Solar Radiation on Photovoltaic Systems Deployed near Obscuring Walls

Abstract

The deployment of solar photovoltaic (PV) systems on rooftops in urban environments utilizes the rooftop areas for electricity generation. Rooftops may provide a large amount of empty space that can reduce the use of land for large PV plant installations and other purposes. These deployments may encounter shading on the PV collectors from surrounding building walls, thus reducing the incident direct beam radiation on the PV collectors, resulting in shading losses. Moreover, walls and collector rows block part of the visible sky, reducing the incident diffuse radiation on the collectors, resulting in masking losses. The present study complements previous studies by the authors (see the references) by calculating the incident beam, diffuse and global radiation, and their distribution across the collector rows for four configurations of PV systems installed near obscuring walls. In addition, the article quantifies the shading problem by simulating the shading dimensions and their patterns caused by walls and collector rows. The article is of practical importance for designers of PV systems in urban environments. The simulation results indicate an almost uniform distribution of the incident radiation between the collector rows. On the other hand, the losses may reach 8 percent for a wall height of 4 m for the parameters used in the study.

1. Introduction

As the penetration of photovoltaic (PV) systems has significantly increased from rural spaces to build environmental areas, rooftop areas are utilized for electricity generation, and thus are becoming distributed energy sources. The advantage of distributed energy is the generation of electricity right at the consumer location. The benefit of local electricity generation to a county’s economy includes avoiding or delaying the construction of new power plants and transmission lines. Rooftop PV systems may encounter shading on the collectors from surrounding building walls. Wall shading losses, in the present article, refer to the loss incident solar radiation from the PV systems and depends on the height of the wall, the distance of the wall to the PV collectors, and the orientation of the walls relative to the PV collectors. Inter-row shading (mutual shading) losses of the PV collectors stem from shading on the second and the subsequent collector rows by the collectors row in front. These losses depend on inter-row spacing, collector height, and on the collector’s inclination and azimuth angles. The combined shading losses (wall and inter row) stem from the reduction of incident direct beam radiation on the collectors. In addition, walls and collectors reduce the incident diffuse radiation on the collectors, resulting in masking losses. The present study complements the studies in [1,2] by calculating the incident beam, diffuse and global solar radiation, and their distribution across the collector rows for PV systems installed near obscuring walls in an urban environment. The study is based on a developed mathematical expression for wall and inter-row shadings in [1], and on expressions for sky view factors for the diffuse radiation losses in [2]. Since the incident solar radiation is reduced by building walls, the economics of the PV systems may not be justified or may become justified depending on the distance between the walls and collectors. In addition, the construction of new high-rise buildings near existing PV systems may reduce the revenue from the PV system to the owner and may require, for example, compensation for the loss of income.

