Project2_Solution.Rmd

---
title: "Wind Power Project"
author: "By Elliot Cohen, Mike Waite, Joe Woo and Vijay Modi"
date: "October 20, 2014"
output: html_document
---

# Part 1: An exploration of wind energy data
Suppose you are hired as a consultant to evaluate the potential for wind energy production on a small chain of islands in the Atlantic Ocean. The islands already have one wind farm up and running, with 7 Vestas wind turbines (v52-850Kw) rated at 850kW each. The combined rated capacity of the 7 turbines is `r 850*7` kW, which is roughly `r round((850*7/7926.5)*100, digits=1)`% of the annual average power demand for the island! This is a huge penetration of wind! Pairwise observations of [10-minute average wind speed and cummulative wind energy production](https://github.com/Ecohen4/ECREEE/blob/master/WIND_VXE_2013_ORIGINAL.xls) data are provided by the utility for the most recent year, in .xls format (click "view raw" to download). [Hourly demand](https://github.com/Ecohen4/ECREEE/blob/master/VXE_Estimated_Hourly_Load.csv) was estimated in-house from the load profile of a nearby island, and is provided in .csv format.

Using this data, prepare the following material for an investor meeting coming up next week. Be sure to document any assumptions, equations or algorithims implimented, and include all underlying R code in a well-commented [Rmarkdown](http://rmarkdown.rstudio.com/) file. [Knit](http://yihui.name/knitr/) the .rmd into a clean .html or .pdf file for review.

1. Visually inspect the data! 
    + Does it make sense? 
    + Are there any obvious issues, such as missing data, incomplete records or suspect values?
2. Compute summary statistics to make sure the values make sense (use benchmark comparisons!).
3. Clean the data. There are two schools of thought on this:
    + Option 1: Remove all records containing suspect/questionable data. For example:
        + Can you have negative energy values?
        + Can you produce 400kw at 0 windspeed? 
        + If the wind and/or power data at a given timestamp are not reasonable on their own *or* do not make sense together as a pairwise observation, omit them.
        + Use textbook knowledge of wind energy systems as one check (hint: Betz Limit!)
        + Keep track of how many observations you remove!
    + Option 2: Data correction, using textbook knowledge (hint: turbine power curve!).
    + After implementing step 3 (option 1 or 2), you should have a "clean" dataset. Now make engineering computations with confidence! Proceed to steps 4-9 (and 10-11 if you're extra ambitious!)
4. Total wind energy produced last year (e.g. KWh delivered to the grid).
5. Capacity factor of the current system.
6. Total wind energy that was *possible* given the observed windspeeds and the [turbine power curve](https://github.com/Ecohen4/ECREEE/blob/master/v52-850KW-power-curve.csv). This is the uncurtailed power output.
7. Uncurtailed capacity factor using the result from 6 (this assumes the grid can accept the full uncurtailed power output). 
8. Turbine efficiency (hint: you will have to compute the kinetic energy contained in the wind!):
    + average efficiency for the entire windfarm; and
    + efficiency as a function of windspeed.
    + compare with Betz Limit.
9. Characterization of the wind resource using a Weibull distribution.  
10. **BONUS** Randomly generate a year's worth of new windspeed data according to the fitted Weibull distrubution. Using the new windspeeds, predict what the resulting wind energy production may look like next year.  
11. **BONUS** Repeat step 10 500 times and compute the capacity factor each time. Boxplot the results of the 500 simulations. This is called an ensemble forecast.  

****************
## First Things First
```{r global.options, include=FALSE}
# load libraries
## The following function will load the packages required for this tutorial.  If a package cannot be found in your instance of Rstudio, it will automatically be insalled.
## require() returns (invisibly) a logical indicating whether the required package is available.

# if (!require(rgdal)) install.packages('rgdal')
load_install<-function(lib){
  if(! require(lib, character.only=TRUE)) install.packages(lib) 
  library(lib, character.only=TRUE, quietly=TRUE)
}

## the required libraries (e.g. packages)
Thelib<-c("knitr", "rmarkdown", "plyr", "ggplot2", "scales", "reshape2", "stats", "fitdistrplus", "lubridate")

## apply the function
lapply(Thelib, load_install)

# set global options for html output
opts_chunk$set(fig.path="figs/", echo=TRUE, cache=TRUE, tidy=TRUE, fig.align="center")

# avoid using factors until you become more familiar with object classes
options(stringsAsFactors=FALSE)
```

