# A Probabilistic Conductor Size Selection Framework for Active Distribution Networks

^{*}

## Abstract

**:**

## 1. Introduction

- New technologies—new load types, such as EVs and ESS, and distributed generation from PV, wind, and arbitrage systems introduce new considerations in the network planning process [1].
- Combination of uncertainties—the capacity and location, often termed allocation, of DERs, is out of the control of DNOs and is mostly random. Furthermore, stochastic factors influence the DER power exports and imports.
- Dynamic operation of DERs—the time-of-use characteristic for DERs, mainly EVs and ESSs, is more complicated than for standard residential loads

## 2. A Stochastic-Probabilistic CSS Methodology

#### 2.1. Input Modelling

#### 2.1.1. Distribution Feeder

#### 2.1.2. Customer Loads

_{nom}, of 230 V is used.

^{2}−σ

^{2})/Cσ

^{2}

^{2}−σ

^{2})/Cσ

^{2}

_{nom}/1000)

#### 2.1.3. Distributed Energy Resources

_{PV}is modeled based on Equation (5). The PV system is modeled to operate at a unity power factor, which concurs with the current planning standards in South Africa.

^{2}) − 1)

_{PV}= (P

_{Pv}× µ

_{PV)}/(0.5 × V

_{nom})

#### 2.2. Process Modelling

#### 2.2.1. Grid Impact Assessment Study

_{max}. P

_{max}refers to the maximum allowable penetration while j refers to a level of penetration between 0 and the maximum. This feeder is designed based on the winter peak load, passive configuration conditions as set out in Section 2.1.1. or from iterative design trials to be discussed later.

- Voltage-rise (VR);
- Voltage-drop (VD);
- Conductor loading (CL);
- Transformer loading (TL), calculated as the three-phase aggregated loading;
- Voltage unbalance factor (VUF), calculated as the ratio between the maximum voltage deviation across the phases and the mean node voltage.

#### 2.2.2. Conductor Validation Analysis

- i.
- Based on the load flow results obtained from the grid assessment study, we identify the penetration level recording the maximum impact on voltage, p_v, and conductor loading, p_i.
- ii.
- At these maximum impact penetration levels, we characterize using beta PDFs the stochastic performance of the conductor loading, CL
^{B}_{p_i}, and node voltage, VR^{N}_{p_q}, results corresponding to the 1000 MCS DER allocation scenarios, for all branches, B, and nodes, N. - iii.
- From the beta PDFs, risk-adjusted performance indices are determined by extracting the 97.5th percentile, reflecting a 2.5% risk level. Compounded with the 2.5% risk applied in the HBE load flow analysis, the overall design process’s risk becomes 5%, which satisfies feeder design standards [43]. The calculation of the conductor performance index (CPI) and the node performance index (NPI) are given in Equations (7) and (8).

^{B}

_{p_i}, 97.5)/CL

_{limit}

_{;}for all branches, B

^{N}

_{p_v}, 97.5)/VR

_{limit}; for all nodes, N

#### 2.2.3. Output Processing–Iterative Resizing

- i.
- We estimate the resistance per km of the replacement conductor by scaling down the initial (or previous) per km resistance, R
_{0}, using the CPI and NPI indices. Equations (9) and (10) show the calculation of the resized conductor resistance, R_{RC-V,}and R_{RC-I}. R_{RC-V}and R_{RC-I}indicate the resized conductors based on voltage and conductor loading performance, respectively.R_{RC-V}= R_{0}/NPIR_{RC-I}= R_{0}/CPI - ii.
- A replacement conductor with the closest and lower resistance per km to that of the resized conductors from Equations (9) and (10) is selected from a directory of available conductors, such as Table 2.
- iii.
- Once a conductor is selected, the feeder model is updated with the new conductor properties (R
_{1}, X_{1}) and fed back as modified inputs for another iteration of the CSS simulation. The stop criterion is based on the compliance of all components (NPI, CPI ≤ 1) or the convergence of conductor selection, resulting from the unavailability of appropriate conductors in the available directory.

## 3. Case Study Simulation, Results, and Analysis

#### 3.1. Feeder Model

#### 3.2. Load and DER Models

#### 3.3. Initial Feeder Design

#### 3.4. Grid Impact Assessment and Conductor Validation Analysis for the Initial Design

#### 3.4.1. Interpretation of Scatterplots

#### 3.4.2. Discussion of the Performance of the Initial System Design

- i.
- Based on voltage impacts (Figure 6a), the feeder can accommodate penetration up to 100% of FMD (81.5 kWp) and between 205% (162 kWp) and the simulated maximum (162 kWp). The feeder violates voltage limits between 125% and 205% penetration, which indicates that future penetration scenarios falling in this range have a high likelihood for overvoltage conditions.
- ii.
- Based on conductor loading impacts (Figure 6b), the feeder can accommodate a PV capacity of up to 160% of FMD (130.5 kWp). All penetration scenarios above this limit violate thermal loading limits.
- iii.
- Combining the limits, the feeder can only host penetrations up to 100% (81.5 kWp), which corresponds to 48.5% of the desired penetration.

