# 3D Solar Irradiance Model for Non-Uniform Shading Environments Using Shading (Aperture) Matrix Enhanced by Local Coordinate System

^{*}

## Abstract

**:**

## 1. Introduction

^{2}and a battery size of 40 kWh. Assuming a vehicle-integrated photovoltaic (VIPV) of a capacity of 1 kW, 70% of cars (traveling less than 30 km/day) are likely to operate on solar energy [4,5,6]. The likely sales would be 50 GW/year [7]. In these expectations, the solar cells are assumed to have been stabilized with little or no degradation. On the other hand, certain solar cells, such as amorphous Si [8], perovskite [9], crystalline Si [10], and Si modules [11], degrade over time. This aspect should be considered in the energy value in the lifetime of the product. These devices, when mounted or integrated with the car body, are commonly referred to as vehicle-integrated photovoltaics (VIPVs). Moreover, VIPV systems are effective in providing auxiliary power and extending the range of vehicles, including hybrid vehicles [12] and battery electric vehicles (BEVs) [13]. They also serve as a research area for innovation in the utilization of solar energy [14], particularly in auxiliary power applications [15]. Several renowned car manufacturers, including Ford [16], Toyota [17], Karma [18], Hanergy [19], and Nissan [20], have developed impressive demonstration cars showcasing the integration of vehicle-integrated photovoltaic (VIPV) technology. Several demonstration programs have validated the use of PV as an energy source for EVs. University educational programs have proposed several innovative designs for SEVs. Sierra and Reinders examined the integration of PV charging stations [21], and Kanz et al. designed lightweight cars [22]. Several authors have independently analyzed the impact of SEVs on society and environment, including the use of energy in disaster zones [23]. Several design advances have been achieved, such as the introduction of lightweight PV modules [24], interconnection designs [25], and vehicle patterns [26]. Kanz et al. analyzed the environmental impact [22], Mallon discussed the advantages of battery lifetime [27], and Pinto examined the range extension [13]. The overall state was reviewed by Conti [28]. Lightyear [29] and Sono Motors [30] are on the threshold of developing a new market for VIPVs.

- The curved surface [31];
- Frequent variation in the orientation angle of a VIPV during driving [31];
- Relatively higher probability for VIPVs to be shaded by objects [31];
- Relatively small size of shading objects, such as street trees and signals, and the consequent high influence of partial shading loss on these [31];
- Temperature variation (parking and driving) [31];
- Dynamic fluctuation of the solar spectrum [31];
- Rapid fluctuations in solar irradiance (in milliseconds) owing to dynamic partial shading [31].

- Solar irradiance onto the VIPV (orthogonal five orientations-roof and four sides, front, left, tail, and right);
- Distribution of shaded objects, estimation of solar irradiance, and irradiation on an arbitrary reference plane tangential to the curved surface of the VIPV.

- Partial shading impact;
- Dynamic shading impact (only for the VIPV);
- Testing the VIPV indoors or outdoors for rating;
- I–V curve measurement during driving (only for the VIPV);
- Influence of power on the curved surface;
- Transient output of the modules, which is affected by the capacitance and other transient characteristics of the device (only for the VIPV);
- Temperature measurements;
- Spectrum correction.

- Solar irradiance in other zones such as residential and open zones;
- Solar irradiance on car sides;
- Simultaneous validation of the five orthogonal orientations (i.e., evidence on only one axis may not be substantial);
- Pathway to the energy rating of the VIPV;
- Consistency among different shading environments, car orientations, and car sides.

