Contents Introduction 介绍
The power grid is one of the foremost examples of a large-scale man-made network. The nodes of the associated graph are the grid buses and its edges are its transmission lines. It is therefore natural to see measurements from the power grid as graph signals [3] and model power grid measurements using tools from the theory of graph signal processing (GSP) whose goal is to extend fundamental insights that come from the frequency analysis for time series to the domain of signals indexed by graphs [3]–[5]. One of the factors that motivate the development of GSP for the power grid is the abundance of high-quality data that can be acquired using phasor measurement units (PMU), the sensors producing estimates of the voltage and current phasors [6]. With that, classical signal processing questions pertaining to sampling, interpolation, denoising and compression and questions that hinge on the underlying structure of the voltage phasors graph signal arise.
电网是大型人造网络的典型例子之一。相关图的节点是电网母线 ，其边是传输线 。因此，将电网测量结果视为图信号 [3] 并使用图信号处理 (GSP) 理论工具对电网测量结果进行建模是很自然的，GSP 的目标是将来自时间序列频率分析的基本见解扩展到图 [3] – [5] 索引的信号域。推动电网 GSP 发展的因素之一是使用相量测量单元 (PMU) 可以获取大量高质量数据，相量测量单元是产生电压和电流相量估计值的传感器 [6] 。随之而来的是与采样、插值、去噪和压缩有关的经典信号处理问题，以及取决于电压相量图信号底层结构的问题。
The overarching goal of this paper is to develop GSP based models for power systems from first principles by building upon the existing system-level knowledge of power systems to create a solid foundation to analyze power-grid measurements using tools from GSP. This is named the Grid-GSP framework. By identifying the correct graph shift operators (GSO), we extend well-known results in GSP to power system data without losing the associated physical interpretation.
本文的总体目标是，基于现有的电力系统系统级知识，从基本原理出发，开发基于 GSP 的电力系统模型，为使用 GSP 工具分析电网测量数据奠定坚实的基础。该框架被称为 Grid-GSP 框架。通过识别正确的图移位算子（GSO），我们将 GSP 中众所周知的结果扩展到电力系统数据，同时又不丢失相关的物理解释。
The core idea is to rewrite the differential algebraic equations (DAE) [7], in a way often done in transient stability analysis of power systems, to reveal that the inherent structure in voltage phasors can be explained using a linear low-pass graph filter as a generative model, whose inputs are the generator voltages. This input signal is the generators’ response to electric load in the grid. Through this model the paper shows also that the temporal dynamics of the input signal, i.e. the generator voltages, can be explained using a non-linear GSP model defined via another GSO derived from the generator-only Kron-reduced network. This is done utilizing the well-known classical swing equations [8], [9]. This spatio-temporal generative model supports the empirical observation that voltage data obtained using PMUs tend to be confined to a much smaller dimension compared to the size of the data record in both space and time [10], [11]. Many papers have leveraged the empirical observation of the low-rank of phasor data for the interpolation of missing data [11], correcting bad data [12] and to detect faulty events [10], [13]–[15]. Importantly, our framework explicitly puts forth the structure of this low-dimensional subspace using our GSP-based generative model, directly tying this subspace to the graph Fourier domain of the GSO.
其核心思想是重写微分代数方程 (DAE) [7] ，这种方式在电力系统暂态稳定性分析中很常见，从而揭示电压相量的固有结构可以用线性低通图滤波器作为生成模型来解释，其输入是发电机电压。该输入信号是发电机对电网电负荷的响应。通过该模型，本文还表明，输入信号（即发电机电压）的时间动态可以用非线性 GSP 模型来解释，该模型通过另一个源自仅包含发电机的 Kron 约化网络的 GSO 定义。这是利用众所周知的经典摆动方程 [8] 、 [9] 来实现的。该时空生成模型支持以下经验观察：与空间和时间上的数据记录大小相比，使用 PMU 获得的电压数据往往被限制在小得多的维度上 [10] 、 [11] 。许多论文利用相量数据低秩的经验观测来插值缺失数据 [11] 、校正不良数据 [12] 以及检测故障事件 [10] 、 [13] – [15] 。重要的是，我们的框架使用基于 GSP 的生成模型明确地提出了该低维子空间的结构，并将该子空间直接绑定到 GSO 的图傅里叶域。
A. Literature Review A. 文献综述
We review prior works by dividing the most relevant literature related to this paper into three categories: 1) a general survey of works that use concepts from graph theory and GSP in power systems in the areas of sensor placement, interpolation and network inference, 2) False Data Injection (FDI) attack detection and 3) literature pertaining to compression of PMU data.
我们通过将与本文最相关的文献分为三类来回顾先前的研究：1）在传感器放置、插值和网络推理等领域使用图论和 GSP 概念在电力系统中的研究的一般调查，2）虚假数据注入 (FDI) 攻击检测和 3) 与 PMU 数据压缩有关的文献。
Graph theory for power systems: Several papers have used insights from spectral and algebraic graph theory. A few applications include optimal placement [16], [17] and generating statistically accurate topologies [18]. Grid topology identification is a network inference problem and has been studied by several works such as in [19]–[23]. GSP concepts have been leveraged in [24], [25] to detect FDI attacks. Prior work in [26] dealt with performance limits on fault localization with inadequate number of PMUs and connected it with graph signal sampling theory and optimal placement of PMUs for best possible resolution of fault localization in this under-sampled regime.
电力系统图论： 已有多篇论文运用了谱图论和代数图论的洞见。一些应用包括最优布局 [16] 、 [17] 和生成统计上准确的拓扑 [18] 。电网拓扑识别是一个网络推理问题，已有多项研究对其进行了研究，例如 [19] – [23] 。 [24] 、 [25] 中已利用 GSP 概念来检测 FDI 攻击。 [26] 中的先前研究探讨了在 PMU 数量不足的情况下故障定位的性能限制，并将其与图信号采样理论和 PMU 的最优布局相结合，以便在这种欠采样情况下获得最佳的故障定位分辨率。
The Kron-reduced network among the generator buses and the associated properties are used in [27], [28] to detect low-frequency oscillations as well as the resulting islanding patterns. In [29], the authors have shed light on the relationship that exists between graph Laplacian and modes in power systems. Recently, a comprehensive review of graph-theoretical concepts in power systems was presented in [7].
在 [27] 和 [28] 中，作者利用发电机母线之间的 Kron 约化网络及其相关特性来检测低频振荡以及由此产生的孤岛效应。在 [29] 中，作者阐明了图拉普拉斯算子与电力系统中模式之间的关系。最近，在 [7] 中，作者对电力系统中的图论概念进行了全面的回顾。
Additionally, there have been several papers adopting graphical models for state estimation [30], topology estimation [31] and optimal power flow [32]. While the modeling approach is valid, the graphical models capture correlation whereas our method models the underlying cause for that correlation structure thereby opening the door for statistical and non-statistical approaches.
此外，已有多篇论文采用图模型进行状态估计 [30] 、拓扑估计 [31] 和最优潮流 [32] 。虽然建模方法有效，但图模型捕捉的是相关性，而我们的方法则模拟了该相关性结构的根本原因，从而为统计和非统计方法打开了大门。
Note that with the exception of [24], [25], no other papers make the connection with GSP and even in the aforementioned work, GSP is used in an empirical manner. On the other hand our preliminary work in [1], [2] established a case for GSP in a more rigorous manner.
请注意，除 [24] 和 [25] 外，其他论文均未提及 GSP，即使在上述研究中，GSP 也以实证方式被运用。另一方面，我们在 [1] 和 [2] 中的前期工作以更严谨的方式为 GSP 提供了论据。
Detection of FDI attacks: While anomaly detection can be broadly applied to identify various events, a large body of prior research has focused on FDI attacks that can bypass classical bad data detection (BDD) mechanisms [33]–[36] and trigger incorrect decisions or hide line overflows and contingencies [37]–[39]). FDI attacks to PMUs can, for instance, exploit the vulnerability of GPS signals to spoofing attacks [40], [41].
检测 FDI 攻击： 虽然异常检测可广泛应用于识别各种事件，但之前大量的研究主要关注能够绕过经典不良数据检测 (BDD) 机制 [33] – [36] 并触发错误决策或隐藏线路溢出和意外事件 [37] – [39] 的 FDI 攻击。例如，针对 PMU 的 FDI 攻击可以利用 GPS 信号易受欺骗攻击的弱点 [40] , [41] 。
Recent papers on PMU data integrity have proposed leveraging the low rank spatio-temporal nature of PMU data not only to help with erasures but also to strengthen conventional BDD mechanisms [42]. In [38] the authors have suggested an FDI attack strategy that can pass the aforementioned BDD approach in [42] by generating false samples that approximately preserve the original subspace structure.
最近关于 PMU 数据完整性的论文提出，利用 PMU 数据的低秩时空特性，不仅可以辅助擦除，还可以增强传统的 BDD 机制 [42] 。在 [38] 中，作者提出了一种 FDI 攻击策略，该策略可以通过生成近似保留原始子空间结构的虚假样本来绕过 [42] 中提到的 BDD 方法。
Compression of PMU data: Due to their relatively high sampling rate and their wide deployment, PMU data have have called for compression. The performance of several off-the-shelf lossless encoding techniques applied to PMU data were investigated in [43]. In [44], a lossless compression called slack referenced encoding (SRE) method of PMU voltage measurements is introduced, by identifying a slack-bus and differentially encoding the difference between slack-bus measurements in time and all other buses. Similarly, in [45], phasor angle data is encoded in a lossless manner by preprocessing data using techniques from [44] and then using Golomb-Rice entropy encoding.
PMU 数据压缩： 由于 PMU 数据采样率相对较高且部署广泛，因此亟需对其进行压缩。 [43] 研究了几种应用于 PMU 数据的现成无损编码技术的性能。 [44] 介绍了一种称为松弛参考编码 (SRE) 的 PMU 电压测量无损压缩方法，该方法通过识别松弛母线 (slack-bus) 并对松弛母线测量值与所有其他母线测量值之间的时间差异进行差分编码。类似地， [45] 通过使用 [44] 中的技术对数据进行预处理，然后使用 Golomb-Rice 熵编码，对相量角数据进行无损编码。
The idea of slow-variation with time in PMU data such as phase angles is used in [46] to firstly transform data into frequency domain (in time) and then using a ‘reverse water-filling’ technique to encode frequency components in the difference measurements. Many lossy compression techniques utilize the low-rank structure inherent in voltage phasor data [47]–[49]. Several other wavelet based compression algorithms exist in the literature as well [50].
在 [46] 中，利用 PMU 数据（例如相位角）随时间缓慢变化的原理，首先将数据变换到频域（时间域），然后使用“反向注水”技术对差分测量中的频率分量进行编码。许多有损压缩技术利用电压相量数据固有的低秩结构 [47] – [49] 。文献中也存在其他几种基于小波的压缩算法 [50] 。
B. Contributions B. 贡献
The aim of this paper is to establish the framework of Grid-GSP and elucidate properties of power grid signals using tools from GSP. In particular, we:
本文旨在建立 Grid-GSP 框架，并利用 GSP 工具阐明电网信号的特性。具体而言，我们：
Establish that PMU voltage measurements from the power grid are result of an excitation to a low-pass graph filter whose graph shift operator (GSO) is defined using a function of the system admittance matrix. This was partly explored in our previous work in [1], [2] and used for blind community detection.
确定电网的 PMU 电压测量值是对低通图滤波器激励的结果，该滤波器的图移位算子 (GSO) 使用系统导纳矩阵的函数定义。我们之前在 [1] 、 [2] 中的工作中对此进行了部分探索，并将其用于盲社区检测。Study the spatio-temporal structure of the excitation that, at a fast time scale, is dominated by the generators dynamics. It is shown that this excitation can be modeled as an auto-regressive graph filter [51] (GF-AR (2)) for the input signals from the generator internal bus. The spatial properties in quasi-steady state are captured by defining another GSO using the ‘generator-only’ or Kron-reduced [9] network for the generator buses.
