Assessing the Potential of Rooftop Photovoltaics by Processing High-Resolution Irradiation Data, as Applied to Giessen, Germany

Abstract

In recent years, prices for photovoltaics have fallen steadily and the demand for sustainable energy has increased. Consequentially, the assessment of roof surfaces in terms of their suitability for PV (Photovoltaic) installations has continuously gained in importance. Several types of assessment approaches have been established, ranging from sampling to complete census or aerial image analysis methodologies. Assessments of rooftop photovoltaic potential are multi-stage processes. The sub-task of examining the photovoltaic potential of individual rooftops is crucial for exact case study results. However, this step is often time-consuming and requires lots of computational effort especially when some form of intelligent classification algorithm needs to be trained. This often leads to the use of sampled rooftop utilization factors when investigating large-scale areas of interest, as data-driven approaches usually are not well-scalable. In this paper, a novel neighbourhood-based filtering approach is introduced that can analyse large amounts of irradiation data in a vectorised manner. It is tested in an application to the city of Giessen, Germany, and its surrounding area. The results show that it outperforms state-of-the-art image filtering techniques. The algorithm is able to process high-resolution data covering 1 km${}^2$ within roughly 2.5 s. It successfully classifies rooftop segments which are feasible for PV installations while omitting small, obstructed or insufficiently exposed segments. Apart from minor shortcomings, the approach presented in this work is capable of generating per-rooftop PV potential assessments at low computational cost and is well scalable to large scale areas.

1. Introduction

The ongoing shift from utilization of fossil fuels to renewable energy generation has led to building rooftops of favourable size and orientation being an increasingly valuable resource as they can host solar energy installations. Continuously falling investment costs for photovoltaics (PV) [1] keep reducing the levelised costs of energy and encouraging an increase in decentralised renewable power and heat generation. This trend has already been developing over several years as it has been observed by Zhang et al. [2] as early as in 2014. Worldwide installed PV capacity is increasing steadily [3] and various scenarios predict an exponentially increasing demand of solar energy by 2050 in order to meet the goal of establishing renewable, carbon-free power and heat supply. International agreements, such as the Kyoto Protocol [4], the Paris Agreement [5] and other regulations and directives [6] have been put into effect to support this trend. For these reasons, estimating the technical and economical solar energy potential on rooftop surfaces has advanced into a highly investigated area of study. Having information on the extent of solar energy potentials usable in rural or urban regions is advantageous for planning electrical grids or district heating and can help with drafting political measures for renewable energy facilitation.

Various methodological approaches have already been published. Approaches are usually developed with respect to availability and resolution of data, scale of area of study and requirements for level of detail. PV assessment approaches can be differentiated into several categories [7,8] and classified with regard to the level of detail [9]. Main categories include sampling approaches and complete census methodologies as well as Machine Learning (ML)-based and GIS-based (Geographic Information System) approaches. Key publications will be highlighted below with regard to these categories and their respective levels of detail. It is important to note that approaches can differ from one another depending on the data amount and quality available. In principle, the entire process of PV rooftop potential estimation consists of several successive steps, as has been described by Assouline [8] (p. 174) or Kodysh [10]. An interpretation of the entire process is depicted in Figure 1.

