Simulations of Transformer Inrush Current by Using BDF-Based Numerical Methods

This paper describes three different ways of transformer modeling for inrush current simulations. The developed transformer models are not dependent on an integration step, thus they can be incorporated in a state-space form of stiff differential equation systems. The eigenvalue propagations during simulation time cause very stiff equation systems. The state-space equation systems are solved by using A- and L-stable numerical differentiation formulas (NDF2) method. This method suppresses spurious numerical oscillations in the transient simulations. The comparisons between measured and simulated inrush and steady-state transformer currents are done for all three of the proposed models. The realized nonlinear inductor, nonlinear resistor, and hysteresis model can be incorporated in the EMTP-type programs by using a combination of existing trapezoidal and proposed NDF2 methods

Modeliranje nelinearnog induktiviteta u proračunima prijelaznih pojava

U radu je nelinearna karakteristika (struja magnetiziranja – magnetski tok) reaktancije praznog hoda transformatora linearno aproksimirana na konačan broj pravaca, pri čemu je utvrđen dovoljan broj pravaca za što točniju aproksimaciju pri različitim prijelaznim pojavama (uklapanje neopterećenog transformatora pri različitim konfiguracijama električnog kruga, isklapanje naponskog mjernog transformatora). Na osnovi matematičkog modela za električni krug s nelinearnim induktivitetom razvijen je kompjutorski program za praćenje valnih oblika varijabli stanja. Rezultati dobiveni tim programom uspoređeni su s namjenskim programom Power System Blockset, koji predstavlja specijalizirani dio Matlaba 5.3 za elektromagnetske prijelazne pojave. Također su sprovedena eksperimentalna mjerenja prilikom uklapanja malog energetskog transformatora te pokazano je da se rezultati pokusa dobro slažu s rezultatima razvijenog programa. Dodatno je razvijen program koji uvažava nelinearnost aktivne otpornosti izazvane Joulovim gubicima u željezu transformatora. Simulacijama su utvrđene električne prilike prilikom uklapanja neopterećenog transformatora. Pri tome su ustanovljena kritična stanja, odnosno najnepovoljnije prilike s obzirom na trenutke uklopa prekidaèa i snage kratkog spoja elektroenergetskog sustava. Na kraju su ispitana energetska naprezanja metal-oksidnih odvodnika prenapona uzrokovana povišenjima napona kod uklapanja neopterećenog transformatora izazvanog prelaskom u zasićeno područje krivulje magnetiziranja nelinearnog induktiviteta. Pokazano je da približavanjem rezonantnim uvjetima dolazi do znatnih energetskih preopterećenja odvodnika prenapona što znaèi da za pravilno dimenzioniranje odvodnika prenapona treba voditi računa i o ovoj vrsti privremenih prenaponskih stanja.Nonlinear curve of non-load transformer reactance (magnetizing current-magnetic flux) was linearized with a finite number of linear functions determined by approximation accuracy at different transients (non-loaded transformer switching at different circuit configurations, voltage transformer switching). State-variable waveform monitoring software based on mathematical model of nonlinear inductive circuit was developed . Results generated by this software were compared to results obtained using Power System Blockset, electromagnetic transient software, integrated in Matlab 5.3. Measurements obtained in the experiment of small power transformer switching in confirmed software results. Software incorporating a nonlinear resistance caused by Joule's losses in a transformer iron core was developed additionally. Conducted simulations established electrical conditions at non-loaded transformer switching in including critical conditions, that is the worst case in the term of switching moment and energy system short circuit power. Energetic stress of metal-oxide surge-arrester caused by overvoltages at non-load transformer switching, as a consequence of transition to saturated area of magnetizing curve, was examined. It was showed that the surge-arrester overload appears approaching to the resonance. According to the previous, it is necessary to take into account such temporary overvoltage conditions in the surge-arrester design

Numerički proračun niskofrekvencijskih elektromagnetskih prijelaznih pojava u transformatorima

