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Abstract
Photoacoustic (PA) spectroscopic technique has become a popular tool for trace gas detection and is especially suitable for in situ measurement of sulfur hexafluoride (SF6) decomposition components in gas insulated switchgear (GIS). However, the concentrations of SF6 decomposition components are generally very low and the resulting PA signals are too weak to be accurately retrieved with traditional methods. In this study, we proposed a Lyapunov exponent based chaotic oscillator algorithm to retrieve the weak PA signals of SF6 decomposition components. Retrieval of weak PA signals from strong noise background was achieved for both simulation and measurement perspectives. The results were compared with those based on phase-locked amplification technique. Both simulation and measurement results concluded that the proposed chaotic oscillator algorithm is superior to the phase-locked amplification in terms of accuracy, sensitivity and stability. Since most trace gases have weak absorption signatures in the atmosphere (below 1%), this study can provide valuable insights in dealt with such weak signals in remote sensing of atmosphere.
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
SF6 enclosed switchgear with its small size and excellent technical performance is advanced high-voltage electrical distribution equipment since early 1970s. Such device is also referred to as Gas-Insulator Switchgear (GIS), which consists of circuit breaker, bus bar, isolating switch, current transformer, voltage transformer and surge arrester casing. SF6 is often used as the insulating medium in each GIS device since its insulation capacity and arc blow-out performance are much better than the air. Stimulated by partial discharge (PD) and superheat, the insulating medium SF6 will decompose into many gaseous components, which can react with O2 and H2O, and further generate SO2, SOF2, SO2F2, CO and CF4, etc. [1,2]. Previous studies concluded that the decomposition components of SF6 are different in species and concentration for different PD insulating accidents. In the presence of moisture, the discharge and thermal decomposition process mainly decompose SF6 into SO2, SOF2, HF and SO2F2. However, in dry conditions, CO, CF4 and CO2 are also produced. As a result, measuring the species and concentration of SF6 decomposition components can establish a link between SF6 decomposition and PD accidents, which is very useful in evaluation of the cause and risk level of each PD insulating accident.
Photoacoustic (PA) spectroscopic device is typically characterized with simple structure, high sensitivity, high accuracy and good stability. PA spectroscopic technique has become a popular tool for trace gas detection and is especially suitable for in situ measurement of SF6 decomposition components in GIS [3–10]. PA spectroscopic technique is based on PA effect defining as a process of continuous transformation and interaction of light, heat and sound. The PA signal produced by the gas absorption is proportional to the gas concentration, and thus the gas concentration can be retrieved by the PA signal measurement. The accuracy of PA spectroscopy technique is dependent on the accuracy in separation of PA signal from noise. Generally, the concentrations of SF6 decomposition components are very low, and the amplitude of the PA signal is in the order of nV. Furthermore, the PA signal is also affected by many interfering factors such as the environmental fluctuations and the limited PA resolution. In order to separate the weak PA signal from the strong noise background, special skills have to be used. The cross-correlation theory based phase-locked amplification technique is one of the widely used methods for detection of weak periodic signals. Since the advent of the lock-in-amplifier in the 1960s, it has been verified to be a qualified device in weak signal detection. By inter-correlating the measured signal to the reference signal, the target signal with a specific frequency can be retrieved from the noise, while the signals with other frequencies are suppressed [11]. However, the phase-locked amplification technique has limited capacity in signal integration and noise suppression. The signals to be detected by the phase-locked amplification technique should have adequate signal-to-noise ratio (SNR). For a low SNR signal due to weak absorption, the traditional phase-locked amplification technique is hard to separate PA signal from strong noise background.
In recent years, chaotic oscillator detection theory opens up new opportunity for weak signal detection [12–18]. Chaotic system is extremely sensitive to periodic signal and immune to noise, and is thus very useful in weak signal detection. Although chaotic oscillator detection theory has been widely used in detection of weak signal, it is rarely used for weak PA signal detection, especially for detecting SF6 decomposition components with non-resonant PA spectroscopy. In this study, the chaotic oscillator algorithm based on Lyapunov exponent was used to retrieve the weak PA signal of SF6 decomposition components. Retrieval of weak PA signals from strong noise background was achieved by both simulation and measurement perspectives. The results were compared with those based on phase-locked amplification technique. Since most trace gases have very weak absorption signatures in the atmosphere (below 1%), this study can provide valuable insights in dealt with such weak signals in remote sensing of atmosphere.
