COORDINATE TRANSFORMATIONS OF THREE-PHASE VARIABLES USING QUATERNIONS

DOI: 10.47026/1810-1909-2022-1-65-72

УДК 621.3.025.3

ББК 31.27-01

АLEKSANDR V. KOROVIN, IVAN V. ALEKSANDROV

Key words

three-phase AC systems, E. Clarke and R.H. Park coordinate transformations, hypercomplex space, current and voltage quaternions, state-space coordinates

Abstract

Among the variety of modern approaches to the mathematical description of the power quality indicators during the processes of transmission, distribution, conversion and calculation of the ac electric power, the representation of three-phase models in the form of a purely imaginary quaternion located in a separate subspace of the four-dimensional hypercomplex space allows, in relation to the generally accepted method of analyzing linear circuits, for example, symmetrical components with the selection of a direct, reverse and zero phase sequence for the fundamental harmonic, to take into a more complete account the features of energy consumption, especially in the presence of distortion in the modified forms of harmonic signals. In addition, the division of the quaternion into scalar (real) and partial (imaginary) makes it possible to significantly simplify the subsequent analytical processing of synthesis of a power converters control signals for active filtering and power supply of autonomous loads of an arbitrary type, including a single-phase configuration, by extracting from its composition individual components responsible for both the amplitude-phase asymmetry and the nonlinearity of the characteristics.

The main algorithmic principles of organizing control structures as part of three-phase systems of various functional purposes, as a rule, are based on the conversion of reference signals and current values ​​of measured currents and voltages into state coordinates obtained by rotating the three-dimensional space plane by a given angle. At the same time, the calculated ratios for the numerical determination of the initial variables transformed by rotation in the quaternion basis are a function of only four kinematic parameters, which, other things being equal, leads to a simplification of the control law in relation to the traditional vector-matrix approach using nine direction cosines with six connection equations. In this regard, this paper is devoted to the applied problems of implementing linear transformations by E. Clarke and R.H. Park in terms of four-dimensional hypercomplex numbers, in compliance with the additional requirement of the invariance of scalar quantities after the transition.

