In a recent 10 MW distributed photovoltaic project at an industrial park, I observed that the traditional fixed installation of solar panels could not effectively utilize the rooftop space, leading to a generation output far below the expected target. The local solar radiation exhibits significant seasonal fluctuations: the summer monthly average total radiation is substantially higher than that in winter. Fixed installations with a uniform tilt angle throughout the year fail to account for the time-varying nature of radiation, causing increased reflection losses during low-radiation seasons and limiting overall energy capture efficiency. To address these challenges, I propose a dynamic optimization algorithm that leverages the spatiotemporal distribution of solar radiation, coupled with a three-dimensional (3D) radiation reception model that incorporates inter-layer shading effects. By integrating neural network predictions with particle swarm optimization (PSO), this method dynamically adjusts the azimuth and tilt sequences of double-layer solar panels to maximize the net revenue over the entire system life cycle. Simulation results show that this approach increases annual power generation by 18.3%, enhances effective radiation intensity by 18.6%, shortens the payback period, and significantly improves spatial utilization and economic viability.
Core Problem of Solar Panel Installation Based on Spatiotemporal Solar Radiation Distribution
From a spatial perspective, the introduction of a double-layer photovoltaic structure exacerbates the complexity of radiation allocation. The installation angle of the upper-layer solar panels not only directly influences their own radiation reception efficiency but also significantly affects the irradiance conditions of the lower-layer components through shading effects. Therefore, it is necessary to employ a high-precision 3D radiation transmission model to quantify inter-layer shading losses under different geometric configurations. Coupled optimization of the spatial layout and tilt angles of both layers must be performed to enhance the overall radiation utilization rate of the system.
In traditional schemes, the contradiction between the unit area installed capacity (120 W/m²) and the total project capacity target (10 MW) creates a significant spatial efficiency challenge. While dynamic tracking solutions can improve radiation adaptation accuracy, they are costly and complex to maintain. Manual coarse adjustment approaches suffer from insufficient precision and delayed response, making them inefficient for utilizing radiation resources. To resolve these contradictions, I developed a data-driven dynamic optimization algorithm based on the spatiotemporal distribution of solar radiation. This algorithm works in conjunction with a 3D radiation reception model to simultaneously optimize the installation azimuth, radiation response strategy, structural spatial constraints, and operational feasibility of solar panels. The goal is to maximize energy capture across all time and spatial domains.
Model Construction and Algorithm Design for Solar Panel Installation
Spatiotemporal Distribution of Solar Radiation
The local solar radiation exhibits significant distribution differences in both time and space. These characteristics directly influence the reception efficiency and installation optimization strategy of solar panels. The specific features are analyzed as follows:
Seasonal variation: Summer (June–August) has the highest total radiation, accounting for approximately 33.3% of the annual total; winter (December–February) monthly average radiation is only 16.3% of the annual total, mainly due to the lower solar altitude angle and reduced sunshine duration.
Diurnal variation: The diurnal radiation intensity shows a single-peak distribution, with the peak occurring around noon (12:00–13:00) when instantaneous radiation intensity can reach 800–1000 W/m². After dusk (after 18:00), the radiation intensity rapidly decays to near zero.
Spatial heterogeneity: Due to the influence of building layout and orientation, radiation intensity varies spatially. The eastern area, affected by building shading, has an average annual radiation intensity of 5180 MJ/m², while the southwestern area, with minimal shading, reaches 5320 MJ/m², yielding a spatial coefficient of variation of about 2.7%.
For the double-layer solar panel system studied here, the seasonal variation of solar altitude and azimuth angles further causes differences in inter-layer shading duration. 3D geometric model simulations reveal that the average daily shading duration is about 1.5 hours in summer and extends to approximately 3.0 hours in winter. This shading effect can be mitigated by dynamically adjusting the azimuth and tilt angles of solar panels.
