View Factors of Flat Collectors, Including Photovoltaics, Visible to Partial Sky

Abstract

The sky view factor of collectors in photovoltaic (PV) fields is a parameter that determines the amount of incident diffuse radiation. The diffuse radiation may contribute significantly to the generated electric energy. PV systems are deployed in multiple rows and separated from each other and from nearby structures. Rows in front and nearby structures may block part of the visible sky to the collectors, thus decreasing the sky view factor. The distance between collectors and objects is an important parameter for the design of photovoltaic systems. The present study develops the expression for sky view factors for cases encountered in deployments of PV fields and presents numerical values for the sky view factors for distances between collectors and from obscuring structures. Sky view factors for flat collectors in the presence of adjacent collectors or structures have not been treated in the literature, besides for simple or common cases. Based on the values of the sky view factor, the PV system designer may assess the diffuse radiation losses, which are a dominant component of the global radiation losses. For example, for a collector installed at a distance 2.75 m from a building, the sky view factor is reduced from 0.97 to 0.85, i.e., a reduction of 12.4% in the incident diffuse radiation on the collector.

1. Introduction

In photovoltaic (PV) systems, the sky view factor is linked to the horizontal diffuse radiation emanating from the sky and reaching the collector; therefore, the view factor is defined between the collector and the sky. The sky view factor depends on geometrical parameters, i.e., collector width, collector inclination angle, and spacing between collector rows, and on nearby obscuring structures. A horizontal collector “sees” the whole sky; hence, its sky view factor is 1. In general, the sky view factor of collectors in PV systems is less than 1, entailing a reduction in the incident diffuse radiation of the PV system. The diffuse radiation may comprise a significant part of the global radiation; therefore, the sky view factor and, consequently, the incident diffuse radiation on the collectors are important factors of the generated electric energy of the PV system. The sky view factors for collectors mounted in multiple rows on horizontal and on inclined planes were first investigated in References [1,2]. A more recent article dealt with view factors of flat collectors [3]. The local sky view factor is calculated in article in Reference [4], along the width of the collector and its effect on the I–V characteristic. A general analytical expression for the view factor for a collector deployed on a sloped plane for which the plane and the azimuth of the collector do not coincide was developed in Reference [5].

Reference [6] deals with view factors to sky, view factors between opposite collectors, and view factors between collectors to shaded and not shaded grounds. View factors of solar collectors deployed on horizontal, inclined, and step-like planes are reported in Reference [7]. The work in Reference [8] analyzes the sky view factor for isotropic and anisotropic diffuse and albedo radiation. The article in Reference [9] uses the Monte Carlo Method to calculate the view factor. The research in Reference [10] uses 2D and 3D numerical analysis to calculate the view factors of a solar PV field. The study applies isotropic and anisotropic diffuse radiation models to determine the solar irradiance incident on the PV field. The method in Reference [11] is established for determining sky view factors for isotropic sky and the diffuse radiations incident on collectors at different slopes, facing any orientation. The view factor to sky is used in Reference [12] to calculate the incident diffuse radiation on collectors. All the above references deal with flat collectors.

A single inclined collector is visible to partial sky depending on the collector’s inclination angle. Collectors in a PV field deployed with multiple rows are obscured by rows in front; therefore, the sky view factors may even be further decreased. Moreover, PV collectors nearby structures may obscure part of the sky, thus decreasing the sky view factor and eventually leading to a decrease in the incident diffuse radiation on the PV collectors. Obscuring part of the diffuse sky light by any means results in power losses, termed as masking losses. Shading losses in PV systems are usually of concern and are mainly reduced by increasing the spacing between the collectors and by distancing the collectors from obscuring objects. However, masking losses may exceed the shading losses and ought to be of more concern. Shading losses take place in the three winter months, whereas masking losses prevail for the entire year. The sky view factor and, hence, the masking losses depend, among others, on the spacing between the collectors and on the height of the obscuring objects, both in front and behind the collectors. These dependencies were not investigated in the past, to the best of our knowledge. Hence, the aim of the present study was to investigate the effect of spacing on the sky view factor for different PV types of installations and to assess the amount of the incident diffuse radiation on the PV system. The effects of the collector and the plane inclination angles on the sky view factor were also studied.

