Some Renovations in Transient Analysis of Transmission Lines by State-space Techniques

A method for transient analysis of single phase transmission lines based on state-space technique is presented. Transmission lines are considered as the interconnection of many lumped parameter sections. By this approach state equations are formulated for the system by choosing the capacitor voltages and inductor currents as the state variables. These equations are solved by state space techniques to compute steady-state and transient responses of transmission lines for various source and load connections. A computer program called LPTLAP (Lumped Parameter Transmission Line Transient Analysis Program) has been prepared for both formulation and solution steps. Transmission faults and switching operations in power systems cause sudden changes in voltage and current. Transient overvoltages must be known to indicate system isolation level in the planning stage, to protect system equipment and for the design of protective devices. Voltage and current at any point on the line can be represented by both space and time dependent partial differential equations. Analytic solutions of these equations are available for a few cases. For this reason, in the past Transient Network Analyzer (TNA) was used for the prediction of overvoltages and many applications are referred [1]. Later, by the use of digital computers numerical methods have got importance and several methods have been developed. These methods may generally be classified as time domain and frequency domain methods. Dommel combined the method of characteristics for transmission lines and trapezoidal integration for lumped parameters to solve power system transients [2]. The program based on this method is called EMTP (Electromagnetic Transients Program). Discrete time steps are important in the evaluation of numerical integration which may cause errors, and the computations must be carried for all time steps starting from the initial time to calculate the state of the system at any time. These are unwanted conditions for long duration transients. Transform methods (Laplace or Fourier) can be used to predict transient overvoltages[3-6]; computation is needed for wide range of frequencies to take the inverse transform numerically to obtain time domain responses. Frequency dependent parameters are naturally included in the calculations but it is difficult to implement nonlinear elements and switching operations. In tins study, state-space technique is used for the solution of steady-state and transient voltages or currents at any point on the line. State equations are formulated from the lumped parameter representation of transmission line. Numerical integration solutions for lumpedparameter transmission line model are available in the literature [7-9], but, in this study explicit formulas are used to solve these equations. This way starting from the initial state and initial time the state of the system at any time can be calculated directly without the need of the computations between that time and the initial time. Switching operations, lumped parameters and lossy distributed parameters can easily be cooperated. Nonlinear elements can also be included in the calculations [9]. One limitation of using lumped parameter model of transmission line, as well as the other time domain methods, is that frequency dependent line parameters can not be dealt with directly. In a uniform single phase transmission line four electrical characteristics r, 1, g, and care distributed perfectly along the line. An approximation for this distributed nature is to represent the transmission line as an interconnection of many lumped parameter identical sections. Each section may be in the form of n,T, 'I or r and contains a series resistance and inductance, and a shunt conductance and capacitance as seen in Fig. 1. R, L, G, and C are the total resistance, inductance, conductance and capacitance of each section of the transmission line, respectively; r, 1, g" and c appearing in the first line represent these parameters for per unit length. The resistance R for each section is determined by dividing the total resistance of the line by the number of sections n. L , C and G can be determined in the same manner [1]. Figure 1.Different lumped parameter networlnnodels of each section of a transmission line; a) IT-network, b) T-nernurk, c) 'I-network crnetwork can be constructed by connecting CoG branch of 'I -model to the left side of R-L branch). When nT-sections are connected in cascade and some series elements are combined, the transmission line model shown in Fig. 2 is obtained. Similar line models can be obtained by using other type of sections. These models will be the fundamental basis for the state-space analysis of the transmission line. It is not always necessary to use a detailed representation of an overall source configuration of a network. The form of source and load representations should be chosen depending on the objectives of the particular study carried out. In view of this, for the studies carried out here simplified lumped parameter equivalent circuits of source and load side networks are used. For the source models source types are assumed in one of the forms shown in Fig. 3. Vs(t) 8-~ Vs~) R L V