#### 3.4.3. Conductor Validation Analysis

#### 3.5. Conductor Resizing and Performance Validation

#### Discussion of the Performance of the Final System Design

## 4. Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 2.**Illustrating uncertainty representation (

**a**), the variation in beta probability distribution function curves with different shape parameters (

**b**).

**Figure 7.**Probability distributions showing (

**a**) conductor currents at the maximum loading impact penetration and (

**b**) feeder voltages at the maximum voltage impact penetration.

**Figure 8.**Illustrating the feeder node voltage level and feeder conductor loading performance and the respective performance indices (NPI, CPI) used in the resizing process.

**Figure 9.**Illustration of the response of the resized feeder. (

**a**) Voltage rise (

**b**) conductor loading.

**Figure 10.**Illustrating the performance response of 6 kWp, 10 kWp, and 12 kWp systems: (

**a**–

**c**) illustrates voltage initial and final design performance while (

**d**–

**f**) illustrate conductor loading initial and final performance.

Technical Constraint | Constraint Limit | Confidence Intervals |
---|---|---|

Voltage drop | >0.9 pu | 2.5% |

Voltage rise | <1.1 pu | 97.5% |

Loading of conductors | <1 pu | 2.5% |

**Table 2.**A representative section of the conductor inventory based on aluminum conductor steel reinforced (ACSR) conductors.

Cable Size | Current Rating | Resistance | Reactance | Impedance |
---|---|---|---|---|

mm^{2} | (A) | Ω/km | Ω/km | Ω/km |

25 | 105 | 1.200 | 0.096 | 1.493 |

35 | 144 | 0.868 | 0.096 | 1.082 |

50 | 183 | 0.641 | 0.090 | 0.801 |

70 | 228 | 0.443 | 0.089 | 0.557 |

95 | 277 | 0.320 | 0.086 | 0.406 |

120 | 322 | 0.253 | 0.084 | 0.325 |

150 | 350 | 0.206 | 0.082 | 0.269 |

Input Type | Input Model Parameters | ||||||
---|---|---|---|---|---|---|---|

Period | Probabilistic Model | ADMD (kVA) | pf | ||||

α | β | C (A) | σ (A) | ||||

Load | Winter | 1.67 | 4.07 | 60 | 10.50 | 4.015 | 0.95 |

Summer | 0.58 | 8.78 | 60 | 4.423 | 0.855 | 0.95 | |

PV | Summer | 255.5 | 255.5 | 6.96 | 0.150 | 0.800 | 1 |

Conductor Properties | Feeder Section | |||||||
---|---|---|---|---|---|---|---|---|

1 | 2 | 3 | 4 | 5 | 6 | 7 | ||

Case 1: (Initial design) | Cross-section (mm^{2}) | 50 | 50 | 35 | 35 | 25 | 25 | 25 |

Resistance (ohms/km) | 0.641 | 0.641 | 0.868 | 0.868 | 1.20 | 1.20 | 1.20 | |

Case 2: (Final design) | Cross-section (mm^{2}) | 70 | 50 | 35 | 35 | 35 | 35 | 35 |

R_{RC-I} (Ω/km) | 0.610 | 0.712 | 0.894 | 1.095 | 1.44 | 2.07 | 3.13 | |

R_{RC-V} (Ω/km) | 0.697 | 0.674 | 0.894 | 0.876 | 1.192 | 1.173 | 1.16 | |

Resistance (ohms/km) | 0.443 | 0.641 | 0.868 | 0.868 | 0.868 | 0.868 | 0.868 | |

Deterministic–active design | Cross-section (mm^{2}) | 70 | 70 | 50 | 50 | 35 | 35 | 35 |

Feeder Design | Permissible Penetration Range (% of FMD) | ||
---|---|---|---|

Voltage Rise | Cond. Loading | Overall | |

Case 1 (Initial, passive design) | 0–100% | 0–160% | 0–100% |

205%–full uptake | |||

Case 2 (Final, active design) | Full range | Full range | Full range |

Capacity (kWp) | Conductor Size (mm^{2}) | ||||||
---|---|---|---|---|---|---|---|

1 | 2 | 3 | 4 | 5 | 6 | 7 | |

Passive Design | 50 | 50 | 35 | 35 | 25 | 25 | 25 |

6 kWp | 50 | 50 | 35 | 35 | 25 | 25 | 25 |

8 kWp | 70 | 50 | 35 | 35 | 35 | 35 | 35 |

10 kWp | 95 | 70 | 50 | 35 | 35 | 50 | 50 |

12 kWp | 120 | 95 | 70 | 50 | 35 | 50 | 50 |

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## Share and Cite

**MDPI and ACS Style**

Waswa, L.; Chihota, M.J.; Bekker, B.
A Probabilistic Conductor Size Selection Framework for Active Distribution Networks. *Energies* **2021**, *14*, 6387.
https://doi.org/10.3390/en14196387

**AMA Style**

Waswa L, Chihota MJ, Bekker B.
A Probabilistic Conductor Size Selection Framework for Active Distribution Networks. *Energies*. 2021; 14(19):6387.
https://doi.org/10.3390/en14196387

**Chicago/Turabian Style**

Waswa, Lewis, Munyaradzi Justice Chihota, and Bernard Bekker.
2021. "A Probabilistic Conductor Size Selection Framework for Active Distribution Networks" *Energies* 14, no. 19: 6387.
https://doi.org/10.3390/en14196387