2. Materials and Method

The length of the collector row is much larger than the collector width, $H$; therefore, the cross-string rule by Hottel [13] may be applied to develop the sky view factors (see Figure 1).

The view factor between surface $A_1$ and surface $A_2$ is given by the following:

$$V{F}_{A1\to A2}=\frac{CE+BD-BC-ED}{2\cdot BE}$$

(1)

The view factor of surface $A_2$ to the sky BC is given by the following:

$$V{F}_{sky}=\frac{CD+BC-BD}{2\cdot CD}$$

(2)

2.1. Single Collector

The sky view factor of a single collector, $H$, mounted with an inclination angle, $\beta$, with respect to a horizontal plane (see Figure 2), or the first collector row in a multiple row field, is given by the following [14,15]:

$$V{F}_{sky}=\left(1+cos\beta \right)/2$$

(3)

A single inclined collector, $\beta$, of width, $H$, deployed on a slopped plane, $\epsilon$, is shown in Figure 3.

The collector sky view factor is given by the following [5]:

$$V{F}_{sky}=\left[1+cos(\beta -\epsilon \right)]/2$$

(4)

2.2. Multiple Rows of Collectors

Figure 4 shows two inclined collectors of width, $H$, inclined with an angle, $\beta$, and separated by spacing, $D$, in a multiple-row horizontal PV field. Based on Hottel [13], $V{F}_{sky}=\frac{H+L_2-L_3}{2H}$, where $L_2=D+Hcos\beta$, $L_3={[D^2+{(Hsin\beta )}^2]}^{1/2}$, obtaining the following:

$$V{F}_{sky}=\frac{H+D+Hcos\beta -{[D^2+{(Hsin\beta )}^2]}^{1/2}}{2H}$$

(5)

Figure 5 shows collectors of width, $H$, deployed on triangle shapes, with inclination angles ($\beta$ and $\mu$) on a flat plane. The triangles are separated by a spacing, $D$.

The collector sky view factor is given by the following: $V{F}_{sky}=\frac{H+L_2-L_3}{2H}$, where $L_2=D+Hcos\beta +Hcos\mu ,\hspace{1em}{L}_3={[{(D+Hcos\mu )}^2+{(Hsin\beta )}^2]}^{1/2}$, obtaining the following:

$$V{F}_{sky}=\frac{H+D+Hcos\beta +Hcos\mu -{[{(D+Hcos\mu )}^2+{(Hsin\beta )}^2]}^{1/2}}{2H}$$

(6)

Figure 6 depicts collectors, $H$, with inclination angles ($\beta$) deployed on an inclined plane, $\epsilon$, and separated by a horizontal spacing, $D$.

The sky view factor is given by the following:

$$V{F}_{sky}=\frac{H+L_2-L_3}{2H},$$

where

$$L_2={[B^2+{(Hcos\beta +D)}^2]}^{1/2},\hspace{0.17em}{L}_3={[{(Hsin\beta -B)}^2+D^2]}^{1/2},\hspace{0.17em}B=\left(Hcos\beta +D\right)tan\epsilon ,$$

obtaining

$$V{F}_{sky}=\frac{H+{[B^2+{(Hcos\beta +D)}^2]}^{1/2}-{[{(Hsin\beta -B)}^2+D^2]}^{1/2}}{2H}$$

(7)

2.3. Collector Obscured by a Building

PV collectors may be deployed near buildings (or near any structure) in PV fields or on rooftops. The obscuring object affects the incident diffuse radiation on the collectors. Therefore, it is necessary to investigate the variation of the collector sky view factor with the distance from the building.

2.3.1. A building in Front of Collector

An inclined collector, $H$, deployed on a horizontal plane and obscured by a building in front of the collector is shown in Figure 7. The building is a specific height, $I$, and separated by spacing, $D$, from the collector.

The collector view factor is given by the following:

$$V{F}_{sky}=\frac{H+{[{(D+Hcos\beta )}^2+{(I-Hsin\beta )}^2]}^{1/2}-{[\left(D^2+I^2\right)]}^{1/2}}{2H}$$

(8)

An inclined collector, $H$, deployed on an inclined plane, $\epsilon$, and obscured by a building in front of the collector is shown in Figure 8. The building is of a specific height, $I$, and separated by a certain distance, $D$, from the inclined plane.

