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Determination of multiple solutions of load flow equations

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Abstract

This paper is concerned with the problem of finding all the real solutions (all components of the solution vector must be real values) of load flow equations. Solutions in which some of the components are complex values are of no interest as they have no physical significance as a load flow solution. This problem is significant not only because of its theoretical challenge but also, its relationship with several system behavior related issues. Approaches suggested so far for solving this problem are rather ad hoc, computationally demanding and have been demonstrated only on very small systems. Further, it has been subsequently shown by others that many of these methods are not capable of finding all solutions. In this work a new approach is proposed which is more systematic and seems to have the potential to handle even large problems. We show that for any system it is possible to find the multiple load flow solutions (MLFS) corresponding to a given operating point extremely easily, starting from a set of points that are referred to as zero load solutions (ZLS) in this paper. It is shown that the complete set of ZLS is unique for a system and MLFS for any other operating point can be obtained starting from these ZLS using only the Newton’s load flow method. The set of procedures for implementing the proposed scheme are illustrated and their features are highlighted by considering several sample systems.

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References

  1. Korsak A J 1972 On the question of uniqueness of stable load-flow solutions. IEEE Trans. Power Apparat. Syst. PAS-91(3): 1093–1100

  2. Johnson B K 1977 Extraneous and false load flow solutions. IEEE Trans. Power Apparat. Syst. 96(2): 524–534

    Article  Google Scholar 

  3. Baillieul J and Byrnes C I 1982 Geometric critical point analysis of lossless power system models. IEEE Trans. Circ. Syst. 29(11): 724–737

    Article  MathSciNet  MATH  Google Scholar 

  4. Sauer P W and Pai M A 1990 Power system steady-state and the load-flow jacobian. IEEE Trans. Power Syst. 5(4): 1374–1383

    Article  Google Scholar 

  5. Tamura Y, Mori H and Iwamoto S 1983 Relationship between voltage instability and multiple load flow solutions in electric power systems. IEEE Trans. Power Apparat. Syst. PAS-102(5): 1115–1125

  6. Yorino N, Harada S and Cheng H 1997 A method to approximate a closest loadability limit using multiple load flow solutions. IEEE Trans. Power Syst. 12(1): 424–429

    Article  Google Scholar 

  7. Tamura Y, Iba K and Iwamoto S 1980 A method for finding multiple load-flow solutions for general power systems. Proceedings of IEEE/PES Winter Meeting, New York, paper A-80-043-0

  8. Salam F, Ni L, Guo S and Sun X 1989 Parallel processing for the load flow of power systems: the approach and applications. Proceedings of the 28th IEEE Conference on Decision and Control, vol. 3, pp. 2173–2178

  9. Zhao S, Liu H, Cheng S and Chen D 1993 A new method for calculating power system multiple load flow solutions. IEE 2nd International conference on Advances in Power System Control, Operation and Management, APSCOM-93, pp. 274–278

  10. Ma W and Thorp S 1993 An efficient algorithm to locate all the load flow solutions. IEEE Trans. Power Syst. 8(3): 1077–1083

    Article  Google Scholar 

  11. Zhigang W, Yao Z, Chun T S, Wennan S and Yixin Y 2000 A new method to calculate multiple power flow solutions. Proceedings of the 5th International Conference on Advances in Power System Control, Operation and Management, APSCOM 2000, pp. 491–495, Hong Kong

  12. Iba K, Suzuki H, Egawa M and Watanabe T 1990 A method for finding a pair of multiple load flow solutions in bulk power systems. IEEE Trans. Power Syst. 5(2): 582–591

    Article  Google Scholar 

  13. Iwamoto S and Tamura Y 1981 A load flow calculation method for ill-conditioned power systems. IEEE Trans. Power Apparat. Syst. PAS-100(4): 1736–1743

  14. Overbye T J and Klump R P 1996 Effective calculation of power system low-voltage solutions. IEEE Trans. Power Syst. 11(1): 75–82

    Article  Google Scholar 

  15. Liu C-W, Chang C-S, Jiang J-A and Yeh G-H 2005 Toward a cpflow-based algorithm to compute all the type-1 load-flow solutions in electric power systems. IEEE Trans. Circuits Syst. 52(3): 625–630

    Article  Google Scholar 

  16. Rawal K 2014 Load flow loci enumeration and application to voltage stability of radial system ME Thesis, Department of Electrical Engineering, Indian Institute of Science, Bangalore, India

  17. Kusic G 2008 Computer-aided power systems analysis, CRC Press, Taylor and Francis Group, ISBN 978-1-4200-6106-2

  18. Ajjarapu V and Christy C 1992 The continuation power flow: A tool for steady state voltage stability analysis. IEEE Trans. Power Syst. 7(1): 416–423

    Article  Google Scholar 

  19. Price G B 1984 A generalized circle diagram approach for global analysis of transmission system performance. IEEE Trans. Power Apparat. Syst. PAS-103(10): 2881–2890

  20. Molzahn D K, Lesieutre B C and Chen H 2012 Counter example to a continuation-based algorithm for finding all power flow solutions. IEEE Trans. Power Syst. 28(1): 1–2

    Google Scholar 

  21. Rajathy R 2011 Investigations on power system operation and management in restructured market, Ph.D. thesis, Department of Electrical and Electronics Engineering, Pondicherry University, Pondicherry, India Available: http://shodhganga.inflibnet.ac.in:8080/jspui/handle/10603/5247

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Authors and Affiliations

  1. Department of Electrical Engineering, Indian Institute of Science, Bangalore, 560012, India

    Pankaj Mahata & P S Nagendra Rao

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  1. Pankaj Mahata
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  2. P S Nagendra Rao
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Correspondence to Pankaj Mahata.

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Mahata, P., Rao, P.S.N. Determination of multiple solutions of load flow equations. Sādhanā 41, 855–867 (2016). https://doi.org/10.1007/s12046-016-0524-5

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  • DOI: https://doi.org/10.1007/s12046-016-0524-5

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