U radu je dat pregled modela linearnih i nelinearnih električnih elemenata sa naglaskom na modelu transformatora u postojećim bibliotekama dvaju poznatih programa: EMTP-ATP-a i MATLAB/PSB-a. Identificirani su uzroci postojanja numeričkih oscilacija i totalne numeričke nestabilnosti u transformatora sa negativnom reaktancijom namota analizom svojstvenih vrijednosti matrica sustava. Pokazan je originalni način modeliranja nelinearnih elemenata u proračunima niskofrekvencijskih prijelaznih pojava u transformatorima. Izgrađen je algoritam baziran na hiperstabilnoj backward differentiation formulas-BDFp p-tog reda numeričkoj metodi za proračune niskofrekvencijskih prijelaznih pojava u jednofaznim i trofaznim transformatorima. Pokazane su prednosti razvijenog algoritma u pogledu eliminiranja efekta “prebačaj” i mogućnosti reduciranja ili izbjegavanja numeričkih oscilacija i totalne numeričke nestabilnosti u proračunima niskofrekvencijskih prijelaznih pojava u jednofaznim i trofaznim transformatorima. Izgrađen je model dinamičke histerezne petlje sa prednostima u odnosu na postojeći model Hysteretic Inductor-Type 96 u EMTP-u. Također su provedena eksperimentalna mjerenja prijelaznih pojava uklapanja i ferorezonancije kod jednofaznog transformatora kao i uklapanja kod trofaznog trostupnog transformatora. Rezultati pokusa vrlo dobro se slažu s rezultatima proračuna razvijenog programa.In this work is presented the model of linear and nonlinear electrical elements with accent on transformer model in existing well-known software libraries: EMTP-ATP and MATLAB/PSB. Causes of existance of numerical oscillations and total numerical instability in transformers with negative winding reactance are identified by analysis of eigenvalues of system matrix. Original method of nonlinear elements modeling in calculation of low frequency transients in transformers is showed in this work. The algorithm based on hyper stable backward differentiation formulas-BDFp p-th order numerical procedure for calculation of low frequency transients in single-phase and three-phase transformers is developed additionally. The advantages of developed algoritham in a way of elimination of overhooting effect and possibility of reduction or avoiding numerical oscillations and total numerical instability in low frequency transients calculations in single-phase and three-phase transformers are presented. Model of dynamic hysteretic loop with advantages in comparison with existing Hysteretic Inductor-Type 96, EMTP model is developed. Experimental measurements of transients: switching-on and ferroresonance in single-phase transformer and switching-on in three-phase transformer are obtained. Measurements obtained in the experiment confirmed calculation results of developed software

Numerički proračun niskofrekvencijskih elektromagnetskih prijelaznih pojava u transformatorima

U radu je dat pregled modela linearnih i nelinearnih električnih elemenata sa naglaskom na modelu transformatora u postojećim bibliotekama dvaju poznatih programa: EMTP-ATP-a i MATLAB/PSB-a. Identificirani su uzroci postojanja numeričkih oscilacija i totalne numeričke nestabilnosti u transformatora sa negativnom reaktancijom namota analizom svojstvenih vrijednosti matrica sustava. Pokazan je originalni način modeliranja nelinearnih elemenata u proračunima niskofrekvencijskih prijelaznih pojava u transformatorima. Izgrađen je algoritam baziran na hiperstabilnoj backward differentiation formulas-BDFp p-tog reda numeričkoj metodi za proračune niskofrekvencijskih prijelaznih pojava u jednofaznim i trofaznim transformatorima. Pokazane su prednosti razvijenog algoritma u pogledu eliminiranja efekta “prebačaj” i mogućnosti reduciranja ili izbjegavanja numeričkih oscilacija i totalne numeričke nestabilnosti u proračunima niskofrekvencijskih prijelaznih pojava u jednofaznim i trofaznim transformatorima. Izgrađen je model dinamičke histerezne petlje sa prednostima u odnosu na postojeći model Hysteretic Inductor-Type 96 u EMTP-u. Također su provedena eksperimentalna mjerenja prijelaznih pojava uklapanja i ferorezonancije kod jednofaznog transformatora kao i uklapanja kod trofaznog trostupnog transformatora. Rezultati pokusa vrlo dobro se slažu s rezultatima proračuna razvijenog programa.In this work is presented the model of linear and nonlinear electrical elements with accent on transformer model in existing well-known software libraries: EMTP-ATP and MATLAB/PSB. Causes of existance of numerical oscillations and total numerical instability in transformers with negative winding reactance are identified by analysis of eigenvalues of system matrix. Original method of nonlinear elements modeling in calculation of low frequency transients in transformers is showed in this work. The algorithm based on hyper stable backward differentiation formulas-BDFp p-th order numerical procedure for calculation of low frequency transients in single-phase and three-phase transformers is developed additionally. The advantages of developed algoritham in a way of elimination of overhooting effect and possibility of reduction or avoiding numerical oscillations and total numerical instability in low frequency transients calculations in single-phase and three-phase transformers are presented. Model of dynamic hysteretic loop with advantages in comparison with existing Hysteretic Inductor-Type 96, EMTP model is developed. Experimental measurements of transients: switching-on and ferroresonance in single-phase transformer and switching-on in three-phase transformer are obtained. Measurements obtained in the experiment confirmed calculation results of developed software

