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Model and experimental verification of SOP transient in OPGW based on direct strike lightning

Kaijing Hu, Fei Tong, Weihua Lian, and Wei Li

Open Access Open Access

Abstract

We introduce inductive lightning and direct lightning that affect the state of polarization (SOP) of signal light in optical ground wire (OPGW) of power transmission line, deduce the helical component of direct lightning on OPGW, and establish the transient model of signal SOP in OPGW based on direct lightning. We also monitored actual optical transport network (OTN) systems deployed on OPGW in thunderstorm-prone areas for three months, recorded error codes and SOP transients of more than 1.5 Mrad/s using a bit error rate tester and a polarimeter accordingly. The highest SOP transient rate was up to 43 Mrad/s. Finally, we verify the accuracy of the model through simulation and experiment comparison.

© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement

1. Introduction

OPGW, which integrates the function of fiber-optic communication and lightning protection, have been widely used in power utilities. As a kind of aerial link exposed to a dynamic environment, SOP of the transmitted optical signal in the fiber is expected to transient due to climatic conditions like wind or lightning strikes as shown in Fig. 1. Although the swing caused by wind can lead to relatively slow SOP rotations, and the lightning induced strong electromagnetic field can lead to very fast SOP rotations, both of which have no adverse impact on the early generations of optical transmission systems based on intensity modulation and direct detection (IM-DD), which are polarization independent [1].

However, when higher modulation formats, polarization division multiplexing (PDM), and coherent detection are adopted to meet the demand for higher capacity in recent years, the fast SOP rotations becomes a non-negligible issue, because the conventional multiple-input-multiple-output (MIMO) equalization algorithm could not keep tracking it [2].

Previously, parts of the research on the SOP transient in OPGW were carried out in the laboratory. The results showed that the SOP transient was mainly caused by the Faraday effect [3], and that the speed of SOP transient, depended on the intensity of lightning current and the length of OPGW [4,5], could reach 7.727 Mrad/s. The other parts were mainly conducted in actual environments, and recorded the fastest the SOP transient of 5.1 Mrad/s [6].

For the theoretical study of SOP transients in OPGW, on the one hand, a model is established for inductive lightning [2], and on the other hand, a transmission line model is used to study the variation of lightning current on OPGW with transmission line [7]. At the same time, it is assumed that all lightning currents propagate along the spiral direction. Our model takes into account the lightning current on the OPGW: one part is the spiral component flowing along the spiral structure of the OPGW, and the other part is the direct component flowing directly from the surface of the OPGW, which is more in line with the actual situation [3].

In this paper, we proposed a SOP transient model of the signal light in OPGW based on the spiral component of direct lightning strikes on OPGW. we conducted a three-month monitoring of the SOP in the long-distance OPGW of the actual 100G-OTN systems in areas with frequent lightning strikes. We recorded the error rate and SOP transients exceeding 1.5 Mrad/s using a bit error rate tester and a polarimeter, and simultaneously used the lightning monitoring system to monitor lightning strikes within a radius of 2 kilometers of the OPGW in real-time. In the actual OTN system, we have monitored the maximum SOP transient reaching 43 Mrad/s, and the model can also match it.

 figure: Fig. 1.

Fig. 1. Lightning strike OPGW

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2. Principle analysis

2.1 Inductive lightning and direct lightning

2.1.1 Inductive lightning

Inductive lightning stroke refers to lightning stroke caused by Electrostatic induction of thunderstorm clouds or electromagnetic induction during discharge. It is characterized by that the place where electronic equipment is punctured or burned is far away from the place where lightning strike occurs, ranging from hundreds of meters to several kilometers or tens of kilometers.

For the Faraday effect produced by the induced lightning stroke on the optical fiber, the SOP transient of the signal light in the optical fiber can be expressed as [2]

$$\theta \textrm{ = }\frac{{V{\mu _0}I}}{2}$$
where V represents the Verdet constant of the optical fiber as 0.53 rad/(T·m), and µ0 represents the vacuum dielectric constant as 4π×10−7 (T·m)/A, I represents the magnitude of the induced current. From the above formula, it can be seen that the polarization rotation angle is only related to the magnitude of the induced lightning current and is not related to other factors. Assuming that the rise time of lightning current is 1µs. At this point, the SOP transient rate is 0.166 Mrad/s. Through calculation, it can be seen that the induced lightning strike has little effect on the SOP of the signal light in OPGW, which is also consistent with the actual situation.

2.1.2 Direct lightning

Direct lightning stroke refers to the discharge phenomenon between the charged cloud layer and a point on the ground, that is, the lightning directly acts on the OPGW, which is far more harmful than the inductive lightning stroke. The peak value of lightning current can reach tens of kA to hundreds of kA. The impact of direct lightning stroke on the SOP of signal light in the OPGW is far greater than the inductive lightning.

At present, there are two kinds of lightning current models in common use, which are double exponential model $i(t) = \frac{{{i_0}}}{\xi }({e^{ - \frac{\tau }{{{\tau _2}}}}} - {e^{ - \frac{\tau }{{{\tau _1}}}}})$ and Heidler model $i(t) = \frac{{{i_0}}}{\eta }\frac{{{{(\frac{t}{{{\tau _1}}})}^n}}}{{1 + {{(\frac{t}{{{\tau _1}}})}^n}}}{\textrm{e}^{ - \frac{t}{{{\tau _2}}}}}$ [8,9]. Where i0 represents the peak current, τ1 and τ2 represent the wave head and tail constants, respectively, ξ and η represent the correction factors. In the above two lightning current models, the double exponential function model is the most used one, but the first derivative of its initial time is not 0, which is inconsistent with the actual situation. The current waveform of Heidler function model is close to the observed current waveform, which can reflect the characteristic value of lightning current well, but its function model does not have an analytical integral, which is not ideal for the analytical integration presented in this work.