The potential of utilizing rooftops, walls, and windows for electricity generation in cities is investigated in [3], and it is estimated that the generation can cover about 74% of the electricity consumption in the city of Melbourne. The study in [4] mentions that many cities could potentially supply their own electricity needs with building-integrated photovoltaics (BIPV) in urban areas. The article in [5] examined three basic building shapes—squares, circles, and triangles—for the deployment of PV panels on the building envelope, in a humid tropical climate in Indonesia. The square-shaped building is the most effective in capturing the solar radiation. An overview of the parameters affecting PV systems in urban areas is presented in [6], stating that shading, along with air temperature, pollution, and soiling, is a significant issue of the performance of BIPV systems. The following three approaches were considered in [7] to increase the electricity generation of photovoltaic systems installed in urban areas: avoiding the shadow, tilting angle optimization, and redirecting additional light by mirrors. The study in [8] investigates solar potential (PV and thermal) in urban residential buildings. The effects of three major design parameters (i.e., building aspect ratio, azimuth, and site coverage) on solar potential are evaluated. The results show that increasing the building aspect ratio tends to raise solar potential, and so does increasing site coverage. The article in [9] describes a method for the estimation of the solar potential of roofs and facades of buildings in an urban landscape. The method calculates hourly roof and facade shadow maps for estimating direct beam radiation, and sky view factor maps for roofs and facades for the estimation of diffuse radiation. Results show that the radiation reaching facades is lower than for roofs; however, facades have a significant impact on the solar potential of buildings in an urban area. The paper in [10] presents a model for shading losses of BIPV systems, as shading is the major source of electricity losses. The method is based on covering a grid over the PV surface and defining coordinate points (azimuth and elevation) of shading segments caused by the inter-row shading. It is worth mentioning the developed mathematical expressions for inter-row shadings in [11,12]. A simple mathematical model for estimating PV array power reduction resulting from shading has been presented and experimentally verified in [13]. The model takes into account the shaded fraction of the array area affected by shading. Software programs for simulating the performance of PV systems are in [14]. Some of these software programs include 3-D shading tools for simulating obscuring objects, such as PVSYS, PVSol, and Solar Pro. An analysis of shading factors using Solar Pro software 4.5 estimates the performance of a PV plant near a specific building to obtain zero shading, which is presented in [15]. The mentioned articles [3,4,5,6,7,8] generally estimate the potential of generating electricity by PV systems in urban environments, but do not present analytical expressions for shading calculation that affect the output of the generated electricity. References [9,10,11,12,13] treat PV systems with respect to the energy yield of shaded PV systems in open space (without obscuring objects) stemming from inter-row shading. However, these studies do not deal with PV systems erected near building walls, causing shadows on the PV collectors. As shading poses a significant problem by reducing the electric generated energy, the present article addresses and quantifies numerically the shading problem by analytically formulating and simulating the shading dimensions and the patterns caused by walls and by collector rows, thus enabling the determination of the incident solar radiation on the PV systems. The incident beam, diffuse and global radiation, and the percentage of shading and masking losses are calculated, in the present study, for four possible configurations of deployments of the PV systems erected near walls of different heights and different wall and collector azimuth angles.

2. Methods and Materials

The mathematical expressions for the shadows and the sky view factors for PV systems deployed near obscuring walls are developed in [1, 2]. This section mentions only the final mathematical expressions used for the analysis of the PV systems. The analysis pertains to the northern hemisphere and for wall dimensions that are longer than the dimensions of the PV system. Figure 1 describes a general deployment of PV modules forming collectors, on a horizontal solar field in the presence of a vertical wall of a building erected on the west side of the collectors. The PV system comprises $K$ collectors of length $L_C$ and width $H_C$ each, deployed in an east–west direction and facing a southern direction with an azimuth angle ${\gamma }_C$ with respect to the south, and inclined with an angle $\beta$ concerning the horizontal plane. A building with height $H_W$ oriented with an azimuth angle ${\gamma }_W$ with respect to the south, is at a distance $R(1)$ from the first collector. The wall casts shadows on the collectors in a right triangle shape marked in pale blue. The height of the triangle is $H_{W,sh}$, its length is $L_{W,sh}$, and its angle is $\psi$. The inter-row shading is rectangular, marked in pale red; the height and length of the shadows are labeled $H_{C,sh}$ and $L_{C,sh}$, respectively.

2.1. Wall Shading on Collector $K$

The shadow equations (see also Figure 1) are given in reference [1].

The shadow angle $\psi$ is as follows:

$$\tan \psi ={\displaystyle \frac{1}{\tan {\gamma }_W}}\left({\displaystyle \frac{\cos ({\gamma }_S-{\gamma }_C)}{\tan \alpha }}\sin \beta \right)+{\displaystyle \frac{\left|\sin ({\gamma }_S-{\gamma }_C)\right|}{\tan \alpha }}\sin \beta$$

(1)

The shadow length $OB$ is as follows:

$$L_{w,sh}=H_w\left|{\displaystyle \frac{\cos ({\gamma }_S-{\gamma }_C-{\gamma }_W)}{\tan \alpha \cdot \sin {\gamma }_W}}\right|-R(K)$$