References

1. Nos O.V. Analiz razlichnykh form predstavleniya kinematicheskikh parametrov v zadachakh lineinogo preobrazovaniya trekhfaznykh peremennykh [Analysis of various forms of representation of kinematic parameters in problems of linear transformation of three-phase variables]. Izvestiya vuzov. Elektromekhanika, 2012, no. 5, pp. 22–28.
2. Nos O.V. Matematicheskie modeli preobrazovaniya energii v asinkhronnom dvigatele [Mathematical models of energy conversion in an asynchronous motor]. Novosibirsk, 2008, 168 p.
3. Nos O.V. Metody analiza i sinteza trekhfaznykh sistem s aktivnymi silovymi fil’trami v giperkompleksnom prostranstve: dis. … d-ra tekhn. nauk [Methods of analysis and synthesis of three-phase systems with active power filters in a hypercomplex space. Doct. Diss]. Novosibirsk, 2015, 385 p.
4. Nos O.V. Primenenie algebry kvaternionov v matematicheskikh modelyakh elektricheskikh mashin peremennogo toka [Application of quaternion algebra in mathematical models of AC electrical machines]. In: Avtomatizirovannye elektromekhanicheskie sistemy: sb. nauch. tr. [Automated electromechanical systems. Proceedings]. Novosibirsk, 2011, pp. 16–32.
5. Nos O.V. Primenenie matematicheskogo apparata giperkompleksnykh chisel pri lineinom preobrazovanii tipa “vrashchenie” [Application of the mathematics of hypercomplex numbers in a linear transformation of the “rotation” type]. In: Materialy 10 mezhdunarodnoi konferentsii “Aktual’nye problemy elektronnogo priborostroeniya”, APEP-2010 [Proc. of 10^th Conf. on actual Problems of Electronic Instrument Engineering Proceedings APEIE–2010]. Novosibirsk, 2010, vol. 7, pp. 46–50.
6. Nos O.V. Sintez algoritma upravleniya avtonomnoi sistemoi energosnabzheniya s ispol’zovaniem kvaternionov [Synthesis of an algorithm for controlling an autonomous power supply system using quaternions]. Izvestiya Tomskogo politekhnicheskogo universiteta. Inzhiniring georesursov, 2022, vol. 333, no. 1. pp. 7–14.
7. Nos O.V. Sistema upravleniya poluprovodnikovym ustroistvom kompen-satsii kvaterniona mgnovennoi neeffektivnoi moshchnosti [Control system for a semiconductor device for compensating the quaternion of instantaneous inefficient power]. In: 8 mezhdunarodnaya (19 Vserossiiskaya) konferentsiya po avtomatizirovannomu elektroprivodu. AEP-2014 [Proc. of 8^th (19^th Russ.) Conf. on Automated Electric Drive]. Saransk, Mordovia State University Publ, 2014, vol. 1, pp. 229–234.
8. Contreras-Hernandez J.L., Almanza-Ojeda D.L., Ledesma-Orozco S., Garcia-Perez A., Romero-Troncoso R.J., Ibarra-Manzano M.A. Quaternion signal analysis algorithm for induction motor fault detection. IEEE Trans. Ind. Electron., vol. 66, no. 11, pp. 8843–8850, Nov. 2019.
9. Duesterhoeft W.C., Schulz M.W., Clarke E. Determination of instantaneous currents and voltages by means of alpha, beta, and zero components. Transactions of the American Institute of Electrical Engineers, 1951, vol. 70, no. 2, pp. 1248–1255.
10. Hamilton W.R. Lectures on quaternions: containing a systematic statement of a new mathematical method. Hodges and Smith, Dublin, 1853, 736 p.
11. Nos O.V. Control strategy of shunt active power filter based on an algebraic approach. In: 16^th Conf. of Young Specialists on micro/nanotechnologies and electron devices (EDM): [proc.], Altai, Erlagol, 29 June – 3 July 2015. IEEE, 2015, pp. 459–463.
12. Nos O.V., Brovanov S.V., Dybko M.A. Development of active filtering algorithms for higher harmonics in electrical power circuits. Optoelectronics, Instrumentation and Data Processing, 2016, vol. 52, no. 6, pp. 557–562.
13. Nos О.V. Linear transformations in mathematical models of an induction motor by quaternions. In: The 13^th Conf. and Seminar on Micro/Nanotechnologies and Electron Devices EDM–2012: Proceedings. Erlagol, Altai, 2012, pp. 295–298.
14. Nos O.V. The quaternion model of doubly-fed induction motor. In: 11^th Forum on strategic technology (IFOST 2016): Proc., Novosibirsk, 1–3 June 2016. Novosibirsk, 2016, part 2, pp. 32–36.
15. Park R.H. Two-reaction theory of synchronous machines. Transactions of the American Institute of Electrical Engineers, 1929, vol. 48, no. 3, pp. 716–727.

Information about the authors

Aleksandr V. Korovin – Academic Degree Candidate, Technological Machinery Design Department, Novosibirsk State Technical University, Russia, Novosibirsk (a_v_k87@bk.ru).

Ivan V. Aleksandrov – Post-Graduate Student, Department of Technological Machinery Design, Novosibirsk State Technical University, Russia, Novosibirsk (alexandrov.i2018@gmail.com; ORCID: https://orcid.org/0000-0002-3976-349X).

For citations

Korovin А.V., Aleksandrov I.V. COORDINATE TRANSFORMATIONS OF THREE-PHASE VARIABLES USING QUATERNIONS. Vestnik Chuvashskogo universiteta, 2022, no. 1, pp. 65–72. DOI: 10.47026/1810-1909-2022-1-65-72 (in Russian).

 Download the full article