Pipeline Walkthrough

Method Efficiency

The method is currently quite slow. Since the slow-down comes from the column conversions, future versions may include a quicker option that simply checks a selection of values in the column for non-numerics and then forces numeric on identified columns. However, the current methodology is preferred in order to keep as much data as possible and avoid misidentifying columns.

X-by-X Power Prediction

Selecting a time period to subset by is an important step; choose to model over days, weeks, or months based on the data being modeled as well as what modeling will be performed on the overall data set. This is what is called the “X-by-X” method: creating models by a certain time period, and analyzing PLR across those models. Below is an example using one of the power predictive models, the data-driven XbX model, on a week-by-week basis.

test_xbx_wbw_res <- plr_xbx_model(test_dfc, var_list, by = "week", data_cutoff = 30, predict_data = NULL)

# Generate Table
knitr::kable(test_xbx_wbw_res[1:5, ], caption = "XbX Model: Week-by-Week Implementation")

The data is subset by week, using the columns created during the cleaning step. Each week is fitted to a least-squares linear regression model. These weekly models are checked to see if they have a minimum number of data points; here, we specified to filter them out if there are 30 or fewer.

Representative Conditions

The weekly models are then fitted to predicted representative conditions. This can be passed to the function as predict_data. If not, the function calculates values based on the data as follows: Daily max irradiance is found, from which the the lowest value over 300 watts/meter squared is used; temperature is averaged over all of the data; and wind speed is averaged over all of the data. It is from this that the power and std_error columns are calculated (excluding the 6k; see model comparison). The sigma column is the standard deviation, calculated from the standard error.

Outlier Removal

Finally, entries are marked as outliers in the last column using Tukey’s fences. It is often desirable to remove these outliers; to do so, use the plr_remove_outliers method. This is a simple operation but common enough that a function was made for it.

test_xbx_wbw_res_no_outliers <- plr_remove_outliers(test_xbx_wbw_res)

rows_before <- nrow(test_xbx_wbw_res)
rows_after <- nrow(test_xbx_wbw_res_no_outliers)
number_outliers <- rows_before - rows_after

Standard (un)weighted Regression

In the more standard regression scheme, a linear model of predicted power and time is fit to the data. PLR is calculated using the formula \[PLR = \frac{m}{b} (py) (100)\], where \(m\) is the slope and \(b\) is the intercept of the regression line, and \(py\) is the conversion between the modeled time period and years. In the case of day-by-day power prediction, for example, \(py = 365\). In the function call, this appears as per_year. It functions similarly to the ‘by’ parameter in other functions, but takes numeric values instead so as to offer greater flexibility (for reference, there’s 12 months, 52 weeks, or 365 days in a year).

# example weighted regression
xbx_wbw_plr <- plr_weighted_regression(test_xbx_wbw_res_no_outliers, power_var = 'power_var', time_var = 'time_var', model = "xbx", per_year = 52, weight_var = 'sigma')

xbx_wbw_plr
#>            plr    error model   method
#> tvar -1.050375 0.353685   xbx weighted

In this example, the regression is weighted by sigma, the standard deviation of each point. Weighted regression has a number of problems - outliers and points with very high or low uncertainty can lead to skewed regressions, for example, and seasonal patterns such as high numbers of accurate readings in the summer can also cause biases in the results. If you prefer to calculate PLR without weightings, simply input NA for weight_var.

# example unweighted regression
xbx_wbw_plr_unweighted <- plr_weighted_regression(test_xbx_wbw_res_no_outliers, power_var = 'power_var', time_var = 'time_var', model = "xbx", per_year = 52, weight_var = NA)

xbx_wbw_plr
#>            plr    error model   method
#> tvar -1.050375 0.353685   xbx weighted

Uncertainty for linear model evaluated PLR is calculated using the variance of the fitting coefficients and converting it to a PLR range. A helper function has been included for this.

mod <- lm(power_var ~ time_var, data = test_xbx_wbw_res_no_outliers)

plr_sd <- plr_var(mod, per_year = 52)

Year-on-Year Regression

The other option given by the package is year-on-year regression, a technique that examines points exactly one year apart to determine PLR’s. This method was developed by E. Hasselbrink et. al.[3]. The median of these yearly PLR’s is identified as the total system PLR. This method avoids issues with outliers and seasonality that the previous method encounters, at the cost of needing long-term data in order to be meaningful.

