A. Semlyen

• University of Toronto

Summary form only given. We would like to thank the discussers for their clarification and appreciate their elegant derivation. We already discovered that we were incorrect in claiming the half-size test matrix to be applicable to unsymmetrical models. The error was clarified in a subsequent paper. We would like to mention that we have also derived...

Rational models (not including explicit sources as in the case of generators) must be passive in order to guarantee stable simulations. Passivity tests for use with perturbation-type passivity enforcement methods are usually based on frequency sweeping or calculation of the eigenvalues of a Hamiltonian matrix derived from the model's parameters. Re...

One major difficulty in the rational modeling of linear systems is that the obtained model can be nonpassive, thereby leading to unstable simulations. The model's passivity properties are usually assessed by computing the eigenvalues of a Hamiltonian matrix, which is derived from the model parameters. The purely imaginary eigenvalues represent cros...

This paper describes a new transmission-line model for frequency-dependent modeling of untransposed overhead lines and underground cables. The nodal admittance matrix is decomposed into two blocks that, respectively, represent the open- and short-circuit conditions of a half-length line obtained by ldquofoldingrdquo about the middle. By subjecting...

Rational models must be passive in order to ensure stable time domain simulations. The assessment of passivity properties is usually done via a Hamiltonian matrix that is associated with the state-space model, allowing precise characterization of passivity violations from its imaginary eigenvalues. The calculation of eigenvalues can be time consumi...

Summary form only given. It is known that an untransposed open-ended line causes a negative sequence component in the current and the generator reacts through third and higher harmonic voltages. The currents and voltages can be greatly amplified by resonance to any of these harmonics. The phenomenon of machine-line interaction has traditionally bee...

Accurate modeling of power system components for the purpose of electromagnetic transient calculations requires the frequency dependence of components to be taken into account. In the case of linear components, this can be achieved by identification of a terminal equivalent based on rational functions. This paper addresses the problem of approximat...

A methodology for including generators as PV buses in a Newton-Type harmonic power flow program is presented. The construction of the Jacobian for the generator and its insertion into the network equations is described in detail. The method does not require an iterative procedure for satisfying the generator constraints and preserves the quadratic...

This paper proposes an algorithm for obtaining the periodic steady-state solution of a multiphase network including nonlinear, switching, and frequency dependent elements. Unlike existing methods which deal with nonlinear and switching elements in the time domain, the approach presented is entirely in the harmonic domain. The method will be used fo...

This paper presents a new concept called harmonic domain dynamic transfer function (HDDTF), which characterizes the dynamics of a nonlinear, time-periodic network as seen from a port (or multiple ports), in terms of the frequency response of harmonic perturbations superimposed on its underlying periodic steady state. It pertains to the transient be...

A methodology is presented for the order reduction of the dynamic model of a linear weakly periodic system obtained by linearization about the nonsinusoidal periodic steady state. It consists of two stages. First, the time-invariant part of the original full-order system is approximated by a reduced system by using singular value decomposition tech...

The methodology for the order reduction of the dynamic model of a linear weakly periodic system presented in the first part of this paper is applied here to the case when the power network includes frequency dependent transmission lines. The technique for obtaining the external equivalent of the network is described. A comparison is given between s...

This paper deals with the calculation of a sparse network equivalent (SNE) for the analysis of electromagnetic transients in large systems. The methodology and the mathematical details are presented in the companion paper . This sequel presents application examples to demonstrate the accuracy and computational efficiency of the SNE. Equivalents for...

This paper deals with the calculation of a sparse network equivalent (SNE) for the analysis of electromagnetic transients in large systems. The main feature of the new approach is the enforcement of sparsity, stability, passivity, and accuracy at specific frequencies of the equivalent. The procedure is based on time-domain fitting with quadratic pr...

The frequency spectrum of an external system used in the simulation of electromagnetic transients shows many peaks due to resonance effects of the leading transmission lines. Therefore, the two-level equivalent we developed contains simplified lines for the leading part of the system and lumped elements represented by rational functions as a correc...