Sampling-based approaches aim to deliver an estimate of PV potential on large scales. Usually, statistical data are extracted from a well known area which can be examined in detail. Other approaches focus on extracting relevant data from a low resolved and scattered data base [11]. These statistics can then be extrapolated to regions of arbitrary scales. For example, Liu et al. use land use maps and meteorological and aerial data to estimate PV generation for the province of Jiangsu, China [12]. Izquierdo et al. estimate the rooftop area available for PV installation based on statistically representative data about common building types in Spain [13]. Using statistical data on buildings in the European Union, Defaix et al. assess the potential for rooftop photovoltaic on a near-continental scale [14]. Usually, statistics consulted for such assessments include information on population, building stock and global irradiation. The choice of which statistics to use more often than not depends on which data are readily available. Arnette, for example, bases his assessment [15] of rooftop PV generation potential on a per-capita estimate of available rooftop area. Defaix et al. choose an equivalent approach utilising existing data on floor area per capita and absolute population numbers [14]) to estimate rooftop area available for PV installation. Meteorological and data on temperature and irradiation is then typically used for PV yield calculations on resulting large scale rooftop area potentials, as has been done, for instance, by Liu et al. using spatially resolved information on yearly global irradiation [12]. Most of the aforementioned publications estimate the overall available rooftop area by some form of rooftop utilization factors. Since these approaches do not require large amounts of high resolution data, they are computationally inexpensive and well scalable. On the downside, they do not deliver spatially resolved results. Additionally, extrapolating data onto large scale regions naturally yields uncertainties regarding the quality of the results.

Complete census approaches are usually applied when a small area of study has to be examined in great detail. They are dependant on high-resolution irradiation data based on LiDAR (Light Detection And Ranging) data, Digital Elevation Models (DEM) and/or building footprint data. These are either readily available or are generated using one (or several) of the following techniques. Mavsar et al. obtained yearly irradiation data by performing a Hillshade analysis on a LiDAR data set [16]. Building footprints are most commonly obtained either by feature extraction techniques (feature extraction generally describes the process of reducing the dimensionality of image data; in this case, it is used to obtain rooftop and building footprints from aerial imagery and omit the rest of the data) on aerial imagery (see, e.g., the works of Khan and Arsalan [17] and Kumar [18]) or by analysing LiDAR data, as has been done, for instance, by Margolis [19], Li and Davey [20] and Nguyen and Pearce [21]. Based on these data sources, it is possible to also include shading calculations in the assessment; see, e.g., the work of Pinna and Massidda [22]. Given sufficiently resolved input data, complete census approaches generate estimates of PV rooftop potential on building scale. To account for different inclination and orientations, rooftops are sometimes subdivided into segments depending on the slope and orientation of different parts of the roof. Different approaches of rooftop segmentation are presented in the works of Palmer et al. [23], de Vries et al. [24] and Jacques et al. [25]. The hosting capacity of individual rooftop segments is then usually estimated by means of some form of utilization factor; see, e.g., Byrne et al. [26] or Horan et al. [27]. More sophisticated approaches, however, have successfully employed module fitting techniques to assess the potential of each segment by finding optimal PV panel layouts. This has, for example, been implemented in the works of de Vries et al. [24] and Kumar [18]. Due to the high spatial resolution and quality of results, complete census methodologies are comparatively resource-intensive. They do not scale well and are usually applied to urban areas of interest, although some publications have covered larger scales (see, e.g., Jacques et al (city and surrounding region of Leeds, UK) [25] and Horan et al. (applicable even on national scale) [27]).

Geostatistics-based methodologies utilize spatial data to assess PV rooftop potential in large scale areas of study (e.g., Bodis et al., European Union [28] and Bergamasco and Asinari, region of Piedmont in Italy [29]). These approaches are able to deliver large scale results in comparatively high resolution, as they yield spatially explicit representations of the entire study area. Depending on the data available, assessment methods employed can range from sampling [29] and working with rooftop utilization factors [28] to approaches similar to complete census methodologies including rooftop segmentation and shading calculations [30]. In a sense, utilization of geographical information can be viewed as an enhancement to other methodologies. Some data-driven approaches process aerial imagery to extract spatial representations of rooftop features on a large scale [31]. Depending on the methods employed, these approaches can be computation-expensive [32] and difficult to transfer to other scales.

