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WANG Jiansheng, MAO Huafeng, MAO Zhiwei, XU Liuwei. Analysis of Harmonic Instability of Tokamak Power System Caused by Rectifier Transformer Bias[J]. SOUTHERN ENERGY CONSTRUCTION, 2022, 9(2): 70-76. doi: 10.16516/j.gedi.issn2095-8676.2022.02.009
Citation: WANG Jiansheng, MAO Huafeng, MAO Zhiwei, XU Liuwei. Analysis of Harmonic Instability of Tokamak Power System Caused by Rectifier Transformer Bias[J]. SOUTHERN ENERGY CONSTRUCTION, 2022, 9(2): 70-76. doi: 10.16516/j.gedi.issn2095-8676.2022.02.009

Analysis of Harmonic Instability of Tokamak Power System Caused by Rectifier Transformer Bias

doi: 10.16516/j.gedi.issn2095-8676.2022.02.009
  • Received Date: 2021-10-27
  • Rev Recd Date: 2022-04-14
  • Available Online: 2022-05-30
  • Publish Date: 2022-06-25
  •   Introduction  The paper aims to effectively suppress the disturbance of non characteristic harmonics to the power grid in the tokamak power supply system, the paper studies the mechanism of low-frequency harmonics in the tokamak power supply system.   Method  By introducing an improved switching function model, selected ITER fusion device as an example and established its corresponding AC/DC equivalent circuit model combined with its corresponding parameters, the corresponding output second harmonic was obtained through calculation and simulation.   Result  The ratio of output to input is calculated, and the stability factor value almost tends to zero.   Conclusion  It is determined that the DC bias of rectifier transformer do not cause harmonic instability of the system.
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    [7] 吴亚楠. Tokamak电源及无功补偿系统与电网的兼容性分析与研究[D]. 北京: 中国科学院大学, 2014.

    WU Y N. Analysis and research on compatibility between Tokamak power supply and reactive power compensation system and power grid[D]. Beijing: University of Chinese Academy of Sciences, 2014.
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    CHEN Z W, BAI B D, CHEN D Z, et al. Study on the formation mechanism of DC magnetic bias in power transformer and a suppression measure [J]. Journal of Electrotechnics, 2015, 30(14):208-214. DOI:  10.19595/j.cnki.1000-6753.tces.2015.14.028.
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    [16] 李冰, 王泽忠, 刘海波, 等. 直流偏磁下500 kV单相变压器振动噪声的试验研究 [J]. 电工技术学报, 2021, 36(13):2801-2811. DOI:  10.19595/j.cnki.1000-6753.tces.200689.

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Analysis of Harmonic Instability of Tokamak Power System Caused by Rectifier Transformer Bias

doi: 10.16516/j.gedi.issn2095-8676.2022.02.009

Abstract:   Introduction  The paper aims to effectively suppress the disturbance of non characteristic harmonics to the power grid in the tokamak power supply system, the paper studies the mechanism of low-frequency harmonics in the tokamak power supply system.   Method  By introducing an improved switching function model, selected ITER fusion device as an example and established its corresponding AC/DC equivalent circuit model combined with its corresponding parameters, the corresponding output second harmonic was obtained through calculation and simulation.   Result  The ratio of output to input is calculated, and the stability factor value almost tends to zero.   Conclusion  It is determined that the DC bias of rectifier transformer do not cause harmonic instability of the system.

WANG Jiansheng, MAO Huafeng, MAO Zhiwei, XU Liuwei. Analysis of Harmonic Instability of Tokamak Power System Caused by Rectifier Transformer Bias[J]. SOUTHERN ENERGY CONSTRUCTION, 2022, 9(2): 70-76. doi: 10.16516/j.gedi.issn2095-8676.2022.02.009
Citation: WANG Jiansheng, MAO Huafeng, MAO Zhiwei, XU Liuwei. Analysis of Harmonic Instability of Tokamak Power System Caused by Rectifier Transformer Bias[J]. SOUTHERN ENERGY CONSTRUCTION, 2022, 9(2): 70-76. doi: 10.16516/j.gedi.issn2095-8676.2022.02.009
  • 工业领域随着可再生能源发电、柔性直流输电FACTS技术、静止同步补偿器STATCOM等电力电子新技术的日益大规模应用,交直流混联电网的谐波问题日益突出[1-2]。电网谐波的频率除3次、5次、7次等特征次谐波外,2次谐波等非特征次谐波也存在较大含量[3]