Aiming at the shortcomings of these two models, an improved lightning current model [10] is applied. The mathematical expression of this model is as follows:

$$I(0,t) = \frac{{{I_0}}}{\varepsilon }{[1 - exp(\frac{t}{{{\tau _1}}})]^n}exp( - \frac{t}{{{\tau _2}}})$$
where I0 is the peak return current at the bottom of the channel, τ1 and τ2 is still the wave head time constant and wave tail time constant, with n being the order, ε is the correction factor.

When the value of n is not too large, this model is very close to the double Exponential function model, so it can be regarded as a modification of the double Exponential function model to a certain extent. The modified model not only overcomes the problem that the first derivative of the double exponential model is not zero at the initial time, but also is an integrable function, which overcomes the defects of the Heidler function model, Therefore, this model is ideal for the analytical integration presented in this work.

2.2 Analysis of OPGW structure and generated magnetic field

A typical OPGW cable consists of optical units and metal conductors. The internal optical unit contains dozens of optical fibers and a buffer tube with fillers to protect the fibers, while the external conductor serves as the ground wire. OPGW is generally divided into two types: central tube type and layer stranded type, as shown in Fig. 2. Metal conductors are wound in parallel layers at large intervals around the center. It is worth noting that for the central tube OPGW, the optical unit is located at the symmetrical center, but for the layered type, it is spirally twisted around the central layer.

 figure: Fig. 2.

Fig. 2. OPGW cables:(a) central tube type; (b) layer-stranded type; (c) cross section of the OPGW used in the work. The left one is central tube type and the right one is layer-stranded type.

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When direct lightning strikes the OPGW, due to the twisted structure outside the OPGW, the lightning current will be divided into two parts: one is the spiral component flowing along the spiral structure of the OPGW, and the other is the direct component flowing directly from the surface of the OPGW. We use COMSOL software to create a new model wizard, select the three-dimensional space dimension for solving and calculating, use the built-in geometry module for geometric modeling of OPGW, use the electrostatic module and current module to set the boundary conditions for the electric field model, set the magnetic field model, and perform grid division for the overall geometric structure, as shown in Fig. 3. After completing the grid generation, the physical fields created are solved successively, and the electric and magnetic field distributions are obtained. When lightning current flows through OPGW in a helical direction, a magnetic field parallel to the OPGW axis will be generated inside, as shown in Fig. 4.

 figure: Fig. 3.

Fig. 3. Mesh of OPGW

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 figure: Fig. 4.

Fig. 4. Simulation of magnetic field distribution in OPGW under direct lightning strike. The red arrow indicates the direction of the magnetic field. The direction of the arrow in the right image is perpendicular to the surface of the paper.

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Earlier studies have shown the time delay between lightning currents measured at different distances along transmission lines. The waveforms of lightning currents are not the same for different transmission line [11,12]. In the case of lightning strike OPGW, the most critical factor affecting the error code is that the lightning strike just falls on the point of OPGW, at this time, the lightning current is the largest, resulting in the fastest transient speed of SOP, which is easy to cause error code. In order to simplify the analysis, we do not consider the time delay between the lightning currents measured at different distances, and assume that the lightning current does not change with distance, but only with its own characteristics, and the magnitude of the lightning current struck at that point shall prevail.

According to the distribution characteristics of lightning current on OPGW during lightning stroke, OPGW can be equivalent to the parallel model of resistance and inductance [13]. The current flowing through inductance represents the spiral component of lightning current, and the current flowing through resistance represents the direct component of lightning current, as shown in Fig. 5.

 figure: Fig. 5.

Fig. 5. Equivalent Circuit Model of OPGW

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Assuming that the direct lightning current is known, it can be considered as a current source. If the equivalent resistance R and equivalent inductance L of OPGW are known, the spiral current component IL can be calculated.

Combined with circuit knowledge, the main circuit voltage can be obtained through Laplace transform:

$$U(s) = R{I_R}(s) = sL{I_L}(s)$$
$$I(s) = {I_R}(s) + {I_L}(s)$$

Combine the above two equations:

$${I_L}(s) = I(s)\frac{R}{{R + sL}}$$

According to the nature of Laplace transform, the corresponding time domain is:

$${I_L}(t) = I(t) \ast \frac{{Ru(t)}}{L}{\textrm{e}^{ - \frac{{Rt}}{L}}}$$
where * represents convolution, u(t) is Heaviside step function, τ= R/L represents the time constant of OPGW, reflecting its electrical characteristics under direct lightning strikes. Therefore, only the equivalent resistance R and equivalent inductance L of OPGW are required.

According to electrical knowledge, the inductance L of a solenoid can be expressed as:

$$L = \mu {n^2}V$$
µ is the average magnetic permeability of the solenoid, n = 1/d is the number of turns per unit length of the coil, and V is the volume of the solenoid, so there is:
$$L = \mu \frac{{\pi {r^2}l}}{{{d^2}}}$$

In the formula, r is the radius of the entire OPGW, and l is the length of the OPGW. According to geometric characteristics, the equivalent magnetic permeability of the wire part (aluminum clad steel + aluminum alloy) can be solved first µc. Reuse µ Calculate the equivalent magnetic permeability of the entire OPGW. Taking the OPGW structure in Fig. 6 as an example, analyze and calculate the process of average magnetic permeability. Where r is the outer radius of OPGW, r0 is the inner radius of OPGW, the diameter of the steel strand is Zs, and the thickness of the aluminum cladding is Za, µc and µ0 represents the magnetic permeability of steel and aluminum respectively, and the pitch of OPGW stranded wire is d.

 figure: Fig. 6.

