In snowy regions, the combined effects of wind and snow loads are critical in the design of photovoltaic support structures. However, research specifically focusing on the combination factors of wind and snow loads for solar panels is limited. Current design codes and standards do not provide dedicated combination factors for solar panel structures, often leading to conservative designs. This study addresses this gap by investigating the combination factor of wind and snow loads on solar panels in Harbin, China, based on the characterization of joint wind-snow hazards. A single-axis solar tracker is used as a case study to demonstrate the methodology.
The research methodology involves several steps. First, a multi-layer snowmelt model is employed to simulate ground snow pressure using meteorological data. Second, various probability models are applied to fit wind speed and snow pressure samples from different data pairs, with the best-fitting model selected based on statistical tests. Third, linear regression analysis assesses the correlation between wind speed and snow pressure samples, leading to the construction of joint wind-snow hazard contours. Finally, finite element analysis is used to compute the structural load effects of the solar panel support system, from which the combination factor for wind and snow loads is derived.
The multi-layer snowmelt model simulates the accumulation and melting processes of snow on the ground. The energy balance equation for the i-th snow layer at time t is given by:
$$ \frac{dU_i(t)}{dt} = L_n(t) + S_n(t) + H_s(t) + E_l(t) + Q_c(t) + Q_p(t) + Q_g(t) $$
where ( L_n ) is net longwave radiation, ( S_n ) is shortwave radiation, ( H_s ) is sensible heat, ( E_l ) is latent heat, ( Q_c ) is energy transferred between adjacent snow layers, ( Q_p ) is energy from precipitation, and ( Q_g ) is energy from ground heat flux. The mass balance equations account for snowfall, rainfall, snowmelt, and sublimation. For the top snow layer (m-th layer):
$$ \frac{dW_m(t)}{dt} = P_{rain}(t) + P_{snow}(t) – M_{out}(t) – W_e(t) $$
For internal and bottom layers (1 ≤ i ≤ m-1):
$$ \frac{dW_i(t)}{dt} = M_{out,i+1}(t) – M_{out,i}(t) $$
Here, ( W ) represents snow water equivalent, ( P_{rain} ) and ( P_{snow} ) are rainfall and snowfall rates, ( M_{out} ) is melt outflow, and ( W_e ) is sublimation or evaporation. The model inputs include hourly meteorological data such as precipitation, temperature, wind speed, and relative humidity. For example, in Harbin, daily data are converted to hourly data using linear interpolation for temperature and averaging for other parameters.
Four methods are used to obtain data pairs of wind speed and ground snow pressure:
- Method 1: Maximum ground snow pressure between adjacent snowfall events and the maximum wind speed during the same period.
- Method 2: Maximum ground snow pressure between snowfall events and the maximum wind speed within three days after a snowfall event.
- Method 3: Maximum ground snow pressure during a snow accumulation event and the maximum wind speed during the same event.
- Method 4: Maximum ground snow pressure and maximum wind speed over the entire winter.
These methods yield different samples for statistical analysis. Probability models including Gumbel, Lognormal, and Generalized Extreme Value (GEV) distributions are fitted to the wind speed and snow pressure samples. Parameter estimation methods include Maximum Likelihood Estimation (MLE), Method of Moments (MOM), and Least Squares Method (LSM). The Kolmogorov-Smirnov (K-S) test and Akaike Information Criterion (AIC) are used to select the best-fitting model. The K-S test statistic is defined as:
$$ D_n = \max |F_n(x) – F(x)| $$
where ( F_n(x) ) is the empirical distribution function and ( F(x) ) is the theoretical distribution. The critical value ( K_\theta ) for significance level ( \theta = 0.05 ) is ( 1.36/\sqrt{N} ) for sample size N > 35. The AIC is calculated as:
$$ AIC = N \ln(RSS) + 2k $$
where RSS is the residual sum of squares and k is the number of parameters. Lower AIC values indicate better models.