## 2. Methods

#### 2.1. Local Coordinate System

#### 2.2. Solar Irradiance Measurement Influenced by Shading Objects

#### 2.3. Measurement Methods of the Distribution of Shading Objects

- A fisheye image of the sky was captured. Without blue-colored structures, such as blue signs and walls, the best capturing condition was a clear-sky day (no clouds) with certain shading objects shading the sun. In this case, a fisheye video or camera were used to generate the RGB images. The alternative timings were sunrise or sunset (Figure 4). The recommended fisheye video system is the model WV-S4550L made by Panasonic, Japan;
- A median filter was applied to remove spot-like image noise;
- If obtained under blue-sky conditions, decomposition into red, green, and blue images were effective. A differential image matrix (gray-scale matrix), such as 2
**B**− (**G**+**R**), converted the sky into white and building walls into black regardless of whether these reflected sunlight (Figure 4).**B**,**G**, and**R**represented the image matrices decomposed from RGB images. A median image filter could erase spot-like noise by applying the median number of five adjacent elements. Note that differential image matrix calculations, such as 2**B**− (**G**+**R**), induced errors in the image, including bright-blue walls or signs. A typical mistake was considering bright-blue objects as part of the sky. In such cases, the image may have been replaced with the ones captured at low sun height (dark sky condition), as shown in Figure 4; - The filtered fisheye images were then binarized. The best threshold was determined conveniently using the median point of the two peaks of the brightness histogram (one peak corresponded to the open sky and the other corresponded to shading objects) (Figure 5);
- The aperture matrix
**E**was generated using a 2D histogram. The matrix elements ranged from zero to one (0: shaded, 1: unshaded). Shading implied shading a point in a hemispherical sky, instead of “shading the sun”. That is, the (i, j) elements,**E**_{i,j}, were the densities of the unshaded points in the 2D bin of the elevation angles [i°, (i + 1)°] (i = 0, 1, …. 89) and orientation angles [4j°, 4(j + 1)°] (j = 0, 1, …., 89). The image matrix could be conveniently converted from a polar coordinate system to an orthogonal coordinate system before the elements were counted in 2D bins (Figure 4). Occasionally, the reflection by a window was identified as the sky. However, it may have been filled in black (or zero in the matrix) manually, or erosion may have been applied using image processing.

#### 2.4. Definition of the Aperture (Shading) Matrix

**E**was a 90 × 90 matrix containing the unshaded probabilities. The matrix elements ranged from zero to one (0: shaded, 1: unshaded). Shading implied shading a point in a hemispherical sky instead of “shading the sun”. That is, the (i, j) elements,

**E**

_{i,j}, were the densities of the unshaded points in the 2D bin of the elevation angles [i°, (i + 1)°] (i = 0, 1, …., 89) and orientation angles [4j°, 4(j + 1)°] (j = 0, 1, …., 89). The values of the elements

**E**

_{i,j}ranged from zero to one. The structure of the aperture matrix

**E**is shown in Figure 6. Furthermore, the orientation angle indicated the local orientation angle defined by φ in Equation (1) and Figure 1.

**x**

_{i,j},

**y**

_{i,j}) used in the aperture matrix

**E**was converted using Equations (2)–(6):

**E**was calculated using Equations (7)–(9):

**θθ**and

**φφ**were vectors of the bins of the local elevation angle and local orientation angle, respectively.

**Bde**was a binarized image of the hemispherical sky (1: bright, 0: dark).

**E**, the sky-view factor (SVF) was calculated using Equation (10):

**M**) calculated the arithmetic mean of all the elements in matrix

**M**.

#### 2.5. 3D Solar Irradiance Calculation Using the Aperture Matrix

**E**.

**θ**was the grazing angle from the car roof;

**φ**was the orientation angle (origin: south, clockwise); MM and NN were the number of divisions including the edge side of elevation and orientation angles, respectively; $\alpha \Phi $ was the local orientation angle of the car (origin: south, clockwise);

**RD**was the vertical direct sunlight to the opposite side; $S{I}_{f}$, $S{I}_{r}$, $S{I}_{b}$, and $S{I}_{l}$ were the solar irradiance of the front, right, back, and left sides, respectively, given by Equation (21); and ${R}_{v}$ was the reflectance of shading objects (=0.25).

_{side}was the irradiance on the vehicle side, $S{I}_{side}$ was the diffused irradiance on the vehicle side, $D{I}_{side}$ was the direct irradiance on the vehicle side, $R{I}_{side}$ was the reflected irradiance on the vehicle side from vertically shaded objects, and $RI{S}_{side}$ was the reflected irradiance on the vehicle side from the street surface.

