# Application of a Numerical Simulation to the Estimation of Wind Loads on Photovoltaic Panels Installed Parallel to Sloped Roofs of Residential Houses

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## Abstract

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## 1. Introduction

_{panel}between the panel’s lower surface and the roof surface was about 150 mm. The geometric scale of the wind-tunnel model was 1/100. The value of H

_{panel}was changed from 0.25 mm to 3 mm (from 25 mm to 300 mm at full scale). The results indicated that the net wind loads on the PV panels were substantially lower than the external loads on the roof surface. Furthermore, it was found that the effect of H

_{panel}on the net wind loads was relatively small. This feature may be related to such a high roof pitch as β = 42°. Aly and Bitsuamlak [23] measured the wind forces on PV panels installed on gable roofs with two distinct pitches—i.e., 14° and 22.6°—in a wind tunnel. The geometric scale of the models was 1/15. The value of H

_{panel}was fixed to 0.1524 m at full scale. They tested three kinds of panel dimensions (small: 0.9144 m × 1.524 m, medium: 1.524 m × 2.4384 m, and big: 1.524 m × 2.7432 m) and four kinds of panel arrangements; the PV panels were installed almost all over the roof. Focus was on the effects of panel dimension and arrangement on the net wind loads of PV panels. Results indicated that the cladding loads on individual panel were smaller or larger in magnitude than those on the corresponding area of a bare roof (roof without PV panels), depending on the location and dimension of panels, as well as on the roof pitch β. Panels located close to the roof corners and edges were generally subjected to lower net pressures than the external pressures on the bare roof. Aly and Bitsuamlak recommended to avoid mounting PV panels in these zones, because the PV panels would be subjected to high suctions. Stenabough et al. [24] made a similar wind tunnel experiment using a gable roof model with β = 30°. The geometric scale of the model was 1/20. The PV panels were modeled by flat panels with equivalent full-scale dimensions of 50 cm width, 145.5 cm length and 6 cm thickness. Focus was on the effects of the horizontal gap G between PV panels and H

_{panel}on the area-averaged wind loads of PV panels. It was found that larger G values and smaller H

_{panel}values yielded lower net wind loads due to pressure equalization, which resulted in the magnitude of net wind loads typically being lower than those for the bare roof surface. Naeiji et al. [25] investigated the net wind loads of PV panels installed on flat, gable and hip roofs using large models in a large wind tunnel, the cross-section of which is 6 m wide and 4 m high. To the authors’ best knowledge, only this study dealt with a hip roof. The geometric scale of the models was 1/6. The panel was 2 m long, 1 m wide and 0.15 m thick at full scale. An investigation was conducted on the effects of building height H, panel clearance distance H

_{panel}and panel tilt angle β

_{panel}on the area-averaged wind loads of PV panels. Note that β

_{panel}was not necessarily equal to the roof pitch β. The value of H

_{panel}was either 0.3 m or 0.45 m at full scale, which is much larger than that of practical PV systems generally used in Japan. The results indicated that the critical wind direction generating the worst maximum or minimum peak wind force coefficient depended on the roof shape as well as on β

_{panel}.

_{panel}were changed from 10° to 40° and from 30 mm to 150 mm at full scale, respectively. Based on the results, they proposed positive and negative wind force coefficients for designing PV panels. It was noted that the magnitude of negative wind force coefficients might become larger than that of the proposed values when H

_{panel}> 100 mm.

_{PV}and the distance D

_{cavity}between the lower surface of PV panels and the roof surface are both as small as several centimeters. When the geometric scale of the wind tunnel model is 1/100, the values of t

_{PV}and D

_{cavity}should be less than 1 mm, such as 0.6 mm, for example. Furthermore, we have to measure the pressures on both the upper and lower surfaces of PV panels simultaneously for evaluating the net wind pressures (wind forces) on PV panels, which also makes it difficult to make wind tunnel models of PV panels. Indeed, some deformed models of PV panels are often used in wind tunnel experiments [26].

