﻿ 电能质量信号重构的广义–正则化正交匹配追踪算法

# 电能质量信号重构的广义–正则化正交匹配追踪算法A Generalized-Regularized Orthogonal Matching Pursuit Algorithm for Power Quality Signal Reconstruction

Abstract: A generalized-regularized orthogonal matching pursuit (GROMP) algorithm is proposed based on compressed sensing theory, and it is used for power quality signal reconstruction. Firstly, the GROMP algorithm selects atoms based on the generalized orthogonal matching pursuit algorithm to form the original support set. Then, the regularization method is added to select atoms from original support set to form the final support set. At last, the least square method is employed to update the residual and reconstruct the original signal. The proposed GROMP algorithm not only compensates for the low accuracy of the generalized orthogonal matching pursuit algorithm, but also overcomes the disadvantages of the poor stability and large computational complexity of the regularized orthogonal matching pursuit algorithm. The simulation results of transient and steady-state power quality signal reconstruction show that the newly proposed GROMP algorithm has good adaptability to a variety of measurement matrices. Compared with the traditional gener-alized orthogonal matching pursuit algorithm and the regularized orthogonal matching pursuit algorithm, the new GROMP algorithm has high reconstruction accuracy for various power quality signals and good stability under small compression ratio.

1. 引言

2. 压缩采样和信号重构

2.1. 压缩采样和信号重构的原理

$y=\Phi x=\Phi \Psi \alpha =\Theta \alpha$ (1)

$\mathrm{min}{‖{\Psi }^{\text{T}}x‖}_{0}\text{ }\text{\hspace{0.17em}}\text{s}\text{.t}\text{.}\text{\hspace{0.17em}}\text{\hspace{0.17em}}y=\Theta \alpha =\Phi \Psi \alpha$ (2)

2.2. 稀疏基和测量矩阵的选择

3. 广义–正则化正交匹配追踪算法

3.1. 广义正交匹配与正则化正交匹配追踪算法

GOMP算法首先给定初始残差(如可将观测信号 $y$ 视为初始残差)，然后每次选取传感矩阵 $\Theta$ 中与残差内积最大的S个原子作为支撑集，并通过最小二乘法更新残差。选取原子时每次只选择与残差相关最大的 $S\left(S 个。

ROMP则是在选取原子时每次选取与残差最相关的K列，再通过正则化的能量筛选选出传感矩阵 $\Theta$ 中满足要求的列作为支撑集，再通过最小二乘更新残差。虽然选取K列提高了精度，但同时增加了算法的复杂度与运行时间。

3.2. 广义–正则化正交匹配追踪算法

1) 初始化：测量矩阵 $\Phi$ ，稀疏基 $\Psi$ ，初始残差 ${r}_{0}=y$ ( $y$ 为观测信号)，循环次数k，传感矩阵 $\Theta$ ，索引集合 ${A}_{k}$ 存储第k次循环中传感矩阵 $\Theta$ 被选择的列序号， ${A}_{0}=\varnothing$${C}_{k}$ 表示第k次循环中依照索引 ${A}_{k}$$\Theta$ 中选取的列集合。

2) 相关性选择：计算相关性 $u=|{\Theta }^{\text{T}}{r}_{k-1}|$ ，从中选取最大的S个值，对应矩阵 $\Theta$ 的列序号构成原始支撑集 $J$

3) 正则化选择：按照正则化原理在 $J$ 中选择最终支撑集 ${J}_{0}$ ，即 $\forall i,j\in {J}_{0}$ 满足 $|u\left(i\right)|\le 2|u\left(j\right)|$ 且能量 ${\sum |u|}^{2}$ 最大。

4) 合并选择：令 ${A}_{k}={A}_{k-1}\cup {J}_{0}$ 来更新存储列序号的集合，对于所有 $j\in {J}_{0}$${C}_{k}={C}_{k-1}\cap {\Theta }_{j}$ 来更新选择的列集合(其中 ${\Theta }_{j}$ 表示 $\Theta$ 的j列)。

5) 更新残差：利用 $y={C}_{k}{\theta }_{k}$ ，通过最小二乘法得到初步估计 ${\theta }_{k}={\left({C}_{k}^{\text{T}}{C}_{k}\right)}^{-1}{C}_{k}^{\text{T}}y$ ，并以此更新残差

6)则跳出循环，否则返回步骤2)。

7) 最终的即为所求的

4. 能质量信号重构的仿真与分析

(3)

(4)

(5)

Table 1. Seven kinds of power quality signals

4.1. 测量矩阵的适应度仿真

Figure 1. Reconstruction of harmonic signal based on three measurement matrices

Figure 2. Matrix error comparison of harmonic signal based on three measurement matrices

4.2. 电能质量信号重构与误差分析

4.2.1. 稳态信号重构误差分析

Figure 3. Reconstruction of two steady state signals

Figure 4. Three performance index comparison of harmonic signals

Figure 5. Three performance index comparison of inter-harmonic signals

Table 2. Performance index of three kinds steady state power quality signals

4.2.2. 暂态信号重构仿真与误差分析

Figure 6. Reconstruction of voltage bump signal

Figure 7. Three performance index comparison of voltage bump signal

4.2.3. 复杂信号重构与误差分析

Table 3. Performance index of four kinds transient power quality signals

4.3. 三种算法的用时比较

Figure 8. Reconstruction of mixed signal

Figure 9. Three performance index comparison of mixed signal

5. 结论

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