Power Systems Electromagnetic Transients Simulation
Electromagnetic transients simulation (EMTS) has become a universal tool for the analysis of power system electromagnetic transients in the range of nanoseconds to seconds. This book provides a thorough review of EMTS and many simple examples are included to clarify difficult concepts. This book will be of particular value to advanced engineering students and practising power systems engineers.
Inspec keywords: transformers; transmission lines; continuous systems; difference equations; FORTRAN; underground cables; EMTP; discrete systems; power electronics; power system CAD; mathematics computing; integration
Other keywords: transformer; Matlab code; EMTDC; numerical integrator substitution; transient simulation; frequency dependent network equivalent; power electronic system; EMT simulation; underground cable; discrete system; mixed timeframe simulation; state variable analysis; PSCAD; FORTRAN code; root matching method; difference equation; continuous system; rotating plant; transmission lines; system identification
Subjects: Differential equations (numerical analysis); Numerical integration and differentiation; Power convertors and power supplies to apparatus; Differential equations (numerical analysis); Transformers and reactors; Numerical integration and differentiation; Power transmission lines and cables; Power electronics, supply and supervisory circuits; Power engineering computing; Power system planning and layout
 Book DOI: 10.1049/PBPO039E
 Chapter DOI: 10.1049/PBPO039E
 ISBN: 9780852961063
 eISBN: 9780863419836
 Page count: 448
 Format: PDF

Front Matter
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1 Definitions, objectives and background
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The operation of an electrical power system involves continuous electromechanical and electromagnetic distribution of energy among the system components. During normal operation, under constant load and topology, these energy exchanges are not modelled explicitly and the system behaviour can be represented by voltage and current phasors in the frequency domain. However, following switching events and system disturbances the energy exchanges subject the circuit components to higher stresses, resulting from exces sive currents or voltage variations, the prediction of which is the main objective of power system transient simulation.

2 Analysis of continuous and discrete systems
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With the exceptions of a few auxiliary components, the electrical power system is a continuous system, which can be represented mathematically by a system of differential and algebraic equations. A convenient form of these equations is the state variable formulation, in which a system of n firstorder linear differential equations results from an n order system. The state variable formulation is not unique and depends on the choice of state variables. The following state variable realisations have been described in this chapter: successive differentiation, controller canonical, observer canonical and diagonal canonical. Digital simulation is by nature a discrete time process and can only provide solutions for the differential and algebraic equations at discrete points in time, hence this requires the formulation of discrete systems. The discrete representation can always be expressed as a difference equation, where the output at a new time point is calculated from the output at previous time points and the inputs at the present and previous time points.

3 State variable analysis
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In the state variable solution it is the set of first order differential equations, rather than the system of individual elements, that is solved by numerical integration. The most popular numerical technique in current use is implicit trapezoidal integration, due to its simplicity, accuracy and stability. Solution accuracy is enhanced by the use of iterative methods to calculate the state variables. State variable is an ideal method for the solution of system components with timevarying nonlinearities, and particularly for power electronic devices involv ing frequent switching. This has been demonstrated with reference to the static a.c.d.c. converter by an algorithm referred to as TCS (Transient Converter Simu lation). Frequent switching, in the state variable approach, imposes no overhead on the solution. Moreover, the use of automatic step length adjustment permits optimising the integration step throughout the solution.

4 Numerical integrator substitution
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A continuous function can be simulated by substituting a numerical integration formula into the differential equation and rearranging the function into an appropriate form. Among the factors to be taken into account in the selection of the numerical integrator are the error due to truncated terms, its properties as a differentiator, error propagation and frequency response. Numerical integration substitution (NIS) constitutes the basis of Dommel's EMTP , which, as explained in the introductory chapter, is now the most generally accepted method for the solution of electromagnetic transients.