研究在快速时间尺度上受发电机动态控制的激励的时空结构。结果表明，该激励可以建模为一个自回归图滤波器 [51] (GF-AR (2))，用于处理来自发电机内部母线的输入信号。通过为发电机母线定义另一个 GSO，并使用“仅发电机”或 Kron 约简 [9] 网络来捕捉准稳态下的空间特性。
这些模型为在信号处理 PMU 数据环境中重新审视已知的基于 GSP 的采样和重构、插值和去噪以及网络推理算法奠定了基础。我们利用 GFT 进行特征提取，检测异常（具体来说，是 FDI 攻击 [2] ），并利用 GFT 信号的稀疏性，推导出一种有损 PMU 电压数据压缩算法。通过阐明电力系统数据建模的所有步骤，从图的构建、信号模型、识别负责低维表示的低通结构，到信号去噪、网络推理和异常检测，我们展示了如何在其他应用领域，尤其是那些可以建模为低通图信号输出的数据领域，类似地开发基于 GSP 的模型 [52] 。
C. Paper Organization C. 论文组织
Section II reviews concepts from GSP focusing on complex-valued graph signals and applicable more broadly to bandpass signals whose signal models rely on complex envelopes or phasors. It also reviews measurements and parameters pertaining to the grid. Section III lays the foundation for Grid-GSP mapping the physical laws to a spatio-temporal generative model for voltage signals. Through these lens, in Section IV, the paper revisits algorithms and tools from GSP for PMU data pertaining to sampling and reconstruction along with optimal placement of PMUs, interpolation of missing samples and network inference. Section V highlights applications of Grid-GSP to detect FDI attacks and for sequential lossy voltage data compression. The algorithms and methods are tested numerically in Section VI. Section VII summarizes conclusions and future research directions.
Section II 回顾了 GSP 中的概念，这些概念侧重于复值图信号，更广泛地适用于信号模型依赖于复包络或相量的带通信号。此外，本文还回顾了与电网相关的测量和参数。 Section III 为 Grid-GSP 将物理定律映射到电压信号的时空生成模型奠定了基础。通过这些视角，在 Section IV 中，本文重新探讨了 GSP 中用于 PMU 数据的算法和工具，这些算法和工具涉及采样和重建、PMU 的最优放置、缺失样本的插值和网络推理。 Section V 重点介绍了 Grid-GSP 在检测 FDI 攻击和顺序有损电压数据压缩中的应用。 Section VI 对这些算法和方法进行了数值测试。 Section VII 总结了结论和未来的研究方向。
Notation: Boldfaced lowercase letters are used for vectors,
符号 ：粗体小写字母表示向量
Preliminaries 准备工作
A. Graph Signal Processing (GSP) in a Nutshell
图信号处理（GSP）简介
Consider an undirected graph
考虑一个无向图
Definition 1:
A graph shift operator (GSO) is a linear neighborhood operator, so that each entry of the shifted graph signal is a linear combination of the graph signal neighbors’ values [5].
定义 1：
图移位算子 (GSO) 是一个线性邻域算子，因此移位后的图信号的每个条目都是图信号邻居值 [5] 的线性组合。
The linear combinations can use complex-valued weights
\begin{equation*}
\left[\mathbf {S}\right]_{i,j} = {\begin{cases}\sum _{k \in \mathcal {N}_{i}} s_{i,k}, \, i = j \\
-s_{i,j}, \, i \ne j \end{cases}} \tag{1}
\end{equation*}
线性组合可以使用复值权重
\begin{equation*}
\left[\mathbf {S}\right]_{i,j} = {\begin{cases}\sum _{k \in \mathcal {N}_{i}} s_{i,k}, \, i = j \\
-s_{i,j}, \, i \ne j \end{cases}} \tag{1}
\end{equation*}
In this work, we focus on complex symmetric GSOs,
在本文中，我们重点研究复对称 GSO，
Definition 2:
Given a GSO
\begin{equation*}
{\mathcal H}: \boldsymbol{x} \mapsto \boldsymbol{v},\,\,\,\,\Longleftrightarrow \,\,\,\,{\mathcal H}: \mathbf {S}\boldsymbol{x} \mapsto \mathbf {S}\boldsymbol{v}. \tag{2}
\end{equation*}
定义 2：
给定一个 GSO
\开始{方程式*}
{\mathcal H}:\boldsymbol{x}\mapsto\boldsymbol{v},\,\,\,\,\Longleftrightarrow\,\,\,\,{\mathcal H}:\mathbf {S}\boldsymbol{x}\mapsto\mathbf {S}\boldsymbol{v}.\tag
\begin{equation*}
{\mathcal H}: \boldsymbol{x} \mapsto \boldsymbol{v},\,\,\,\,\Longleftrightarrow \,\,\,\,{\mathcal H}: \mathbf {S}\boldsymbol{x} \mapsto \mathbf {S}\boldsymbol{v}. \tag{2}
\end{equation*}
\end{方程*}
Linear shift-invariant operators must be matrix polynomials of the GSO
\begin{equation*}
\textstyle {\boldsymbol v}={\mathcal H}(\mathbf {S})\boldsymbol{x},\,\, {\mathcal H}\left(\mathbf {S}\right)\!=\! \textstyle \sum \limits _{k=0}^{K-1}h_{k} \mathbf {S}^{k} \tag{3}
\end{equation*}
Additionally, linear shift-invariant graph filters satisfy the condition: 线性移位不变算子必须是 GSO
\begin{equation*}
\textstyle {\boldsymbol v}={\mathcal H}(\mathbf {S})\boldsymbol{x},\,\, {\mathcal H}\left(\mathbf {S}\right)\!=\! \textstyle \sum \limits _{k=0}^{K-1}h_{k} \mathbf {S}^{k} \tag{3}
\end{equation*}
此外，线性移位不变图过滤器满足条件： Consider the following eigenvalue decomposition of the complex symmetric GSO
\begin{align*}
\mathbf {S} &= \mathbf {U} \boldsymbol{\Lambda } \mathbf {U}^{\top }, & \mathbf {U}^{\top }\mathbf {U} = \mathbf {U}\mathbf {U}^{\top } = \mathbb {I}. \tag{4}
\end{align*}
考虑复对称 GSO
\begin{align*}
\mathbf {S} &= \mathbf {U} \boldsymbol{\Lambda } \mathbf {U}^{\top }, & \mathbf {U}^{\top }\mathbf {U} = \mathbf {U}\mathbf {U}^{\top } = \mathbb {I}. \tag{4}
\end{align*}
Here
这里，
Graph frequencies are the eigenvalues of the GSO and the order of frequencies is based on the total variation (TV) criterion [5], [55] defined using the discrete
\begin{align*}
S_{1}\!\!\left(\boldsymbol{x}\right)\!=\! \left\Vert \mathbf {S}\boldsymbol{x} \right\Vert _{1} \Rightarrow S_{1}\!\left(\boldsymbol{u}_{i}\right) = \left\Vert \mathbf {S}\boldsymbol{u}_{i} \right\Vert _{1} = \vert \lambda _{i} \vert \left\Vert \boldsymbol{u}_{i} \right\Vert _{1} \tag{5}
\end{align*}
After normalizing the eigenvectors such that 图频率是 GSO 的特征值，频率的顺序基于总变差 (TV) 标准 [5] 、 [55] ，使用离散
\begin{align*}
S_{1}\!\!\left(\boldsymbol{x}\right)\!=\! \left\Vert \mathbf {S}\boldsymbol{x} \right\Vert _{1} \Rightarrow S_{1}\!\left(\boldsymbol{u}_{i}\right) = \left\Vert \mathbf {S}\boldsymbol{u}_{i} \right\Vert _{1} = \vert \lambda _{i} \vert \left\Vert \boldsymbol{u}_{i} \right\Vert _{1} \tag{5}
\end{align*}
对特征向量进行归一化，使得 The Graph Fourier Transform (GFT) basis is the complex orthogonal basis
\begin{align*}
{\kern-2.0pt}\textstyle \mathcal {H}\,\left(\mathbf {S}\right)&\!=\!\! \textstyle \sum \limits _{k=0}^{K} h_{k} \mathbf {S}^{k} \!=\! \mathbf {U} \mathtt {diag} (\tilde{\boldsymbol{h}}) \mathbf {U}^{\top },\tag{6}
\\
{[\tilde{\boldsymbol{h}}]}_{i} &\!= \!\mathrm{H}(\lambda _i),\,\,\,\, \mathrm{H}(\lambda):=\sum \limits _{k=0}^{K}h_{k} \lambda ^{k} \tag{7}
\end{align*}
\begin{align*}
{\boldsymbol v} = {\mathcal H}(\mathbf {S})\boldsymbol{x} \,\,\rightarrow \,\,\tilde{ \boldsymbol{v} } = \mathtt {diag} (\tilde{\boldsymbol{h}}) \tilde{\boldsymbol{x}}, \tag{8}
\end{align*}
图傅里叶变换 (GFT) 基是 (4) 中的复正交基
\begin{align*}
{\kern-2.0pt}\textstyle \mathcal {H}\,\left(\mathbf {S}\right)&\!=\!\! \textstyle \sum \limits _{k=0}^{K} h_{k} \mathbf {S}^{k} \!=\! \mathbf {U} \mathtt {diag} (\tilde{\boldsymbol{h}}) \mathbf {U}^{\top },\tag{6}
\\
{[\tilde{\boldsymbol{h}}]}_{i} &\!= \!\mathrm{H}(\lambda _i),\,\,\,\, \mathrm{H}(\lambda):=\sum \limits _{k=0}^{K}h_{k} \lambda ^{k} \tag{7}
\end{align*}
\begin{align*}
{\boldsymbol v} = {\mathcal H}(\mathbf {S})\boldsymbol{x} \,\,\rightarrow \,\,\tilde{ \boldsymbol{v} } = \mathtt {diag} (\tilde{\boldsymbol{h}}) \tilde{\boldsymbol{x}}, \tag{8}
\end{align*}
这类似于时域信号的卷积定理。这自然而然地引出了诸如低通、高通和带通滤波器等概念的扩展，以及采样和插值方案的核心信号。B. GSP for Time Series of Graph Signals
B. 图形信号时间序列的 GSP
So far, only the nodal index for the graph signal
到目前为止，我们仅考虑了图信号
In order to characterize graph signal process
\begin{align*}
\textstyle \boldsymbol{V}(z)=\sum\limits _{t=0}^{+\infty }\boldsymbol{v}_tz^{-t}, \widetilde{\boldsymbol{V}}(z) = \mathbf {U}^T\boldsymbol{V}(z), \tag{9}
\end{align*}
We refer to
\begin{align*}
{\mathcal H}_t(\mathbf {S}) = \sum _{k=0}^{K} h_{k,t}\mathbf {S}^{k}, \,\boldsymbol{v}_t = \sum _{\tau = 0}^{t} {\mathcal H}_{t-\tau }(\mathbf {S}) \boldsymbol{x}_{\tau }, \tag{10}
\end{align*}
\begin{align*}
\!\!\!\!\boldsymbol{V}(z)\!=\!{\mathcal H}(\mathbf {S}\!\otimes \! z)\boldsymbol{X}(z), \text{where}\, {\mathcal H}(\mathbf {S}\!\otimes \! z)\!:=\! \sum _{t=0}^{+\infty }\!{\mathcal H}_t(\mathbf {S})z^{-t} \tag{11}
\end{align*}
\begin{equation*}
\textstyle {\mathcal H}_t(\mathbf {S})=\sum\limits _{k=0}^{K} h_{k,t}\mathbf {S}^{k}\,\,\leftrightarrow \,\, {\mathcal H}(\mathbf {S}\otimes z)= \sum \limits _{k=0}^{K}H_k(z)\mathbf {S}^{k}. \tag{12}
\end{equation*}
\begin{equation*}
\textstyle [\tilde{\boldsymbol h}_t]_i=\mathrm{H}_t(\lambda _i),\,\,\,\,\mathrm{H}_t(\lambda):=\sum\limits _{k=0}^{ K}h_{k,t}\lambda ^k \tag{13}
\end{equation*}
\begin{equation*}
\textstyle [\tilde{\boldsymbol h}(z)]_i=\mathbb {H}(\lambda _i,z),\,\,\mathbb {H}(\lambda,z)= \sum\limits _{t=0}^{+\infty }\sum\limits _{k=0}^{K}h_{k,t}\lambda ^k z^{-t} \tag{14}
\end{equation*}
\begin{align*}
\tilde{\boldsymbol{V}}(z)= \mathtt {diag} (\tilde{\boldsymbol h}(z)) \tilde{\boldsymbol{X}}(z), \tag{15}
\end{align*}
为了表征图信号过程
\begin{align*}