The paper presents a general model for the representation of a thyristor-controlled reactor (TCR) in the frequency domain. This model underlies the implementation of a Newton-Raphson procedure to determine the periodic steady state of the power system with fast convergence and high accuracy. The TCR model is used as a building block to represent mo...

Transmission lines with nonparallel conductors or significant sags and vertical structures such as towers can be viewed and modeled as nonuniform lines (NUL). The parameters of NULs are distance dependent. This paper presents a mathematical model for time domain simulation of electromagnetic transients in multiphase NULs taking into account the fre...

This paper presents a general model for the representation of a thyristor-controlled reactor (TCR) in frequency domain. The model undelines the implementation of a Newton-Raphson procedure to determine the periodic steady state of the power system with fast convergence and high accuracy. The TCR model is used as a building block to represent more s...

This paper presents a new concept called harmonic domain dynamic transfer function (HDDTF), which characterizes the dynamics of a nonlinear time-periodic network as seen from a port (or multiple ports) in terms of the frequency response of harmonic perturbations superimposed on its underlying periodic steady state. It pertains to the transient beha...

A new solution methodology for the constant matrix, decoupled power flow problem is presented in this paper. The proposed method uses the conjugate gradient method instead of the traditional direct solution of Ax = b for updating the power flow variables. The conjugate gradient method is accelerated with an approximate inverse matrix preconditioner...

This paper presents a methodology for the identification of a reduced-order dynamic equivalent of a nonlinear power network for the simulation of electromagnetic transients. The equivalent is deduced from the observer companion form of the state equations with periodic coefficients and includes the effects of the nonlinearity of the power network a...

Electromagnetic transients on overhead transmission lines are strongly affected by the resistivity ¿ of ground return, which, however, is often poorly known. Therefore, the calculation of transients with very high accuracy for a given ¿ may not make good engineering sense if it entails loss of computational efficiency. The effect of ¿ on the tra...

Nonuniform line (NL) is the designation of a transmission line segment where the conductors are not parallel or where the sag is significant. Transformers and parts of metallic towers can be viewed as NLs. The propagation characteristics of NLs are examined. They appear as generalizations of concepts from uniform transmission lines (UL). It was fou...

The paper presents an approach for network equivalent calculation for the analysis of electromagnetic transients in power systems. It is based on time-domain fitting and enforces some degree of sparsity while preserving the accuracy of the equivalent. The calculated equivalent is appropriate for direct interface with the rest of the system in time-...

This article presents an approach for a network equivalent calculation for the analysis of electromagnetic transients in power systems. It is based on time-domain fitting and enforces some degree of sparsity while preserving the accuracy of the equivalent. The calculated equivalent is appropriate for direct interface with the rest of the system in...

The author discusses the paper "Enforcing passivity for admittance matrices approximated by rational functions" (see ibid., vol.16, no.1, p.97-104, 2001), requesting some additional information on the passivity enforcement of rational function approximations. The original authors' response is also included in this paper.

First Page of the Article

We present a quasi-Newton power flow methodology that incorporates several strategies to obtain substantial computing savings. Newton steps are combined with constant Jacobian (or “simple”) steps and partial Jacobian updates to get an efficient quasi-Newton method. The methodology proposed includes the possibility of selecting the next best step by...

We present a quasi-Newton power flow methodology that incorporates several strategies to obtain substantial computing savings. Newton steps are combined with constant Jacobian (or "simple") steps and partial Jacobian updates to get an efficient quasi-Newton method. The methodology proposed includes the possibility of selecting the next best step by...

A linear power system component can be included in a transient simulation as a terminal equivalent by approximating its admittance matrix Y by rational functions in the frequency domain. Physical behavior of the resulting model entails that it should absorb active power for any set of applied voltages, at any frequency. This requires the real part...

