An application of the shortest path algorithm for the identification of weak MHD mode

1.   Introduction

Magnetohydrodynamic (MHD) modes widely exist in magnetized plasma [1]. The frequency, amplitude, and pattern are the three basic characteristics of MHD modes. The characterization of MHD modes is crucial for the investigation of mode behaviors [2, 3] and plasma control, e.g., disruption prevention [4]. Major disruptions in tokamaks occur frequently following the formation of the locked tearing mode, which is indicated by a fast drop in the mode frequency and an increase in the mode amplitude [5, 6]. The prediction and prevention of disruption depends strongly on mode evolution information. In addition, active control of MHD modes is of practical significance for plasma control in fusion devices [7, 8]. It requires the fast and accurate acquisition of mode information, especially the mode frequency.

However, noise will significantly degrade the performance of frequency identification. Noise can originate in the measurement process and in plasma fluctuations [9] which are not concerned. Weak amplitudes of modes and strong noises will both lower the signal-to-noise ratio (SNR) of the signal. In many cases, the detection of weak MHD modes and highly noisy modes is crucial for applications. Spectrum analysis is the most commonly used method for mode frequency identification. Low SNR can introduce large errors and even mistakes when the Fourier or wavelet transform is applied directly to the signal. On the one hand, to attenuate the influence of noise, one can preprocess the raw signal with a digital filter before calculating the spectrum. Optimal filters such as the Wiener filter are available to some extent by leveraging the statistical properties of the noise [10]. When the coherent noise reference signal is obtained, the adaptive filter [11] is very effective in suppressing noise in these quasi-stationary signals. On the other hand, methods such as the cross-spectrum, the singular value decomposition, etc [12–15] can be applied to attenuate the influence of noise by utilizing two or more channels of signals. These methods manage to highlight the mode by obtaining more information about it. They can usually recognize the pattern of the mode at the same time. To a large extent, the above approaches work well on noisy signals, but their common limitation is that they require extra channels of the signal to get coherent information and then attenuate the influence of noise, which does not always happen in a variety of circumstances.

Regarding the problem of identifying the mode frequency with a single channel of low SNR signals, a novel method named 'shortest-path-algorithm-based mode identification (SPAMI)' is proposed in this paper. SPAMI employs the time-frequency spectrum diagram of the signal to generate a directed weighted graph. The shortest path of this directed weighted graph is then solved as an estimation of the frequency evolution of the mode. Unlike previous approaches, SPAMI does not need additional knowledge of signals from extra measurements. Instead, SPAMI takes advantage of the past and future information of the signal to attenuate the influence of noise. This is the reason that SPAMI can identify the mode frequency from just one channel of the signal, even if the SNR of the signal is low. This paper will show how SPAMI works and how it was successfully applied for the weak mode frequency identification on the J-TEXT tokamak. In the rest of this paper, the problem and principle are described in section 2 based on an example. Section 3 demonstrates the implementation of this method, and section 4 shows the results. Section 5 deals with the uncertainty of this method. Section 6 is the conclusion.

2.   Principle

A typical example is the frequency identification of the tearing mode on the J-TEXT tokamak [16]. Figure 1(a) shows the magnetic signal of a Mirnov probe [17] in the process of rotation recovery of a forced locked mode [18, 19]. Figure 1(b) shows the time-frequency power density spectrum of the signal by the short-time Fourier transform (STFT) [20]. The mode starts to rotate from a static state at 0.47 s and its frequency increases to 7 kHz over time. From 0.57 s, the mode amplitude decreases and the frequency increases slightly. In the period from 0.49 to 0.57 s, the mode amplitude is large at a frequency of 5 kHz. Its second, third, and fourth harmonics can be observed at frequencies of 10 kHz, 15 kHz, and 20 kHz. The identification of high-frequency harmonics is of great benefit for studying the nonuniformity of mode rotation. However, after 0.57 s, the harmonics are hardly observed except for the fundamental frequency. The problem to be solved is to identify the frequency evolution of all harmonics. An intuitive idea to estimate the frequency evolution of the mode is to take the maximum of S(f, t) in a certain frequency range at each time segment, where S(f, t) is the power density obtained by the STFT. For example, between 0.5 s and 0.55 s, one can take the frequency that corresponds to the maximum power density in the frequency range of 2 kHz and 7 kHz in the spectrum at each time segment as the estimation of the frequency evolution of the mode. This is also the most common method. However, for areas A and B marked by the dashed ellipses in figure 1(b), this method is not available. In area A, there are two vague band-like structures of high power density, indicating the frequency evolution of the third and fourth harmonics. Due to the strong background noise, the frequency of the actual maximum power density does not lie on the band-like structure at several time segments. This implies that the power density distribution should be considered globally to avoid errors caused by the local maximum at each segment. In area B, there is expected to be a second harmonic, but it is challenging to obtain its frequency evolution with such a weak power density. SPAMI is proposed to address these obstacles.