Fig. 6. Cross section and longitudinal section of OPGW

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Magnetic permeability of steel wire µc>>µ0, then the average magnetic permeability of the aluminum clad steel wire µc is:

$${\mu _c} = \frac{{{Z_S} + {Z_a}}}{{\frac{{{Z_S}}}{{{\mu _S}}} + \frac{{{Z_a}}}{{{\mu _0}}}}} = {\mu _0}\left( {1 + \frac{{{Z_s}}}{{{Z_a}}}} \right)$$

The average magnetic permeability of OPGW is µ:

$$\mu = {\mu _c}\left( {\frac{{{r^2} - r_0^2}}{{{r^2}}}} \right) + {\mu _0}\frac{{r_0^2}}{{{r^2}}}$$

We take r = 5.7 mm, r0 = 2.5 mm, Zs = 1.34 mm, Za = 0.26 mm, and take it into the above equation to obtain µ≈5µ0. In this way, the inductance L can be solved. In order to solve for the current in the inductance, it is also necessary to know the size of the resistance R. Due to the variation of lightning current over time, there is a high-frequency component (which can be obtained by Fourier transform of the time-domain waveform of lightning current). Therefore, there is a skin effect in the current in the conductor, and the current is not uniformly distributed on the cross-section of the conductor, but concentrated on the surface, resulting in an increase in the equivalent resistance. Skin depth δ Given by the following equation:

$$\delta = \sqrt {\frac{{2\rho }}{{{\mu _0}\omega }}} $$

In the equation ρ is the resistivity of the conductor, ω is the Angular frequency of the current. So the equivalent resistance R of OPGW is:

$$R = \frac{{\rho l}}{{2\pi r\delta }}$$

By introducing the equivalent resistance and inductance into the OPGW time constant, it can be obtained that:

$$\tau = 4{\pi ^3}\frac{{{\mu ^2}}}{{{\mu _0}}}\frac{{{r^6}}}{{{d^4}\rho }}$$

Taking r = 5.7 mm, d = 16.5 cm, the conductivity of aluminum is 2.83 × 10−8 Ω/m, the equivalent structural time constant of OPGW can be obtained as τ=6.4µs. By incorporating the improved lightning current model into Eq. (6), the expression of the spiral lightning current component can be obtained as follows:

$$\begin{aligned} \textrm{ }{I_L}(t) &= \frac{{2{l_0}{\tau _1}{\tau _2}exp( - \frac{{t({\tau _1} + {\tau _2})}}{{{\tau _1}{\tau _2}}})}}{{\tau {\tau _1} + \tau {\tau _2} - {\tau _1}{\tau _2}}} - \frac{{{l_0}{\tau _1}{\tau _2}exp(\frac{{t({\tau _1} + 2{\tau _2})}}{{{\tau _1}{\tau _2}}})}}{{\tau {\tau _1} + 2\tau {\tau _2} - {\tau _1}{\tau _2}}} - \frac{{{l_0}{\tau _2}exp(\frac{t}{{{\tau _2}}})}}{{\tau - {\tau _2}}}\\ & + \frac{{2{l_0}{\tau ^2}\tau _2^3exp(\frac{{t({\tau _1} + {\tau _2})}}{{{\tau _1}{\tau _2}}})}}{{(\tau - {\tau _2})(\tau {\tau _1} + \tau {\tau _2} - {\tau _1}{\tau _2})(\tau {\tau _1} + 2\tau {\tau _2} - {\tau _1}{\tau _2})}} \end{aligned}$$

It can be seen from the above formula that the amplitude of the spiral current is correspondingly lower than that of the original lightning current, but the duration will be prolonged. The component of the spiral current is jointly determined by the equivalent time constant of the OPGW, the wave head time and the wave tail time of the lightning current, that is, when the lightning current waveform is fixed, the spiral current waveform is determined by the time constant of the OPGW τ inductance structure, and τ is related to the parameters of OPGW itself. As shown in Fig. 7, the wave head constant of lightning current τ1 = 2µs and τ2 = 10µs, order n = 2. When the peak current is 226 kA, the waveform of direct lightning strike and spiral current are plotted.

 figure: Fig. 7.

Fig. 7. Lightning current waveform and spiral current waveform

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We equate the spiral current to a combination of several annular currents, and perform a simple analysis of the magnetic field generated at the center, where R represents the radius of the annular current and r represents the distance from the annular current to a point x on the central axis, θ Represents the angle between the magnetic field dB and the y-axis, as shown in Fig. 8.

 figure: Fig. 8.

Fig. 8. Schematic diagram of ring current

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We divide the annular current into several small segments, one of which dl generates a magnetic field at point x as follows:

$$dB = \frac{{{\mu _0}}}{{4\pi }}\frac{{Idl}}{{{r^2}}}$$

From geometric symmetry, it can be seen that:

$${B_y} = {B_z} = 0$$

The magnetic field generated by the entire annular current at x point can be expressed as:

Combining Faraday rotation effect θ=VBL and definite integral knowledge:

$$\mathop \int \nolimits_{ - \infty }^\infty \frac{1}{{{{({R^2} + {x^2})}^{3/2}}}}dx = \frac{2}{{{R^2}}}$$

We can derive:

$$\theta = \mathop \int \nolimits_{ - \infty }^\infty V{B_x}dx = \frac{{V{\mu _0}I{R^2}}}{2}\mathop \int \nolimits_{ - \infty }^\infty \frac{1}{{{{({R^2} + {x^2})}^{3/2}}}}dx = V{\mu _0}I$$

If there are N turns of ring current, the polarization rotation angle is:

$$\theta = NV{\mu _0}I$$

N is related to the pitch of the OPGW wire (number of turns per unit distance) as well as on the length of the interaction region [7]. In summary, whether it is induced lightning or direct lightning, the magnetic field component that affects the SOP of light in OPGW is the same as the direction of light propagation, and the magnetic field component is generated by lightning current. Therefore, the biggest factor affecting the polarization rotation angle is the spiral current on the OPGW.

2.3 Optical fiber transmission model in OPGW

The influence of the Faraday rotation effect caused by lightning strikes on the SOP of signal light in optical fibers has been concluded from both theoretical [14] and experimental [2], that is, it exhibits pure circular birefringence; Reflecting the Faraday rotation effect on the Poincaré sphere, the Stokes vectors S1 S2 change, S3 remains unchanged, and the trajectory of the SOP rotates around the equator. The Jones matrix can be expressed as:

$$R[\theta (I)] = \left( {\begin{array}{{cc}} {\cos \theta (I)}&{ - \sin \theta (I)}\\ {\sin \theta (I)}&{\cos \theta (I)} \end{array}} \right)$$
θ indicates the change in polarization rotation angle caused by lightning strikes, typically in the Mrad/s order.