2. Methodologies
In nonlinear dynamic system, a periodic perturbation in certain parameters can cause an essential change in its state. Using chaotic oscillator algorithm to separate a target signal from noise usually takes the target signal as a periodic perturbation force superimposed on a nonlinear chaotic system. In this chaotic system, even strong noise interference don’t affect its state but a weak periodic signal can cause a dramatic change in its state [12–18]. Specifically, the phase of a chaotic system is very sensitive to the perturbation of periodic PA signal but is immune to irregular noise; regardless of amplitude, the noise only impacts the trajectory of a chaotic system rather than the phase. As a result, we can take the advantage that the sensitivity of the chaotic phase to PA signal is much higher than that to noise to separate weak PA signal from strong noise background. In this section, we generate a numerical chaotic system and simulate its different chaotic states, and then demonstrate how to separate weak PA signal from strong noise background with the chaotic oscillator algorithm.
Duffing equation is one of the most popular nonlinear dynamic systems for generating chaotic phenomena [12–18]. The Duffing equation which is often used to describe the weak damping motion of a soft spring oscillator is a two-order differential equation with a cubic term. Although the Duffing equation is evolved from a simple physical model, its results show broad representative. Perturbed by periodic external forces, the Duffing equation can be expressed as Eq. (1):
where a is the linear recovery coefficient, b is the nonlinear recovery coefficient, β is the amplitude of the periodic driving force, k is the damping coefficient, ω is the dynamic angular frequency, x(t) is the chaotic time series, $\dot{x}(t)$ and $\ddot{x}(t)$ are the 1st and 2nd derivatives of the chaotic time series, respectively. To deal with the Duffing equation properly, the influence of a and b on signal to noise ratio (SNR) should be considered extensively, and the optimal coefficients for the restoring force should be selected [19,20]. In this study, we have used the empirical restoring force coefficients of previous studies, i.e., -a = b=−1, to initialize the Duffing equation. In this case, Eq. (1) becomes a Holmes type Duffing equation, which can be written as Eq. (2): where the state of Eq. (2) can be expressed by Eq. (3):For a fixed damping coefficient k, the dynamic behavior of the chaotic system described by Eq. (3) is only affected by the amplitude of the periodic driving force. Figures 1–7 show the simulated variations of the chaotic system with respect to different amplitudes of periodic driving force. In all these simulations, the initial condition, the angular frequency of the driving force and the damping coefficient are set to be $[x, \dot{x}] = [1,1]$, 1.0 rad/s and 0.5, respectively.
Fig. 1. Variation of the chaotic system simulated by Matlab 12a. The amplitude of the driving force β=0, initial values are $[x, \dot{x}] = [1,1]$. (a) Phase portrait; (b) time domain waveform.
As Figs. 1–7 show, for a fixed damping coefficient of 0.5, the chaotic system varies incrementally from the states of convergence, harmonic oscillation, homoclinic oscillation, periodic bifurcation, chaotic, critical to large-scale periodic oscillation and large-scale periodic oscillation with an increase in driving force β from 0 to 0.8266. In detail:
- (1) For the driving force β = 0, the chaotic system eventually converges to one of the two focal points (±1, 0) or the saddle point (0, 0) of the phase portrait. The initial condition determines which saddle point the chaotic system will converge. For the initial value of [1,1] as shown in Fig. 1, the chaotic system eventually converges to the saddle points (1, 0).
- (2) For the driving force β > 0, the behaviors of the chaotic system are much complicated than the case of β = 0, and different amplitudes result in different dynamic states. If the driving force β is small, the phase trajectory of the chaotic system oscillates around the focal point of the phase portrait with a same frequency as the driving force, which is referred to as harmonic oscillation as shown in Fig. 2. As the amplitude of the driving force increases further, the phase trajectory of the chaotic system changes to the states of homoclinic oscillation (Fig. 3), periodic bifurcation (Fig. 4), chaotic (Fig. 5), critical state (Fig. 6) and large-scale periodic oscillation (Fig. 7). In the state of homoclinic oscillation (Fig. 3), the chaotic system characterizes as an oscillation around one of the two stable focuses of the nonlinear system, and its oscillation frequency is the same as that of the periodic driving force. In the state of periodic bifurcation (Fig. 4), the nonlinear portion of the chaotic system starts to affect the phase trajectory of the chaotic system, and the period doubling bifurcation occurs. The distribution of phase trajectory points in chaotic state is constrained in a certain range and ergodic in this range (Fig. 5). The large-scale periodic oscillation state represents the stable state of the chaotic system. In this stable state, the phase trajectories are symmetrical around the origin, the rotation period is the same as the reference signal period, and the phase trajectories are distributed in a narrow closed circular distribution (Fig. 7). The state of the chaotic system changes quickly before chaotic state but slowly afterwards. In this study, the chaotic system keeps at the chaotic state for a long time till the driving force is about 0.8265, when the chaotic system changes to the state of large-scale periodic oscillation. Here 0.8265 represents a critical state indicating the transition between the states of chaotic to large-scale periodic oscillation (Fig. 6).