(2)

and the shadow height $OC$ is

$$H_{w,sh}=L_{w,sh}/\tan \psi$$

(3)

The distance of the collector number $K$ to the wall is as follows:

$$R(K)=R(1)+(K-1)(Hc\cdot \cos \beta +D)\tan (90°-{\gamma }_W+{\gamma }_C),$$

(4)

The sun’s altitude angle $\alpha$ is given by the following:

$$\sin \alpha =\sin \varphi \sin \delta +\cos \varphi \cos \delta \cos \omega$$

(5)

where $\varphi$ is the collector latitude, $\delta$ is the sun declination angle, ${\gamma }_S$ is the solar azimuth angle, ${\gamma }_W$ is the wall angle, and $\omega$ is the hour angle.

The shaded triangle area (Equations (2) and (3)) is $0.5\times H_{W,sh}{L}_{W,sh}$.

2.2. Inter-Row Shading on the Second Collector, $K=2$

The equations for the inter-row shadow height and length, respectively, on the second and subsequent collectors cast by the preceding collectors are given in reference [1]:

$$H{c}_{sh}=Hc\cdot (1-{\displaystyle \frac{D+Hc\cdot \cos \beta }{Hc\cdot \cos \beta +Hc\cdot \sin \beta \cos ({\gamma }_S-{\gamma }_C)/\tan \alpha }})$$

(6)

$$L{c}_{sh}=Lc-(D+Hc\cdot \cos \beta ){\displaystyle \frac{\sin \beta \sin \left|({\gamma }_S-{\gamma }_C\right|/\tan \alpha }{\cos \beta +\sin \beta \cos ({\gamma }_S-{\gamma }_C)/\tan \alpha }}$$

(7)

2.3. Sky View Factors for Diffuse Radiation

The sky view factor of a horizontal plane $V{F}_R^h$ for the collector number $K$ at a distance $R(K)$ (see Figure 1) from the collectors in the presence of a wall, is given in reference [2]:

$$V{F}_R^h={\displaystyle \frac{L_C+{\left\{{[L_C+R(K)]}^2+H_W^2\right\}}^{1/2}-[{(R{(K)}^2+H_W^2]}^{1/2}}{2\times L_C}}$$

(8)

where

$$R(K)=R(1)+(K-1)(H_C\times \cos \beta +D)\times \tan (90°-{\gamma }_W+{\gamma }_C)$$

(9)

The sky view factor [16] of the first collector, $K=1$ (see Figure 1) is:

$$V{F}^C=(1+\cos \beta )/2$$

(10)

The inter-row sky view factor (in multiple collector rows) of the second and subsequent rows is obtained based on the “cross-string rule” by Hottel [17]:

$$V{F}^C={\displaystyle \frac{H_C+D+H_C\cos \beta -{[D^2+{(H_C\sin \beta )}^2]}^{1/2}}{2\times H_C}}$$

(11)

where $D$ is the inter-row spacing.

The combined sky factor $V{F}^{comb.}$ of a collector comprises the sky view factor caused by the wall with respect to the horizontal plane $V{F}_R^h$, and the inter-row sky view factor of the collector $V{F}^C$, i.e.,

$$V{F}^{comb.}=V{F}^C\times V{F}_R^h$$

(12)

The incident diffuse radiation $G_d^{c,w}$ on a collector, in the presence of a wall (see Equation (13)) thus becomes:

$$G_d^{c,w}=V{F}^{comb.}{G}_{dh}$$

(13)

where $G_{dh}$ is the diffuse radiation on a horizontal plane, and the diffuse radiation losses (masking losses) are as follows:

$$\Delta G_C^W=(1-V{F}^{comb.})\cdot G_{dh}$$

(14)

2.4. Incident Beam Radiation

The in-plane (incident) beam radiation on a collector is given by:

$$G_{beam}=G_b\cos \theta \times (H_C\times L_C-S_{sh}),$$

(15)

where $\theta$ is the angle between solar rays and the normal to the collector surface.

