#
Stochastic Approach for Increasing the PV Hosting Capacity of a Low-Voltage Distribution Network^{ †}

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^{†}

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## Abstract

**:**

## 1. Introduction

#### 1.1. Current State of PV Systems

- asymmetry in the voltage and current flow of neutral conductors in low-voltage distribution systems, which is caused by single-phase production/consumption devices;
- thermal effects caused by the high current in phase or neutral conductors, which increase losses.

#### 1.2. Photovoltaic Hosting Capacity

- the power of the installed PV;
- type of network, voltage level, whether there is a 3- or 4-wire system;
- PV connection, single or three-phase;
- PV type, concentrated or distributed;
- global irradiation, wind conditions, ambient temperature variations;
- load type;
- network characteristics such as rural or urban network, line impedances, feeder lengths, overhead power lines, or cable lines;
- BESS integration;
- transformer rating, construction, and regulation capabilities for tap changing [32].

- Peak load is the ratio of the PV system’s maximum capacity to the feeder’s peak load. It is estimated that 47% of all PVHC studies use this definition.
- TR rating represents the ratio between the sum of PV production to the TR rated capacity [33]. This is used in around 20% PVHC studies.
- PoC ratio is defined as the ratio of PoC equipped with PV to the total number of PoC [34]. PoC ratio is used in around 20% of PVHC studies.
- Energy ratio is the ratio between yearly energy generation by PV system to energy consumption from all PoC [35]. It is used in around 7% of PVHC studies.
- Active power ratio is the ratio of PV power output to the active power of load at PoC. Around 5% of PVHC studies use this definition.
- Roof space PV is the roof space of PoC with PV installation in one feeder. Around 2% of PVHC studies use this definition.

#### 1.2.1. Limitations of Photovoltaic Hosting Capacity

- power quality standards;
- thermal limits of conductors and TR;
- element fault current.

#### Voltage Violations

#### Voltage Unbalance

#### Harmonics and Supraharmonics

- Thermal limit of phase conductors—increased power flow from significant PV penetration results in reverse current flow in the phase conductor.
- Thermal limit of neutral conductor—this is an issue only in a 4-wire LV network with a higher voltage unbalance. The neutral conductor is commonly designed with lower ampacity as in phase conductors.
- Thermal limit of TR—can be violated in the case of reverse flow of power due to PC installations, e.g., from LV network to MV network.

#### 1.2.2. Increasing PV Hosting Capacity

- (a)
- Grid upratingGrid uprating can effectively decrease the voltage rise in the PoC. The main drawback of grid reinforcement is the high cost associated with this method [50].
- (b)
- Dynamic line ratingDynamic line rating (DLR) is represented by the reevaluation of the distribution line ampacity, taking the actual, real-time, meteorological data throughout the year into consideration, causing its allowable current rating to be a function of time. Therefore, it is sometimes referred to as real-time thermal rating [52]. This solution is only applicable to overhead power lines.
- (c)
- Reactive power managementReactive power control by PV is a cost-effective solution, but it has limited potential for increasing the PVHC [53,54]. The possible mitigation of the voltage drop is lower in electrical grids with ground cables compared to grids with OPL because of the lower inductance per unit length values. Injecting reactive power into the grid increases the losses. The inverters that can support the grid with reactive power are not standard for residential PV systems. Another aspect discussed is that the change of PF at the PoC from 0.8 leading (generating reactive power) to 0.8 lagging (absorbing reactive power) can increase PVHC by nearly 7 times compared with the case with a PV inverter generating reactive power. More case studies regarding power quality improvements can be found in [31].
- (d)
- Active distribution transformersThe obvious choice, in this case, would be the on-load tap-changers (OLTCs) on supply transformers, which in this configuration are sometimes referred to as active transformers. Furthermore, modern solid-state OLTCs can operate in a range from 100 ms to seconds, which is very close to the standard BESS. The utilization of active MV/LV transformers in LV grids with high PV penetration is still uncommon.
- (e)
- Battery energy storage systemsBESS can be utilized to store a part of the PV-generated energy and minimize the amount of injected active power back into the grid, thus, overcoming overvoltage issues. This stored energy can be used later when electricity prices are high or during emergencies. The main problem with BESS is the initial investment. The main potential of customer-owned BESS as a distributed control strategy is in fully utilizing the active power potential of BESS. At the same time, centralized BESS are an efficient means to improve the PVHC in remote areas [55].