Figure  1.  (a) Time trace of the Mirnov signal of J-TEXT's discharge 1070071. (b) Time-frequency power density spectrum of the Mirnov signal. Areas A and B are the areas where the mode frequency is difficult to identify.

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An important assumption in the following discussion is that the frequency evolution of the target mode is continuous, which is reasonable for most MHD modes. Considering a noised time series of a mode with continuous frequency evolution, the power density spectra at times t₀, t₁, and t₂ are obtained, and the frequencies of the peak power density are f_true, f_false, and f_true, respectively. From the knowledge of mode frequency f_true at times t₀ and t₂, it can be concluded that the expected mode frequency at time t₁ should also be f_true given the continuity of the frequency evolution. However, the frequency corresponding to the maximum power density at time t₁ is f_false due to the influence of noise. Using the peak power density frequency as the estimated mode frequency will yield incorrect results. To address this issue, a proper cost can be defined such that the cost of f_true is less than that of f_false when estimating the mode frequency. Considering the continuity of the frequency evolution of the mode, the cost should be positively correlated with the frequency deviation between adjacent time segments. In addition, for the consideration of globality mentioned before, the estimation of frequency at one time should not be determined only by the current and adjacent time segments, but also by all the time segments. SPAMI does this cleverly, by using the graph method. The above two requirements of continuity and globality are naturally included in the optimal path planning of graphs.

3.   Implementation of SPAMI

3.1   Generation of the directed weighted graph

The graph in graph theory is composed of points and the edges connecting them. In the graph G = (E, V), V represents the set of points, and E represents the set of edges. Each edge in E can be directional or nondirectional and can have different weights (distances). SPAMI treats the power density spectrum as a 2D graph G₁, as shown in figure 2. The time and frequency form the two-tuple (t, f) as the set of points V₁. The set of edges E₁ naturally connects one point to its eight surrounding points. Graph G₁ is supposed to be a directional graph due to the time arrow. Edges of E₁ on the time axis point in the direction of time increase, while edges of E₁ on the frequency axis can point in both directions of frequency increase and decrease. A valid path is supposed to go through edges along its direction from a start point to an end point. The weight (or the cost) of this path equals the sum of the weights of the edges it passes by. By setting the weight of E₁ as follows, one can estimate the frequency evolution of the mode by finding the path with the smallest weight in the graph.

Figure  2.  Directed weighted graph generated from the time-frequency spectrum. The horizontal direction is the time direction, and the vertical direction is the frequency direction.

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An example is employed to illustrate the details of the set of weights. Considering an MHD mode that occurs at time t₀ and frequency f₀, and disappears at time t_end and frequency f_end, the start point and the end point can be specified as (t₀, f₀) and (t_end, f_end) in the graph, respectively. At the next time t₁, the mode frequency is more likely to be f₀ than other frequencies due to the continuity. When the edge weight is set to be positive, going from (t₀, f₀) to (t₁, f_{i, i ≠ 0}) costs more than going from (t₁, f₀), because the former path must pass by more edges with positive weight. With the increase in the frequency deviation, the cost will also increase. In addition, it is reasonable to assume that the point with greater power density S(t_i, f_i) is more likely to be the actual mode frequency, so the edges connecting to this point should have lower weights. It means that the weights of the edges are supposed to be negatively related to the power density of the point they connect. The requirements of continuity and globality are included in the process of shortest path search by setting weights in this manner. The weights of the edges are expressed as

where W_T, W_F and W_diag represent the weight of edges in the time direction, the frequency direction, and the diagonal direction, respectively. M is an arbitrary function that can transform an increasing sequence into a decreasing sequence. k_T, k_F, and k_diag are constant weight coefficients, representing the difficulty of moving in the time direction, the frequency direction, and the diagonal direction when searching for the shortest path. k_diag is supposed to be more than k_T and k_F since it connects points with a larger deviation. From point (t_i, f_i) to point (t_i+1, f_i+1), it can travel directly by the diagonal edge or pass by the point (t_i, f_i+1). In most cases, the weight of the former path is supposed to be equal to or less than the latter. It yields

We have established a directed weighted graph representation of the time-frequency spectrum, the next problem is to determine the shortest path in the graph.