However, in actual power OPGW, the SOP of the signal light in the fiber is not solely affected by the Faraday rotation effect. The natural birefringence of optical fibers, the shaking of OPGW caused by wind oscillation, the deformation of OPGW caused by icing, and temperature changes caused by sunlight can all affect the SOP of signal light in OPGW, including circular birefringence and linear birefringence, namely elliptical birefringence. So the Jones matrix of three parameters can be used to represent its changes:

$$U(t) = \left[ {\begin{array}{{cc}} {cos\kappa (t){e^{j\xi (t)}}}&{ - sin\kappa (t){e^{j\eta (t)}}}\\ {sin\kappa (t){e^{ - j\eta (t)}}}&{cos\kappa (t){e^{ - j\xi (t)}}} \end{array}} \right]$$
κ,ξ,η represent three independent polarization rotation angles. The polarization rotation caused by wind pendulum and stress is generally at a rate of krad/s, which is much smaller than the polarization rotation caused by lightning strikes.

The SOP transmission model of signal light in OPGW based on direct lightning strike not only includes the model of rapid change of SOP in a very short time caused by direct lightning strike, but also includes the model of slow change of SOP in OPGW under natural conditions. It is combined with polarization mode dispersion and phase noise into the nonlinear Schrödinger equation of optical fiber transmission to get the final model.

The transmission process of optical signal in an ideal optical fiber can be given by the nonlinear Schrödinger equation (NLSE), and its most general form is:

$$\frac{{\partial A}}{{\partial T}} + \frac{{i{\beta _2}}}{2}\frac{{{\partial ^2}A}}{{\partial {T^2}}} + \frac{\alpha }{2}A = i\gamma {|A |^2}A$$
where A is the pulse envelope, α, β2 and γ are the fiber loss coefficient, group velocity dispersion and nonlinear coefficient respectively, z is the direction of pulse propagation, and T = t−β1z is the introduced delay system.

The transmission of light pulses in elliptical birefringence can be modified from the above equation, as shown in

$$\begin{array}{l} \frac{{\partial {A_x}}}{{\partial z}} + {\beta _{1x}}\frac{{\partial {A_x}}}{{\partial T}} + \frac{{i{\beta _2}}}{2}\frac{{{\partial ^2}{A_x}}}{{\partial {T^2}}} + \frac{\alpha }{2}{A_x} = i\gamma ({|{{A_x}} |^2} + B{|{{A_y}} |^2}){A_x}\\ \frac{{\partial {A_y}}}{{\partial z}} + {\beta _{1y}}\frac{{\partial {A_y}}}{{\partial T}} + \frac{{i{\beta _2}}}{2}\frac{{{\partial ^2}{A_y}}}{{\partial {T^2}}} + \frac{\alpha }{2}{A_y} = i\gamma ({|{{A_y}} |^2} + B{|{{A_x}} |^2}){A_y} \end{array}$$
$B = \frac{{2 + 2si{n^2}\theta }}{{2 + 2co{s^2}\theta }}$, θ Is the elliptical angle of an elliptical birefringent fiber.

For such transmission equations, without considering nonlinearity, taking unmodulated signal light as an example, we can simplify the overall channel model [9] as shown in Fig. 9. The meaning of xi signal representation is shown in Table 1.

 figure: Fig. 9.

Fig. 9. Simplified Channel Model

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Tables Icon

Table 1. The meaning represented by the xi signal

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1) The transmission baseband signal x(t) undergoes RSOP1 action in the time domain

For RSOP1 under other natural factors, because it is generated in the time domain, we use the three-parameter Jones matrix in the time domain to simulate its action, and multiply the baseband signal x(t) in the time domain to get:

$${x_1}(t) = U(t)x(t) = \left( {\begin{array}{{cc}} {cos\kappa (t){e^{j\xi (t)}}}&{ - sin\kappa (t){e^{j\eta (t)}}}\\ {sin\kappa (t){e^{ - j\eta (t)}}}&{cos\kappa (t){e^{ - j\xi (t)}}} \end{array}} \right)x(t)$$

2) The transmission signal x1(t) undergoes PMD action in the frequency domain

For the polarization mode dispersion (PMD) in optical fibers, as it occurs in the frequency domain, we first perform Fourier transform (FFT) on it to transform the signal into the frequency domain:

$${x_2}(\omega ) = FFT[{x_1}(t)]$$

For high-order polarization mode dispersion, it can be equivalent to the total polarization mode dispersion of a long fiber composed of multiple short fiber cascades. The birefringence generated by each short fiber segment can be considered as uniform birefringence. The differential group delay (DGD) generated by the i-th fiber is Δτi. Each short fiber segment can be regarded as a phase retarder, and the Jones transmission matrix of phase delay can be expressed as:

$${J_i}(\omega ,\mathrm{\Delta }{\tau _i}) = \left( {\begin{array}{{cc}} {\exp (j(\frac{{\omega \mathrm{\Delta }{\tau_i}}}{2}))}&0\\ 0&{\exp ( - j(\frac{{\omega \mathrm{\Delta }{\tau_i}}}{2}))} \end{array}} \right)$$

The direction of the main state can be represented by a Rotation matrix:

$${D_i}({\alpha _i},{\delta _i}) = \left( {\begin{array}{{cc}} {cos{\alpha_i}}&{ - {e^{ - j{\delta_i}}}sin{\alpha_i}}\\ {{e^{j{\delta_i}}}sin{\alpha_i}}&{cos{\alpha_i}} \end{array}} \right)$$

By combining the phase delay matrix with the principal state rotation, the Jones transfer matrix of polarization mode dispersion in the i-th segment of the fiber is obtained as

$${U_i} = D_i^{ - 1}({\alpha _i},{\delta _i}){J_i}(\omega ,\mathrm{\Delta }{\tau _i}){D_i}({\alpha _i},{\delta _i})$$

