Having a better understanding of how locational marginal prices (LMPs) change helps in price forecasting and market strategy making. This paper investigates the fundamental distribution of the congestion part of LMPs in high-dimensional Euclidean space using an unsupervised approach. LMP models based on the lossless and lossy DC optimal power flow (DC-OPF) are analyzed to show the overlapping subspace property of the LMP data. The congestion part of LMPs is spanned by certain row vectors of the power transfer distribution factor (PTDF) matrix, and the subspace attributes of an LMP vector uniquely are found to reflect the instantaneous congestion status of all the transmission lines. The proposed method searches for the basis vectors that span the subspaces of congestion LMP data in hierarchical ways. In the bottom-up search, the data belonging to 1-dimensional subspaces are detected, and other data are projected on the orthogonal subspaces. This procedure is repeated until all the basis vectors are found or the basis gap appears. Top-down searching is used to address the basis gap by hyperplane detection with outliers. Once all the basis vectors are detected, the congestion status can be identified. Numerical experiments based on the IEEE 30-bus system, IEEE 118-bus system, Illinois 200-bus system, and Southwest Power Pool are conducted to show the performance of the proposed method.

翻译：更好地理解区域边际价格（LMP）的变化规律有助于价格预测和市场策略制定。本文采用无监督方法研究了高维欧几里得空间中LMP拥堵部分的基本分布特性。通过分析基于无损和有损直流最优潮流（DC-OPF）的LMP模型，揭示了LMP数据具有子空间重叠特性。LMP的拥堵部分由功率传输分布因子（PTDF）矩阵的特定行向量张成，研究发现LMP向量的子空间属性唯一反映了所有输电线路的瞬时拥堵状态。所提方法通过分层方式搜索张成拥堵LMP数据子空间的基向量：在自底向上搜索中，首先检测属于1维子空间的数据，其余数据则投影到正交子空间上，该过程迭代执行直至所有基向量被找到或出现基间隙；自顶向下搜索通过含异常值的超平面检测来处理基间隙问题。当所有基向量被检测后，即可识别拥堵状态。基于IEEE 30节点系统、IEEE 118节点系统、伊利诺伊200节点系统及西南电力池的数值实验验证了所提方法的性能。