Generally, both types of shadow patterns may occur on a collector during the day. The shaded area $S_{sh}$ on a collector is determined according to the marked areas in Figure 1. The wall height shadow $H_{W,sh}$ may exceed the collector width $H_C$, and overlapping of the shaded areas (wall and inter-row) may take place during the day. The calculation of the effective shaded areas considers these events. The wall shading losses are calculated as follows:

$$\Delta G_b^w=G_b\cos \theta \times S_{sh}$$

(16)

The global incident solar radiation on the PV system (neglecting the wall and ground albedo) comprises the direct beam Equation (15) and the diffuse Equation (13) radiation.

3. Results

The study on the solar radiation losses (wall and inter-row shading, and wall and inter-row masking) and the incident solar radiation of photovoltaic systems deployed near obscuring walls, pertains to solar PV fields comprising $K=20$ collector rows with collector length and height $L_C=20\mathrm{m}$, $H_C=2.12\mathrm{m}$, respectively. The collectors’ inclination angle is $\beta =20°$, and the inter-row distance is $D=1.05\mathrm{m}$ (based on “no shading” on a winter solstice day at solar noon). The distance between the wall and the collectors is $R(1)=2\mathrm{m}$, and the wall height $H_W$ is a variable parameter, see Figure 1. The incident solar radiation is based on 10 min solar radiation data (direct beam and diffuse radiation, average data for years 2014–2023, Israel, Meteorological Service–IMS) for Tel Aviv, at latitude $\varphi =32°6^{\prime } \mathrm{N}$ and longitude $34°{51}^{\prime } \mathrm{E}$. The solar radiation losses and the incident solar radiation on the collectors are calculated for four possible configurations of PV system deployments near walls. The effect of wall height on the direct beam and diffuse and global radiation is investigated, in addition to the variation of shadow height and length on the collectors caused by the obscuring walls and collectors. As the distribution of the incident radiation among the different collectors is of utmost importance, the distribution is also examined. The parameters of the PV field in the present study include the following: $H_C, L_C,\beta ,K,D,{\gamma }_C,{\gamma }_W,R(1),H_W$. The variable parameters are ${\gamma }_C,{\gamma }_W,H_W$, and the remaining parameters are assumed to be constant values.

3.1. Collectors Deployed with ${\gamma }_C=0°$ and a Wall with ${\gamma }_W=90°$

A deployment of PV collectors near a vertical wall is depicted in Figure 2. The collectors are facing south (${\gamma }_C=0°$), and a wall is erected on the west side of the collectors with an azimuth angle ${\gamma }_W=90°$.

As the wall shading is affected by the direct beam radiation, Figure 3 shows the daily variation of the incident beam radiation, in W/m², on a collector on August 1, for a wall height $4\mathrm{m}$ (in blue) and for a wall height $10\mathrm{m}$ (in black), as compared to no wall shading in red. As the wall is placed on the west side of the PV collectors, the shading occurs in the afternoon hours. The calculation of the wall shading is based on Equations (1) to (3). The figure clearly shows a reduction in the incident beam radiation, due to wall shading, with the increase of wall height.

It is interesting to observe the variation of the wall shadow height and length during the day. The variation is depicted in Figure 4 on 21 June for the parameters $L_C=20\mathrm{m}$, wall height $H_W=2\mathrm{m}$, and $H_C=2.12\mathrm{m},\beta =20°,D=1.05\mathrm{m},R(1)=2\mathrm{m}$ using the average solar radiation data. At 16:00, the shadow forms a triangle shape, marked in yellow, where $H_{W,sh}=1.43\mathrm{m}$ and $L_{W,sh}=2.26\mathrm{m}$; at 17:00, $H_{W,sh}=3.1\mathrm{m}$ and $L_{W,sh}=2.3\mathrm{m}$ marked in pale yellow. At 17:00, the shadow height exceeds the collector height $H_C=2.12\mathrm{m}$; therefore, the shaded height $2.12\mathrm{m}$ is taken into account for the calculation of the shaded area of the collector.