The Jones transmission matrix of the total polarization mode dispersion of the whole length fiber composed of N short fibers is:

$${U_{PMD}}(\omega ) = \mathop \prod \limits_{i = 1}^N D_i^{ - 1}({\alpha _i},{\delta _i}){J_i}(\omega ,\mathrm{\Delta }{\tau _i}){D_i}({\alpha _i},{\delta _i})$$

Total length of the entire fiber optic cable Δτ with for each small fiber segment Δτi the relationship between is:

$$< \mathrm{\Delta }{\tau ^2} > = \mathop \sum \limits_{i = 1}^N < \mathrm{\Delta }\tau _i^2 > $$

At this point, the transmission signal through PMD is:

$${x_3}(\omega ) = {U_{PMD}}(\omega ){x_2}(t) = \mathop \prod \limits_{i = 1}^N D_i^{\textrm{ - }1}({\alpha _i},{\delta _i}){J_i}(\omega ,\mathrm{\Delta }{\tau _i}){D_i}({\alpha _i},{\delta _i}){x_2}(\omega )$$

3) The transmission signal x3(t) undergoes RSOP2 action in the time domain

For RSOP2 caused by lightning current, it is also generated in the time domain. We then change the signal from the frequency domain back to the time domain, that is, perform an inverse Fourier transform on the signal x3(t):

$${x_4}(t) = IFFT[{x_3}(\omega )]$$

The RSOP Jones matrix caused by lightning current can be expressed as:

$$R[\theta (t)] = \left( {\begin{array}{{cc}} {\cos \theta (t)}&{ - \sin \theta (t)}\\ {\sin \theta (t)}&{\cos \theta (t)} \end{array}} \right)$$

The transmission signal of RSOP2 caused by lightning current can be represented as:

$${x_5}(t) = \left( {\begin{array}{{cc}} {\cos \theta (t)}&{ - \sin \theta (t)}\\ {\sin \theta (t)}&{\cos \theta (t)} \end{array}} \right){x_4}(t)$$

The mathematical expression for the total process can be expressed as:

$$\begin{aligned} q(t) &= {U_{RSOP2}}IFFT\{ {U_{PMD}}FFT\{ {U_{RSOP1}}x(t)\} \} \\ &\textrm{ = }\left( {\begin{array}{{cc}} {\cos \theta (t)}&{ - \sin \theta (t)}\\ {\sin \theta (t)}&{\cos \theta (t)} \end{array}} \right)IFFT\{ \mathop \prod \limits_{i = 1}^N D_i^{\textrm{ - }1}({\alpha _i},{\delta _i}){J_i}(\omega ,\mathrm{\Delta }{\tau _i}){D_i}({\alpha _i},{\delta _i})\\ &\textrm{ }FFT\{ \left( {\begin{array}{{cc}} {cos\kappa (t){e^{i\xi (t)}}}&{ - sin\kappa (t){e^{j\eta (t)}}}\\ {sin\kappa (t){e^{ - j\eta (t)}}}&{cos\kappa (t){e^{ - j\xi (t)}}} \end{array}} \right)x(t)\} \end{aligned}$$

3. Simulation comparison

In order to verify the correctness of the signal light SOP transient model in OPGW based on direct strike lightning, we simulated it. Figure 10 shows the simulation platform built by us. The 1550 nm wavelength polarized signal light generated by the DFB laser enters the OPGW fiber, and without considering the loss and nonlinearity, the SOP is affected by CD, RSOP1, PMD and RSOP2, and finally the SOP is monitored by the polarimeter. In the whole simulation process, the total fiber length is 100 km, the PMD coefficient is 0.1ps/$\sqrt {\textrm{km}}$, and the fiber group velocity dispersion is -16.9 × 10−24s2 /km. The polarization rotation speed of RSOP1 is set to fluctuate around 1 Mrad/s, RSOP2 is a single lightning strike, the peak current is 227 kA, the duration is 40µs, the whole SOP monitoring time is 100µs, and the lightning starts to act from 20µs.

 figure: Fig. 10.

Fig. 10. Simulation platform diagram

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We also analyze the simulation from four aspects:

(1) Degree of polarization (DOP)

The signal light used in the simulation is fully polarized light, with a polarization degree of always 1.

(2) Stokes vector curve with time

As can be seen from Fig. 11, the three Stokes vectors change relatively slowly before 20µs when there is no lightning strike. Between 20µs and 60µs, the three Stokes vectors change rapidly with time. At the end of the lightning strike, after 60µs, the three Stokes vectors returned to a slow change.

 figure: Fig. 11.

Fig. 11. Stokes vector curve with time

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(3) Rotation rate curve with time

As can be seen from Fig. 12, when there is no lightning action, that is, between 0 and 20µs and after 60µs, the rotation rate is basically in the range of 1 Mrad/s, while when there is lightning action, that is, 20µs to 60µs, the peak rotation rate can reach 43 Mrad/s.

 figure: Fig. 12.

Fig. 12. Rotation rate curve with time

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(4) Stokes curves on the Poincare sphere

We can see that the trajectory curve on the Poincare sphere is chaotic and irregular.

4. Experimental investigations

We monitored the SOP of the actual power OTN medium and long distance OPGW cable in the 100G-OTN system for a period of three months in the thundery-frequented coastal areas of southern China, and monitored the OPGW with a total length of 154.3 km between the three 500 kV power transformer substation. We recorded the error rate and SOP transients exceeding 1.5 Mrad/s using a bit error rate tester and a polarimeter.

4.1. Experimental setup

We monitored the SOP propagated by the polarization light source generated by the bit error analyzer on the OPGW between the three power stations. The signal light started from site A, transmitted through the two sections of OPGW site B and OPGW site C, and then transmitted back to site A to record the relevant data for 24 hours, as shown in Fig. 14. The link uses standard single-mode fiber (G.652), mainly packaged in OPGW cable. The whole test is carried out in the framework of the real power system 100G-OTN network:

 figure: Fig. 13.

Fig. 13. Stokes curves on the Poincare sphere

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 figure: Fig. 14.

