Wind Power

2023-07-22

Source: vignettes/windpower.Rmd

Wind speed extrapolation

Hellmann constant 

dat <- merra2_mar %>%
  select(-SWGDN,    -ALBEDO) %>% # drop solar radiation
  mutate(h = fH(W10M = W10M, W50M = W50M)) # Hellmann constant

summary(dat$h) # check basic statistics
#>    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
#> 0.00000 0.05406 0.08197 0.12715 0.16302 0.60000
MERRA-2 subset with estimated Hellmann exponent (h)
UTC locid W10M W50M WDIR T10M PRECTOTCORR RHOA h
2010-03-21 00:30:00 1 6.5 9.0 70 -56 0 1.09 0.2021963
2010-03-21 00:30:00 2 6.5 9.0 70 -56 0 1.09 0.2021963
2010-03-21 00:30:00 3 6.5 9.1 60 -56 0 1.09 0.2090620
2010-03-21 00:30:00 4 6.5 9.1 60 -56 0 1.09 0.2090620
2010-03-21 00:30:00 5 6.6 9.1 60 -56 0 1.09 0.1995757
2010-03-21 00:30:00 6 6.6 9.1 60 -56 0 1.09 0.1995757
Figure code 
(h_mean = mean(dat$h))
#> [1] 0.1271491
(h_median = median(dat$h))
#> [1] 0.08197119

# .GeomVline_draw_key <- GeomVline$draw_key
GeomVline$draw_key <- GeomHline$draw_key # adjusting legend

fig <- ggplot(dat) +
  geom_histogram(aes(h, after_stat(density), 
                     fill = "Hellmann", colour = "NA"),
                 alpha = .75, show.legend = F, bins = 50) +
  labs(x = "Hellmann exponent (h)") +
  geom_vline(aes(xintercept = x, linetype = name),
             data = tibble(name = c("mean", "median"),
                           x = c(h_mean, h_median)),
             colour = "blue") +
  scale_linetype(name = "Statistics") +
  theme_bw() + 
  theme(legend.position = c(0.9, .8),
        legend.box.background = element_rect(
          colour = "black", fill = "grey"
          ))
Distribution of Hellmann constant, estimated on MERRA-2 sample

Distribution of Hellmann constant, estimated on MERRA-2 sample

Extrapolation 

# extrapolation
for (i in seq(10, 200, 10)) {
  if (i == 10) cat("Height, m: ")
  cat(i)
  dat[[paste0("w", i)]] <- fWSE(i, dat$W10M, dat$h)
  cat(" ")
}
#> Height, m: 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 160 170 180 190 200

Figure below shows a sample of extrapolated speed of wind from 10 and 50 to up to 200 meters height.

Figure code 
dd <- dat %>% ungroup() %>%
  select(UTC, locid, starts_with("w", ignore.case = F)) %>%
  pivot_longer(cols = starts_with("w"), names_prefix = "w") %>%
  mutate(name = as.integer(name))

set.seed(0)
fig <- ggplot(filter(dd, locid %in% sample(locid, 100), UTC == dd$UTC[1])) +
  geom_line(aes(value, name, group = locid, colour = locid), 
            alpha = .75, show.legend = F, size = .75) +
  scale_colour_distiller(palette = "Spectral") +
  geom_hline(yintercept = 10, linetype = "dashed", colour = "blue", alpha = 1) + 
  geom_hline(yintercept = 50, linetype = "dashed", colour = "blue", alpha = 1) +
  # ylim(0, NA) +
  labs(x = "Wind speed, m/s", y = "Height, m") +
  scale_x_continuous(expand = c(0,0)) + 
  scale_y_continuous(expand = c(0,0), limits = c(0, NA), 
                     breaks = c(0, 10, 50, 100, 150, 200),
                     minor_breaks = c(0, 10, seq(25, 200, by = 25)),
                     labels = c(0, 10, 50, 100, 150, 200)) +
  geom_text(data = tibble(name = c("wind speed at 10m from MERRA-2 (W10M)", 
                                   "wind speed at 10m from MERRA-2 (W50M)"), 
                          x = c(19.2, 19.2), y = c(11, 51)),
                          size = 3, colour = "black", alpha = .85,
             aes(x, y, label = name), vjust = 0, hjust = 1) +
             # nudge_x = 0.25, nudge_y = 7) +
  theme_bw()
#> Warning: Using `size` aesthetic for lines was deprecated in ggplot2 3.4.0.
#> ℹ Please use `linewidth` instead.
#> This warning is displayed once every 8 hours.
#> Call `lifecycle::last_lifecycle_warnings()` to see where this warning was
#> generated.
Wind speed extrapolation using estimated Hellmann constant.

Wind speed extrapolation using estimated Hellmann constant.

Wind speed at 50m 

locid <- merra2ools::locid %>% select(locid, lon, lat) 
dat2 <- select(dat, UTC, locid, w10, w50, w100, w200) %>%
  full_join(locid)
#> Joining with `by = join_by(locid)`
Figure code
Wind speed at 50 meters height, m/s

Wind speed at 50 meters height, m/s

Wind power curve

Figure code 
# Simplified wind power curve
wpc <- tibble( 
  mps = c(0:25, NA, 25.01, 26:30), # wind speed m/s
  af = fWPC(mps) # availability factor [0,1]
)

fig <- ggplot(wpc) + 
  geom_line(aes(x = mps, y = af, colour = "red"), size = 2, show.legend = F) +
  labs(x = "Wind speed, m/s", 
       y = "Wind turbine availability (capacity) factor") +
  theme_bw()
#> Warning: Removed 1 row containing missing values (`geom_line()`).
Simplified wind power curve

Simplified wind power curve

Estimated hourly availabillity factor 


dat2 <- mutate(
  dat2,
  af10 = fWPC(w10),
  af50 = fWPC(w50),
  af100 = fWPC(w100),
  af200 = fWPC(w200)
)

summary(dat2$w50); summary(dat2$af50)
summary(dat2$af50[dat2$w50 > 25])

plot_merra(dat2, "w50", 1, map.border = NA)
plot_merra(dat2, "af50", 1, limits = c(0, 1), 
          palette = "Blues", direction = 1, legend.name = "AF")
plot_merra(dat2, "af10", 1, limits = c(0, 1),
          palette = "Blues", direction = 1, legend.name = "AF")
plot_merra(dat2, "af100", 1, limits = c(0, 1),
          palette = "Blues", direction = 1, legend.name = "AF")
plot_merra(dat2, "af200", 1, limits = c(0, 1), 
          palette = "Blues", direction = 1, legend.name = "AF")

gif_merra(dat2, "af50", 1, limits = c(0, 1), fps = 12, 
         palette = "PuBu", direction = 1,
         legend.name = "AF",
         filename = "merra_wind_af50_12fps.gif")
Hourly capacity factor for a generic wind turbine with 50 meters hub height.

Hourly capacity factor for a generic wind turbine with 50 meters hub height.