\textstyle \boldsymbol{V}(z)=\sum\limits _{t=0}^{+\infty }\boldsymbol{v}_tz^{-t}, \widetilde{\boldsymbol{V}}(z) = \mathbf {U}^T\boldsymbol{V}(z), \tag{9}
\end{align*}
我们将
\begin{align*}
{\mathcal H}_t(\mathbf {S}) = \sum _{k=0}^{K} h_{k,t}\mathbf {S}^{k}, \,\boldsymbol{v}_t = \sum _{\tau = 0}^{t} {\mathcal H}_{t-\tau }(\mathbf {S}) \boldsymbol{x}_{\tau }, \tag{10}
\end{align*}
。
\begin{align*}
\!\!\!\!\boldsymbol{V}(z)\!=\!{\mathcal H}(\mathbf {S}\!\otimes \! z)\boldsymbol{X}(z), \text{where}\, {\mathcal H}(\mathbf {S}\!\otimes \! z)\!:=\! \sum _{t=0}^{+\infty }\!{\mathcal H}_t(\mathbf {S})z^{-t} \tag{11}
\end{align*}
当输入为
\begin{equation*}
\textstyle {\mathcal H}_t(\mathbf {S})=\sum\limits _{k=0}^{K} h_{k,t}\mathbf {S}^{k}\,\,\leftrightarrow \,\, {\mathcal H}(\mathbf {S}\otimes z)= \sum \limits _{k=0}^{K}H_k(z)\mathbf {S}^{k}. \tag{12}
\end{equation*}
这里
\begin{equation*}
\textstyle [\tilde{\boldsymbol h}_t]_i=\mathrm{H}_t(\lambda _i),\,\,\,\,\mathrm{H}_t(\lambda):=\sum\limits _{k=0}^{ K}h_{k,t}\lambda ^k \tag{13}
\end{equation*}
并且
\begin{equation*}
\textstyle [\tilde{\boldsymbol h}(z)]_i=\mathbb {H}(\lambda _i,z),\,\,\mathbb {H}(\lambda,z)= \sum\limits _{t=0}^{+\infty }\sum\limits _{k=0}^{K}h_{k,t}\lambda ^k z^{-t} \tag{14}
\end{equation*}
由此，我们得到以下输入输出关系：
\begin{align*}
\tilde{\boldsymbol{V}}(z)= \mathtt {diag} (\tilde{\boldsymbol h}(z)) \tilde{\boldsymbol{X}}(z), \tag{15}
\end{align*}
通过将 GFT 应用于 (11) 中的 In this work, we focus on a class of graph-temporal filters called GF-ARMA
\begin{align*} \begin{split} & {\boldsymbol v}_t\!-\! {\mathcal A}_1 (\mathbf {S}) {\boldsymbol v}_{t-1}\!\cdots \! -\! {\mathcal A}_q (\mathbf {S}) {\boldsymbol v}_{t-q} \!\! =\!\! {\mathcal B}_0 (\mathbf {S}) {\boldsymbol x}_t \!\!+ \cdots \!+\! {\mathcal B}_r (\mathbf {S}) {\boldsymbol x}_{t\!-\!r}, \\
& \mathtt {diag}(\tilde{\boldsymbol a}(z)) \tilde{\boldsymbol{V}}(z)= \mathtt {diag} \left(\tilde{\boldsymbol b}(z) \right) \tilde{\boldsymbol{X}}(z), \end{split} \end{align*}
where 在本研究中，我们重点研究了一类图时域滤波器，称为 GF-ARMA
\begin{align*} \begin{split} & {\boldsymbol v}_t\!-\! {\mathcal A}_1 (\mathbf {S}) {\boldsymbol v}_{t-1}\!\cdots \! -\! {\mathcal A}_q (\mathbf {S}) {\boldsymbol v}_{t-q} \!\! =\!\! {\mathcal B}_0 (\mathbf {S}) {\boldsymbol x}_t \!\!+ \cdots \!+\! {\mathcal B}_r (\mathbf {S}) {\boldsymbol x}_{t\!-\!r}, \\
& \mathtt {diag}(\tilde{\boldsymbol a}(z)) \tilde{\boldsymbol{V}}(z)= \mathtt {diag} \left(\tilde{\boldsymbol b}(z) \right) \tilde{\boldsymbol{X}}(z), \end{split} \end{align*}
其中 C. Measurements and Parameters of the Electric Grid
C. 电网的测量和参数
The electric grid network can be represented by an undirected graph
\begin{align*}
\left[\boldsymbol{Y}\right]_{i,j} = {\begin{cases}\sum _{k \in \mathcal {N}_{i}} y_{i,k}, \, i = j \\
-y_{i,j}, \, i \ne j \end{cases}} \tag{16}
\end{align*}
where
\begin{align*}
\boldsymbol{Y} = \begin{bmatrix}\boldsymbol{Y}_{gg} & \boldsymbol{Y}_{g\ell } \\
\boldsymbol{Y}^{\top }_{g\ell }& \boldsymbol{Y}_{\ell \ell } \end{bmatrix}, \tag{17}
\end{align*}
电网网络可以用无向图
\begin{align*}
\left[\boldsymbol{Y}\right]_{i,j} = {\begin{cases}\sum _{k \in \mathcal {N}_{i}} y_{i,k}, \, i = j \\
-y_{i,j}, \, i \ne j \end{cases}} \tag{16}
\end{align*}
其中，
\begin{align*}
\boldsymbol{Y} = \begin{bmatrix}\boldsymbol{Y}_{gg} & \boldsymbol{Y}_{g\ell } \\
\boldsymbol{Y}^{\top }_{g\ell }& \boldsymbol{Y}_{\ell \ell } \end{bmatrix}, \tag{17}
\end{align*}
其中， The state of the system, from which all other physical quantities of interest can be derived, are the voltage phasors at each bus. In the following we assume that a PMU installed on node/bus
\begin{align*}
{\kern-11.00006pt}\left(\!\boldsymbol Y + \mathtt {diag}\! \left(\begin{bmatrix}\boldsymbol{y}_{sh}^{g} \\
\boldsymbol{y}_{sh}^{\ell } \end{bmatrix}\right)\!\right)\boldsymbol{v}_{t} \! =\! \boldsymbol{i}_{t}, \text{where}\, \boldsymbol{v}_{t} = \begin{bmatrix}\boldsymbol{v}^{g}_{t} \\
\boldsymbol{v}^{\ell }_{t} \end{bmatrix},\! \boldsymbol{i}_{t}\! =\! \begin{bmatrix}\boldsymbol{i}^{g}_{t} \\
\boldsymbol{i}^{\ell }_{t} \end{bmatrix} \tag{18}
\end{align*}
To describe the operating conditions of the system we introduce a few more quantities. In power systems transient dynamic analysis the impact of generating units is modeled as an internal bus characterized by a generator impedance (or admittance)
\begin{equation*}
\boldsymbol{i}^g_{t}= \mathtt {diag}(\boldsymbol y_g)\left(\boldsymbol{e}_{t}-\boldsymbol{v}^{g}_{t}\right) \tag{19}
\end{equation*}
系统状态是每条母线的电压相量，所有其他感兴趣的物理量都可以从中推导出来。下面我们假设安装在节点/母线
\begin{align*}
{\kern-11.00006pt}\left(\!\boldsymbol Y + \mathtt {diag}\! \left(\begin{bmatrix}\boldsymbol{y}_{sh}^{g} \\
\boldsymbol{y}_{sh}^{\ell } \end{bmatrix}\right)\!\right)\boldsymbol{v}_{t} \! =\! \boldsymbol{i}_{t}, \text{where}\, \boldsymbol{v}_{t} = \begin{bmatrix}\boldsymbol{v}^{g}_{t} \\
\boldsymbol{v}^{\ell }_{t} \end{bmatrix},\! \boldsymbol{i}_{t}\! =\! \begin{bmatrix}\boldsymbol{i}^{g}_{t} \\
\boldsymbol{i}^{\ell }_{t} \end{bmatrix} \tag{18}
\end{align*}
为了描述系统的运行条件，我们引入了其他一些量。在电力系统暂态动态分析中，发电机组的影响被建模为内部母线，其特征是发电机阻抗（或导纳）
\begin{equation*}
\boldsymbol{i}^g_{t}= \mathtt {diag}(\boldsymbol y_g)\left(\boldsymbol{e}_{t}-\boldsymbol{v}^{g}_{t}\right) \tag{19}
\end{equation*}
如第 I 节所述，发电机响应电网中的电力负荷。为了模拟发电机的响应，一种常用的近似方法是，在负荷母线 Graph Signal Processing for the Grid
网格的图形信号处理
Having described the relevant GSP concepts and introduced grid quantities and parameters of interest, we are ready to introduce the Grid-GSP framework.4 Firstly, we define the GSO for the grid, then support the definition by introducing the graph-filter model for voltage phasors, and finally characterize the temporal dynamics. All of the above yields a GSP generative model for the voltage phasor measurements as a low-pass GSP model, as detailed next.
在描述了相关的 GSP 概念并介绍了感兴趣的电网量和参数之后，我们准备介绍 Grid-GSP 框架。 4 首先，我们定义电网的 GSO，然后通过引入电压相量的图滤波模型来支持该定义，最后表征其时间动态。以上所有步骤最终生成一个用于电压相量测量的 GSP 生成模型，即低通 GSP 模型，下文将详细介绍。
A. Grid Graph Generative Model
A.网格图生成模型
Grid-GSP for voltage phasors data relies on the following definitions:
电压相量数据的 Grid-GSP 依赖于以下定义：
Definition 3:
The grid graph shift operator (GSO) is a complex symmetric matrix equal to a diagonal perturbation of the system admittance matrix with generator admittance values,
\begin{align*}
\mathbf {S} \triangleq \boldsymbol{Y} + \mathtt {diag} \left(\begin{bmatrix}\boldsymbol{y}_{g} + \boldsymbol{y}_{sh}^{g} \\
\boldsymbol{y}_{sh}^{\ell } \end{bmatrix}\right) \tag{20}
\end{align*}
定义 3：
网格图移位算子 (GSO) 是一个复对称矩阵，等于具有发电机导纳值的系统导纳矩阵的对角扰动，
\开始{对齐*}
\mathbf {S} \triangleq \boldsymbol{Y} + \mathtt {diag} \left(\begin{bmatrix}\boldsymbol{y}_{g} + \boldsymbol{y}_{sh}^{g} \\
\boldsymbol{y}_{sh}^{\ell } \end{bmatrix}\right) \tag{20}
\结束{对齐*}
From the definition of the GSO it follows that:
根据 GSO 的定义可以得出：
Definition 4:
The grid Graph Fourier Transform (GFT) basis for voltage phasors is the orthogonal matrix
\begin{align*}
\mathbf {S} = \mathbf {U} \boldsymbol{\Lambda } \mathbf {U}^{\top }, \,\, \left|\lambda _{\text{min}}\right| > 0 \tag{21}
\end{align*}
定义 4：
电压相量的网格图傅里叶变换 (GFT) 基是正交矩阵
\开始{对齐*}
\mathbf {S} = \mathbf {U} \boldsymbol{\Lambda } \mathbf {U}^{\top }, \,\, \left|\lambda _{\text{min}}\right| > 0 \标签{21}
\结束{对齐*}
Here, the GSO
这里，GSO
With the GSO
\begin{align*}
&\left(\!\boldsymbol Y \!+\! \mathtt {diag}\! \left(\begin{bmatrix}\boldsymbol{y}_{sh}^{g} \\
\boldsymbol{y}_{sh}^{\ell } \end{bmatrix}\right)\!\right)\boldsymbol{v}_{t} \! =\! \begin{bmatrix}\mathtt {diag}(\boldsymbol y_g)\left(\boldsymbol{e}_{t}-\boldsymbol{v}^{g}_{t}\right) \\
\boldsymbol{i}^{\ell }_t \end{bmatrix} \tag{22}
\end{align*}
From now on with slight abuse of notation we denote
\begin{align*}
\boxed{\boldsymbol{v}_{t} = \mathcal {H} (\mathbf {S}) \begin{bmatrix}\mathtt {diag}(\boldsymbol y_g)\boldsymbol{e}_t\\
\boldsymbol{i}^{\ell }_t \end{bmatrix} + \boldsymbol{\eta }_{t}} \tag{23}
\end{align*}
由于 GSO
\begin{align*}
&\left(\!\boldsymbol Y \!+\! \mathtt {diag}\! \left(\begin{bmatrix}\boldsymbol{y}_{sh}^{g} \\
\boldsymbol{y}_{sh}^{\ell } \end{bmatrix}\right)\!\right)\boldsymbol{v}_{t} \! =\! \begin{bmatrix}\mathtt {diag}(\boldsymbol y_g)\left(\boldsymbol{e}_{t}-\boldsymbol{v}^{g}_{t}\right) \\