Fig. 14. Experimental installation drawing

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The polarized light source is a 10 G unmodulated signal generated by the SDH/OTN analyzer EXFO FTBx-88200NGE. The analyzer measures the quality of channel operation 24 hours a day and records the number of code errors and outages caused by lightning strikes. The polarimeter uses the Novoptel PM1000, which samples the Stokes vector at a rate of 125 MHz. The maximum Angle of change in the SOP of light measured on the Poincare sphere is π, so the maximum speed of change in the SOP measured by the polarimeter is 78.5 Mrad/s. To avoid interference due to changes in SOP due to non-lightning factors throughout the recording process, we set thresholds for the polarimeter. Data can only be recorded if the rate of change of the SOP exceeds 1.5 Mrad/s. The sampling interval is 40 ns, and the total recording time of a single lightning strike is 40.880µs. The data signal received by power station a is sent back to the headquarters computer terminal through the MSTP special line.

During the whole process, we recorded the lightning current size data within the 2 km diameter of the cable where the OPGW cable carrying OTN is located through the lightning strike detection system.

4.2. Experimental results

The polarimeter measuring instrument recorded a total of 131 groups of data with SOP transients of more than 1.5 Mrad/s during the three-month measurement period. We divided SOP rotation speed into 14 gears and plotted it in Fig. 15. with occurrence times as the vertical coordinate.

 figure: Fig. 15.

Fig. 15. SOP maximum rotation speed distribution diagram

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From Fig. 15, we can see that the SOP rotation speed caused by lightning is mainly concentrated between 2 Mrad/s and 3 Mrad/s, accounting for 26.7% of the total recording times, and the maximum SOP rotation speed even exceeds 40 Mrad/s.

During a 3-month test, the lightning monitoring system detected a total of 5175 lightning strikes within a radius of 2 kilometers of OPGW, as shown in Fig. 16. The magnitude of lightning current is mainly concentrated between 10-30 kA, accounting for 72.79% of the entire recording frequency. The probability of lightning current exceeding 100 kA is extremely low, and the maximum lightning current we have monitored can reach 379.3 kA.

 figure: Fig. 16.

Fig. 16. Lightning current size distribution diagram

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We made a more detailed analysis of this recorded data with SOP rotation speed of 43 Mrad/s, which is divided into four aspects:

(1) Degree of polarization

The degree of polarization reflects the proportion of the total intensity of the fully polarized light:

$$DOP = \frac{{\sqrt {S_1^2 + S_2^2 + S_3^2} }}{{S_0^2}}$$

We put the three parameters S1-S2-S3 collected by the polarimeter and S0, which represents the light intensity, into Eq. (36) to obtain the curve of polarization degree change with time during the sampling time:

As can be seen from Fig. 17, the variation range of its polarization degree DOP is not large, fluctuating between 0.99994 and 0.99995, and the main reason for the fluctuation may be caused by external noise during acquisition. Therefore, we can think that the polarization degree of the signal light is unchanged under the entire lightning strike, that is, the signal light is still fully polarized. This is consistent with the DOP of the light source in the simulation.

 figure: Fig. 17.

Fig. 17. Polarization degree over time curve

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2) Stokes vector curve with time

As can be seen from Fig. 18, when there is no lightning action, that is, before 23µs, the three Stokes vectors change relatively slowly; Under lightning, between 23 and 40µs, the three Stokes vectors change rapidly with time. It is basically consistent with Fig. 11.

 figure: Fig. 18.

Fig. 18. Stokes vector curve with time

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(3) Rotation rate curve with time

Since the sampling interval of the polarimeter is 40 ns, we make the difference between the front and back S1-S2-S3 vectors, and calculate the Angle between the two vectors, divide by the sampling interval, and obtain the curve of SOP rotation speed change with time at each moment, which is expressed by the equation as follows:

$$RSOP = \frac{{\arcsin \left( {\sqrt {{{({{S_1}(t + \mathrm{\Delta }T) - {S_1}(t)} )}^2} + {{({{S_2}(t + \mathrm{\Delta }T) - {S_2}(t)} )}^2} + {{({{S_3}(t + \mathrm{\Delta }T) - {S_3}(t)} )}^2}} } \right)}}{{\mathrm{\Delta }T}}$$
ΔT is the sampling time. As can be seen in Fig. 19, when there is no lightning strike, that is, between 0 and 23µs, the rotation speed is basically in the range of 0-1 Mrad /s; while when there is lightning strike, that is, between 23 and 40µs, the rotation speed of SOP rises sharply, forming multiple peaks, among which the highest peak can reach 43 Mrad/s. The time domain waveform of the lightning current of a single lightning strike generally rises abruptly and then gradually declines, and di/dt generally has two peaks, one large and one small, in the whole time period. For the occurrence of multiple peaks, we believe that it may be caused by lightning striking back several times in a very short time. It is basically consistent with Fig. 12. Due to the fact that the lightning current may have multiple returns, there is a certain error.

 figure: Fig. 19.

Fig. 19. Rotation rate curve with time

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(4) Stokes curves on the Poincare sphere

We represent the S1-S2-S3 of each sampling moment in three-dimensional space to obtain the variation curve of Stokes vector on Poincare sphere in Fig. 20. Firstly, it can be seen that all the points on the curve are on the Poincare sphere, which indicates that the signal light is basically polarized light, which is consistent with the conclusion obtained in the figure. Secondly, we can see that the trajectory curve on the Poincare ball is chaotic and irregular, while the rapid changes make the trajectory spread throughout the Poincare ball. It is basically consistent with Fig. 13.

 figure: Fig. 20.

Fig. 20. Stokes curves on the Poincare sphere

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In summary, the simulation is basically consistent with the actual situation of each reference figure, because the measurement may be caused by instruments or other reasons, the curve will have some jitter, the general trend is the same. It can be concluded that the model is in good agreement with the actual test.