The collector sky view factor is given by $V{F}_{sky}=\frac{H+L_2-L_3}{2H}$, i.e.,

$$V{F}_{sky}=\frac{H+{[{(I-Hsin\beta -B)}^2+{(D+Hcos\beta +B/tan\epsilon )}^2]}^{1/2}-{[{(I-B)}^2+{(D+B/tan\epsilon )}^2]}^{1/2}}{2H}$$

(9)

2.3.2. A Building behind a Collector

An inclined collector deployed on a horizontal plane and obscured by a building of a specific height, $I$, behind the collector is shown in Figure 9. The collector is of a certain width ($H$) and separated by a distance, $D$, from the building. The calculation of the sky view factor is based on Equation (1), where point F is at a far distance from the collector, obtaining the following:

$$V{F}_{sky}=\frac{Hcos\beta +L_2-L_3}{2H}$$

resulting in

$$V{F}_{sky}=\frac{Hcos\beta +{[{(D+Hcos\beta )}^2+I^2]}^{1/2}-{[{(I-Hsin\beta )}^2+D^2]}^{1/2}}{2H}$$

(10)

For $I\le Hsin\beta$, the building is not blocking the visible sky to the collector, and the view factor is determined by Equation (3).

2.4. Overhang Collectors

Overhangs on south-facing walls in northern latitudes are protruding structures used for shading windows and doors by blocking the incoming sunlight during the summer months. At the same time, PV collectors may be deployed on the overhangs to generate electrical energy. Two overhang collectors projected from a vertical wall and inclined with angles ($\epsilon$) with respect to the wall are depicted in Figure 10. The sky view factors of the collectors are calculated based on Equations (1) and (2).

2.4.1. Top Collector

The calculation of the sky view factor of the top collector is based on Equation (1), where point F is at a far distance from the collector, obtaining the following:

$$V{F}_{sky}=\frac{Hcos(90-\epsilon )+ad-S}{2H},$$

resulting in

$$V{F}_{sky}=\frac{Hcos({90}^{\mathrm{\circ }}-\epsilon )+{[{(Hcos\epsilon +S)}^2+{(Hsin\epsilon )}^2]}^{1/2}-S}{2H}$$

(11)

2.4.2. Bottom Collector

The sky view factor of the bottom collector is given by $V{F}_{sky}=\frac{H+ac-ab}{2H}$, resulting in the following:

$$V{F}_{sky}=\frac{H+D-{[{(D-Hcos\epsilon )}^2+\left(Hsin\epsilon \right)]}^{1/2}}{2H}$$

(12)

2.5. A Flat Collector Facing a Convex Roof

PV collectors may be mounted near buildings with curved surfaces. An example is an agriculture field occupying greenhouses where nearby PV systems may be installed.

2.5.1. A Convex Roof in Front of Collector

Figure 11 shows the right part of a convex roof’s height ($I$) and span ($K$) separated by a distance, $D$, from a flat collector of a specific width, $H$. The convex surface may be presented by an ellipse.

The equation of an ellipse shifted to a center ($Hcos\beta +D+K$) is given by the following:

$$\frac{{(y-Hcos\beta -D-K)}^2}{A^2}+\frac{z^2}{B^2}=1$$

(13)

For an ellipse of a particular height ($I$) and span ($K$), the coefficients become $A=K \mathrm{and}\hspace{1em}B=I$, leading to the ellipse equation:

$$\frac{{(y-Hcos\beta -D-K)}^2}{K^2}+\frac{z_{ellipse}^2}{I^2}=1$$

(14)

or

$$z_{ellipse}\left(y\right)=I\times {\left[1-\frac{{(y-Hcos\beta -D-K)}^2}{K^2}\right]}^{1/2},Hcos\beta +D+K\ge y\ge Hcos\beta +D$$

(15)

The equation of the flat collector is given by the following:

$$z_H=-ytan\beta +Hsin\beta$$

(16)

