## 1. Introduction

Cities in developed countries are undergoing fast growth due to different factors such as the increase in the global population [

1] or migration from rural environments and underdeveloped countries, among others [

2,

3]. In fact, more than half of the world’s population now lives in cities and this number is expected to reach 66% by 2050 [

4,

5]. As a result of this urban population increase, major problems related to the sustainability of cities are emerging, such as the worsening of air quality as a consequence of the use of fossil fuels for transportation or heating of buildings [

3,

5,

6,

7]. Specifically, in terms of energy supply, it is estimated that 75% of total energy is consumed in cities [

3] and that this consumption will double over the next three decades [

8,

9]. These circumstances, together with the environmental problems associated with fossil fuels [

10], have encouraged research and development of renewable energies in order to improve energy efficiency and sustainability in cities [

5,

9]. Among these renewable energy sources, solar energy stands out as a source of clean energy, abundant and available, to a greater or lesser extent, throughout the Earth [

11].

In addition, urban planning has always searched, as the main goal, an ideal integration of living spaces (buildings and squares), communication systems (roads and streets) and land area (topography) [

12]. However, the lack of available space in cities has caused the appearance of new neighborhood configurations, which only give importance to the rise of population density [

13]. Therefore, nowadays, parameters such as population density, street width, and accessibility, determine the typology of new neighborhoods [

14]. Nevertheless, despite the harnessing of solar energy in cities becoming an obligation in new dwellings to make them sustainable, solar radiation levels are not frequently taken into account when making decisions about urban planning.

In fact, to develop energy efficiency measures in new buildings it is necessary to know the levels of solar radiation reaching every piece of the building which could be used to install solar panels or thermal collectors [

15,

16]. In addition, this information about solar radiation can also be used for the estimation of natural lighting on its windows and, consequently, to guarantee solar rights [

17], especially in cities with a great presence of skyscrapers. Daylight also has a positive influence on the health and human behavior of the residents [

18,

19] and it contributes to improving the indoor climate, increasing thermal comfort, and, consequently, reducing the energy demand of a dwelling [

19,

20,

21,

22,

23,

24]. For all these reasons, an in-depth knowledge of the level of available solar radiation on the façades of buildings in cities is necessary [

25].

Furthermore, complete knowledge of the solar radiation on façades in complex cities also allows developing new passive solar building designs [

26], making it easier to choose the ideal materials for windows (transparent or translucent polymer) and their layout in each case [

27]. These techniques turn out to be especially interesting in areas where the heating loads in buildings represent an important part of the electricity bill. With this in mind, Building Performance Simulation (BPS) tools compare different design alternatives related to the efficiency and energy consumption in buildings, providing useful and quick information to the technicians [

28]. Owing to the importance of the level of solar irradiance on façades and roofs [

29,

30], architects should consider it during the early phases of their projects. In this line of work, Tang Minfang [

31] studied the effect of the azimuth angle and the height of the main façades of a building on the available solar radiation and Salazar Trujillo [

32] described the influence of solar radiation on the temperatures inside the rooms in order to improve energy efficiency.

However, in cities, this analysis can prove complex [

23], due to several interactions existing, including those with neighboring buildings or the effect of the trees [

33] and the fact that each neighborhood must be studied independently [

34].

Geographic Information System (GIS) techniques allow representing complex cities and can be used for the estimation of the most appropriate parts of a building for the installation of PV panels [

35] or identifying the zones of optimal solar energy potential [

36]. Besides, using these techniques, the results may be scalable and automated, in comparison with points-based methods [

37]. A good example of a methodology based in GIS is the Solar Energy Planning (SEP) developed by Gadsden et al. [

38], which not only can predict the energy consumed by dwellings but also the achievable power saving when using PV systems, solar-assisted hot water or passive solar design.

Several software applications have been developed to study the distribution of solar radiation in complex cities. One of the most important is the Heliodon. This tool, designed by Benoit Beckers and Luc Masset, graphically represents the solar irradiance reaching building façades. However, to minimize the computation time, it only considers the direct component of solar radiation [

39]. Additionally, Solene software, designed by the Centre de Recherche Méthodologique d’Architecture (CERMA) analyzes sunlight in cities [

40]. It allows determining shadows between buildings as well as daylight both inside and outside a building. Accordingly, it is quite useful as a tool for architects who can easily simulate daylight when deciding window distribution on façades and roofs.

In this paper, a new characterization of the solar radiation reaching the building façades of neighborhoods of different typologies is presented. As an innovation, it takes into account not only direct solar radiation but also the diffuse and reflected components. In that way, a new framework for characterizing solar radiation reaching building façades in urban environments is provided. This framework is applied in two neighborhoods with different typologies in Cordoba (Spain) in order to determine the influence of the neighborhood morphology on solar access. Finally, a new correlation to estimate the solar radiation on the façades of the buildings of each neighborhood has been determined.

## 4. Discussion and Regressions

As explained before, the USC value has been calculated for the 121 points considered (

Figure 7) on each façade of neighborhoods A and B. Specifically, 14,036 USC values for neighborhood A and 13,673 USC values for neighborhood B have been obtained. To simplify the graphical representation and visualization of the data, an auxiliary variable, USC

_{100}, is defined according to Equation (6).

Figure 12 shows the USC

_{100} absolute frequency histogram for neighborhood A. For this representation, consecutive classes of index

i have been defined so that

i meets Equation (7).

This condition is equivalent to define

$i$ according to Equation (8).

Thus, as an example, if USC = 0.341, it will belong to class 35.

Similarly,

Figure 13 shows the

${\mathrm{USC}}_{100}$ absolute frequency histogram for neighborhood B.

