Machine Learning Load Forecasting

🎯 Lab Objectives

• Understand load forecasting as a supervised-learning problem.
• Construct a simple electrical-load dataset.
• Understand linear regression mathematically.
• Estimate model parameters in MATLAB.
• Predict electrical load from temperature.
• Calculate MAE, MSE, and RMSE.
• Visualize actual and predicted load.

1. Load Forecasting Problem

Electrical load forecasting estimates future power demand using historical measurements and explanatory variables.

In this first example, we intentionally use only one input:

2. Supervised Learning

For each training sample, we know both the input and the desired output.

In this laboratory:

3. Example Dataset

4. Enter the Data in MATLAB

temperature = [10 12 14 16 18 20 22 24 26 28]';

loadMW = [105 110 116 121 128 134 141 147 154 162]';

The apostrophe converts each vector into a column vector.

5. Linear Regression Model

Assume a linear relationship:

where:

• ŷ = predicted electrical load
• x = temperature
• β₀ = intercept
• β₁ = slope

6. Prediction Error

For sample i:

A good model should make these residual errors small.

7. Least-Squares Principle

Linear regression selects the parameters that minimize the sum of squared errors:

Therefore:

8. Matrix Form

Write the model for all samples:

with:

and:

9. Deriving the Optimal Parameters

The squared-error function is:

Differentiate with respect to β:

At the minimum:

therefore:

which gives the normal-equation solution:

10. MATLAB Implementation

X = [ones(size(temperature)) temperature];

beta = X \ loadMW;

MATLAB's backslash operator solves the least-squares problem directly and is generally preferable to explicitly computing a matrix inverse.

beta0 = beta(1);
beta1 = beta(2);

11. Make Predictions

Once β₀ and β₁ are known:

MATLAB:

predictedLoad = X * beta;

12. Mean Absolute Error — MAE

MAE = mean(abs(loadMW - predictedLoad));

MAE gives the average magnitude of the prediction error in MW.

13. Mean Squared Error — MSE

MSE = mean((loadMW - predictedLoad).^2);

Squaring makes larger errors more influential.

14. Root Mean Squared Error — RMSE

RMSE = sqrt(MSE);

15. Actual vs Predicted Load

figure

plot(temperature,loadMW,'o','LineWidth',1.6)
hold on

plot(temperature,predictedLoad,'-','LineWidth',1.8)

grid on

xlabel('Temperature [°C]')
ylabel('Electrical Load [MW]')

legend('Actual Load','Predicted Load','Location','best')

title('Electrical Load Forecasting')

16. Forecast a New Operating Point

Suppose tomorrow's expected temperature is:

The predicted load is:

Tnew = 30;

forecastLoad = beta0 + beta1*Tnew;

fprintf('Forecast at %.1f C = %.2f MW\n', ...
        Tnew,forecastLoad);

17. Complete MATLAB Program

clear; clc; close all;

% -----------------------------------------------
% Machine Learning Load Forecasting
% Linear Regression
% -----------------------------------------------

% Dataset
temperature = [10 12 14 16 18 20 22 24 26 28]';

loadMW = [105 110 116 121 128 134 141 147 154 162]';

% -----------------------------------------------
% Build regression matrix
% y = beta0 + beta1*x
% -----------------------------------------------

X = [ones(size(temperature)) temperature];

% Least-squares training
beta = X \ loadMW;

beta0 = beta(1);
beta1 = beta(2);

% -----------------------------------------------
% Prediction on training data
% -----------------------------------------------

predictedLoad = X * beta;

% Residual error
error = loadMW - predictedLoad;

% -----------------------------------------------
% Performance indices
% -----------------------------------------------

MAE = mean(abs(error));

MSE = mean(error.^2);

RMSE = sqrt(MSE);

% -----------------------------------------------
% Display model
% -----------------------------------------------

fprintf('Learned model:\n')
fprintf('Load = %.4f + %.4f * Temperature\n\n', ...
        beta0,beta1)

fprintf('MAE  = %.4f MW\n',MAE)
fprintf('MSE  = %.4f MW^2\n',MSE)
fprintf('RMSE = %.4f MW\n\n',RMSE)

% -----------------------------------------------
% Forecast new temperature
% -----------------------------------------------

Tnew = 30;

forecastLoad = beta0 + beta1*Tnew;

fprintf('Forecast for T = %.1f C: %.2f MW\n', ...
        Tnew,forecastLoad)

% -----------------------------------------------
% Plot actual and predicted load
% -----------------------------------------------

figure

plot(temperature,loadMW,'o','LineWidth',1.6)
hold on

plot(temperature,predictedLoad,'-','LineWidth',1.8)

grid on

xlabel('Temperature [°C]')
ylabel('Electrical Load [MW]')

legend('Actual Load','Predicted Load', ...
       'Location','best')

title('Machine Learning Load Forecasting')

% -----------------------------------------------
% Plot prediction errors
% -----------------------------------------------

figure

stem(temperature,error,'LineWidth',1.5)

grid on

xlabel('Temperature [°C]')
ylabel('Prediction Error [MW]')

title('Load Forecasting Residuals')

18. What Does β₁ Mean Physically?

Its approximate unit is:

For this educational dataset, a positive β₁ means that load tends to increase as temperature increases.

19. Training Data vs Test Data

Evaluating a model only on the same data used for training can give an overly optimistic impression.

20. Simple Train/Test Experiment

Use the first seven samples for training and the last three for testing:

xTrain = temperature(1:7);
yTrain = loadMW(1:7);

xTest = temperature(8:10);
yTest = loadMW(8:10);

Xtrain = [ones(size(xTrain)) xTrain];

betaTest = Xtrain \ yTrain;

Xtest = [ones(size(xTest)) xTest];

yPrediction = Xtest * betaTest;

testError = yTest - yPrediction;

RMSEtest = sqrt(mean(testError.^2));

fprintf('Test RMSE = %.4f MW\n',RMSEtest)

21. Extending the Model

Real load forecasting rarely depends on temperature alone.

A more realistic input vector may be:

The model becomes:

22. Experiment 1 — Add Measurement Noise

Replace the original load data by:

loadNoisy = loadMW + 3*randn(size(loadMW));

Train the model again and observe how the regression line and RMSE change.

23. Experiment 2 — Remove Training Samples

Train using only a small number of samples and compare the prediction.

For example:

temperatureSmall = temperature(1:4);
loadSmall = loadMW(1:4);

24. Experiment 3 — Nonlinear Load Behavior

Suppose the real relationship becomes nonlinear:

A straight line may no longer represent the relationship sufficiently well.

25. Lab Tasks

1. Enter the temperature and load dataset.
2. Plot the raw data.
3. Construct the regression matrix X.
4. Calculate β₀ and β₁.
5. Explain the physical meaning of β₁.
6. Calculate predicted load values.
7. Calculate MAE, MSE, and RMSE.
8. Forecast the load at 30°C.
9. Perform the train/test experiment.
10. Add measurement noise and repeat the training.
11. Explain when linear regression may become inadequate.

📝 Review Questions

1. Why is load forecasting a supervised-learning problem?
2. What do β₀ and β₁ represent?
3. Why do we minimize squared errors?
4. How is the least-squares solution obtained?
5. What is the difference between MAE, MSE, and RMSE?
6. Why should test data not be used for training?
7. What does a large test RMSE indicate?
8. Why can temperature alone be insufficient for realistic load forecasting?
9. When would a nonlinear model be preferable?