一種三相對稱系統快速諧波檢測算法

葉宗彬 侯 波 張延澳 秦嘉盛 張旭隆

一種三相對稱系統快速諧波檢測算法

葉宗彬1侯 波1張延澳1秦嘉盛1張旭隆2

（1. 中國礦業大學電氣與動力工程學院 徐州 221116 2. 徐州工程學院電氣與控制工程學院 徐州 221018）

有源電力濾波器（APF）在諧波治理方面有著廣泛應用，其諧波補償性能在很大程度上取決于諧波檢測環節的性能。為了提升諧波檢測速度，該文提出一種基于滑窗離散傅里葉變換（SDFT）的快速諧波檢測算法。相較于傳統滑窗離散傅里葉算法存在一個基波周期延時，該文提出的快速諧波檢測算法利用三相系統的對稱性，替換傳統滑窗離散傅里葉變換諧波檢測算法的梳狀濾波器構成環節，能在1/6個基波周期內實現諧波的有效檢測，降低檢測延遲。該文首先分析傳統滑窗離散傅里葉變換諧波檢測算法的優缺點；其次推導所提出的快速諧波檢測算法的Z域表達式并分析其特點；最后通過仿真及小功率縮比有源電力濾波器樣機實驗平臺驗證了所提出的快速諧波檢測算法的有效性。

離散傅里葉變換 特定次諧波檢測 有源電力濾波器 滑窗迭代

0 引言

為了早日實現“碳達峰，碳中和”的目標，新能源分布式發電比重不斷提高，但是其中的高頻電力電子裝置等非線性設備會造成電網中的諧波污染日益嚴重[1-2]。為有效治理諧波污染，設計了多種補償裝置，如統一電能質量調節器（Unified Power Quality Conditioner, UPQC）[3]、有源濾波器（Active Power Filter, APF）。其中，有源電力濾波器以其補償靈活、容量大等特點，在諧波治理領域有著廣泛應用[4-8]。由于負載電路含有非線性器件，使得負載電流產生諧波分量，通過諧波檢測環節檢測出負載電流中的諧波分量并控制變流器產生與諧波分量幅值相同、相位相反的電流，從而避免諧波電流注入電網。因此，有源電力濾波器的功能實現主要由諧波檢測和諧波跟蹤兩部分組成，而諧波檢測部分能否快速準確地檢測出諧波分量是影響APF性能的一個重要因素[4-5]。在諧波治理系統中，諧波檢測方法主要包含時域和頻域兩個方面。

離散傅里葉變換（Discrete Fourier Transform, DFT）[19-20]作為一種典型的頻域諧波檢測方法可以實現對指定次諧波檢測，但是存在計算量大、延時長等缺點。為減少實時系統計算負擔，文獻[5, 21-24]將滑窗迭代算法引入離散傅里葉變換中，提出滑窗迭代離散傅里葉（Sliding-window Discrete Fourier Transform, SDFT）諧波檢測算法。SDFT的輸出頻率單元數據與輸入數據速率相同，即輸入增加一個點的同時輸出也增加一個點。因此在實時計算系統中，SDFT相較于Goertzel算法所需計算量更少[25]，當只檢測某一或幾個頻率分量時，SDFT比傳統的基2時間抽取快速傅里葉變換（Fast Fourier Trans- form, FFT）法[26]更簡便。但是SDFT依舊存在一個基波周期的延遲，不利于實現諧波分量實時快速檢測。文獻[27]提出一種改進的方法，二次采樣遞歸離散傅里葉變換（Twice Sampling Recursive Discrete Fourier Transform, TS-RDFT），可根據輸入信號的頻譜分布對梳狀濾波器重新設計，去除不需要的零點，降低檢測算法的延遲。

本文介紹了DFT算法原理和提出的快速諧波檢測算法原理的詳細推導過程，通過仿真及實驗對快速諧波檢測算法的可行性進行驗證。

1 算法原理

1.1 DFT算法原理

在三相系統中，三相畸變信號可以表示為

令

將式（3）代入（2）中得

滑窗迭代算法模型如圖1所示。采樣數據與其對應的旋轉因子乘積存儲在連續的存儲空間內，當采樣數據更新為最新采樣點時，通過數據運算循環指針定位當前數據存儲位置，用最新數據代替老數據以實現數據更新，計算量減少至一個復數乘法運算。