Effective Radiation Calculation Model
The actual radiation received by a solar panel includes direct, diffuse, and reflected components, but not all can be effectively converted to electricity. I constructed an effective radiation calculation model to filter out ineffective radiation and quantify the effective radiation intensity Geff for a given installation orientation. The total solar radiation Gtotal can be decomposed as:
$$ G_{total} = G_b + G_d + G_r $$
where Gb is direct beam radiation, Gd is sky diffuse radiation, and Gr is ground-reflected radiation. For a solar panel at the k-th layer (k=1 for the upper layer, k=2 for the lower layer), the effective radiation intensity is:
$$ G_{eff,k} = G_{b,k} \cos\theta_k + G_{d,k} \frac{1+\cos\alpha_k}{2} + G_{r,k} \rho \frac{1-\cos\alpha_k}{2} $$
where θk is the incidence angle (angle between the sun’s rays and the panel normal), αk is the tilt angle of the k-th layer solar panel, and ρ is the ground reflectance. To validate the model’s accuracy, I compared measured and modeled values at noon on typical days in summer and winter. The results are shown in Table 1, with relative errors all below 5%, confirming that the model can effectively support subsequent installation optimization.
| Date | Time | Layer | Measured (W/m²) | Modeled (W/m²) | Relative Error (%) |
|---|---|---|---|---|---|
| 2023-07-15 | 12:00 | Upper | 826 | 800 | 3.1 |
| 2023-07-15 | 12:00 | Lower | 728 | 705 | 3.2 |
| 2024-01-15 | 12:00 | Upper | 385 | 367 | 4.7 |
| 2024-01-15 | 12:00 | Lower | 198 | 190 | 4.0 |
Three-Dimensional Radiation Reception Model for Solar Panels
Traditional planar photovoltaic systems can only capture radiation from a single direction. The 3D radiation reception model proposed here achieves synergistic capture of multi-directional radiation through a three-dimensional layout (upper array, lower array, and vertical array), which is especially suitable for industrial rooftop scenarios with building obstructions. The system configuration and generation characteristics are summarized in Table 2.
| Structural Layer | Installation Height (m) | Primary Function | Best Generation Period | Generation Characteristics |
|---|---|---|---|---|
| Upper Array | 3.0 | Main generation | 09:00–15:00 | Primarily receives direct radiation; minimally affected by shadows |
| Lower Array | 1.5 | Auxiliary generation | 08:00–10:00 & 14:00–16:00 | Receives reflected light from upper layer and diffuse radiation |
| Vertical Array | Along wall | Supplementary generation | Early morning and late afternoon | Utilizes low-angle solar radiation, extending system generation hours |
After accounting for inter-layer shading, the direct radiation received by each layer is corrected as:
$$ G_{b,k} = G_b \times \max(0, \cos\theta_k) \times S_k(\alpha, \beta, t) $$
where Sk(α, β, t) is the shading loss coefficient of the k-th layer at time t, calculated via a 3D ray-tracing method and ranging from 0 to 1. Preliminary model runs show the following benefits:
- Inter-layer shading losses are reduced by 30%–50% through angle adjustments.
- The system can still maintain partial power output under building shading conditions.
- In winter, by increasing the tilt angle, the generation of the lower array increases by about 40%.
- The total generation period is extended by 3–4 hours compared to traditional schemes.
Dynamic Optimization Algorithm Design
Based on the spatiotemporal radiation distribution and the reception model, I propose a dynamic optimization algorithm that maximizes the total net generation revenue over the system lifecycle to determine the optimal adjustment sequence of azimuth angle βk(t) and tilt angle αk(t) for each layer. The objective is to maximize the total net revenue Rtotal:
$$ R_{total} = \sum_{t=1}^{T} \left[ P_{gen}(t) \times p_{elec}(t) – C_{adj}(t) \right] $$
where Pgen(t) is the total system power generation at time t, pelec(t) is the electricity price (¥/kWh), and Cadj(t) is the operational cost of angle adjustments at time t.
To predict future radiation intensity, I established a back-propagation neural network with two hidden layers. The network configuration is as follows:
- Input layer: 8 nodes: month, hour, solar altitude angle, solar azimuth angle, cloud cover, ambient temperature, layer index (1 or 2), and inter-layer relative angle.
- Hidden layers: first layer: 12 neurons; second layer: 8 neurons.
- Output layer: 2 nodes, corresponding to the effective radiation intensity of the upper and lower layers (Geff,1, Geff,2).
- Training parameters: learning rate η = 0.01, number of iterations N = 1000, loss function: mean squared error (MSE).
The optimization solution process is as follows:
- Collect historical radiation data and shading simulation data, split into training and test sets covering a full year.