5 The rootmatching method
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An alternative to the difference equation using the trapezoidal integration developed in Chapter 4 for the solution of the differential equations has been described in this chapter. It involves the exponential form of the difference equation and has been developed using the rootmatching technique. The exponential form offers the following advantages: 1) Eliminates truncation errors, and hence numerical oscillations, regardless of the step length used. 2) Can be applied to both electrical networks and control blocks. 3) Can be viewed as a Norton equivalent in exactly the same way as the difference equation developed by the numerical integration substitution (NIS) method. 4) It is perfectly compatible with NIS and the matrix solution technique remains unchanged. 5) Provides highly efficient and accurate time domain simulation. The exponential form can be implemented for all series and parallel RL, RC, LC and RLC combinations, but not arbitrary components and hence is not a replacement for NIS but a supplement.

6 Transmission lines and cables
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In this chapter, the authors discusses the travelling wave transmission line models. The frequency dependent transmission line model was presented. Details of transmission line geometry and conductor data are required in order to calculate accurately the frequencydependent electrical parameters of the line. The simulation time step was based on the shortest response time of the line. The chapter also presents the phasedomain models which is the most accurate and robust for detailed transmission line representation. Calculation of electrical parameters for overhead power transmission lines and underground power cables were also demonstrated.

7 Transformers and rotating plant
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The basic theory of the singlephase transformer has been described, including the derivation of parameters, the modelling of magnetisation nonlinearities and its numerical implementation in a form acceptable for electromagnetic transients programs. The need for advanced models has been justified, and a detailed description made of UMEC (the Unified Magnetic Equivalent Circuit), a general transformer model developed for the accurate representation of multiphase, multiwinding arrangements. UMEC is a standard transformer model in the latest version of PSCAD/EMTDC. Rotating machine modelling based on Park's transformation is reasonably stan dard, the different implementations relating to the way of interfacing the machine to the system. A state variable formulation of the equations is used but the solution, in line with EMTP philosophy, is carried out by numerical integrator substitution.

8 Control and protection
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The control equations are solved separately from the power system equations though still using the EMTP philosophy, thereby maintaining the symmetry of the conductance matrix. The main facilities developed to segment the control, as well as devices or phenomena which cannot be directly modelled by the basic network components, are TACS and MODELS (in the original EMTP package) and a CMSF library (in the PSCAD/EMTDC package). The separate solution of control and power system introduces a timestep delay, however with the sample and hold used in digital control this is becoming less of an issue. Modern digital controls, with multiple time steps, are more the norm and can be adequately represented in EMT programs. The use of a modular approach to build up a control system, although it gives greater flexibility, introduces timestep delays in data paths, which can have a detrimental effect on the simulation results. The use of the zdomain for analysing the difference equations either generated using NIS, with and without timestep delay, or the rootmatching technique, has been demonstrated. Interpolation is important for modelling controls as well as for the nonlinear surge arrester, if numerical errors and possible instabilities are to be avoided. A description of the present state of protective system implementation has been given, indicating the difficulty of modelling individual devices in detail. Instead, the emphasis is on the use of realtime digital simulators interfaced with the actual protection hardware via digitaltoanalogue conversion.

9 Power electronic systems
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The computer implementation of power electronic devices in electromagnetic transient programs has taken much of the development effort in recent years, aiming at preserving the elegance and efficiency of the EMTP algorithm. The main feature that characterises power electronic devices is the use of frequent periodic switching of the power components under their control. The incorporation of power electronics in EMT simulation is discussed in this chapter with reference to the EMTDC version but appropriate references are made, as required, to other EMTPbased algorithms. This is partly due to the fact that the EMTDC program was specifically developed for the simulation of HVDC transmission and partly to the authors' involvement in the development of some of its recent components. A concise description of the PSCAD/EMTDC program structure is given in Appendix A. This chapter also describes the state variable implementation of a.c.d.c. converters and systems, which offers some advantages over the EMTP solution, as well as a hybrid algorithm involving both the state variable and EMTP methods.

10 Frequency dependent network equivalents
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Frequency dependent network equivalents are important for modelling modern power systems due to their size and complexity. The first stage is to determine the response of the portion of the network to be replaced by an equivalent, as seen from its boundary busbar(s). This is most efficiently performed using frequency domain techniques to perform a frequency scan. Once determined, a rational function which is easily implemented can be fitted to match this response.

