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The loss of small signal stability under parameter variation (e.g., fluctuation of loads and wind powers) can be ascribed to local bifurcations, i.e., saddle-node, Hopf, singularity-induced, and limit-induced bifurcations. Classic bifurcation calculation methods like the direct method and continuation method can only provide a bifurcation point or two-parameter bifurcation curve. Based on Galerkin method and the implicit function theorem, this paper proposes accurate polynomial approximations of bifurcation hypersurfaces in the multi-dimensional parameter space of interest. The proposed method can ensure high accuracy in this parameter space, and thus is preferable to the existing Taylor expansion-based local approximation method, given the intrinsic large-variation characteristic of wind powers. Acquired bifurcation hypersurfaces are immediately used to construct the small signal stability region of the continuous system, and then compute the stability margin for the operating point subjected to uncertainty of wind generation. Besides, the validity scope of the proposed method is analyzed, and the handling of limit-triggered equation switching as well as its computational difficulty is discussed. Computational results on the two-parameter 11-bus two-area and six-parameter IEEE 145-bus test systems validate the high accuracy and effectiveness of the proposed method. [ABSTRACT FROM AUTHOR]

The imbalance of generation and load caused by the increasing integration of volatile generations poses challenges on frequency regulation. AGC is required to respond to the generation fluctuations without violating operational and security constraints. Explicit model predictive control (EMPC) provides an approach to reaching such requirements, which calculates the control laws of MPC in an explicit form, allowing for offline validation of the controller and enabling fast online computation. However, the partition number of EMPC's piecewise affine control laws grows exponentially with the number of constraints and prediction/control horizons, which hinders its application in large systems. In this paper, we propose an alternative explicit control approach for AGC by approximating the control laws of EMPC using Legendre polynomial series expansions, thus entirely eliminating the partition issue. The Galerkin method is applied to the KKT conditions of EMPC's multiparametric quadratic programming (mp-QP) problem to compute the approximation. Case studies in an illustrative system and IEEE 118-Bus System verify the performance and efficiency of the proposed controller. [ABSTRACT FROM AUTHOR]

Parametric problems in the transient analysis refer to the investigations of the relationship between parameters and dynamic behavior. To solve parametric problems, we propose a Galerkin method-based polynomial approximation. The proposed method uses a series of polynomials combined with coefficients to approximately represent state variables with parameters, where the Galerkin method is introduced to determine the coefficients. Thereby, the relationship between parameters and dynamic behavior is established through the polynomial expressions. Because models in power system transient analysis are a set of complicated differential-algebraic equations, practical issues related to the non-polynomial model terms, parameterization of initial operation point are discussed in this paper. The proposed method is tested on a three-machine system and the IEEE 145-bus test system. The case studies show that compared to trajectory sensitivities, the proposed method can maintain high accuracy in cases of large parameter variations and strong non-linearity. We also illustrate that the validity of the proposed method for parametric problems like trajectory approximation, finding the critical parameter values and uncertainty quantification. [ABSTRACT FROM AUTHOR]

A novel method for globally approximating the static voltage stability region boundaries (SVSRBs) of power systems is proposed by applying parametric polynomial approximation to the criterion equation which defines the SVSRB. Known as the Galerkin method, the implicit function portraying the SVSRB is described as a linear combination of basis polynomial functions, and the coefficients of the basis functions are obtained by projecting the SVSRB criterion onto each basis function. The approximation guarantees high precision globally in the whole domain of the parameter of interest rather than only in the neighborhood of a point, and the error can be controlled by the degree of polynomial basis functions. In the meantime, analytical expression of the left or right eigenvector of the system's Jacobian matrix corresponding to the zero eigenvalue is obtained in the form of polynomial, which provides valuable information for online voltage stability control or monitoring. Case studies in a 10-bus test system and IEEE 118-bus test system verifies the validity, accuracy, and flexibility of the proposed method. [ABSTRACT FROM AUTHOR]