Time Domain Methods for the Calculation of Harmonic Propagation and Distortion

Numerical methods for the computation of harmonic propagation and distortion differ in the manner in which they represent the harmonic sources and the system impedance. Iterative methods (e.g. harmonic load flow) use a phasor representation of these parameters. Time domain methods, on the other hand, use a time representation of the system elements and the harmonic sources. Thus, they are generally more accurate than the iterative methods. The simplest system modeling for harmonic calculations considers rigid harmonic sources and linear system impedance [1-3]. A rigid harmonic source produces harmonics of only characteristic orders with a pre-defined and constant magnitude and phase. The linear impedance is the result of mainly transmission lines and compensation devices. A model which includes rigid harmonic sources and linear elements can be solved by an iterative method with the same accuracy as a time domain simulation. This is because a linear model has a phasor representation and the principle of superposition is applicable. The presence of non-linear and time varying elements in the system model can significantly change the manner by which harmonic currents and voltages propagate through and interact with the network. Some of the effects that may appear and which are better studied by time simulation are [3-7]: a) Under ideal conditions, harmonic devices (converters, transformers, etc.) produce harmonics of characteristic orders. For example, in symmetric saturation a transformer produces all odd orders, if its terminal voltage is near sinusoidal. An ac/dc 6-pulse converter produces orders such as 1,5,7,11,13.., if its terminal voltage is nearly sinusoidal and balanced and its dc current is nearly free of ripple. Most of these devices will produce uncharacteristic orders if their terminal conditions are not ideal. Examples are the inrush current in a transformer and converters operating with unbalanced voltages. b) The switching function of power converters is equivalent to a modulation\demodulation between ac and dc quantities [8,9]. This results in interactions between harmonics of different order. Such interactions are not predictable by linear time invariant models. The study of this phenomenon is important, especially in systems likely to have significant harmonic distortion because a path is provided through the converter for unrelated harmonics to interact. c) The gate control of power converters may interact with harmonics on the system through the synchronizing loop. This interaction in combination with the modulation\demodulation property of power converters can provide a feedback loop with significant gain for the amplification of harmonics. Extreme phenomena that can result from these interactions include limit cycles and harmonic instability. The following sections summarize the methods for the simulation in the time domain of non-linear and time varying systems. The methods discussed are suitable for widely used programs such as EMTP.