Apparent impedance analysis: A small-signal method for stability analysis of power electronic based systems
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In this paper a new method for power system stability analysis is introduced. The method is based on injection of a small voltage or current in an arbitrary point of a power system. The apparent impedance is then defined as the ratio between the voltage and current at the injection point. It is shown that the apparent impedance can be used to estimate the eigenvalues of the system that are observable from the injection point. The eigenvalues are obtained by applying system identifi- cation techniques to the measured set of apparent impedances. The method is similar to the well established impedance-based stability analysis based on source and load impedance models. However, while the source/load impedance ratio is viewed as the minor-loop gain, the apparent impedance can be viewed as a closed-loop transfer function. It can also be expressed as the parallel connection of the source and load impedance. It is shown in the paper how the system eigenvalues can be extracted based on a set of apparent impedance values. The apparent impedance holds therefore complementary information compared with the existing impedance-based stability analysis. The method can also be used as a tool to validate analytically derived state-space models. In this paper the method is presented as a simulation tool, while further work will extend it to include experimental setups. Two case-studies are presented to illustrate the method. 1) A DC-case with a buck converter feeding a constant power load, and 2) a three-phase grid-connected VSC with current controller and PLL. The estimated (apparent) eigenvalues of the studied systems are equal to those obtained from the analytic state-space model.