A linear power system component can be included in a transient simulation as a terminal equivalent by approximating its admittance matrix Y by rational functions in the frequency domain. Physical behavior of the resulting model entails that it should absorb active power for any set of applied voltages, at any frequency. This requires the real part...

The generalisation and refinement of a very efficient modular harmonic power flow (MHPF) methodology, initially available for simple components only, is presented. Its capabilities have been extended to three-phase systems that may include: nonlinear elements, such as generators; components with distributed and frequency-dependent parameters, as in...

The paper describes a general methodology for the fitting of measured or calculated frequency domain responses with rational function approximations. This is achieved by replacing a set of starting poles with an improved set of poles via a scaling procedure. A previous paper (Gustavsen et al., 1997) described the application of the method to smooth...

The paper presents a new power transmission line model suitable for the calculation of electromagnetic transients on overhead lines and underground cables. This is achieved by expanding the fitted modes of a conventional line model with constant transformation matrices into simple fractions on the diagonal, and by extending the transformation matri...

This paper presents an efficient methodology for transient modeling of power transformers based on measured or calculated frequency responses. The responses are approximated with rational functions using the method of vector fitting. The resulting model is dynamically stable since only negative poles are used, and input-output stability is assured...

The purpose of this note is twofold. First, it brings to the attention of the readers the work by A. Semlyen (see IEEE Trans. Power Appar. Syst., Vol.PAS-93, no.2, p.676-84, 1974) with essentially the same subject matter as the work of J.S. Thorp et al. (see ibid., vol.18, no.1, p.49-50, 1998), namely the representation of a large power system-from...

This paper introduces a fast and robust method for rational fitting of frequency domain responses, well suited for both scalar and vector transfer functions. Application of the new method results in increased computational efficiency for transmission line models using modal decomposition with frequency dependent transformation matrices. This is due...

This paper introduces two highly accurate transmission line models. In the first one, particularly suitable for overhead lines, the matrix transfer functions for both characteristic admittance and propagation are fitted directly in the phase domain using the method of vector fitting by optimal scaling. In the second model we use for propagation a m...

The well established power flow methods-Gauss-Seidel, Newton-Raphson, and fast decoupled load flow-are all based an major, classical methodologies of applied mathematics. The Krylov subspace power flow (KSPF) method presented in this paper uses a newer, very successful approach-the Krylov subspace methodology-developed in applied linear algebra for...

This paper presents improved and new methodologies for the calculation of critical eigenvalues in the small-signal stability analysis of large electric power systems. They augment the robustness and efficiency of existing methods and provide new alternatives. The procedures are implementations of Newton's method, inverse power and Rayleigh quotient...

The basic principles of an efficient new methodology for the calculation of the nonsinusoidal periodic steady state in power systems with nonlinear and time-varying components are described. All linear parts, including the network and part of the loads, are represented in the frequency domain, while nonlinear and time-varying components, mainly loa...

This paper presents a new method for the simulation of electromagnetic transients on transmission lines. Instead of using convolutions of the input variables only, we perform short convolutions with both input and output variables. The result is a method of two-sided recursions (TSR), which is comparable in efficiency with the existing recursive co...

The paper presents a methodology for the calculation of a selected set of eigenvalues, considered critical in the small signal stability analysis of power systems. It analyzes several alternatives for refining a preliminary rough solution obtained by subspace iterations. These alternatives range from constant-matrix iterative refinement to Newton's...

The paper describes a procedure for the representation of hysteresis in the laminations of power transformers in the simulation of electromagnetic transient phenomena. The model is based on the recognition that in today's iron cores the hysteresis loops are narrow and therefore the modeling details are only important in relation to the incurred los...

A direct method based on the linear least squares approach is applied for obtaining reduced order approximations of the propagation and characteristic admittance transfer functions of transmission lines for use in calculation of transients. The practical level of the acceptable error is discussed in terms of the precision of the frequency domain in...

A frequency dependent resistance model for the representation of eddy current losses in the windings of three-phase, two or three windings, transformers is presented. It consists of a low order R, L Foster circuit connected in series with any existing transformer model to produce the realistic damping that results from increased losses during trans...