## 2. Wind Tunnel Experiment

#### 2.1. Wind Tunnel Model

_{L}= 1/75 using acrylic plates of 2 mm thickness. These models are named A, B0 and B1, respectively. Model A is not equipped with PV panels on the roof. Model B0 and B1 are equipped with PV panels almost all over a roof surface; the difference between Models B0 and B1 will be described later. The PV panels are also modeled by 2 mm thick acrylic plate. Therefore, the model is somewhat thicker than the practical ones considering that the geometric scale of the model is λ

_{L}= 1/75. However, this is a limitation in making models of PV panels. Similar modified models of PV panels were used in previous studies [26].

_{L}= 1/75, it is replaced by a circular hole of 1 mm diameter, the area of which is almost equal to that of the gap. The circular hole is placed at the center of gap (see Figure 6). In Model B0, PV panels are arranged with no gap between them; that is, the PV panels are in contact with each other. In practice, Model B0 is obtained by covering the circular holes drilled on the PV panel model of Model B1 with thin adhesive tape.

#### 2.2. Wind Tunnel Flow

_{z}and turbulence intensity I

_{z}at the location of the model’s center without model are plotted in Figure 8a. The power-law exponent α for the mean wind speed profile is about 0.27 and the turbulence intensity I

_{H}at the mean roof height H (=0.126 m) is about 0.17. Figure 8b shows the normalized power spectral density function, $f{S}_{u}(f)/{\sigma}_{u}{}^{2}$, of fluctuating wind speed at a height of z = 100 mm, where S

_{u}(f) = power spectral density function, f = frequency, σ

_{u}= standard deviation of fluctuating wind speed, and L

_{x}= integral length scale of turbulence. It is found that the general shape of S

_{u}(f) agrees well with that of the Karman-type spectrum with L

_{x}= 0.2 m (solid line in Figure 8b). According to the AIJ Recommendations for Loads on Buildings [37], the values of α, I

_{H}and L

_{x}for Terrain Category III (typical of suburban exposure) are specified as 0.20, 0.26 and 58 m at full scale, respectively. Comparing the above-mentioned values of the wind tunnel flow with these specified values, we can find that α is larger, while I

_{H}and L

_{x}are smaller. In particular, the value of L

_{x}is as small as about 1/4 of the target value, considering that the geometric scale is 1/75. In wind tunnel experiments for low-rise buildings, it is generally difficult to satisfy the similarity for all of these parameters, particularly for L

_{x}. Tieleman et al. [38,39,40] investigated the effects of these parameters on the characteristics of wind pressures on the roofs of low-rise buildings. They found that the fluctuating wind pressures were minutely affected by α. By comparison, the effect of I

_{H}was significant. Furthermore, the fluctuating pressures were less sensitive to L

_{x}. Regarding the effect of L

_{x}on the roof pressures they mentioned that the wind tunnel experiment could reproduce the fluctuating wind pressures almost satisfactorily, provided that the L

_{x}value of the wind tunnel flow was larger than 0.2 times the target value and the maximum length of the wind tunnel model. The wind tunnel flow used in this study satisfies this criterion. That is, the value of L

_{x}($\approx $0.2 m) of the wind tunnel flow is larger than 0.2 times the target value (=58 m/75 = 0.77 m) and the largest size of the building (=10.7/75 = 0.14 m). Tieleman [39] also found that a small-scale turbulence parameter S defined by Equation (1) played an important role on the flow simulation in wind tunnel experiments for low-rise buildings:

_{B}= characteristic length of the building. The wind pressure fluctuations were sensitive to S up to about 300, but they were minutely affected by S when S > 300. Therefore, the value of S should be larger than about 300. The value of S of the wind tunnel flow used in the present study was about 470, which was larger than 300. Note that U

_{H}and H are used for U and L

_{B}in Equation (1). Although the I

_{H}value of the wind tunnel flow is somewhat smaller than the target value, such a disagreement can be acceptable, because the main purpose of the present study is not to estimate the design wind loads on PV panels precisely but to investigate the application of the numerical simulation to the wind load estimation of PV panels.