Figure 12 shows a displacement of USC values in neighborhood A with respect to values in neighborhood B (

Figure 13), which implies better access to solar resources in neighborhood A in general. This effect is linked to the lower height of the buildings in neighborhood A, as well as to the distribution in a simpler geometry. The intertwined geometry of neighborhood B favors the existence of north-facing walls that are also obstructed in all directions. Normally the lowest points of this type of façades are associated with the lowest values of the USC index. In both neighborhoods, as expected, the maximum USC values are reached at the highest points of the façades that are best oriented to the south and have a low level of obstruction. The value of the maximums of USC is slightly higher in neighborhood B, which could be explained by the greater height of the buildings and their better South orientation.

Table 2 shows the values of the descriptive statistics of USC for both distributions, from which significant differences in the mean and the median have been observed. Both parameters indicate that access to the solar resource is about 20% higher in neighborhood A than in neighborhood B.

The exposed methodology also allows mapping the USC

_{100} variable in façades. This enables us to deepen the details of the differences in access to the solar resource at each point of the same façade.

Figure 14 and

Figure 15 show the variability of USC

_{100} in representative façades of neighborhoods A and B respectively. They also allow quantifying, in specific façades, the dependence of the USC

_{100} gradient on the height. It is worth highlighting that in the façades analyzed, the increase of this gradient is greater on the highest points than on the lowest ones. This behavior is more evident as the height of the building increases.

Finally, with the USC data of all the façades and buildings, a regression analysis has been performed for each neighborhood in order to analyze the variables with the most significant influence on the USC value.

Studying the influence of façade facing, street width and height of the studied point on the outcome of the USC factor at each point studied, linear regression is proposed (Equation (9)).

where:

$\mathrm{H}$: Height of the studied point over the ground (in meters)

$D$: Street width (in meters)

$\theta $: Façade facing (in grades)

${c}_{1},{c}_{2},{c}_{3},{c}_{4}$: Constants

Equation (10) shows the result of the regression for neighborhood A. It is observed that the height of the studied point over the floor and the street width keep a direct relationship with the USC factor while in the case of the cosine of the orientation of the façade is reversed. Its correlation coefficient has a value of 0.919.

On the other hand, the regression for neighborhood B (Equation (11)) has a correlation value of 0.86 which is lower than the value obtained for neighborhood A. In this case, as in the previous one, the influence of the height of the point considered and the street width is direct and the orientation of the façade reverse.

These regressions allow knowing the value of the USC factor, and therefore the annual radiation received at any point on the façade was chosen, knowing only the typology of the neighborhood and the annual radiation on a horizontal surface.

## 5. Conclusions

In this paper, a novel model for calculating the solar radiation that reaches the façade of a building in a complex neighborhood is presented. The equation proposed allows obtaining accurate results for the irradiance, considering how the position of the adjacent buildings affects to the diffuse and reflected irradiance received on the façade. For this purpose, the Sky View Factor (SVF) is calculated. Considering the obstacles posed by surrounding buildings, it represents the portion of celestial vault viewed from each point of the façade. Accordingly, the SVF depends on the position of the point on the façade of the building but remains constant as long as the geometry of the neighborhood is not altered.

Four subroutines, programmed in Visual Basic Excel environment, have been developed to solve the problem of shading, the quantification of the obstacles seen, and the radiation received on a complete façade. Therefore, these subroutines allow the calculation of the solar radiation reaching any point of a selected façade of a neighborhood. From it, it is possible to display the results of the radiation on the façade in an intuitive way, through a solar radiation contour map.

The model and the program created have been used to characterize two typologically different neighborhoods through their capacity for solar energy harnessing. To this aim, the Urban Solar Coefficient (USC) has been defined, relating the annual radiation on horizontal surface of the study area and the annual radiation received at a chosen point on a façade. Calculating this factor on 121 points on each façade and for all the façades compounding the neighborhood, the distribution of the USC values for each neighborhood and the two different histograms that characterize the radiation according to the urban typology have been obtained. A comparative statistical analysis of the USC data for each neighborhood shows that the maximum values of USC are registered in neighborhood B (0.540) since its buildings are higher than the ones in neighborhood A, with a relative maximum USC value of 0.528. On the contrary, the relative minimum value in neighborhood A (0.063) is higher than in neighborhood B (0.012) due to the fact that the geometry of the neighborhood A is simpler, and its buildings are lower and more separated which reduces the effect of obstacles. On the whole, the mean value is higher in neighborhood A (0.317) than in neighborhood B (0.260) which implies the best access to solar resources in the first area than in the second one.

Finally, with the results obtained, one regression for each neighborhood has been proposed to determine the dependence of USC on the geometry of the buildings. These regressions allow calculating easily the amount of radiation received on the points of the façades, in neighborhoods that meet the characteristics described in this paper. In this way, the methodology and the tool proposed provide the calculation of the solar irradiance incident on any point of the façade of a given neighborhood. However, the tool presents some limitations and it is planned to be improved in upcoming works with the inclusion of a new subroutine that represents, in 3D contour maps, the values of irradiance in all the facades of the buildings of a certain neighborhood. Likewise, a network of sensors is being designed to automatically monitor the experimental irradiance received on a given façade in order to validate the methodology and tool proposed in this work.

Despite these limitations, the methodology and tool proposed could be very useful, among other applications, to plan urban designs of new neighborhoods that guarantee the solar rights and favor an optimum harnessing of the solar resource, whether for natural lighting or for the generation of energy from renewable sources, which will have a positive impact on the sustainability of cities.