# Probabilistic Methodology for Calculating PV Hosting Capacity in LV Networks Using Actual Building Roof Data

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## Abstract

**:**

## 1. Introduction

## 2. Methodology

#### 2.1. Input Data Preparation

- electrical network data;
- measurements from smart meters (not essential);
- LIDAR;
- GIS (land register); and
- solar irradiance data.

**A**is area matrix of roof surfaces in m

^{2}for every tilt-orientation combination and for every building roof polygon, where r is the number of roofs in LV network. The following matrix represents tilt-orientation information for every roof:

- ${\mathit{I}}_{\mathit{m}}\left[\frac{W}{{m}^{2}}\right]$ represents the GHI measurements; and
- ${\mathit{I}}_{\mathit{m},\mathit{d}}\left[\frac{W}{{m}^{2}}\right]$ represents the DHI measurements, for the last few years and with an hourly resolution.

**X**.

#### 2.2. Consumption and PV Generation Modeling

- selecting roofs with high PV potential; and
- accurate modeling of stochastic PV generation on different roof surfaces.

#### 2.2.1. Solar Irradiance on Tilted and Oriented Surfaces

#### 2.2.2. Selecting Roofs with High PV Potential

**I**and

_{A},_{m}**I**are calculated from

_{A}_{,m,d}**I**and

_{m}**I**, transposed and stacked vertically.

_{m}_{,d}**X**has 8760 columns, as analyzed data have hourly resolution and columns in a matrix

_{A}**X**represent annual data (365 days × 24 h).

_{A}**X**for all possible combinations of tilt and orientation using 5° step (n combinations).

_{A}**I**are summarized for every row separately to obtain annual solar radiant exposure for every tilt-orientation combination. The result is represented as a column vector

_{A}**H**.

**H**is normalized.

**H**is further changed in such way that only surfaces having normalized annual solar radiant exposure greater than 0.8 are selected, which means that in further simulations PV systems will be placed only on the most suitable roof surfaces. This vector represents a metric of the PV potential for every surface (n combinations) depending on how much annual solar radiant exposure a particular surface receives.

_{norm}**H**is calculated from

_{metric}**H**having n rows for all tilt-orientation combinations and equals to 1 for all suitable tilt-orientation combinations and to zero for other surfaces. The resulting vector

_{norm}**H**serves as a metric for selecting only the most suitable roof surfaces in further analysis (step C and D in Figure 1).

_{metric}#### 2.2.3. Stochastic PV Generation Modeling using Actual Roofs Surfaces

**I**is calculated from

**X**using the Klucher model for all n tilt-orientation combinations and only for summer days (the highest solar irradiance in Slovenia is in summer), where s denotes the number of summer days in a dataset

**X**

_{1,1}is the GI on a tilt 0° and an orientation 0° for first summer solar irradiance measurement, whereas i

_{n}

_{,s}is the GI on a tilt 90° and an orientation 180° calculated from the last summer solar irradiance measurement.