The inter-row shading expressions are given in Equations (6) and (7). Figure 5 describes, for example, the variation in the shadow height and length with time (in the afternoon) on 21 June, 21 November, and 21 December, on the second and subsequent collectors, see Figure 1. There is no shading on the collectors on 21 June. The shadow height and length (and area) are indicated by the green rectangle for 15:00 on 21 December.

The masking losses (diffuse radiation losses) and the direct beam losses are given in Equations (14) and (16), respectively, adding up to the global radiation losses. Figure 6 depicts the percentage of monthly global losses for different wall heights $H_W$. The global losses are less evident in summer months compared to winter months. The effect of wall height is apparent in the figure.

The Incident Solar Radiation on the PV System

The annual incident solar radiation (beam, diffuse, and global) in kWh of the PV system deployed with ${\gamma }_C=0°$, and a wall in a north–south direction,${\gamma }_W=90°$, on the west side of the collector (see Figure 2), is tabulated in

Table 1. Annual beam, diffuse, and global incident radiation on the PV system, for R = 2 m and $K=20,H_C=2.12\mathrm{m},L_C=20\mathrm{m},D=1.05\mathrm{m},\beta =20°,{\gamma }_C=0°,{\gamma }_W=90°$.

${\mathit{H}}_{\mathit{W}}=2\mathbf{m}$ Row	Annual Beam Radiation [kWh]	Annual Diffuse Radiation [kWh]	Annual Global Radiation [kWh]
1	59,117	24,325	83,441
2	58,828	22,988	81,816
All System	1,176,853	461,089	1,637,942
${\mathit{H}}_{\mathit{W}}\mathbf{=}\mathbf{3}\mathbf{m}$ Row	Annual Beam Radiation [kWh]	Annual Diffuse Radiation [kWh]	Annual Global Radiation [kWh]
1	58,070	23,913	81,983
2	57,828	22,599	80,427
All System	1,156,802	453,288	1,610,090
${\mathit{H}}_{\mathit{W}}\mathbf{=}\mathbf{4}\mathbf{m}$ Row	Annual Beam Radiation [kWh]	Annual Diffuse Radiation [kWh]	Annual Global Radiation [kWh]
1	56,869	23,474	80,342
2	56,661	22,183	78,845
All System	1,133,435	444,956	1,578,391

for the system parameters listed in the table. The table contains three parts for each wall height $Hw=2,3,4\mathrm{m}$. Collector row number 1 is subject to wall shading (beam radiation) and masking (diffuse radiation using Equation (10). Collector row number 2 and the subsequent collector rows are subject to wall and inter-row shading and masking as presented in Equation (11). The last line of each part in the table lists the “all system” (for $K=20$) annual incident radiation (beam, diffuse, and global). The table shows that the incident radiations decrease for higher walls. The incident radiation is evenly distributed (see Equation (9), $R(K)=R(1)$) across the collector rows of the system for row numbers $K$ $\ge 2$.

3.2. Collectors Deployed with ${\gamma }_C=30°$ and a Wall with ${\gamma }_W=90°$

A possible deployment of PV collectors near a vertical wall is depicted in Figure 7. The collectors are facing southward with an azimuth angle ${\gamma }_C=30°$ and a wall is erected on the west side of the collectors with an azimuth angle ${\gamma }_W=90°$.

In this section, we do not report on the daily variation of the solar radiation since the variation is similar to Figure 3 in Section 3.1. Here we report only the annual incident solar radiation.

The annual incident radiation (beam, diffuse, and global) on each collector row, in kWh, is shown in

Table 2. Annual beam, diffuse, and global incident radiation on the PV system for $R(1)=2\mathrm{m}$ and $K=20$, $H_C=2.12\mathrm{m},L_C=20\mathrm{m},D=1.05\mathrm{m},\beta =20°,{\gamma }_C=30°,{\gamma }_W=90°$.