# example YoY regression
xbx_wbw_yoy_plr <- plr_yoy_regression(test_xbx_wbw_res_no_outliers, power_var = 'power_var', time_var = 'time_var', model = "xbx", per_year = 52, return_PLR = TRUE)

xbx_wbw_yoy_plr
#>          plr   plr_sd model       method
#> 1 -0.7995984 1.992959   xbx year-on-year

Note that the example data included in this package is not well-suited at all to year-on-year regression - it is a randomly taken 1% sample of a larger data file, so few points are exactly one year apart from each other.

Bootstrap Uncertainty

The following method samples from the data before putting it through a power predictive model and PLR Regressions. It is written so that it can be used flexibly with data that may have been put through other power predictive models, so time_var and power_var parameters are requested. Within this package, those will always be ‘time_var’ and ‘.fitted’, respectively. If desired, predicted data and nameplate power for the 6k model can be passed as well.

# samples data before applying PLR regression
xbx_wbw_plr_uncertainty <- plr_bootstrap_uncertainty(test_dfc, n = 2, fraction = 0.65, by = 'week', power_var = 'power_var', time_var = 'time_var', var_list = var_list, model = "xbx", data_cutoff = 10, np = NA, pred = NULL)

knitr::kable(xbx_wbw_plr_uncertainty, caption = "XbX Week-by-Week Bootstrapped Uncertainty")

Note that the number of samples, n, is set rather low here. That is so the vignette can be processed by R more quickly; in practical use, it is best to set n to much higher values, e.g. 1000. Increasing the sample count makes the mean and standard error more precise and meaningful.

A helper function is included to resample data from each individual time segment, either days, weeks, or months. Resampling from the enitre dataset would bias certain time segments with more or less data than others.

dfc_resampled <- mbm_resample(test_dfc, fraction = 0.65, by = "week")

Bootstrap Output

An alternative method, plr_bootstrap_output, first puts the data through both PLR regression schemes, then bootstraps from the output.

xbx_wbw_plr_output_uncertainty <- plr_bootstrap_output(test_dfc, var_list, model = "xbx", fraction = 0.65, n = 10, power_var = 'power_var', time_var = 'time_var', ref_irrad = 900, irrad_range = 10, by = "week", np = NA, pred = NULL)

knitr::kable(xbx_wbw_plr_output_uncertainty, caption = "XbX Week-by-Week Bootstrapped Output Uncertainty")

This method will typically give more meaningful results for Year-on-Year regression: that method can be biased by random sampling prior to regression, since it relies on data points exactly a year apart.

Bootstrap after Modeling

The methods above both incorporate power prediction, but sometimes it is preferred to bootstrap data that has already been through power prediction. In this case, make use of plr_bootstrap_output_from_results.K

xbx_wbw_plr_result_uncertainty <- plr_bootstrap_output_from_results(test_xbx_wbw_res_no_outliers, power_var = 'power_var', time_var = 'time_var', weight_var = 'sigma', by = "week", model = 'xbx', fraction = 0.65, n = 10)

knitr::kable(xbx_wbw_plr_result_uncertainty, caption = "XbX Week-by-Week Bootstrapped Output Uncertainty From Results")

Note that here, defining power_var, time_var, and weight_var takes on extra importance since the data has already been through power prediction. If you’re unsure, examine the colnames of your data. The model parameter should be a string, and is only passed through to the result for the purposes of consistency; it does not impact the function at all.