\boldsymbol{i}^{\ell }_t \end{bmatrix} \tag{22}
\end{align*}
从现在开始，我们将
\begin{align*}
\boxed{\boldsymbol{v}_{t} = \mathcal {H} (\mathbf {S}) \begin{bmatrix}\mathtt {diag}(\boldsymbol y_g)\boldsymbol{e}_t\\
\boldsymbol{i}^{\ell }_t \end{bmatrix} + \boldsymbol{\eta }_{t}} \tag{23}
\end{align*}
电压相量测量的 Grid-GSP 生成模型 Remark 1:
Shift-invariance of
备注 1：
由于 GSO 的逆变换，
To visualize this more explicitly, consider
\begin{align*}
\mathcal {H}_{k} (\mathbf {S}) &\triangleq \! {\mathbf U} \,\mathtt {diag}\left(\tilde{\boldsymbol{h}}_{k}\right) {\mathbf U}^{\top }, \left[\tilde{\boldsymbol{h}}_{k}\right]_{i} \!\!=\!\! {\begin{cases}\lambda _{i}^{-1}, \,i \in \mathcal {K} \\
0, \,\, \text{else} \end{cases}} \\
{\boldsymbol v}_t &\approx \mathcal {H}_{k} (\mathbf {S}) \begin{bmatrix}\mathtt {diag}(\boldsymbol y_g)\boldsymbol{e}_t\\
\boldsymbol{i}^{\ell }_t \end{bmatrix} +{\boldsymbol \eta }_t, \tag{24}
\end{align*}
where 为了更直观地理解这一点，假设
\begin{align*}
\mathcal {H}_{k} (\mathbf {S}) &\triangleq \! {\mathbf U} \,\mathtt {diag}\left(\tilde{\boldsymbol{h}}_{k}\right) {\mathbf U}^{\top }, \left[\tilde{\boldsymbol{h}}_{k}\right]_{i} \!\!=\!\! {\begin{cases}\lambda _{i}^{-1}, \,i \in \mathcal {K} \\
0, \,\, \text{else} \end{cases}} \\
{\boldsymbol v}_t &\approx \mathcal {H}_{k} (\mathbf {S}) \begin{bmatrix}\mathtt {diag}(\boldsymbol y_g)\boldsymbol{e}_t\\
\boldsymbol{i}^{\ell }_t \end{bmatrix} +{\boldsymbol \eta }_t, \tag{24}
\end{align*}
其中 To provide insights on the temporal dynamics of the voltage phasors, we need to capture the structure of the excitation term. As a matter of fact,
为了深入了解电压相量的时间动态，我们需要捕捉激励项的结构。事实上，
B. A GSP Model for Generator Dynamics: et
B. 发电机动力学的 GSP 模型： et
The excitation term corresponding to generator currents has elements as
对应于发电机电流的励磁项包含来自每个发电机
Our model, relies on two steps. First, we model the dynamics of a signal
\begin{align*}
\boldsymbol{x}_t &\triangleq (\mathtt {diag}(\boldsymbol{m}))^{\frac{1}{2}}\ln (\boldsymbol{e}_t) \\
\boldsymbol{\delta }_t&= (\mathtt {diag}(\boldsymbol{m}))^{-\frac{1}{2}} \Im \lbrace \boldsymbol{x}_t \rbrace, \left|\boldsymbol{e}\right|_t \!=\! (\mathtt {diag}(\boldsymbol{m}))^{-\frac{1}{2}} \Re \lbrace \boldsymbol{x}_t \rbrace \tag{25}
\end{align*}
where the vector
\begin{align*}
&\boldsymbol{Y}_{\text{all}}\!=\! \begin{bmatrix}
\mathtt{diag}\left(\boldsymbol{y}_{{g}} \!+\! \boldsymbol{y}_{sh}^{{g}} \right) & -\begin{bmatrix}
\!\mathtt{diag}\left(\boldsymbol{y}_{{g}}\! +\! \boldsymbol{y}_{sh}^{{g}} \right) \!&\! \boldsymbol{0}
\end{bmatrix}\\
-\begin{bmatrix}
\!\mathtt{diag}\left(\boldsymbol{y}_{{g}} \!+\! \boldsymbol{y}_{sh}^{{g}} \right) \!&\! \boldsymbol{0}
\end{bmatrix}^{\top}& \mathbf{S}_{\text{ci}}\\
\end{bmatrix} \\
&\mathbf{S}_{\text{ci}} \triangleq \mathbf{S}+ \!\mathtt{diag}\left (\begin{bmatrix} \boldsymbol 0 &
\boldsymbol{y}_{\ell}^{\top} \end{bmatrix}\right)
\end{align*}
\begin{equation*}
\mathfrak {Sh}(\boldsymbol {Y}_{\text{all}}, \mathbf {S}_{\text{ci}}) =j\boldsymbol {Y}_{\text{red}}+{\boldsymbol E}_{\text{red}} \tag{26}
\end{equation*}
我们的模型依赖于两个步骤。首先，我们对通过内部发电机电压进行以下非线性变换获得的信号
\begin{align*}
\boldsymbol{x}_t &\triangleq (\mathtt {diag}(\boldsymbol{m}))^{\frac{1}{2}}\ln (\boldsymbol{e}_t) \\
\boldsymbol{\delta }_t&= (\mathtt {diag}(\boldsymbol{m}))^{-\frac{1}{2}} \Im \lbrace \boldsymbol{x}_t \rbrace, \left|\boldsymbol{e}\right|_t \!=\! (\mathtt {diag}(\boldsymbol{m}))^{-\frac{1}{2}} \Re \lbrace \boldsymbol{x}_t \rbrace \tag{25}
\end{align*}
其中，向量
\begin{align*}
&\boldsymbol{Y}_{\text{all}}\!=\! \begin{bmatrix}
\mathtt{diag}\left(\boldsymbol{y}_{{g}} \!+\! \boldsymbol{y}_{sh}^{{g}} \right) & -\begin{bmatrix}
\!\mathtt{diag}\left(\boldsymbol{y}_{{g}}\! +\! \boldsymbol{y}_{sh}^{{g}} \right) \!&\! \boldsymbol{0}
\end{bmatrix}\\
-\begin{bmatrix}
\!\mathtt{diag}\left(\boldsymbol{y}_{{g}} \!+\! \boldsymbol{y}_{sh}^{{g}} \right) \!&\! \boldsymbol{0}
\end{bmatrix}^{\top}& \mathbf{S}_{\text{ci}}\\
\end{bmatrix} \\
&\mathbf{S}_{\text{ci}} \triangleq \mathbf{S}+ \!\mathtt{diag}\left (\begin{bmatrix} \boldsymbol 0 &
\boldsymbol{y}_{\ell}^{\top} \end{bmatrix}\right)
\end{align*}
为了对
\begin{equation*}
\mathfrak {Sh}(\boldsymbol {Y}_{\text{all}}, \mathbf {S}_{\text{ci}}) =j\boldsymbol {Y}_{\text{red}}+{\boldsymbol E}_{\text{red}} \tag{26}
\end{equation*}
为实数对角占优矩阵，虚部 Definition 5:
A GSO is defined for the Kron-reduced generator only network as
\begin{equation*}
\mathbf {S}_{\text{red}}= (\mathtt {diag}(\boldsymbol m))^{-\frac{1}{2}} \boldsymbol {Y}_{\text{red}}(\mathtt {diag}(\boldsymbol m))^{-\frac{1}{2}} \in \mathbb {R}^{\left|\mathcal {N}_G\right|} \tag{27}
\end{equation*}
with the following eigenvalue decomposition,
\begin{align*}
\mathbf {S}_{\text{red}} = \mathbf {U}_{\text{red}} \boldsymbol {\Lambda }_{\text{red}} \mathbf {U}_{\text{red}}^{\top } \tag{28}
\end{align*}
定义 5：
对于 Kron 简化的仅生成网络，GSO 定义为
\开始{方程式*}
\mathbf {S}_{\text{red}}= (\mathtt {diag}(\boldsymbol m))^{-\frac
\begin{align*}
\mathbf {S}_{\text{red}} = \mathbf {U}_{\text{red}} \boldsymbol {\Lambda }_{\text{red}} \mathbf {U}_{\text{red}}^{\top } \tag{28}
\end{align*}
\begin{align*}
\mathbf {S}_{\text{red}} = \mathbf {U}_{\text{red}} \boldsymbol {\Lambda }_{\text{red}} \mathbf {U}_{\text{red}}^{\top } \tag{28}
\end{align*}
\begin{align*}
\mathbf {S}_{\text{red}} = \mathbf {U}_{\text{red}} \boldsymbol {\Lambda }_{\text{red}} \mathbf {U}_{\text{red}}^{\top } \tag{28}
\end{align*}
且正交 GFT 基为 We introduce the GSP based dynamical model for the complex-valued generator internal voltages
我们通过图时间滤波器 GF-AR (2) 引入基于 GSP 的复值发电机内部电压
GSP-based dynamics for generator internal voltages
\begin{equation*}
\boldsymbol {e}_t=\exp \left((\mathtt {diag}(\boldsymbol {m}))^{-\frac{1}{2}}\boldsymbol {x}_t\right) \tag{29}
\end{equation*}
where
\begin{align*}
\mathtt {diag}\left(\tilde{\boldsymbol a}(z) \right) \tilde{\boldsymbol {X}}(z)&= \tilde{\boldsymbol {W}}(z) \tag{30}
\\
\tilde{\boldsymbol a}(z) &=1-\tilde{\boldsymbol a}_1 z^{-1}-\tilde{\boldsymbol a}_2 z^{-2} \tag{31}
\end{align*}
基于 GSP 的发电机内部电压动力学
\begin{equation*}
\boldsymbol {e}_t=\exp \left((\mathtt {diag}(\boldsymbol {m}))^{-\frac{1}{2}}\boldsymbol {x}_t\right) \tag{29}
\end{equation*}
其中
\begin{align*}
\mathtt {diag}\left(\tilde{\boldsymbol a}(z) \right) \tilde{\boldsymbol {X}}(z)&= \tilde{\boldsymbol {W}}(z) \tag{30}
\\
\tilde{\boldsymbol a}(z) &=1-\tilde{\boldsymbol a}_1 z^{-1}-\tilde{\boldsymbol a}_2 z^{-2} \tag{31}
\end{align*}
The GSP based dynamical model takes inspiration from swing equation for generator angles and we empirically choose to model generator internal voltage magnitude
\begin{align*}
\mathtt {diag}\left(\boldsymbol {m} \right)\ddot{\boldsymbol {\delta }} + \mathtt {diag}\left(\boldsymbol {d} \right)\dot{\boldsymbol {\delta }} = \overline{\boldsymbol {w}} - \boldsymbol {Y}_{\text{red}} \boldsymbol {\delta }, \tag{32}
\end{align*}
where 基于 GSP 的动态模型灵感源自发电机角的摆动方程，我们根据经验选择也使用 GF-AR (2)模型来模拟发电机内部电压幅值
\begin{align*}
\mathtt {diag}\left(\boldsymbol {m} \right)\ddot{\boldsymbol {\delta }} + \mathtt {diag}\left(\boldsymbol {d} \right)\dot{\boldsymbol {\delta }} = \overline{\boldsymbol {w}} - \boldsymbol {Y}_{\text{red}} \boldsymbol {\delta }, \tag{32}
\end{align*}
其中， Proposition 1:
Let
\begin{equation*}
\mathtt {diag}\left(\boldsymbol {d} \right)\mathtt {diag}\left(\boldsymbol {m} \right)^{-1}\approx \chi \mathbb {I},
\end{equation*}
i.e. the homogeneous simplification as in [64], [65], the dynamics of
\begin{align*}
&\Im \lbrace \boldsymbol x_t\rbrace -{\mathcal A}_1(\mathbf {S}_{\text{red}})\Im \lbrace \boldsymbol x_{t-1}\rbrace -{\mathcal A}_2(\mathbf {S}_{\text{red}})\Im \lbrace \boldsymbol x_{t-2}\rbrace \approx \Im \lbrace \boldsymbol {w}\rbrace, \\
&{\mathcal A}_1(\mathbf {S}_{\text{red}}):= (2\!-\!\chi)\mathbb {I}-\mathbf {S}_{\text{red}},\,\,{\mathcal A}_2(\mathbf {S}_{\text{red}}):=(\chi \!-\!1)\mathbb {I} \tag{33}
\end{align*}
命题 1：
令
\开始{方程式*}
\mathtt {diag}\left(\boldsymbol {d} \right)\mathtt {diag}\left(\boldsymbol {m} \right)^{-1}\approx \chi \mathbb {I},
\end{方程*}
即 [64] 、 [65] 中的均质简化，
\begin{align*}
&\Im \lbrace \boldsymbol x_t\rbrace -{\mathcal A}_1(\mathbf {S}_{\text{red}})\Im \lbrace \boldsymbol x_{t-1}\rbrace -{\mathcal A}_2(\mathbf {S}_{\text{red}})\Im \lbrace \boldsymbol x_{t-2}\rbrace \approx \Im \lbrace \boldsymbol {w}\rbrace, \\
