Power Predictive Model Comparison

Equation

\(P_{pred.} = \beta_{0} + \beta_{1} G + \beta_{2} T + \epsilon\)

This model uses a simplistic, linear equation relating power to irradiance (G) and temperature (T) with constant coefficients and an error term (\(\epsilon\)). It is this simplicity that enables its use for change point PLR, as well as the options to use POA or GHI irradiance and module or air temperature.

Implementation

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

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

Due to its data-driven nature, this method will typically show the strongest seasonality of those offered in this package. This is likely due to extrapolation done when predicting representative conditions affects seasonality in the data. Higher irradiance filters will reduce this seasonality, but also reduce the amount of data being used.

Determining UTC

Data is subset to a narrow irradiance range so that the effect of temperature on generated power can be studied in isolation. A linear regression is performed between temperature and power, giving the change in power per degree celsius. This is then converted to percent power loss per degree celcius (\(\gamma_{T}\)).

Equation

\(P_{cor} = \frac{P_{obs}}{1 + \gamma_{T} (T_{obs} - T_{rep}) (\frac{G_{obs}}{G_{rep}})}\)

\(P_{cor} = \beta_{0} + \beta_{1} G + \epsilon\)

Note the similarity in the second equation to that in the XbX method;

Implementation

test_xbxutc_wbw_res <- plr_xbx_utc_model(test_dfc, var_list, by = "week", data_cutoff = 30, predict_data = NULL, ref_irrad = 900, irrad_range = 10)

knitr::kable(test_xbxutc_wbw_res[1:5, ], caption = "XbX + UTC: Week-by-Week Implementation")

Equation

\(P = G_{POA} (\beta_{0} + \beta_{1} G + \beta_{2} T_{amb} + \beta_{3} W)\)

It is worth noting that the method requires POA irradiance and ambient temperature specifically - it lacks that flexibility found in the XbX method.

Implementation

test_pvusa_wbw_res <- plr_pvusa_model(test_dfc, var_list, by = "week", data_cutoff = 30, predict_data = NULL)

knitr::kable(test_pvusa_wbw_res[1:5, ], caption = "PVUSA: Week-by-Week Implementation")

See this paper as well for an in-depth look at uncertainties in the model.

Equation

\(G^{\prime} = G_{POA} / G_{STC}\)

\(T^{\prime} = T_{mod} / T_{STC}\)

\(P = G^{\prime} * (P_{NP} + k_1 \log(G^{\prime}) + k_2 \log(G^{\prime})^2 + k_3 T^{\prime} + k_4 T^{\prime} log(G^{\prime}) + k_5 T^{\prime} (G^{\prime})^2 + k_6 (T^{\prime})^2)\)

The \(G^{\prime}\) and \(T^{\prime}\) terms are defined for the purposes of temperature and irradiance correction.

Implementation

test_6k_wbw_res <- plr_6k_model(test_dfc, var_list, nameplate_power = 230, by = "week", data_cutoff = 30, predict_data = NULL)

knitr::kable(test_6k_wbw_res[1:5, ], caption = "6k: Week-by-Week Implementation")

Another important distinction between the 6k and other models is its linear regression scheme. While the others use standard least-squares linear regression through the base R ‘lm()’ function, the 6k implements the Levenberge-Marquardt method through minpack.lm (specifically, the ‘nlsLM()’ function). This causes several disparities. Firstly, the returned dataframe lacks .se.fit and sigma columns. Secondly, the model attempts to fit the data by adjusting its six constants, so the fit improves dramatically with a greater quantity of data.

The 6k model has the greatest uncertainty of any of the models as well as the most extreme outliers, and so it is recommended that data given to it is filtered as little as possible. In particular, an SDLE publication found that the lowest deviation was achieved when irradiance is filtered only below 0 watts per meter squared.

One final note on the 6k model: since it uses nameplate power in its fitting, if it is given standard conditions (STC) for predict_data, it will simply return the nameplate power value at all points in time. Be sure to either let predicted data be calculated by the model, or supply values specific to the actual location of the system.

# This code should create an error
test6k_dbd_res <- plr_6k_model(test_dfc, var_list, nameplate_power = 230, by = "day", data_cutoff = 30, predict_data = NULL)
#> Error in `dplyr::mutate()`:
#> ! Problem while computing `tidy = purrr::map(fit, tidy)`.
#> Caused by error in `chol2inv()`:
#> ! 'size' cannot exceed nrow(x) = 2