This study presents a probabilistic load flow analysis of a radial distribution network using the Three-Point Estimation Method (3-PEM) to evaluate the impact of load uncertainty on voltage performance. Backward-Forward Sweep was used for the load flow analysis. The analysis was implemented using MATLAB R2022, and the Ayepe 11-kV, 34-bus radial distribution network in Ibadan, Nigeria was adopted as the test system. The results indicate that the minimum bus voltage obtained from deterministic load flow analysis was 0.8609 p.u., while the minimum mean voltage obtained using the 3-PEM was 0.8598 p.u. The maximum voltage standard deviation across the network was 0.0328 p.u., reflecting the extent of voltage variability due to uncertain loading conditions. The findings confirm that voltage violation probability increases with increasing load uncertainty. Notably, buses 6–9 and 26–28, which satisfied voltage limits under deterministic load flow analysis, experienced voltage violations when probabilistic effects were considered. This clearly demonstrates the limitation of deterministic analysis and its tendency to underestimate voltage instability risk in radial distribution networks. Furthermore, a top-10 worst-bus ranking based on voltage violation probability was developed, with buses ordered in descending order of violation risk. Bus 18 exhibited the highest voltage violation probability, while bus 13 recorded the lowest among the critical buses. This ranking provides valuable insight into priority locations for voltage control, reactive power compensation, and reliability enhancement in radial distribution systems.

The results demonstrate that the Three-Point Estimation Method effectively captures the impact of load uncertainty on system performance with significantly reduced computational burden compared to Monte Carlo simulation. The close agreement between deterministic and probabilistic mean values validates the accuracy of the method, while the additional information on variance and violation probability provides deeper insight into system reliability. For Nigerian radial distribution networks, where load demand is highly uncertain due to irregular consumption patterns and embedded generation, probabilistic load flow analysis using 3-PEM offers a practical and efficient tool for planning and operational studies.

1. A. Y. Abdelaziz, E. S. Ali, & R. A. El-Sehiemy. Probabilistic assessment of voltage stability in distribution networks with high penetration of renewable energy sources. International Journal of Electrical Power & Energy Systems, 148, 2023. 2. R. N. Allan & R. Billinton. Probabilistic assessment of power systems. IET Press, 2013. 3. R. Billinton & R. N. Allan. Reliability evaluation of power systems (3rd ed.). Springer, 2019. 4. B. Borkowska. Probabilistic load flow analysis. IEEE Transactions on Power Apparatus and Systems, PAS-93(3), 752–759, 2017. 5. G. Carpinelli, C. Noce, & A. Russo. Probabilistic voltage profile assessment in radial distribution systems using point estimation methods. Electric Power Systems Research, 189, 106712, 2020. 6. J. J. Grainger & W. D. Stevenson. Power system analysis (2nd ed.). McGraw-Hill Education, 2016. 7. H. P. Hong. An efficient point estimate method for probabilistic analysis. Reliability Engineering & System Safety, 59(3), 261–267, 2019. 8. IEEE Power & Energy Society. IEEE guide for probabilistic methods applied to power system planning. IEEE Standards Association, 2024. 9. W. H. Kersting. Distribution system modeling and analysis (4th ed.). CRC Press, 2018. 10. P. Kundur, N. J. Balu, & M. G. Lauby. Power system stability and control. McGraw-Hill Education, 2019. 11. J. M. Morales, A. J. Conejo, H. Madsen, P. Pinson, & M. Zugno. Integrating renewables in electricity markets: Operational problems. Springer, 2014. 12. R. Singh & B. C. Pal. Uncertainty modeling in power systems with renewable energy sources. IEEE Transactions on Power Systems, 36(2), 1335–1345, 2021. 13. Z. Wang & J. Liu. Voltage stability assessment of radial distribution systems with distributed generation under uncertainty. IET Generation, Transmission & Distribution, 14(18), 3790–3799, 2020. 14. Y. Zhang, H. Chen, & X. Wang. Improved three-point estimation method for probabilistic power flow analysis. International Journal of Electrical Power & Energy Systems, 134, 107383, 2022. 15. M. Aien, M. Fotuhi-Firuzabad, & F. Aminifar. Probabilistic load flow in correlated uncertain environment using unscented transformation. IEEE Transactions on Power Systems, 27(4), 2233–2241, 2011. 16. B. Borkowska. Probabilistic load flow. IEEE Transactions on Power Apparatus and Systems, PAS-93(3), 752–759, 1974. 17. Y. Y. Hong & C. S. Wu. A point estimate method for probabilistic load flow. International Journal of Electrical Power & Energy Systems, 32(6), 527–533, 2010. 18. J. M. Morales, J. Pérez-Ruiz, & A. J. Conejo. A simple probabilistic approach to incorporate uncertainty in power system analysis. IEEE Transactions on Power Systems, 24(4), 1811–1820, 2009. 19. P. Zhang, S. T. Lee, & F. Li. Probabilistic load flow computation using point estimate method. Electric Power Systems Research, 107, 10–18, 2014. 20. F. A. Hossain. Probabilistic load flow–based optimal placement and sizing of distributed generators. Energies, 14(23), Article 7857, 2021. 21. B. Li, R. Abbas, M. Shahzad, & N. Safdar. Probabilistic load flow analysis using nonparametric distribution. Sustainability, 16(1), Article 240, 2023. 22. V. Singh, T. Moger, & D. Jena. A modified point estimate-based probabilistic load flow approach for improving tail accuracy in wind-integrated power systems. Electric Power Systems Research, 245, 111606, 2025. 23. J. Wang & P. Wu. An improved three-point method for power flow calculation based on Halton sequence. Frontiers in Computing and Intelligent Systems, 4(1), 10–16, 2023. 24. F. Zishan. Analysis of probabilistic optimal power flow. Cogent Engineering, 2024.

© 2025 International Journal of Engineering and Techniques (IJET).

Digital Object Identifier (DOI)

IJET assigns a unique DOI to every accepted article via Zenodo for persistent linking and citation. Your article’s DOI: https://doi.org/{{doi}}

Open DOI About Zenodo DOI

Close