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The continuation method is a powerful mathematical approach for the solution of systems of equations or of optimization problems, which tracks the (optimum) solution as a function of the varying parameter. The applications of this methodology to power system optimization problems are presented. Some implementations for the real power secure-economic dispatch, redispatching and planning are shown. The method provides the optimum trajectories of the system variables as a function of the varying parameter. This can be : system load, generation or transmission limits, generation parameters, line parameters, etc. The particular nature of power systems makes this approach very promising when solving economy-security functions. Because the continuation method proves very reliable, cases of difficult convergence with other optimization techniques may be resolved. Furthermore infeasible cases can be systematically detected.