Three-phase modeling of unbalanced distribution systems during open conductors and/or shunt fault conditions using the bus impedance matrix

The transient stability of a distribution system containing rotating machines is usually evaluated by considering a symmetrical disturbance applied to a balanced network. Hence, most, if not all, available algorithms and stability programs use the bus admittance matrix for the positive sequence only. However, a typical distribution system may contain untransposed feeders, single-and/or three-phase unbalanced static shunt loads, single-and/or three-phase dynamic loads (such as induction motors), co-generators, transformers and capacitor banks. Furthermore, even if the network is balanced, unsymmetrical faults introduce unbalance. The effects of unbalances on the transient behavior of a single induction motor in a two-bus system using simultaneous nodal voltage equations has been recently investigated. It was concluded that a significant error may occur by replacing an unbalanced shunt load by an averaged balanced load and by assuming an unbalanced feeder to be an equivalent balanced feeder. This paper extends the previous investigation to large scale networks using a generalized three-phase bus admittance matrix [Ybus] method which takes care of the unbalances referred to above. Developing the three-phase [Ybus] itself is not new, but the novelty of the proposed method lies in the combination of the three-phase [Ybus] with the dynamics of rotating machines in transient stability studies of large networks using step-by-step integration of nonlinear differential equations. The method utilizes phase frame representation of network and machine elements. In order to be general, the computer program allows the user to include synchronous generators, induction motors, transformers, feeders and static loads.

Three-phase modeling for transient stability of large scale unbalanced distribution systems

The effects of an unbalanced distribution system, unbalanced single-phase loads, and unbalanced network faults on the transient behavior of a three-phase induction motor are evaluated. Results are presented to illustrate that a significant error can be made under certain conditions by ignoring unbalance or even by simulating such a system by an equivalent balanced one. >

The effects of unbalanced networks and unbalanced faults on induction motor transient stability

Because the Fourier transform causes the convolution operation to become a simple complex product, it has been used to solve power system problems. A similar convolution property of the Hartley transform is used to calculate transients and nonsinusoidal waveshape propagation in electric power systems. The importance of this type of calculation relates to the impact of loads, particularly electronic loads, whose demand currents are nonsinusoidal. An example is given in which the Hartley transform is used to assess the impact of an electronic load with a demand which contains rapidly changing current. The authors also present a general introduction to the use of Hartley transforms for electric circuit analysis. A brief discussion of the error characteristics of discrete Fourier and Hartley solutions is presented. Because the Hartley transform is a real transformation, it is more computationally efficient then the Fourier or Laplace transforms. >

Application of the Hartley transform for the analysis of the propagation of nonsinusoidal waveforms in power systems

A new method for the calculation of subtransmission and distribution system voltages is proposed. The method is appropriate for the calculation of bus voltage transients due to the operation of large industrial loads: in this sense, the method is for power quality calculations. In particular, the method may be used for load currents which are electronically switched. The method is based on the Fourier transform of Ohm's law Vi(?)=Zij(?Ij(?) where Zij is the transfer impedeance between the busses i and j which are the busses at which the voltage transient is to be calculated and the bus at which high load currents occur, respectively. The Fourier transform is valid for most types of power system transients of practical interest, and this transform may be readily approximated by the fast Fourier transform (FFT). The FFT is a computationally efficient transform which is widely used in communications engineering and certain other areas. In this paper, a new method is described for the calculation of Zij(?). The method is based on a decoupling procedure whereby the power network is decomposed into two circuits. One of these circuits is composed of resistive and inductive elements, and the other is composed of conductive and capacitive elements. The former is a series network which has frequency dependence given by R + j?L and the latter is a shunt network with the admittance characteristic G + j?C. An interesting matrix version of the parallel impedance formula is developed and used to find Zij,(?).

A new method for the calculation of subtransmission and distribution system transients based on the FFT

This paper details an accurate technique to solve the three-phase power system equations under faulted conditions. The technique makes use of frequency domain simulation to provide a means of representing the transmission line with frequency dependent parameters. Faults can be simulated at any location along the line and at any time of fault inception. The technique enables a more accurate representation of synchronous machines to be combined with a transmission line model having distributed frequency dependent parameters. A typical power system example along with the modeling details illustrate the practicality and accuracy of the method.

E. B. Makram

Papers

An Accurate Computer Method for Obtaining Boundary Conditions in Faulted Power Systems