3.2   Dijkstra's shortest path algorithm

The shortest path algorithm is a classic algorithm in graph theory. It is utilized to find the path between any two points with a minimized weight of the edges in the graph. There are several shortest path algorithms, such as the Dijkstra's algorithm, the Bellman–Ford algorithm, the Floyd–Warshall algorithm, etc [21]. Dijkstra's algorithm is the best candidate for the above problem.

Dijkstra's algorithm is an algorithm based on breadth-first search and greedy strategy, utilized to solve the shortest path from all points to a single source point in a nonnegative weight graph. The main steps of Dijkstra's algorithm are as follows. (1) Maintain a set S, which stores the points that have been calculated for the shortest distance. Maintain a vector d with a length equal to the number of elements of points in set V to store the path distance. (2) Initialize S so that it contains only the source point. Initialize d as the distance from each point to the source point. If a point is not directly connected to the source point, the distance is initialized to infinity. (3) The following operation is performed in a loop until S = V: find the element v in the vector d that does not correspond to the element in S and has the smallest distance, record its distance as d_v. Add the corresponding point of v to set S and perform the relaxation operation. The relaxation operation is that, for each point in S, the distance recorded in d is updated according to the law:

where c is the distance from point v to i. After the loop ends, the shortest distances from each point to the source point are saved in d. From the above steps, it can be realized that the basic idea of Dijkstra's algorithm is to find the points closest to the point set whose shortest distance is known one by one by the relaxation operation. For our purpose, the shortest path is needed instead of the value of the shortest distance. It can be easily solved by maintaining a vector to record the index of the closest point after the relaxation.

4.   Results

SPAMI transfers the time-frequency spectrum to a directed weighted graph and utilizes the Dijkstra's algorithm to solve the shortest path. The applications of this approach are demonstrated as follows.

4.1   Comparison with traditional methods

In the following case, the function M takes the inverse proportional function. The parameters k_T = 1, k_F = 1, and k_diag = 1.4. Figure 3 demonstrates the performance of SPAMI in estimating the mode frequency of the signal depicted in figure 1, in comparison to three traditional methods. In the results for SPAMI, the source point is at (0.55 s, 0 kHz) where the mode occurs. Four end points are respectively taken as (0.67 s, 10 kHz), (0.67 s, 20 kHz), (0.53 s, 35 kHz), and (0.58 s, 25 kHz), corresponding to the fundamental, second, third, and fourth harmonics of this mode. These four shortest paths are marked with black lines in figure 3(b). The shortest path indicates the frequency evolution of each harmonic well. Even if the background noise is strong, SPAMI obtains a smooth and correct frequency evolution in the time around 0.48 s of the third and fourth harmonics (corresponding to area A in figure 3(a)). For the second harmonic after 0.57 s (corresponding to area B in figure 3(a)), SPAMI also calculates a path of frequency evolution even if the second harmonic is so weak that it almost disappears. Since the fundamental frequency is obvious and the frequency of the second harmonic is twice the fundamental frequency, the estimation of the evolution of the second harmonic by SPAMI can be examined. The white line in figure 3(b) shows twice the fundamental frequency, and it is pretty close to the evolution of the second harmonic estimated by SPAMI. Figure 3(c) shows the mode frequency estimation using peak detection of the density power. Choosing the frequency window for peak detection is difficult for the signal with multiple and varying frequencies in this case. Here, the windows are taken by using results from the SPAMI (black lines) as the central frequency with a ± 3 kHz width. It can be observed that peak detection can figure out frequencies in the region with high SNR, despite the fact that its result is not as continuous as that of SPAMI. In the region with low SNR, peak detection is not available. The points with peak density power are extremely discrete, and the resulting frequency estimation is wrong. To improve the continuity of the results of peak detection, one can limit the frequency difference between every two adjacent time segments when detecting the maximum density power. It is called local peak detection. The results of the local peak detection are shown in figure 3(d). The frequency difference is limited to ± 2Δf, where Δf is the frequency resolution. It can obtain continuous frequency estimation, but the estimation is incorrect. Compared with SPAMI, local peak detection meets the requirement of continuity but lacks globality. This leads to the incorrect results of local peak detection. Figure 3(e) shows the estimation of the mode frequency using linear prediction coefficients (LPC) [22]. LPC is a model-based method and does not need to calculate the time-frequency spectrum. A 70th-order model is applied here to calculate the coefficients of the linear predictor. The frequency peak is calculated with a bandwidth less than 550 Hz. It can be observed that LPC is also unable to obtain available frequencies in areas A and B. These comparisons demonstrate that SPAMI can obtain much better results in weak mode identification than traditional methods. According to the principle of SPAMI, its performance is also closely related to the quality of the original time-frequency spectrum. As a result, depending on the properties of the target signals, other time-frequency analysis methods, such as wavelet and Hilbert–Huang transforms, may be candidates for calculating the original time-frequency spectrum. In our case, SPAMI can also obtain good results with the wavelet spectrum.