## Data Cleaning

### Import data from .csv
```{r read.data, cache=TRUE}
setwd("~/github/ECREEE")
data.raw<-read.csv(file="WIND_VXE_2013_original.csv", header=TRUE)
```

### Assign succint, descriptive column names
```{r column.names, results='hide'}
# assign new column names to be more succint (avoid spaces, use _ or . to seperate words)
new.names<-c("date_time","T1_Possible_Power","T2_Possible_Power","T3_Possible_Power","T4_Possible_Power","T5_Possible_Power","T6_Possible_Power","T7_Possible_Power","T1_Total_Active_Power","T2_Total_Active_Power","T3_Total_Active_Power","T4_Total_Active_Power","T5_Total_Active_Power","T6_Total_Active_Power","T7_Total_Active_Power","mean_wind_mps", "min_wind_mps", "max_wind_mps", "cum_energy_delivered_kwh")

# make sure the new names lineup properly with the ones they're replacing...
cbind(names(data.raw), new.names)

# if yes, replace column names with new names
names(data.raw)<-new.names

# Let's create a data frame that we'll work on as we go, so that we can always come back and reference the raw data if we'd like to
data <- data.raw
```

### Format timestamp
```{r}
# Use "UTC" to avoid daylight savings
data$date_time <- as.POSIXct(data$date_time,format="%m/%d/%y %H:%M",tz="UTC")
```

### Energy Delivered
We know our energy data is coming in as a cumulative meter. We can check this with a time series plot.
```{r echo=FALSE}
plot(data$date_time[-is.na(data$cum_energy_delivered_kwh)],data$cum_energy_delivered_kwh[-is.na(data$cum_energy_delivered_kwh)],xlab="",ylab="Energy Meter (kWh)")
```

```{r}
# Calculate energy delivered at each time step
data$energy_delivered_kwh <- c(data$cum_energy_delivered_kwh[2:nrow(data)]-data$cum_energy_delivered_kwh[1:(nrow(data)-1)],0)
# We know negative values with this approach will be the actual value minus 10^7
data$energy_delivered_kwh <- ifelse(data$energy_delivered_kwh < 0,data$energy_delivered_kwh+10^7,data$energy_delivered_kwh)
```

### Check for missing values
We can check for NAs and other things we don't particularly like as we go, but to make it quicker later, let's write a custom function that easily conveys information regarding missing values.  It wraps several useful checks into a single function:
```{r na.check}
check<-function(df){
  # count NA (missing values)
  NAs<-sum(is.na(df))
  print(paste("Missing Values:", NAs))
  
  # count incomplete records (rows containing missing values)
  ok<-complete.cases(df)
  print(paste("Incomplete Records:", sum(! ok)))
  
  # Show incomplete records (if less than 100 NAs). 
  if(NAs > 0 & NAs <= 100) print( df[which(! complete.cases(df)), ] )
  
  # If more than 100, show column-wise distribution of NAs.
  if (NAs > 100) hist(which(is.na(df), arr.ind=TRUE)[,2], xlab="Column", freq=TRUE, breaks=1:dim(df)[2], main="Column-wise distribution of missing values")
  }
```

Similarly, we can write another quick function to show how many records are removed in any subset operation:
```{r omit.count}
removed<-function(nrow, nrow1){
  print(paste("number of records REMOVED:", nrow-nrow1, sep=" "))
  print(paste("number of records REMAINING:", nrow1, sep=" "))
}
```

Now let's take our `check` function for a test drive!

```{r na.check.1}
check(data)  # NAs present. Column-wise distribution is relatively uniform (besides wind with relatively few)
summary(data)
```

### Remove missing values
Apply the `na.omit` function {stats package} to create a new dataframe called `dat` with all NAs removed.  
Apply the `removed` function {user defined} to compare the two data frames.
```{r na.omit}
nrow.orig<-nrow(data)[1] # record the dimensions of the data (before removing anything!)
data<-na.omit(data) # omit rows containing missing values
nrow.new<-dim(data)[1] # record the new dimensions of the data (after removing NAs)
removed(nrow.orig, nrow.new) # check how many records have been removed
summary(data)
```