    关于非特征次谐波的研究,过去数十年间国内外相关领域的专家做了大量的研究。1967年英国人Ainsworth通过对换流器和非特征次谐波相互作用的研究,首次提出了谐波不稳定的概念。随后于1977年又确定了铁心饱和型谐波不稳定的产生机理。随后数年的时间里,Yacamini和Oliveria又依次提出了交流侧和直流侧互补谐振的概念。这一系列概念的提出,奠定了谐波理论的基础[4-6]

    在Tokamak电源系统中,整流变压器同样会产生较为可观的二次谐波分量[7],此二次谐波可能会造成整流变压器的偏磁问题,进而导致出现谐波不稳定现象[8-11],对聚变电源系统的正常运行产生一定的影响。本文引入了一种改进型的开关函数模型,并选取ITER电源系统中谐波不稳定的产生机理,完成了对ITER电源系统这一实例的计算和仿真,进而确定了最终的影响。

    • 对于Tokamak磁体电源系统,变流器负载多为数十毫亨至数亨量级的大电感,可以认为直流侧电流为平滑直线基本无脉动。考虑低频谐波计算的复杂性,首先忽略换相过程,采用调制函数法进行分析,而后对换相过程影响进行说明。假设并联运行时均流系数为1,每桥承担$ {I_{\text{d}}}/2 $

      设调制函数$f(t) = \varepsilon (t - {\alpha _{n}}) - \varepsilon (t - {\alpha _{n}} - {\alpha _{\text{v}}})$,由Fourier计算公式得:

      $$ f(\omega t,{\alpha _0}) = A +\displaystyle \sum\limits_{{k} = 1}^\infty {{A_{k}}\cos (k\omega t - {\varphi _{k}})} $$ (1)

      式中:

      $A = {\alpha _{\text{v}}}/2{\text{π}}$${A_k} = 2\sin \dfrac{{k{\alpha _{\text{v}}}}}{2}/k{\text{π}}$${\varphi _k} = k{\alpha _0} + \dfrac{{k{\alpha _{\text{v}}}}}{2}$

      $ {\alpha _{\text{v}}} $——晶闸管导通区间;

      $ \alpha $ ——触发角;

      $ {\alpha _0} $——导通时刻;

      ${\alpha _{n}}$——第n个晶闸管的触发时刻,当晶闸管导通时,调制函数取值为1,关闭时取值为0。

      共阴极和共阳极的晶闸管按顺序依次导通120°,如图1所示。

      Figure 1.  Thyristor conduction sequence modulation function

      对于三相全控桥调制函数${\alpha _{\text{v}}} = \dfrac{2}{3}{\text{π}}$${\alpha _{n}} = \dfrac{{(2n - 1){\text{π}} }}{6} + \alpha$$ \alpha $相过零点为时间起点,将其代入式(1)可得:

      $$ \begin{split} &{f_{n}} = \frac{1}{3} +\displaystyle \sum\limits_{{k} = 1}^\infty {\dfrac{2}{{k{\text{π}} }}} \sin \dfrac{{k{\text{π}} }}{3}\cos \left[ {k\omega t - k\alpha - \dfrac{{(2n + 1)k{\text{π}}}}{6}} \right]\\&(n = 1,2,3,4,5,6) \end{split} $$ (2)

      A相交流电流是由调制函数$ {f_1} $$ {f_4} $对直流电流$ {I_{\text{d}}} $调制的结果:

      $$ \begin{split} &{i_{\text{a}}} = ({f_1} - {f_4}){I_{\text{d}}} = \displaystyle \sum\limits_{{k} = 1}^\infty {{I_{\text{d}}} \cdot \dfrac{4}{{k{\text{π}}}}} \cos \dfrac{{k{\text{π}}}}{6}\sin k(\omega t - \alpha )\\&(k = 1,5, \cdots ,6n \pm 1) \end{split} $$ (3)

      图2所示三相全控桥并联运行,副边线电压差30°,两桥所产生的含量较高的5,7次谐波相位刚好相差180°,当幅值相等时,可以相互抵消。设$ {\stackrel{·}{U}}_{\text{A}1\text{B}1} $$ {\stackrel{·}{U}}_{\text{A}2\text{B2}} $滞后30°,原副边绕组的电流瞬时值关系:

      Figure 2.  Three phase fully controlled bridge parallel operation

      $$ \left\{ \begin{array}{l} \displaystyle{i_{{\text{A1}}}} = \sum\limits_{{k} = 1}^\infty {\frac{{{I_{\text{d}}}}}{2} \cdot \frac{4}{{k{\text{π}}}} \cdot \cos \frac{{k{\text{π}}}}{6} \cdot \sin k(\omega t - \alpha )} \\ \\ \displaystyle{i_{{\text{A2}}}} = \sum\limits_{{k} = 1}^\infty {\frac{{{I_{\text{d}}}}}{2} \cdot \frac{4}{{k{\text{π}}}} \cdot \cos \frac{{k{\text{π}}}}{6} \cdot \sin k\left( {\omega t - \alpha + \frac{{\text{π}}}{6}} \right)} \\ \\ \displaystyle{i_{{\text{B2}}}} = \sum\limits_{{k} = 1}^\infty {\frac{{{I_{\text{d}}}}}{2} \cdot \frac{4}{{k{\text{π}}}} \cdot \cos \frac{{k{\text{π}}}}{6} \cdot \sin k\left( {\omega t - \alpha - \frac{5}{6}{\text{π}}} \right)} \\ \\ (k = 6n \pm 1) \end{array} \right. $$ (4)

      故可推出:${i_{\text{A}}} = \displaystyle \sum\limits_{{k} = 1}^\infty \dfrac{{2\sqrt 3 }}{{k\text{π}}}{I_{\text{d}}} \cdot \sin k(\omega t - \alpha ), k = 12n \pm 1 。$

    • 调制函数的影响,可以视为基本分量${s_{\text{n}}}$、修正分量${s_{\text{m}}}$和换相分量的叠加。基本分量${s_{\text{n}}}$、修正分量${s_{\text{m}}}$为幅值为1、宽度分别为$2{\text{π }}/3$$\theta $的矩形波,前者反映不存在换相过程且严格按照触发脉冲导通的情况下变流器调制动作,后者则用于修正因变流器导通时刻偏移而引起的调制函数波形的变化[12-13]。对应的电压电流开关函数为:

      $$ \left\{ \begin{array}{l} {s_{{\rm{ua}}}} = {s_{\rm{n}}}(\omega t) + {s_{{\rm{u}}\mu {\rm{A}}}} + {s_{{\rm{mA}}}}\\ {s_{{\rm{ub}}}} = {s_{\rm{n}}}\left( {\omega t - \dfrac{2}{3}\text{π}} \right) + {s_{{\rm{u}}\mu {\rm{B}}}} + {s_{{\rm{mB}}}}\\ {s_{{\rm{uc}}}} = {s_{\rm{n}}}\left( {\omega t + \dfrac{2}{3}\text{π}} \right) + {s_{{\rm{u}}\mu {\rm{C}}}} + {s_{{\rm{mC}}}} \end{array} \right. $$ (5)
      $$ \left\{ \begin{gathered} {s_{{\text{ia}}}} = {s_{\text{n}}}(\omega t) + {s_{{\text{i}} \mu {\text{A}}}} + {s_{{\text{m}} {\text{A}}}} \\ {s_{{\text{ib}}}} = {s_{\text{n}}}\left( {\omega t - \dfrac{2}{3}{\text{π }}} \right) + {s_{{\text{i}} \mu {\text{B}}}} + {s_{{\text{mB}}}} \\ {s_{{\text{ic}}}} = {s_{\text{n}}}\left( {\omega t + \dfrac{2}{3}{\text{π }}} \right) + {s_{{\text{i}} \mu {\text{C}}}} + {s_{{\text{mC}}}} \\ \end{gathered} \right. $$ (6)

      式中:

      $$ \left\{ \begin{gathered} {s_{{\text{u}}\mu {\text{A}}} } = {s_{{\text{u}}} \mu }\left( {{\mu _{{\text{ab}}}},\omega t - \dfrac{{\text{π }}}{3} - {\theta _{{\text{ab}}}}} \right) - {s_{{\text{u}}} \mu }\left( {{\mu _{{\text{ca}}}},\omega t + \dfrac{{\text{π }}}{3} - {\theta _{{\text{ca}}}}} \right) \\ {s_{{\text{u}}\mu \text{B}}} = - {s_{{\text{u}}} \mu }\left( {{\mu _{{\text{bc}}}},\omega t - \dfrac{{\text{π }}}{3} - {\theta _{{\text{bc}}}}} \right) - {s_{{\text{u}}} \mu }\left( {{\mu _{{\text{ab}}}},\omega t - \dfrac{{\text{π }}}{3} - {\theta _{{\text{ab}}}}} \right) \\ {s_{{\text{u}}\mu \text{C}}} = - {s_{{\text{u}}} \mu }\left( {{\mu _{{\text{ca}}}},\omega t + \dfrac{{\text{π }}}{3} - {\theta _{{\text{ca}}}}} \right) - {s_{{\text{u}}} \mu }\left( {{\mu _{{\text{bc}}}},\omega t - \dfrac{{\text{π }}}{3} - {\theta _{{\text{bc}}}}} \right) \\ \end{gathered} \right. $$ (7)
      $$ \left\{ \begin{gathered} {s_{{\text{mA}}}} = {s_{\text{m}}}\left( {{\theta _{{\text{ab}}}},\omega t - \dfrac{{\text{π }}}{3}} \right) - {s_{{\text{u}}} \mu }\left( {{\theta _{{\text{ca}}}},\omega t + \dfrac{{\text{π }}}{3}} \right) \\ {s_{{\text{mB}}}} = {s_{\text{m}}}\left( {{\theta _{{\text{bc}}}},\omega t} \right) - {s_{\text{m}}}\left( {{\theta _{{\text{ab}}}},\omega t - \dfrac{{\text{π }}}{3}} \right) \\ {s_{{\text{mC}}}} = {s_{\text{m}}}\left( {{\theta _{{\text{ca}}}},\omega t + \dfrac{{\text{π }}}{3}} \right) - {s_{\text{m}}}\left( {{\theta _{{\text{bc}}}},\omega t} \right) \\ \end{gathered} \right. $$ (8)
      $$ \left\{ \begin{split} &{s_{{\text{i}}\mu {\text{A}}}} = {s_{{\text{i}}} \mu }\left( {{U_{{\text{ab}}}},{\alpha _{{\text{ab}}}},{\mu _{{\text{ab}}}},\omega t - {\theta _{{\text{ab}}}} - \dfrac{{\text{π }}}{3}} \right) -\\& \qquad\;\;{s_{{\text{i}}} \mu }\left( {{U_{{\text{ca}}}},{\alpha _{{\text{ca}}}},{\mu _{{\text{ca}}}},\omega t - {\theta _{{\text{ca}}}} + \dfrac{{\text{π }}}{3}} \right) \\& {s_{{\text{i}\mu {\text{B}}}}} = {s_{{\text{i}}} \mu }\left( { {U_{{\text{bc}}}},{\alpha _{{\text{bc}}}},{\mu _{{\text{bc}}}},\omega t - {\theta _{{\text{bc}}}} - \dfrac{{\text{π }}}{3}} \right) -\\& \qquad\;\;{s_{{\text{i}}} \mu }\left( {{U_{{\text{ab}}}},{\alpha _{{\text{ab}}}},{\mu _{{\text{ab}}}},\omega t - {\theta _{{\text{ab}}}} - \dfrac{{\text{π }}}{3}} \right) \\& {s_{{\text{i}} \mu {\text{C}}}} = {s_{{\text{i}}} \mu }\left( {{U_{{\text{ca}}}},{\alpha _{{\text{ca}}}},{\mu _{{\text{ca}}}},\omega t - {\theta _{{\text{ca}}}} - \dfrac{{\text{π }}}{3}} \right) -\\& \qquad\;\;{s_{{\text{i}}} \mu }\left( {{U_{{\text{bc}}}},{\alpha _{{\text{bc}}}},{\mu _{{\text{bc}}}},\omega t - {\theta _{{\text{bc}}}} - \dfrac{{\text{π }}}{3}} \right) \end{split} \right. $$ (9)

      通过计算分析换相角对2次谐波影响小,为简化低次谐波建模,使模型具备工程实用价值,忽略换相不平衡造成的影响。

    • 本节主要通过推导交流侧2次谐波与直流侧基频电流之间的关系,分析了ITER PF变流器建立直流偏磁等值参数电路,并得到整流变压器铁心饱和情况下谐波在变流器两侧以及整流变压器两侧的传递过程,如图3所示[14-16]

      Figure 3.  Harmonic transmission diagram of rectifier transformer core saturation

