You can check out the code in this article in the interactive Jupyter notebook here (Google Colab).
Introduction
A popular query in Google is about load-duration curves. Some of the questions are:
- What is a load-duration curve?
- What is the importance of a load-duration curve?
- How do you calculate a load-duration curve?
- What is the shape of a load-duration curve?
- How to plot a load-duration curve?
This article will address these questions from the perspective of the electric power industry, with which I am familiar. It will also show how to calculate and plot a load-duration curve using Python and its NumPy, Pandas, and Seaborn modules.
What is a Load-Duration Curve?
A load-duration curve is a plot of an electrical system’s load (demand), in units of power versus the percentage of the time for which this load or a load is greater than it existed. The load is plotted on the y-axis and the percentage of time on the x-axis. The load is not plotted in chronological order but in descending order of magnitude.
The Shape of a Load-Duration Curve
The highest load is on the left of the graph, with the load decreasing towards the right and the lowest load on the far right. The resulting curve, therefore, slopes downward from the maximum load to the minimum load and is generally shaped like a reverse sigmoid function ()).
The Importance of a Load-Duration Curve
Load-duration curves are plotted for daily, weekly, or annual periods and are used to conduct technical and economic analyses of an electrical system. It presents load information in an intuitive form and can be used to determine the maximum load and its duration.
They can be used to determine the duration of a specific load, that is, the amount of time that a load was equal to or greater than a particular value. Therefore, it can be used to determine the amount of time that various generating capacities must be available. The load is greater than or equal to the minimum load 100 percent of the time. This minimum load is referred to as the baseload. The capacity to meet this load must be always available. The area under the curve represents the total energy generated in this period.
How to Plot a Load-Duration Curve
Electric utilities usually record the system load at hourly intervals. This data can be used to develop the load duration for a selected period. The loads for each recording interval are sorted in descending order of magnitude and multiplied by the percentage of the period that the load was equal to or above this value. The load is then plotted on the y-axis and the percentage of time that it was equal to or above that value on the x-axis.
Plotting a Load-Duration Curve Using Python
The above procedure is illustrated using a Google Collaboratory Notebook, using Python and its NumPy, Matplotlib, Pandas, and Seaborn modules. This will be done using made-up data. For brevity, a period of one day will be used. However, the period could be a week, a month, or a year.
In the first cell, the required modules were loaded. The data are then loaded. These can be loaded from *.csv or Excel files or databases. For this illustration, a Python dictionary data structure was used. The loads and the hour at which they occurred were put in a dictionary as shown in the second cell below.
#load the required modules import numpy import matplotlib.pyplot as plt import pandas as pd import seaborn as sb
# initialise a dictionary with the load data.
load_data = {'Hour':[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23], 'Load':[39.0, 39.2, 37.6, 36.8, 36.4, 36.2, 37.4, 41.4, 45.6, 48.8, 53.0, 53.6, 53.2, 54.0, 53.8, 52.8, 50.8, 44.4, 46.8, 47.4, 46.4, 44.0, 42.4, 40.6]}
print(load_data)
In the next cell, the dictionary is converted to a Pandas DataFrame with the hour and load as columns. A graph of the load versus the hour is plotted using the lineplot() function of the Seaborn module. The load, in this case, is in chronological order and the graph is known as a (daily) load profile.
# Convert dictionary to DataFrame and print it load_df = pd.DataFrame(load_data) print(load_df)
# Plot the load profile
sb.set(rc={"figure.figsize":(10, 7)})
p = sb.lineplot(x = "Hour", y = "Load", data = load_df)
plt.ylim(0, None)
plt.xlim(0, None)
p.set_title("Load Profile", fontsize = 30)
p.set_xlabel("Time (Hrs)", fontsize = 20)
p.set_ylabel("Load (MW)", fontsize = 20)
plt.show()
The next step is to add a column with the time interval for which the loads were recorded. In this case, the loads were recorded in one-hour intervals.
# Add a column for the time interval for which the loads were recorded load_df['interval'] = 1 print(load_df)
The next step is to rearrange the loads from being in chronological order to one of descending order of magnitude. This is done by sorting the DataFrame by the load column using the sort_values() function with the ascending parameter set to False. The printout of the DataFrame confirms that the loads are now in descending order of magnitude.
# Sort the DataFrame by the loads, in descending order of magnitude load_df_sorted = load_df.sort_values(by=['Load'], ascending = False) print(load_df_sorted)
The next step is to add a column with the amount of time the system load was equal to or greater than the load in each row of the DataFrame. This is achieved by applying the cumsum() function to the interval column and assigning the results to a new column name duration. The cumsum() function calculates the cumulative sum over an axis of a Pandas DataFrame. In this case, it’s over the column axis. The printout verifies that the operation was successful.
# Use the cumsum() function to to add a column with the duration # for which the system load is greater than or equal to each load load_df_sorted['duration'] = load_df_sorted['interval'].cumsum() print(load_df_sorted)
The x-axis of the load duration curve is expressed as a percentage of the total time. So, the next step is to add a column with the duration expressed as a percentage of the total time. This is done in the next cell with the result printed out.
# Calculate the percentage of time for which # the system load is greater than or equal to each load load_df_sorted['percentage'] = load_df_sorted['duration']*100/24 print(load_df_sorted)
It’s now time to plot the load duration curve. As was done for the load profile, it was plotted using the lineplot() function of the Seaborn module. The load is on the y-axis and the percentage on the x-axis. The plot shows a curve that slopes downward from the maximum load to the minimum load and has some resemblance to a reverse sigmoid curve.
# Plot the load_duration curve (Load vs Percentage of time)
p = sb.lineplot(x = "percentage", y = "Load", data = load_df_sorted)
plt.ylim(0, None)
plt.xlim(0, None)
p.set_title("Load-Duration Curve", fontsize = 30)
p.set_xlabel("Time (%)", fontsize = 20)
p.set_ylabel("Load (MW)", fontsize = 20)
plt.show()
Closing Comments
Having read this article, you now know what a load duration curve is, its importance, how it is calculated, its shape, and how to plot it using python and its NumPy, Pandas, and Seaborn modules. I hope that this article is a help to you.
The full notebook with the code is available here, on GitHub.