# Stochastic Planning and Operational Constraint Assessment of System-Customer Power Supply Risks in Electricity Distribution Networks

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

- A new reliability modelling methodology that incorporates different expected lifetimes for each power component and different network operational modes;
- Use of probabilistic curve fitting to model overloading violations and maintenance actions in distribution network operation;
- A new monetary reliability index to assess the impact of different network operation modes on customer interruption costs; and
- Reliability cost-benefit analysis of operating the network under different operational constraints.

## 2. Risk and Reliability Modelling

#### 2.1. Time-Sequential Simulation

#### 2.2. Power Component Ageing

#### 2.3. Impact of Time-Varying Failure Rates

#### 2.4. Impact of Overloading Violations

#### 2.5. Impact of Maintenance Actions

#### 2.6. Reliability Cost-Benefit Analysis

#### 2.6.1. Estimation of Customer Interruption Cost

#### 2.6.2. Formulation of the Cost-Based Index

## 3. Validation and Network Modelling

#### 3.1. Network Design

#### 3.2. Network Scenarios

#### 3.2.1. S-1 Constant Failure Rates

#### 3.2.2. S-2 Base Case

#### 3.2.3. Violations Due to Overloading

- A.
- S-3 Non-linear Method Modelled by Skewing Bathtub

- B.
- S-4 Linear method modelled by reducing PC lifetime

#### 3.2.4. Maintenance Actions

- A.
- S-5 High-frequency maintenance modelled by a linear method of lowering bathtub

- B.
- S-6 Low-frequency maintenance modelled by a nonlinear method of sawtooth bathtub distribution

## 4. Reliability Performance Assessment

#### 4.1. Constant vs. Time-Varying Failure Rates

#### 4.2. Impact of Overloading Violations

#### 4.2.1. Interruption Duration and Frequency

#### 4.2.2. Energy Not Supplied and Interruption Cost

#### 4.3. Impact of Maintenance Actions

#### 4.3.1. Interruption Duration and Frequency

#### 4.3.2. Energy Not Supplied and Interruption Cost

## 5. Conclusions

- Novel component lifetime modelling of failure rate distributions;
- Modelling of different network operational conditions using probabilistic curve fitting of the bathtub distribution; and
- Reliability cost-benefit analysis of operating the distribution network under different operational constraints.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

