##
Transformatör Sargılarında Oluşan Hızlı Değişimli Geçici Olayların İncelenmesi Ve Enerji İletim Sistemlerinin Modellenmesinde Yeni Bir Yaklaşım

Transformatör Sargılarında Oluşan Hızlı Değişimli Geçici Olayların İncelenmesi Ve Enerji İletim Sistemlerinin Modellenmesinde Yeni Bir Yaklaşım

##### Dosyalar

##### Tarih

1985

##### Yazarlar

Soysal, A. Oğuz

##### Süreli Yayın başlığı

##### Süreli Yayın ISSN

##### Cilt Başlığı

##### Yayınevi

Fen Bilimleri Enstitüsü

Institute of Science and Technology

Institute of Science and Technology

##### Özet

Gerilim darbeleri ve açma-kapama işlemleri, transforma törlerde yüksek frekanslı salınımlara yol açarlar. Bu tür ge çici salınımlar, genellikle geometrik boyutlardan 'hareketle hesaplanan R, L, C parametrelerinin oluşturduğu eşdeğer devre ler yardımıyla incelenmektedir. Tezde geliştirilen yeni bir yöntem, sargıların uç empedans karakteristiğinden hareketle modellenraesine olanak vermiştir. Bu yöntem, sargı iç yapısı nın bilinmesine gerek göstermediğinden, özellikle, enerji ile tim ve dağıtım sistemlerinde yer alan transformatörlerde ortaya çıkan geçici olayların incelenmesinde kolaylık sağla maktadır. Tezde yapılan çalışmalar aşağıdaki gibi özetlenebilir: Tezin ikinci bölümünde, yüksek frekanslı alan değişimle rinde demir çekirdeğin davranışı incelenmiştir. Hem teorik, hem de deneysel yollardan yapılan incelemeler, nötr noktası topraklanmış bir sargıda ortaya çıkan geçici olayları belirle mek amacıyla yapılan çalışmalarda demir çekirdeğin etkilerinin ihmal edilebileceğini göstermiştir. Üçüncü bölümde, literatürde yaygın olarak kullanılan bir toplu parametreli eşdeğer devre ele alınmış ve bu devreye ilişkin durum denklemleri elde edilmiştir. Dördüncü bölümde, transformatör sargısı iki uçlu bir ele man gibi düşünülerek uç empedans fonksiyonu tanımlanmış, bu fonksiyonun geçici rejim davranışı açısından anlamı tartışıl mıştır. Beşinci bölümde, sargıya ilişkin uç empedans fonksiyonunu sağlayan kanonik devrelerin sentezi gösterilmiş; elde edilen Cauer ve Poster biçimindeki devrelerin durum denklemleri ya zılmıştır. - I - Altıncı bölümde, transformatörlerle birlikte enerji iletim sistemlerini oluşturan, - enerji iletim hattı, güç anahtarı, filtre, parafudr gibi temel elemanların matematiksel modelleri veril miştir. Genel halde bir enerji iletim ve dağıtım sisteminin model- lenmesi için geliştirilen sistematik bir yöntem, yedinci bölüm de açıklanmıştır. Bu yöntemin pratikteki uygulamalarını göster mek üzere iki örnek gözönüne alınmıştır. Bunlardan birincisi, laboratuarda, bir deney transformatörü ve bir hat modeli ile oluşturulan sistemdir. Bu sistemde, çeşitli basamak girişler için transformatör uçlarında gözlenen gerilim değişimleri, he saplanan değerlerle karşılaştırılmıştır. Diğer örnek ise, bir çelik fabrikasında ark ocaklarını besleyen elektriksel sistemdir. Söz konusu sistemin tümüne ilişkin matematiksel modelin kurulması gösterilmiş, tipik çözüm örnekleri verilmiş tir.

