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 photovoltaic structures is limited. This paper investigates the combination factor of wind and snow loads on photovoltaic panels based on the characterization of joint wind-snow hazards, using a single-axis photovoltaic tracker as a case study in Harbin. We employ a multi-layer snowmelt model to calculate ground snow pressure, fit wind speed and snow pressure samples using various probability models, assess their correlation through linear regression, construct joint hazard contours, and compute structural load effects via finite element analysis to derive the combination factor. The study finds that when both wind and snow loads act as pressure and the design considers column axial force and main beam bending moment, the recommended combination factor for wind and snow loads on photovoltaic panels is 0.7.
The rapid advancement of technology and growing environmental awareness have propelled solar energy as a safe and clean alternative, leading to a booming global photovoltaic industry. Photovoltaic systems, comprising support structures and mounted components, are lightweight and prone to damage under combined wind and snow loads. Thus, the combination of wind and snow loads is a crucial consideration in the design of photovoltaic structures. While some studies have examined wind and snow loads on photovoltaic panels separately, there is a lack of research on their joint effects, particularly for photovoltaic applications. Current load codes and standards do not provide specific combination factors for photovoltaic structures, and existing design guidelines for photovoltaic supports often adopt the same factors as building structures, which may yield conservative results. This study addresses this gap by developing a methodology to determine the combination factor for wind and snow loads on photovoltaic panels, leveraging joint hazard characterization and structural analysis.
We begin by simulating ground snow pressure using a multi-layer snowmelt model, which accounts for energy and mass exchanges during snow accumulation and melting. The energy balance equation for a 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 transfer between adjacent snow layers, \( Q_p \) is energy from precipitation, and \( Q_g \) is energy from ground conduction, assumed to be zero due to constant boundary temperature. The mass balance equations for the top snow layer (layer m) and internal layers (layers 1 to m-1) are:
$$ \frac{dW_m(t)}{dt} = P_{\text{rain}}(t) + P_{\text{snow}}(t) – M_{\text{out}}(t) – W_e(t) $$
$$ \frac{dW_i(t)}{dt} = M_{\text{out}, i+1}(t) – M_{\text{out}, i}(t) $$
where \( W \) is snow water equivalent, \( P_{\text{rain}} \) and \( P_{\text{snow}} \) are rainfall and snowfall, \( M_{\text{out}} \) is melt outflow, and \( W_e \) is sublimation or evaporation. Using hourly meteorological data from Harbin (1951–2016), including precipitation, temperature, wind speed, and relative humidity, we simulate ground snow pressure. For instance, the 2015–2016 winter data is converted to hourly values, and the snowmelt model outputs ground snow pressure evolution, as shown in the simulated results for Harbin.
To obtain wind speed and ground snow pressure data pairs, we define four methods based on snow events and daily maximum wind speed. Method 1 uses the maximum ground snow pressure between adjacent snowfall events and the corresponding maximum wind speed. Method 2 considers the maximum wind speed within three days after a snowfall event and the maximum ground snow pressure between events, as snow particles are assumed not to be wind-driven after three days. Method 3 takes the maximum ground snow pressure and maximum wind speed during a snow accumulation event. Method 4 uses the maximum ground snow pressure and maximum wind speed over the entire winter. These methods provide different datasets for probabilistic analysis.
We fit wind speed and ground snow pressure samples from each data pair using three probability distributions: Gumbel (extreme value type I), Lognormal, and Generalized Extreme Value (GEV). Parameter estimation methods include maximum likelihood estimation (MLE), method of moments (MOM), and least squares method (LSM). The Kolmogorov-Smirnov (K-S) test is applied to assess goodness-of-fit, with the test statistic \( D_n \) defined as:
$$ D_n = \max |F_n(x) – F(x)| $$
where \( F_n(x) \) is the empirical distribution and \( F(x) \) is the theoretical distribution. For a sample size N > 35 and significance level θ = 0.05, the critical value \( K_θ = 1.36 / \sqrt{N} \). If \( D_n \leq K_θ \), the distribution is accepted. Additionally, the Akaike Information Criterion (AIC) is used to select the best model, calculated as:
$$ \text{AIC} = N \ln(\text{RSS}) + 2k $$
where RSS is the residual sum of squares, and k is the number of parameters. Lower AIC values indicate better models. The results for Harbin show that for Methods 1 and 2, the best-fitting distributions for both wind speed and ground snow pressure are GEV (MLE). For Method 3, Lognormal (MLE) is optimal for both, and for Method 4, Lognormal (MLE) for wind speed and GEV (MLE) for ground snow pressure are selected.
| Data Pair Method | Best Distribution for Wind Speed | Best Distribution for Ground Snow Pressure |
|---|---|---|
| Method 1 | GEV (MLE) | GEV (MLE) |
| Method 2 | GEV (MLE) | GEV (MLE) |
| Method 3 | Lognormal (MLE) | Lognormal (MLE) |
| Method 4 | Lognormal (MLE) | GEV (MLE) |
Linear regression analysis reveals low correlation between wind speed and ground snow pressure samples, with coefficients of determination R² below 0.24 for all methods, indicating independence. Thus, the joint exceedance probability for independent variables is:
$$ P(X > x, Y > y) = [1 – F_X(x)] \cdot [1 – F_Y(y)] $$
and the joint return period T is:
$$ T = \frac{1}{[1 – F_X(x)] \cdot [1 – F_Y(y)]} $$
Using the optimal probability models and return periods of 10, 25, 50, and 100 years, we construct joint wind-snow hazard contours. For a 25-year return period, relevant to photovoltaic support design, the contours show combinations of wind speed and ground snow pressure. Method 4 yields the highest values due to using annual maxima, while Methods 1 and 2 show similar trends with Method 1 having higher wind speeds.