- Dzobo, O.; Gaunt, C.; Herman, R. Investigating the use of probability distribution functions in reliability-worth analysis of electric power systems. Int. J. Electr. Power Energy Syst.
**2012**, 37, 110–116. [Google Scholar] [CrossRef] - Peyghami, S.; Fotuhi-Firuzabad, M.; Blaabjerg, F. Reliability Evaluation in Microgrids with Non-Exponential Failure Rates of Power Units. IEEE Syst. J.
**2020**, 14, 2861–2872. [Google Scholar] [CrossRef] - Alvehag, K. Impact of Dependencies in Risk Assessments of Power Distribution Systems. Ph.D. Thesis, KTH Royal Institute of Technology, Stockholm, Sweden, 2008. [Google Scholar]
- Dzobo, O.; Gaunt, C.T.; Herman, R. Customer interruption cost for composite reliability analysis. In Proceedings of the Probabilistic Methods Applied to Power Systems (PMAPS), Istanbul, Turkey, 10–14 June 2012. [Google Scholar]
- Brown, R.; Marshall, M. Budget constrained planning to optimize power system reliability. IEEE Trans. Power Syst.
**2000**, 15, 887–892. [Google Scholar] [CrossRef] - Bowles, J. Commentary—caution: Constant failure-rate models may be hazardous to your design. IEEE Trans. Reliab.
**2002**, 51, 375–377. [Google Scholar] [CrossRef] - Retterath, B.; Venkata, S.; Chowdhury, A. Impact of time-varying failure rates on distribution reliability. Int. J. Electr. Power Energy Syst.
**2005**, 27, 682–688. [Google Scholar] [CrossRef] - Hernando-Gil, I.; Ilie, I.; Djokic, S.Z. Reliability planning of active distribution systems incorporating regulator requirements and network-reliability equivalents. IET Gener. Transm. Distrib.
**2016**, 10, 93–106. [Google Scholar] [CrossRef] [Green Version] - Ndawula, M.B.; Djokic, S.Z.; Hernando-Gil, I. Reliability Enhancement in Power Networks under Uncertainty from Distributed Energy Resources. In Proceedings of the 18th IEEE International Conference on Environment and Electrical Engineering, Palermo, Italy, 12–15 June 2018. [Google Scholar]
- Ilie, I.-S.; Hernando-Gil, I.; Djokic, S.Z. Risk assessment of interruption times affecting domestic and non-domestic electricity customers. Int. J. Electr. Power Energy Syst.
**2014**, 55, 59–65. [Google Scholar] [CrossRef] - Ndawula, M.B. Aggregated Impact of Smart Grid Technologies on the Quality of Power Supply. Ph.D. Thesis, University of Bath, Bath, UK, 2021. [Google Scholar]
- Wang, P.; Billington, R. Reliability cost/worth assessment of distribution systems incorporating time-varying weather condi-tions and restoration resources. IEEE Power Eng. Rev.
**2001**, 21, 63. [Google Scholar] [CrossRef] - Bhargava, C.; Murty, P. Reliability evaluation of radial distribution system using analytical and time sequential techniques. In Proceedings of the 2016 7th India International Conference on Power Electronics (IICPE), Patiala, India, 17–19 November 2016; pp. 1–6. [Google Scholar]
- Agarwal, U.; Jain, N. Reconfiguration of Radial Distribution Network for Reliability Enhancement considering Renewal Energy Sources. In Proceedings of the 2020 International Conference on Electrical and Electronics Engineering (ICE3), Gorakhpur, India, 14–15 February 2020; pp. 162–167. [Google Scholar]
- Rashid, N. Short-Time Overloading of Power Transformers. Ph.D. Thesis, Royal Institute of Technology KTH Stockholm, Stockholm, Sweden, 2011. [Google Scholar]
- Bertling, L.; Allan, R.; Eriksson, R. A Reliability-Centered Asset Maintenance Method for Assessing the Impact of Maintenance in Power Distribution Systems. IEEE Trans. Power Syst.
**2005**, 20, 75–82. [Google Scholar] [CrossRef] - Li, F.; Brown, R. A Cost-Effective Approach of Prioritizing Distribution Maintenance Based on System Reliability. IEEE Trans. Power Deliv.
**2004**, 19, 439–441. [Google Scholar] [CrossRef] - Piasson, D.; Biscaro, A.; Leão, F.B.; Mantovani, J.R.S. A new approach for reliability-centered maintenance programs in electric power distribution systems based on a multiobjective genetic algorithm. Electr. Power Syst. Res.
**2016**, 137, 41–50. [Google Scholar] [CrossRef] [Green Version] - Afzali, P.; Keynia, F.; Rashidinejad, M. A new model for reliability-centered maintenance prioritisation of distribution feeders. Energy
**2019**, 171, 701–709. [Google Scholar] [CrossRef] - Logan, D.M.; Papic, M. A Survey of Industry Practices in Probabilistic Assessment and Composite System Reliability Analysis. In Proceedings of the 2020 International Conference on Probabilistic Methods Applied to Power Systems (PMAPS), Liege, Belgium, 18–21 August 2020; pp. 1–6. [Google Scholar]
- Bagen, B.; Moura, J.; Jefferies, K. Probabilistic reliability assessment of North American Electric Power Systems. In Proceedings of the 2014 International Conference on Probabilistic Methods Applied to Power Systems (PMAPS), Durham, UK, 7–10 July 2014; pp. 1–6. [Google Scholar]
- Gil, I.H. Integrated Assessment of Quality of Supply in Future Electricity Networks; The University of Edinburgh: Edinburgh, UK, 2014. [Google Scholar]