A detailed time domain model for the eddy current losses in the windings of a transformer is presented. The basic elements for the derivation of the model are the turns which may be combined into sections. The model is expressed as an R-matrix. Its diagonal elements were fitted using a series Foster circuit while for the off-diagonal elements the a...

A general methodology is presented for the state equation approximation of a multiple input/output linear system from transfer matrix data. A complex transformation matrix, obtained by eigenanalysis at a fixed frequency, is used for diagonalization of the transfer matrix over the whole frequency range. A scalar estimation procedure is applied for i...

A complete, three phase transformer model for the calculation of electromagnetic transients is presented. The model consists of a set of state equations solved with the trapezoidal rule of integration in order to obtain an equivalent Norton circuit at the transformer terminals. Thus the transformer model can be easily interfaced with an electromagn...

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The paper presents a new method for optimal power flow calculations, using the generalized power balance constraint, obtained from the power flow equations by elimination of all state variables. The reduced optimization problem in the control variables is subsequently decoupled into two stages: a primary stage in the primary control variables (gene...

When computing transients with the EMTP (electromagnetic transients program) a single time step is selected for the trapezoidal integration all over the system. When this step is small because of the presence of a fast component, the simulation may be significantly slowed down. The authors present a procedure by which the computations at selected f...

A widely applicable, general methodology for estimation of transfer function parameters from frequency response data is presented. The procedure is based on the solution of a linear least squares problem by the singular value decomposition (SVD). The condition of the problem is discussed and approaches referred to as shifting and scaling are introd...

The standard EMTP (Electromagnetic Transients Program) is based on time domain calculations as these are eminently appropriate for simulation purposes. An approach to direct frequency-domain representation of an external system of any size or complexity is presented. The requirements for the external system are that it must be linear and time invar...

Eddy current effects are included in a model of a power transformer for the study of electromagnetic transients. Existing analytical formulae for the calculation of losses in the windings are evaluated. Various equivalent circuits are fitted to represent in the time domain the damping produced by eddy currents in the windings. A frequency dependent...

The authors extend the analysis of torsional oscillations into the nonlinear range and investigate the Hopf bifurcation phenomenon of the torsional dynamics of a turbine generator in a series capacitor compensated power system. The transition of growing torsional oscillations into nonlinear oscillations with limit cycles is demonstrated. Based on p...

Very efficient procedures for computing elementary parameters (turn leakage inductances and capacitances) in a transformer are presented. The turns are used as a calculation base to permit modeling at very high frequencies. Turn-to-turn (or loop) leakage inductances are obtained by an image method. The charge simulation method is used for finding t...

A complete model for transformers is derived on the basis of very efficiently calculated elementary (turn-to-turn) parameters. A high-order turn-to-turn model is constructed for the windings. This model is reduced to a lower order by operating on the resulting matrices. An electric equivalent circuit for the core is obtained from the principle of d...

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The extreme loading condition (XLC) of a power system is defined by assuming a load increase (according to a predefined pattern for both active and reactive powers) until a maximum is reached for one of the loads. The XLC is significant for the assessment of voltage stability. Its calculation, as presented, is based on increasing the load admittanc...

An efficient method for calculating the load flow solution of weakly meshed transmission and distribution systems is presented. Its essential advantages over a previous approach are the following: (1) It uses active and reactive powers as flow variables rather than complex currents, thus simplifying the treatment of P , V buses and reducing the rel...

A general package for harmonic-domain computation is described. It consists of a set of routines which can be used by developers of programs for power system harmonic applications. The most basic routines have been listed. The package represents nonlinear characteristics by fitting the characteristic with a polynomial, for which special harmonic do...

Two sparsity-based eigenvalue simultaneous iterations and the modified Arnoldi method are presented and their application to the small signal stability analysis of large power systems is discussed. An algorithm utilizing these two methods is proposed for calculating the eigenvalues around a fixed point which can be placed at will in various parts o...

