extremes will be obtained step-by-step showing the intermediate outputs. However, a convenience function, the loads.extreme.short_term_extreme function, can be be used directly in a single step. For this usage jump to the last section. It is recommended that you go over this entire example to understand what is going on "under the hood" prior to using this convenience function.
The quantity (load) of interest can be anything related to the WEC, such as position (e.g. PTO extension, heave displacement), velocity, structural load, mooring loads, etc. For this example we will use the wave elevation as the quantity of interest, since such time-series can be easily created without having to run a WEC-modeling software. The short-term period of interest will be 3-hours.
An important pre-processing step is making the time-series zero-mean. In this case, since wave elevation is zero-mean no pre-processing is needed.
We first define the short-term period (3 hours).
% short-term period in seconds
t_st = 3.0 * 60.0 * 60.0
t_st = 10800
The methods based on peaks distributions require a single time series of any length while the methods based on block maxima require N time series of length equal to the short-term time period.
We will now create N=10 time series of length equal to the short-term period (3 hours). For block maxima methods we will use all 10 and for peaks distribution methods we will only use the first hour of the first time-series. For the block maxima we use N=10 for speed considerations, but note that this would be inadequately small for most real applications.
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 code below is simply creating the synthetic data (wave elevation time series). In a realistic case the 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 the quantity of interest there are ten 3-hours time-series available.
We start by creating the ten 3-hour time-series of the quantity of interest. We do this with a list comprehension and using the wave.resource.surface_elevation function. The result is a list of 10 different time-series. Alternatively we could have created a single time-series of length N times the short-term period (30 hours) and used the extreme.block_maxima function to split it into blocks and return the block maxima.
% JONSWAP spectrum for creating short-term time-series
T_min = 1; % s
Tp = 8; % s
Hs = 1.5; % m
df = 1/t_st;
f_max = 1/T_min;
Nf = f_max/df + 1;
f = linspace(0.0, f_max, Nf);
S = jonswap_spectrum(f, Tp, Hs);
 
% time in seconds
time = linspace(0, t_st, 2*Nf+1);
 
% qoi timeseries
qoi_timeseries = surface_elevation(S, time);
qoi = qoi_timeseries.elevation;
We can view the different time-series. Change value of i below to plot a different time-series. Below we plot both the full 3 hours time series and the first 2 minutes to get a better look at the behavior of the response.
figure;
plot(time, qoi)
title("Full 3 hours"); xlabel("time [s]"); ylabel("elevation [m]")
figure;
plot(time(1:120), qoi(1:120))
title("First 2 minutes"); xlabel("time [s]"); ylabel("elevation [m]")

Block Maxima

The short-term extreme distribution can be estimated directly from N time-series of short-term period. These time-series can be obtained, for instance, from N distinct simulations of the quantity of interest, or alternatively by dividing a long time-series of the quantity of interest into blocks of length equal to the short-term period. In this example we will divide our 3 hour time series into 10-minute segments for speed considerations, but this would be inadequately small for most real applications.

1. Find the maximum of each block

The first step is to get the maximum in each block, i.e. in each of the 10-minute time series. We can use the block_maxima funtion to do this for us.
% split time series into 10-minute (600s) segments
block_max = block_maxima(time, qoi', 600);
We can also visualize this result, which represents the maximum elevation of each 10-minute segment for the 3 hour period we are analyzing.
figure;
plot(block_max, 'o'); grid on; xticks(0:1:19); ylim([0,2])
xlabel('block #'); ylabel('block max elevation [m]'); title('Block maxima')

