In the realm of renewable energy, off grid solar systems play a critical role in providing electricity to remote locations where grid connectivity is unavailable. The design of such systems requires meticulous planning, particularly in optimizing the tilt angle of photovoltaic (PV) modules to maximize energy capture while ensuring reliable power supply to loads. Unlike grid-tied systems, where the primary goal is to maximize annual energy yield, off grid solar systems must balance energy production with load demands, battery storage limitations, and local climatic conditions. This paper presents a comprehensive methodology for determining the optimal tilt angle of PV arrays in off grid solar systems, leveraging advanced simulation tools and mathematical models to enhance system reliability and battery longevity.
The installation of PV modules in off grid solar systems typically employs fixed mounting structures due to their cost-effectiveness, simplicity, and low maintenance requirements. However, the fixed tilt angle must be carefully selected to align with seasonal variations in solar radiation and load profiles. Traditional approaches for tilt angle optimization in grid-connected systems focus solely on maximizing the annual global solar irradiance incident on the PV array. In contrast, off grid solar systems necessitate a holistic approach that accounts for monthly irradiance fluctuations, load consumption patterns, battery state of charge, and depth of discharge cycles. Failure to consider these factors can lead to insufficient power generation during low-sunlight periods, accelerated battery degradation, and compromised system autonomy. Thus, this study introduces a simulation-based framework that integrates irradiance modeling, load profiling, and battery performance analysis to derive the optimal tilt angle for off grid solar systems.
To quantify the solar energy harvested by a tilted PV array, the total irradiance incident on the surface must be computed. The total irradiance on a tilted plane, denoted as \( H_T \), comprises three components: direct beam irradiance \( H_{bT} \), diffuse sky irradiance \( H_{dT} \), and ground-reflected irradiance \( H_{rT} \). The relationship is expressed as:
$$ H_T = H_{bT} + H_{dT} + H_{rT} $$
For a surface oriented towards the equator, the direct beam irradiance \( H_{bT} \) can be calculated using the following equation:
$$ H_{bT} = H_b \left[ \frac{\cos(\phi – \beta) \cos \delta \cos \omega_{sT} + \frac{\pi}{180} \omega_{sT} \sin(\phi – \beta) \sin \delta}{\cos \phi \cos \delta \cos \omega_s + \frac{\pi}{180} \omega_s \sin \phi \sin \delta} \right] $$
where \( \beta \) is the tilt angle, \( \phi \) is the local latitude, \( \delta \) is the solar declination angle, \( \omega_s \) is the sunset hour angle for a horizontal surface, and \( \omega_{sT} \) is the sunset hour angle for the tilted surface. The solar declination angle \( \delta \) varies throughout the year and can be approximated by:
$$ \delta = 23.45^\circ \sin \left( \frac{360}{365} (284 + n) \right) $$
where \( n \) is the day of the year. The sunset hour angles are derived from the latitude and declination, with \( \omega_s = \cos^{-1}(-\tan \phi \tan \delta) \) and \( \omega_{sT} = \min(\omega_s, \cos^{-1}(-\tan(\phi – \beta) \tan \delta)) \).
The diffuse sky irradiance \( H_{dT} \) is modeled using Hay’s anisotropic model, which accounts for the circumsolar and isotropic diffuse components. The formula is given by:
$$ H_{dT} = H_d \left[ \frac{H – H_d}{H_o} H_b + 0.5 (1 + \cos \beta) \left(1 – \frac{H – H_d}{H_o} \right) \right] $$
where \( H_d \) is the diffuse irradiance on a horizontal surface, \( H \) is the global horizontal irradiance, and \( H_o \) is the extraterrestrial irradiance on a horizontal surface, calculated as:
$$ H_o = \frac{24 \times 3600 G_{sc}}{\pi} \left[ 1 + 0.033 \cos \left( \frac{360 n}{365} \right) \right] \left( \cos \phi \cos \delta \sin \omega_s + \frac{\pi \omega_s}{180} \sin \phi \sin \delta \right) $$
Here, \( G_{sc} \) is the solar constant, approximately 1367 W/m². The ground-reflected irradiance \( H_{rT} \) is estimated by:
$$ H_{rT} = \frac{1}{2} \rho H (1 – \cos \beta) $$
where \( \rho \) is the ground albedo, typically set to 0.2 for common surfaces. These equations form the basis for evaluating the solar energy potential at different tilt angles, which is essential for designing efficient off grid solar systems.
Given the complexity of these calculations and the need for location-specific meteorological data, photovoltaic simulation software is indispensable for practical design. Several tools are available for modeling off grid solar systems, each with unique features and databases. Below, we compare four prominent software options used in the industry for tilt angle optimization and system simulation.
| Software | Advantages | Disadvantages | Suitability for Off Grid Solar Systems |
|---|---|---|---|
| PVsyst | Comprehensive analysis capabilities; extensive component databases; detailed hourly simulations. | Limited Chinese language support; sparse data for Chinese regions. | High, due to advanced modeling of off-grid components and battery storage. |
| RETScreen | Free to use; user-friendly Excel-based interface; global meteorological data coverage. | No hourly simulation capability; simplified models may lack precision. | Moderate, suitable for preliminary feasibility studies of off grid solar systems. |
| PV & SOL | Intuitive interface; precise load profiling; integrated 3D shading analysis; detailed financial reports. | No Chinese language support; slow performance with large databases. | Excellent, due to robust off-grid simulation and battery life cycle analysis. |
| PVwatts | Free online access; no installation required; simple input parameters. | Limited functionality; basic models insufficient for complex off grid solar systems. | Low, best for rough estimates of grid-tied systems. |
From the comparison, PV & SOL emerges as the preferred tool for designing off grid solar systems due to its ability to model detailed load profiles, battery performance, and shading effects. It supports custom input for daily and weekly load variations, aligning with real-world consumption patterns in off grid solar systems. The software incorporates meteorological data from over 8,000 stations worldwide and includes databases for more than 12,000 PV modules and 2,600 inverters, facilitating accurate system sizing and tilt angle optimization.