### Pairwise observations: Power vs Windspeed
Now let's check if the windspeed and energy measurements make sense together (e.g. pairwise observations).  
Recall this is timeseries data, so at every timestamp there is a windspeed and energy measurement.

```{r benchmark.plot, fig.width=8, fig.height=5, warning=FALSE}
# Plot the wind farm power curve with benchmark comparisons
ggplot(data, aes(x=mean_wind_mps), colour=variable) + 
  geom_point(aes(y=energy_delivered_kwh, color="Energy Delivered")) +
  scale_y_continuous(name="Energy Delivered (KWh in 10min intervals)", limit=c(0, max(data$energy_delivered_kwh))) + 
  scale_x_continuous(name="Windspeed (mps)", limit=c(0, max(data$mean_wind_mps)))  +
    labs(title="Empirical Power Curve") + 
    theme_classic() +
    theme(legend.position="")
```

### Overlay Energy Benchmarks: Kinetic energy in the wind, Betz limit and Turbine Power Curve

``` {r}
# compute kinetic energy in the wind at each windspeed
# Wind Power = (1/2)*rho*area*(velocity)^3 = [kg/m^3]*[m^2]*[m/s]^3 = [kg*m^2/s^3] = [kg*m^2/s^2][1/s] = [Newton-meter]/[second] = [Joules/second] = [Watts]
turbine.rating <- 850
rho <- 1.225 # density of wind (kg/m^3)
area <- 2124 # sweep area of wind turbines (m^2)
n.turbines <- 7 # number of turbines
data$wind_power_kw<-(1/2)*rho*area*(data$mean_wind_mps)^3*n.turbines/1000 # kW avg power
data$wind_energy_10min_kwh<-data$wind_power_kw*(10/60) # kWh in 10 min

# compute betz limit
betz.coef<- 16/27
data$betz_limit_10min_kwh<-data$wind_energy_10min_kwh*betz.coef

# Turbine Power Curve
power.curve <- read.csv("v52-850KW-power-curve.csv")
# Keep power curve only for wind speeds below cut out speed (25 m/s)
power.curve <- power.curve[which(power.curve$windspeed_mps<=25),]
power.curve$energy_10min_kwh <- power.curve$power_kW*(10/60)*n.turbines
splinefun.power.curve <- splinefun(power.curve$windspeed_mps,power.curve$energy_10min_kwh)
data$power_curve_10min_kwh <- splinefun.power.curve(data$mean_wind_mps)
# Enrsure now power curve values are below zero or above maximum
data$power_curve_10min_kwh[which(data$power_curve_10min_kwh<0)] <- 0
data$power_curve_10min_kwh[which(data$power_curve_10min_kwh>turbine.rating*(10/60)*n.turbines)] <- turbine.rating*(10/60)*n.turbines
```

``` {r echo=FALSE}
# Add benchmarks to empirical power curve plot
ggplot(data, aes(x=mean_wind_mps)) + 
  geom_point(aes(y=energy_delivered_kwh, color="Energy Delivered")) +
    geom_line(aes(y=wind_energy_10min_kwh, color="Total Wind Energy")) +
    geom_line(aes(y=betz_limit_10min_kwh, color="Betz Limit")) +
    geom_line(aes(y=power_curve_10min_kwh, color="Power Curve")) +
  scale_y_continuous(name="Energy Delivered (KWh in 10min intervals)", limit=c(0, max(data$energy_delivered_kwh))) + 
  scale_x_continuous(name="Windspeed (mps)", limit=c(0, max(data$mean_wind_mps)))  +
    labs(title="Empirical Power Curve") + 
    theme_classic() +
    theme(legend.position="bottom",legend.title=element_blank())
```

***********
## 3. Clean the data
Assume the energy delivered measurements are correct, and that all the error is contained in the windspeed measurements.  