2. Fit a distribution to the block maxima

We can now get the short-term extreme distribution by fitting either a generalized extreme value distribution or a Gumbel (right) distributions to these N=18 maxima. We will compare both methods.
% generalized extreme value distribution
x = linspace(0,3,1000);
ste_gev_pdf = ste_block_maxima_gev(block_max, x, "pdf");
ste_gev_cdf = ste_block_maxima_gev(block_max, x, "cdf");
ste_gum_pdf = ste_block_maxima_gumbel(block_max, x, "pdf");
ste_gum_cdf = ste_block_maxima_gumbel(block_max, x, "cdf");
The outputs of these functions are probability distributions. Here we will plot the CDF and PDF and print the 95% interval for both methods.
% plot cdf and pdf
subplot(1,2,1); hold on;
plot(x,ste_gev_pdf); xlabel('elevation [m]'); ylabel('PDF')
plot(x,ste_gum_pdf); xlabel('elevation [m]'); ylabel('PDF')
legend('GEV','Gumbel')
subplot(1,2,2); hold on;
plot(x,ste_gev_cdf); xlabel('elevation [m]'); ylabel('CDF')
plot(x,ste_gum_cdf); xlabel('elevation [m]'); ylabel('CDF')
fprintf('GEV:\n 95 percent interval: (%.3f m, %.3f m)',ste_block_maxima_gev(block_max, 0.025, "ppf"),ste_block_maxima_gev(block_max, 0.975, "ppf"))
GEV: + 95 percent interval: (1.081 m, 1.459 m)
fprintf('GUMBEL:\n 95 percent interval: (%.3f m, %.3f m)',ste_block_maxima_gumbel(block_max, 0.025, "ppf"),ste_block_maxima_gumbel(block_max, 0.975, "ppf"))
GUMBEL: + 95 percent interval: (1.084 m, 1.483 m)

Peaks Distribution

The block maxima approach is the most direct way to approximate the short-term extreme but it requires a lot of data (N timeseries of length equal to the short-term period). An alternative approach approximates the short-term extreme from a timeseries of arbitrary length, even shorter than the short-term period, by fitting a distribution to the peaks of the timeseries.
To demonstrate these methods we will approximate the 3-hour short-term extreme distribution from a single 1-hour timeseries.
We start by generating this 1-hour response timeseries.
t_end = 1.0 * 60.0 * 60.0;
qoi_1hr = qoi(1:t_end);
time_1hr = time(1:t_end);
% visualize
figure;
plot(time_1hr, qoi_1hr)
title("1 hour"); xlabel("time [s]"); ylabel("elevation [m]")

1. Find the global peaks

The global peaks are the maximum between succesive zero-upcrossings. We use the extreme.global_peaks function to find these peaks.
[t_peaks, qoi_peaks] = global_peaks(time_1hr, qoi_1hr);
npeaks = length(qoi_peaks)
npeaks = 305
We will plot the entire time-series and the global peaks and then zoom in on the first 2 minutes.
figure; hold on;
plot(time_1hr, qoi_1hr)
title("1 hour"); xlabel("time [s]"); ylabel("elevation [m]")
scatter(t_peaks, qoi_peaks)
yline(0)
figure; hold on;
plot(time_1hr, qoi_1hr)
title("1 hour"); xlabel("time [s]"); ylabel("elevation [m]")
scatter(t_peaks, qoi_peaks, 'filled')
yline(0)
xlim([0,120])