To demonstrate the application of this methodology, we consider a case study of an off grid solar system in a northwestern region. The system powers a load with a daily energy consumption of approximately 9.8 kW, operating in two shifts: 4 hours in the morning and 5 hours in the evening. The battery bank comprises lead-acid batteries with a capacity of 1000 Ah, and the PV array is sized at around 30 kW peak power. The azimuth angle is set to 0° (true south). The goal is to determine the optimal tilt angle that maximizes energy reliability and battery life.
Using PV & SOL, we simulate the system performance at tilt angles of 35°, 40°, 45°, 50°, and 55°, with PV configurations of 9 modules in series and 13, 14, or 15 strings in parallel (denoted as 9×13, 9×14, 9×15). Key performance indicators include the photovoltaic supply fraction \( \sigma \) (percentage of load demand met by PV), average annual battery charge percentage \( \eta_1 \), and minimum monthly battery charge percentage \( \eta_2 \). The results are summarized below:
| Tilt Angle | PV Configuration | Supply Fraction \( \sigma \) (%) | Average Charge \( \eta_1 \) (%) | Minimum Charge \( \eta_2 \) (%) | Critical Month |
|---|---|---|---|---|---|
| 35° | 9×13 | 96 | 81 | 45 | November |
| 40° | 9×13 | 96 | 80 | 53 | November |
| 45° | 9×13 | 96 | 78 | 59 | November |
| 50° | 9×13 | 95 | 76 | 56 | April/November |
| 55° | 9×13 | 93 | 71 | 47 | April |
| 35° | 9×14 | 97 | 85 | 60 | November |
| 40° | 9×14 | 97 | 85 | 66 | November |
| 45° | 9×14 | 97 | 84 | 69 | November |
| 50° | 9×14 | 97 | 82 | 70 | April |
| 55° | 9×14 | 96 | 79 | 59 | April |
| 35° | 9×15 | 98 | 88 | 68 | November |
| 40° | 9×15 | 98 | 88 | 71 | November |
| 45° | 9×15 | 98 | 87 | 73 | November |
| 50° | 9×15 | 98 | 86 | 75 | April |
| 55° | 9×15 | 97 | 84 | 73 | April |
The data indicates that for most configurations, the supply fraction exceeds 95%, meeting the load requirements. However, the battery charge percentage varies significantly with tilt angle. Notably, as the tilt angle increases from 45° to 50°, the critical month shifts from November to April, and the minimum charge percentage \( \eta_2 \) reaches a local maximum in this range. This suggests that the optimal tilt angle lies between 45° and 50° for this off grid solar system. To refine the selection, we simulate additional angles of 46°, 47°, 48°, and 49° for the same PV configurations.
| Tilt Angle | PV Configuration | Supply Fraction \( \sigma \) (%) | Average Charge \( \eta_1 \) (%) | Minimum Charge \( \eta_2 \) (%) |
|---|---|---|---|---|
| 46° | 9×13 | 96 | 78 | 59.9 |
| 47° | 9×13 | 96 | 78 | 60.5 |
| 48° | 9×13 | 95 | 77 | 59.4 |
| 49° | 9×13 | 95 | 76 | 58.1 |
| 46° | 9×14 | 97 | 84 | 68.9 |
| 47° | 9×14 | 97 | 83 | 69.3 |
| 48° | 9×14 | 97 | 83 | 69.7 |
| 49° | 9×14 | 97 | 83 | 70.1 |
| 46° | 9×15 | 98 | 87 | 74.0 |
| 47° | 9×15 | 98 | 87 | 74.4 |
| 48° | 9×15 | 98 | 87 | 74.6 |
| 49° | 9×15 | 98 | 86 | 74.8 |
Based on the highest minimum battery charge percentage \( \eta_2 \), the optimal tilt angle is 49° for the 9×13 configuration, 47° for the 9×14 configuration, and 49° for the 9×15 configuration. Comparing the configurations, the 9×14 setup offers a significant improvement in battery charge over 9×13, while 9×15 provides marginal gains at higher cost. Thus, for this off grid solar system, the 9×14 PV array at a tilt angle of 47° is selected to balance performance and economics.
The simulation outputs from PV & SOL further illustrate the daily energy balance and battery dynamics. For instance, the battery state of charge (SOC) over a 24-hour period can be modeled using the equation:
$$ SOC(t) = SOC(t-1) + \frac{P_{PV}(t) – P_{load}(t)}{V_{batt} \cdot C_{batt}} \cdot \eta_{batt} $$
where \( P_{PV}(t) \) is the PV power output at time \( t \), \( P_{load}(t) \) is the load power, \( V_{batt} \) is the battery voltage, \( C_{batt} \) is the battery capacity, and \( \eta_{batt} \) is the battery efficiency. This equation highlights the interplay between energy generation and consumption in off grid solar systems, emphasizing the need for tilt angles that minimize SOC fluctuations.
In conclusion, the design of tilt angles for off grid solar systems requires a multifaceted approach that integrates solar geometry, load characteristics, and battery health. By employing simulation tools like PV & SOL, designers can optimize tilt angles to enhance system reliability and longevity. The proposed methodology, demonstrated through a case study, underscores the importance of moving beyond mere irradiance maximization to holistic energy management in off grid solar systems. Future work could explore the integration of machine learning algorithms for real-time tilt adjustment or the impact of climate change on long-term performance of off grid solar systems.