- (1) Before incorporating PA signal to the chaotic system, we adjust the amplitude of the driving force to ensure that the chaotic system is in a critical state. We save the amplitude of the corresponding driving force as β1;
- (2) Then, the mixture of PA signal and noise is taken as part of driving force and incorporated into the chaotic system, which will change the phase of the chaotic system. Since the chaotic system is sensitive to the PA signal which has a same frequency as β1 and is immune to noise, this change in phase could be solely caused by the PA signal rather than the noise;
- (3) We re-adjust the amplitude of the driving force until the chaotic system is in critical state again, and save the amplitude of the corresponding driving force as β2;
- (4) The amplitude of the PA signal is thus calculated as:${\beta _s} = |{{\beta_1} - {\beta_2}} |$.
The accuracy of βs is determined by the accuracies of both β1 and β2, which are further determined by that if the two aforementioned critical states can be identified accurately. The simplest method for identifying the critical state is to observe intuitionally if the phase trajectories of the chaotic system are changed dramatically. However, this method may lead to misjudgment and result in a large error in PA signal separation since the transition threshold between the states of chaotic and large-scale periodic oscillation is not fixed, which is hard to quantify accurately by eye. To avoid this deficiency, this study set up a maximal Lyapunov exponent (MLE) criterion to identify the critical state of the chaotic system.
3. Chaotic state identification by MLE criterion
Lyapunov exponent is defined as the averaged exponential rate of the divergence or convergence states of nearby orbits in phase space. The Lyapunov exponent of a chaotic system is a measure of the extent of separation of infinitesimally close trajectories [22], where the MLE quantifies a chaotic system’s predictability and sensitivity to changes in its initial condition. A positive MLE indicates that the chaotic system is in chaotic state, and vice versa. We follow the method of A. Wolf [23] to judge the state of the chaotic system and define the chaotic system to be at a critical state if MLE=0. If the motion equation of a chaotic system is explicitly known, the MLE can be calculated by using the definition of the Lyapunov exponent. If the motion equation of a chaotic system is unknown [22], the phase space reconstructed time series can be used to calculate the Lyapunov exponent.
Phase space reconstruction was originally used for recovering chaotic attractors in high-dimensional phase space. The theoretical basis for recovering chaotic attractors in high-dimensional phase space is that the evolution of any component in the chaotic system is not determined by itself but all components which have connections with it. Therefore, the mutual information among all components can be extracted from the evolution of either component. As a result, the original state of a chaotic system can be restored based on the time series of a specific component.
Packard et al. (1980) [24]concluded that the phase space can be reconstructed using the delay coordinates of some variables in the original system [25]. If the embedding dimension m satisfies the condition of $m \ge 2d + 1$ (d is the dimension of the chaotic system), the attractor can be recovered in this m-dimensional space. Assume the chaotic time series are ${x_1},{x_2},\ldots ,{x_k},\ldots {x_N}$, the phase space reconstructed time series can be expressed as Eq. (4) after embedding the time series into the m-dimensional Euclidean space:
where τ is the time delay and m is the embedding dimension.Determination of embedding dimension m and time delay τ is the most important procedures in phase space reconstruction. In this study, a correlation integral method (namely the C-C method) which is verified to be a time saving and reliable method for small datasets is adopted. The C-C method was proposed by H.S. Kim et al. (1999) [26] in 1999 and can simultaneously estimate the time delay τ and the embedding dimension m. After reconstructing the phase space with the embedding dimension m and time delay τ, Eq. (5) and Eq. (6) are used to find the nearest neighbor point for each point on each given orbit.
where P is the average period of the reconstructed time series. The value of P is estimated as the reciprocal of the average frequency of the energy spectrum. ${Y_{\hat{j}}}$ and Yj in Eq. (6) represent the points on different trajectories. As a result, the MLE can be estimated as Eq. (7) by calculating the divergence rate mean of the nearest neighbor points of each point on the basic orbit [25].Taken the logarithm on both sides of Eq. (8), the results become:
It shows that the MLE is approximately equal to the slope of the set of lines in Eq. (9). As a result, the MLE can be obtained directly by the least squares approximation of the set of lines [12,22]. The results are as follows:
After the MLE of the system is calculated by the phase space reconstruction method, the state of the chaotic system is identified.
