Feature Construction and Selection for PV Solar Power Modeling: A Machine Learning Framework

Table of Contents

1. Introduction & Overview

The integration of solar photovoltaic (PV) power into industrial processes is a key strategy for reducing greenhouse gas emissions and enhancing sustainability. However, the inherent intermittency and variability of solar energy pose significant challenges for grid stability and reliable energy supply. Accurate short-term prediction of PV power generation is therefore critical for effective energy management, load balancing, and operational planning.

This paper presents a novel machine learning framework for 1-hour ahead solar power prediction. The core innovation lies in its approach to feature engineering. Instead of relying solely on raw historical data and weather variables, the method constructs a higher-dimensional feature space using Chebyshev polynomials and trigonometric functions. A subsequent feature selection scheme coupled with constrained linear regression is then employed to build a robust and interpretable predictive model tailored to different weather types.

2. Methodology

2.1 Data and Input Features

The model utilizes a combination of temporal, meteorological, and autoregressive inputs:

• Meteorological Variables: Irradiance, temperature, dew point, humidity, wind speed.
• Weather Type Classification: Inputs are categorized based on prevailing weather conditions (e.g., clear, cloudy, rainy).
• Autoregressive Term: The solar power generation from the previous time step (e.g., 15 minutes prior) is included to capture temporal dependencies.

2.2 Feature Construction with Chebyshev Polynomials

The raw input features are transformed into a richer, higher-dimensional space. For a given input variable $x$, Chebyshev polynomials of the first kind, $T_n(x)$, are used. These polynomials are defined by the recurrence relation:

$T_0(x) = 1$

$T_1(x) = x$

$T_{n+1}(x) = 2xT_n(x) - T_{n-1}(x)$

Features are constructed as $T_n(x)$ for $n$ up to a specified order, and may also include cross-terms (e.g., $T_i(x) \cdot T_j(y)$) and trigonometric functions (e.g., $\sin(\omega t)$, $\cos(\omega t)$) to capture periodic patterns.

2.3 Feature Selection Scheme

A wrapper method is employed to select the most relevant features from the expanded set. This process is performed separately for each weather type to account for the varying influence of factors under different conditions. The selection aims to balance model complexity and predictive power, avoiding overfitting.

2.4 Constrained Linear Regression Model

After feature selection, a linear regression model is built: $\hat{y} = \mathbf{w}^T \mathbf{x} + b$, where $\mathbf{x}$ is the vector of selected features. To enhance physical plausibility and stability, the regression is formulated as a constrained least squares problem. Constraints may include non-negativity on certain coefficients (e.g., irradiance should have a non-negative impact on power output) or bounds on coefficient magnitudes.

3. Experimental Results & Performance

3.1 Experimental Setup

The proposed framework was tested on historical PV plant data. The dataset was split into training and testing sets, with performance evaluated using Mean Squared Error (MSE) and potentially other metrics like Mean Absolute Error (MAE).

3.2 Comparison with Baseline Models

The paper compares its method against several established machine learning benchmarks:

• Support Vector Machine (SVM)/Support Vector Regression (SVR)
• Random Forest (RF)
• Gradient Boosting Decision Tree (GBDT)

Key Finding: The proposed Chebyshev polynomial-based regression model with feature selection achieved a lower MSE than all the compared classical methods.

3.3 Performance Across Weather Conditions

The weather-type-specific modeling approach likely showed superior adaptability. For instance, under highly variable cloudy conditions, the model's selected features (perhaps higher-order polynomial terms capturing non-linear irradiance effects) would differ from those selected for stable clear-sky conditions, leading to more accurate predictions across the board.

4. Technical Details & Mathematical Formulation

The core optimization problem can be summarized as:

1. Feature Expansion: Create an expanded feature vector $\mathbf{\Phi}(\mathbf{z}) = [T_0(z_1), T_1(z_1), ..., T_n(z_m), \text{ cross-terms}, \text{ trig terms}]$ from the original input vector $\mathbf{z}$.
2. Feature Selection: Find a subset $\mathbf{x} \subset \mathbf{\Phi}(\mathbf{z})$ that minimizes prediction error for a specific weather type $k$.
3. Constrained Regression: Solve for weights $\mathbf{w}$:
   $\min_{\mathbf{w}} ||\mathbf{y} - \mathbf{X}\mathbf{w}||^2_2$
   subject to: $\mathbf{A}\mathbf{w} \leq \mathbf{b}$ (linear inequality constraints, e.g., $w_i \geq 0$).

5. Analysis Framework: A Non-Code Example

Consider a simplified scenario for predicting power at noon on a partly cloudy day. The raw inputs are: Irradiance ($I=600 W/m^2$), Temperature ($T=25^\circ C$), and previous power ($P_{t-1}=300 kW$).

