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The uncertainty of wind power brings many challenges to the operation and control of power systems, especially for the joint operation of multiple wind farms. Therefore, the study of the joint probability density function (JPDF) of multiple wind farms plays a significant role in the operation and control of power systems with multiple wind farms. This research was innovative in two ways. One, an adaptive bandwidth improvement strategy was proposed. It replaced the traditional fixed bandwidth of multivariate nonparametric kernel density estimation (MNKDE) with an adaptive bandwidth. Two, based on the above strategy, an adaptive multi-variable non-parametric kernel density estimation (AMNKDE) approach was proposed and applied to the JPDF modeling for multiple wind farms. The specific steps of AMNKDE were as follows: First, the model of AMNKDE was constructed using the optimal bandwidth. Second, an optimal model of bandwidth based on Euclidean distance and maximum distance was constructed, and the comprehensive minimum of these distances was used as a measure of optimal bandwidth. Finally, the ordinal optimization (OO) algorithm was used to solve this model. The scenario results indicated that the overall fitness error of the AMNKDE method was 8.81% and 11.6% lower than that of the traditional MNKDE method and the Copula-based parameter estimation method, respectively. After replacing the modeling object the overall fitness error of the comprehensive Copula method increased by as much as 1.94 times that of AMNKDE. In summary, the proposed approach not only possesses higher accuracy and better applicability but also solved the local adaptability problem of the traditional MNKDE.

Concerning a time-invariant, autonomous and polynomial system, we propose a new approach to estimate the domain of attraction (DOA) around an asymptotically stable equilibrium. Special emphasis is laid on elaborating the connections between modern results of real algebraic geometry and Lyapunov's stability theory, namely between the positive definite polynomials and the direct method of Lyapunov. The estimation problem can thereby be reduced to solving a sequence of low non-convexity-rank bilinear matrix inequalities (BMI) optimization problems. The BMI problem enables the calculation of the inner approximation to the relevant region of the DOA. We illustrate the presented approach with an example.

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