Voltage surges due to atmospheric discharges and switching operations cause high frequency transient oscillations of very- short duration in electrical transmission and distribution systems. In the course of such transient phenomena, various points of the network and transformer windings are subjected to overvoltage stresses. An accurate calculation of these voltage- stresses is of prime importance for the design of power trans formers. On the other hand, some recent transient phenomena causing failures in power systems have shown that the transient behavior must be taken into account to provide a suitable insulation coordination and protection for high voltage networks [15], [16]. Since the beginning of the century, extensive works of theoretical and experimental basis have been carried out to in vestigate the surge response of transformers. In the past, attempts were made to find analytical solutions for integro- differential equations written for distributed-parameter winding models of the transformers. Unfortunately, even with many assumptions to simplify the solution of the problem, complete calculations were laborious and unsuitable for routine design. Wide-spread use of digital computers in recent years made it possible to solve the large number of system equations in the time domain. Thus, various transformer winding models with lumped parameters were developed and the transient behavior of the transformers has been determined in terms of the state equations. Computer aided analysis is obviously the most convenient and powerful method for determining the transient phenomena which occur in power transmission and distribution systems. - Ill - in general, such a method consists of building a mathematical model for the entire system involving transmission lines and transformers; then solving the state equations in the time do main. All possible switching and fault conditions can be simu lated in this way, in order to predict the complete behavior of the system: Methods of modelling for transmission lines and trans formers are extensively covered in the litterature. For trans mission lines, lumped parameter models based on inductance, capacitance and resistance per unit length are available. The conventional equivalent circuits developed for transformers consist of RLC1 ladder networks similar to those representing transmission lines; the main difference being the presence of a series capacitances and mutual inductances between sections. The network parameters are normally calculated from. the geomet ric dimensions of windings, it is obvious that the determination of those parameters is not difficult at the design stage of a transformer, since all the constructional details and material properties are known. However, when the task of building a model of a complex electrical power system for high speed switching or impulse surge analysis arises, one must think of two important problems: a- Generally, no sufficient information is available about the internal structure, winding geometry and material properties of transformers already manufactured and operating in the net work. In such a case, conventional transformer models become unsuitable since it is not possible to determine the parameters. b- Electric power supply systems consist of several transfor mers interconnected by transmission lines. When transformers are represented by eauivalent circuits which consist of large number of tandem sections as it has been proposed in early works in literature, the number of the system dynamic equations to be solved reaches easily several hundreds. Since the system - IV - matrix is generally plain for these equations, the solution in computer recmires large memory and long computation time. In this thesis, a new method is proposed to build a mathe matical model of power transformers by means of terminal im pedance measurements. This method offers two advantages in the study of switching surges in power systems. First, all the parameters related to transformer models are determined from the terminal measurements, and a detailed knowledge of the in ternal structure is not required. These measurements can be performed easily on any transformer existing in electric power systems. Second, the canonic structure of the network permits the use of sparse matrix techniques and offers advantages in numerical integration of state equations. In the 2 nd. chapter, the effect of the iron core is exa mined both in theoretical and experimental ways. By the rela tions derived from the Maxwell equations, it is shown that at high frequencies, the magnetic flux is attenuated in the inter nal regions of the core, and condensed on the surface. Besides, experimental works also have shown that an iron core winding behaves frequency-dependent up to 10 kHz; however, for higher frequencies, self and mutual inductances approach asymptotically to the values of the air core winding. Time domain measurements on a power transformer agreed with the above observations and have shown clearly that, for a winding with the neutral point earthed, the iron core has a negligible effect on the high speed