The assessment of transients in electrical systems requires numerical solving of systems of ordinary differential equations with the main purpose of localizing the relative extrema of state variables and of characteristic quantities in closely correlation with the selected state variables. Searching for relative extrema is traditionally performed immediately after the numerical integration is accomplished by sequentially processing a large file containing the amount of data recorded at all steps of numerical integration. Considering that, within the available software environments, the state variables time-related derivatives are usually discarded during numerical integration, finding approximations for the relative maxima and minima of each characteristic quantity could become a complex problem related to mathematical optimization. In this context, the present paper advances a simple procedure for real-time localizing of relative extrema of state variables and of the various characteristic quantities correlated with state variables by employing a high-order explicit linear multistep method. Computer experiments will be carried out by considering the more elaborate case of a synchronous generator subjected to sudden three-phase short circuit fault.