#### 2.3. Experimental Procedure

_{H}at the mean roof height H (=9.45 m) is calculated based on the AIJ Recommendations for Loads on Buildings [37]. The ‘Basic wind speed’ U

_{0}is set to 35 m/s as a representative value for the Main Island of Japan. The terrain category is assumed to be III. In practice, U

_{H}is calculated as 27.8 m/s. The value of U

_{H}of the wind tunnel flow is set to 8 m/s. The Reynolds number Re, defined in terms of U

_{H}and H, is approximately 6.7 × 10

^{4}. The blockage ratio Br of the model with respect to the wind tunnel’s cross-section is about 1.9% at a maximum. The values of Re and Br satisfy the experimental criteria recommended in Wind Tunnel Testing for Buildings and other Structures [41]; that is, Re > 1.1 × 10

^{4}and Br < 5%. The velocity scale λ

_{V}is approximately 1/3.5. Considering that the geometric scale is λ

_{L}= 1/75, the time scale λ

_{T}(=λ

_{L}/λ

_{V}) is calculated as approximately 1/22.

_{p}is defined by

_{s}= static pressure in the wind tunnel; and q

_{H}= mean velocity pressure (=${\scriptscriptstyle \frac{1}{2}}\rho {U}_{H}{}^{2}$, with $\rho $ being the air density) of approach flow at the mean roof height H. The wind force (net wind pressure) acting on PV panel is provided by the difference between the pressures, P

_{t}and P

_{b}, on the upper and lower surfaces. The lower surface pressure P

_{b}is equal to the layer pressure P

_{l}under the PV panel. Therefore, the wind force coefficient, or the net wind pressure coefficient, C

_{f}, is provided by the following equation:

_{pt}represents the wind pressure coefficient on the upper surface of PV panel; and C

_{pl}is called “layer pressure coefficient” in this paper. When evaluating the statistical values of wind pressure and force coefficients, we apply ensemble averaging to the results of the 10 measurements conducted under the same condition. The distortion of measured fluctuating pressures due to tubing is compensated by using the frequency response function of the measuring system in the frequency domain. No moving average is applied to the time histories of wind pressure and force coefficients.

#### 2.4. Experimental Results

#### 2.4.1. Wind Pressures on the Roof without PV Panels

_{pe}during a period of 10 min at full scale, which can be obtained from the time history of C

_{pe}. Large positive pressures occur near the windward eaves at both wind directions. However, the magnitude is not so large. On the other hand, large suctions occur near the windward eaves when θ = 0° and near the ridge when θ = 45°. These suctions are induced by flow separation at the windward eaves or ridge. Figure 12a,b, respectively show the distributions of the most critical maximum and minimum peak pressure coefficients, ${\hat{C}}_{pe,\mathrm{cr}}$ and ${\stackrel{\u02c7}{C}}_{pe,\mathrm{cr}}$ irrespective of wind direction, i.e., the maximum value of ${\hat{C}}_{pe}$ and the minimum value of ${\stackrel{\u02c7}{C}}_{pe}$ among all wind directions. The maximum value of ${\hat{C}}_{pe,\mathrm{cr}}$ among all pressure taps is 0.66 when θ = 45° and the minimum value of ${\stackrel{\u02c7}{C}}_{pe,\mathrm{cr}}$ among all pressure taps is −4.1 when θ = 0°. These values occur at pressure taps marked by white circles in Figure 10b and Figure 11a. The present results for the wind pressure distributions on the square roof are consistent with those of previous studies investigating the wind pressure distributions on hip roofs [42,43,44,45,46,47,48].

#### 2.4.2. Wind Pressures and Forces on PV Panels

_{f}

_{,area}is calculated for each PV panel. Because the resolution of pressure taps on the wind tunnel model is relatively coarse, we apply the cubic spline function to the experimental data of C

_{f}in order to obtain the C

_{f}values at the lattice points of a fine grid, from which the value of C

_{f}

_{,area}is computed for each panel. Then, the maximum and minimum peak area-averaged wind force coefficients, ${\hat{C}}_{f,\mathrm{area}}$ and ${\stackrel{\u02c7}{C}}_{f,\mathrm{area}}$, are obtained from the time history of C

_{f}

_{,area}for each PV panel. Figure 17 shows the minimum value of ${\stackrel{\u02c7}{C}}_{f,\mathrm{area}}$ among all panels at each wind direction, in which the results for Models B0 and B1 are plotted in order to investigate the effect of gaps between panels on ${\stackrel{\u02c7}{C}}_{f,\mathrm{area}}$. Table 1 shows the PV panel that provides the minimum value of ${\stackrel{\u02c7}{C}}_{f,\mathrm{area}}$ among all panels at each wind direction. Note that the same wind direction provides the minimum values for both Models B0 and B1. It can be seen that the results for Models B0 and B1 show a similar trend. The minimum values occurred on panels located near the eaves or ridge. In particular, panels located near the windward eaves exhibited large-magnitude negative values (upward) when θ = 0–30°. The largest one was −3.7 for Model B0 and −3.3 for Model B1, which occurred on Panel 4 at θ = 0°. It is found that the values for Model B1 are generally smaller in magnitude than those for Model B0. The largest difference between these two models is about 0.4 at θ = 0°. This feature may be due to the pressure equalization caused by the gaps between PV panels, as Stenabough et al. [24] indicated.

