一種三相對(duì)稱系統(tǒng)快速諧波檢測(cè)算法

葉宗彬 侯 波 張延澳 秦嘉盛 張旭隆

一種三相對(duì)稱系統(tǒng)快速諧波檢測(cè)算法

葉宗彬1侯 波1張延澳1秦嘉盛1張旭隆2

（1. 中國(guó)礦業(yè)大學(xué)電氣與動(dòng)力工程學(xué)院 徐州 221116 2. 徐州工程學(xué)院電氣與控制工程學(xué)院 徐州 221018）

有源電力濾波器（APF）在諧波治理方面有著廣泛應(yīng)用，其諧波補(bǔ)償性能在很大程度上取決于諧波檢測(cè)環(huán)節(jié)的性能。為了提升諧波檢測(cè)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(cè)算法。相較于傳統(tǒng)滑窗離散傅里葉算法存在一個(gè)基波周期延時(shí)，該文提出的快速諧波檢測(cè)算法利用三相系統(tǒng)的對(duì)稱性，替換傳統(tǒng)滑窗離散傅里葉變換諧波檢測(cè)算法的梳狀濾波器構(gòu)成環(huán)節(jié)，能在1/6個(gè)基波周期內(nèi)實(shí)現(xiàn)諧波的有效檢測(cè)，降低檢測(cè)延遲。該文首先分析傳統(tǒng)滑窗離散傅里葉變換諧波檢測(cè)算法的優(yōu)缺點(diǎn)；其次推導(dǎo)所提出的快速諧波檢測(cè)算法的Z域表達(dá)式并分析其特點(diǎn)；最后通過(guò)仿真及小功率縮比有源電力濾波器樣機(jī)實(shí)驗(yàn)平臺(tái)驗(yàn)證了所提出的快速諧波檢測(cè)算法的有效性。

離散傅里葉變換 特定次諧波檢測(cè) 有源電力濾波器 滑窗迭代

0 引言

為了早日實(shí)現(xiàn)“碳達(dá)峰，碳中和”的目標(biāo)，新能源分布式發(fā)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(shè)備會(huì)造成電網(wǎng)中的諧波污染日益嚴(yán)重[1-2]。為有效治理諧波污染，設(shè)計(jì)了多種補(bǔ)償裝置，如統(tǒng)一電能質(zhì)量調(diào)節(jié)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(bǔ)償靈活、容量大等特點(diǎn)，在諧波治理領(lǐng)域有著廣泛應(yīng)用[4-8]。由于負(fù)載電路含有非線性器件，使得負(fù)載電流產(chǎn)生諧波分量，通過(guò)諧波檢測(cè)環(huán)節(jié)檢測(cè)出負(fù)載電流中的諧波分量并控制變流器產(chǎn)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(wǎng)。因此，有源電力濾波器的功能實(shí)現(xiàn)主要由諧波檢測(cè)和諧波跟蹤兩部分組成，而諧波檢測(cè)部分能否快速準(zhǔn)確地檢測(cè)出諧波分量是影響APF性能的一個(gè)重要因素[4-5]。在諧波治理系統(tǒng)中，諧波檢測(cè)方法主要包含時(shí)域和頻域兩個(gè)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(cè)方法可以實(shí)現(xiàn)對(duì)指定次諧波檢測(cè)，但是存在計(jì)算量大、延時(shí)長(zhǎng)等缺點(diǎn)。為減少實(shí)時(shí)系統(tǒng)計(jì)算負(fù)擔(dān)，文獻(xiàn)[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(cè)算法。SDFT的輸出頻率單元數(shù)據(jù)與輸入數(shù)據(jù)速率相同，即輸入增加一個(gè)點(diǎn)的同時(shí)輸出也增加一個(gè)點(diǎn)。因此在實(shí)時(shí)計(jì)算系統(tǒng)中，SDFT相較于Goertzel算法所需計(jì)算量更少[25]，當(dāng)只檢測(cè)某一或幾個(gè)頻率分量時(shí)，SDFT比傳統(tǒng)的基2時(shí)間抽取快速傅里葉變換（Fast Fourier Trans- form, FFT）法[26]更簡(jiǎn)便。但是SDFT依舊存在一個(gè)基波周期的延遲，不利于實(shí)現(xiàn)諧波分量實(shí)時(shí)快速檢測(cè)。文獻(xiàn)[27]提出一種改進(jìn)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(jù)輸入信號(hào)的頻譜分布對(duì)梳狀濾波器重新設(shè)計(jì)，去除不需要的零點(diǎn)，降低檢測(cè)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(cè)算法原理的詳細(xì)推導(dǎo)過(guò)程，通過(guò)仿真及實(shí)驗(yàn)對(duì)快速諧波檢測(cè)算法的可行性進(jìn)行驗(yàn)證。