4. Results and discussion
Retrieval of weak PA signal by the chaotic oscillator algorithm with the MLE criterion is shown in Fig. 9. Flow chart for MLE calculation based on phase space reconstruction is shown in Fig. 10.
We applied an iterative scheme to judge the state of the chaotic system. For both before and after the incorporation of the PA signal, we iteratively adjust the magnitude of the driving force till MLE = 0. We then use the methodology described in section 2 to retrieve the PA signal. We applied both simulated signals and measured signals to verify the performance of the chaotic oscillator algorithm. We performed a series of tests and found that the optimal operating frequency of the non-resonant PA cell in our non-resonant PA spectrometer is 28 Hz. As a result, the frequencies of the following simulated signals and measured signals are kept at 28 Hz.
4.1 Simulation perspective
As shown in Fig. 11, we generated four groups of numerical sinusoidal signals to stimulate the PA signals. All these signals have the same frequency of 28 Hz and amplitude of 0.1 but different SNRs ranging from −6.0 dB to −20 dB. Before incorporating these simulated PA signals into the chaotic system, we adjust the magnitude of the driving force (with a frequency of 28 Hz) iteratively with a step size of 1.0 × 10−6. We stop the adjusting procedures once the MLE is less than 1×10−5. The chaotic system is, in the meantime, assumed to be in a critical state. Figure 12 shows that the chaotic system is in critical state, where the corresponding amplitude of the driving force β1=1.009000. We then incorporate the simulated PA signals shown in Fig. 11 into the chaotic system and re-adjust the magnitude of the driving force till the chaotic system in critical state again. Figure 13 shows that the chaotic system retain to the critical state after incorporating the simulated PA signals. Finally, the amplitude of each PA signal can be deduced with the methodology described in section 2.
Fig. 12. Phase trajectory of the chaotic system before incorporating the PA signal, where the frequency f=28 Hz and β1=1.009000
Fig. 13. Phase trajectories of the chaotic system after incorporating four groups of PA signals shown in Fig. 11, where the frequency f=28 Hz and β1=1.009000
Retrieval results for the simulated PA signals with the chaotic oscillator algorithm and the lock-in amplification technique are summarized in Table 1. Retrieval of the simulated PA signals with the lock-in amplification technique is following the methodology of Yun (2008) [27]. The relative error for both methods is calculated by Eq. (11).
where Ucal represents the retrieval results using either the chaotic oscillator algorithm or phase-locked amplification technique, and Ureal is the real signal amplitude, i.e., 0.1 V in this study.
Table 1. Retrieval results for the simulated PA signals with the chaotic oscillator algorithm and the lock-in amplification techniquea
We can see from Table 1 that the chaotic oscillator algorithm can retrieve weak PA signal from the strong noise background, and its retrieval accuracy is better than the phase-locked amplification method. For the cases that the SNRs are too low (e.g., worse than −10.0 dB) to be detected by phase-locked amplification technique, the chaotic oscillator algorithm can still retrieve the PA signal with an accuracy of less than 10%.
4.2 Measurement perspective
Figure 14 shows the structure of the non-resonant PA detection device, including an infrared (IR) light source, a chopper, an optical filter wheel, a PA sample cell, a detector and a data processor. The IR light source is a glowing metal filament heated to 1175 K, generating an infrared optical signal from 1.5µm to 10µm. The optical filter wheel controlled by the chopper motor can filter and modulate the optical signal simultaneously. It includes four filters to filter the optical signal at a certain frequency band. Different optical filters are selected in sequence to modulate the optical signal. The reference filter used for adjusting instrument drift has a pass band at which there is no absorption by any gases or only negligible absorption occurs. Because the sound pressure in the non-resonant PA cell is equal everywhere, the design is flexible. In this study, a cylindrical structure with simple structure and symmetrical distribution is selected, which is coaxial with the infrared light source. For the non-resonant PA cell, when the cell body is in the open state, the sound pressure at the open port is approximately to zero, which leads to the sharp decrease of the PA signal in the PA cell, so the non-resonant PA cell can only work in the closed state. The sample gas is pumped into sample cell at a constant speed (limited at ∼50ML/min.) with the capillary and pump to ensure the stability of the pressure in the cell. The temperature of the sample cell is maintained at 343 K by the temperature controller. The gas path is switched by the magnetic valve. The detector consists of two microphone devices available from Sens-Array Infrared, USA. It offers good response performance between 1µm and 10µm.