#### 1.3. Our Contribution

## 2. Materials and Methods

#### 2.1. Determining Photovoltaic Hosting Capacity

#### Description of General PVHC Calculation Model

#### 2.2. Optimal Data

## 3. Description of Case Study

- steady state time-series calculations are made for one year at 1-h time steps;
- the LV network has static topology and is connected to a 22 kV MV system through two winding 22/0.4 kV 250 kVA DETC power transformers;
- the model is built and calculated as a typical 4-wire LV European system with 3-phase conductors and one neutral conductor;
- the model consists of 139 PoC separated into eight categories by their tariff, where each category is defined by a different yearly load shape;
- the model combines probabilistic and deterministic parameters described in detail in the following section.

#### 3.1. Analyzed Scenarios

- V0
- basic state of the model;
- V1
- influence of different PV power factors on PVHC;
- V2
- influence of tap change on DETC power factor on PVHC;
- V3
- influence of BESS parallel to PV on PVHC.

#### 3.2. Model Topology and Electrical Parameters

- every bus between two lines of the primary feeder, with connection details shown in Figure 3;
- the output terminal of the unbalanced load;
- grounding of the LV side of the power transformer.

#### 3.3. Loads and Load Shape Curves

#### 3.4. PV System

- topological position (parallel connection to specific PoC), which is randomly selected in every VR;
- installed power of PV system and type (single or 3-phase), which are selected according to discreet probabilities, Figure 7;
- installed PV systems range from 0.2 up to 10 kW per single PoC, with only one PV per PoC possible in the model;
- PF of an inverter is a constant, 1, for the basic state scenario V0;
- in the case of a single-phase PV, a random phase is selected for the connection.

#### 3.5. PVHC Limitations

- 230 V ± 10% for all 10 min average values for 95% of the time in a week;
- 230 V + 10%–15% for all 10 min average values.

- max. U (phase-neutral) 253 V, with an acceptable 1% violation probability$$\mathbb{P}({U}_{L-neutral}>1.1{U}_{n})\le 0.01$$
- min. U (phase-neutral)
- –
- 207 V, with an acceptable 5% violation probability$$\mathbb{P}({U}_{L-neutral}<0.9{U}_{n})\le 0.05$$
- –
- 195.5 V, with acceptable 0% violation probability$$\mathbb{P}({U}_{L-neutral}<0.85{U}_{n})\le 0$$

- max. voltage unbalance ${\alpha}_{VUF}$ 2%, with acceptable 5% violation probability$$\mathbb{P}({\alpha}_{VUF}>2\%)\le 0.05$$

#### 3.6. HC Definition and Calculation

## 4. Results

#### 4.1. Scenario V0

- static tap −5% on 22/0.4 kV DETC power transformer;
- PF of PV systems equals 1;
- DPI varies from 0 to 0.5 in steps of 0.1.

- (a)
- A box plot of voltage distribution for increasing DPI. In the case with no PV in the system, all values are within the voltage limit range. Increasing DPI does not cause undervoltages.
- (b)
- CDF calculated for overvoltage in the network. PVHC defined by 1% probability of overvoltage is 13.6%. This is equal to the sum of 79.9 kW installed PV systems in the network. Continuous CDF was calculated from discreet data using a second-degree polynomial fit. In all CDF calculations, the residual sum of squares (RSS) is below 0.002.
- (c)
- A box plot of yearly losses, lines, and TR for increasing DPI. An average decrease of 15.6% in losses was calculated for DPI 0.3. Losses as a function of DPI form a typical bathtub curve. If PV represents up to 30% of consumed energy, a positive effect in lowering power losses is observed.
- (d)
- Power flow and relative frequency through the transformer. Increasing PV penetration causes significant reverse power flow. However, the TR power limit was not violated even for high DPI.