11 Steady state applications
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Modelling of voltage sags and voltage interruptions requires accurate representation of the dynamic characteristics of the main system components, particularly the synchronous generators and induction motors, power electronic equipment and their protection and control. The EMT programs meet all these requirements adequately and can thus be used with confidence in the simulation of sag characteristics, their effects and the role of sag compensation devices. Subject to the unpredictability of the arc furnace characteristics, EMT simulation with either deterministic or stochastic models of the arc behaviour can be used to investigate possible mitigation techniques. Flicker penetration can also be predicted with these programs, although the derivation of the IEC short and longterm flicker indices is currently computationally prohibitive. However, realtime digital simulators should make this task easier.

12 Mixed timeframe simulation
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The use of a single time frame throughout the simulation is inefficient for studies involving widely varying time constants. A typical example is multimachine transient stability assessment when the system contains HVDC converters. In such cases the stability levels are affected by both the long time constant of the electromechanical response of the generators and the short time constant of the converter's power electronic control. It is, of course, possible to include the equations of motion of the generators in the electromagnetic transient programs to represent the electromechanical behaviour of multimachine power systems. However, considering the different time constants influencing the electromechanical and electromagnetic behaviour, such approach would be extremely inefficient. Electromagnetic transient simulations use steps of (typically) 50 μs, whereas the stability programs use steps at least 200 times larger.

13 Transient simulation in real time
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Advances in digital parallel processing, combined with the ability of power systems to be processed by means of subsystems, provides the basis for realtime transient simulation. Simulation in realtime permits realistic testing of the behaviour of control and protection systems. This requires the addition of digital to analogue and analogue to digital converters, as well as analogue signal amplifiers.The original, and at present still the main application in the market, is a simula tor based on dedicated architecture called RTDS (realtime digital simulator). This unit practically replaced all the scaledown physical simulators and can potentially represent any size system, The development of multipurpose parallel computing is now providing the basis for realtime simulation using standard computers instead of dedicated architectures, and should eventually provide a more economical solution.

Appendix A: Structure of the PSCAD/EMTDC program
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PSCAD/EMTDC version 2 consists of a set of programs which enable the efficient simulation of a wide variety of power system networks. EMTDC (Electromagnetic Transient and DC), although based on the EMTP method, introduced a number of modifications so that switching discontinuities could be accommodated accurately and quickly, the primary motivation being the simulation of HVDC systems. PSCAD (Power Systems Computer Aided Design) is a graphical Unixbased user interface for the EMTDC program. PSCAD consists of software enabling the user to enter a circuit graphically, create new custom components, solve transmission line and cable parameters, interact with an EMTDC simulation while in progress and to process the results of a simulation.

Appendix B: System identification techniques
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System identification techniques used in Power Systems Electromagnetic Transients Simulation.

Appendix C: Numerical integration
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A numerical integration algorithm is either explicit or implicit. There are various ways of developing numerical integration algorithms, such as manipulation of Taylor series expansions or the use of numerical solution by polynomial approximation. Among the wealth of material from the literature, only a few of the classical numerical integration algorithms have been selected for presentation.

Appendix D: Test systems data
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The CIGRE benchmark model consists of weak rectifier and inverter a.c. systems resonant at the second harmonic and a d.c. system resonant at the fundamental frequency. Both a.c. systems are balanced and connected in starground. The HVDC link is a 12pulse monopolar configuration with the converter transformers connected starground/star and starground/delta. Impedance scans of the a.c. and d.c. systems. A phase imbalance is created in the inverter a.c. system by inserting typically a 5.0 per cent resistance into one phase in series with the a.c. system.

Appendix E: Developing difference equations
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This example illustrates the use of the rootmatching technique to develop a difference equation.

Appendix F: MATLAB code examples
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The section shows MATLAB code examples for Power Systems Electromagnetic Transients Simulation.

Appendix G: FORTRAN code for state variable analysis
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This program demonstrates the state variable analysis technique for simulating the dynamics of a network. The results of this program are presented.

Appendix H: FORTRAN code for EMT simulation
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This sections shows the use of FORTRAN code for EMT simulation.

Back Matter
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