# DEFINITION OF FLOW DISTRIBUTION IN NETWORKS WITH THE PREDOMINANT TREE STRUCTURE OF THE GRAPH ON THE BASIS OF POTENTIAL VALUES AT THE MIDPOINT OF BRANCH-CHORDS

Èlektron. model. 2018, 40(2):03-16
https://doi.org/10.15407/emodel.40.02.003

### ABSTRACT

An algorithm RP is proposed for calculating the flow distribution in distributive networks with graph G of the dominant tree structure, in which the number of cycles h does not exceed the root from the number of its nodes V, with linear dependences of the potential change on the current. The algorithm is based on reducing the graph to a tree by breaking the branches-chords and determining the value of the potential at their midpoint. The algorithm provides for the calculation of two currents for fixed potentials, the computational complexity of which is T (E) =O (E), where E is the number of branches of the graph. To determine the unknown potentials at the midpoints h of the branches-chords, a system of linear equations of the order h is formed, the coefficients and right-hand parts of which are formed from the results of h additional calculations of currents for
different variants of fixed values of the potentials. The computational complexity of determining the unknown potentials and currents is of the order O (hE* + E + h3), where E* is the number of branches of the equivalent graph G*, i.e., the subgraph G obtained on the basis of the folding of the hanging nodes. For h of magnitude no higher than O (V3/2), and the amount of necessary memory is proportional to the number of nodes in the graph. For h will be the order of magnitude no higher than O (V3/2), and the amount of necessary memory is proportional to the number of nodes in the graph. A method for analyzing the structure of the graph of the distribution system is proposed, which allows identifying the branches of the graph, the removal of which leads to the decomposition of the graph G* into components whereby the system of linear equations of order h can be divided into subsystems.

### KEYWORDS

distribution network, flow distribution, equivalence, convolution, brancheschords.

### REFERENCES

1. Bun, R.A., Vasiliev, E.D. and Semotyuk, V.N. (1991), Modelirovanie elektricheskikh tsepey metodom podskhem, Otv. red. Gritsyk, V.V., AN Ukrainy, Fiziko-mekhanicheskiy in-t. [Simulation of electrical circuits by subcircuits, Ed. Grytsyk, V.V., Academy of Sciences of Ukraine, Physical-Mechanical inst.], Naukova dumka, Kiev, Ukraine.
2. Maksimovich, N.G. (1961), Lineinye elektricheskie tsepi i ikh preobrazovaniya [Linear circuits and their conversion], Gosenergoizdat, Moscow-Leningrad, Russia.
3. Pukhov, G.Ye. (1967), Metody analiza i sinteza kvazi analogovykh elektronnykh tsepey [Methods of analysis and synthesis of the quasi analog electronic circuits], Naukova Dumka, Kiev, Ukraine.
4. Seshu, S. and Rid, M.B. (1971), Lineinye grafy i elektricheskie tsepi, Per. s angl., pod red. P.A. Ionkina, Uchebnoe posobie dlya vuzov spetsialnostey radiotekhnika, elektronnaya tekhnika, elektropriborostroenie i avtomatika [Line graphs and circuits, Transl. from English., Ed. Ionkin, P.A., Textbook for Universities radio engineering specialties, electronic engineering, electrical instrumentation and automation], Vysshaya shkola, Moscow, Russia.
5. Sigorskiy, V.P. and Petrenko, A.I. (1970), Algoritmy analiza elektronnykh skhem [Algorithms analysis of electronic circuits], Tekhnika, Kiev, Ukraine.
6. Shakirov, I.A. (1980), “Universal conversion and diakoptics circuits”, Abstract of Dr. Sci. (Tech.) dissertation, Leningrad, Russia.
7. Vynnychuk, S.D. (2016), “Definition of stream distribution in networks with a tree-like graph”, Elektronnoe modelirovanie, Vol. 38, no. 4, pp. 65-80.
8. Merenkov, A.P. and Khasilev, V.Ya. (1985), Teoriya gidravlicheskikh tsepey [Theory of hydraulic circuits], Nauka, Moscow, Russia.
9. Akopyan, S.G. (1993), “Electrical theory of hydraulic circuits and methodical bases of modes analysis and optimal design of gas transmission systems”, Abstract of Dr. Sci. (Tech.) dissertation, 05.13.12, 05.15.13. Gosudarstvennyi inzhenernyi universitet, Yerevan, Armenia.
10. Shargin, Yu.M. and Merkuriev, A.G. (2003), “Calculation of electrical modes of the method of equivalent transformation”, Elektrichestvo, no. 4, pp. 53-55.
11. Vynnychuk, S.D. and Samoylov, V.D. (2015), “Determination of the currents in the switching structures of the electrical energy networks with tree graph’s structure”, Elektronnoe modelirovanie, Vol. 37, no. 5, pp. 89—104.
12. Bessonov, L.A. (1978), Teoreticheskie osnovy elektrotekhniki: Elektricheskie tsepi. Uchebnik dlya studentov elektrotekhnicheskikh, energeticheskikh i priborostroitelnykh spetsialnostey vuzov, 7-ye izd., pererabotannoye i dop. [Theoretical fundamentals of electrical engineering: Electric circuits. Textbook for students of electrical, power and instrument engineering specialties of universities, 7-th ed., revised and additional], Vysshaya shkola, Moscow, Russia.

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