&{\mathcal A}_1(\mathbf {S}_{\text{red}}):= (2\!-\!\chi)\mathbb {I}-\mathbf {S}_{\text{red}},\,\,{\mathcal A}_2(\mathbf {S}_{\text{red}}):=(\chi \!-\!1)\mathbb {I} \tag{33}
\end{align*}
Proof:
Simple algebra on (32) allows us to recast the equations in the following form:
\begin{align*}
\Im \lbrace \ddot{\boldsymbol x}\rbrace + \chi \Im \lbrace \dot{\boldsymbol x}\rbrace = \mathtt {diag}\left(\boldsymbol {m} \right)^{-\frac{1}{2}}\overline{\boldsymbol {w}} - \mathbf {S}_{\text{red}} \Im \lbrace \boldsymbol x\rbrace . \tag{34}
\end{align*}
证明：
对 (32) 进行简单的代数运算，我们可以将方程重新表述为以下形式：
\开始{对齐*}
\Im \lbrace \ddot{\boldsymbol x}\rbrace + \chi \Im \lbrace \dot{\boldsymbol x}\rbrace = \mathtt {diag}\left(\boldsymbol {m} \right)^{-\frac
\begin{align*}
\Im \lbrace \ddot{\boldsymbol x}\rbrace + \chi \Im \lbrace \dot{\boldsymbol x}\rbrace = \mathtt {diag}\left(\boldsymbol {m} \right)^{-\frac{1}{2}}\overline{\boldsymbol {w}} - \mathbf {S}_{\text{red}} \Im \lbrace \boldsymbol x\rbrace . \tag{34}
\end{align*}
{2}}\overline{\boldsymbol {w}} - \mathbf {S}_{\text{red}} \Im \lbrace \boldsymbol x\rbrace .\tag{34}
\结束{对齐*}
Assuming that the sampling rate is fast enough, and normalizing it to 1, the finite difference approximations for the derivatives are
假设采样率足够快，并将其归一化为 1，则导数的有限差分近似值为
The model that we introduce is simply extending the GF-AR (2) model to capture both the real and imaginary part of
\begin{align*}
\tilde{\boldsymbol {x}}_{t} = \mathtt {diag}\left(\tilde{\boldsymbol {a}}_{1} \right) \tilde{\boldsymbol {x}}_{t-1} + \mathtt {diag}\left(\tilde{\boldsymbol {a}}_{2} \right) \tilde{\boldsymbol {x}}_{t-2} + \tilde{\boldsymbol {w}}_{t} \tag{35}
\end{align*}
such that the impulse response of the filter at graph-frequency
\begin{align*}
\textstyle \left[\tilde{\boldsymbol {a}}_{1}\right]_{i} = \sum\limits _{k=0}^{K_1-1} a_{k,1} \lambda _{\text{red},i}^{k}, \,\left[\tilde{\boldsymbol {a}}_{2}\right]_{i} = \sum\limits _{k=0}^{K_2-1} a_{k,2} \lambda _{\text{red},i}^{k}, \tag{36}
\end{align*}
我们引入的模型只是对 GF-AR (2) 模型进行了简单的扩展，使其能够同时捕捉
\begin{align*}
\tilde{\boldsymbol {x}}_{t} = \mathtt {diag}\left(\tilde{\boldsymbol {a}}_{1} \right) \tilde{\boldsymbol {x}}_{t-1} + \mathtt {diag}\left(\tilde{\boldsymbol {a}}_{2} \right) \tilde{\boldsymbol {x}}_{t-2} + \tilde{\boldsymbol {w}}_{t} \tag{35}
\end{align*}
使得滤波器在图频率
\begin{align*}
\textstyle \left[\tilde{\boldsymbol {a}}_{1}\right]_{i} = \sum\limits _{k=0}^{K_1-1} a_{k,1} \lambda _{\text{red},i}^{k}, \,\left[\tilde{\boldsymbol {a}}_{2}\right]_{i} = \sum\limits _{k=0}^{K_2-1} a_{k,2} \lambda _{\text{red},i}^{k}, \tag{36}
\end{align*}
使用图频率来表征 AR 系统的极点。从 (33) 可知，在大多数情况下，一阶多项式足以表征 C. Load Dynamics: iℓt iℓt
There are several papers in the literature that deal with load forecasting and modeling [66]. We adopt a simple AR-2 model per node or load bus to describe the dynamics of the load,
\begin{align*}
\boldsymbol {i}^{\ell }_{t} = \mathtt {diag} \left(\boldsymbol {b}_{1} \right) \boldsymbol {i}^{\ell }_{t-1} + \mathtt {diag} \left(\boldsymbol {b}_{2} \right) \boldsymbol {i}^{\ell }_{t-2} + \boldsymbol {\epsilon }_{t} \tag{37}
\end{align*}
where parameters 文献中已有多篇论文探讨了负荷预测和建模 [66] 。我们采用简单的 AR-2 模型（每个节点或负荷母线）来描述负荷的动态变化。
\begin{align*}
\boldsymbol {i}^{\ell }_{t} = \mathtt {diag} \left(\boldsymbol {b}_{1} \right) \boldsymbol {i}^{\ell }_{t-1} + \mathtt {diag} \left(\boldsymbol {b}_{2} \right) \boldsymbol {i}^{\ell }_{t-2} + \boldsymbol {\epsilon }_{t} \tag{37}
\end{align*}
其中参数 
Block diagram showing generative model for voltage phasor measurements.
显示电压相量测量生成模型的框图。
The unique nature of voltage phasor measurements allows us to describe a similar model for any subset of measurements on a graph. This is discussed next.
电压相量测量的独特性质使我们能够在图表上为任何测量子集描述类似的模型。下文将对此进行讨论。
D. Low-Pass Property of Down-Sampled Voltage Graph Signal
D. 下采样电压图信号的低通特性
Let
令
Lemma 1:
Let
\begin{align*}
\boldsymbol {v}_{\mathcal {M}} = \mathcal {H}\left(\mathbf {S}_{\text{red},\mathcal {M}} \right) \boldsymbol {\varphi } \tag{38}
\end{align*}
where the GSO for the reduced-graph is given by Kron-reduction of 引理 1：
令
\开始{对齐*}
\boldsymbol {v}_{\mathcal {M}} = \mathcal {H}\left(\mathbf {S}_{\text{red},\mathcal {M}} \right) \boldsymbol {\varphi } \tag{38}
\结束{对齐*}
其中，简化图的 GSO 由 Proof:
Consider a graph signal
\begin{align*}
\boldsymbol {v} = \mathcal {H} \left(\mathbf {S} \right) \boldsymbol {x} = \mathbf {S}^{-1} \left(\mathbf {S} \mathcal {H} \left(\mathbf {S} \right) \boldsymbol {x}\right) \tag{39}
\end{align*}
The GSO
\begin{align*}
\mathbf {S} = \begin{bmatrix}\mathbf {S}_{\mathcal {M}\mathcal {M}} & \mathbf {S}_{\mathcal {M}\mathcal {M}^{c}} \\
\mathbf {S}_{\mathcal {M}\mathcal {M}^{c}}^{\top } & \mathbf {S}_{\mathcal {M}^{c}\mathcal {M}^{c}} \end{bmatrix}, \tag{40}
\end{align*}
\begin{align*}
\boldsymbol {v}_{\mathcal {M}} &= \mathbf {S}_{\text{red},\mathcal {M}}^{-1} \overbrace{\begin{bmatrix}\mathbb {I}_{\left|\mathcal {M}\right|} & - \mathbf {S}_{\mathcal {M}\mathcal {M}^{c}} \mathbf {S}_{\mathcal {M}^{c}\mathcal {M}^{c}}^{-1} \end{bmatrix} \left(\mathbf {S} \mathcal {H} \left(\mathbf {S} \right) \boldsymbol {x}\right) }^{\boldsymbol {\varphi }} \tag{41}
\end{align*}
\begin{align*}
\!\!\! \mathbf {S}_{\text{red},\mathcal {M}}\!=\!\mathfrak {Sh}\!\left(\mathbf {S}, \!\mathbf {S}_{\mathcal {M}^{c}\mathcal {M}^{c}}\! \right) \!=\!\mathbf {S}_{\mathcal {M}\mathcal {M}}\!-\!\mathbf {S}_{\mathcal {M}\mathcal {M}^{c}}\mathbf {S}_{\mathcal {M}^{c}\mathcal {M}^{c}}^{-1} \mathbf {S}_{\mathcal {M}\mathcal {M}^{c}}^{\top }
\end{align*}
证明：
考虑图形信号 v 具有关于 GSO 的任意图形频率响应 S , v=H(S)x= S −1 (SH(S)x) (39) 查看源代码 GSO S 被重写为 2×2 块状形式 S=[ S 毫米 S ⊤ M M c S M M c S M c M c ]， (40) 查看源代码 和 S −1 可以用分块矩阵的逆公式来表示。当图像信号 v 是下采样的，仅 M 在（39）的两边都考虑行。因此，我们有 v M = S −1 红色,M [ I |中| − S M M c S −1 M c M c ](SH(S)x) φ (41) 查看源代码 在哪里 S 红色,M 是块的 Schur 补 S M c M c 在 GSO S IE， S 红色,M =Sh(S, S M c M c )= S 毫米 − S M M c S −1 M c M c S ⊤ M M c 查看源代码 ■Lemma 1 translates to an interesting self-similarity/fractional property for voltage graph signals in that the down-sampled version
\begin{align*}
\boldsymbol {v} = \mathbf {S}^{-1} \boldsymbol {i}, \,\, \boldsymbol {v}_{\mathcal {M}} = \mathbf {S}_{\text{red},\mathcal {M}}^{-1} \left(\boldsymbol {i}_{\mathcal {M}} \!-\! \mathbf {S}_{\mathcal {M}\mathcal {M}^{c}} \mathbf {S}_{\mathcal {M}^{c}\mathcal {M}^{c}}^{-1} \boldsymbol {i}_{\mathcal {M}^{c}} \right) \tag{42}
\end{align*}
In the power grid, this property has been illustrated empirically in several papers [12], [67] that highlight low-dimensionality of measurements from a subset of buses. Although the reduced-graph is denser compared to the original graph, it still helps to infer faults or events that occurred in a subset of nodes where sensors are not installed as long as correct placement strategies are devised i.e. that of choosing the subset 引理 1 转化为电压图信号一个有趣的自相似/分数特性，即下采样版本
\begin{align*}
\boldsymbol {v} = \mathbf {S}^{-1} \boldsymbol {i}, \,\, \boldsymbol {v}_{\mathcal {M}} = \mathbf {S}_{\text{red},\mathcal {M}}^{-1} \left(\boldsymbol {i}_{\mathcal {M}} \!-\! \mathbf {S}_{\mathcal {M}\mathcal {M}^{c}} \mathbf {S}_{\mathcal {M}^{c}\mathcal {M}^{c}}^{-1} \boldsymbol {i}_{\mathcal {M}^{c}} \right) \tag{42}
\end{align*}
在电网中，多篇论文 [12] 、 [67] 已通过实证研究证明了这一特性，这些论文强调了母线子集测量值的低维性。尽管简化后的图比原始图更密集，但只要设计出正确的放置策略（例如选择子集），它仍然有助于推断未安装传感器的节点子集中发生的故障或事件 Revisiting Algorithms From GSP for PMU Data
重新审视 GSP 中针对 PMU 数据的算法
In this section we study some of the implications Grid-GSP has while understanding sampling, optimal placement of measurement devices in power systems, interpolation of missing samples and network inference. The underlying generative model responsible for low-rank nature of data that has been established in the previous section helps explaining the success that many past works, such as [12], [68], [69], have attained in recovering missing PMU data using matrix completion methods. The low-pass nature of the voltage graph signals discussed in Section III provides the theoretical underpinning that support the arguments made in the literature.