An efficient method is described for the solution of the short-term hydro-thermal dispatch problem including optimal power flow (OPF) as the mathematical model of the thermal subsystem. This approach has the capability of taking into account the following effects: coupling of cascaded multichannel reservoirs, water time delays, reservoir head varia...

A novel approach for the investigation of the inrush phenomenon in single and three-phase transformers is presented. It consists of the calculation of a sequence of quasi-steady-state images using harmonic domain computation. Thus, lengthy-time domain simulations with the related round-off and truncation errors are avoided. The advantage of the met...

Although the magnetising nonlinearity of power transformers is recognised as a source of harmonic distortion, its effect is often considered negligible, and the analytical models used to assess such effect are extremely simple. A new and more general power system frame of reference which models the coupling between phases and between harmonics is p...

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The author's consider a new formulation of optimal power flow (OPF) that makes it eminently suitable for accurate incremental modeling. G. Zoutendijk's method of feasible directions (1960) for solving the nonlinear programming problems is adapted for the solution of the OPF. Hydraulic modeling of systems with a considerable share of hydraulic gener...

The authors give a detailed analysis of the space and time distribution of the field variables B , H , and E inside saturating laminations, using a harmonic domain modeling method. A simple way has been devised for the representation of hysteresis. It was found that hysteresis affects to a small degree the variation of H and B but has no noticeable...

The steady-state modeling of a laminated saturating iron core is discussed. The modeling is in the harmonic domain, i.e. in terms of phasors for the different harmonics due to the nonlinear magnetization characteristic. The procedure is based on a polynomial approximation of the magnetization characteristic. In the harmonic domain the computations...

Computer algorithms of the Newton-Raphson type are derived for the harmonic analysis of systems containing nonlinear dynamic components in periodic steady state. The problem is formulated in the complex conjugate multiharmonic space that inherently represents the harmonic coupling between the different harmonic frequencies. The theory is applied to...

A efficient method is reported for the calculation of the complete set of eigenvalues and eigenvectors associated with the state variables in a specified study zone of a large power system, linearized for the purpose of small-signal stability studies. The computation is performed in two stages. Stage 1 starts with a set of first guesses for the eig...

A power flow method is described for solving weakly meshed distribution and transmission networks, using a multiport compensation technique and basic formations of Kirchoff's laws. This method has excellent convergence characteristics and is robust. A computer program implementing this scheme was developed and successfully applied to several practi...

The harmonic voltage distortion caused by static convertors can be greatly affected by the presence of local salient-pole synchronous generators or compensators. Such machines act as harmonic converters and are sensitive to the sequence of the fundamental and harmonic frequencies. An accurate equivalent circuit for the synchronous machine in the mu...

A generalized loss formula (GLF) is developed for the system losses around an operating point. It is a quasi-oscillatory approximation of the exact losses by having the same first and approximately the same second derivatives at the operating point, with respect to three sets of variables: generator powers Pg, generator voltage magnitudes Vg, and t...

The harmonic analysis of a power system requires appropriate models of all system components. Synchronous machines act as harmonic converters, sensitive to the sequence of the fundamental and harmonic frequencies. This paper describes the derivation of a harmonic model of the machine in the form of a three-phase complex admittance matrix and its ap...

In the analysis of harmonic power flows, the transformer is generally represented as a passive component. However, due to the magnetisation nonlinearity, the transformer constitutes a source of harmonic currents which can aggregate and cause substantial waveform distortion. This paper describes the development of a Norton equivalent circuit which a...

The synchronous machine dynamic equations are expanded into a multi-harmonic set suitable for the analysis of steady state voltage and current power system harmonics. A generalized Norton Equivalent is then developed to represent the machine under any type of external steady state system conditions affecting the stator or the rotor. Various harmoni...

The paper describes a rather comprehensive set of computational experiments for the instruction of senior students in the field of power systems. The experiments are based on a program package called FORCEPS (FORtran Codes for Education in Power Systems). These replace previous laboratory experiments which were only partially adequate to demonstrat...

