A new procedure to determine the stability of a steady-state solution based on a sequential continuation scheme, Floquet theory and the limit cycle method is presented in this paper. The continuation scheme relies on an efficient predictor-corrector scheme, where the correction is accomplished through the limit cycle method. Furthermore, a comprehensive variable frequency transformer (VFT) Park model is proposed to study the stability of asynchronous links. The VFT Park allows the simulation of multiunit VFTs operated in parallel in order to increase the power transfer between two electric power networks. Each VFT unit consists of a wound-rotor asynchronous machine, a dc motor, and a control system, which provides power transfer regulation using power and speed controllers. The steady-state operating point of a VFT Park consisting of three 100-MW VFT units is computed with the limit cycle method for a set of heterogeneous operating scenarios. Stability diagrams are reported for changes of the VFT parameters, power transfer, and frequency on both sides of the asynchronous link. Even though the VFT is operated within its rated power transfer limits, simulation results reveal that a VFT may become unstable not only for changes of power transfer but also for variations of frequency.Index Terms-Floquet theory, limit cycle method, sequential continuation scheme, stability analyses, variable-frequency transformer (VFT), wound-rotor induction machine. NOMENCLATUREFlux linkage. Stator leakage reactance. Rotor leakage reactance. Magnetizing reactance. Stator angular velocity. Rotor angular velocity. Base angular velocity. Synchronous reference frame. Stator reference frame. Rotor reference frame. Quadrature-axis quantity. Transition matrix. Manuscript received

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This book chapter presents a flexible approach to incorporate mathematical models of FACTS devices into the Power Flow (PF) and the Optimal Power Flow (OPF) analysis tools, as well as into the standard OPF Market-Clearing (OPF-MC) procedure. The proposed approach uses the Matlab Optimization Toolbox because it allows to easily: (a) implement a given optimization model, (b) include different objective functions using distinct equality and inequality constraints and (c) modify and reuse an optimization model that has been previously implemented. The conventional OPF model is the main core of the proposed approach, which is easily implemented and adapted to include the mathematical models of FACTS devices. The resulting implementation of the OPF model featuring FACTS devices can be easily modified and adjusted to obtain the implementation of both the PF and the OPF-MC models which includes such devices. It should be mentioned that with the flexible approach proposed here, the complexity as well as the implementation time of optimized models featuring embedded FACTS devices is significantly reduced, since it is not necessary to define the expressions associated with the hessian matrix and the gradient vector. The flexibility and reliability of the proposed approach are demonstrated by means of several study cases using test as well as real power systems.

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