1 算法原理

1.1 DFT算法原理

在三相系統(tǒng)中，三相畸變信號(hào)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(shù)據(jù)與其對(duì)應(yīng)的旋轉(zhuǎn)因子乘積存儲(chǔ)在連續(xù)的存儲(chǔ)空間內(nèi)，當(dāng)采樣數(shù)據(jù)更新為最新采樣點(diǎn)時(shí)，通過(guò)數(shù)據(jù)運(yùn)算循環(huán)指針定位當(dāng)前數(shù)據(jù)存儲(chǔ)位置，用最新數(shù)據(jù)代替老數(shù)據(jù)以實(shí)現(xiàn)數(shù)據(jù)更新，計(jì)算量減少至一個(gè)復(fù)數(shù)乘法運(yùn)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(yīng)曲線

1.2 一種快速諧波檢測(cè)算法

因此，為了改善檢測(cè)算法的動(dòng)態(tài)性能，本文提出一種快速諧波檢測(cè)方法。在三相信號(hào)對(duì)稱時(shí)，只需要1/6個(gè)基波周期即可獲得檢測(cè)信號(hào)的有效結(jié)果，三相信號(hào)不對(duì)稱時(shí)，只需要1/3個(gè)基波周期即可獲得檢測(cè)信號(hào)的有效結(jié)果。

將式（5）代入式（11）得

在三相系統(tǒng)中偶次諧波分量含量很少，可以忽略不計(jì)，根據(jù)奇諧函數(shù)的性質(zhì)，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(shù)為

其中

以上分析是基于標(biāo)準(zhǔn)復(fù)輸入序列DFT，因此根據(jù)1.1節(jié)的分析，當(dāng)輸入序列為實(shí)信號(hào)時(shí)，可將式（19）改寫為

等效傳遞函數(shù)為

2 仿真與實(shí)驗(yàn)結(jié)果分析

2.1 仿真結(jié)果分析

為驗(yàn)證本文算法的有效性，分別對(duì)SDFT算法和本文提出的快速諧波檢測(cè)算法仿真模型性能進(jìn)行測(cè)試。在仿真測(cè)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(dòng)桿菌；居前三位的革蘭陽(yáng)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(jiàn)，占檢出真菌的55.10%；具體見(jiàn)表2。

1）仿真測(cè)試1：SDFT和本文提出的快速諧波檢測(cè)算法動(dòng)態(tài)性能。

在測(cè)試1中，輸入信號(hào)為三相對(duì)稱信號(hào)，標(biāo)幺值為1.0(pu)。通過(guò)Matlab/Simulink平臺(tái)進(jìn)行仿真證明，結(jié)果如圖3所示。仿真結(jié)果表明，SDFT和本文提出的算法都能夠?qū)崿F(xiàn)對(duì)目標(biāo)頻率分量的有效檢測(cè)。但是，采用SDFT需要約20ms才能獲得有效的檢測(cè)結(jié)果。而采用本文提出的方法僅需約3.3ms，即1/6個(gè)SDFT延遲時(shí)間就可以獲得有效的檢測(cè)結(jié)果。