1. Feature Construction: For irradiance $I$, generate Chebyshev terms up to order 2: $T_0(I)=1$, $T_1(I)=600$, $T_2(I)=2*600*600 - 1 = 719,999$. Similar expansions are done for $T$ and $P_{t-1}$. Cross-terms like $T_1(I)*T_1(T)$ are also created.
2. Feature Selection (for "Partly Cloudy" model): The selection algorithm might retain $T_1(I)$ (linear irradiance), $T_2(I)$ (capturing a non-linear saturation effect), $T_1(T)$, and $P_{t-1}$, while discarding many other constructed features as irrelevant for this weather type.
3. Prediction: The final prediction is a linear combination: $\hat{P} = w_1*600 + w_2*719,999 + w_3*25 + w_4*300 + b$, where $w_1, w_2 \geq 0$ due to constraints.

6. Core Insight & Analyst's Perspective

Core Insight: This paper's real breakthrough isn't a new black-box algorithm, but a disciplined, physics-aware feature engineering pipeline. It recognizes that the relationship between weather and PV output isn't merely linear or easily captured by standard decision trees. By explicitly constructing a basis space (Chebyshev polynomials) known for excellent function approximation properties and then applying sparsity-inducing selection, the method builds interpretable, high-performance models tailored to specific operational regimes (weather types). This is a smarter use of ML than brute-force application of deep learning, especially in data-limited industrial settings.

Logical Flow: The logic is sound: 1) Acknowledge problem complexity (non-linear, weather-dependent). 2) Systematically expand input space to represent potential complex relationships. 3) Prune back aggressively with domain-informed (weather-typed) selection to avoid overfitting. 4) Apply simple, constrained linear models on the refined features for stability and insight. This pipeline mirrors best practices in modern ML, reminiscent of the philosophy behind basis expansion in generalized additive models or feature learning in structured domains.

Strengths & Flaws:
Strengths: The approach is interpretable—you can see which polynomial terms matter for which weather. It's computationally lighter than training massive ensembles or neural nets for each weather type. The constraints enforce physical realism, a step often missing in pure data-driven models. Outperforming RF and GBDT on its own dataset is a strong result, as these are powerful benchmarks.
Flaws: The major limitation is reliance on accurate, real-time weather typing, which is itself a prediction problem. The method may struggle with rapidly evolving or mixed weather conditions not cleanly captured in the training categories. Furthermore, while better than benchmarks here, the ultimate performance ceiling of a linear model on selected features may be lower than a perfectly tuned, ultra-complex model for very large datasets, as seen in domains like computer vision where models like CycleGAN (Zhu et al., 2017) thrive on raw pixel data without manual feature construction.

Actionable Insights: For industry practitioners, the takeaway is clear: Invest in feature engineering before model complexity. Before deploying a neural network, try a systematic expansion of your inputs with orthogonal polynomials or Fourier terms. Implement weather- or regime-specific models. Always consider adding simple constraints to align models with domain knowledge. For researchers, the next step is to hybridize this approach: use automated feature construction/selection as an input processor to more advanced models (e.g., the selected features become inputs to a recurrent neural network for sequence modeling), or integrate the weather classification step directly into an end-to-end learning framework.

7. Future Applications & Research Directions

• Integration with Deep Learning: The feature construction layer could be integrated as a custom layer in a neural network, allowing the model to learn the optimal combination of basis functions.
• Probabilistic Forecasting: Extend the constrained regression framework to produce prediction intervals, crucial for risk-aware grid management. Techniques like Gaussian Process regression with custom kernels inspired by Chebyshev polynomials could be explored.
• Transfer Learning Across Sites: Investigate if the feature selection patterns (which polynomials are important for "cloudy" weather) are transferable between different geographic locations with similar climates, reducing data needs for new PV installations.
• Real-time Adaptive Selection: Develop online learning versions of the algorithm that can adapt the feature set dynamically as weather patterns shift, moving beyond static weather-type buckets.
• Broader Energy Applications: Apply the same feature construction/selection philosophy to other intermittent renewable forecasts, like wind power, or to related problems like building energy load forecasting.

8. References

1. Yang, Y., Mao, J., Nguyen, R., Tohmeh, A., & Yeh, H. (Year). Feature Construction and Selection for PV Solar Power Modeling. Journal/Conference Name.
2. Zhu, J., Park, T., Isola, P., & Efros, A. A. (2017). Unpaired Image-to-Image Translation using Cycle-Consistent Adversarial Networks. Proceedings of the IEEE International Conference on Computer Vision (ICCV).
3. International Energy Agency (IEA). (2023). Renewables 2023: Analysis and forecast to 2028. IEA Publications. [External Source on renewable energy growth]
4. Mason, K., & Ghanem, R. (2021). Statistical Learning for Renewable Energy Forecasting. Wiley.
5. National Renewable Energy Laboratory (NREL). (n.d.). Solar Forecasting. Retrieved from https://www.nrel.gov/grid/solar-forecasting.html [Authoritative external source on solar forecasting research]