2. Fit a distribution to the peaks

MHKiT has three methods for doing this.
  1. Fit a Weibull distribution to all the peaks.
  2. Weibull tail fit: seven different weibull distributions are fit to successively smaller subsets of the highest peaks, and the peak distribution is taken as the average of these sevenn
  3. Peaks over threshold: a generalized Pareto distribution is fit to all peaks above a certain threshold, typically taken as 1.4 standard deviations above the mean.
We will now compare the results from these three methods. For the peaks over threshold we will take the threshold (variable thresh below) to be 1.4 standard deviations above the mean. Note in this example, we are also storing the python statistical object. This is something we will need in the step!
% Weibull
[peaks_w_pdf, peaks_w] = peaks_distribution_weibull(qoi_peaks, x, "pdf");
peaks_w_cdf = peaks_distribution_weibull(qoi_peaks, x, "cdf");
% Weibull tail fit
[peaks_wtf_pdf, peaks_wtf] = peaks_distribution_weibull_tail_fit(qoi_peaks, x, "pdf");
peaks_wtf_cdf = peaks_distribution_weibull_tail_fit(qoi_peaks, x, "cdf");
% peaks over threshold
thresh = mean(qoi_peaks) + 1.4 * std(qoi_peaks);
[peaks_pot_pdf, peaks_pot] = peaks_distribution_peaks_over_threshold(qoi_peaks, x, "pdf", "threshold", thresh);
peaks_pot_cdf = peaks_distribution_peaks_over_threshold(qoi_peaks, x, "cdf", "threshold", thresh);
We will plot both the PDF and CDF of the peaks distribution, and obtain the 95% interval for each distribution. Note however that for the peaks over threshold method the distribution is only defined for values over the threshold, and many of the functionalities that require the entire distribution are therefore not available.
% plot cdf and pdf
subplot(1,2,1); hold on;
plot(x,peaks_w_pdf); xlabel('elevation [m]'); ylabel('PDF')
plot(x,peaks_wtf_pdf); xlabel('elevation [m]'); ylabel('PDF')
plot(x,peaks_pot_pdf,'green'); xlabel('elevation [m]'); ylabel('PDF')
legend('Weibull','WTF','POT')
subplot(1,2,2); hold on;
plot(x,peaks_w_cdf); xlabel('elevation [m]'); ylabel('CDF')
plot(x,peaks_wtf_cdf); xlabel('elevation [m]'); ylabel('CDF')
plot(x,peaks_pot_cdf, 'green'); xlabel('elevation [m]'); ylabel('CDF')
s1 = sprintf('Weibull:\n 95 percent interval: (%.3f m, %.3f m)\n',peaks_distribution_weibull(qoi_peaks,0.025,"ppf"),peaks_distribution_weibull(qoi_peaks,0.975,"ppf"));
s2 = sprintf('Weibull tail fit:\n 95 percent interval: (%.3f m, %.3f m)',peaks_distribution_weibull_tail_fit(qoi_peaks,0.025,"ppf"),peaks_distribution_weibull_tail_fit(qoi_peaks,0.975,"ppf"));
disp([s1, s2])
Weibull: + 95 percent interval: (0.055 m, 1.122 m) +Weibull tail fit: + 95 percent interval: (0.049 m, 1.101 m)

3. Obtain the short-term extreme distribution from the peaks distribution

We do this using the peaks distribution and an approximation for the number of peaks per short-term period. We use the extreme.ste_peaks function to obtain an approximation for the number of peaks in a short-term period.
N_st = number_of_short_term_peaks(npeaks, t_end, t_st);
fprintf("Approximate number of peaks in a (3-hours) short-term period: %d",N_st)
Approximate number of peaks in a (3-hours) short-term period: 915
%
ste_w_pdf = ste_peaks(peaks_w, N_st, x, "pdf");
ste_w_cdf = ste_peaks(peaks_w, N_st, x, "cdf");
ste_wtf_pdf = ste_peaks(peaks_wtf, N_st, x, "pdf");
ste_wtf_cdf = ste_peaks(peaks_wtf, N_st, x, "cdf");
ste_pot_pdf = ste_peaks(peaks_pot, N_st, x, "pdf");
ste_pot_cdf = ste_peaks(peaks_pot, N_st, x, "cdf");
 
% plot cdf and pdf
figure;
subplot(1,2,1); hold on;
plot(x,ste_w_pdf); xlabel('elevation [m]'); ylabel('PDF')
plot(x,ste_wtf_pdf); xlabel('elevation [m]'); ylabel('PDF')
plot(x,ste_pot_pdf, 'green'); xlabel('elevation [m]'); ylabel('PDF')
legend('Weibull','WTF','POT')
subplot(1,2,2); hold on;
plot(x,ste_w_cdf); xlabel('elevation [m]'); ylabel('CDF')
plot(x,ste_wtf_cdf); xlabel('elevation [m]'); ylabel('CDF')
plot(x,ste_pot_cdf, 'green'); xlabel('elevation [m]'); ylabel('CDF')

The short_term_extreme function

This example showed step-by-step how to estimate the short-term extreme using different methods. However for most use cases the results from the intermediate steps are not of interest. The estimated short-term extreme distribution can be obtained directly from a time-series of the response using the short_term_extreme function. For the methods based on fitting the peaks the time-series can be of arbitrary length, while for methods based on block maxima it should be many times longer than the short-term period.
Note: This function is provided for convenience but the user should be aware of what is going on "under the hood" and ensure the method they choose is appropriate. For instance if using a block maxima method it is good practice to have at least 20 short-term periods worth of data.
As an example the Weibull tail fit approximation of the short-term extreme distribution can be obtained in a single step. The function takes the time series (time, response), the desired short-term period, and the method as inputs.
ste_wtf = short_term_extreme(time_1hr, qoi_1hr, t_st, "peaks_weibull_tail_fit", x, "pdf");
+
+ +
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