圖3 動(dòng)態(tài)響應(yīng)的仿真結(jié)果

2）仿真測(cè)試2：三相畸變信號(hào)檢測(cè)。

為更好地驗(yàn)證算法的可行性，將兩種檢測(cè)方法用于檢測(cè)三相不控整流電路產(chǎn)生的諧波信號(hào)。

圖4 用SDFT進(jìn)行諧波檢測(cè)的仿真波形

圖5 用本文提出方法進(jìn)行諧波檢測(cè)的仿真波形

2.2 實(shí)驗(yàn)結(jié)果分析

將本文提出的方法與SDFT應(yīng)用于APF系統(tǒng)，補(bǔ)償由三相不控整流電路產(chǎn)生的諧波，從而驗(yàn)證兩種檢測(cè)方法對(duì)APF動(dòng)態(tài)性能的影響。

實(shí)驗(yàn)裝置由三相并聯(lián)有源電力濾波器（APF）和三相不控整流橋帶純電阻負(fù)載組成，APF系統(tǒng)電路及控制策略如圖6所示。

圖6 APF系統(tǒng)電路及控制策略

表1 實(shí)驗(yàn)的系統(tǒng)參數(shù)

圖7 負(fù)序5次諧波電流分量檢測(cè)的實(shí)驗(yàn)波形和 APF補(bǔ)償之后的A相電網(wǎng)電流波形

圖8 發(fā)生負(fù)載突變時(shí)負(fù)序5次諧波電流分量檢測(cè)的實(shí)驗(yàn)波形和APF補(bǔ)償之后的A相電網(wǎng)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(wǎng)電流的FFT結(jié)果

圖10 采用SDFT檢測(cè)算法全補(bǔ)償動(dòng)態(tài)波形

圖11 采用本文提出的檢測(cè)算法全補(bǔ)償動(dòng)態(tài)波形

3 結(jié)論

SDFT諧波檢測(cè)算法存在長(zhǎng)延時(shí)的缺陷，為此本文提出一種適用于三相對(duì)稱系統(tǒng)中的快速諧波檢測(cè)算法，能夠?qū)崿F(xiàn)特定次諧波快速有效的檢測(cè)。仿真結(jié)果表明，本文提出的方法具有更快的動(dòng)態(tài)響應(yīng)，只需要1/6個(gè)基波周期就可以實(shí)現(xiàn)特定次諧波檢測(cè)。將本文提出的方法應(yīng)用于APF系統(tǒng)中可以改善系統(tǒng)的動(dòng)態(tài)性能，更快實(shí)現(xiàn)對(duì)目標(biāo)頻率分量的補(bǔ)償。

圖12 用SDFT和本文提出方法進(jìn)行補(bǔ)償之后并網(wǎng)電流達(dá)到穩(wěn)態(tài)的動(dòng)態(tài)過(guò)程

附 錄

將式（A1）代入式（6）可得

令

式（A2）可簡(jiǎn)化為

令

對(duì)式（A8）進(jìn)行變換得

傳遞函數(shù)可表示為

其中

對(duì)式（18）的變換只需要證明

又因?yàn)?/p>

式（18）的變換表達(dá)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?；究蒲袠I(yè)務(wù)費(fèi)專項(xiàng)資金資助項(xiàng)目（2019XKQYMS36）。

2022-03-14

2022-06-23

葉宗彬 男，1983年生，博士，副教授，碩士生導(dǎo)師，研究方向?yàn)殡姍C(jī)驅(qū)動(dòng)控制、變流器控制技術(shù)、電能質(zhì)量治理。

E-mail: yezongbin@163.com（通信作者）

侯 波 男，1999年生，碩士，研究方向?yàn)樽兞髌骺刂萍夹g(shù)。

E-mail: houbo0113@163.com

（編輯 陳 誠(chéng)）