### Option 1: Remove all records containing suspect/questionable data

```{r}
# Remove all data points to the left of the power curve
data.opt1 <- data[which(data$energy_delivered_kwh<=data$power_curve_10min_kwh),]
nrow.orig <- nrow(data)
nrow.new <- nrow(data.opt1)
removed(nrow.orig, nrow.new) # check how many records have been removed (a lot)
```

``` {r echo=FALSE}
# Plot data.opt1 and power curve
ggplot(data.opt1, aes(x=mean_wind_mps)) + 
  geom_point(aes(y=energy_delivered_kwh, color="Energy Delivered")) +
    geom_line(aes(y=power_curve_10min_kwh, color="Power Curve")) +
  scale_y_continuous(name="Energy Delivered (KWh in 10min intervals)", limit=c(0, max(data.opt1$energy_delivered_kwh))) + 
  scale_x_continuous(name="Windspeed (mps)", limit=c(0, max(data.opt1$mean_wind_mps)))  +
    labs(title="Empirical Power Curve - Data Cleaning Option 1") + 
    theme_classic() +
    theme(legend.position="bottom",legend.title=element_blank())
```

### Option 2: Move wind speeds to the left of the power curve to the power curve values
```{r}
data.opt2 <- data
# Sort power curve in ascending order of electricity generated
power.curve.sorted <- power.curve[order(power.curve$energy_10min_kwh),]
# Calculate the "projected" wind speed from the power curve for each entry's energy delivered
data.opt2$wind_mps_projected <- power.curve.sorted$windspeed_mps[findInterval(x=data.opt2$energy_delivered_kwh, vec=power.curve.sorted$energy_10min_kwh, all.inside=TRUE, rightmost.closed=TRUE)]
# Choose the greater of the wind speed data value and the projected wind speed as the corrected wind speed
# Do not move values off 0 kWh
data.opt2$wind_mps_corrected <- ifelse(data.opt2$energy_delivered_kwh==0,data.opt2$mean_wind_mps,pmax(data.opt2$mean_wind_mps,data.opt2$wind_mps_projected))
# Recalculate power curve values corresponding to corrected wind speeds in data frame
data.opt2$corrected_power_curve_10min_kwh <- splinefun.power.curve(data.opt2$wind_mps_corrected)
data.opt2$corrected_power_curve_10min_kwh[which(data.opt2$corrected_power_curve_10min_kwh<0)] <- 0
# Recalculate total wind energy
data.opt2$corrected_wind_energy_10min_kwh <- ((1/2)*rho*area*(data.opt2$wind_mps_corrected)^3*n.turbines/1000)*(10/60)
```

``` {r echo=FALSE}
# Plot data.opt2 and power curve
ggplot(data.opt2, aes(x=wind_mps_corrected)) + 
  geom_point(aes(y=energy_delivered_kwh, color="Energy Delivered")) +
    geom_line(aes(y=corrected_power_curve_10min_kwh, color="Power Curve")) +
  scale_y_continuous(name="Energy Delivered (KWh in 10min intervals)", limit=c(0, max(data.opt2$energy_delivered_kwh))) + 
  scale_x_continuous(name="Corrected Windspeed (mps)", limit=c(0, max(data.opt2$wind_mps_corrected)))  +
    labs(title="Empirical Power Curve - Data Cleaning Option 2") + 
    theme_classic() +
    theme(legend.position="bottom",legend.title=element_blank())
```

4. Total wind energy produced last year

```{r}
total.energy.kwh.raw <- data.raw$cum_energy_delivered_kwh[nrow(data.raw)] - data.raw$cum_energy_delivered_kwh[1] + 2*10^7
total.energy.kwh.clean <- sum(data$energy_delivered_kwh)
total.energy.kwh.opt1 <- sum(data.opt1$energy_delivered_kwh)
total.energy.kwh.opt2 <- sum(data.opt2$energy_delivered_kwh)
data.frame(total.energy.kwh.raw,total.energy.kwh.clean,total.energy.kwh.opt1,total.energy.kwh.opt2)
```

5. Capacity Factor

```{r}
cf.raw <- total.energy.kwh.raw/(nrow(data.raw)*(10/60)*turbine.rating*n.turbines)
cf.clean <- total.energy.kwh.clean/(nrow(data)*(10/60)*turbine.rating*n.turbines)
cf.opt1 <- total.energy.kwh.opt1/(nrow(data.opt1)*(10/60)*turbine.rating*n.turbines)
cf.opt2 <- total.energy.kwh.opt2/(nrow(data.opt2)*(10/60)*turbine.rating*n.turbines)
data.frame(cf.raw,cf.clean,cf.opt1,cf.opt2)
```