在本节中，我们将探讨 Grid-GSP 在理解采样、电力系统中测量设备的最佳配置、缺失样本的插值以及网络推断方面的一些意义。上一节中建立的、用于解释数据低秩特性的底层生成模型，有助于解释许多以往的研究（例如 [12] 、 [68] 、 [69] ）在使用矩阵补全方法恢复缺失的 PMU 数据方面所取得的成功。 Section III 中讨论的电压图信号的低通特性，为文献中的论证提供了理论基础。
A. Sampling and Recovery of Grid-Graph Signals
A. 网格图信号的采样和恢复
From the approximation in (24) we see that voltage graph signals have graph frequency content that drops as
\begin{align*}
\boldsymbol {\mathcal {B}}_{\mathcal {K}} \boldsymbol {v}_{t} = \mathbf {U}_{\mathcal {K}} \mathbf {U}^{\top }_{\mathcal {K}} \boldsymbol {v}_{t} \tag{43}
\end{align*}
Similarly, a vertex limiting operator (with 从 (24) 中的近似可以看出，电压图信号的图频率成分会随着
\begin{align*}
\boldsymbol {\mathcal {B}}_{\mathcal {K}} \boldsymbol {v}_{t} = \mathbf {U}_{\mathcal {K}} \mathbf {U}^{\top }_{\mathcal {K}} \boldsymbol {v}_{t} \tag{43}
\end{align*}
类似地，顶点限制算子（具有 1) Sampling 1）采样
The optimal placement of
Power systems topologies exhibit naturally a community structure that is reflected in the system admittance matrix
电力系统拓扑自然地表现出一种社群结构 ，由于人口密度或负载集群，这种结构反映在系统导纳矩阵
2) Reconstruction 2）重建
Voltage data samples are obtained down-sampling in space after the optimal placement of PMUs. At time
\begin{align*}
\left[\boldsymbol {v}_t\right]_{\mathcal {M}} \approx \boldsymbol {P}_{{\mathcal {M}}}^{\top } \mathbf {U}_{\mathcal {K}} \tilde{\boldsymbol {v}}_{t} \tag{44}
\end{align*}
where
\begin{align*}
\hat{\boldsymbol {v}}_{t} = \mathbf {U}_{\mathcal {K}} \left(\boldsymbol {P}_{{\mathcal {M}}}^{\top } \mathbf {U}_{\mathcal {K}} \right)^{\dagger } \left[\boldsymbol {v}_t\right]_{\mathcal {M}} \tag{45}
\end{align*}
在 PMU 实现最优放置后，电压数据样本在空间中通过下采样获得。在时间
\begin{align*}
\left[\boldsymbol {v}_t\right]_{\mathcal {M}} \approx \boldsymbol {P}_{{\mathcal {M}}}^{\top } \mathbf {U}_{\mathcal {K}} \tilde{\boldsymbol {v}}_{t} \tag{44}
\end{align*}
其中
\begin{align*}
\hat{\boldsymbol {v}}_{t} = \mathbf {U}_{\mathcal {K}} \left(\boldsymbol {P}_{{\mathcal {M}}}^{\top } \mathbf {U}_{\mathcal {K}} \right)^{\dagger } \left[\boldsymbol {v}_t\right]_{\mathcal {M}} \tag{45}
\end{align*}
B. Interpolation of Missing Samples
B.缺失样本的插值
When voltage measurements are missing or corrupted, denoising and interpolation of such data can be cast as a graph signal recovery problem by regularizing the total variation, (TV). Overall, the problem resembles time-vertex graph signal recovery [59]. Let
\begin{align*}
\min _{\mathbf {V}} \,\, &\textstyle \Vert \mathbb {P}_{\Omega }(\hat{\mathbf {V}}\!-\!\mathbf {V}) \Vert _{F}^{2}\! +\! c_g \sum\limits _{t=1}^{T} \left\Vert \mathbf {S}\boldsymbol {v}_{t} \right\Vert _{1} \\
&+ \textstyle c_t\sum\limits _{t=2}^{T} \left\Vert \boldsymbol {v}_{t} - \boldsymbol {v}_{t-1} \right\Vert _{2}^{2} \tag{46}
\end{align*}
where the two regularizing terms measure the variation in the graph and time domain and 当电压测量值缺失或损坏时，此类数据的去噪和插值可以通过正则化总变分 (TV) 转化为图信号恢复问题。总体而言，该问题类似于时间顶点图信号恢复 [59] 。令
\begin{align*}
\min _{\mathbf {V}} \,\, &\textstyle \Vert \mathbb {P}_{\Omega }(\hat{\mathbf {V}}\!-\!\mathbf {V}) \Vert _{F}^{2}\! +\! c_g \sum\limits _{t=1}^{T} \left\Vert \mathbf {S}\boldsymbol {v}_{t} \right\Vert _{1} \\
&+ \textstyle c_t\sum\limits _{t=2}^{T} \left\Vert \boldsymbol {v}_{t} - \boldsymbol {v}_{t-1} \right\Vert _{2}^{2} \tag{46}
\end{align*}
其中两个正则项测量图和时域中的变化， C. Network Inference as Graph Laplacian Learning
C. 网络推理作为图拉普拉斯学习
The problem of estimation of GSO
\begin{align*}
& \min _{\mathbf {S}}\,\,\,\, \textstyle \sum\limits _{t=1}^{T} \left\Vert \mathbf {S}\boldsymbol {v}_{t} \right\Vert _{1} + \gamma \left\Vert \mathbf {S} - \mathtt {diag}\left(\mathtt {Diag}\left(\mathbf {S}\right) \right) \right\Vert _{F}^{2} \\
&\,\,\,\,\,\,\,\,+ \sum _{t=1}^{T} \left\Vert \mathbf {S}\boldsymbol {v}_{t} - \boldsymbol {i}_{t} \right\Vert _{2}^{2} \tag{47}
\\
&\text{subject to} \,\, \Re {\left[\mathbf {S}\right]_{i,j}} = \Re {\left[\mathbf {S}\right]_{j,i}}, \Im {\left[\mathbf {S}\right]_{i,j}} = \Im {\left[\mathbf {S}\right]_{j,i}}, i\ne j, \tag{48}
\\
& \Re {\text{Tr}\left(\mathbf {S}\right)} = \alpha \left|\mathcal {N}\right|, \Im {\text{Tr}\left(\mathbf {S}\right)} = \beta \left|\mathcal {N}\right| \tag{49}
\end{align*}
Additional constraints on the GSO can be imposed based on the properties of complex-symmetry (see (48)), sparse off-diagonal entries via the term 根据电压相量测量值估计 GSO
\begin{align*}
& \min _{\mathbf {S}}\,\,\,\, \textstyle \sum\limits _{t=1}^{T} \left\Vert \mathbf {S}\boldsymbol {v}_{t} \right\Vert _{1} + \gamma \left\Vert \mathbf {S} - \mathtt {diag}\left(\mathtt {Diag}\left(\mathbf {S}\right) \right) \right\Vert _{F}^{2} \\
&\,\,\,\,\,\,\,\,+ \sum _{t=1}^{T} \left\Vert \mathbf {S}\boldsymbol {v}_{t} - \boldsymbol {i}_{t} \right\Vert _{2}^{2} \tag{47}
\\
&\text{subject to} \,\, \Re {\left[\mathbf {S}\right]_{i,j}} = \Re {\left[\mathbf {S}\right]_{j,i}}, \Im {\left[\mathbf {S}\right]_{i,j}} = \Im {\left[\mathbf {S}\right]_{j,i}}, i\ne j, \tag{48}
\\
& \Re {\text{Tr}\left(\mathbf {S}\right)} = \alpha \left|\mathcal {N}\right|, \Im {\text{Tr}\left(\mathbf {S}\right)} = \beta \left|\mathcal {N}\right| \tag{49}
\end{align*}
可以基于复对称性（参见 (48) ）、通过项 Applications of Grid-GSP Grid-GSP 的应用
The goal of this section is to showcase the benefits of casting problems in the Grid-GSP framework through two exemplary applications, namely anomaly detection and data compression. The common thread between them is the use of the Grid-GFT as a tool to extract informative features from PMU data.
本节旨在通过两个示例应用（异常检测和数据压缩）展示在 Grid-GSP 框架中处理铸造问题的优势。这两个应用的共同点在于都使用 Grid-GFT 作为工具，从 PMU 数据中提取信息特征。
A. Detection of FDI Attacks on PMU Measurements
A. 检测针对 PMU 测量的 FDI 攻击
This application is based on our preliminary work in [2]. Note that, even though we cast the problem as that of FDI attacks detection, the idea can be easily extended to unveil sudden changes due to physical events (like fault-currents, or topology changes) that similarly excite high GF content. We assume that we have access to PMU measurements of voltage and current from the buses they are installed on. Let
\begin{align*}
\underbrace{\begin{bmatrix}\hat{\boldsymbol {i}}_{\mathcal {A}} \\
\hat{\boldsymbol {v}}_{\mathcal {A}} \end{bmatrix}}_{\boldsymbol {z}_t} = \underbrace{\begin{bmatrix}\boldsymbol {Y}_{\mathcal {A}\mathcal {A}} & \boldsymbol {Y}_{\mathcal {A}\mathcal {U}} \\
\mathbb {I}_{\left|\mathcal {A} \right|} & \boldsymbol {0} \end{bmatrix}}_{\boldsymbol {H}} \underbrace{\begin{bmatrix}\boldsymbol {v}_{\mathcal {A}} \\
\boldsymbol {v}_{\mathcal {U}} \end{bmatrix}}_{\boldsymbol {v}} + \boldsymbol {\varepsilon }_t \tag{50}
\end{align*}
The attacker follows the strategy of FDI attack to manipulate both current and voltage on the set of malicious buses,
\begin{align*}
\delta \boldsymbol {v}^{T}_t = \begin{bmatrix}\delta \boldsymbol {v}_{\mathcal {C} }^{T} & \boldsymbol {0}_{\left|\mathcal {P}\right|+\left|\mathcal {U}\right|}^{T} \end{bmatrix}, \,\text{such that}\, \boldsymbol {Y}_{\mathcal {P}\mathcal {C}} \delta \boldsymbol {v}_{\mathcal {C}} = \boldsymbol {0} \tag{51}
\end{align*}
\begin{align*}
\boldsymbol{z}_t \!=\!\! \begin{cases}
\!{H}_{0}: \boldsymbol{H} \mathcal{H}_{k} (\mathbf{S})\!\! \begin{bmatrix}
\left(\mathtt{diag}(\boldsymbol y_g)\boldsymbol{e}_t\right)^{\top} &\
\left(\boldsymbol{i}^{\ell}_t\right)^{\top}
\end{bmatrix}^{\top} \!\!+\! \boldsymbol{\varepsilon}_t \\
\!{H}_{1}: \boldsymbol{H}\mathcal{H}_{k} (\mathbf{S})\!\! \begin{bmatrix}
\left(\mathtt{diag}(\boldsymbol y_g)\boldsymbol{e}_t\right)^{\top} &\
\left(\boldsymbol{i}^{\ell}_t\right)^{\top}
\end{bmatrix}^{\top}\!\!\!\! +\!\! \boldsymbol{H}\delta \boldsymbol{v}_t\!+\! \boldsymbol{\varepsilon}_t
\end{cases}
\tag{52}
\end{align*}
\begin{align*}
\boldsymbol {\Pi }_{\perp \boldsymbol {H} \mathcal {H}_{k} (\mathbf {S}) } \triangleq \mathbb {I} - \left(\boldsymbol {H}\mathcal {H}_{k} (\mathbf {S})\right)\left(\boldsymbol {H}\mathcal {H}_{k} (\mathbf {S})\right)^{\dagger } \tag{53}
