多变量几何拉普拉斯变换与分布式换流器线路的故障检测
Multivariable Geometric Laplace Transform and Fault Detection in Distributed-Converter Lines
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中文总结 AI 辅助
本文提出基于几何代数的二维几何拉普拉斯变换，构建分布式换流器线路模型与故障诊断链，可检测定位分类多种故障，区分设备停机与电缆缺陷，成果由开源实现复现。
中文摘要 AI 辅助
监测带有大量分布式电力换流器的直流线路是一个典型的时空问题：局部故障的信息会沿整个导体传播，到达少量测量点时会与线路自身的动态特性混合。本文在几何代数Cl(4,0)的可交换子代数上构建了二维几何拉普拉斯变换(t,x)→(s_t,s_x)，该子代数同构于Segre双复数，其中两个二重向量B_t和B_x分别作为时间和空间相位的独立虚单位。由于两个相位处于代数上可区分的平面，位置x_f处的故障会留下一个变换后的残差，可分解为F_f(s_t)e^{-s_x x_f}：其时间特性保留在第一个因子中，位置可作为第二个因子的几何参数读取。基于该表示，本文构建了换流器线路的传输线模型及其时空色散关系、通过导纳整形实现的分布式控制，包括对离散换流器站点的精确处理（空间采样、混叠，以及亚奈奎斯特带内精确的单换流器下垂实现），还构建了故障诊断链，可检测、定位和分类注入损耗、并联、传感器及本地控制器故障，扩展至具有自动阶数选择的多个同时故障，区分设备停机与电缆缺陷。该变换作为整体对象在双复数分析中已为人熟知，单变量时退化为复拉普拉斯变换；本文的贡献在于其几何嵌入及其在分布式换流器网络故障诊断中的操作应用，所有结果均通过配套的开源实现复现。
英文摘要
Monitoring a DC line with many distributed power converters is a genuinely spatio-temporal problem: the information about a localized fault travels along the whole conductor and reaches a few measurement points mixed with the dynamics of the line itself. This paper develops a two-dimensional geometric Laplace transform (t,x) -> (s_t,s_x) over a commutative subalgebra of the geometric algebra Cl(4,0), isomorphic to Segre's bicomplex numbers, in which two bivectors B_t and B_x act as independent imaginary units for the temporal and the spatial phase. Because the two phases live in algebraically distinguishable planes, a fault at position x_f leaves a transformed residual that factorizes as F_f(s_t) e^{-s_x x_f}: its temporal nature stays in the first factor and its location can be read as a geometric argument of the second. On this representation we build a transmission-line model of the converter line and its space-time dispersion relation, a distributed control by admittance shaping, including an exact treatment of discrete converter sites (spatial sampling, aliasing, and a per-converter droop realization that is exact on the sub-Nyquist band), and a fault diagnosis chain that detects, localizes and classifies injection-loss, shunt, sensor and local-controller faults, extends to multiple simultaneous faults with automatic order selection, and distinguishes the outage of a plant from a cable defect. As an integral object the transform is known in bicomplex analysis, and with a single independent variable it reduces to the complex Laplace transform; the contribution lies in its geometric embedding and in its operational use for fault diagnosis in distributed-converter networks. All results are reproduced by an accompanying open implementation.
发表机构
- University of Almeria（阿尔梅里亚大学）
- SINTEF（辛特夫研究所）
机构由 AI 辅助整理，请以论文原文为准。