Figure  3.  (a) Power density spectrum of the Mirnov signal (copy of figure 1(b)). (b) Estimation of the mode frequency by SPAMI, four black lines represent the evolution of the fundamental, second, third, and fourth harmonics of this mode. The white lines represent the expected frequency evolutions of the second, third, and fourth harmonics by multiplying the fundamental frequency accordingly. (c) Estimation of the mode frequency by the peak detection. The frequencies of the peak power density (magenta points) are taken in the windows which use results from the SPAMI (black lines) as the central frequency with a ± 3 kHz width. (d) Estimation of the mode frequency by the local peak detection. (e) Estimation of the mode frequency by LPC.

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4.2   Influence of weight coefficients

Parameters k_T, k_F, and k_diag can substantially influence the performance of SPAMI. To investigate the influence of these parameters, we scan the k_T/k_F and k_diag/k_F in the above case and calculate the root-mean-square deviations between the result of SPAMI and the expected value. Figure 4 shows the deviation map with k_T/k_F in the range of 0–2 and k_diag/k_F in the range of 1–3. It can be observed that the region with the minimum deviation (10^1.6 Hz) is located near k_T/k_F ~1 and k_diag/k_F ~1.5. If parameters k_T, k_F, and k_diag are determined within this region, SPAMI can obtain good results. If the parameters k_T, k_F, and k_diag vary within a certain range, the deviation will not vary much. However, if the parameters k_T, k_F, and k_diag vary beyond this certain range, the deviation will vary significantly, as shown in the region of 10^2.0–10^2.4 in figure 4. This indicates that SPAMI will make mistakes with improper k_T, k_F, and k_diag.

Figure  4.  Deviation map with respect to k_T/k_F and k_diag/k_F.

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It should be noted that there is no universally optimal setting of parameters k_T, k_F, and k_diag for different signals. The characteristics of the mode affect the optimal determination of parameters. Machine learning approaches [23, 24] can be utilized to classify characteristics of modes and aid in the determination of the parameters. In future work, the optimal parameters of SPAMI can be calculated, aiming at general MHD modes in magnetized plasma.

4.3   Application of SPAMI for plasma physics

Two additional cases are presented in figure 5. The mode identifications of these cases are crucial in the research of plasma physics. Figure 5(a) shows the spectrum of the Mirnov signal with the process of the mode couple on the J-TEXT tokamak [25]. There are an m/n = 2/1 mode with a frequency of 8.5 kHz and a 3/1 mode with an initial frequency of 15 kHz. The frequency of the 3/1 mode decreases gradually due to the torque of the 2/1 mode. At 0.4 s, the 3/1 mode is coupled to the 2/1 mode and locked to the frame of the 2/1 mode. After that, the frequencies of the 2/1 mode and 3/1 mode become the same and quickly drop. A major disruption is eventually induced at 0.43 s by the mode couple. To understand the dynamics of the mode couple and avoid disruption in applications, finding the evolution of mode frequency is essential. The difficulty is that the frequencies of two modes that will couple with each other are usually close, and the power density of each mode is disturbed by the other mode. SPAMI is applied to solve this problem as shown in the dashed lines in figure 5(a). Two start points (0.26 s, 8.5 kHz), (0.26 s, 15 kHz) and one end point (0.45 s, 0 kHz) are set. The parameters k_T, k_F, and k_diag are still the same as in the above case. It can be observed that SPAMI obtains good results for both the 2/1 mode and the 3/1 mode.