In recent years, ML has advanced to become an effective tool for rooftop PV potential assessment. It is able to deliver high-resolution results with accurate outcomes and can be applied to the calculation of many features. ML approaches are also easily scalable and adaptive. Similar to complete census or sampling approaches, most applications either aim to extract rooftop features out of some form of imagery, as has been conducted by Mainzer et al. [33] and Huang et al. [34], for example, or to classify buildings into several categories or obtain a set of statistical characteristics for sampling methods [35]. ML can also compliment geostatistics-based methods by building large-scale population density or irradiation maps [28]. Other approaches do not explicitly employ ML methods but are still data-driven in some form. These include the application of genetic algorithms [36] or the construction of DEM from LiDAR data; see, e.g., Kodysh et al. [10] and Lukač and Žalic [37]. However, they require extensive amounts of training data in order to produce accurate results and are very computationally expensive. Some publications, for example, still rely on rooftop utilization factors instead of using ML to recognize unsuitable rooftop segments [8]. Training neuronal networks or support vector machines to extract rooftop features from satellite or LiDAR data is a difficult task as rooftops come in a large variety of shapes. Thus, although they yield a lot of theoretical potential, these approaches are hard to apply in many cases.

This paper aims to develop a complete census methodology based on readily available high resolution irradiation data stemming from LiDAR measurements and incorporating building footprints. The methodology will be applied to the area of the city of Giessen, Germany and the results will be validated by comparison to already existing rooftop PV installations. It has been pointed out that across all four categories of PV assessment approaches, the usage of utilization factors is common when assessing the rooftop area potential. This is mainly because the recognition of geometrically unfeasible rooftop segments is difficult to implement. Module fitting techniques, which generate the most precise results, are computationally expensive and also rely on some sort of information about feasible and unfeasible rooftop segments. The approach presented in this paper features a high-performance filtering approach based on available irradiation data which is able to classify rooftop segments into feasible and unfeasible parts. The main novel aspect is the way the irradiation data are processed in a vectorised manner using neighbourhood patterns that cover and investigate certain cells surrounding a center cell in the raster data set. This drastically reduces computational costs compared to existing methods. This approach can replace the usage of generalised rooftop area utilization factors due to its low computation expense. With respect to Figure 1, this paper focuses on the sub-processes of estimating the rooftop area potential and technical potential.

It is worth noting that some of the aforementioned works have employed very similar methodologies. However, the studies carried out by Boz et al. [30], de Vries et al. [24] and Huang et al. [34] implement some form of rooftop segmentation which relies on additional computations compared to raw raster data analysis. Furthermore, depending on the implementation, rooftop segmentation may be blind to smaller obstacles within rooftop segments because slope and azimuth values are often statistically grouped resulting in loss of information. The works of Li and Davey [20], Kodysh et al. [10] and Margolis [19] employ sophisticated methodology to extract rooftop features from LiDAR or irradiation data. However, given the focus of their respective works, no further segmentation of rooftops is carried out despite sufficiently detailed raster data may be available as a result of the previous analysis [10,19]. To the best of the authors’ knowledge, the only study to develop and implement a methodology analogous to the one proposed in this work is that of Palmer et al. [23]. Instead of rooftop segmentation, rooftop segments are assessed purely on the basis of raster data processing, saving lots of computational effort and delivering comparable results. The raster cell neighbourhood check employed by Palmer et al., the Rooke’s connectivity, is inverse to one of the neighbourhood patterns, the Von Neumann neighbourhood, utilised in this paper. Palmer et al. performed their operations using ArcGIS and georeferenced data, stating that the computing time is “acceptable”. we assume that our approach requires lower computation times by transforming the georeferenced data into purely numerical arrays and the resulting degrees of freedom with respect to vectorisation.