- Randomly choose one column for a previously chosen hour from matrix
**I**. Here we utilize random sampling with replacement from a finite population, which means that one column can be selected more than once. The result is a vector $\mathit{D}\in {\mathit{R}}^{nx1}$ holding irradiance for every tilt-orientation combination on a particular day and represents one weather scenario. - Select roofs with high PV potential using
**H**and utilize PV systems (μ_{metric}_{PV}) and inverter efficiency (μ_{inverter}) to derive PV generation for all tilt-orientation combinations from vector**D**(**P**_{g}has unit [W/m^{2}])$${\mathit{P}}_{\mathit{g}}={\mu}_{PV}\xb7{\mu}_{inverter}\xb7{\mathit{H}}_{\mathit{metric}}\xb7\mathit{D},where{\mathit{P}}_{\mathit{g}}\in {\mathit{R}}^{nx1}$$ - Let matrix
**B**represent a subset of matrix**A**, where**B**holds only the selected columns (randomly chosen roofs during PV hosting capacity calculations) and the total number of chosen roofs is denoted as i. For every column j in a matrix**B**(every selected roof), we calculate$${P}_{g,roof,j}={\displaystyle \sum}_{i=1}^{n}\left({\mathit{B}}_{\mathit{i},\mathit{j}}\xb7{\mathit{P}}_{\mathit{g}}\right),where{P}_{g,roof,j}\in \mathit{R}$$ - Which means that PV generation is summarized for all surfaces in a particular roof to create PV generation on a particular day.After calculating PV generation for every roof in step 3, the results are stacked vertically to create a PV generation vector.$${\mathit{P}}_{\mathit{g},\mathit{P}\mathit{V}}=\left[\begin{array}{c}{P}_{g,roof,1}\\ \vdots \\ {P}_{g,roof,i}\end{array}\right],where{\mathit{P}}_{\mathit{g},\mathit{P}\mathit{V}}\in {\mathit{R}}^{ix1}$$
- Output of the procedure in step 4 is PV generated power for every chosen roof on a particular day, which is further used in load flow calculations.

#### 2.3. Calculating PV Hosting Capacity Using the Monte Carlo Method

_{n}). The maximally allowed voltage at MV and LV levels were 110% of U

_{n}. The maximally allowed cable loading was set to 75% nominal apparent power (S

_{n}; due to cable installation factors) and 100% S

_{n}for the transformer.

- Choose real number K which denotes the number of iterations and equals to a size of a final set C.
- Set i = 0, where i denotes the total number of chosen roofs and create empty matrix
**B**, which later holds the information about the roof data. - Randomly select one column from a matrix
**A**holding roofs data and add a selected column to**B**. Here, we utilize random sampling without replacement, which means that every roof (location) has the same chance of being chosen and that every roof is chosen only once (until the network violation occur and all the locations are reset). - Generate stochastic load (described in Section 2.2) for all consumers and form a load matrix
**P**_{l}of size r (number of roofs and consumers).$${\mathit{P}}_{\mathit{l}}=\left[\begin{array}{c}{P}_{l,consumer1}\\ \vdots \\ {P}_{l,consumerr}\end{array}\right],where{\mathit{P}}_{\mathit{l}}\in {\mathit{R}}^{rx1}$$ - Generate stochastic PV generation (described in Section 2.2.3) for all chosen roofs and form a PV generation matrix
**P**using roof matrix_{g,PV}**B**from step 3. - Calculate load flow. Here, we solve the following equations:$${P}_{Gk}-{P}_{Dk}={V}_{k}{\displaystyle \sum}_{j=1}^{N}{V}_{j}\left[{G}_{kj}\mathrm{cos}\left({\mathsf{\delta}}_{k}-{\mathsf{\delta}}_{j}\right)+{B}_{kj}\mathrm{sin}\left({\mathsf{\delta}}_{k}-{\mathsf{\delta}}_{j}\right)\right];k=1,2,\dots N\phantom{\rule{0ex}{0ex}}{Q}_{Gk}-{Q}_{Dk}={V}_{k}{\displaystyle \sum}_{j=1}^{N}{V}_{j}\left[{G}_{kj}\mathrm{sin}\left({\mathsf{\delta}}_{k}-{\mathsf{\delta}}_{j}\right)-{B}_{kj}\mathrm{cos}\left({\mathsf{\delta}}_{k}-{\mathsf{\delta}}_{j}\right)\right]\phantom{\rule{0ex}{0ex}}{G}_{kj}+j{B}_{kj}=\left(k,j\right)elementofthebusadmittancematrix$$
_{Gk}is active power generation, P_{Dk}is active power demand, Q_{Gk}is reactive power generation, Q_{Dk}is reactive power demand, V_{k}is voltage, and δ_{k}is voltage angle for a particular node k in a network. V_{j}is voltage and δ_{j}is voltage angle for a node j which is close to the node k. In our case, P_{Gk}, P_{Dk}, Q_{Gk}, and Q_{Dk}are random variables representing consumer demand and PV generation. P_{Dk}equals to P_{l}(calculated in step 7), Q_{Dk}= 0.33 P_{l}, P_{Gk}for every bus equals to**P**(calculated in step 8) and Q_{g}_{,PV}_{Gk}= 0. - Check network violations (voltage limits and element loadings). If there are no violations, return to a step 3. If there is at least one violation, go to the next step.
- Aggregate the nominal power of all PVs in the network without the last one (there were no violations until the last PV was added). If P
_{g}_{,j}denotes installed power of j-th added PV system in the network and C_{k}is one value in a set C of PV hosting capacity values then C_{k}is calculated as follows:$${C}_{k}={\displaystyle \sum}_{j=1}^{i-1}{P}_{g,j}$$ - Add C
_{k}to a set C which holds PV hosting capacities.