## 3. Results

#### 3.1. Shading Probability Distribution

**E**(Figure 7). Because the distribution of shaded objects is not axially symmetrical along the street (along the x-direction), the distribution curve varies in the x- and y-directions (Figure 7).

#### 3.2. Model Validation by Measurements

## 4. Discussion

#### 4.1. Categorizing the Shading Environment

#### 4.2. Estimation of Annual Solar Irradiance Using Shading Probability Distribution

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

2D | two-dimensional |

3D | three-dimensional |

BEV | battery electric vehicle |

BIPV | building-integrated photovoltaics |

DC | direct current |

DNI | direct normal irradiance |

ECU | electronic control unit |

EV | electric vehicle |

GHI | global horizonal irradiance |

GPS | global positioning system |

IEC | International Electrotechnical Commission |

I-V | current-voltage |

MPPT | maximum power point tracking |

PV | photovoltaic |

RGB | red-green-blue |

SEV | solar electric vehicle |

Si | silicon |

SVF | sky view factor |

VIPV | vehicle-integrated photovoltaics |

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**Figure 2.**Local coordinate system around the car body for VIPVs [23].

**Figure 3.**Measurement system for 3D solar irradiance on five orientations by the local coordinate system [23].

**Figure 5.**Image processing and 2D histogram [23].

**Figure 6.**Structure of the aperture matrix [23].

**Figure 9.**Bar chart comparing the measured and calculated solar irradiance on five orthogonal orientations with regard to global horizontal irradiance (GHI) without shading (stationary measurement on the roof of the building (eight floors)).

**Figure 10.**Comparison of the plot in Figure 9 in various shading environments and car orientations with regard to sky view factor (SVF).

**Figure 11.**Comparison between the measured and calculated values when distributed along the 45° line.

**Figure 14.**Typical sky images (fisheye camera) of the three zones: (

**a**) Open zone; (

**b**) Residential zone; and (

**c**) Building zone.

**Figure 15.**Shading probabilities of the three zones (open, residential, and building zones). The red and blue lines indicate the probability in the y-direction (front–rear direction) and x-direction (left–right direction), respectively [23].

**Figure 16.**Distribution of solar irradiance on car roofs in buildings zone, residential zone, and open zone.

Conditions | |
---|---|

Pyranometer mount | The angular error should be at most 1°. The pyranometers should be fixed tightly to the vehicle structure, with no vibration resonance during driving. The absorber of pyranometers should capture the entire hemisphere and not be shaded by the vehicle body. |

Pyranometer performance | The time constant should be at most 5 s. The temperature error should be at most ±5%/K. Class B or better |

**Table 2.**Parameter list of the shading probability function. Typical sky view factors (SVF) for the open, residential, and building zones are 0.95, 0.75, and 0.55, respectively.

Category | Orientation | SVF | ${\mathit{\mu}}_{\mathit{s}}$ | $\mathit{s}$ |
---|---|---|---|---|

Open zone | X | 0.95 | 2.8° | 1.2° |

Y | 2.2° | |||

Residential zone | X | 0.75 | 24.0° | 7.9° |

Y | 19.1° | |||

Building zone | X | 0.55 | 45.1° | 10.3° |

Y | 36.0° |

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Araki, K.; Ota, Y.; Nagaoka, A.; Nishioka, K.
3D Solar Irradiance Model for Non-Uniform Shading Environments Using Shading (Aperture) Matrix Enhanced by Local Coordinate System. *Energies* **2023**, *16*, 4414.
https://doi.org/10.3390/en16114414

**AMA Style**

Araki K, Ota Y, Nagaoka A, Nishioka K.
3D Solar Irradiance Model for Non-Uniform Shading Environments Using Shading (Aperture) Matrix Enhanced by Local Coordinate System. *Energies*. 2023; 16(11):4414.
https://doi.org/10.3390/en16114414

**Chicago/Turabian Style**

Araki, Kenji, Yasuyuki Ota, Akira Nagaoka, and Kensuke Nishioka.
2023. "3D Solar Irradiance Model for Non-Uniform Shading Environments Using Shading (Aperture) Matrix Enhanced by Local Coordinate System" *Energies* 16, no. 11: 4414.
https://doi.org/10.3390/en16114414