\end{align*}
\begin{align*}
d(\boldsymbol {z}_t) \triangleq \left\Vert \boldsymbol {\Pi }_{\perp \boldsymbol {H} \mathcal {H}_{k} (\mathbf {S}) } \boldsymbol {z}_t \right\Vert _{2}^{2} \,\, \mathop {\gtreqless }_{{H}_{0}}^{{H}_{1}}\,\, \tau \tag{54}
\end{align*}
此应用基于我们在 [2] 中的初步工作。需要注意的是，即使我们将问题定义为 FDI 攻击检测，该思路也可以轻松扩展，以揭示由类似地激发高 GF 含量的物理事件（例如故障电流或拓扑变化）引起的突变。我们假设我们可以访问安装 PMU 的母线的电压和电流 PMU 测量值。令
\begin{align*}
\underbrace{\begin{bmatrix}\hat{\boldsymbol {i}}_{\mathcal {A}} \\
\hat{\boldsymbol {v}}_{\mathcal {A}} \end{bmatrix}}_{\boldsymbol {z}_t} = \underbrace{\begin{bmatrix}\boldsymbol {Y}_{\mathcal {A}\mathcal {A}} & \boldsymbol {Y}_{\mathcal {A}\mathcal {U}} \\
\mathbb {I}_{\left|\mathcal {A} \right|} & \boldsymbol {0} \end{bmatrix}}_{\boldsymbol {H}} \underbrace{\begin{bmatrix}\boldsymbol {v}_{\mathcal {A}} \\
\boldsymbol {v}_{\mathcal {U}} \end{bmatrix}}_{\boldsymbol {v}} + \boldsymbol {\varepsilon }_t \tag{50}
\end{align*}
攻击者遵循 FDI 攻击策略，通过引入扰动来操纵恶意总线上的电流和电压，
\begin{align*}
\delta \boldsymbol {v}^{T}_t = \begin{bmatrix}\delta \boldsymbol {v}_{\mathcal {C} }^{T} & \boldsymbol {0}_{\left|\mathcal {P}\right|+\left|\mathcal {U}\right|}^{T} \end{bmatrix}, \,\text{such that}\, \boldsymbol {Y}_{\mathcal {P}\mathcal {C}} \delta \boldsymbol {v}_{\mathcal {C}} = \boldsymbol {0} \tag{51}
\end{align*}
其中
\begin{align*}
\boldsymbol{z}_t \!=\!\! \begin{cases}
\!{H}_{0}: \boldsymbol{H} \mathcal{H}_{k} (\mathbf{S})\!\! \begin{bmatrix}
\left(\mathtt{diag}(\boldsymbol y_g)\boldsymbol{e}_t\right)^{\top} &\
\left(\boldsymbol{i}^{\ell}_t\right)^{\top}
\end{bmatrix}^{\top} \!\!+\! \boldsymbol{\varepsilon}_t \\
\!{H}_{1}: \boldsymbol{H}\mathcal{H}_{k} (\mathbf{S})\!\! \begin{bmatrix}
\left(\mathtt{diag}(\boldsymbol y_g)\boldsymbol{e}_t\right)^{\top} &\
\left(\boldsymbol{i}^{\ell}_t\right)^{\top}
\end{bmatrix}^{\top}\!\!\!\! +\!\! \boldsymbol{H}\delta \boldsymbol{v}_t\!+\! \boldsymbol{\varepsilon}_t
\end{cases}
\tag{52}
\end{align*}
因此，我们将
\begin{align*}
\boldsymbol {\Pi }_{\perp \boldsymbol {H} \mathcal {H}_{k} (\mathbf {S}) } \triangleq \mathbb {I} - \left(\boldsymbol {H}\mathcal {H}_{k} (\mathbf {S})\right)\left(\boldsymbol {H}\mathcal {H}_{k} (\mathbf {S})\right)^{\dagger } \tag{53}
\end{align*}
并且在无攻击假设
\begin{align*}
d(\boldsymbol {z}_t) \triangleq \left\Vert \boldsymbol {\Pi }_{\perp \boldsymbol {H} \mathcal {H}_{k} (\mathbf {S}) } \boldsymbol {z}_t \right\Vert _{2}^{2} \,\, \mathop {\gtreqless }_{{H}_{0}}^{{H}_{1}}\,\, \tau \tag{54}
\end{align*}
其中 Isolation of compromised buses or estimate of
\begin{align*}
{\kern-6.00006pt} \min _{\delta {\boldsymbol {v}_t}} \, \left\Vert \boldsymbol {\Pi }_{\perp \boldsymbol {H} \mathcal {H}_{k} (\mathbf {S}) } ({\boldsymbol z}\!-\! \boldsymbol {H}\delta {\boldsymbol {v}}) \right\Vert ^{2}_{2} \,\, \text{subject to} \,\, \left\Vert \delta {\boldsymbol {v}_t} \right\Vert _{1} \!\leq \! \mu \tag{55}
\end{align*}
Constraint on the 隔离受损公交车或估算
\begin{align*}
{\kern-6.00006pt} \min _{\delta {\boldsymbol {v}_t}} \, \left\Vert \boldsymbol {\Pi }_{\perp \boldsymbol {H} \mathcal {H}_{k} (\mathbf {S}) } ({\boldsymbol z}\!-\! \boldsymbol {H}\delta {\boldsymbol {v}}) \right\Vert ^{2}_{2} \,\, \text{subject to} \,\, \left\Vert \delta {\boldsymbol {v}_t} \right\Vert _{1} \!\leq \! \mu \tag{55}
\end{align*}
对 B. Compression of PMU Measurements
B. PMU 测量的压缩
The proposed compression algorithm leverages both (23) and (35). The measure of distortion we use is the mean-squared error (MSE):
\begin{align*}
\textstyle d(\boldsymbol {v},\hat{\boldsymbol {v}}) \triangleq (\left|\mathcal {N}\right|T)^{-1} \sum\limits _{t=0}^{T} \left\Vert \boldsymbol {v}_{t} - \hat{\boldsymbol {v}}_{t} \right\Vert _{2}^{2} \tag{56}
\end{align*}
where 建议的压缩算法同时利用了 (23) 和 (35) 。我们使用的失真度量是均方误差 (MSE)：
\begin{align*}
\textstyle d(\boldsymbol {v},\hat{\boldsymbol {v}}) \triangleq (\left|\mathcal {N}\right|T)^{-1} \sum\limits _{t=0}^{T} \left\Vert \boldsymbol {v}_{t} - \hat{\boldsymbol {v}}_{t} \right\Vert _{2}^{2} \tag{56}
\end{align*}
其中 Algorithm 1: Encoding Algorithm for Compression.
算法1：压缩的编码算法。
Input:
for
对于
Voltage estimate:
\begin{align*}
{\kern-3.00003pt}\hat{\boldsymbol {v}}^{0}_{t} \!=\! \mathcal {H} (\mathbf {S})\! \begin{bmatrix}\mathtt {diag}\!\left(\boldsymbol {y}_{g}\right) \exp \left\lbrace \!\mathtt {diag}(\boldsymbol {m})^{-\frac{1}{2}}\mathbf {U}_{\text{red}} \hat{\tilde{\boldsymbol {x}}}_{t} \! \right\rbrace \\
\hat{\boldsymbol {i}}^{\ell }_{t} \end{bmatrix} \tag{57}
\end{align*}
电压估算：
\begin{align*}
{\kern-3.00003pt}\hat{\boldsymbol {v}}^{0}_{t} \!=\! \mathcal {H} (\mathbf {S})\! \begin{bmatrix}\mathtt {diag}\!\left(\boldsymbol {y}_{g}\right) \exp \left\lbrace \!\mathtt {diag}(\boldsymbol {m})^{-\frac{1}{2}}\mathbf {U}_{\text{red}} \hat{\tilde{\boldsymbol {x}}}_{t} \! \right\rbrace \\
\hat{\boldsymbol {i}}^{\ell }_{t} \end{bmatrix} \tag{57}
\end{align*}
Compute GFT of modeling error:
计算建模误差的 GFT：
Quantize:
Update states,
\begin{align*}
&\hat{\tilde{\boldsymbol {x}}}_t \leftarrow \mathbf {U}_{\text{red}}^{\top } \mathtt {diag}(\boldsymbol {m})^{\frac{1}{2}} \ln (\hat{\boldsymbol {e}}_{t}), \hat{\boldsymbol {i}}^{\ell }_{t} \leftarrow \left[\mathbf {S} \hat{\boldsymbol {v}}_t\right]_{\mathcal {N}_L} \\
&\text{where} \,\hat{\boldsymbol {e}}_{t} = \left(\mathtt {diag}(\boldsymbol {y}_g) \right)^{-1} \left[\mathbf {S} \hat{\boldsymbol {v}}_t \right]_{\mathcal {N}_G} \tag{58}
\end{align*}
更新状态，
\begin{align*}
&\hat{\tilde{\boldsymbol {x}}}_t \leftarrow \mathbf {U}_{\text{red}}^{\top } \mathtt {diag}(\boldsymbol {m})^{\frac{1}{2}} \ln (\hat{\boldsymbol {e}}_{t}), \hat{\boldsymbol {i}}^{\ell }_{t} \leftarrow \left[\mathbf {S} \hat{\boldsymbol {v}}_t\right]_{\mathcal {N}_L} \\
&\text{where} \,\hat{\boldsymbol {e}}_{t} = \left(\mathtt {diag}(\boldsymbol {y}_g) \right)^{-1} \left[\mathbf {S} \hat{\boldsymbol {v}}_t \right]_{\mathcal {N}_G} \tag{58}
\end{align*}
end for 结束于
Output:
\begin{align*}
&{\boldsymbol {v}}_{t} = \mathcal {H} (\mathbf {S}) \begin{bmatrix}\mathtt {diag}\left(\boldsymbol {y}_{g}\right)\exp \left\lbrace \mathtt {diag}(\boldsymbol {m})^{-\frac{1}{2}}\mathbf {U}_{\text{red}} \left({\tilde{\boldsymbol {x}}}^{0}_{t} +\tilde{\boldsymbol {w}}_{t}\right) \right\rbrace \\
{\boldsymbol {i}}^{\ell,0}_{t} + \boldsymbol {\epsilon }_{t} \end{bmatrix} \\
& \text{where}\,\,\,\, \tilde{\boldsymbol {x}}_{t}^{0} = {\mathtt {diag} \left(\boldsymbol {a}_{1}\right) {\tilde{\boldsymbol {x}}}_{t-1}\! + \mathtt {diag} \left(\boldsymbol {a}_{2} \right) } {\tilde{\boldsymbol {x}}}_{t-2} \tag{59}
\\
&\,\,\,\,\,\,\,\,\,\,\,{\boldsymbol {i}}^{\ell,0}_{t} = \mathtt {diag} \left(\boldsymbol {b}_{1}\right) \hat{\boldsymbol {i}}_{t-1}^{\ell }\! + \mathtt {diag} \left(\boldsymbol {b}_{2} \right) \hat{\boldsymbol {i}}_{t-2}^{\ell } \tag{60}
\end{align*}
Thus,
\begin{align*}
{\kern-8.00003pt}{\boldsymbol {v}}_{t}\!\!\approx \! \mathcal {H} (\mathbf {S}) \!\left(\!\begin{bmatrix}\!\mathtt {diag}\left(\!\boldsymbol {y}_{g}\!\right)\exp \left\lbrace \!\mathtt {diag}(\boldsymbol {m})^{-\frac{1}{2}}\mathbf {U}_{\text{red}} {\tilde{\boldsymbol {x}}}^{0}_{t} \right\rbrace \\
{\boldsymbol {i}}^{\ell,0}_{t} \end{bmatrix} \!\!\!+\! \boldsymbol {\xi }_t \!\!\right) \tag{61}
\end{align*}
Note that, the vector 因此，
\begin{align*}
{\kern-8.00003pt}{\boldsymbol {v}}_{t}\!\!\approx \! \mathcal {H} (\mathbf {S}) \!\left(\!\begin{bmatrix}\!\mathtt {diag}\left(\!\boldsymbol {y}_{g}\!\right)\exp \left\lbrace \!\mathtt {diag}(\boldsymbol {m})^{-\frac{1}{2}}\mathbf {U}_{\text{red}} {\tilde{\boldsymbol {x}}}^{0}_{t} \right\rbrace \\
{\boldsymbol {i}}^{\ell,0}_{t} \end{bmatrix} \!\!\!+\! \boldsymbol {\xi }_t \!\!\right) \tag{61}
\end{align*}
请注意，向量 Algorithm 2: Decoding Algorithm for Reconstruction.