${\mathit{H}}_{\mathit{W}}=4\mathbf{m}$ Row No.	Annual Beam Radiation [kWh]	Annual Diffuse Radiation [kWh]	Annual Global Radiation [kWh]
1	57,633	23,474	81,106
2	57,660	22,602	80,262
3	58,141	22,842	80,983
4	58,444	22,989	81,432
5	58,626	23,084	81,710
6	58,744	23,151	81,895
7	58,844	23,198	82,043
8	58,899	23,234	82,133
9	58,953	23,262	82,215
10	58,996	23,283	82,279
11	59,018	23,301	82,318
12	59,045	23,315	82,360
13	59,062	23,327	82,389
14	59,078	23,336	82,414
15	59,099	23,345	82,444
16	59,109	23,352	82,461
17	59,117	23,358	82,475
18	59,122	23,363	82,485
19	59,130	23,368	82,498
20	59,138	23,372	82,510
All System	1,175,856	464,554	1,640,410

for parameters listed in the table, and for a wall height of $4\mathrm{m}$. The results show uneven distribution of the radiation among the different collector rows because the distance of each collector to the wall increases with the collector number (see Equation (9)). The difference in radiation between collector no. 2 and collector no. 20 is 2.8%; however, the difference in the radiation between successive collectors is less than one percent (depending on the parameter values of $R(K)$ in Equation (9)).

3.3. Collectors Deployed with ${\gamma }_C=0°$ and a Wall with ${\gamma }_W=30°$

A possible deployment of PV collectors near a vertical wall is depicted in Figure 8. The collectors are facing south (${\gamma }_C=0°$) and a wall is erected on the west side of the collectors with an azimuth angle ${\gamma }_W=30°$.

In this section, we do not report again on the daily variation of the solar radiation, and we report only on the annual incident solar radiation of the entire system. The annual incident radiation (beam, diffuse, and global), in kWh, is shown in

Table 3. Annual beam, diffuse, and global incident radiation on the PV system, $K=20,H_C=2.12\mathrm{m},L_C=20\mathrm{m},D=1.05\mathrm{m},\beta =20°,{\gamma }_C=0°,{\gamma }_W=30°,R(1)=2\mathrm{m}$.

All System	Annual Beam Radiation [kWh]	Annual Diffuse Radiation [kWh]	Annual Global Radiation [kWh]
2 m	1,183,774	468,249	1,652,023
3 m	1,180,268	466,606	1,646,875
4 m	1,175,856	464,554	1,640,410

for the system parameters listed in the table and for a wall height of $2,3,4\mathrm{m}$. The incident solar radiation on the different collectors is unevenly distributed across the PV rows; however, the difference in the radiation between successive collectors is less than one percent (depending on the parameter values of $R(K)$ in Equation (9)). The results show a reduction in the annual solar radiation with wall height.

3.4. Collectors Deployed with ${\gamma }_C=-30°$ and a Wall with ${\gamma }_W=30°$

A possible deployment of PV collectors near a vertical wall is depicted in Figure 9. The collectors are facing southward with an azimuth angle ${\gamma }_C=-30°$ and a wall is erected on the west side of the collectors with an azimuth angle ${\gamma }_W=30°$ (collectors are in parallel with the wall).

In this section, we report again only on the incident solar radiation of the entire system. The annual incident radiation (beam, diffuse, and global), in kWh, of the PV system is shown in

Table 4. Annual beam, diffuse, and global incident radiation on the PV system, $K=20,H_C=2.12\mathrm{m},L_C=20\mathrm{m},D=1.05\mathrm{m},\beta =20°,{\gamma }_C=-30°,{\gamma }_W=30°,R(1)=2\mathrm{m}$.

All System	Annual Beam Radiation [kWh]	Annual Diffuse Radiation [kWh]	Annual Global Radiation [kWh]
2 m	1,081,768	468,249	1,550,017
3 m	1,073,494	466,606	1,540,100
4 m	1,060,302	464,554	1,524,856

for the system parameters listed in the table for a wall height $2,3,4\mathrm{m}$. The incident solar radiation across the different collector rows is evenly distributed (depending on the parameter values of $R(K)$ in Equation (9)). The results show a reduction in the solar radiation with wall height.

