Rapid calculation of the periodic steady state for electronically switched, time varying power system loads

Linear time invariant loads, upon being energized by a power distribution system, often exhibit a transient operating state followed by a steady state operation. The circuit voltages and current in both the transient and steady states are calculated by a variety of methods including transform methods, numerical integration, and mathematical analysis. The case of time varying loads, however, is not as easily analyzed by any of these means as the time invariant case. Because of the power control possibilities of certain time-varying, electronic devices, the time varying case is of considerable interest. If X denotes the state vector that describes these loads, the periodic steady state (PSS) is said to occur when X(t) ≈ X(t + T) for all t > TPSS; when T is minimal, it is the period of X. The notation t > Tpss refers to sufficiently large time that the transient has decayed and the periodic steady state occurs. For switched loads and certain other power electronic loads, some states x(t) may be of the form x(t) = x_a(t) + x_b (t) where x_a is periodic with period T_a and x_b is periodic with period T_b. If T_a/T_b is rational, x(t) is periodic; however, if T_a/T_b is not rational, x(t) is not periodic. We will refer to the latter case as quasi-periodic. If the state vector X(t) is composed of quasi-periodic states for t > T_PSS, the system is in the quasi-periodic steady state. The analysis of distribution system loads usually includes (and is often focused on) the periodic or quasi-periodic steady state. In many applications, it is desired to know the spectrum of the demand current, the harmonic distortion of the load current or distribution bus voltage, or any number of several indices which quantify the voltage or current distortion. When the load is characterized by a time varying model Ẋ = A(t)X + U(t) the desired solution states may not be so easy to calculate. For example, a three phase distribution primary energizing a low-loss distribution transformer that is loaded by an electronically switched load could have a transient period which is measured in seconds or minutes. If the detailed solution of the system is needed in the periodic steady state, numerical integration could be cumbersome due to the incongruity of the small AT needed for accuracy and the long time horizon needed to obtain the periodic steady state. Transform and other mathematical methods may also be inconvenient because they generally give the entire solution, transient period included. There are two distinct classes of cases for which the linear system above (assumed stable) exhibits PSS behavior: when the system has a periodic input U(t); or when the system matrix A(t) is periodic; or both (provided the period of A. T_a, and that of U, T_u, form rational T_a/T_u). When T_a/T_u is not rational, the system will generally exhibit the quasiperiodic steady state. In this paper, a new technique is presented for the calculation of the PSS of linear, time varying systems that possess periodic or quasi-periodic solutions. The method is based on the properties of the discrete Fourier transform and a discrete form of the given system state equations. The technique is termed the gain shift formula (GSF); Table 1 shows a comparison of this method with selected other techniques. The GSF method is found to be especially applicable to distribution circuit solutions with power electronic loads that are modeled as linear time varying devices. The main advantage of the GSF method is that it is not necessary to calculate the transient solution that precedes the PSS.