**1. Introduction**

The analysis of transient regimes in electric networks is a complex problem due to great number of elements, some nonlinear, as well as, due to the nonsinusoidal variation of currents and voltages. Numerical simulation can solve properly such a problem

There are available a lot of simulation programs for transients. One of the first of them, dedicated to electro energetic systems, was initiated by H.W.Dommel. Electromagnetic Transient Program (EMTP) was bought by Bonneville Power Administration (USA) (Dommel, 1995). Later on, from this initial program were developed several versions, such as MicroTrans, EMTP-RV, ATP (Alternative Transients Program), PSCAD (Chuco, 2005; DeCarlo & Lin, 2001).

Comparing some of the main simulation programs is not very conclusive, because each one of them are used in some versions trying to solve a large group of problems (Iordache & Mandache, 2004; Istrate et al., 2009).

Interface becomes more and more friendly, the library becomes larger and the facilities for creating own models are simpler to use. The speed of calculations and the storage capacity do no more represent a problem. It is possible now to implement some complicated algorithms able to solve problems with high speed in the variation of variables, able to solve all kinds of nonlinearities or to handle eigenvalues highly distanced one from another (large stiffness ratio dynamical systems).

An important criterion for the selection is represented by the integration of complex dynamical systems like rotating electric machines, relays, power electronic devices, FACTS, controllers etc. in the simulation. Some of the programs, depending on the way that were conceived, allow a better integration of these subsystems.

© 2012 Toader et al., licensee InTech. This is an open access chapter distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. © 2012 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Choosing one or other of the simulation programs depends on the previous experience of the user, as well as, on the manner the program responds to the specific demands (Chuco, 2005; Danyek et al., 2002; Foltin et al., 2006; Karlsson, 2005; Rashid & Rashid, 2006).

Numerical Methods for Analyzing the Transients in Medium Voltage Networks 443

Analyzing transient regimes in such networks is the same as the analysis of any other complex electric circuit (Dessaint, et al. 1999; Mandache & Topan, 2009; SIMULINK, 1997).

Generally, an electric circuit, no mater how complex it is, is described by an algebraic linear system of equations, obtained by applying the Kirchhoff's laws. This system of equations reflects the circuit's topology. To complete the model, the voltage-current equation at the

The inductors and the capacitors are described by constitutive relations in which are involved the derivatives of currents, respectively voltages. As a result, the mathematic model of an

In the electric circuit theory it is demonstrated that each current or voltage can be expressed as function of inductors currents and capacitors voltages. By other words if the inductors currents and the capacitors voltages are known, all remaining currents and voltages are

The state-variable equations are obtained by elimination of the algebraic equations in the

The modelization of an electric circuit made using state-equations has several important

First of all, electric circuits can be integrated with other dynamic systems, of completely different physical nature, as long as the last ones are described also by state-equations.

By the other hand, the procedures for integrating state-equations have a well-established theoretic support, being the domain of interest for quite a long period of time for famous

Finally, but not last, the method can be applied to linear circuits, as well as, to nonlinear

Of course there is a price to be paid for all these advantages: a certain difficulty in the

In order to give an example for the basics of the state-variable method, we will analyze the simple RLC series connection circuit supplied by an ideal voltage source *u*(*t*). The Kirchhoff

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*RLC*

*RLC*

*it it it ut u t u t u t u t Ri t*

 

*R*

*ut L*

*it C*

*L*

*C*

,

(1)

( ) ( ) ( ),

*L*

*dt du t*

*di t*

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.

equations and the constitutive relations for such a circuit are, respectivelly:

terminals of each element of circuit (called also constitutive relations) must be added.

electric circuit consists of a system of algebraic and differential equations (DAE).

uniquely determined and because of this they are called state-variables.

elimination of the algebraic equations from the system.

initial DAE.

advantages.

mathematicians.

circuits.

No mater what program is used, the numerical analysis of transients eventually must be able to solve a set of differential – algebraic equations that models the physical system. By accounting the stage the numerical integration takes place at, there are two main classes of simulation programms.

One class consists of those programs that make the integration at element level, meaning that for each step of time discretization the differential equations associated to dynamic elements are transformed in finite difference relations. Several ways of approximation can be used, more frequently the trapeze rule being used. All these algebraic relations are then assembled and as a result an algebraic system of equations is obtained and consequently solved using specific algorithms. This procedure is repeated at each iteration step, the parameters involved in the equations being modified by the results obtained at the previous step. Usually the integration step is fixed, but if this is required by a lack of convergence, the integration step can be split in half and the computation process is resumed. This method is known under the name of the implicit integration method and is used in programs such as SPICE and EMTP (Blume, 1986).

A second class of programs uses a two step procedure. In the first step the mathematic model is expressed as a system of first order differential equations, known as the system's state equations. During the next step this system of equations is integrated using algorithms with fixed step or with variable steps, depending on the systems particularities. The advantage of this method, called the state variables method, is to treat in the same way, in a unitary manner, electric networks, electric machines, drives, control devices or any other device that allows state equations. A representative of this method is SimPowerSystemsTM which is an extension of Simulink® with tools for modeling and simulating of electrical power systems. It provides models of many components used in these systems, including three-phase machines, electric drives, and libraries of application-specific models such as Flexible AC Transmission Systems (FACTS) and wind-power generation. Harmonic analysis, calculation of Total Harmonic Distortion (THD), load flow, and other key power system analyses are automated (Mathworks®).

The precision of the results obtained by integration of the equations describing the simulated system is quite remarkably high. It is obvious that none of the simulation programs, no mater which of the methods is using, cannot be more precise than the mathematic models used for the simulated components. The accuracy of the values of the parameters of these models is of high importance.