3.5. Annual Incident Solar Radiation and Shading Losses of the PV Systems

A comparison of the annual global radiation and the annual shading losses for the four different configurations of the PV systems is presented in this section. The annual incident global radiation on a PV system, without the presence of a wall, includes inter-row shading and masking losses. The annual incident global radiation on a PV system, with the presence of a wall, includes, in addition, wall shading and wall masking losses. The percentage of global radiation losses of the PV system is defined by the difference between these two annual radiations (with and without a wall), relative to the annual incident global radiation without the presence of a wall. The annual incident (beam, diffuse, and global) radiation on the PV system, without the presence of a wall is calculated for the following system parameters in the study: $K=20,H_C=2.12\mathrm{m},L_C=20\mathrm{m},D=1.05\mathrm{m},\beta =20°,{\gamma }_C=0°$, and the results are as follows: beam—1,191,231 kWh, diffuse—469,752 kWh, and global—1,660,983 kWh. The annual incident global radiation on the different deployments of the PV systems and for different heights of the wall $H_W$ (Section 3.1, Section 3.2,Section 3.3 and Section 3.4) is presented in

Table 5. Annual incident global radiation on PV systems. $K=20,H_C=2.12\mathrm{m},L_C=20\mathrm{m},D=1.05\mathrm{m},\beta =20°$.

Deployment	Global Radiation [kWh] ${\mathit{H}}_{\mathit{W}}=2\mathbf{m}$	Global Radiation [kWh] ${\mathit{H}}_{\mathit{W}}=3\mathbf{m}$	Global Radiation [kWh] ${\mathit{H}}_{\mathit{W}}=4\mathbf{m}$
0	1,660,983	1,660,983	1,660,983
${\gamma }_C=0°,{\gamma }_W=90°$	1,637,942	1,610,090	1,578,391
${\gamma }_C=30°,{\gamma }_W=90°$	1,652,023	1,646,875	1,640,410
${\gamma }_C=0°,{\gamma }_W=30°$	1,655,624	1,648,022	1,638,534
${\gamma }_C=-30°,{\gamma }_W=30°$	1,550,017	1,540,100	1,524,856

. Deployment “0” refers to the deployment without a wall for ${\gamma }_C=0°$. The percentage of annual global radiation losses of the different PV systems is presented in

Table 6. Percentage of annual global radiation losses of PV systems. $K=20,H_C=2.12\mathrm{m},L_C=20\mathrm{m},D=1.05\mathrm{m},\beta =20°$.

Deployment	Percentage Global Radiation Loss [%] $({\mathit{H}}_{\mathit{W}}=2\mathbf{m})$	Percentage Global Radiation Loss [%] $({\mathit{H}}_{\mathit{W}}=3\mathbf{m})$	Percentage Global Radiation Loss [%] $({\mathit{H}}_{\mathit{W}}=4\mathbf{m})$
${\gamma }_C=0°,{\gamma }_W=90°$	1.39%	3.06%	4.97%
${\gamma }_C=30°,{\gamma }_W=90°$	0.54%	0.85%	1.24%
${\gamma }_C=0°,{\gamma }_W=30°$	0.32%	0.78%	1.35%
${\gamma }_C=-30°,{\gamma }_W=30°$	6.68%	7.28%	8.20%

. Deployments where ${\gamma }_C=30°,{\gamma }_W=90°$ and ${\gamma }_C=0°,{\gamma }_W=30°$ suffer low losses, whereas deployments where ${\gamma }_C=0°,{\gamma }_W=90°$ and ${\gamma }_C=-30°,{\gamma }_W=30°$ suffer considerable losses.