- Cha, J.; Park, J.; Choi, J.; Jung, Y.; Yun, Y. Determination of a deterministic reliability criterion for composite power system expansion planning. In Proceedings of the 2009 IEEE Power & Energy Society General Meeting, Calgary, AB, Canada, 26–30 July 2009; pp. 1–6. [Google Scholar] [CrossRef]
- Heylen, E.; Ovaere, M.; Deconinck, G.; Van Hertem, D. Fair Reliability Management: Comparing Deterministic and Probabilistic Short-Term Reliability Management. In Proceedings of the 2018 IEEE Power & Energy Society General Meeting (PESGM), Portland, OR, USA, 5–10 August 2018; pp. 1–5. [Google Scholar] [CrossRef]
- Edimu, M.; Gaunt, C.; Herman, R. Using probability distribution functions in reliability analyses. Electr. Power Syst. Res.
**2010**, 81, 915–921. [Google Scholar] [CrossRef] - Pan, J.; Wang, Z.; Lubkeman, D. Condition based failure rate modeling for electric network components. In Proceedings of the 2009 IEEE/PES Power Systems Conference and Exposition, Seattle, WA, USA, 15–18 March 2009; pp. 1–6. [Google Scholar]
- Rubinstein, R.Y.; Kroese, D.P. Simulation and the Monte Carlo Method, 3rd ed.; John Wiley & Sons: Hoboken, NJ, USA, 2011; ISBN 9781118631980. [Google Scholar]
- Hadjsaïd, N.; Sabonnadière, J.C. Electrical Distribution Networks; John Wiley & Sons: Hoboken, NJ, USA, 2013. [Google Scholar]
- Ilie, I.; Hernando-Gil, I.; Djokic, S.Z. Theoretical interruption model for reliability assessment of power supply systems. IET Gener. Transm. Distrib.
**2014**, 8, 670–681. [Google Scholar] [CrossRef] - IEEE Guide for Electric Power Distribution Reliability Indices. In IEEE Std 1366–2012 (Revision of IEEE Std 1366–2003); IEEE: New York, NY, USA, 2012; pp. 1–43. [CrossRef]
- Brown, R.E. Electric Power Distribution Reliability, 2nd ed.; CRC Press: New York, NY, USA, 2009; ISBN 978-8493-7567-5. [Google Scholar]
- Moon, J.-F.; Kim, J.-C.; Lee, H.-T.; Lee, S.-S.; Yoon, Y.T.; Song, K.-B. Time-varying failure rate extraction in electric power distribution equipment. In Proceedings of the 2006 International Conference on Probabilistic Methods Applied to Power Systems, Stockholm, Sweden, 11–15 June 2006; pp. 1–6. [Google Scholar]
- Sen, P.; Pansuwan, S. Overloading and loss-of-life assessment guidelines of oil-cooled transformers. In Proceedings of the 2001 Rural Electric Power Conference, Little Rock, AR, USA, 29 April–1 May 2001. [Google Scholar]
- Fu, W.; McCalley, J.; Vittal, V. Risk assessment for transformer loading. IEEE Trans. Power Syst.
**2001**, 16, 346–353. [Google Scholar] [CrossRef] [Green Version] - Nadarahan, S.; Kotz, S. The beta-exponential distribution. Reliab. Eng. Syst. Saf.
**2006**, 91, 689–697. [Google Scholar] [CrossRef] - Hilber, P. Component Reliability Importance Indices for Maintenance Optimization of Electrical Networks. Master’s Thesis, The Royal Institute of Technology Stockholm, Stockholm, Sweden, 2005. [Google Scholar]
- Majid, A.S.N.A.; Salim, N.A.; Mohamad, H.; Yasin, Z.M. Assessment of Expected Customer Interruption Cost Due to Power System Contingency by Sensitivity Analysis. In Proceedings of the 2020 IEEE International Conference on Power and Energy (PECon), Penang, Malaysia, 7–8 December 2020; pp. 171–175. [Google Scholar]
- Küfeoğlu, S.; Lehtonen, M. A review on the theory of electric power reliability worth and customer interruption costs assessment techniques. In Proceedings of the 2016 13th International Conference on the European Energy Market (EEM), Porto, Portugal, 6–9 June 2016; pp. 1–6. [Google Scholar]
- Chowdhury, A.A.; Koval, D.O. Value-based distribution system reliability planning. IEEE Trans. Ind. Appl.
**1999**, 35, 305–311. [Google Scholar] [CrossRef] - Quality of Service Guaranteed Standards. OFGEM. Available online: https://www.ofgem.gov.uk/licences-codes-and-standards/standards/quality-service-guaranteed-standards (accessed on 15 February 2021).
- Allan, R.; Billinton, R.; Sjarief, I.; Goel, L.; So, K. A reliability test system for educational purposes-basic distribution system data and results. IEEE Trans. Power Syst.
**1991**, 6, 813–820. [Google Scholar] [CrossRef] - Hernando-Gil, I.; Hayes, B.; Collin, A.; Djokic, S. Distribution network equivalents for reliability analysis. Part 2: Storage and demand-side resources. In Proceedings of the IEEE PES ISGT Europe 2013, Lyngby, Denmark, 6–9 October 2013. [Google Scholar]
- Collin, A.; Hernando-Gil, I.; Acosta, J.L.; Djokic, S.Z. An 11 kV steady state residential aggregate load model. Part 1: Aggregation methodology. In Proceedings of the 2011 IEEE Trondheim PowerTech, Trondheim, Norway, 19–23 June 2011; pp. 1–8. [Google Scholar] [CrossRef]
- Babu, S. Reliability Evaluation of Distribution Systems; Considering Failure Modes and Network Configuration; Ener-giforsk AB, KTH Royal Institute of Technology: Stockholm, Sweden, 2017. [Google Scholar]