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Lumped parameter computer models of transmission lines as well as Transient Network Analyzers (TNA) require the representation of the frequency dependent longitudinal impedance of a three-phase line in the form of R, L branches or Ordinary Differential Equations (ODE), respectively. The advantage of discretizing a distributed parameter line consist...

A frequency dependent transmission line model, developed in 1975 and referred to in the BPA Electro-Magnetic Transients Program (EMTP) as 'Semlyen Setup', has persistently exhibited an anomaly frustrating to its users. When the line model is isolated from the rest of the power system model by switch action it responds with an apparently linearly in...

Powerful simulation programs are available for large electric and electronic circuits with lumped parameter elements. State equation models are used for each component. This paper presents the methodology for obtaining a finite order state equation model for transmission lines, despite the distributed nature of their parameters. The lines considere...

Powerful simulation programs are available for large electric and electronic circuits with lumped parameter elements. State equation models are used for each component. This paper presents the methodology for obtaining a finite order state equation model for transmission lines, despite the distributed nature of their parameters. The lines considere...

A multiphase transmission line is modelled by sinusoidally distributed charges on a cylindrical surface. In its inside, the field is uniform and outside the cylinder it can be obtained by a rotating equivalent dipole. Total charge, capacitance, required number of conductors and natural power are calculated for this idealized multiphase line model,...

The paper presents a methodology for the closed form calculation of electromagnetic transients in power system networks, resulting from sequential switching operations. The system is viewed from the terminals of the switch poles where a compensating voltage source is applied for closing and a compensating current source is applied for opening of a...

The economic scheduling of generation in power systems was traditionally performed by solving the equations of coordination while satisfying the constraint of power balance between load and total generation. Later, optimal load flow programs were developed to take into account generator voltages as control variables and different operational constr...

Natural frequencies of transmission lines are identified by the poles obtained from line transfer function matrices. The procedure is based on an analysis in terms of the complex frequency s which has been introduced in the expressions of line parameters and transfer functions using the concept of complex depth.

This paper gives the outline of a new method for the calculation of electromagnetic transients on transmission lines above homogeneous ground. The basic idea is that line transfer functions which normally are calculated in terms of hyperbolic functions of omega are now expressed in terms of s = sigma + j omega . This is possible due to the existenc...

The paper is based on analytical expressions of line impedance and admittance in terms of the complex variable s = a + iw. Therefore, transfer functions of lines with homogeneous earth return, such as terminal admittances or two-port parameters, also become functions of s. Their poles have been obtained by the secant method and then the residues ha...

The paper addresses the problem of induction effects of a three phase transmission line on a parallel circuit, in the presence of homogeneous earth. It is based on the simple expression' Z¿= K¿ln(D"/D') of a mutual impedance (where D¿" is complex) which approximates well earth return effects as calculated by Carson2. This expression is used to d...

For modelling current return in homogeneous ground, the paper introduces the concept of an ideal (superconducting) current return plane placed below the ground surface at a complex distance p equal to the complex penetration depth for plane waves. This "complex" plane appears as a mirroring surface, so that conductor images can be used to derive ve...

Wave propagation with ground return is studied in the classical literature, by Carson andothers, for frequencies up to a critical upper limit, where displacement currents in the earth start becoming predominant. This paper derives simple expressions for the additional ground impedances, valid in the super-critical range. Problems when these are rel...

Recursive convolutions are believed to be the basic approach for digital calculation of electromagnetic transients on transmission systems. They require step responses expressed by means of exponential functions. This paper presents the theory for obtaining an arbitrary number of exponential components - with real or complex exponents directly from...

Today's power systems tend to cover whole continents and the density of transmission facilities is continuously increasing. A disturbance produced in one point propagates to remote points of the system, it may be reflected at the boundaries and standing waves may build up. To obtain better insight into such phenomena, the complex power system is re...