AI-Based PV Power Forecasting

🎯 Lab Objectives

• Understand PV-power forecasting as a supervised-learning problem.
• Use solar irradiance and temperature as input features.
• Build a multiple linear-regression model.
• Interpret model parameters physically.
• Forecast PV output for new weather conditions.
• Evaluate predictions using MAE, MSE, and RMSE.
• Understand the limitations of a simple linear model.

1. PV Forecasting Problem

Photovoltaic power depends strongly on solar irradiance and is also affected by temperature.

where:

• G = solar irradiance [W/m²]
• T = temperature [°C]
• PPV = PV electrical power [kW]

2. Why Use Two Inputs?

3. Example Dataset

4. Enter Data in MATLAB

G = [200 300 400 500 600 700 800 900 1000 850]';

T = [15 18 20 22 25 27 30 32 35 38]';

Ppv = [18 27 36 44 52 60 67 74 81 68]';

5. Multiple Linear-Regression Model

We assume:

where:

• β₀ = intercept
• β₁ = irradiance coefficient
• β₂ = temperature coefficient

6. Matrix Form

with:

and:

7. Least-Squares Training

We minimize:

The normal equation is:

In MATLAB, use the more numerically stable backslash operator:

X = [ones(size(G)) G T];

beta = X \ Ppv;

8. Extract Model Parameters

beta0 = beta(1);
beta1 = beta(2);
beta2 = beta(3);

The learned model becomes:

9. Physical Interpretation of β₁

β₁ approximates the sensitivity of PV power to irradiance:

Its approximate unit is:

10. Physical Interpretation of β₂

β₂ approximates the effect of temperature:

If β₂ is negative, the model indicates that increasing temperature tends to reduce PV electrical output when irradiance is held fixed.

11. Prediction on Training Data

Ppred = X * beta;

The residual error is:

error = Ppv - Ppred;

12. Performance Indices

MAE = mean(abs(error));

MSE = mean(error.^2);

RMSE = sqrt(MSE);

13. Forecast a New Weather Condition

Assume:

Then:

Gnew = 750;
Tnew = 28;

Pforecast = beta0 + beta1*Gnew + beta2*Tnew;

fprintf('Forecast PV Power = %.2f kW\n',Pforecast);

14. Plot Actual and Predicted PV Power

figure

plot(Ppv,'o-','LineWidth',1.6)
hold on

plot(Ppred,'s-','LineWidth',1.6)

grid on

xlabel('Sample')
ylabel('PV Power [kW]')

legend('Actual PV Power','Predicted PV Power', ...
       'Location','best')

title('PV Power Forecasting')

15. Plot Prediction Error

figure

stem(error,'LineWidth',1.5)

grid on

xlabel('Sample')
ylabel('Prediction Error [kW]')

title('PV Forecasting Residuals')

16. Complete MATLAB Program

clear; clc; close all;

% ------------------------------------------------
% AI-Based PV Power Forecasting
% Multiple Linear Regression
% ------------------------------------------------

% Dataset
G = [200 300 400 500 600 700 800 900 1000 850]';

T = [15 18 20 22 25 27 30 32 35 38]';

Ppv = [18 27 36 44 52 60 67 74 81 68]';

% ------------------------------------------------
% Build regression matrix
% Ppv = beta0 + beta1*G + beta2*T
% ------------------------------------------------

X = [ones(size(G)) G T];

% Train model
beta = X \ Ppv;

beta0 = beta(1);
beta1 = beta(2);
beta2 = beta(3);

% ------------------------------------------------
% Prediction
% ------------------------------------------------

Ppred = X * beta;

error = Ppv - Ppred;

% ------------------------------------------------
% Performance
% ------------------------------------------------

MAE = mean(abs(error));

MSE = mean(error.^2);

RMSE = sqrt(MSE);

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

fprintf('Learned PV model:\n')
fprintf('Ppv = %.4f + %.6f*G + %.4f*T\n\n', ...
        beta0,beta1,beta2)

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

% ------------------------------------------------
% Forecast new weather condition
% ------------------------------------------------

Gnew = 750;
Tnew = 28;

Pforecast = beta0 + beta1*Gnew + beta2*Tnew;

fprintf('New irradiance = %.1f W/m^2\n',Gnew)
fprintf('New temperature = %.1f C\n',Tnew)
fprintf('Forecast PV power = %.2f kW\n',Pforecast)

% ------------------------------------------------
% Plot actual and predicted PV power
% ------------------------------------------------

figure

plot(Ppv,'o-','LineWidth',1.6)
hold on

plot(Ppred,'s-','LineWidth',1.6)

grid on

xlabel('Sample')
ylabel('PV Power [kW]')

legend('Actual PV Power','Predicted PV Power', ...
       'Location','best')

title('PV Power Forecasting')

% ------------------------------------------------
% Residual errors
% ------------------------------------------------

figure

stem(error,'LineWidth',1.5)

grid on

xlabel('Sample')
ylabel('Prediction Error [kW]')

title('PV Forecasting Residuals')

17. Why Multiple Regression?

If we used irradiance alone:

the model would ignore temperature effects.

Adding T allows the model to separate part of the variation caused by temperature:

18. Experiment 1 — Remove Temperature

Train a simpler model:

X1 = [ones(size(G)) G];

beta1D = X1 \ Ppv;

Ppred1D = X1 * beta1D;

RMSE1D = sqrt(mean((Ppv-Ppred1D).^2));

Compare RMSE with the two-input model.

19. Experiment 2 — Add Measurement Noise

PpvNoisy = Ppv + 2*randn(size(Ppv));

Train the model using the noisy measurements and compare the learned coefficients.

20. Experiment 3 — Train/Test Split

Gtrain = G(1:7);
Ttrain = T(1:7);
Ptrain = Ppv(1:7);

Gtest = G(8:10);
Ttest = T(8:10);
Ptest = Ppv(8:10);

Xtrain = [ones(size(Gtrain)) Gtrain Ttrain];

betaTT = Xtrain \ Ptrain;

Xtest = [ones(size(Gtest)) Gtest Ttest];

PtestPred = Xtest * betaTT;

RMSEtest = sqrt(mean((Ptest-PtestPred).^2));

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

21. From Linear Regression to Neural Networks

Real PV behavior may be nonlinear:

For example, interactions between irradiance and temperature may not be well represented by a simple linear equation.

22. Engineering Caution

A forecasting model should not be judged only by training accuracy.

• Weather conditions may change.
• Sensors may contain errors.
• PV modules may age.
• Shading patterns may change.
• New operating conditions may not exist in the training data.

23. Lab Tasks

1. Enter the PV dataset in MATLAB.
2. Construct matrix X.
3. Train the two-input regression model.
4. Record β₀, β₁, and β₂.
5. Explain the physical meaning of β₁ and β₂.
6. Calculate MAE, MSE, and RMSE.
7. Forecast PV power at G=750 W/m² and T=28°C.
8. Plot actual and predicted PV power.
9. Remove temperature and compare RMSE.
10. Add measurement noise and retrain.
11. Perform the train/test experiment.

📝 Review Questions

1. Why is irradiance important for PV forecasting?
2. Why can temperature influence PV electrical power?
3. What is the meaning of β₁?
4. What is the meaning of β₂?
5. Why do we use multiple regression?
6. What does RMSE represent?
7. Why can a model perform well on training data but poorly on test data?
8. When might a neural network outperform this linear model?