# The general solution of lumped parameter ladder circuits, and a new approach in analyzing electrical line transient states

### Significance

Partial differential equations are commonly used to represent the behaviors of currents and voltages along a power transmission line. Several techniques have been used in solving such equations including the use of modulators and large range frequency sinusoidal oscillations. The solutions techniques are, however, limited for the analysis of transmission lines that operate under steady-state conditions.

For systems with uniformly distributed parameters, lumped parameter ladder circuits are preferred. Due to their physical and mathematical model representation, they can be used for various analyses such as heat transfer and the behavior of electronics and electrical systems. Lumped parameter circuits represent the quantity values of currents and voltages as a function of time, by reducing the partial differential equations into ordinary differential equations as well as the state space representation of the system into finite dimensions. The resulting analytical solution, well-known General line equations, is however not sufficiently accurate for the certain cases of ground fault analysis in an HV network. Therefore, more effort has been put in place over the past decades to model transmission lines for ground fault current analysis.

A research by Professor Ljubivoje M. Popović at the J. P. Elektrodistribucija-Beograd in Serbia
enables a new approach for the analysis of electrical line transient states and resonant phenomena by using relatively simple equations. The primary objective was to develop the equations for correct ground fault analysis at any point along a transmission line. These equations Professor Popović named as the General equations of the lumped parameters ladder circuits. The research work is currently published in the research journal, Electrical Power and Energy Systems.

Through summation of the specially formed finite and infinite geometric series and the principle of superposition, accurate lumped parameter model for the systems with distributed parameters was developed. The resultant relatively simple equations describe the relationships between the currents and voltages and their behaviors in both steady and transient conditions at any point of a transmission line and enable these quantities analyses along the whole line length.

Since the developed procedure successively follows one by one each phase of the actual physical process which precedes establishing the steady state in a line it is very convenient for the analysis of electrical line transient states and resonant phenomena. Additionally, it is important to say that the developed analytical procedure at the same time represents an alternative way of obtaining the already existing and well-known General line equations and as such it is very convenient for the educational purposes as well.

Consequently, the developed analytical procedures by Professor Ljubivoje M. Popović introduce a new approach for the analysis of the transient processes in the power transmission lines. The desired accuracy is achievable by using the relatively simple mathematical operations which do not require the use of computers. For example, the maximum over-voltages resulting from the power frequency transients of a power line can be determined. It is a crucial consideration because its underestimation may lead to ground faults and damages of the insulation and protective device while its overestimation may also result to over cost of the whole systems such as power insulation.

Ljubivoje M. Popović was born in Markovac (at Mladenovac), Serbia, in 1944, graduated (1969) and received Master (1983) and Doctor (1991) degrees, all at the School of Electrical Engineering, University of Belgrade.
In 1969 joined the Electric Power Distribution Company of Belgrade, where he stayed until retirement in 2007. In 1999 he was elected an associated professor at the School of Electrical Engineering, University of Belgrade and in 2010 he was elected an IEEE R8 Industry Lecturer (Industry Continuing Education Program).
At the beginning of his professional carrier, he worked on the design of different electric-power installations, including the first 110/10 kV substations in the power distribution network of Belgrade. At the end of the seventies, he moved to the Development and Research Department of this company.

His research work has been mainly focused on the following topics:

• Grounding systems of HV substations located in urban areas,
• Ground fault current distribution along the overhead and cable feeding lines,
• Fault locator algorithms,
• Resonant phenomena in the transmission lines and transformer windings,
• Influence of metal installations surrounding the feeding line on the ground fault current distribution,
• Influence of electric-power lines on surrounding metal installations,
• Influence of surrounding metal installations on the transfer characteristics of distribution lines.

In addition to realization of the numerous studies and projects, in the area of electric-power delivery he published: over 50 research papers in international journals and proceedings of international conferences, 2 chapters in two international scientific books and two scientific books under the following titles: ”Actual Parameters of Power Lines Passing through Urban Areas” (2015) and ”Practical Methods for Analysis and Design of HV Installation Grounding Systems” (2018).

Some of his papers and research results have had an impact on the following IEC publications: Technical Report IEC 60909-2, Ed 1(1992-09) and IEC standard 60909-3, Ed 2 (2003-09)), while some other were especially highlighted by: ”Vertical News”, ”High-beam Research”, ”High-beam Business”, ”Goliath Business News” and ”News-edge”. Also, two of his papers published in 2014 and one in 2017 have been selected by ”Renewable Energy Global Innovations” as the key scientific articles.

He was a member of the IEC Technical Committee – IEC/TC73 – Short Circuit Currents from 2004. Also, a member of IEEE since 1987, he became Senior Member in 1991 and was the Chair of IEEE PES Serbia and Montenegro Chapter from 2002 until 2009.
He received: national ”Nikola Tesla” Award in 2006, IEEE PES Chapter Outstanding Engineer Award, and Certificate of Appreciation for Notable Services and Contributions towards the advancement of IEEE and Engineering Professions.

### Reference

Popović, Lj. (2018). General equations of lumped parameter ladder circuits and a special approach to analyzing electrical line transient states. International Journal of Electrical Power & Energy Systems, 95, 568-576