Abstract | S obzirom na to da su današnji trendovi u projektiranju električnih strojeva takvi da se ide na sve veća specifična opterećenja pa tako i magnetske indukcije u željezu, induktiviteti se ovisno o razini magnetskog zasićenja, odnosno o radnoj točki mijenjaju. Zbog toga opis električnih strojeva, pa tako i sinkronih strojeva temeljen na konstantnim induktivitetima neće dati točne rezultate u širem rasponu područja rada. Jedan od načina kako se može riješiti ovaj problem je uvođenje promjenjivih induktiviteta. Međutim, još jednostavniji način za opis je pomoću strujno ovisnih funkcija ulančenih tokova. U radu je obrađena problematika proračuna stacionarnih stanja sinkronih strojeva pomoću strujno ovisnih funkcija ulančenih tokova u direktnoj i poprečnoj osi armaturnog namota. U radu je predstavljen model sinkronog stroja za opis stacionarnih stanja temeljen na strujno ovisnim funkcijama ulančenih tokova, njegova identifikacija i upotreba za proračun proizvoljnih stacionarnih radnih točaka kao i za proračun granica u pogonskoj karti. Kako bi se odredile te funkcije potrebno je raspolagati sa skupom podataka ulančenih tokova armaturnog namota u d i q osi zatim sa skupom podataka struja armaturnog namota u d i q osi kao i strujom uzbude dobivenih obradom radnih točaka u stacionarnom stanju. Ako se strujno ovisne funkcije ulančenih tokova modeliraju polinomima, proces identifikacije svodi se na rješavanje linearnog sustava što je brzo i jednostavno. Nadalje, predloženi model je testiran na tri sinkrona stroja te su dobivene vrijednosti uspoređene s mjerenim vrijednostima i s vrijednostima dobivenima upotrebom vektorsko - fazorskog dijagrama s konstantnim sinkronim reaktancijama. Predloženi model temeljen na strujno ovisnim funkcijama ulančenih tokova daje bolje rezultate u usporedbi s mjerenjima u odnosu na rezultate koji se dobivaju pomoću vektorsko - fazorskog dijagrama. Za proračun skupa podataka koji se mogu koristiti za identificiranje funkcija strujno ovisnih ulančenih tokova u okviru ovog rada razvijene su dvije metode. To su metoda za iterativni proračun stacionarnih stanja sinkronih strojeva pomoću tranzijentne metode konačnih elemenata i metoda za proračun stacionarnih stanja sinkronih strojeva temeljena na reluktantnim mrežama i konformnim preslikavanjima. U okviru metode za iterativni proračun stacionarnih stanja sinkronih strojeva pomoću tranzijentne metode konačnih elemenata obrađeno je nekoliko metoda za računalno učinkovit proračun armaturnog napona i faznog kuta između armaturnog napona i struje. Nadalje, dan je detaljan prikaz načina modeliranja i korekcija koje je potrebno provesti kako bi se na pravilan način podesio model sinkronog stroja u okviru metode konačnih elementa. Također su razvijene dvije iterativne metode za proračun stacionarnih stanja pomoću metode konačnih elemenata. Tri različite razvijene metode uspoređene su za proračun stacionarnih stanja sinkronih strojeva pomoću metode konačnih elemenata i uspoređene su s mjerenjima provedenim na dva sinkrona stroja instaliranima u elektranama. Također je u radu razvijena i metoda za proračun stacionarnih stanja sinkronih strojeva temeljena na reluktantnim mrežama i konformnim preslikavanjima. Metoda koristi mnoge novitete u modeliranju kao što su proračun magnetskih otpora pomoću konformnih preslikavanja, modeliranje dva ekvivalentna pola u magnetskoj shemi, korekcija magnetskih otpora s obzirom na raspodjelu protjecanja u zračnom rasporu, proračun broja magnetski aktivnih broja zubi i utora, proračun ekvivalentne permeabilnosti željeza u polnoj papuči u zoni prigušnog kaveza te u modeliranju efekta poprečne magnetizacije između d i q osi. Točnost metode testirana je na proračunu dva sinkrona stroja te su dobivene vrijednosti uspoređene s mjerenjima i s rezultatima dobivenim proračunom pomoću metode konačnih elemenata. |

Abstract (english) | Current trends in the area of the design of the electrical machines are proceeding in a way of increasement of the specific loads and more intensive utilization of materials which leads to higher levels of the magnetic saturation of the iron core in the machine. Moreover, saturation level is also changing with the operating point, especially for synchronous machines. Therefore, the description of electrical machines using constant inductances will lead to the wrong results when considering wide operational range. To solve that problem variable inductances can be introduced in the model. However, it has been proven within this thesis that it is easier and more appropriate to use current dependent flux linkage functions instead. This thesis addresses the calculation of the steady - state conditions for synchronous machines using current dependent flux linkage functions in direct - d and quadrature - q axis. Thesis present the derivation of the synchronous machine steady - state model based on current dependent flux linkage functions. Also the identification of the current dependent flux linkage function is covered in the thesis. A suggestion in this thesis is to use polynomial functions with armature current in d and q axis and field current as independent variables to model current dependent flux linkage functions. Identification of the current depended flux linkage function requires a set of known values of armature current in d and q axis and field current and armature flux linkages in d and q axis