For wind speed and snow pressure samples in Harbin, the optimal probability models are determined as follows:
| Variable | Method 1 | Method 2 | Method 3 | Method 4 |
|---|---|---|---|---|
| Wind Speed | GEV (MLE) | GEV (MLE) | Lognormal (MLE) | Lognormal (MLE) |
| Snow Pressure | GEV (MLE) | GEV (MLE) | Lognormal (MLE) | GEV (MLE) |
Linear regression analysis shows low correlation between wind speed and snow pressure samples, with coefficients of determination (R²) below 0.24 for all methods. Thus, the variables are treated as independent. The joint hazard contours for different return periods (10, 25, 50, 100 years) are constructed using the formula for independent variables:
$$ T = \frac{1}{[1 – F_X(x)] [1 – F_Y(y)]} $$
where ( F_X(x) ) and ( F_Y(y) ) are the cumulative distribution functions of wind speed and snow pressure, respectively. The contours represent combinations of wind speed and snow pressure that correspond to specific return periods.
The structural analysis focuses on a single-axis solar tracker with a length of 100 m, height of 1.5 m, and tilt angle of 30°. The support structure consists of a main beam and 13 steel columns spaced 8 m apart. Wind loads on the solar panels are calculated as:
$$ w_k = \beta_z \mu_z \mu_s w_0 $$
where ( \beta_z = 1.2 ) is the wind vibration coefficient, ( \mu_z = 1.0 ) is the height variation coefficient for roughness category B, ( \mu_s ) is the shape coefficient, and ( w_0 = 0.5 \rho v_0^2 ) is the basic wind pressure with air density ( \rho = 1.25 \, \text{kg/m}^3 ). The shape coefficients for a 30° tilted surface are derived from mono-slope roof values: ( \mu_{s1} = 1.4 ) and ( \mu_{s2} = 0.6 ) for windward pressure, and ( \mu_{s3} = -1.4 ) and ( \mu_{s4} = -0.6 ) for leeward suction, with linear interpolation for intermediate values. The wind load distribution is trapezoidal, decomposed into uniform load and torque effects.
Snow loads are computed as:
$$ s_k = \mu_r s_0 $$
where ( \mu_r = 0.85 ) is the snow distribution coefficient for a 30° mono-slope roof, and ( s_0 ) is the ground snow pressure. Snow load is assumed uniformly distributed, neglecting wind-induced snow drift.
Two load cases are considered:
- Load Case 1: Wind pressure and snow pressure both act downward.
- Load Case 2: Wind suction acts upward while snow pressure acts downward.
Finite element analysis is performed to determine the load effects on the solar panel support structure, including maximum column axial force ( P_{\max} ), maximum main beam bending moment ( M_{\max} ), and maximum main beam torque ( T_{\max} ). For Load Case 1, the combined load effects are significant, whereas for Load Case 2, the effects oppose each other, reducing the need for combination.
The combination factor ( \psi ) for wind and snow loads is derived from the load effect combination:
$$ \psi = \frac{S_m}{S_{w,25} + S_{s,25}} $$
where ( S_m ) is the maximum load effect under combined wind and snow loads, and ( S_{w,25} ) and ( S_{s,25} ) are the load effects due to wind and snow loads at 25-year return period, respectively. The 25-year return period is chosen as the design benchmark for solar panel structures.
For Harbin, the combination factors for column axial force and main beam bending moment are calculated based on the joint hazard contours from the four methods. The results are summarized below:
| Combination Factor | Method 1 | Method 2 | Method 3 | Method 4 | Average |
|---|---|---|---|---|---|
| \( \psi_P \) (Axial Force) | 0.7 | 0.7 | 0.6 | 0.6 | 0.7 |
| \( \psi_M \) (Bending Moment) | 0.6 | 0.7 | 0.6 | 0.7 | 0.7 |
The analysis shows that when wind and snow loads both act as pressure (Load Case 1), and the design considers column axial force and main beam bending moment, the combination factor for solar panels is 0.7. For Load Case 2, no combination is needed due to the opposing effects. The main beam torque is solely influenced by wind load, thus not requiring combination with snow load.
This study provides a framework for determining wind-snow load combination factors for solar panels based on joint hazard characterization. The proposed factor of 0.7 can inform revisions to design codes and standards, ensuring safer and more economical solar panel structures in snowy regions. Future work should consider wind-induced snow drift effects and the influence of solar panel design parameters such as tilt angle and row spacing on the combination factor.
In conclusion, the integration of multi-layer snowmelt modeling, probabilistic analysis, and finite element simulation enables a comprehensive assessment of joint wind-snow hazards on solar panels. The methodology demonstrated for Harbin can be applied to other regions with similar climatic conditions, enhancing the reliability of solar energy infrastructure worldwide.