Study on prediction-correction homotopy method of tracking Hopf bifurcation point

Abstract

A new method named prediction-correction homotopy method is proposed to calculate Hopf bifurcation points associated with the parameter dependent differential algebraic equations (DAE) which are used to model power systems dynamics. It uses the secant prediction method to track the Hopf bifurcation homotopy path. Compared with the tangent prediction method, the computation load is much less because it is not related to the matrix inversion computation. At the same time, the automatic step-size control strategy ensures the calculation accuracy and speed to effectively implement the stepby-step correction. Finally, it is proved that this algorithm can be used accurately and effectively through WSCC 3-machine 9-bus system and NewEngland 39-bus system.

Accelerating the solving of nonlinear equations using the homotopy method: application on finding the operating point of complex circuits

“…(02)) to two other methods; the first method is the VGNH (variable gain Newton homotopy) implemented with the SPICE software [7,8] while the second is the method (Alg. (01)) proposed in [12]. The parameters used in Alg.…”

“…(02) is faster than the two other methods, namely the method used in SPICE and the one used in [12]. From the average line, Alg.…”

“…If there is a problem in the convergence in the correction process, half of the previous step (∆λ/2) should be used to find a new prediction. An algorithm proposed in [12] …”

“…The continuation method has widely been used in the literature to solve the homotopy equation (Eq. (2)) [12,14]. First, the homotopy parameter λ should be adjusted by choosing a step size ∆λ (0 < ∆λ <1); then the predictor-corrector method (illustrated in Figure 1) is used.…”

“…The predictor delivers a new value of the continuation parameter at each step of the method and predicts an approximate solution of the corresponding new system in the homotopy. To calculate the predictor we can use either the secant or Euler methods [12]. The predicted approximate solution is then corrected by applying the corrector, e.g., the NE method.…”