for a wide range operation of synchronous machines. This data set can be obtained from measurements and from finite element calculation. If the current dependent flux linkage functions are modeled as polynomials then identification process of determination of polynomial coefficients using least square method leads to solving linear system of equations which is fast and simple. Once identified and calculated, current dependent flux linkage functions can be used for calculation of the arbitrary steady - state operating point and limits in capability P - Q diagram of synchronous machine. Moreover, the proposed model has been tested and verified on three different synchronous machines and obtained values have been compared with measured values and also with the values obtained by using vector - phasor diagram. As expected, model based on current dependent flux linkage functions shows better agreement with the measurements compared to the classical model based on vector - phasor diagram with the constant values of synchronous reactances. For the calculation of the data set required for identification of the current dependent flux linkage functions two different approaches were developed. These are the methodology for iterative calculation of the steady - state condition using transient time-stepping finite element method with rotor motion taken into account the and method for calculation of the steady - state conditions based on releuctance networks and conformal mapping. Within the methodology for the iterative calculation of the steady - state operating conditions for synchronous machines using transient finite element method several different methods were developed for computationally efficient calculation of the armature voltage and power angle (i.e. the phase angle between armature voltage and current). Additionally, the detail description of the procedures and corrections for setting synchronous machine model using finite element method has been provided in order to achieve high accuracy of calculation. Two different iterative methods for calculation of the steady - state conditions for synchronous machines using finite elements have been developed. Those are relaxation parameter iterative method and Newton's iterative method. Two different developed methods have been compared with measurements conducted for two different synchronous machines installed in power plants. Method for calculation of steady - state conditions based on reluctance equivalent network and conformal mapping has also been developed within this thesis. Synchronous machine with salient poles is subdivided into flux tubes where the direction of the magnetic flux is assumed. Magnetic resistance is then calculated for every flux tube using conformal mapping or analytical expression. A reluctance network is formed with magneto-motive force for field winding and armature winding in d and q axis separately. Afterwards, a reluctance network is solved in order to obtain magnetic fluxes in the network. Armature flux linkages in d and q axis can be obtained from those fluxes. Proposed method uses many innovations in modeling such as calculation of the magnetic resistances using conformal mapping, modeling of the machine using two equivalent poles in magnetic equivalent network, correction of the magnetic resistances due to different distribution of the magnetomotive force in the air gap, calculation of the number of the magnetically active teeth and slots, calculation of the equivalent permeability in the pole shoe in the zone of the damper winding as well as the modeling of the cross-magnetization effect. The accuracy of the proposed method has been verified by comparing the obtained results with the measurements and with values obtained by finite element method. There are three original science contributions of this thesis. All scientific contributions are related to new approaches in the modeling techniques and the calculation of the steady - state operating conditions for synchronous machines. Original scientific contributions of this thesis are: - Steady - state model of the synchronous machines based on current dependent flux linkage functions in direct and quadrature axis and application of the proposed model to calculation of the limits in P-Q capability diagram of synchronous machine} - Model of the synchronous machine with salient poles for the calculation of the steady - state operation based on reluctance equivalent network and conformal mapping} - Methodology for the iterative calculation of the steady - state operating conditions for synchronous machines using transient finite element method} Thesis has been divided into seven chapters. Chapter 1 - Introduction describes the motivation for development of the methods and methodologies described in this thesis. Also describes the original scientific contributions and provide the description of the structure of the thesis. Chapter 2 - Classical steady - state model of the synchronous machine provides complete derivation of the dynamical model which is used for derivation of the steady - state model of the synchronous machine. The reason for embedding of this chapter within this thesis is to ensure completeness due to many references in thesis to the classical steady-state model of synchronous machine based on constant values of synchronous reactances and vector - phasor diagram. Chapter 3 - Modeling of the steady - state conditions of the synchronous machines using current dependent flux linkage functions. After literature review, a detailed derivation of the proposed steady - state model of synchronous machines based on current dependent flux linkage functions has been given. Also the identification of the current dependent flux linkage functions have been described. If those functions are modeled with polynomials identification process comes down to solving linear system which is fast and simple. Once identified current dependent flux linkage functions can be used for calculation of the arbitrary operating points and for calculation of the operational limits in P-Q capability diagram. Proposed method has been tested and verified on three different synchronous machines (500 MVA turbo-generator from Bull Run power plant, 35 MVA hydro-generator G1 from Vinodol hydro power plant and 83 kVA synchronous machines SG 30 from Laboratory of electric machines located at Department of Electric Machines, Drives and Automation). Obtained results are compared with measurements and with the values obtained with classical vector - phasor diagram with constant synchronous reactances. As expected, model based on current dependent flux linkage functions predicted more correctly steady - state operating conditions than the classical model based on vector - phasor diagram with constant synchronous reactances when compared with measured values. Chapter 4 - Calculation of the steady - state conditions of synchronous machines using transient finite element method. After literature review, this chapter provides modeling procedures and corrections for setting finite element model for synchronous machines for obtaining hight accuracy. Methodology for calculation of the sinusoidal symmetrical load conditions for synchronous machines using time-stepping transient calculation with rotor motion taken into account has been described. Several approaches has been described for the computationally efficient calculation of the armature voltage and power angle (i.e. the phase angle between armature voltage and current). Two different iterative methods have been proposed for calculation of the steady - state conditions using transient finite element method. Proposed methods have been implemented and tested for the calculation of the steady - state conditions for two synchronous machines installed in power plants and the results were compared with measurements. Those synchronous machines are 35 MVA hydro-generator G1 from Vinodol hydro power plant and 247 MVA turbo-generator from Plomin 2 thermal power plant. Chapter 5 - Calculation of the steady - state conditions of synchronous machines using reluctant networks and conformal mappings. In this chapter, after literature review a detailed description of the model based on reluctant networks and conformal mappings for calculation of the steady - state conditions has been presented. Proposed method uses many novelties such as calculation of the magnetic resistances using conformal mapping, modeling of the machine using two equivalent poles in magnetic equivalent network, correction of the magnetic resistances due to different distribution of the magneto-motive force in the air gap, calculation of the number of the magnetically active teeth and slots, calculation of the equivalent permeability in the pole shoe in the zone of the damper winding as well as the modeling of the cross-magnetization effect. The proposed method uses analytical expressions and conformal mapping for calculation of reluctances of flux tubes. Those flux tubes are used to form magnetic equivalent network (or reluctance network) together with magneto-motive forces of field and armature windings. The network is solved afterwards and magnetic fluxes are obtained. Flux linkages of armature winding in d and q axis can be calculated using magnetic fluxes obtained from solving the magnetic equivalent circuit. The accuracy of the proposed method has been verified on two synchronous machines (35 MVA hydro-generator G1 from Vinodol hydro power plant and 2 MW synchronous machine with fractional slot winding designed only for purpose of this thesis). Obtained results with the measurements (for 35 MVA hydro-generator G1 from Vinodol hydro power) and with values obtained by finite element method (for 2 MW synchronous machine with fractional slot winding). Chapter 6. – Conclusion. In conclusion a short survey of the thesis has been presented together with the conclusions drawn from the thesis. At the end the proposal for the future research activities have been provided. Chapter 7. - Appendix contains two appendices. In the first appendix an analytical derivation of the calculation of the Mobius transformation of the un-concentric circles into concentric circles which is used for calculation of the magnetic resistances in the air gap. In the second appendix an analytical derivation of the calculation of the length of the flux line for arc shaped air gap for un-concentric air gap has been provided. Obtained expressions were used for calculation of the correction of the magnetic resistance values due to different distribution of the magneto-motive force in the air gap. This PhD thesis has been carried out at the University of Zagreb Faculty of Electrical Engineering and Computing at Department of Electric Machines, Drives and Automation. |