4. Discussion

The present article complements the studies in [1, 2] and calculates the incident beam and diffuse and global radiation and their distribution across the collector rows for PV systems installed near obscuring walls in an urban environment. Article [1] formulates mathematical expressions for shadows caused by walls and by the collectors, and calculates the combined losses affected by wall height, distance from walls to the collectors, and length and azimuth angles of collectors. Article [2] formulates mathematical expressions for sky view factors of PV systems deployed near obscuring walls and investigates the variation of the sky view factors affected by wall height, distance of the wall to the collectors, collector length, and collector and wall azimuth angles. The sky view factors are associated with the incident diffuse radiation on the collectors. The mathematical expressions for shading parameters contain the following solar field and collector rows design parameters: $H_C, L_C,\beta ,K,D,{\gamma }_C,{\gamma }_W,R(1),H_W$; therefore, PV systems may be analyzed by considering any of the above parameters. The present article calculates the incident beam, diffuse and global radiation, and the shading losses for four PV system configurations, considering the collector azimuth angle ${\gamma }_C$, wall azimuth angle ${\gamma }_W$, and wall height $H_W$ as variable parameters, and the remaining parameters being constant values. The height $H_C\times \sin \beta$ of the collector dictates the minimum wall height. The incident solar radiation is calculated for walls erected on the west side of the PV collectors; however, the calculations may also pertain to walls on the east side, as well as on both sides of the PV collectors (with some modifications of the mathematical expressions). In addition, the mathematical expressions (algorithms) for the shadows are valid for any PV system at any desired location in the built environment. The study in [3] estimates that about 74% of the electricity consumption of the city of Melbourne can be generated by photovoltaic panels, and cities may supply their own electricity by BIPV technology, as stated in [4]. Shading of different kinds on the PV collectors may reduce about 20% of the generated electricity, as reported in [6]. In a high-density populated area, the mutual shading by buildings may reduce the PV output energy by up to 47.5%, as stated in [8]. PV systems deployed in open space suffer much lower shading losses caused by the collector rows. The experimental study in [13] showed about 5% shading losses. The mentioned studies do not deal analytically with PV systems erected near building walls that cause shadows on the PV collectors. Therefore, the present article adds insight into the incident solar radiation and shading losses of PV systems. The calculation of the incident radiation on PV systems is based on the mathematical expressions developed in [1, 2] for wall and inter-row shading losses and for wall and inter-row diffuse radiation (masking) losses. In addition, the article analyses shading patterns and uniformity of the incident solar radiation across the different collector rows. Some software programs include shading tools for simulating obscuring objects; however, these algorithms (mathematical expressions) are not apparent (visible) to the user of the programs for judging the correctness and accuracy of the expressions.

5. Conclusions

The deployment of solar PV systems on rooftops in urban environments utilizes the rooftop areas for electricity generation; however, they may encounter shading on the PV collectors from surrounding building walls. In addition, the PV system suffers from inherent inter-row shading. The combined shading losses (wall and inter-row) may result in high shading losses. Moreover, the sky view factors caused by the wall and the inter-row sky view factor reduce the incident diffuse radiation on the collectors, thus causing masking losses. Wall shading and masking losses depend on the height of the wall, the distance of the wall to the PV collectors, and the orientation of the wall relative to the PV collectors. Since the incident of solar radiation is reduced, the economics of the PV system installation may become questionable and justify investigation. The article calculates the incident beam, diffuse and global radiation, and the percentage of shading and masking losses of PV systems installed near western obscuring walls, for four configurations of deployments of the PV systems. In addition, the article analyses the shading patterns and the uniformity of the incident solar radiation on the different collector rows. The simulation results indicate an almost uniform distribution of the incident radiation between the collector rows for the four configurations of the deployments. On the other hand, the radiation losses may reach as high as 8 percent for deployments where ${\gamma }_C=-30°,{\gamma }_W=30°$, for a wall height of 4 m above the collector surface. Wall shading losses dominate the inter-row losses. The article presents an insight into the incident solar radiation and the radiation losses (shading and masking) for PV systems deployed near obscuring walls. The provided mathematical expressions (algorithms) are valid for any PV system at any desired location and, therefore, are of practical importance for designers of PV systems in urban environments.