Modeliranje nelinearnog induktiviteta u proračunima prijelaznih pojava

U radu je nelinearna karakteristika (struja magnetiziranja – magnetski tok) reaktancije praznog hoda transformatora linearno aproksimirana na konačan broj pravaca, pri čemu je utvrđen dovoljan broj pravaca za što točniju aproksimaciju pri različitim prijelaznim pojavama (uklapanje neopterećenog transformatora pri različitim konfiguracijama električnog kruga, isklapanje naponskog mjernog transformatora). Na osnovi matematičkog modela za električni krug s nelinearnim induktivitetom razvijen je kompjutorski program za praćenje valnih oblika varijabli stanja. Rezultati dobiveni tim programom uspoređeni su s namjenskim programom Power System Blockset, koji predstavlja specijalizirani dio Matlaba 5.3 za elektromagnetske prijelazne pojave. Također su sprovedena eksperimentalna mjerenja prilikom uklapanja malog energetskog transformatora te pokazano je da se rezultati pokusa dobro slažu s rezultatima razvijenog programa. Dodatno je razvijen program koji uvažava nelinearnost aktivne otpornosti izazvane Joulovim gubicima u željezu transformatora. Simulacijama su utvrđene električne prilike prilikom uklapanja neopterećenog transformatora. Pri tome su ustanovljena kritična stanja, odnosno najnepovoljnije prilike s obzirom na trenutke uklopa prekidaèa i snage kratkog spoja elektroenergetskog sustava. Na kraju su ispitana energetska naprezanja metal-oksidnih odvodnika prenapona uzrokovana povišenjima napona kod uklapanja neopterećenog transformatora izazvanog prelaskom u zasićeno područje krivulje magnetiziranja nelinearnog induktiviteta. Pokazano je da približavanjem rezonantnim uvjetima dolazi do znatnih energetskih preopterećenja odvodnika prenapona što znaèi da za pravilno dimenzioniranje odvodnika prenapona treba voditi računa i o ovoj vrsti privremenih prenaponskih stanja.Nonlinear curve of non-load transformer reactance (magnetizing current-magnetic flux) was linearized with a finite number of linear functions determined by approximation accuracy at different transients (non-loaded transformer switching at different circuit configurations, voltage transformer switching). State-variable waveform monitoring software based on mathematical model of nonlinear inductive circuit was developed . Results generated by this software were compared to results obtained using Power System Blockset, electromagnetic transient software, integrated in Matlab 5.3. Measurements obtained in the experiment of small power transformer switching in confirmed software results. Software incorporating a nonlinear resistance caused by Joule's losses in a transformer iron core was developed additionally. Conducted simulations established electrical conditions at non-loaded transformer switching in including critical conditions, that is the worst case in the term of switching moment and energy system short circuit power. Energetic stress of metal-oxide surge-arrester caused by overvoltages at non-load transformer switching, as a consequence of transition to saturated area of magnetizing curve, was examined. It was showed that the surge-arrester overload appears approaching to the resonance. According to the previous, it is necessary to take into account such temporary overvoltage conditions in the surge-arrester design

Numerički proračun niskofrekvencijskih elektromagnetskih prijelaznih pojava u transformatorima

U radu je dat pregled modela linearnih i nelinearnih električnih elemenata sa naglaskom na modelu transformatora u postojećim bibliotekama dvaju poznatih programa: EMTP-ATP-a i MATLAB/PSB-a. Identificirani su uzroci postojanja numeričkih oscilacija i totalne numeričke nestabilnosti u transformatora sa negativnom reaktancijom namota analizom svojstvenih vrijednosti matrica sustava. Pokazan je originalni način modeliranja nelinearnih elemenata u proračunima niskofrekvencijskih prijelaznih pojava u transformatorima. Izgrađen je algoritam baziran na hiperstabilnoj backward differentiation formulas-BDFp p-tog reda numeričkoj metodi za proračune niskofrekvencijskih prijelaznih pojava u jednofaznim i trofaznim transformatorima. Pokazane su prednosti razvijenog algoritma u pogledu eliminiranja efekta “prebačaj” i mogućnosti reduciranja ili izbjegavanja numeričkih oscilacija i totalne numeričke nestabilnosti u proračunima niskofrekvencijskih prijelaznih pojava u jednofaznim i trofaznim transformatorima. Izgrađen je model dinamičke histerezne petlje sa prednostima u odnosu na postojeći model Hysteretic Inductor-Type 96 u EMTP-u. Također su provedena eksperimentalna mjerenja prijelaznih pojava uklapanja i ferorezonancije kod jednofaznog transformatora kao i uklapanja kod trofaznog trostupnog transformatora. Rezultati pokusa vrlo dobro se slažu s rezultatima proračuna razvijenog programa.In this work is presented the model of linear and nonlinear electrical elements with accent on transformer model in existing well-known software libraries: EMTP-ATP and MATLAB/PSB. Causes of existance of numerical oscillations and total numerical instability in transformers with negative winding reactance are identified by analysis of eigenvalues of system matrix. Original method of nonlinear elements modeling in calculation of low frequency transients in transformers is showed in this work. The algorithm based on hyper stable backward differentiation formulas-BDFp p-th order numerical procedure for calculation of low frequency transients in single-phase and three-phase transformers is developed additionally. The advantages of developed algoritham in a way of elimination of overhooting effect and possibility of reduction or avoiding numerical oscillations and total numerical instability in low frequency transients calculations in single-phase and three-phase transformers are presented. Model of dynamic hysteretic loop with advantages in comparison with existing Hysteretic Inductor-Type 96, EMTP model is developed. Experimental measurements of transients: switching-on and ferroresonance in single-phase transformer and switching-on in three-phase transformer are obtained. Measurements obtained in the experiment confirmed calculation results of developed software

