Enhancement of thermal unit commitment using immune algorithms based optimization approaches

The method of substituting the trapezoidal integrator, developed by Dommel, is generally applied in electromagnetic transient simulations. The trapezoidal rule is based on a truncated Taylor series and therefore contains truncation errors. These truncation errors cause numerical oscillations when the time step is large relative to some of the time constants in the network. Various techniques have been developed to reduce the numerical oscillations, each with strengths and drawbacks. In this paper an exponential form of the difference equation is developed, using root-matching techniques, that eliminates the truncation error and provides a highly efficient and accurate time domain regardless of the time step used. This exponential form of the difference equation generates a solution at each time point that is exact for the step response and a good approximation for an arbitrary forcing function. The exponential form is compatible with Dommelʹs method.