圖1 滑窗迭代算法模型

Fig.1 Sliding window iterative algorithm model

其中

其中

圖2 的幅頻響應曲線

1.2 一種快速諧波檢測算法

因此，為了改善檢測算法的動態性能，本文提出一種快速諧波檢測方法。在三相信號對稱時，只需要1/6個基波周期即可獲得檢測信號的有效結果，三相信號不對稱時，只需要1/3個基波周期即可獲得檢測信號的有效結果。

將式（5）代入式（11）得

在三相系統中偶次諧波分量含量很少，可以忽略不計，根據奇諧函數的性質，則

式（12）改寫為

即

將式（20）代入式（19），得到本文提出方法的等效傳遞函數為

其中

以上分析是基于標準復輸入序列DFT，因此根據1.1節的分析，當輸入序列為實信號時，可將式（19）改寫為

等效傳遞函數為

2 仿真與實驗結果分析

2.1 仿真結果分析

為驗證本文算法的有效性，分別對SDFT算法和本文提出的快速諧波檢測算法仿真模型性能進行測試。在仿真測試中基波頻率為50Hz，采樣頻率為15kHz。

2.2 病原菌排位 2012-2016年病原菌的排位中，居前三位的革蘭陰性菌依次是大腸埃希菌、肺炎克雷伯菌、鮑曼不動桿菌；居前三位的革蘭陽性菌依次是金黃色萄萄球菌、凝固酶陰性葡萄球菌、腸球菌，這六種菌占檢出菌的38.01%；真菌以白色念珠菌最常見，占檢出真菌的55.10%；具體見表2。

1）仿真測試1：SDFT和本文提出的快速諧波檢測算法動態性能。

在測試1中，輸入信號為三相對稱信號，標幺值為1.0(pu)。通過Matlab/Simulink平臺進行仿真證明，結果如圖3所示。仿真結果表明，SDFT和本文提出的算法都能夠實現對目標頻率分量的有效檢測。但是，采用SDFT需要約20ms才能獲得有效的檢測結果。而采用本文提出的方法僅需約3.3ms，即1/6個SDFT延遲時間就可以獲得有效的檢測結果。

圖3 動態響應的仿真結果

2）仿真測試2：三相畸變信號檢測。

為更好地驗證算法的可行性，將兩種檢測方法用于檢測三相不控整流電路產生的諧波信號。

圖4 用SDFT進行諧波檢測的仿真波形

圖5 用本文提出方法進行諧波檢測的仿真波形

2.2 實驗結果分析

將本文提出的方法與SDFT應用于APF系統，補償由三相不控整流電路產生的諧波，從而驗證兩種檢測方法對APF動態性能的影響。

實驗裝置由三相并聯有源電力濾波器（APF）和三相不控整流橋帶純電阻負載組成，APF系統電路及控制策略如圖6所示。

圖6 APF系統電路及控制策略

表1 實驗的系統參數

圖7 負序5次諧波電流分量檢測的實驗波形和 APF補償之后的A相電網電流波形

圖8 發生負載突變時負序5次諧波電流分量檢測的實驗波形和APF補償之后的A相電網電流波形

圖9 用SDFT和本文提出方法消除特定次諧波之后 A相電網電流的FFT結果

圖10 采用SDFT檢測算法全補償動態波形

圖11 采用本文提出的檢測算法全補償動態波形

3 結論

SDFT諧波檢測算法存在長延時的缺陷，為此本文提出一種適用于三相對稱系統中的快速諧波檢測算法，能夠實現特定次諧波快速有效的檢測。仿真結果表明，本文提出的方法具有更快的動態響應，只需要1/6個基波周期就可以實現特定次諧波檢測。將本文提出的方法應用于APF系統中可以改善系統的動態性能，更快實現對目標頻率分量的補償。