Modeliranje nelinearnog induktiviteta u proračunima prijelaznih pojava

U radu je nelinearna karakteristika (struja magnetiziranja – magnetski tok) reaktancije praznog hoda transformatora linearno aproksimirana na konačan broj pravaca, pri čemu je utvrđen dovoljan broj pravaca za što točniju aproksimaciju pri različitim prijelaznim pojavama (uklapanje neopterećenog transformatora pri različitim konfiguracijama električnog kruga, isklapanje naponskog mjernog transformatora). Na osnovi matematičkog modela za električni krug s nelinearnim induktivitetom razvijen je kompjutorski program za praćenje valnih oblika varijabli stanja. Rezultati dobiveni tim programom uspoređeni su s namjenskim programom Power System Blockset, koji predstavlja specijalizirani dio Matlaba 5.3 za elektromagnetske prijelazne pojave. Također su sprovedena eksperimentalna mjerenja prilikom uklapanja malog energetskog transformatora te pokazano je da se rezultati pokusa dobro slažu s rezultatima razvijenog programa. Dodatno je razvijen program koji uvažava nelinearnost aktivne otpornosti izazvane Joulovim gubicima u željezu transformatora. Simulacijama su utvrđene električne prilike prilikom uklapanja neopterećenog transformatora. Pri tome su ustanovljena kritična stanja, odnosno najnepovoljnije prilike s obzirom na trenutke uklopa prekidaèa i snage kratkog spoja elektroenergetskog sustava. Na kraju su ispitana energetska naprezanja metal-oksidnih odvodnika prenapona uzrokovana povišenjima napona kod uklapanja neopterećenog transformatora izazvanog prelaskom u zasićeno područje krivulje magnetiziranja nelinearnog induktiviteta. Pokazano je da približavanjem rezonantnim uvjetima dolazi do znatnih energetskih preopterećenja odvodnika prenapona što znaèi da za pravilno dimenzioniranje odvodnika prenapona treba voditi računa i o ovoj vrsti privremenih prenaponskih stanja.Nonlinear curve of non-load transformer reactance (magnetizing current-magnetic flux) was linearized with a finite number of linear functions determined by approximation accuracy at different transients (non-loaded transformer switching at different circuit configurations, voltage transformer switching). State-variable waveform monitoring software based on mathematical model of nonlinear inductive circuit was developed . Results generated by this software were compared to results obtained using Power System Blockset, electromagnetic transient software, integrated in Matlab 5.3. Measurements obtained in the experiment of small power transformer switching in confirmed software results. Software incorporating a nonlinear resistance caused by Joule's losses in a transformer iron core was developed additionally. Conducted simulations established electrical conditions at non-loaded transformer switching in including critical conditions, that is the worst case in the term of switching moment and energy system short circuit power. Energetic stress of metal-oxide surge-arrester caused by overvoltages at non-load transformer switching, as a consequence of transition to saturated area of magnetizing curve, was examined. It was showed that the surge-arrester overload appears approaching to the resonance. According to the previous, it is necessary to take into account such temporary overvoltage conditions in the surge-arrester design

Simulations of transformer inrush current by using BDF-based numerical methods

This paper describes three different ways of transformer modeling for inrush current simulations. The developed transformer models are not dependent on an integration step, thus they can be incorporated in a state-space form of stiff differential equation systems. The eigenvalue propagations during simulation time cause very stiff equation systems. The state-space equation systems are solved by using A- and L-stable numerical differentiation formulas (NDF2) method. This method suppresses spurious numerical oscillations in the transient simulations. The comparisons between measured and simulated inrush and steady-state transformer currents are done for all three of the proposed models. The realized nonlinear inductor, nonlinear resistor, and hysteresis model can be incorporated in the EMTP-type programs by using a combination of existing trapezoidal and proposed NDF2 methods