算法2：重构的解码算法。
Input:
输入 ：
for
对于
end for 结束于
Output:
需要注意的是，与文献中的其他方案不同，本文提出的压缩方案是顺序的。随着数据的及时收集，可以进行一些修正，例如参数
Numerical Results 数值结果
The numerical results in this section are mostly obtained using data from the synthetic ACTIVSg2000 case [77], a realistic model emulating the ERCOT system, which includes 2,000 buses-with 432 generators and the rest non-generator buses. The ACTIVSg2000 case data include a realistic PMU data time series, in which 392 generators are dispatched to meet variable load demand. The sampling rate, as for real PMUs, is 30 samples per second. As all the system related parameters are known, it is easier to verify the proposed modeling strategy through the ACTIVSg2000 PMU data set. Fig. 2 shows the support of the graph Laplacian or the
本节中的数值结果主要基于合成的 ACTIVSg2000 案例 [77] 的数据获得。该案例是一个模拟 ERCOT 系统的真实模型，包含 2000 辆公交车（其中 432 辆为发电机组公交车），其余为非发电机组公交车。ACTIVSg2000 案例数据包含真实的 PMU 数据时间序列，其中调度 392 辆发电机以满足可变的负载需求。采样率为每秒 30 次，与真实 PMU 相同。由于所有系统相关参数均为已知，因此更容易通过 ACTIVSg2000 PMU 数据集验证所提出的建模策略。 Fig. 2 显示了按发电机组和非发电机组公交车排序时对图拉普拉斯算子或
Grid-GSP model: In Fig. 3, magnitude of GFT of voltage graph signal
电网-GSP 模型： 在 Fig. 3 中，电压图信号
Magnitude of Graph Fourier Transform (GFT) for voltage graph signal,
电压图信号
In Fig. 4, magnitude of GFT of the down-sampled voltage graph signal,
在 Fig. 4 中，下采样电压图信号的 GFT 幅度，
Magnitude of GFT for spatially down-sampled voltage graph signal,
空间下采样电压图信号的 GFT 幅度，
To highlight the temporal variation in the GFT domain of input exponent,
为了突出输入指数
![Fig. 5. - AR model fit to the GFT of $\boldsymbol {x}_{t}$. Component corresponding to smallest graph frequency, $\lambda _{\text{red},1}$, $[\tilde{\boldsymbol {x}}_{t}]_1$ shown.](/mediastore/IEEE/content/media/78/9307529/9415125/ramak5-3075145-large.gif)
AR model fit to the GFT of
AR 模型拟合了
AR model for load bus,
负载母线的 AR 模型，
To emphasize the temporal nature of the input, the 2-dimensional frequency response (in both graph and time domains) is plotted for the input
为了强调输入的时间特性，绘制了 Fig. 7 中输入
Revising GSP tools: sampling and optimal placement Fig. 8 shows the placement of
修订 GSP 工具： 采样和最优放置 Fig. 8 显示了在有序 Grid-GSO 的支持上叠加
Optimal placement of
![Fig. 9. - Reconstruction performance after optimal placement [70] of $|\mathcal {M}|$ PMUs. Number of frequencies used: $|\mathcal {K}|=|\mathcal {M}|$. For random placement, $|\mathcal {K}| =100$ used.](/mediastore/IEEE/content/media/78/9307529/9415125/ramak9-3075145-large.gif)
To illustrate that the proposed modeling holds and algorithms work well also for real PMU data, in the next numerical experiments we used a real-world dataset of measurements from 35 PMUs placed in ISO New-England grid (ISO-NE) [78]. The data corresponds to a period of 180 seconds when a large generator near Ln:2 and Ln:4 introduces oscillations in the system. We decimated in time the PMU signals down to sampling frequency 1 sample/s.
为了说明所提出的模型成立且算法也适用于真实的 PMU 数据，在接下来的数值实验中，我们使用了来自 ISO 新英格兰电网（ISO-NE） [78] 中 35 个 PMU 的真实测量数据集。该数据对应于 Ln:2 和 Ln:4 附近的大型发电机在系统中引入振荡的 180 秒周期。我们及时将 PMU 信号抽取至采样频率 1 样本/秒。
Network inference: As the underlying GSO is unknown, it is estimated via (47) with the goal of recovering the underlying reduced-GSO. Since admittance values are not given, we only compare the support of the estimated GSO with the community of PMUs in the network. Fig. 10 shows the support of the estimated GSO and compares it with the map of PMUs highlighting a few clusters of correspondence. From Fig. 10 we see that the block-diagonal nature of the estimated GSO captures the community structure in the map.
网络推断： 由于底层 GSO 未知，我们通过 (47) 进行估计，目的是恢复底层的简化 GSO。由于未给出导纳值，我们仅将估计的 GSO 的支持度与网络中的 PMU 社群进行比较。 Fig. 10 显示了估计的 GSO 的支持度，并将其与 PMU 图进行比较，突出显示了一些对应关系聚类。从 Fig. 10 中我们可以看出，估计的 GSO 的块对角线特性能够捕捉到图中的社群结构。
![Fig. 10. - The map of PMUs placed in ISO-NE test case 3 [78] (left) and the support of estimated GSO via (47) (right) shown. Note that the community structure corresponds to groups of PMUs in the actual system as highlighted in the figure for a few clusters.](/mediastore/IEEE/content/media/78/9307529/9415125/ramak10-3075145-large.gif)
The map of PMUs placed in ISO-NE test case 3 [78] (left) and the support of estimated GSO via (47) (right) shown. Note that the community structure corresponds to groups of PMUs in the actual system as highlighted in the figure for a few clusters.
图中显示了 ISO-NE 测试用例 3 [78] 中的 PMU 分布图（左），以及通过 (47) 估算的 GSO 支持度（右）。请注意，社区结构与实际系统中的 PMU 组相对应，如图中突出显示的几个集群所示。
Interpolation of missing measurements: Once the GSO is estimated, we consider the interpolation problem in (46) for the same ISO-NE dataset. We delete data at random and add noise. We solve the problem in (46) to recover missing measurements. In Fig. 11 we compare the original, corrupted and recovered measurements. Corrupted measurements have missing samples not just at random but also contiguous in time. The normalized MSE,
\begin{align*}
\min _{\mathbf {V}} \Vert \mathbb {P}_{\Omega }(\hat{\mathbf {V}}-\mathbf {V})\Vert _{F}^{2} \,\,\text{subject to} \,\, \text{rank}\left({\mathbb {H}(\mathbf {V})}\right) = r \tag{62}
\end{align*}
The plot comparing the two methods is shown in Fig. 12. As seen, the GSO based method outperforms the AM-FIHT for this dataset, indicating that the regularization using the GSO is more effective at capturing the low-rank nature of the data, compared to seeking an arbitrary low rank structure in the the Hankel matrix of the data.缺失测量值的插值： 估算出 GSO 后，我们考虑 (46) 中针对相同 ISO-NE 数据集的插值问题。我们随机删除数据并添加噪声。我们解决 (46) 中的问题以恢复缺失的测量值。在 Fig. 11 中，我们比较了原始测量值、损坏的测量值和恢复的测量值。损坏的测量值不仅随机地缺失样本，而且在时间上是连续的。归一化 MSE
\begin{align*}
\min _{\mathbf {V}} \Vert \mathbb {P}_{\Omega }(\hat{\mathbf {V}}-\mathbf {V})\Vert _{F}^{2} \,\,\text{subject to} \,\, \text{rank}\left({\mathbb {H}(\mathbf {V})}\right) = r \tag{62}
\end{align*}
两种方法的比较图如 Fig. 12 所示。可以看出，基于 GSO 的方法在此数据集上的表现优于 AM-FIHT，这表明使用 GSO 进行正则化能够更有效地捕捉数据的低秩特性，而不是在数据的 Hankel 矩阵中寻找任意的低秩结构。![Fig. 11. - Interpolation of missing measurements for an ISO-NE case [78] using GSO based regularization. Note the contiguous missing of samples and our ability to interpolate. The relative noise level used is, $(|\mathcal {M}|T)\sigma ^2/\Vert \mathbf {V} \Vert _{F}^{2} = 10^{-4}$ Normalized MSE for this run is $6.22 \times 10^{-4}$.](/mediastore/IEEE/content/media/78/9307529/9415125/ramak11-3075145-large.gif)
Interpolation of missing measurements for an ISO-NE case [78] using GSO based regularization. Note the contiguous missing of samples and our ability to interpolate. The relative noise level used is,
使用基于 GSO 的正则化对 ISO-NE 案例 [78] 的缺失测量值进行插值。请注意样本的连续缺失以及我们的插值能力。使用的相对噪声水平为
![Fig. 12. - Comparison of AM-FIHT algorithm in [79] ($r=10, n1=3, \beta =0$) and the proposed GSO based interpolation with 50% of missing measurements in the ISO-NE dataset [78].](/mediastore/IEEE/content/media/78/9307529/9415125/ramak12-3075145-large.gif)
Detection of FDI attacks: Fig. 13 shows the magnitude of the projection of the received measurement
FDI 攻击检测： Fig. 13 显示了接收到的测量值
Components of projection of received measurement
将接收到的测量结果
Empirical ROC curve for different
不同
![Fig. 16. - Empirical ROC curve for methods proposed here and by Drayer & Routtenberg [25] when all voltage measurements are available, $|\mathcal {A}| = 2,000$. A percent of the measurements, $|{\mathcal {C}}|/|{\mathcal {A}}| \!\times \! 100$ are malicious. The relative noise level is $10^{-2}$.](/mediastore/IEEE/content/media/78/9307529/9415125/ramak16-3075145-large.gif)
Empirical ROC curve for methods proposed here and by Drayer & Routtenberg [25] when all voltage measurements are available,
当所有电压测量值均可用时，本文提出的方法以及 Drayer & Routtenberg 提出的方法的经验 ROC 曲线为 [25] ，
Fig. 15 shows the reconstruction of magnitude of the attack vector
Fig. 15 显示了当有 500 个测量值可用时攻击向量的幅度
Compression based results:
For voltage data compression, we compared with two schemes: scalar quantization and singular value thresholding (SVT) from [49]. Fig. 17 plots the empirical rate-distortion (RD) curve and shows the comparison between all 3 schemes. As expected, scalar quantization does poorly compared to the other schemes. The SVT scheme simply uses few of the largest singular vectors for data reconstruction. Considering that it is indicative of voltage graph signal lying in a low-dimensional subspace, it is not surprising that the SVT scheme does well. However, the SVT curve rate-distortion curve eventually saturates. Note that the performance of the proposed method are comparable to those of the SVT. However, the latter is a batch method, while the proposed method is sequential, which has important implication for the online communications of PMU data.
基于压缩的结果：
对于电压数据压缩，我们与两种方案进行了比较：来自 [49] 的标量量化和奇异值阈值 (SVT)。 Fig. 17 绘制了经验率失真 (RD) 曲线并显示了这 3 种方案之间的比较。正如预期的那样，标量量化与其他方案相比表现不佳。SVT 方案仅使用少数最大的奇异向量进行数据重建。考虑到它表明电压图信号位于低维子空间中，SVT 方案表现良好也就不足为奇了。然而，SVT 曲线率失真曲线最终会饱和。请注意，所提出方法的性能与 SVT 相当。然而，后者是一种批处理方法，而所提出的方法是顺序的，这对 PMU 数据的在线通信具有重要意义。
Empirical rate distortion (RD) curve for the proposed compression method compared with singular value thresholding and quantization.
与奇异值阈值和量化相比，所提出的压缩方法的经验率失真（RD）曲线。
Conclusion 结论
In this paper, we proposed the framework of Grid-GSP for the power grid that highlights the inherent spatio-temporal structure in the voltage phasors by employing concepts from GSP. Grid-GSP revisits the concepts of sampling and reconstruction, interpolation, network inference and applications, to detection of FDI attacks and a lossy sequential data compression, were introduced using the lens of GSP. The resulting algorithms were tested on data from both synthetic and real-world datasets. The paper opens the door to leverage the GSP foundations for all types of grid data analytical tasks.
本文提出了适用于电网的 Grid-GSP 框架，该框架运用 GSP 的概念，突出了电压相量的固有时空结构。Grid-GSP 重新审视了采样和重构、插值、网络推理及应用等概念，并结合 GSP 的视角介绍了 FDI 攻击检测和有损顺序数据压缩。最终的算法已在来自合成数据集和真实数据集的数据上进行了测试。本文为利用 GSP 基础进行各种类型的电网数据分析任务打开了大门。
ACKNOWLEDGMENT 致谢
The authors would like to thank the anonymous reviewers and the editor for their comments to improve the quality of the paper. The views expressed in the material are those of the authors and do not necessarily reflect those of the sponsors.
作者谨感谢匿名审稿人和编辑提出的改进论文质量的建议。文中表达的观点仅代表作者本人，并不一定反映资助方的观点。