6. Total Possible Energy

```{r}
total.possible.kwh.clean <- sum(data$power_curve_10min_kwh)
total.possible.kwh.opt1 <- sum(data.opt1$power_curve_10min_kwh)
total.possible.kwh.opt2 <- sum(data.opt2$corrected_power_curve_10min_kwh)
data.frame(total.possible.kwh.clean,total.possible.kwh.opt1,total.possible.kwh.opt2)
```

7. Uncurtailed Capacity Factor

```{r}
cf.uncurtailed.clean <- total.possible.kwh.clean/(nrow(data)*(10/60)*turbine.rating*n.turbines)
cf.uncurtailed.opt1 <- total.possible.kwh.opt1/(nrow(data.opt1)*(10/60)*turbine.rating*n.turbines)
cf.uncurtailed.opt2 <- total.possible.kwh.opt2/(nrow(data.opt2)*(10/60)*turbine.rating*n.turbines)
data.frame(cf.uncurtailed.clean,cf.uncurtailed.opt1,cf.uncurtailed.opt2)
```

8. Turbine Efficiency

```{r}
data$turbine.eff <- data$energy_delivered_kwh/data$wind_energy_10min_kwh
data$turbine.eff[which(data$turbine_eff == "NaN")] <- 0
turbine.eff.clean <- total.energy.kwh.clean/sum(data$wind_energy_10min_kwh)
data.opt1$turbine.eff <- data.opt1$energy_delivered_kwh/data.opt1$wind_energy_10min_kwh
data.opt1$turbine.eff[which(data.opt1$turbine_eff == "NaN")] <- 0
turbine.eff.opt1 <- total.energy.kwh.opt1/sum(data.opt1$wind_energy_10min_kwh)
data.opt2$turbine.eff <- data.opt2$energy_delivered_kwh/data.opt2$corrected_wind_energy_10min_kwh
data.opt2$turbine.eff[which(data.opt2$turbine_eff == "NaN")] <- 0
turbine.eff.opt2 <- total.energy.kwh.opt2/sum(data.opt2$corrected_wind_energy_10min_kwh)
data.frame(turbine.eff.clean,turbine.eff.opt1,turbine.eff.opt2)
```

``` {r}
# Plot turbine efficiency vs. wind speed for uncorrected data
ggplot(data, aes(x=mean_wind_mps)) + 
  geom_point(aes(y=turbine.eff, color="Turbine Efficiency")) +
    geom_hline(aes(yintercept=betz.coef, color="Betz Limit")) +
  scale_y_continuous(name="Turbine Efficiency", limit=c(0, 1)) + 
  scale_x_continuous(name="Windspeed (mps)", limit=c(0, max(data$mean_wind_mps)))  +
    labs(title="Turbine Efficiency vs. Wind Speed - No Correction") + 
    theme_classic() +
    theme(legend.position="bottom",legend.title=element_blank())
```


``` {r}
# plot turbine efficiency vs wind speed for correction option 1
ggplot(data.opt1, aes(x=mean_wind_mps)) + 
  geom_point(aes(y=turbine.eff, color="Turbine Efficiency")) +
    geom_hline(aes(yintercept=betz.coef, color="Betz Limit")) +
  scale_y_continuous(name="Turbine Efficiency", limit=c(0, 0.6)) + 
  scale_x_continuous(name="Windspeed (mps)", limit=c(0, max(data.opt1$mean_wind_mps)))  +
    labs(title="Turbine Efficiency vs. Wind Speed - Correction Option 1") + 
    theme_classic() +
    theme(legend.position="bottom",legend.title=element_blank())
```

``` {r}
# plot turbine efficiency vs wind speed for correction option 2
ggplot(data.opt2, aes(x=wind_mps_corrected)) + 
  geom_point(aes(y=turbine.eff, color="Turbine Efficiency")) +
    geom_hline(aes(yintercept=betz.coef, color="Betz Limit")) +
  scale_y_continuous(name="Turbine Efficiency", limit=c(0, 0.6)) + 
  scale_x_continuous(name="Windspeed (mps)", limit=c(0, max(data.opt2$wind_mps_corrected)))  +
    labs(title="Turbine Efficiency vs. Wind Speed - Correction Option 2") + 
    theme_classic() +
    theme(legend.position="bottom",legend.title=element_blank())
```

