The maximum electrical power output from a hydroelectric plant is defined by:

\[P = \eta \cdot \rho \cdot g \cdot Q \cdot H\]

To model power generation, we generate a random dataset based on flow rate (\(Q\)) and hydraulic head (\(H\)):

set.seed(42)
n <- 150

Q <- runif(n, min = 20, max = 100)  # Flow rate (m^3/s)
H <- runif(n, min = 10, max = 50)   # Hydraulic Head (m)

eta_rho_g <- 0.85 * 1000 * 9.81 / 1e6
noise <- rnorm(n, mean = 0, sd = 0.5)

Power_MW <- (eta_rho_g * Q * H) + noise


hydro_data <- data.frame(
  Flow_Rate = Q,
  Head_Height = H,
  Power_Output = Power_MW
)

head(hydro_data, 3)

##   Flow_Rate Head_Height Power_Output
## 1  93.18448    38.77514     30.10867
## 2  94.96603    22.96344     17.40838
## 3  42.89116    41.15238     15.30165

This graph is specifically showing the residual power output against the fitted to show that it is in a health balance at around the zero fitted line

We formulate the hydropower system as a Multiple Linear Regression model:

\[Y_i = \beta_0 + \beta_1 X_{1i} + \beta_2 X_{2i} + \varepsilon_i, \quad \varepsilon_i \overset{\text{iid}}{\sim} N(0, \sigma^2)\]

Where \(Y_i\) is Power Output (\(\text{MW}\)), \(X_{1i}\) is Flow Rate (\(Q\)), and \(X_{2i}\) is Hydraulic Head (\(H\)).

This report is to show the general guidelines of power output of hydro power as a function of its parts in graphical formats. This is to show how data can be collected and previewed to report on numourous things.