5. Conclusion

In this paper, two types of lightning strikes affecting signal light in OPGW are analyzed, and the effect of direct lightning strikes is far greater than that of induced lightning. Then, the expression of polarization rotation Angle is obtained by solving the helical current component of direct lightning strike on OPGW, and the SOP transient model of signal light in OPGW based on lightning strike is proposed and simulated. In order to verify the accuracy of the model simulation, we monitored the SOP of the actual power OTN medium and long distance OPGW optical cable in the area with frequent lightning occurrence in the 100G-OTN system for a period of three months, and used the lightning monitoring system to monitor the lightning strike events within a radius of 2 kilometers of OPGW in real time. We recorded error codes and SOP transients of more than 1.5 Mrad/s using a bit error rate tester and a polarimeter, and an unprecedented SOP transient rate of 43 Mrad/s was detected in the experiment. Through the combination of theory and experiment, we verify the accuracy of the simulation model, which can better describe the transient of polarization state in OPGW, and thus provide help to solve the signal error in OPGW in thunderstorm weather [15].

Funding

China Southern Power Grid.

Disclosures

The authors declare no conflicts of interest.

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

References

1. S. M. Pietralunga, J. Colombelli, A. Fellegara, et al., “Fast polarization effects in optical aerial cables caused by lightning and impulse current,” IEEE Photonics Technol. Lett. 16(11), 2583–2585 (2004). [CrossRef]  

2. P. M. Krummrich, D. Ronnenberg, W. Schairer, et al., “Demanding response time requirements on coherent receivers due to fast polarization rotations caused by lightning events,” Opt. Express 24(11), 12442–12457 (2016). [CrossRef]  

3. M. Kurono, K. Isawa, and M. Kuribara, “Transient state of polarization in optical ground wire caused by lightning and impulse current,” Proc. SPIE 2873, 242–245 (1996). [CrossRef]  

4. F. Pittalà, C. Stone, D. Clark, et al., “Laboratory Measurements of SOP Transients due to Lightning Strikes on OPGW Cables,” in Optical Fiber Communication Conference (OFC, 2018), paper M4B.5.

5. F. Pittalà, C. Xie, D. Clark, et al., “Effect of Lightning Strikes on Optical Fibres Installed on Overhead Line Conductors,” in 2018 34th International Conference on Lightning Protection, (ICLP, 2018), pp. 1–5.

6. D. Charlton, S. Clarke, D. Doucet, et al., “Field measurements of SOP transients in OPGW, with time and location correlation to lightning strikes,” Opt. Express 25(9), 9689–9696 (2017). [CrossRef]  

7. William C. Snider, R. C. Moore, A. J. Erdman, et al., “Lightning-Induced State of Polarization Change in OPGW Using a Transmission Line Model,” 2021 Optical Fiber Communications Conference and Exhibition (OFC) (2021): 1–3.

8. C. E. R. Bruce and R. H. Golde, “The lightning discharge. Journal of the Institution of Electrical Engineers - Part II: Power Engineering,” 88(6), 487–505 (1941).

9. F. Heidler, J. M. Cvetic, and B. V. Stanic, “Calculation of lightning current parameters,” IEEE Trans. Power Delivery 14(2), 399–404 (1999). [CrossRef]  

10. W. Zhao, J. Zhao, H. Zhang, et al., “Channel modeling and compensation of ultra-fast spike-shaped RSOP caused by lightning,” in 2021 Asia Communications and Photonics Conference (ACP, 2021), pp. 1–3.

11. C. T. Mata, M. I. Fernandez, V. A. Rakov, et al., “EMTP modeling of a triggered-lightning strike to the phase conductor of an overhead distribution line,” IEEE Trans. Power Delivery 15(4), 1175–1181 (2000). [CrossRef]  

12. V. A. Rakov, M.A. Uman, M.I. Fernandez, et al., “Direct lightning strikes to the lightning protective system of a residential building: triggered-lightning experiments,” IEEE Trans. Power Delivery 17(2), 575–586 (2002). [CrossRef]  

13. D. E. N. G. Huihua, Z. H. A. N. G. Ruiqi, C. H. E. N. Junwu, et al., “Research on Principles and Methods of Lightning Strike Location Based on OPGW Light Polarization State,” Insulators and Surge Arresters | Insul Surg Arres 2018(01), 148–153 (2018).

14. Jeunhornme Luc B, Single-Mode Fber Optics: Principles and Applications.Marcel Deller (Inc, 1983), Chap. 2.

15. W. Lian, F. Tong, W. Li, et al., “Two-stage equalization for ultra-fast RSOP and inter symbol interference compensation based on a simplified Kalman filter,” Opt. Express 31(20), 33355–33368 (2023). [CrossRef]  

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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Figures (20)
Fig. 1.
Fig. 1. Lightning strike OPGW

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Fig. 2.
Fig. 2. OPGW cables:(a) central tube type; (b) layer-stranded type; (c) cross section of the OPGW used in the work. The left one is central tube type and the right one is layer-stranded type.

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Fig. 3.
Fig. 3. Mesh of OPGW

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Fig. 4.
Fig. 4. Simulation of magnetic field distribution in OPGW under direct lightning strike. The red arrow indicates the direction of the magnetic field. The direction of the arrow in the right image is perpendicular to the surface of the paper.

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Fig. 5.
Fig. 5. Equivalent Circuit Model of OPGW

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Fig. 6.
Fig. 6. Cross section and longitudinal section of OPGW

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Fig. 7.
Fig. 7. Lightning current waveform and spiral current waveform

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Fig. 8.
Fig. 8. Schematic diagram of ring current

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Fig. 9.
Fig. 9. Simplified Channel Model

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Fig. 10.
Fig. 10. Simulation platform diagram

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Fig. 11.
Fig. 11. Stokes vector curve with time

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Fig. 12.
Fig. 12. Rotation rate curve with time

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Fig. 13.
Fig. 13. Stokes curves on the Poincare sphere

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Fig. 14.
Fig. 14. Experimental installation drawing

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Fig. 15.
Fig. 15. SOP maximum rotation speed distribution diagram

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Fig. 16.
Fig. 16. Lightning current size distribution diagram

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Fig. 17.
Fig. 17. Polarization degree over time curve

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Fig. 18.
Fig. 18. Stokes vector curve with time

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Fig. 19.
Fig. 19. Rotation rate curve with time