#### 2.4.3. Effect of PV Panels on the Roof Pressures

## 3. Numerical Simulation of Layer Pressures

#### 3.1. Method of Simulation

_{H}= reference velocity pressure (N/m

^{2}); l = distance from the center of a Room to that of the next Room (or to the edge of the Room), parallel to the flow (m);

_{e}C = external pressure coefficient at the gap location; C = layer pressure coefficient; the subscripts of

_{e}C and C represent the Room location; C

_{L}= shape resistance coefficient for the cavity flow in the x or y direction; C

_{L}

_{e}= pressure loss coefficient of the gap (shape resistance coefficient) in the z direction, depending on the gap configuration; and Δp = pressure loss due to friction (N/m

^{2}). When the boundary of a Room corresponds to the periphery of PV panel, the subscript should be replaced by ‘e’ in Equations (4) and (5). Note that Equations (4) and (5) are applied to the cavity flows under PV panels or to the gap flows at the periphery of the PV system, while Equation (6) is applied to the flows through the gaps between PV panels. Although the values of C

_{L}and C

_{Le}depend on the gap configuration in a strict sense, they are assumed to be 1.0 for simplicity. This assumption can be accepted, because the width of gaps and the depth of cavity are relatively large. However, a detailed examination, such as a test with full-scale specimens [50], is required for validation.

_{e}C at the location of gaps in Equations (4)–(6) are obtained from the wind tunnel experiment mentioned in the previous section. Because the location of gaps does not coincide with that of pressure taps on the wind tunnel model, a spatial interpolation using the cubic spline function is applied to the experimental data; the value at the center of each gap is used as a representative value for the gap. Considering the calculation load, the above equations are solved by the 4th order Runge–Kutta method. Assuming the weak compressibility of the air and an adiabatic condition, the internal pressure P in each Room (layer pressure) can be obtained by the following equation (see [49]):

_{0}= atmospheric pressure (N/m

^{2}); V

_{0}= volume of Room (m

^{3}); Q

_{m}= flow rate (m

^{3}/s) at gap m; M = total number of gaps; and t = time (s). The layer pressure at the next time step is calculated by the Euler method with a time increment Δt of 1/8000 s. The sampling interval employed in the pressure measurements (Section 3) was approximately 0.036 s (=1/800 s $\times $ 29) at full scale, which is longer than Δt. Therefore, a temporal interpolation using the cubic spline function is applied to the time histories of wind pressure coefficients obtained from the wind tunnel experiment.

_{f}of PV panel is provided by Equation (3) mentioned above. Here, the values of C

_{pt}are obtained from the above-mentioned wind tunnel experiment, while those of C

_{pl}are provided by the numerical simulation.

#### 3.2. Comparison with the Experimental Results

## 4. Concluding Remarks

## Author Contributions

## Funding

## Informed Consent Statement

## Conflicts of Interest

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**Figure 2.**Square-roof building considered in the present study: (

**a**) General view; (

**b**) Plan; (

**c**) Elevation.

**Figure 3.**Wind tunnel models: (

**a**) Model A; (

**b**) Model B0; (

**c**) Model B1; (

**d**) Close-up view of the installed PV panels.

**Figure 7.**Arrangement of pressure taps on the roof and PV panels: (

**a**) Roof (Model A); (

**b**) Roof (Model B0, B1); (

**c**) PV panel (Model B0, B1).

**Figure 8.**Characteristics of wind tunnel flow at the location of model’s center without model: (

**a**) Profiles of mean wind speed and turbulence intensity; (

**b**) Normalized power spectral density function.

**Figure 10.**Distribution of the maximum peak pressure coefficients ${\hat{C}}_{pe}$ on the roof (Model A): (

**a**) θ = 0°; (

**b**) θ = 45°.

**Figure 11.**Distribution of the minimum peak pressure coefficients ${\stackrel{\u02c7}{C}}_{pe}$ on the roof (Model A): (