9. Characterization of Weibull distribution

``` {r}
# Uncorrected data
# Remove rows with zero wind speed
data.weibull <- subset(data, mean_wind_mps >= 0.1)

# Weibull fit for uncorrected data
# Uses package "fitdistrplus"
weibull.fit<-fitdist(data.weibull$mean_wind_mps, distr="weibull")
summary(weibull.fit)
plot(weibull.fit, demp=TRUE)
```

``` {r}
# Correction Option 1
# Remove rows with zero wind speed
data.opt1.weibull <- subset(data.opt1, mean_wind_mps >= 0.1)

# Weibull fit for corrected data - Option 1
weibull.fit.opt1<-fitdist(data.opt1.weibull$mean_wind_mps, distr="weibull")
summary(weibull.fit.opt1)
plot(weibull.fit.opt1, demp=TRUE)
```

``` {r}
# Correction Option 2
# Remove rows with zero wind speed
data.opt2.weibull <- subset(data.opt2, wind_mps_corrected >= 0.1)

# Weibull fit for corrected data - Option 2
weibull.fit.opt2<-fitdist(data.opt2.weibull$wind_mps_corrected, distr="weibull")
summary(weibull.fit.opt2)
plot(weibull.fit.opt2, demp=TRUE)
```

# Part 2: Grid Integration of Wind Energy

We need to get the hourly wind-generated electricity supply from our Option 2 corrected data:

``` {r}
# Load in the date_time and corrected_wind_energy_10min_kwh from Option 2 for the wind-generated electricity supply
supply.orig <- data.opt2[,c("date_time","corrected_power_curve_10min_kwh")]
# Create columns for dates and hours
supply.orig$date <- as.Date(supply.orig$date_time)
supply.orig$hour <- hour(supply.orig$date_time)
# Get hourly wind-generated electricity
supply.hrly <- ddply(supply.orig, .(date,hour), .fun=summarise,wind.supply.kwh=sum(corrected_power_curve_10min_kwh))
# Create date.time column from "date" and "hour"
supply.hrly$date.time <- as.POSIXct(paste(as.character(supply.hrly$date),as.character(supply.hrly$hour),sep=" "),format="%Y-%m-%d %H",tz="UTC")
```

Create a data frame that will include the wind-generated electricity supply and the island demand:

``` {r}
# Initiate data frame as a sequence of date-times for each hour of the year
start.datetime <- as.POSIXct("2013-01-01 0:00", format="%Y-%m-%d %H:%M", tz="UTC")
end.datetime <- as.POSIXct("2013-12-31 23:00", format="%Y-%m-%d %H:%M", tz="UTC")
data.pt2 <- as.data.frame(seq(from=start.datetime, to=end.datetime, by=3600))
names(data.pt2) <- "date.time"
# Add month and hour columns, as this will be required later
data.pt2$month <- month(data.pt2$date.time)
data.pt2$hour <- hour(data.pt2$date.time)
# Use "match" function to add wind-generated electricity supply to the data frame. If no match, set value to zero (i.e. if we don't have a supply point, assume the supply is zero).
data.pt2$wind.supply.kwh <- supply.hrly$wind.supply.kwh[match(data.pt2$date.time,supply.hrly$date.time,nomatch=NA)]
data.pt2$wind.supply.kwh[is.na(data.pt2$wind.supply.kwh)] <- 0
# Read in demand data and add to data.pt2
demand.data <- read.csv("island_demand_data.csv")
data.pt2$demand.kwh <- demand.data$demand.kWh
```

Calculate the hourly curtailment and capacity factor:

``` {r}
data.pt2$wind.curtail.kwh <- ifelse(data.pt2$wind.supply.kwh<=data.pt2$demand.kwh,0,data.pt2$wind.supply.kwh-data.pt2$demand.kwh)
cf.nostorage <- (sum(data.pt2$wind.supply.kwh)-sum(data.pt2$wind.curtail.kwh))/(nrow(data.pt2)*turbine.rating*n.turbines)
```