圖12 用SDFT和本文提出方法進行補償之后并網電流達到穩態的動態過程

附 錄

將式（A1）代入式（6）可得

令

式（A2）可簡化為

令

對式（A8）進行變換得

傳遞函數可表示為

其中

對式（18）的變換只需要證明

又因為

式（18）的變換表達式為

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A Fast Harmonic Detection Algorithm for Three-Phase Symmetric Systems

11112

（1. School of Electrical Engineering China University of Mining and Technology Xuzhou 221116 China 2. School of Electrical and Control Engineering Xuzhou University of Technology Xuzhou 221018 China）

The active power filter (APF) can compensate harmonics caused by nonlinear equipment such as high-frequency power electronic devices. Whether its harmonic detection algorithm can detect the harmonic component quickly and accurately largely determines the dynamic response and harmonic compensation performance of APF. The traditional discrete Fourier transform (DFT) was a frequency domain harmonic detection method, which can detect specific harmonics. However, it had large computation and long delay and, hence, can not detect harmonics quickly and compensate them in time. This was insufficient to support fast harmonic compensation of APF. Recently, some methods introduced the sliding window iterative algorithm into DFT, but there was still delay of a fundamental period. To address these issues, this paper proposes a new sliding-window discrete Fourier transform (SDFT) algorithm for three-phase symmetric systems. By using the symmetry of three-phase signals, it effectively detects harmonics within 1/6 fundamental cycle.

This method is based on DFT and sliding window iterative algorithm. Firstly, the DFT algorithm needs N complex multiplications to detect a specific order harmonic. The sliding window iterative algorithm updates the datas using cyclic sliding pointer, reducing the calculation to one complex multiplication, thus leading to the delay of one fundamental period. Secondly, the z-domain transfer function of DFT is composed of a comb filter, a complex resonator and gain coefficient. The method proposed in this paper uses a new comb filter, which makes use of the characteristic that the sampling value of B and C phase signals in three-phase symmetric signals can replace the partial sampling value of phase A as the input sequence of DFT calculation. It only needs 1/6 fundamental period to obtain the output sequence of the harmonic components. This way, the problem that SDFT requires one fundamental cycle delay is addressed, and APF can detect and compensate harmonics more quickly.

The test results in the simulation model with the fundamental frequency of 50Hz and sampling frequency of 15kHz show that both SDFT and the proposed method can effectively detect the target frequency components. However, it takes about 20ms to obtain detection results by using SDFT. The proposed method only needs about 3.3ms, that is, 1/6 SDFT delay time. The proposed method and SDFT are applied to the system composed of three-phase shunt APF and three-phase uncontrolled rectifier bridge with resistive load. The negative sequence 5th harmonic current components generated by three-phase uncontrolled rectifier circuit are detected and extracted respectively, and the detection results are used as the reference value of harmonic current in current loop to compensate the negative sequence 5th harmonic current components. The experimental results show that both SDFT and the proposed method can achieve specific harmonic detection, but the proposed method has better dynamic response performance and requires less storage space than SDFT. After the compensation by APF system using SDFT and the proposed method, the 5th harmonic current content in grid current decreases from 23.56% to 0.64% and 0.77% respectively. The experimental results show that the grid-connected current of the APF system with the proposed method can reach the steady state faster.

The following conclusions can be drawn from the simulation and experimental results: The proposed fast harmonic detection algorithm, which is suitable for three-phase symmetric systems, can achieve fast and effective detection of specific harmonics. Compared with SDFT harmonic detection algorithm, the proposed method obtains faster dynamic response, and only needs 1/6 fundamental cycle to detect specific harmonics. Applying the proposed method to APF system can improve the dynamic performance of the system and compensate specific harmonics faster.

Discrete fourier transform, selective harmonic detection, active power filter, sliding- window iterative

TM714

10.19595/j.cnki.1000-6753.tces.220362

中央高校基本科研業務費專項資金資助項目（2019XKQYMS36）。

2022-03-14

2022-06-23

葉宗彬 男，1983年生，博士，副教授，碩士生導師，研究方向為電機驅動控制、變流器控制技術、電能質量治理。

E-mail: yezongbin@163.com（通信作者）

侯 波 男，1999年生，碩士，研究方向為變流器控制技術。

E-mail: houbo0113@163.com

（編輯 陳 誠）