#### 4.2. Scenario V1

- static tap –5% on 22/0.4 kV DETC power transformer;
- PF of PV systems varies from 1 to 0.85 (absorbing reactive power);
- DPI varies from 0 to 0.5 in steps of 0.1.

#### 4.3. Scenario V2

- tap changes from −5% −0% with 2.5% step on 22/0.4 kV DETC power transformer;
- PF of PV systems equals 1;
- DPI varies from 0 to 0.5 in steps of 0.1.

## 5. Discussion

- 230 V ± 10% for all 10 min average values for 95% of the time in a week;
- 230 V + 10%–15% for all 10 min average values.

#### Comparison of the Results

- characteristics of the network, whether urban, rural, or industrial;
- impedances of lines typical of a specific region;
- grounding resistance of the neutral conductor;
- customs in TR ratings;
- type of lines, whether cable, overhead, or hybrid lines;
- latitude;
- estimation value, percentage, or kWp of installed power;
- definition of PVHC;
- probability limits in the case of stochastic studies;
- real synthetic typified networks;
- voltage level.

- A study in 2007 [57] in which 30% PVHC was calculated, with voltage limitation, in an LV of a United Kingdom network.
- A study in 2010 [65] performed on a rural European network, with a calculated maximum installed power of 3.5 kWp per PoC. Overvoltages were also the PVHC limitation identified in this study. Our results, for 5% tolerance, allow connecting 1.2 kWp per PoC on average.
- A study in 2016 [66] performed on a rural system with rooftop PV installations with a calculation of 13% PVHC, with overvoltages estimated as the limitation source.
- A study in 2016 [67] performed on an LV network in Denmark with estimation of 40% PVHC.
- A stochastic study in 2019 [68] performed on an IEEE 123-bus system with rooftop PV installations with a calculation of 16.48% PVHC with PV inverter PF 1 and 5% tolerance for overvoltage. By changing PF to 0.895, the PVHC was increased up to 32.3%.

## 6. Conclusions

- include all deterministic parameters as probabilities defined by mean value and standard deviation for specific time step;
- use a smaller time step in the simulation, e.g., 10 min;
- increase the number of VR per PPF, which would result in finding only the possible extremes in voltage deviation.

- improve the stochastic model;
- implement decentralized and centralized BESS systems;
- implement other DERs, such as EVs, into the model;
- conduct harmonics analyses.

## Author Contributions

## Funding

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

DER | distributed energy resource |

RES | renewable energy sources |

PV | photovoltaic |

WT | wind turbine |

EV | electric vehicle |

CSP | concentrated solar power |

HC | hosting capacity |

PVHC | photovoltaic hosting capacity |

SS | small source |

BESS | battery energy storage systems |

OLTC | on-load tap changer |

DETC | deenergized tap changer |

TR | transformer |

LV | low-voltage network |

MV | medium voltage network |

PoC | point of consumption |

DSO | distribution system operator |

PPF | probabilistic power flow |

DPI | distributed energy resources penetration index |

VR | model variation |

IMS | intelligent metering system |

CDF | cumulative distribution function |

RSS | residual sum of squares |

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**Figure 9.**Results of V0 scenario analysis (

**a**) box plot of voltage distribution, (

**b**) DF of overvoltage, (

**c**) box plot of calculated looses distribution, (

**d**) heat map of power flow through TR 22/04 power TR.

**Figure 14.**Box plot of voltage distribution for increasing DPI for two different tap positions of DETCTR.

Conductor | No. Phases | R [$\Omega $/km] | X [$\Omega $/km] | B [mS/km] | ${\mathit{I}}_{\mathit{max}}$ [A] |
---|---|---|---|---|---|