- Train the neural network to obtain a radiation prediction model.
- Using the predicted radiation as input, combined with the reception model, apply Particle Swarm Optimization (PSO) to determine the optimal installation angle combination for each quarter.
- Output the angle adjustment plan and evaluate the improvement in generation revenue.
Prediction accuracy is evaluated using Mean Absolute Error (MAE) and Root Mean Square Error (RMSE):
$$ \text{MAE} = \frac{1}{n} \sum_{i=1}^{n} |y_i – \hat{y}_i| $$
$$ \text{RMSE} = \sqrt{ \frac{1}{n} \sum_{i=1}^{n} (y_i – \hat{y}_i)^2 } $$
where n is the number of samples, yi is the actual effective radiation intensity, and ŷi is the predicted value. MAE reflects the average deviation between predictions and actual values; RMSE gives higher weight to larger errors, making it more sensitive to significant deviations. Smaller values indicate better prediction performance.
Algorithm Optimization and Simulation Verification
To ensure the practical feasibility of the dynamic optimization algorithm, I further improved the algorithm and verified its effectiveness through simulation. During the PSO solution process, I introduced dynamic inertia weights to balance global exploration and local exploitation, improving convergence speed and accuracy. The objective function (Rtotal) incorporates inter-layer shading losses and economic constraints of angle adjustments. The adjustment cost Cadj(t) is quantified as:
$$ C_{adj}(t) = C_f + \sum_{k=1}^{2} c_k \cdot |\alpha_k(t) – \alpha_k(t-1)| $$
where Cf is the fixed operational base cost (¥/adjustment) covering labor inspection and equipment wear; ck is the marginal cost per degree of angle adjustment for the k-th layer (¥/degree), related to actuator energy consumption and mechanical wear; and |αk(t) − αk(t−1)| is the absolute change in tilt angle between consecutive adjustment periods (degrees).
Simulations were run over a full year of operational data, comparing the traditional fixed installation scheme with the proposed dynamic optimization scheme. The results indicate that the dynamic optimization algorithm increases the annual average effective radiation intensity by approximately 18.6%, and the lifecycle net generation revenue is expected to rise by 15.2%. This validates the algorithm’s effectiveness in addressing spatiotemporal solar radiation heterogeneity and improving the overall performance of double-layer solar panel systems.
Performance Analysis
To comprehensively evaluate the practical benefits of the dynamic optimization algorithm, I conducted a one-year simulation and comparative analysis from four dimensions: generation performance, economic returns, space utilization, and algorithm stability. The detailed comparison with the traditional fixed installation scheme is presented in Table 3, confirming the overall effectiveness of the proposed approach.
| Metric | Traditional Fixed Scheme | Dynamic Optimization Scheme | Improvement |
|---|---|---|---|
| Annual total generation (10,000 kWh) | 1164.5 | 1378.2 | +18.3% |
| Generation per unit area (kWh/m²) | 105.8 | 129.1 | +22.0% |
| Annual average effective radiation intensity (W/m²) | 318.7 | 378.0 | +18.6% |
| Annual generation revenue (10,000 ¥) | 931.6 | 1107.2 | +18.8% |
| Annual operation & maintenance cost (10,000 ¥) | 10.0 | 12.4 | +2.4 (absolute increase) |
| Static payback period (years) | 8.2 | 6.9 | Shortened by 1.3 years |
| Space utilization assessment | Limited to planar extension | Synergistic use of three dimensions | Significantly optimized |
Through the three-dimensional layout and dynamic angle coordination, the effective generation per unit rooftop area increased by about 22%. While meeting the target installed capacity of 10 MW, this alleviates the pressure of insufficient rooftop space faced by early design schemes. Moreover, the algorithm maintains relatively stable generation under various simulated weather scenarios and partial shading conditions by implementing real-time adjustments, demonstrating strong environmental adaptability.
In summary, during the control process of solar panel installation orientation, the focus should be on developing low-cost, robust adjustment mechanisms. Future work should explore intelligent control strategies that integrate solar panels with building-integrated design and energy storage systems to enhance the technology’s versatility and economic competitiveness across diverse industrial scenarios. The real-time optimization capability and adaptability to extreme weather conditions are also valuable directions for further investigation.