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Due to the difficulty in collecting samples of certain effect levels, the class imbalance problem is a challenging issue in vulnerability assessment. The class imbalance problem would cause deviation of likelihood function and thus deviate the result of vulnerability assessment. Conventional assessment method ignoring it would lead to over- or under-estimation. In this article, we propose the improved kernel density estimation (IKDE) to address this issue. The proposed method first calculates the probability density function of each effect level sample. Then, the spatial characteristics are introduced in the likelihood function to reduce the impact of the class imbalance problem. The parameters in IKDE are determined by Markov Chain Monte Carlo and the probabilities of each effect level can be obtained at last. Through simulations with different class imbalance degrees and sample sizes, we illustrate the effectiveness of the proposed IKDE. Finally, a case study is carried out to assess the vulnerability of 10 kV distribution transformer excited by nanosecond electromagnetic pulse.

Large-scale pipeline networks may be subject to wide-area and spatially nonuniform external electromagnetic interference. This paper presents a generalized and efficient algorithm for the calculation of inductive and conductive coupling to pipeline networks based on the two-port equivalent circuit. In the modified transmission line model of the pipeline, the external inteference are characterized by using only longitudinal voltage sources. In addition, the nonlinear polarization of coating breakdown is incorporated into the circuit model of the network. The linear pipe segments can be independently reduced to the equivalent-pi circuit with analytical parameters, which enables model order reduction of the network. Such techniques can be applied to arbitrary nonuniform excitation fields. Several test cases are used to illustrate the response characteristics of pipelines excited by interference with different spatial patterns, including geomagnetic disturbances and HVDC earth return currents.