2. Materials and Methods

For the complete census approach applied in this paper, two distinct data sets are required: High resolution raster data containing yearly global irradiation values (also called irradiation data below) and building footprint layer. These data sets can in principle be extracted from aerial imagery with sufficient resolution by means of the methods described in the literature review. In many cases, the data are readily available. In our case, irradiation values have been computed on the basis of a LiDAR data measurement and implicitly include slopes and orientations of rooftop segments. The area of interest covered in the case study is composed of the city of Giessen (Hesse, Germany) including roughly two thirds of its administrative district. Within it lie roughly 170,000 buildings whose rooftop PV potential is to be determined. The data used have been made available for the authors by the Hessian Ministry of Economics, Energy, Transport and Housing as well as the local land registry office [38,39] (Since January 2022, these data are available for unrestricted use via https://gds.hessen.de/INTERSHOP/web/WFS/HLBG-Geodaten-Site/de_DE/-/EUR/Default-Start, accessed on 18 February 2022). A representation of both data sets is depicted in Figure 2. As the building footprint data contains information on the usage type of buildings (“privat”, “public”, “commercial”), results will be put into context with the respective usage types. The process described in detail below is summarised schematically in Figure 3.

In the first step, the footprint data are used to crop the irradiation data to buildings only. This way, irrelevant data are omitted before processing. Next, it is desirable to also omit any rooftop segments from the data that are insufficiently exposed to global irradiation. Determining which amount of yearly global irradiation is deemed unfeasible for solar energy installations is usually subject to economical considerations. As installation and maintenance costs of PV systems and the return on invest differ depending on building and household characteristics, it is difficult to specify a globally valid cut-off value. However, with regards to the case study area examined in this paper, PV installations for the most part become unfeasible around yearly global irradiation values of 800 $\frac{\mathrm{kWh}}{{\mathrm{m}}^2·\mathrm{a}}$ and below. Thus, for potential estimation, this value is chosen as the irradiation cut-off and applied to the irradiation data set. Any raster points below this cut-off value are set to zero, excluding them from further consideration. It is worth noting that applying this threshold cut-off leads to fragmentation of rooftop segments whose yearly global irradiation is close to the cut-off value chosen. Additionally, remaining rooftop segments may be too small or obstructed to be feasible for PV installation. To exclude these problematic rooftop segments from the analysis the application of neighbourhood-based filtering approaches is proposed. In this case study, two filters are constructed and applied to the irradiation data sequentially. This process can be applied in a vectorised fashion to the entire raster data set at once, resulting in very low memory latency and therefore short computation times. The first filter takes into consideration the expanded Moore neighbourhood (as described for example by Farbod [41]) of every raster data point. The second filter instead examines the Von Neumann neighbourhood. In Figure 4, both neighbourhood concepts are represented in the array on the left side, centered around the middle cell. The Moore neighbourhood is highlighted by a green frame while the Von Neumann neighbourhood is highlighted by a blue frame. Each array cell represents one irradiation data point in the grid.

Each filter scans the respective neighbourhoods of all data points and counts the number of other data points within the neighbourhood that are non-zero. Whenever this number of non-zero neighbouring data points is equal to or below a certain threshold, the irradiation data point under investigation is set to zero as well. If enough neighbouring data points are feasible for PV installations the irradiation value remains unchanged. The thresholds must be chosen for each filter individually. In this case study, a threshold value of 8 non-zero cells (=1/3 of all neighbouring cells) was found to perform best for the Moore neighbourhood. Both the size of the expanded Moore neighbourhood and the threshold value may have to be adapted to other areas of study. A parameter variation with regard to different neighbourhood sizes and thresholds is depicted in Figure 5. For the Von Neumann filter a threshold value of 2 non-zero cells was chosen.

After the application of both filters, fragmented and too small rooftop segments are almost entirely sorted out. In a last step, irradiation data patches with a total surface area of less than 25 m${}^2$ are removed from the data set, omitting any leftover rooftop segments too small for economically viable PV installations. The results of the different stages of this methodology are depicted in Figure 6. As validation will show, the algorithm succeeds in identifying rooftop segments suitable for rooftop PV installations. It does, however, tend to overestimate the area available because area losses due to module fitting are not taken into consideration. To account for this discrepancy a correction factor of $c_{mod}=0.9$ is applied to the resulting PV yield and area values.

