Probabilistic modelling of power systems operation and planning processes depends on data-driven methods, which require sufficiently large datasets. When historical data lacks this, it is desired to model the underlying data generation mechanism as a probability distribution to assess the data quality and generate more data, if needed. Kernel density estimation (KDE) based models are popular choices for this task, but they fail to adapt to data regions with varying densities. In this paper, an adaptive KDE model is employed to circumvent this, where each kernel in the model has an individual bandwidth. The leave-one-out maximum log-likelihood (LOO-MLL) criterion is proposed to prevent the singular solutions that the regular MLL criterion gives rise to, and it is proven that LOO-MLL prevents these. Relying on this guaranteed robustness, the model is extended by assigning learnable weights to the kernels. In addition, a modified expectation-maximization algorithm is employed to accelerate the optimization speed reliably. The performance of the proposed method and models are exhibited on two power systems datasets using different statistical tests and by comparison with Gaussian mixture models. Results show that the proposed models have promising performance, in addition to their singularity prevention guarantees.

翻译：电力系统运行与规划过程的概率建模依赖于数据驱动方法，这需要足够大的数据集。当历史数据不足时，需要将底层数据生成机制建模为概率分布，以评估数据质量并在必要时生成更多数据。基于核密度估计（KDE）的模型是完成此任务的常用选择，但它们难以适应具有不同密度的数据区域。本文采用自适应KDE模型来解决这一问题，该模型中每个核具有独立的带宽。提出使用留一法最大对数似然（LOO-MLL）准则来防止常规MLL准则引起的奇异解，并证明LOO-MLL可避免此类问题。基于这一鲁棒性保证，模型进一步扩展为赋予核可学习的权重。此外，采用改进的期望最大化算法来可靠地加速优化速度。通过不同统计检验及与高斯混合模型的对比，在两个电力系统数据集上展示了所提方法及模型的性能。结果表明，所提模型除具有防止奇异解的保证外，还展现了良好的性能表现。