The local sky view factor is given by the following (see Figure 11):

$$V{F}_{sky}\left(y\right)=\frac{bc+ab-ac}{2\times bc}$$

(17)

where $\overline{ac}$ in Figure 11 is tangent to the ellipse at point $a$ (point $c$ is on the collector). The tangent lines at $a$ points pertain to all $c$ points on the collector. The tangent line at point $a$ is obtained by the derivative of Equation (15):

$${(z_{ellipse}\left(y_a\right))}^{\prime }=-\frac{I}{K^2}\cdot \frac{\left(y_a-Hcos\beta -D-K\right)}{{\left[1-\frac{{(y_a-Hcos\beta -D-K)}^2}{K^2}\right]}^{1/2}}$$

(18)

The coordinates of points $a$ and $c$, respectively, are as follows:

$$\begin{gathered} a\left(y_a,z_a\right)=a\left\{y_a,\hspace{0.17em}I\cdot {\left[1-\frac{{(y_a-Hcos\beta -D-K)}^2}{K^2}\right]}^{1/2}\right\},\hspace{1em}Hcos\beta +D+K \\ \ge y_a\ge Hcos\beta +D \end{gathered}$$

(19)

$$c\left(y_c,z_c\right)=c\left\{y_c,-tan\beta \cdot y_c+Hsin\beta \right\},\hspace{1em}Hcos\beta \ge y_c\ge 0$$

(20)

Equating the tangent line (Equation (18)) with the slope passing through points $a$ and $c$ (Equations (19) and (20)) enables us to determine the value $y_a$:

$$-\frac{I}{K^2}\times \frac{\left(y_a-Hcos\beta -D-K\right)}{{\left[1-\frac{{(y_a-Hcos\beta -D-K)}^2}{K^2}\right]}^{1/2}}=\frac{I\cdot {\left[1-\frac{{(y_a-Hcos\beta -D-K)}^2}{K^2}\right]}^{1/2}-\left[-tan\beta \cdot y_c+Hsin\beta \right]}{y_a-y_c},$$

(21)

finally obtaining

$$y_a=T+2K\frac{I^2\left(y_c-T\right)+{\left\{\left[I^2\left(y_c-T\right){]}^2-\left(I^2-U^2\right)\right[I^2{(y_c-T)}^2+K^2{U}^2\right\}}^{1/2}}{2[I^2\left(y_c-T{)}^2+K^2{U}^2\right]}$$

(22)

where $T=Hcos\beta +D+K,\hspace{1em}U=-y_{c}tan\beta +Hsin\beta$.

Moving along the collector surface (Equation (16)), by a fixed small step, $dy$, the segment $bc$ becomes $bc=dy/cos\beta$, $ab={[{(ad)}^2+{(bd)}^2]}^{\frac{1}{2}},\hspace{1em}\mathrm{and} ac={[{(ce)}^2+{(ae)}^2]}^{1/2}$ (see Equation (17)). Note that $y_c$ is bounded by $Hcos\beta \ge y\ge 0$.

For the given parameters $K$, $H$, $cos\beta$, and $D$, and by assuming a point $y_c$ along the collector surface Equation (16), $y_a$ is calculated by Equation (22) and, thus, $z_{H,c}$ and $z_{ellipse,a}$ are determined by Equations (15) and (16). Referring to Equation (17), the distances for $V{F}_{sky}$ (see Equation (23)) may now be calculated:

$$\mathrm{point} b\left(y_b,z_b\right)=b\left(y_c-dy,z_c+dz\right)$$

$$ac={[{(z_{ellipse,a}-z_{H,c})}^2+{(y_a-y_c)}^2]}^{1/2}$$

(23)

$$ab={[{(z_{ellipse,a}-z_{H,b})}^2+{(y_a-y_b)}^2]}^{1/2}$$

2.5.2. A Convex Roof behind a Collector

Figure 12 shows the right part of a convex roof of a specific height ($I$) and span ($K$), separated by a distance ($D$) from a flat collector of a certain width ($H$). The roof is now located behind the collector. The calculation of the sky view factor is based on Equation (1), where point F is at a far distance from the collector.