**Figure 2.**Failure rate distribution functions for a 33/11 kV primary transformer; (

**a**) skewed bathtub distribution; (

**b**) sawtooth bathtub distribution.

**Figure 4.**Aggregate daily load demand curves for commercial and residential loads [43].

Power Component | λ (/year) | μ (h) | EL (years) |
---|---|---|---|

33 kV bus | 0.08 | 140 | 25 |

11 kV bus | 0.05 | 120 | 25 |

415 V bus | 0.05 | 24 | 25 |

33/11 kV Transformer | 0.01 | 205.5 | 20 |

11/0.4 kV Transformer | 0.002 | 75 | 10 |

Circuit breaker | 0.0033 | 120.9 | 10 |

Fuse (11 kV and LV) | 0.0004 | 35.3 | 20 |

Overhead line | 0.091 * | 9.5 | 25 |

Underground cable | 0.051 * | 56.2 | 25 |

ID | Scenario | Description |
---|---|---|

S-1 | Constant failure rates | Fixed failure rates |

S-2 | Base case | Time-varying failure rates |

S-3 | Overloading violations; non-linear method | Longer wear-out |

S-4 | Overloading violations; linear method | Reduced lifetime |

S-5 | High-frequency maintenance; linear method | Reduced failure rate |

S-6 | Low-frequency maintenance; nonlinear method | Sawtooth bathtub curves |

ID | Scenario | SAIFI (int/cust/yr) | SAIDI (h/cust/yr) | ENS (kWh/cust/yr) | SAICI (£/cust/yr) |
---|---|---|---|---|---|

S-1 | Constant failure rates | 0.153 | 12.731 | 392.63 | 2844.40 |

S-2 | Bathtub beta distribution | 0.132 | 8.060 | 248.02 | 1804.56 |

Percent decrease | 13.70% | 36.70% | 36.83% | 36.55% |

ID | Scenario | SAIFI (int/cust/yr) | SAIDI (h/cust/yr) | ENS (kWh/cust/yr) | SAICI (£/cust/yr) |
---|---|---|---|---|---|

S-2 | Base case | 0.132 | 8.059 | 248.02 | 1804.56 |

S-3 | Overloading violations by skewing bathtub to have longer wear-out | 0.143 | 9.554 | 295.93 | 2274.80 |

Percent increase from the base case | 7.34% | 18.54% | 19.32% | 26.06% | |

S-4 | Overloading violations by reducing PC lifetime | 0.135 | 8.475 | 260.97 | 1887.45 |

Percent increase from the base case | 1.65% | 5.15% | 5.22% | 4.59% |

ID | Scenario | SAIFI (int/cust/yr) | SAIDI (h/cust/yr) | ENS (kWh/cust/yr) | SAICI (£/cust/yr) |
---|---|---|---|---|---|

S-2 | Base case | 0.132 | 8.06 | 248.02 | 1804.56 |

S-5 | High-frequency maintenance actions by lowering bathtub | 0.112 | 7.13 | 219.12 | 1560.41 |

Percent decrease from the base case | 15.25% | 11.57% | 11.65% | 13.53% | |

S-6 | Low-frequency maintenance actions by sawtooth curves | 0.131 | 7.88 | 239.74 | 1721.19 |

Percent decrease from the base case | 1.02% | 2.29% | 3.34% | 4.62% |

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Kisuule, M.; Hernando-Gil, I.; Serugunda, J.; Namaganda-Kiyimba, J.; Ndawula, M.B.
Stochastic Planning and Operational Constraint Assessment of System-Customer Power Supply Risks in Electricity Distribution Networks. *Sustainability* **2021**, *13*, 9579.
https://doi.org/10.3390/su13179579

**AMA Style**

Kisuule M, Hernando-Gil I, Serugunda J, Namaganda-Kiyimba J, Ndawula MB.
Stochastic Planning and Operational Constraint Assessment of System-Customer Power Supply Risks in Electricity Distribution Networks. *Sustainability*. 2021; 13(17):9579.
https://doi.org/10.3390/su13179579

**Chicago/Turabian Style**

Kisuule, Mikka, Ignacio Hernando-Gil, Jonathan Serugunda, Jane Namaganda-Kiyimba, and Mike Brian Ndawula.
2021. "Stochastic Planning and Operational Constraint Assessment of System-Customer Power Supply Risks in Electricity Distribution Networks" *Sustainability* 13, no. 17: 9579.
https://doi.org/10.3390/su13179579