One of the main novel aspects of this work is that the application of the Moore and Von Neumann neighbourhood filters can be vectorised by creating copies of the irradiation data set and arranging them along a third axis. These copies are then shifted by the relative indices of the neighbourhood cells, allowing for parallel array-wide operations on the neighbourhoods of each individual cell, making the approach very time-efficient.

This procedure will be explained in detail assuming the Von Neumann filter is to be applied to a 4000 × 4000 cell array B containing ones for feasible raster cells and zeros for infeasible raster cells. The Von Neumann neighbourhood consists of the four adjacent cells for any array cell. If these cells are index relative to the center cell, we obtain a list of two-dimensional indices such as:

$$idx=\left[\right(0,-1),(-1,0),(0,1),(1,0\left)\right]$$

(1)

A 3-dimensional array A is created that shares its first two dimensions with B. Its third dimension corresponds to the length of the list of indices $idx$. For each tuple i of row and column indices in $idx$, one copy of $B_{x,y}$ is created and assigned to $A_{x,y,i}$. The copy is then shifted by the relative neighbourhood indexing $idx\left(i\right)$. In order to prevent edge cut-off, all arrays are buffered with zero cells accordingly beforehand. Once this step is completed, A is now structured in such a way that along the third axis for any cell in $B_{x,y}$ lie copies of all of its neighbouring cells $\left(B_{(x,y)+idx}\right)$. This arrangement is illustrated on the left side in Figure 4. If any cell $B_{x\ast ,y\ast }$ is queried along the third axis in A, the query returns a list of the values of the neighbouring cells such as:

$$A_{x\ast ,y\ast ,i}=B_{(x\ast ,y\ast )+idx\left(i\right)}$$

(2)

Instead of performing the filter queries sequentially for each cell in B, a vectorised query can now be performed across the entire raster array at once by querying along the third axis of A. Instead of iterating over each individual cell and querying its neighbourhood, the algorithm iterates over the indices of whatever neighbourhood pattern it is passed, creates the three dimensional array and then performs the query in one step. This reduces the amount of iterations from the number of array cells to the number of cells in the neighbourhood pattern. In this example this means that the number of iteration steps is reduced from 16,000,000 to 4. Utilizing this drastically reduces the amount of sum function calls and the numbers of working memory accesses. This approach has been implemented as a Python algorithm and its performance is benchmarked in Section 3. Other state-of-the-art solutions such as SciPy ndimage implement similar operations in order to vectorise the applications of patterns to arrays. The benchmarking, however, shows that the proposed methodology outperforms ndimage by a factor of 50.

3. Results and Discussion

The methodology presented was applied to the area of the city of Giessen and surrounding settlements. In this section, firstly the results obtained will be presented. Secondly, the methodology will be validated using a comparison between existing PV installations and the corresponding rooftop areas designated by the algorithm. Furthermore, the resulting rooftop utilization factor will be computed and set in context with the results of other studies. At last, the performance of the algorithm described will be benchmarked and compared to the performance of other state-of-the-art approaches. The potential analysis was carried out using the parameter specifications listed in

Table 1. Parameter settings chosen for the potential analysis in the city of Giessen and surrounding region.

Parameter	Value
Global irradiation cutoff value	$800 \frac{\mathrm{kWh}}{{\mathrm{m}}^2·\mathrm{a}}$
Moore filter threshold value	1/3 of neighbouring cells
Von Neumann filter threshold value	1/2 of neighbouring cells
surface area size cutoff value	25 m${}^2$

.

These specifications were found to produce the most plausible results in terms of rooftop surface classification. For different areas of study, it is recommendable to carry out a parameter analysis in order to find the best fitting set of parameters. The results of the analysis are summarised in

Table 2. Results of the PV potential analysis carried out in and around the city of Giessen.