The collector view factor is given by the following:

$$V{F}_{sky}=\frac{Hcos\beta +L_2-L_3}{2H}$$

(24)

The Equation of the collector, $H$, is now given by the following:

$$z_H=ytan\beta$$

(25)

and the equation of the ellipse is given in Equation (15). The value of $y_a$ is determined by the same procedure as in Section 2.5.1, obtaining the following:

$$y_a=T+2K\frac{I^2\left(y_c-T\right)+{\left\{\left[I^2\left(y_c-T\right){]}^2-\left(I^2-U^2\right)\right[I^2{(y_c-T)}^2+K^2{U}^2\right\}}^{1/2}}{2[I^2\left(y_c-T{)}^2+K^2{U}^2\right]}$$

(26)

where $T=Hcos\beta +D+K,\hspace{1em}U=Hsin\beta$,

$$L_2={(y_a^2+z_{ellipse.a}^2)}^{1/2}, \mathrm{and} L_3={[{(y_a-Hcos\beta )}^2+{(z_{ellipse,a}-Hsin\beta )}^2]}^{1/2}$$

(27)

By inserting Equation (27) into Equation (24), the collector view factor, $V{F}_{sky}$, is determined. Note that, for $z_{ellipse,a}\le z_{H,b}$, the ellipse is not blocking the visible sky to the collector, and the view factor is determined by Equation (3).

3. Results

The variations of the sky view factors of flat collectors that are visible to partial skies are reported in this section. The results are based on the developed expressions for the sky view factors in Section 2, for the different deployments of the collectors mentioned in the article.

3.1. Single Collector on Horizontal Plane, See Figure 2

The variation of the sky view factor with the inclination angle of the collector based on Equation (3) is shown in Figure 13. The figure shows that, by increasing the inclination angle, $\beta$, the visible sky decreases, and hence the sky view factor decreases.

3.2. Single Collector Deployed on a Sloped Plane, See Figure 3

The variation of the collector sky view factor with the plane slope angle, $\epsilon$, based on Equation (4) is shown in Figure 14. Collectors deployed on steeper planes attain larger sky view factors. Still, larger collector inclination angles ($\beta$) decrease the sky view factor.

3.3. Multiple Rows of Collectors, See Figure 4 and Figure 5

The variation of the collector view factor with the row spacing in a multiple horizontal PV field is depicted in Figure 15 for collector inclination angles $\beta ={20}^{\circ },\beta ={30}^{\circ }$, and $H=2 \mathrm{m}$ (see Figure 4). Higher inclination angles result in a reduced sky view factor.

The variation of the view factor of a collector, $H=2 \mathrm{m}$, with the row spacing on triangle shapes (see Figure 5) on a horizontal plane is shown in Figure 16. The view factor is less sensitive to inter-row spacing. Still, larger collector inclination angles decrease the sky view factor.

The variation of the view factor of a collector, $H=2 \mathrm{m}$, with row spacing for collectors mounted on a plane with an inclination angle, $\epsilon$ (see Figure 6), is depicted in Figure 17. The effect of the plane inclination angle is noticeable.

3.4. A Collector Obscured by a Building

The variation of the collector sky view factor with a distance from a building, where the building is in front of the collector (see Figure 7) is shown in Figure 18 for two heights of the building: $I=2 \mathrm{m}$ and $I=3 \mathrm{m}$; $H=2 \mathrm{m}$. A higher building reduces the sky view factor of the collector.

The results for the collector sky view factor for an inclined collector, $H=2 \mathrm{m}$ and $B=0.3 \mathrm{m}$, deployed on an inclined plane, $\epsilon$, and obscured by a building in front of the collector (see Figure 8) is depicted in Figure 19. A higher building reduces the sky view factor of the collector.

The variation of the sky view factor with a distance from a building behind the collector (see Figure 9) is shown in Figure 20 for two heights, $I=2 \mathrm{m}$ and $I=3 \mathrm{m}$, of the building, $H=2 \mathrm{m}$. A higher building reduces the sky view factor of the collector.

3.5. Overhang Collectors

The variation of the sky view factor of a top collector, $H=2 \mathrm{m}$, on an overhang with an angle, $\epsilon$, from the wall (see Figure 10) is shown in Figure 21. A higher wall reduces the sky view factor.