**a**) θ = 0°; (

**b**) θ = 45°.

**Figure 12.**Distributions of the most critical maximum and minimum peak pressure coefficients, ${\hat{C}}_{pe,\mathrm{cr}}$ and ${\stackrel{\u02c7}{C}}_{pe,\mathrm{cr}}$, irrespective of wind direction on the roof (Model A): (

**a**) ${\hat{C}}_{pe,\mathrm{cr}}$; (

**b**) ${\stackrel{\u02c7}{C}}_{pe,\mathrm{cr}}$.

**Figure 13.**Distributions of the minimum peak pressure coefficients when θ = 0°, 45° and 90°: (

**a**) Model A; (

**b**) Model B0.

**Figure 17.**Minimum peak value of area-averaged wind force coefficient among all panels at each wind direction.

**Figure 19.**Minimum peak pressure coefficients among all pressure taps on the roof at each wind direction.

**Figure 21.**Comparison between simulation and experiment for the distributions of mean layer pressure coefficients (Model B1): (

**a**) θ = 0°; (

**b**) θ = 45°; (

**c**) θ = 90°.

**Figure 22.**Comparison between simulation and experiment for the mean layer pressure coefficients (Model B1) at each pressure tap: (

**a**) θ = 0°; (

**b**) θ = 45°; (

**c**) θ = 90°.

**Figure 24.**Comparison between simulation and experiment for the minimum peak layer pressure coefficients (Model B1) at each pressure tap: (

**a**) θ = 0°; (

**b**) θ = 45°; (

**c**) θ = 90°.

**Figure 25.**Standard deviation of the minimum peak layer pressure coefficient for θ = 0, 45 and 90° (Model B1).

**Table 1.**PV panel number that provides the minimum peak area-averaged wind force coefficient at each wind direction.

θ (deg) | 0 | 15 | 30 | 45 | 60 | 75 | 90 | 105 | 120 | 135 | 150 | 165 | 180 |

Panel | 4 | 3 | 2 | 3 | 3 | 23 | 9 | 23 | 19 | 19 | 9 | 19 | 18 |

**Table 2.**The most critical positive and negative peak wind force coefficients irrespective of wind direction for panels in the edge and internal zones, compared with the specification of JIS C 9855 [27].

Data Source | Positive Value | Negative Value | ||
---|---|---|---|---|

Edge Zone | Internal Zone | Edge Zone | Internal Zone | |

JIS C 8955 (${G}_{f}\times {C}_{f,\mathrm{positive}}$, ${G}_{f}\times {C}_{f,\mathrm{negative}}$) | – | +2.85 | – | −2.81 |

Experiment & simulation (${\hat{C}}_{f,\mathrm{area},\mathrm{cr}}$, ${\stackrel{\u02c7}{C}}_{f,\mathrm{area},\mathrm{cr}}$) | +2.34 | +1.84 | −3.70 | −2.58 |

Prediction from the C_{p} distribution | +1.79 | +1.65 | −4.18 | −2.62 |

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## Share and Cite

**MDPI and ACS Style**

Uematsu, Y.; Yambe, T.; Yamamoto, A.
Application of a Numerical Simulation to the Estimation of Wind Loads on Photovoltaic Panels Installed Parallel to Sloped Roofs of Residential Houses. *Wind* **2022**, *2*, 129-149.
https://doi.org/10.3390/wind2010008

**AMA Style**

Uematsu Y, Yambe T, Yamamoto A.
Application of a Numerical Simulation to the Estimation of Wind Loads on Photovoltaic Panels Installed Parallel to Sloped Roofs of Residential Houses. *Wind*. 2022; 2(1):129-149.
https://doi.org/10.3390/wind2010008

**Chicago/Turabian Style**

Uematsu, Yasushi, Tetsuo Yambe, and Atsushi Yamamoto.
2022. "Application of a Numerical Simulation to the Estimation of Wind Loads on Photovoltaic Panels Installed Parallel to Sloped Roofs of Residential Houses" *Wind* 2, no. 1: 129-149.
https://doi.org/10.3390/wind2010008