All Building Types
Number of buildings	171,062
Number of buildings suitable for PV installation	94,444
Total rooftop area	11,009,428 m${}^2$
Suitable rooftop area	4,785,557 m${}^2$
Sum of yearly irradiation on suitable rooftop area	4585 GWh/a
Residential buildings
Total rooftop area	6,844,726 m${}^2$
Suitable rooftop area	2,749,627 m${}^2$
Sum of yearly irradiation on suitable rooftop area	2649 GWh/a
Commercial buildings
Total rooftop area	3,228,899 m${}^2$
Suitable rooftop area	2,060,404 m${}^2$
Sum of yearly irradiation on suitable rooftop area	1967 GWh/a
Public buildings
Total rooftop area	935,803 m${}^2$
Suitable rooftop area	507,255 m${}^2$
Sum of yearly irradiation on suitable rooftop area	479 GWh/a

.

The results show that roughly 54% of buildings are eligible for PV installation. The rooftop area suitable for PV installations makes up 43% of the total rooftop area. Half of the eligible rooftop area is found on residential buildings. Within the residential category roughly 40% of the total rooftop surface can be utilised for PV generation. Among commercial and public buildings the resulting factor is higher at 62% due to larger scale rooftop segments and a high percentage of flat roofs.

Based on these findings and assuming a reference PV installation, an estimation of total solar electricity yield per year can be performed. In recent years, mono-crystalline modules have seen a steady increase in production numbers. The photovoltaic report published by the Fraunhofer Institute for Solar Energy Systems in 2022 finds that yearly mono-crystalline module production has overtaken that of poly-crystalline roughly around 2018 [42] (p. 7). Benda and Cern [43] point out that the market share of mono-crystalline PV modules was 60% in 2019, twice as large as that of poly-crystalline modules. Considering that mono-crystalline PV modules will most likely be the standard choice for rooftop PV installations in the near future and already achieve efficiency ratios of nearly 25% under lab conditions [42] (p. 7), we assume an overall average efficiency of 20% for entire PV rooftop systems. Under this assumption, a maximum of 1.132 GWh/a electricity generation by means of rooftop PV is possible.

In order to validate the methodology proposed, firstly, the resulting utilization factors will be compared with utilization factors found by other publications. As mentioned in the literature review, various publications construct rooftop utilization factors based on either theoretical assumptions or some form of sample data. Furthermore, utilization factors can be computed from the results of complete census methodologies, as is shown in Equation (3). $A_{PV}$ denotes the portion of rooftop area suitable for hosting PV installations. $A_R$ denotes the entirety of rooftop area within the area of study.

$$f=\frac{A_{PV}}{A_R}$$

(3)

Using the examples obtained in this case study, the resulting utilization factors will be compared to those found by other authors [13,14,27,44,45]. The factors were computed and grouped with regard to the constellations other authors have investigated. For both urban and rural settlements with comparable building and population density, Izquierdo et al. [13] found a utilization factor of 42% (the void fraction coefficient accounting for gaps between buildings is neglected for means of comparability. The data used in this publication was spatially detailed enough to exclude such void fractions beforehand). Peng [45] obtains a utilization factor of 46% for the city of Hong Kong. For the urban area studied in this paper, the utilization factor is 43%. Regarding rural settlements, the results lead to a utilization factor of 44%. Defaix et al. [14] conclude that 64% of all residential and 54% of all commercial rooftop area in the European Union is suitable for hosting PV installations. In comparison to their modelling results, the methodology employed in this paper shows utilization factors of 40% and 62%, respectively. Horan et al. [27] investigated utilization factors on commercial buildings for various countries. They found values ranging from 33% to 64%, indicating large differences across various countries. Comparing utilization factors across all buildings in the area of study yields a utilization factor of 43%. For comparison, Pillai and Banerjee [44] found a utilization factor of 30% for the country of India. Naturally, comparing key figures such as the utilization factor involves some inaccuracies. Different countries have different limitations due to their landscape and different proportions between population, building density and rooftop area due to cultural and socioeconomic causes. However, the figures found in the case study presented compare relatively well to the existing literature, especially when focusing on the distinction between rural and urban settlements. The results are illustrated in Figure 7.