The variation of the sky view factor of a bottom collector, $H=2 \mathrm{m},$ with the collector angle, $\epsilon$, from the wall (see Figure 10) is shown in Figure 22. Larger distances ($D$) between the top and the bottom collectors increase the sky view factor.

3.6. A Flat Collector near a Convex Roof

The variation with the distance of the sky view factor of a collector installed in front of a convex roof (see Figure 11) is depicted in Figure 23. Higher convex roofs $\left(I=3 \mathrm{m}\right)$ result in lower view factors.

The variation with the distance of the sky view factor of a collector installed behind a convex roof (see Figure 12) is depicted in Figure 24. A convex roof of moderate height has little effect on the collector sky view factor.

4. Discussion

The diffuse incident radiation on a PV collector is coupled to the sky view factor, $V{F}_{sky}$. For isotropic skies, the diffuse radiation is given by $V{F}_{sky}\times G_{dh}$, where $G_{dh}$ is the diffuse radiation on a horizontal surface. The diffuse radiation may comprise a significant part of the global solar radiation; therefore, the sky view factor indicates the part of the incident diffuse radiation on the PV collector. PV collectors in solar fields are deployed with an inclination angle in multiple rows. The view factor of the first row, for example, $\beta ={20}^{\circ }$ (see Figure 13), is $0.97$. As the second and subsequent rows are obscured by rows in front, the view factor reduces to $0.917$ for row spacing of $D=1.0 \mathrm{m}$ and $\beta ={20}^{\circ }$ (see Figure 15), i.e., a reduction of $5.7\%$ in the sky view factor, meaning a reduction of $5.7\%$ in the incident diffuse radiation reaching the collector. The reduction in the incident diffuse radiation is known as masking losses. The same applies to collectors deployed near structures that obscure part of the sky. A collector installed at a distance of 2.75 m from a building (see Figure 7), the sky view factor is reduced from 0.97 for $\beta ={20}^{\circ }$ to 0.85 (see Figure 18), meaning that there is a reduction of 12.4% in the incident diffuse radiation reaching the collector.

The cross-string rule by Hottel [13] for calculating the sky view factor may be applied to collector rows with a length, $L$, much larger than its width, $H$. The method used in the present study for calculating the sky view factor lists common obstacles in PV fields; however, other field obstacles may be encountered, including fences and trees. Flat collector rows, flat building roofs and curved roofs are 2D structures for which the cross-string rule are applied. The calculation of the sky view factors for 3D curved surfaces and complex structures would need different approaches, left for further studies. The different methods for calculating the sky view factors in the present study are summarized in

Table 1. Sky view factor: methods and references.

Collectors	Methods	References
Single collector	Hottel, Liu-Jordan	[13,14]
Single collector	Models describing solar radiation from visible sky dome	[8]
Multiple rows	Disintegrating the hemispherical sky vault into small sky elements	[11]
Multiple rows	Hottel (cross-string rule)	[1,2,3,4,6,7,10]
Multiple rows	Vector calculation	[5]
Multiple rows	Monte Carlo Method	[9]

. The Hottel and Liu-Jordan methods [13,14] are used for single collectors (see Equation (3)). Reference [8] analyzes different methods to describe the solar radiation from the sky incident on a single collector; disintegrating the sky vault around multiple collector rows in urban areas by small elements is described in Reference [11]; the useful cross-string rule by Hottel [13] is widely used in the literature for multiple collector rows; the vector calculation approach is used in Reference [5] for multiple collector rows; and the Monte Carlo Method, combined with finite element technique, is used in Reference [9] for multiple collector rows. Sky view factors for flat collectors in the presence of structures have not been treated in the literature, to the best of our knowledge; it was our aim to address them in the present study.

5. Conclusions

The important parameter governing the reduction in the incident diffuse radiation (known as masking losses) on PV collectors is the distance between collectors and the distance between the collectors to obscuring objects. The article developed an analytical expression for collector sky view factors for cases encountered in the deployment of PV fields and presented numerical values for the variation of the collector sky view factor with distances. Based on an acceptable value of the sky view factor (meaning an accepted level of diffuse radiation losses), the distances for PV collector deployments were determined, or for given deployment distances, the sky view factors were defined. The collector sky view factor is therefore an important parameter for the design of PV systems.