transients either in the cases of the secondary winding open or short circuited. According to these observations, it has been assumed that the iron core can be ignored in the ana lysis of high speed transients in order to represent the trans former windings by linear passive RLC networks. In the 4 th. chapter, the transformer winding is considered as a two terminal component and the terminal impedance function is defined with the following relation: - V - vo(t) = z iQ(t) (i) where, v is the terminal voltage and i is the terminal current. Linearity of the system implies that Z does not depend neither of current nor voltage. The terminal (driving point) impedance function can be expressed in the s domain in terms of the coef ficient matrices of the state equations by a rational function in the form: Z(e) = 2İ2İ. (ii) Q(s). '. In the case of a no-loss winding for which all resistances and conductances of the equivalent circuit (fig. 11) are zero, the poles and zeros are pure imaginary and they are located alternatively on the jv axis as shown in fig.14. For the values of angular frequencies corresponding to the poles, the impedance tends to infinity, whereas it goes to zero for the values cor responding to the zeros. Experiments have shown that, when ohmic resistance of the winding is present, the poles and zeros shift leftward in the complex plane, in such a way that imaginary parts remain almost unchanged (fig.15). Thus, in the general case, it may be assumed that the magnitude of winding terminal impedance takes maximum and minimum values for angular frequencies corresponding to the imaginary part of poles and zeros respectively. Synthesis of a passive LC network described by an impe dance function is a solved problem and several algorithms were developed. In order to realise (ii) written in the form %(B\ fc.g-(s2 + ^)(s2^)... (3*,^) (i±i) (s2-iO(s2+w2)... (s2-w22m) for a lossless winding, either Cauer, or Poster synthesis methods can be used. Once the LC network is realised, ohmic - VI - resistances are added to the sections. The resistance values and the scaling factor k are determined by an optimization process in such a way that the deviations between impedance function and the impedance characteristic obtained by terminal measurements be minimum. As an application to the above mentioned procedure, the equivalent circuit of Poster type is developed for a laboratory size model transformer. Optimal gradient algorithm is applied to calculate ' the resistor values and the scaling factors, and the parameters are found as listed in Table-Ill (Chapter 5). It must be noted that the parameters determined for this '.synthetic model' are fictitious values and they do not corres pond to any physical section of the winding. However, the model have the same driving point impedance function as the winding. In the Chapter 6» the constituents of a power supply net work such as transmission lines, power switches, lightning arresters, compensating capacitors and filters are considered as two or three terminal components and their state equations are given. The Chapter 7 deals with the determination of the mathe matical model of a power system comprising several transformers, interconnected by transmission lines. To establish the mathema tical model for such a system, the graph should be constituted by using the terminal graphs corresponding to the components. A proper tree in this graph is next chosen according to the following ruless a- All voltage sources must be taken on branches and all current sources on chords. b- Components for which the terminal current is chosen as input function, should be on branches. Reciprocally, components for which the terminal voltage is the input function, should be on chords. - VII - The components on branches and chords are grouped to write the state equations of the whole system in the following form: d dt LxkJ Ad"BdQ2Gk(xk) BdQlCk '-^4 Ak-Sfc^W, Id Bd% M ° İ (iv) where* the indice d denotes the branches and the indice k denotes the chords. A,B,C are the coefficient matrices of the corresponding state equations. Q q q are the cutset matrices corresponding respectively to the components which comprise reactive elements; resistances and conductances; current and voltage sources. As the system may include non-linear resistors (lightning arresters) G. and R, are taken in the general non linear form. To illustrate the proposed method of modelling power systems and determination of switching surges, two examples are given. The first one corresponds to the model system built in the laboratory with an experimental transformer and a line model. The system was driven by a step function generator and the terminal voltage of the transformer was observed by means of a storage oscilloscope. On the other hand, the state equa tions were solved in digital computer. A good agreement was obtained between measured and calculated values. The second application is an example of the supply network of a steel manufacturing plant in which heavy arc furnaces take place. The mathematical model of this system is obtained and simu lation of switching operations is shown.

