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考虑到实测布里渊谱满足的分布,故采用基于如下式所示的伪Voigt模型的最小二乘拟合法逼近布里渊谱,根据拟合结果来获得布里渊频移:
$ \begin{gathered} g_{\mathrm{B}}(\nu)=g_1 \frac{\left(\Delta \nu_{\mathrm{B}} / 2\right)^2}{\left(\nu-\nu_{\mathrm{B}}\right)^2+\left(\Delta \nu_{\mathrm{B}} / 2\right)^2}+ \\ g_2 \exp \left[\frac{-(4 \ln 2)\left(\nu-\nu_{\mathrm{B}}\right)^2}{\left(\Delta \nu_{\mathrm{B}}\right)^2}\right] \end{gathered} $
(1) 式中,νB为布里渊频移;ν为频率;gB为布里渊增益;ΔνB为线宽;g1和g2分别为洛伦兹和高斯模型布里渊增益峰值。目标函数如下式所示:
$ E=\sum\limits_{n=1}^N\left[g_{\text {в }}\left(\nu_n\right)-g_{\text {в, n }}\right]^2 $
(2) 式中,νn和gB, n分别为第n个扫频点和对应的布里渊增益测量值,E为增益误差的平方和,N为扫频点数。
光纤温度和应变与布里渊频移近似成线性关系,如果考虑到同一时刻架空线路上温度近似为一固定值,则应变可直接由布里渊频移计算,如下式所示:
$ \varepsilon=\left(\nu_{\mathrm{B}}-\nu_{\mathrm{B}, 0}\right) / C_{\nu, \varepsilon} $
(3) 式中, νB, 0为温度固定且无应变下的布里渊频移;Cν, ε为布里渊频移的应变系数,典型值为20 με/MHz;ε为应变。由以上分析可知,根据布里渊频移误差可直接获得应变误差,后续的研究多侧重于布里渊频移误差。考虑到Levenberg-Marquardt算法适合于求解以上的非线性最小二乘问题,选择它来优化(2)式。
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与谱拟合法不同,单斜坡法仅需要获得工作点对应的布里渊增益,无需耗时的扫频过程。设ν0为工作点频率,取光纤沿线布里渊谱的真实参数,即光纤上布里渊频移和线宽的平均值分别为10.83 GHz和149.57 MHz,g1=3.91×10-4,g2=1.67×10-3,ν0分别选择为10.755 GHz和10.905 GHz,且分别位于布里渊频移的左侧和右侧,布里渊频移与工作点增益满足如图 2所示的关系。需要注意的是,实际使用时工作点可以取在布里渊频移的左侧,也可以取在布里渊频移的右侧。这样根据实测的工作点布里渊增益及以上关系即可获得布里渊频移。
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根据第3节中单斜坡法原理可知,工作点与布里渊频移差距及工作点增益的信噪比会影响单斜坡法的准确性,本节中对该问题开展研究。实测布里渊谱模型及噪声服从的模型与仿真采用的模型必然存在差距, 所以采用实测谱数据的研究结果具有更高的可靠性。但由于实测布里渊谱对应参数的精确值未知,如何可靠确定信噪比等参数非常关键。伪Voigt模型在不同入射光脉宽下均可以有效逼近实测布里渊谱,准确性高于洛伦兹模型和高斯模型。因此,本文中通过基于伪Voigt模型的谱拟合法可以获得比较准确的布里渊频移和布里渊谱参数。基于这些数据得到不同工作点与布里渊频移之差下, 信噪比对布里渊频移误差的影响如图 4所示。不同信噪比下工作点与布里渊频移之差对布里渊频移误差的影响如图 5所示。
图 4 工作点增益信噪比对布里渊频移误差的影响
Figure 4. Influence of SNR of working point gain on Brillouin frequency shift error
图 5 布里渊频移与工作点频率之差对布里渊频移误差的影响
Figure 5. Influence of the difference between working point frequency and Brillouin frequency shift on Brillouin frequency shift error
由图 4可知,随着工作点增益信噪比的增加,单斜坡法算得布里渊频移误差逐渐下降。根据二者的关系,采用如下式所示的指数规律进行拟合,拟合结果一并画于图 4中。
$ E_\nu=a \exp (b R) $
(4) 式中,Eν为布里渊频移误差(MHz);a和b为待优化变量;R为信噪比(dB)。
显然,单斜坡法算得布里渊频移误差随工作点增益信噪比增加近似成指数规律下降。由图 5可知,不同信噪比下,随着布里渊频移与工作点频率之差的增加,布里渊频移误差先快速下降,达到谷值后略有增加。基本上当0.55 < (νB-ν0)/ΔνB < 0.95时, 布里渊频移误差取到最小值。为了更加准确获得最佳工作点,根据以上所有数据进行整理获得了工作点与布里渊频移之差与单斜坡法平均误差的关系, 如图 6所示。
图 6 工作点频率对布里渊频移误差的影响
Figure 6. Influence of working point frequency on Brillouin frequency shift error
图 6中工作点频率对误差的影响与图 5类似。由图 6可知,工作点要满足(5)式,这样单斜坡法算得布里渊频移误差接近最小值。
$ \left(\nu_{\mathrm{B}}-\nu_0\right) / \Delta \nu_{\mathrm{B}}=0.5 $
(5) 由于光纤沿线的布里渊频移会变化,而工作点频率应该是一个固定值,因此工作点应该满足下式:
$ \nu_0=\overline{\nu_{\mathrm{B}}}-\Delta \nu_{\mathrm{B}} / 2 $
(6) 式中,$\overline{\nu_{\mathrm{B}}}$为光纤上布里渊频移的平均值。需要注意的是,(6)式中工作点频率始终小于布里渊频移,如果将式中的“-”改为“+”,则工作点频率始终大于布里渊频移。