Secondly, the functionality of identifying suitable rooftop segments will be tested comparing rooftops with existing PV installations to the respective results proposed by the algorithm. The goal of the methodology proposed is to identify rooftop area segments which are both strongly exposed to irradiation and suitable for PV installations in terms of structure and size. In the best case, this leads to isolation of those area segments that would also be considered for PV installation in reality. To qualitatively investigate whether this is the case, individual rooftops with existing PV installations can be investigated. The examples in Figure 8 show that the results highlight the segments which are already utilised for PV hosting. Although the comparison shows that the most promising rooftop segments are determined, there are some inaccuracies due to the following reasons:

• Empty surface fractions on the edges resulting from module fitting are not accounted for.
• Unfeasible rooftop segments which are close to the selected irradiation threshold can not always be removed entirely (see i.e., panel 2 in Figure 8).
• Within building footprints, no distinction between rooftop area and other surfaces is possible. This can lead to false classifications (see i.e., panel 2 in Figure 8).

As mentioned earlier, the application of a correction coefficient should be considered in order to account for these circumstances. Apart from these inaccuracies the estimation seems to be accurate enough to be applied to areas on city and community scales, or even larger areas of interest.

In addition to validating results based on existing PV installations and comparison to existing literature, the algorithm can be investigated by means of a sensitivity analysis. The three parameters that have influence on the resulting suitable rooftop area are (it is worth noting that the Von Neumann filter threshold value also is a parameter. However, since the Von Neumann neighbourhood consists of only four cells, a threshold value of 2 is the only one that produces meaningful results):

• the irradiation cut-off
• the size of the Moore neighbourhood, defined by its edge length
• and the Moore neighbourhood threshold value which is defined by the product of a coefficient $\{c_M\in \mathbf{R}|0\le c_M\le 1\}$

For each of these parameters, a sensitivity analysis was conducted to verify that the results behaved plausibly with respect to the inputs. The irradiation cut-off value determines whether a rooftop segment is sufficiently exposed to global irradiation and therefore included in the analysis. With increasing cut-off values, a decline in suitable rooftop area would be expected. This decline can be more or less linear, depending on the mix of rooftop orientations in the area of study, but should always be monotonously. Increasing values would hint at an error in the code or the methodology. The other parameters behave in a very similar way. Both of them, when increased, lead to an increasing exclusion of raster cells due to increasingly strict evaluation of their respective neighbourhoods. An increasing neighbourhood size generally means that more raster cells outside of the currently evaluated batch will be taken into consideration. Most of these cells tend to be zeros, especially since for large neighbourhoods, the risk of evaluating non-rooftop cells increases. Increasing the Moore threshold coefficient value requires any cell under evaluation to have more non-zero neighbours in order to pass the neighbourhood test. In Figure 9, the results of the sensitivity analysis are depicted. On the x-axis, the parameter variation values are arranged. The fix point is marked by the red vertical line. When varying one parameter, the other two will always have the value that is denoted in the fix point. On the y-axis, the resulting suitable rooftop area for the entire area of study is plotted for each parameter study. As expected, the rooftop area values decrease monotonously when any parameter value is increased. The algorithm is most sensitive towards the irradiation cut-off value. Naturally, when increasing the cut-off to values of 1000 $\frac{\mathrm{kWh}}{{\mathrm{m}}^2·\mathrm{a}}$ or above, only few rooftop segments will actually pass this threshold as the global irradiation is physically limited. The irradiation cut-off supposedly is the parameter that is most interesting for carrying out potential studies. It can be used to exclude certain ranges of irradiation, allowing for potential assessment only on highly or poorly irradiated rooftops, for example. The other parameters, as was mentioned before, are only relevant for the quality of extraction of suitable rooftop segments. Choosing very high values will eventually lead to suitable rooftop segments being omitted during the filtering process. This effect starts to become noticable especially on the two right-most data points $(9,2/3)$ and $(11,3/4)$. Overall, it can be concluded that the results seem plausible with respect to the input parameters and the sensitivity essentially focuses on the parameter of the irradiation cut-off.