Voltage surges due to atmospheric discharges and switching operations cause high frequency transient oscillations of very- short duration in electrical transmission and distribution systems. In the course of such transient phenomena, various points of the network and transformer windings are subjected to overvoltage stresses. An accurate calculation of these voltage- stresses is of prime importance for the design of power trans formers. On the other hand, some recent transient phenomena causing failures in power systems have shown that the transient behavior must be taken into account to provide a suitable insulation coordination and protection for high voltage networks [15], [16]. Since the beginning of the century, extensive works of theoretical and experimental basis have been carried out to in vestigate the surge response of transformers. In the past, attempts were made to find analytical solutions for integro- differential equations written for distributed-parameter winding models of the transformers. Unfortunately, even with many assumptions to simplify the solution of the problem, complete calculations were laborious and unsuitable for routine design. Wide-spread use of digital computers in recent years made it possible to solve the large number of system equations in the time domain. Thus, various transformer winding models with lumped parameters were developed and the transient behavior of the transformers has been determined in terms of the state equations. Computer aided analysis is obviously the most convenient and powerful method for determining the transient phenomena which occur in power transmission and distribution systems. - Ill - in general, such a method consists of building a mathematical model for the entire system involving transmission lines and transformers; then solving the state equations in the time do main. All possible switching and fault conditions can be simu lated in this way, in order to predict the complete behavior of the system: Methods of modelling for transmission lines and trans formers are extensively covered in the litterature. For trans mission lines, lumped parameter models based on inductance, capacitance and resistance per unit length are available. The conventional equivalent circuits developed for transformers consist of RLC1 ladder networks similar to those representing transmission lines; the main difference being the presence of a series capacitances and mutual inductances between sections. The network parameters are normally calculated from. the geomet ric dimensions of windings, it is obvious that the determination of those parameters is not difficult at the design stage of a transformer, since all the constructional details and material properties are known. However, when the task of building a model of a complex electrical power system for high speed switching or impulse surge analysis arises, one must think of two important problems: a- Generally, no sufficient information is available about the internal structure, winding geometry and material properties of transformers already manufactured and operating in the net work. In such a case, conventional transformer models become unsuitable since it is not possible to determine the parameters. b- Electric power supply systems consist of several transfor mers interconnected by transmission lines. When transformers are represented by eauivalent circuits which consist of large number of tandem sections as it has been proposed in early works in literature, the number of the system dynamic equations to be solved reaches easily several hundreds. Since the system - IV - matrix is generally plain for these equations, the solution in computer recmires large memory and long computation time. In this thesis, a new method is proposed to build a mathe matical model of power transformers by means of terminal im pedance measurements. This method offers two advantages in the study of switching surges in power systems. First, all the parameters related to transformer models are determined from the terminal measurements, and a detailed knowledge of the in ternal structure is not required. These measurements can be performed easily on any transformer existing in electric power systems. Second, the canonic structure of the network permits the use of sparse matrix techniques and offers advantages in numerical integration of state equations. In the 2 nd. chapter, the effect of the iron core is exa mined both in theoretical and experimental ways. By the rela tions derived from the Maxwell equations, it is shown that at high frequencies, the magnetic flux is attenuated in the inter nal regions of the core, and condensed on the surface. Besides, experimental works also have shown that an iron core winding behaves frequency-dependent up to 10 kHz; however, for higher frequencies, self and mutual inductances approach asymptotically to the values of the air core winding. Time domain measurements on a power transformer agreed with the above observations and have shown clearly that, for a winding with the neutral point earthed, the iron core has a negligible effect on the high speed transients either in the cases of the secondary winding open or short circuited. According to these observations, it has been assumed that the iron core can be ignored in the ana lysis of high speed