For structural analysis, we consider a single-axis tracking photovoltaic panel array with a length of 100 m, height of 1.5 m, and tilt angle of 30°. The finite element model includes a main beam and 13 steel columns spaced 8 m apart. Wind load on the photovoltaic panel is calculated as \( w_k = β_z μ_z μ_s w_0 \), where \( β_z = 1.2 \) is the wind vibration coefficient, \( μ_z = 1.0 \) is the height variation coefficient for rough terrain category B, \( μ_s \) is the shape coefficient, and \( w_0 = 0.5 ρ v_0^2 \) is the basic wind pressure with air density ρ = 1.25 kg/m³ and basic wind speed \( v_0 \). For a 30° tilt, shape coefficients are \( μ_{s1} = 1.4 \) and \( μ_{s2} = 0.6 \) for windward pressure, and \( μ_{s3} = -1.4 \) and \( μ_{s4} = -0.6 \) for leeward suction, resulting in a trapezoidal wind load distribution that decomposes into uniform load and torque. Snow load is computed as \( s_k = μ_r s_0 \), where \( μ_r = 0.85 \) is the snow distribution coefficient for a 30° single-slope roof, and \( s_0 \) is the basic snow pressure. We assume uniform snow load without wind-induced drift effects.
Two load cases are considered: Case 1 with both wind and snow as pressure, and Case 2 with wind as suction and snow as pressure. For Case 1, load effects include maximum column axial force \( P_{\text{max}} \), maximum main beam bending moment \( M_{\text{max}} \), and maximum main beam torque \( T_{\text{max}} \). Using the 25-year return period hazard contours, we convert wind speed and ground snow pressure to loads and compute load effects via finite element analysis. Results show that for Methods 1-3, maximum axial force and bending moment occur when snow load is near maximum with moderate wind, while for Method 4, they occur at high wind with some snow. Torque depends solely on wind speed. For Case 2, load effects counteract, so combination is not necessary.
| Data Pair Method | \( S_{P,w25} \) (kN) | \( S_{P,s25} \) (kN) | \( S_{P,m} \) (kN) | \( ψ_P \) |
|---|---|---|---|---|
| Method 1 | -3.60 | -6.22 | -6.39 | 0.7 |
| Method 2 | -2.87 | -6.16 | -6.28 | 0.7 |
| Method 3 | -5.16 | -6.17 | -6.47 | 0.6 |
| Method 4 | -9.85 | -7.00 | -10.81 | 0.6 |
| Data Pair Method | \( S_{M,w25} \) (kN·m) | \( S_{M,s25} \) (kN·m) | \( S_{M,m} \) (kN·m) | \( ψ_M \) |
|---|---|---|---|---|
| Method 1 | -2.40 | -3.83 | -3.98 | 0.6 |
| Method 2 | -1.92 | -3.80 | -3.90 | 0.7 |
| Method 3 | -3.45 | -3.80 | -4.07 | 0.6 |
| Method 4 | -6.58 | -4.31 | -7.21 | 0.7 |
The combination factor ψ for wind and snow loads is derived from the maximum load effect \( S_m \) under combined loads and the sum of individual load effects at the 25-year return period:
$$ ψ = \frac{S_m}{S_{w25} + S_{s25}} $$
For axial force and bending moment in Case 1, the average combination factor across methods is 0.7. For torque, no combination is needed as it is wind-dependent. Thus, for photovoltaic panel design in Harbin, when wind and snow loads are both pressure and column axial force or main beam bending moment govern, the combination factor is 0.7. This study provides a framework for evaluating joint hazards and combination factors for photovoltaic structures, enhancing design accuracy and reliability. Future work should consider wind-induced snow drift and parametric effects of photovoltaic panel configuration on loads.
In summary, we have demonstrated a comprehensive approach to determining the combination factor for wind and snow loads on photovoltaic panels. The multi-layer snowmelt model effectively simulates ground snow pressure, and probabilistic analysis with optimal distributions enables the construction of joint hazard contours. Structural analysis of a single-axis photovoltaic tracker reveals that load combination is essential for pressure cases, with a recommended factor of 0.7. This research contributes to the safe and efficient design of photovoltaic systems in snowy regions, addressing a critical gap in current standards. The methodology can be extended to other locations and photovoltaic configurations, promoting the resilience of solar energy infrastructure.