50 AlFe 6 | 1 | 0.774 | 0.403 | 2.980 | 200 |

35 AlFe | 1 | 0.980 | 0.411 | 2.921 | 153 |

RETILENS 3 × 150 | 3 | 0.206 | 0.079 | - | 250 |

RETILENS 1 × 70 | 1 | 0.443 | 0.082 | - | 170 |

AlFe 16/2 | 1 | 1.879 | 0.312 | - | 67 |

25 AlFe | 1 | 1.531 | 0.425 | 2.816 | 122 |

70 AlFe | 1 | 0.506 | 0.272 | - | 241 |

PF | PVHC [%] | PV Power Equivalent [kW] | Improvement over V0 [%] | Limitation |
---|---|---|---|---|

1 | 13.6 | 79.9 | - | overvoltage |

0.95 | 21.4 | 127.7 | 57.35 | overvoltage |

0.9 | 26.4 | 155 | 94.12 | overvoltage |

0.85 | 33.3 | 195.7 | 144.85 | overvoltage |

0.8 | 43.2 | 253.4 | 217.65 | overvoltage |

Tap | PVHC [%] | PV Power Equivalent [kW] | Improvement over V0 [%] | Limitation |
---|---|---|---|---|

−5% | 13.6 | 79.9 | - | overvoltage |

−2.5% | 31.5 | 185.1 | 131.62 | overvoltage |

0% | 0 | 0 | - | undervoltages below 0.85${U}_{n}$ |

PF | PVHC [%] | |
---|---|---|

$\mathbb{P}({\mathit{U}}_{\mathit{L}-\mathit{neutral}}>\mathbf{1.1}{\mathit{U}}_{\mathit{n}})\le \mathbf{0.01}$ | $\mathbb{P}({\mathit{U}}_{\mathit{L}-\mathit{neutral}}>\mathbf{1.1}{\mathit{U}}_{\mathit{n}})\le \mathbf{0.05}$ | |

1 | 13.6 | 24.4 |

0.95 | 21.4 | 37.5 |

0.9 | 26.4 | 47.2 |

0.85 | 33.3 | 60 |

0.8 | 43.2 | 82.2 |

tap | ||

−5% | 13.6 | 24.4 |

−2.5% | 31.5 | 50.6 |

0% | 0 | 0 |

Probabilistic Parameters | Deterministic Parameters |
---|---|

size of PV | size of load |

type of PV (single-phase or 3-phase) | load shapes of power consumption |

phase selection for single-phase PV connection | meteorological parameters |

topological position of PV (Monte Carlo) | loads topological position |

distribution of 3-phase load among phases | - |

PF of each single-phase load | - |

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## Share and Cite

**MDPI and ACS Style**

Bendík, J.; Cenký, M.; Cintula, B.; Beláń, A.; Eleschová, Ž.; Janiga, P. Stochastic Approach for Increasing the PV Hosting Capacity of a Low-Voltage Distribution Network. *Processes* **2023**, *11*, 9.
https://doi.org/10.3390/pr11010009

**AMA Style**

Bendík J, Cenký M, Cintula B, Beláń A, Eleschová Ž, Janiga P. Stochastic Approach for Increasing the PV Hosting Capacity of a Low-Voltage Distribution Network. *Processes*. 2023; 11(1):9.
https://doi.org/10.3390/pr11010009

**Chicago/Turabian Style**

Bendík, Jozef, Matej Cenký, Boris Cintula, Anton Beláń, Žaneta Eleschová, and Peter Janiga. 2023. "Stochastic Approach for Increasing the PV Hosting Capacity of a Low-Voltage Distribution Network" *Processes* 11, no. 1: 9.
https://doi.org/10.3390/pr11010009