Power system static voltage stability region boundary (SVSRB) is a complicated hypersurface in parameter space. Due to inequality constraints such as generator reactive power limits, different critical points on the SVSRB may be related to different groups of active constraints and can be associated with different types of loadability limit, which makes it difficult to approximate the whole SVSRB at once or assemble many local approximations of the SVSRB. To overcome the difficulty, this paper proposes an approach for globally approximating the whole SVSRB considering generator reactive power limits at one go. First, a parametric optimization model incorporating all inequality constraints is formulated to describe the SVSRB, where the optimal solutions are the critical points on the SVSRB. The optimality condition, i.e., parametric Karush-Kuhn-Tucker conditions (KKT conditions), is also derived. Then, by applying the Galerkin method to the parametric KKT conditions in an interior-point-method-like manner, the whole SVSRB is approximated and represented using a single polynomial expression. Case studies in a 10-bus system and IEEE 300-bus system verify the effectiveness and accuracy of the proposed method.

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. As for the transient analysis, models of power systems are a set of complicated differential-algebraic equations (DAEs), practical issues related to the non-polynomial model terms, parameterization of initial operation point are also discussed. 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 stay high accuracy in cases of large parameter variations and strong nonlinearity. We also illustrate the validity of the proposed method for parametric problems like trajectory approximation, finding the critical parameter values and uncertainty quantification.

Commutation failures (CFs) are usually triggered by the voltage dips caused by the faults in the receiving-end systems. To reduce the probability of CFs, counter measures to reduce the voltage dips should be implemented, among which adding dynamic var sources (DVSs) is very helpful. In this paper, an optimal allocation method for DVSs is proposed to maximize the benefit of DVSs to be installed. Firstly, the probability of CFs is quantified based on the concept of area of vulnerability (AOV), where the detailed ac network is considered. In order to find the most effective locations for installing DVSs, a 0-1 nonlinear programming model is formulated based on the quantitative index. Then, we introduce penalty functions to transform the binary programming to an unconstrained optimization problem, which can be solved by the gradient iteration method efficiently. Moreover, the sensitivity analysis in the calculation of gradient provides a better understanding of the relationships between the DVSs and CFs. In case studies, the 37-bus system and 1648-bus system are used to show the effectiveness of the proposed method.

Many problems in power systems depend on parameters, which could be stochastic variables or deterministic system control variables practically, e.g., generation outputs, nodal voltages, etc. Due to the nonlinearity of power systems, the analytical relation between system states and parameters cannot be obtained directly. Using the sampling method to evaluate the influence of parameters on system states is very powerful but time-consuming. One feasible approach is to use polynomial approximations, where the system states are approximately expressed in the form of polynomials in terms of parameters. Galerkin method can be used to identify the approximate solution with high accuracy by solving high-dimensional equations. However, if a large number of parameters are involved, solving these high-dimensional equations becomes a serious challenge. This manuscript proposes an innovative method for resolving these high-dimensional equations in power systems, which constructs a sequence of decoupled equations to determine the polynomial expansion coefficients. This new approach can provide a local approximation in the form of Taylor expansion at a given operation point. Although the method is general, for simplicity and good readability, we introduce the detailed process in its application to load flow problems. Case studies from 6-, 118- and 1648-bus system show that the proposed method provides approximation more efficiently than traditional Galerkin method does, and 3-order polynomials can give very accurate results.