Compute hourly curtailment by month and create table:

```{r}
# Create empty 12 column by 24 row data frame
hrly.curtail.kwh <- as.data.frame(matrix(0,nrow=24,ncol=12))
colnames(hrly.curtail.kwh) <- c(1:12)
rownames(hrly.curtail.kwh) <- c(1:24)
# Fill in data frame with average hourly curtailment values
for (i in 1:12)
{
    data.month <- data.pt2[which(data.pt2$month==i),]
    hrly.curtail.kwh[,i] <- ddply(data.month,.(hour),.fun=summarise,curtail.means=mean(wind.curtail.kwh))[,2]
}
hrly.curtail.kwh
```

Compute hourly wind-generated electricity production by month and create table:

```{r}
# Create empty 12 column by 24 row data frame
hrly.production.kwh <- as.data.frame(matrix(0,nrow=24,ncol=12))
colnames(hrly.production.kwh) <- c(1:12)
rownames(hrly.production.kwh) <- c(1:24)
# Fill in data frame with average hourly curtailment values
for (i in 1:12)
{
    data.month <- data.pt2[which(data.pt2$month==i),]
    hrly.production.kwh[,i] <- ddply(data.month,.(hour),.fun=summarise,production.means=mean(wind.supply.kwh))[,2]
}
hrly.production.kwh
```

# Pumped hydro model

``` {r}
# Pumped hydro storage capacity
ph.cap.kwh <- 5000
# Assume initial amount stored is zero
ph.stor.init.kwh <- 0
# Pump efficiency
eff.pump <- 0.85
# Turbine efficiency
eff.turb <- 0.90
# We'll need to develop a 'for' loop, so we'll add columns to our data frame for the amount of energy stored in the pumped hydro system at the start and end of each time step
data.pt2$ph.stor.start.kwh <- 0
data.pt2$ph.stor.end.kwh <- 0
# For loop to determine, at each hour, how much electricity is stored in the pumped hydro system and how much is drawn from the system.
for (h in 1:nrow(data.pt2))
    {
    ifelse(h==1, data.pt2$ph.stor.start.kwh[h] <- ph.stor.init.kwh, data.pt2$ph.stor.start.kwh[h] <- data.pt2$ph.stor.end.kwh[h-1])
    data.pt2$ph.stor.end.kwh[h] <- ifelse(data.pt2$wind.supply.kwh[h]>data.pt2$demand.kwh[h], min(data.pt2$ph.stor.start.kwh[h] + (data.pt2$wind.supply.kwh[h]-data.pt2$demand.kwh[h])*eff.pump, ph.cap.kwh), max(data.pt2$ph.stor.start.kwh[h] + (data.pt2$wind.supply.kwh[h]-data.pt2$demand.kwh[h])*eff.turb, 0))
        max(min(data.pt2$ph.stor.start.kwh[h] + (data.pt2$wind.supply.kwh[h]-data.pt2$demand.kwh[h]), ph.cap.kwh),0)
    data.pt2$wind.curtail.afterph.kwh[h] <- ifelse((data.pt2$ph.stor.end.kwh[h] - data.pt2$ph.stor.start.kwh[h]) > 0, data.pt2$wind.curtail.kwh[h] - (data.pt2$ph.stor.end.kwh[h] - data.pt2$ph.stor.start.kwh[h]), data.pt2$wind.curtail.kwh[h])
    }
# Calculate capacity factor with pumped hydro
cf.withph <- (sum(data.pt2$wind.supply.kwh)-sum(data.pt2$wind.curtail.afterph.kwh))/(nrow(data.pt2)*turbine.rating*n.turbines)
```

Compare capacity factors to see affects of the pumped hydro storage system:

``` {r}
# Capaity factor without storage
cf.nostorage
# Capacity factor with storage
cf.withph
# Absoluate increase in capacity factor
cf.withph - cf.nostorage
# Relative increase in capacity factor
(cf.withph - cf.nostorage)/cf.nostorage
```

Only a small difference is seen due to the low curtailment levels, due to demand generally being higher than wind-generated electricity supply.