Fig. 14. Structure of the non-resonant PA detection device. MFC is the abbreviation of Mass Flow Controller
At room temperature, we fill up the PA cell with 275.9 µL/L of CO standard gas (SF6 gas as carrier gas). We then start to identify the critical state of the chaotic system once the pressure of the PA cell become stable. We repeated this procedure for five times and thus obtained five groups of PA signals for 275.9 µL/L of CO. The true value of the signal to be measured can be calculated by the calibration equation of CO gas as Eq. (12) [28].
where 2.517 V is the background PA signal corresponding to the CO filter, the PA signal for 275.9µL/L s of CO is 3.0169 V and thus the true value is 0.4999 V.Before incorporating the measured PA signal, the initial critical state of the chaotic system is shown in Fig. 12. After incorporating five groups of measured PA signals and the phase trajectory diagram for the chaotic system at the critical state is shown in Fig. 15. The five groups of sinusoidal PA signals are retrieved by both chaotic oscillator algorithm and phase-locked amplification technique (Software phase-locked amplification was used, and the integration time was set to 1s.). Comparison between the two methods is as summarized in Table 2. It can be seen from Table 2 that the measured PA signal also shows that the chaotic oscillator algorithm is superior to the cross-correlation theory based phase-locked amplification technique in terms of accuracy and stability.
Fig. 15. Phase trajectories of the chaotic system after incorporating five groups of measured PA signals, where the frequency f=28 Hz and β1=1.009000
Table 2. Retrieval results for the measured PA signals with the chaotic oscillator algorithm and the lock-in amplification technique
The detection sensitivity is the minimum concentration that can be detected when the SNR is equal to 1, which is mainly affected by systematic noise such as chopper noise, absorption noise of cell walls and windows, circuit noise, air turbulence noise and environmental noise. To determine the CO detection sensitivity of the PA spectroscopic device, the 1-σ standard deviation of the amplitudes of the five groups of PA signals were regarded as the systematic noise. By using Eq. (13), the detection sensitivity for CO was obtained.
where cmin is the detection sensitivity, c is the known concentration of the target gas and SNR is the signal-to-noise ratio.By using Eq. (13) to estimate the sensitivity, we found that the detection sensitivity is improved by nearly 5.5 times with the chaotic oscillator algorithm.
5. Conclusions
Measuring the species and concentration of SF6 decomposition components in gas insulated switchgear (GIS) can establish a link between SF6 decomposition and partial change (PD) accidents, which is very useful in evaluation of the cause and risk level of each PD insulating accident. Photoacoustic (PA) spectroscopic technique has become a popular tool for trace gas detection and is especially suitable for in situ measurement of SF6 decomposition components in GIS. However, the concentrations of SF6 decomposition components are generally very low and the resulting PA signals are too weak to be accurately retrieved with traditional methods.
In this study, a Lyapunov exponent based chaotic oscillator algorithm was introduced to detect the weak non-resonant PA signal. This algorithm takes the advantage that the sensitivity of the chaotic phase to periodic PA signal is much higher than that to irregular noise. We have performed both simulation and measurement experiments to verify the robustness of the chaotic oscillator algorithm. Both simulation and measurement results concluded that the proposed chaotic oscillator algorithm is superior to the cross-correlation theory based phase-locked amplification method in terms of accuracy and stability. Furthermore, the detection sensitivity is improved by nearly 5.5 times with the chaotic oscillator algorithm. A disadvantage of this method is that, under strong noise interference, the chaotic system is easy to result in a pseudo chaotic state, which interferes state determination and thus decreases the detection accuracy. Future study will explore new method for state determination to address this deficiency.
Funding
the Sino-German Mobility programme (M-0036); National Key Research and Development Program of China (2019YFC0214802); National Natural Science Foundation of China (41705012, 41875040); Natural Science Research Project for Colleges and Universities of Anhui Province (KJ2020A0029).
Disclosures
The authors declare that they have no conflicts of interest.
Data availability
Data underlying the results are available upon reasonable request.
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