## 3. Case Study

#### 3.1. Data Preparation

^{2}cables, whereas the side branches were mainly Al 35 mm

^{2}cables.

#### 3.2. Solar Potential Results for an Analyzed Village

**H**was calculated. Here, the vector

_{norm}**H**is reshaped to a matrix having orientation values on x-axis and tilt values on y-axis, so that the resulting matrix for a chosen area can be plotted (shown in in Figure 3). In order to receive as much radiant exposure as possible, it is important to have a slightly west oriented PV panel. The optimal position around the analyzed weather station was at azimuth 198° and tilt 34°. In order for the surface to receive at least 90% of the maximal yearly radiant exposure (red color), there were a wide range of tilt-orientation (azimuth) combinations available. The authors in Reference [37] performed a national study of solar energy in Slovenia and came up with similar results.

_{norm}#### 3.3. PV Hosting Capacity Results

## 4. Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

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**Figure 3.**Normalized yearly solar radiant exposure on a tilted and oriented surface for an analyzed area, which serves as a basis for selecting appropriate roof surfaces, depending on their solar potential.

**Figure 4.**Solar irradiance on a building roofs enables assessment of the solar potential for an analyzed village.

**Figure 5.**Results of calculating PV hosting capacity with the proposed methodology. Figure shows kernel density estimation of a set C of PV hosting capacity values calculated using the Monte Carlo method.

**Figure 6.**Comparison of the proposed PV hosting capacity methodology using actual roof surfaces (dashed red line) with existing ones, that use the same PV installed power per roof across all buildings (blue and green line).

Variable Name | Unit | Explanation |
---|---|---|

Irradiance | $I\left[\frac{W}{{m}^{2}}\right]$ | radiant flux received by a surface per unit area; |

Radiant exposure | $H\left[\frac{Wh}{{m}^{2}}\right]$ | radiant energy received by a surface per unit area. |

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## Share and Cite

**MDPI and ACS Style**

Grabner, M.; Souvent, A.; Suljanović, N.; Košir, A.; Blažič, B.
Probabilistic Methodology for Calculating PV Hosting Capacity in LV Networks Using Actual Building Roof Data. *Energies* **2019**, *12*, 4086.
https://doi.org/10.3390/en12214086

**AMA Style**

Grabner M, Souvent A, Suljanović N, Košir A, Blažič B.
Probabilistic Methodology for Calculating PV Hosting Capacity in LV Networks Using Actual Building Roof Data. *Energies*. 2019; 12(21):4086.
https://doi.org/10.3390/en12214086

**Chicago/Turabian Style**

Grabner, Miha, Andrej Souvent, Nermin Suljanović, Andrej Košir, and Boštjan Blažič.
2019. "Probabilistic Methodology for Calculating PV Hosting Capacity in LV Networks Using Actual Building Roof Data" *Energies* 12, no. 21: 4086.
https://doi.org/10.3390/en12214086