One of the key advantages of the methodology proposed is its low computational cost. The quality of results produced can not necessarily compete with the quality of image recognition techniques or other AI (Artificial Intelligence) approaches. However, it is significantly faster to operate and requires less data input since no training is involved. A performance benchmark was carried out comparing the methodology proposed to a state of the art image filtering algorithm by scipy. The performance measurement was conducted using the profiling library cProfile in Python. The size of the test irradiation data set was 1.5 GB, split up into 50 test files of equal size. Across these 50 test runs, the benchmarking resulted into the figures presented in

Table 3. Profiling results for the proposed methodology and the image filtering module ‘ndimage’ of SciPy.

	Proposed Methodology	SciPy Ndimage
Min. execution time	0.65 s	114.37 s
Max. execution runtime	11.393 s	147.91 s
Avg. execution time	2.467 s	123.16 s
Avg. sum function calls	6,089,203	7,056,001,886

. In addition to runtime measurements, the amount of primitive sum function calls was tracked as they make up the majority of computational work in both approaches. Both runtime and amount of function calls vary due to the complexity of the respective test files. Taking into consideration the area covered by each test file, it takes 2.5 s on average to process an area as large as 1 km${}^2$ for the methodology presented in this paper.

4. Conclusions

In this paper, a data-efficient approach for assessing rooftop PV potential is proposed. Applying the methodology to the city of Giessen and the surrounding rural area shows that around 44% of all rooftop area can be utilised for PV installations.

The methodology proposed can deliver detailed per-rooftop results regarding PV potential while performing well on the run time scale. Given the necessary input data, it takes around 2.5 s to process an area of 1 km${}^2$. Thus it can easily be scaled to larger areas, as large raster data files can be split up and the computational effort only increases linearly. The methodology may be integrated into complete census methodologies and may replace sampling approaches and the usage of utilization factors whenever the data situation is sufficient without drastically increasing computation time. As mentioned in the introduction, the methodology proposed in this work is not entirely new. Compared to the current state of the art, the main advantages of this methodology are the presented concept of vectorisation by utilising three dimensional arrays and the lack of complex GIS-based operations due to the absence of rooftop segmentation. These lead to resource efficient performance and very short computing times, making the approach well scalable to larger areas of interest and data sets where usually only GIS-based methods or sampling approaches are employed.

As far as limitations are concerned, it is necessary to mention the following restrictions. While the approach consistently produces good and robust results on regular or nearly-regular rooftop types, the quality of results may vary for more complex rooftop geometries. This is mainly because obstacles and glass surfaces can not be classified as unsuitable surface segments due to the absence of intelligent image processing techniques. A reliable analysis of such special cases will usually require the use of more computationally expensive ML and image processing approaches. Additionally, constraints regarding the installation of rooftop PV systems on old buildings need to be taken into account separately as they are not represented in irradiation data sets. The main difficulties in this respect are static properties of old rooftops as well as monumental protection. While not integratable into the methodology itself, such matters can be treated beforehand if corresponding data are available. Alternatively, these constraints maybe incorporated into the results ex ante through the use of statistical methods.

Apart from these shortcomings, the presented methodology, due to its high performance and ability to produce high-quality results, offers an efficient middle ground between sampling and AI approaches. It is suitable for performing high-resolution complete-census investigations, delivering per-rooftop PV potential assessments even on a large scales.