    • 结合图3中的2次谐波传递关系图可知,这是一个正反馈通道,即当线路中产生一个二次谐波扰动时,会经过上述正反馈通道进一步输出一个对应的二次谐波分量作用于变压器交流侧。对应的参数计算情况如下:

    • 交直流系统电路对应的模型如图4所示。其中交流侧等值阻抗计算公式为:

      Figure 4.  AC and DC circuit model

      $$ {Z_{{\text{ac}}2}} = \left[ {({Z_{\text{S}}}//{Z_{{\text{lb2}}}}) + {Z_{{\text{T2}}}}} \right]/{k^2} $$ (10)

      式中:

      k ——变压器变比;

      $ {Z_{\text{S}}} $ ——电源等值阻抗(Ω);

      $ {Z_{{\text{lb2}}}} $ ——滤波器和无功补偿装置的等值阻抗(Ω);

      ${Z_{{\text{T2}}}}$ ——ITER变压器网侧对应的阻抗(Ω)。

    • 参考ITER PF整流变压器参数(如表1所示),接入点等效阻抗根据主变接入点短路容量为922 MVA,故接入点的等效电抗为${X_{\text{S}}} = 14.348\;8\;\Omega $。根据$ {X_{\text{S}}} = 14.348\;8\;\Omega $可求出${R_{\text{S}}} = 1.434\;88\;\Omega $

      变压器名称ITER PF整流变压器
      变压器型号ZHSFP-82000/66
      额定容量2×41 MVA
      额定电压66/1.05 kV
      额定电流2×358.7/2×22 501
      联接组标号网侧延边三角形移相±15°,阀侧角形联接
      短路阻抗15.63%
      空载电流0.27%
      空载损耗53.09 kW
      负载损耗423.92 kW
      额定空载直流电压1 420 V
      额定直流电流2×27 500 A
      总脉波数12

      Table 1.  ITER PF transformer related parameters

      为了简化计算,采用“一”型等效电路,只考虑变压器的励磁支路和负载支路。对应的参数(66 kV侧)关系如下:

      电阻${R_{\text{T}}} = \dfrac{{\Delta {P_{\text{k}}}{U_{{\text{1N}}}}^2}}{{{S_{{\text{TN}}}}^2}} = \dfrac{{423.92 \times {{66}^2}}}{{{{41\;000}^2}}} = 1.098\;5\;\Omega$,电抗${X_{\text{T}}} = \dfrac{{{U_{\text{k}}}\% }}{{100}} \times \dfrac{{{U_{\text{N}}}^2}}{{{S_{\text{N}}}}} = \dfrac{{15.63}}{{100}} \times 106.24 = 16.605\;3\;\Omega$,故对应变压器阻抗${Z_{\text{T}}} = {R_{\text{T}}} + j{X_{\text{T}}} = (1.098\;5 + 16.605\;3j)\Omega$

      ITER无功补偿及滤波系统测试平台安装容量168 Mvar,基波补偿容量83.2 Mvar,电压等级66 kV,等效电阻${R_{\text{L}}} = 0.131\;5\;\Omega $,滤波支路基波下的等效电抗为:

      $$ {X_{\text{C}}} = \dfrac{{{U_{\text{N}}}^2}}{{{S_{{\text{1N}}}}}} = \dfrac{{{{66}^2}}}{{83.2}} = 52.350\;7\;\Omega $$

      其它等效参数如表2所示。

      滤波支路等值参数3支路5支路7支路11支路
      调谐电阻/Ω1.378 90.359 60.422 70.460 1
      电感/mH137.4424.7412.625.11
      电容/μF8.357 216.714 416.714 416.714 4

      Table 2.  Equivalent parameters of filter branch

    • 相控电抗器为三角形连接,在实验过程中,由于容性滤波支路始终投入,那么随着负载电流的变化,TCR提供的补偿容量将在0~83.208 Mvar之间变化。计算过程中,将TCR等效为一个三相星形连接的可变电感,对应的基波等效阻抗${X_{{\rm{TCR}}}} = 52.350\;7\;\Omega \sim \infty$

      以ITER PF变流器集成试验的并联谐振等值回路进行分析,如图5所示。

      Figure 5.  Parallel resonant equivalent circuit for ITER PF converter integration test