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Fig. 20.
Fig. 20. Stokes curves on the Poincare sphere

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Tables (1)

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Table 1. The meaning represented by the xi signal

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Equations (37)

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θ  =  V μ 0 I 2
I ( 0 , t ) = I 0 ε [ 1 − e x p ( t τ 1 ) ] n e x p ( − t τ 2 )
U ( s ) = R I R ( s ) = s L I L ( s )
I ( s ) = I R ( s ) + I L ( s )
I L ( s ) = I ( s ) R R + s L
I L ( t ) = I ( t ) ∗ R u ( t ) L e − R t L
L = μ n 2 V
L = μ π r 2 l d 2
μ c = Z S + Z a Z S μ S + Z a μ 0 = μ 0 ( 1 + Z s Z a )
μ = μ c ( r 2 − r 0 2 r 2 ) + μ 0 r 0 2 r 2
δ = 2 ρ μ 0 ω
R = ρ l 2 π r δ
τ = 4 π 3 μ 2 μ 0 r 6 d 4 ρ
  I L ( t ) = 2 l 0 τ 1 τ 2 e x p ( − t ( τ 1 + τ 2 ) τ 1 τ 2 ) τ τ 1 + τ τ 2 − τ 1 τ 2 − l 0 τ 1 τ 2 e x p ( t ( τ 1 + 2 τ 2 ) τ 1 τ 2 ) τ τ 1 + 2 τ τ 2 − τ 1 τ 2 − l 0 τ 2 e x p ( t τ 2 ) τ − τ 2 + 2 l 0 τ 2 τ 2 3 e x p ( t ( τ 1 + τ 2 ) τ 1 τ 2 ) ( τ − τ 2 ) ( τ τ 1 + τ τ 2 − τ 1 τ 2 ) ( τ τ 1 + 2 τ τ 2 − τ 1 τ 2 )
d B = μ 0 4 π I d l r 2
B y = B z = 0
∫ − ∞ ∞ ⁡ 1 ( R 2 + x 2 ) 3 / 2 d x = 2 R 2
θ = ∫ − ∞ ∞ ⁡ V B x d x = V μ 0 I R 2 2 ∫ − ∞ ∞ ⁡ 1 ( R 2 + x 2 ) 3 / 2 d x = V μ 0 I
θ = N V μ 0 I
R [ θ ( I ) ] = ( cos ⁡ θ ( I ) − sin ⁡ θ ( I ) sin ⁡ θ ( I ) cos ⁡ θ ( I ) )
U ( t ) = [ c o s κ ( t ) e j ξ ( t ) − s i n κ ( t ) e j η ( t ) s i n κ ( t ) e − j η ( t ) c o s κ ( t ) e − j ξ ( t ) ]
∂ A ∂ T + i β 2 2 ∂ 2 A ∂ T 2 + α 2 A = i γ | A | 2 A
∂ A x ∂ z + β 1 x ∂ A x ∂ T + i β 2 2 ∂ 2 A x ∂ T 2 + α 2 A x = i γ ( | A x | 2 + B | A y | 2 ) A x ∂ A y ∂ z + β 1 y ∂ A y ∂ T + i β 2 2 ∂ 2 A y ∂ T 2 + α 2 A y = i γ ( | A y | 2 + B | A x | 2 ) A y
x 1 ( t ) = U ( t ) x ( t ) = ( c o s κ ( t ) e j ξ ( t ) − s i n κ ( t ) e j η ( t ) s i n κ ( t ) e − j η ( t ) c o s κ ( t ) e − j ξ ( t ) ) x ( t )
x 2 ( ω ) = F F T [ x 1 ( t ) ]
J i ( ω , Δ τ i ) = ( exp ⁡ ( j ( ω Δ τ i 2 ) ) 0 0 exp ⁡ ( − j ( ω Δ τ i 2 ) ) )
D i ( α i , δ i ) = ( c o s α i − e − j δ i s i n α i e j δ i s i n α i c o s α i )
U i = D i − 1 ( α i , δ i ) J i ( ω , Δ τ i ) D i ( α i , δ i )
U P M D ( ω ) = ∏ i = 1 N ⁡ D i − 1 ( α i , δ i ) J i ( ω , Δ τ i ) D i ( α i , δ i )
< Δ τ 2 >= ∑ i = 1 N < Δ τ i 2 >
x 3 ( ω ) = U P M D ( ω ) x 2 ( t ) = ∏ i = 1 N ⁡ D i  -  1 ( α i , δ i ) J i ( ω , Δ τ i ) D i ( α i , δ i ) x 2 ( ω )
x 4 ( t ) = I F F T [ x 3 ( ω ) ]
R [ θ ( t ) ] = ( cos ⁡ θ ( t ) − sin ⁡ θ ( t ) sin ⁡ θ ( t ) cos ⁡ θ ( t ) )
x 5 ( t ) = ( cos ⁡ θ ( t ) − sin ⁡ θ ( t ) sin ⁡ θ ( t ) cos ⁡ θ ( t ) ) x 4 ( t )
q ( t ) = U R S O P 2 I F F T { U P M D F F T { U R S O P 1 x ( t ) } }  =  ( cos ⁡ θ ( t ) − sin ⁡ θ ( t ) sin ⁡ θ ( t ) cos ⁡ θ ( t ) ) I F F T { ∏ i = 1 N ⁡ D i  -  1 ( α i , δ i ) J i ( ω , Δ τ i ) D i ( α i , δ i )   F F T { ( c o s κ ( t ) e i ξ ( t ) − s i n κ ( t ) e j η ( t ) s i n κ ( t ) e − j η ( t ) c o s κ ( t ) e − j ξ ( t ) ) x ( t ) }
D O P = S 1 2 + S 2 2 + S 3 2 S 0 2
R S O P = arcsin ⁡ ( ( S 1 ( t + Δ T ) − S 1 ( t ) ) 2 + ( S 2 ( t + Δ T ) − S 2 ( t ) ) 2 + ( S 3 ( t + Δ T ) − S 3 ( t ) ) 2 ) Δ T
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