transients in order to represent the trans former windings by linear passive RLC networks. In the 4 th. chapter, the transformer winding is considered as a two terminal component and the terminal impedance function is defined with the following relation: - V - vo(t) = z iQ(t) (i) where, v is the terminal voltage and i is the terminal current. Linearity of the system implies that Z does not depend neither of current nor voltage. The terminal (driving point) impedance function can be expressed in the s domain in terms of the coef ficient matrices of the state equations by a rational function in the form: Z(e) = 2İ2İ. (ii) Q(s). '. In the case of a no-loss winding for which all resistances and conductances of the equivalent circuit (fig. 11) are zero, the poles and zeros are pure imaginary and they are located alternatively on the jv axis as shown in fig.14. For the values of angular frequencies corresponding to the poles, the impedance tends to infinity, whereas it goes to zero for the values cor responding to the zeros. Experiments have shown that, when ohmic resistance of the winding is present, the poles and zeros shift leftward in the complex plane, in such a way that imaginary parts remain almost unchanged (fig.15). Thus, in the general case, it may be assumed that the magnitude of winding terminal impedance takes maximum and minimum values for angular frequencies corresponding to the imaginary part of poles and zeros respectively. Synthesis of a passive LC network described by an impe dance function is a solved problem and several algorithms were developed. In order to realise (ii) written in the form %(B\ fc.g-(s2 + ^)(s2^)... (3*,^) (i±i) (s2-iO(s2+w2)... (s2-w22m) for a lossless winding, either Cauer, or Poster synthesis methods can be used. Once the LC network is realised, ohmic - VI - resistances are added to the sections. The resistance values and the scaling factor k are determined by an optimization process in such a way that the deviations between impedance function and the impedance characteristic obtained by terminal measurements be minimum. As an application to the above mentioned procedure, the equivalent circuit of Poster type is developed for a laboratory size model transformer. Optimal gradient algorithm is applied to calculate ' the resistor values and the scaling factors, and the parameters are found as listed in Table-Ill (Chapter 5). It must be noted that the parameters determined for this '.synthetic model' are fictitious values and they do not corres pond to any physical section of the winding. However, the model have the same driving point impedance function as the winding. In the Chapter 6» the constituents of a power supply net work such as transmission lines, power switches, lightning arresters, compensating capacitors and filters are considered as two or three terminal components and their state equations are given. The Chapter 7 deals with the determination of the mathe matical model of a power system comprising several transformers, interconnected by transmission lines. To establish the mathema tical model for such a system, the graph should be constituted by using the terminal graphs corresponding to the components. A proper tree in this graph is next chosen according to the following ruless a- All voltage sources must be taken on branches and all current sources on chords. b- Components for which the terminal current is chosen as input function, should be on branches. Reciprocally, components for which the terminal voltage is the input function, should be on chords. - VII - The components on branches and chords are grouped to write the state equations of the whole system in the following form: d dt LxkJ Ad"BdQ2Gk(xk) BdQlCk '-^4 Ak-Sfc^W, Id Bd% M ° İ (iv) where* the indice d denotes the branches and the indice k denotes the chords. A,B,C are the coefficient matrices of the corresponding state equations. Q q q are the cutset matrices corresponding respectively to the components which comprise reactive elements; resistances and conductances; current and voltage sources. As the system may include non-linear resistors (lightning arresters) G. and R, are taken in the general non linear form. To illustrate the proposed method of modelling power systems and determination of switching surges, two examples are given. The first one corresponds to the model system built in the laboratory with an experimental transformer and a line model. The system was driven by a step function generator and the terminal voltage of the transformer was observed by means of a storage oscilloscope. On the other hand, the state equa tions were solved in digital computer. A good agreement was obtained between measured and calculated values. The second application is an example of the supply network of a steel manufacturing plant in which heavy arc furnaces take place. The mathematical model of this system is obtained and simu lation of switching operations is shown.

##### Açıklama

Tez (Doktora) -- İstanbul Teknik Üniversitesi, Fen Bilimleri Enstitüsü, 1985

Thesis (Ph.D.) -- İstanbul Technical University, Institute of Science and Technology, 1985

Thesis (Ph.D.) -- İstanbul Technical University, Institute of Science and Technology, 1985

##### Anahtar kelimeler

Dönüştürücüler,
Enerji iletim sistemleri,
Transformers,
Energy transmission systems