      ITER PF变流器等效为谐波源,忽略主变压器励磁阻抗,等值电路进一步简化为图6所示电路。

      Figure 6.  Simplified parallel resonant equivalent circuit of ITER pf converter

      将相关等值参数折算到66 kV侧,得:${R_{\text{S}}}' = \dfrac{{1.434\;4 + 0.155\;8}}{{{{(115/66)}^2}}} = 0.523\;8\;\Omega$${X_{\text{S}}}' = \dfrac{{14.343\;8 + 15.830\;3}}{{{{(115/66)}^2}}} = 9.938\;6\;\Omega$${R_{{\text{TCR}}}} = 0.26\;\Omega$$ {X_{{\text{TCR}}}} = $$52.350\;7\;\Omega$${R_{\text{L}}} = 0.131\;5\;\Omega$${X_{\text{C}}}' = 52.350\;7\;\Omega$,结合式(10)可以求解出$ {Z_{{\text{ac}}2}} $的值为:

      $$ \begin{split} &{Z_{{\text{ac2}}}} = [ {({Z_{\text{S}}}'//{Z_{{\text{lb2}}}}) + {Z_{{\text{T2}}}}} ]/{k^2} = \dfrac{1}{{{{62.86}^2}}} \times \{ [ {(0.523\;8 + }\\& 9.938\;6j)//(0.26 + 52.350\;7j)// {(0.131\;5 + 52.350\;7j)} ] + \\&{(0.545\;2 + 16.605\;3j)} \} = (0.282\;5 + 7.206\;9j)\Omega \end{split} $$
    • 根据表1主变参数的参数可知:触发角$\alpha = 15^\circ$,按网侧参数来折算,根据触发角$\alpha $和换相角$\mu $的关系,列出对应的关系如下:

      $$ \cos \alpha - \cos (\alpha + \mu ) = {K_{\text{I}}}\frac{{2{I_{\text{d}}}{X_{\text{B}}}}}{{\sqrt 6 E}} $$ (11)

      式中:

      ${K_{\text{I}}}$ ——电容性元件对换相过程影响的修正系数,其取值主要受到特征次谐波的影响,考虑到此处分析的是二次谐波问题,故电容性元件对换相过程影响可忽略不计,所以${K_{\text{I}}} = 1$

      ${I_{\text{d}}}$ ——网侧电流(A);

      ${X_{\text{B}}}$ ——换相电抗(Ω);

      E ——网侧电压(kV)。

    • 折算到66 kV侧,对应的各电抗值如下:

      系统等效电抗${X_{\text{S}}} = 14.348\;8\;\Omega$,主变压器电抗${X_{\text{T}}} = 16.605\;3\;\Omega$,交流母排(包括封闭母线)${X_{{\text{bus}}}} = 1.489\;5\;\Omega$,所以换相电抗的大小为${X_{\text{B}}} = 32.443\;6\;\Omega$

    • 根据表1主变参数中数据可知:电压$E = 66\;{\text{kV}}$${I_{\text{d}}} = 358.7\;{\text{A}}$。将相关值代入式(11)中可求解出μ的值。

    • 图7所示为直流侧等值谐波阻抗的等值电路,其中${Z_{{\text{L}}(m)}}$${Z_{{\text{E}}(m)}}$${Z_{{\text{S}}(m)}}$分别为直流线路、对侧换流器的直流侧和平波电抗器的等值m次谐波阻抗(m为正整数)。

      Figure 7.  DC side equivalent harmonic impedance circuit model

      图7可知:直流侧等值阻抗${Z_{{\text{dc}}1}} = {Z_{{\text{E}}1}} + 2{Z_{{\text{S}}1}} + {Z_{{\text{L}}1}}$。需要说明的是,由于超导电感非常大,可以认为${Z_{{\text{L}}1}} \gg {Z_{{\text{S}}1}}$${Z_{{\text{L}}1}} \gg {Z_{{\text{E}}1}}$。设${L_1} = 1{\text{H}}$,从而得到直流侧阻抗${Z_{{\text{dc}}1}}$的值。

    • 综合上述计算,对应的各参数值如表3所示。

      换相角μ17.6983°
      交流侧二次谐波阻抗${Z_{{\text{ac}}2}}$$(0.282\;5 + j7.206\;9)\;\Omega$
      直流侧基波阻抗${Z_{{\text{dc}}1}}$$j314.159\;\Omega$