综上所述,图 3c中两种算法计算结果差距较小,是因为光纤头部测得布里渊谱信噪比较高,此时两种算法的准确性均较高,因此二者算得布里渊频移比较接近。图 3d中两种算法计算结果差距较大,是因为随着光纤位置逐渐靠近尾端,测得布里渊谱信噪比逐渐下降,两种算法的误差均逐渐增大,尤其是单斜坡法,故二者差距逐渐增大。
基于斜坡法的智能化架空线路应变快速感知
A fast strain sensing for intelligent overhead line based on slope-assisted technique
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摘要: 为了提高输电线路状态感知实时性, 将单斜坡法和基于伪Voigt模型的最小二乘谱拟合法用于架空线路复合光纤应变的测量, 基于实测布里渊谱系统, 研究了工作点增益的信噪比和布里渊频移对单斜坡法准确性的影响。结果表明, 随着工作点增益信噪比增加, 布里渊频移误差近似成指数规律减小; 随着布里渊频移与工作点频率之差的增加, 布里渊频移误差先快速下降, 然后略有增加; 当工作点处布里渊增益的信噪比不小于25 dB且工作点频率始终小于(或大于)布里渊频移时, 单斜坡法应变误差小于60 με; 根据不同布里渊频移下60 με对应的临界信噪比, 可插值获得不同情况下对应的临界信噪比; 单斜坡法的谱测量时间和计算时间分别为谱拟合法的1/161和1/600左右。该工作为提高智能化光纤复合架空线路态势感知实时性奠定了基础。Abstract: In order to improve the real-time performance of state sensing, the slope-assisted techniqueand the least-squares spectrum fit method based on the pseudo-Voigt model were introduced into the strain measurement of optical fiber composite overhead lines. The programs about the least squares spectrum fitting method and the slope-assisted technique based on the pseudo-Voigt model were written and applied to the strain measurement along an optical fiber composite overhead line. Based on the measured Brillouin spectra, the influence of the signal-to-noise ratio (SNR) of the working point gain and the Brillouin frequency shift (BFS) on the accuracy of the slope-assisted technique was systematically investigated. The results reveal that the BFS error decreases exponentially with SNR. The BFS error decreases rapidly and then increases slightly with the difference between BFS and working point frequency. The strain error of the slope-assisted technique is less than 60 με when the SNR of Brillouin gain at the working point is not less than 25 dB and at the same time, the frequency at the working point is always less or larger than the BFS. In addition, the corresponding critical SNR of 60 με is presented for different BFSs, and the critical SNR corresponding to different cases can be obtained by interpolation. The measurement time and computation time of the slope-assisted technique are about 1/161 and 1/600 of that of the spectrum fitting method, respectively. The work provides a reference for improving the real-time performance of sate sensing of intelligent optical fiber composite overhead lines.
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图 3 单斜坡法与谱拟合法计算结果的比较
a—两种方法的计算结果b—两种方法的计算结果之差c—两种方法的计算结果(光纤头部) d—两种方法的计算结果(光纤尾部)
Figure 3. Comparison of calculation results between slope-assisted technique and spectrum fitting method
a—calculation results by two methods b—difference in calculation results by two methods c—calculation results by two methods of head of optical fiber d—calculation results by two methods of tail of optical fiber
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