      Table 3.  Relevant numerical calculation results of ITER PF transformer

      结合图4可知:其会经过一正反馈通道输出一对应的二次谐波电压,该二次谐波电压为${\stackrel{·}{U}}_{{{\rm{ac}}2+}'}$对应的关系式为:

      $$ {\stackrel{·}{U}}_{{\text{ac}2}+'}=36{\stackrel{·}{I}}_{\text{ac}2}{Z}_{\text{ac}2}{}^{2}{\mathrm{sin}}^{2}\dfrac{\mu }{2}/k{\text{π}}^{2}{\mu }^{2}{Z}_{\text{dc}1} $$ (12)

      式中:

      ${I_{{\text{ac}}2}}$ ——二次谐波电流(A);

      ${Z_{{\text{ac}}2}}$ ——交流侧二次谐波阻抗(Ω);

      $\mu $ ——换相角(°);

      k ——变压器变比;

      ${Z_{{\text{dc}}1}}$ ——直流侧阻抗(Ω)。

      表3中数值代入上式,整理可求出${\stackrel{·}{U}}_{{\text{ac}2}'}$

    • $\lambda $为换流变压器铁心饱和型谐波不稳定的稳定因子,当$\lambda > 1$时,扰动将随着时间增大而增大,系统出现不稳定;反之,即$0 < \lambda < 1$时,扰动会随着时间而衰减,系统最终趋于稳定。谐波电压$ {\stackrel{·}{U}}_{{\rm{ac}}2} $经变流器及饱和情况下的换流变压器在交直流两侧传变后,其对应的幅值增益$\lambda =\left|{\stackrel{·}{U}}_{{\text{ac}2}'}/{\stackrel{·}{U}}_{\text{ac2}}\right|=1.993\times {10}^{-10}$。这个结果几乎趋向于0,可以忽略不计。这说明,在ITER整流变压器系统中,因偏磁产生的扰动而导致的低次谐波可忽略不计。

    • ITER PF变流器在合肥集成试验,1、3正向或2、4反向6脉波变流桥并联55 kA稳定运行时,整流变压器阀侧电流波形采用30 kA罗科和数据采集仪(20 kHz采样频率)采集,使用Origin软件进行FFT分析仿真,频谱分析时间窗按照IEC标准采用200 ms,2次谐波电流值和波动较大。图8(a)为1、3桥并联55 kA稳定运行,假负载2串2并(5 mH)时整流变阀侧2次谐波电流,图8(b)为同时刻检测到的直流分量,蓝色为1桥,红色为3桥。表4中列出1、3桥2次谐波电流和直流分量5 s的平均值。

      Figure 8.  1、3 bridge second harmonic current component and DC current component

      桥编号2次谐波
      电流/A
      2次谐波电
      流相角/(°)
      直流分
      量/A
      直流分量
      相角/(°)
      1桥 73.14 无规律变化 176.55 180
      3桥 165.32 无规律变化 186.74 0

      Table 4.  5 seconds mean value of 2nd harmonic and DC components of 1、3 bridges

      图8可以看出1桥2次谐波电流最大达到近600 A且波动较大,3桥2次谐波电流最大仅为100 A且波动较小;可以看出1桥直流分量最大达到430 A且波动较大,3桥直流分量最大为280 A,波动相对较小。

      图9(a)为2、4桥并联55 kA稳定运行,假负载2串2并(5 mH)时整流变阀侧2次谐波电流,图9(b)为同时刻检测到的直流分量,蓝色为2桥,红色为4桥。表5中列出2、4桥2次谐波电流和直流分量5 s的平均值。

      Figure 9.  2、4 bridge second harmonic current component and DC current component

      桥编号 2次谐波
      电流/A
      2次谐波电
      流相角/(°)
      直流分
      量/A
      直流分量
      相角/(°)
      2桥188.48无规律变化183.32180
      4桥59.76无规律变化178.340

      Table 5.  5 seconds mean value of 2nd harmonic and DC components of 2、4 bridges

    • 本文发展了一种改进型的开关函数,并基于这种改进型开关函数建立了ITER电源系统的数学模型,进而推导出整流变压器直流偏磁所产生的二次谐波输出。从实验结果来看,尽管整流变阀侧出现了百安培量级的直流分量,但其与整流变压器阀侧的数万A的比值小于对应的允许值0.033,不足以引起整流变直流偏磁,因此不存在由直流偏磁形成正反馈从而导致二次谐波的恶性循环。ITER PF变流器实际运行时负载为最大亨量级的超导大电感,直流侧不会发生工频串联谐振,可以排除直流偏磁引起的谐波不稳定。

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