Figure  5.  (a) Results of SPAMI (dashed lines) during the mode couple process. (b) Estimation of the frequency of BAE mode by SPAMI.

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Figure 5(b) shows the identification of the BAE mode during the locked mode process [26]. The BAE mode is weakly observed at a frequency of 38 kHz during the locked mode (0.4–0.52 s). Once the locked mode is unlocked and starts to rotate, the BAE frequency drops quickly and disappears at 0.54 s. SPAMI effectively traces the frequency evolution of the BAE mode, with the start point set at (0.4 s, 38 kHz) and the end point set at (0.54 s, 22 kHz). In this case, the end point is not set at the end of the time window (0.55 s) due to the early disappearance of the BAE mode. SPAMI does not know whether there is a mode or not, although it can always find the shortest path. The end point and the start point should be carefully set when applying SPAMI.

However, it is not easy to specify the end point in many cases. Figure 6 shows an alternative solution when the end point cannot be specified. Two series of points X and Y are chosen as end points at 0.61 s and 0.67 s with frequencies evenly spaced at 1 kHz to perform SPAMI. All the shortest paths form tree-like structures with shared roots and different branches. For example, for the second harmonic, the root path is from 0.45 to 0.57 s, and branches are from 0.57 to 0.61 s when end points are X. The root path is from 0.45 to 0.64 s and branches are from 0.64 to 0.67 s when end points are Y. The root path is the most overlapped path, so it can be chosen as the estimation of frequency evolution with the highest confidence. It indicates that the best shortest path can be determined by pruning the shortest path branches when the end point cannot be specified. To what extent the path can be trusted will be discussed in the next section.

Figure  6.  Groups of shortest paths, with two series of end points chosen at 0.61 s and 0.67 s with frequencies evenly spaced at 1 kHz.

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Some of the advantages and disadvantages of SPAMI can be concluded from the above cases. The main advantage is that SPAMI is available for low SNR signals and can obtain a good estimation of the frequency evolution. The disadvantage is that the calculation of SPAMI does not depend on the existence of the modes. Thus, the start point and the end point should be carefully chosen. Additionally, SPAMI is not suitable for signals that carry discontinuous modes.

It is noteworthy that Dijkstra's algorithm is one of the shortest path algorithms with a low time complexity, the time complexity of which can be O(mlogn) or O(m + nlogn) depending on implementation methods. The fast Fourier transform which calculates the spectrum also has a low time complexity of O(nlogn). This indicates that SPAMI has the potential to be applied in quasi-real-time frequency identification. For example, for the 103 × 105 time-frequency graph in our case, the running time of SPAMI for each time step for each shortest path is around 1.3 ms. In the actual implementation, the running time is expected to be shorter. This response time is sufficient for real-time detection of MHD activities and suppression by electron cyclotron wave (ECW) or low hybrid wave (LHW) [27] in the experiments.

5.   Uncertainty of SPAMI

To quantify the uncertainty of SPAMI, the spectrum of the target signal is assumed to consist of stationary random noises N and a mode with constant intensity s in the following model. The probability distribution of N can be estimated from the power density distribution in an unperturbed time interval. Figure 7 shows the power density distribution of noises in the first case, resulting in a gamma distribution Gamma(α, λ) as

Figure  7.  Power density distribution of the noise and the gamma fitting (black line).

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The mathematical expectation of N is α/λ = 1.31, indicating the intensity of the noise. The variation of N influences the uncertainty of the SPAMI results, but the method of analysis is shared. The following results will employ equation (6) as an example, parameters k_T, k_F, and k_diag are also the same as in the above case.

5.1   Uncertainty of the element unit

Figure 8(a) shows the element unit of the directed weighted graph. There are multiple paths from the start point to the end point. The power distribution of points that do not contain a mode is 1/N, while the power distribution of points containing a mode is 1/(N + s). Assuming that the correct path is the path in the horizontal direction, the distances of these paths are:

where L₁ is the distance of the horizontal path, and L_diag is the distance of paths that pass through diagonal edges. Other paths which are not shown in figure 8(a) are neglected since their distances are much greater than L₁ and L_diag. The probability that SPAMI chooses the correct path is P₁(L₁ < L_diag, L₁ < L_diag). If the correct path is the path that passes by diagonal edges, the expressions of L₁ and L_diag will change accordingly. The probability that SPAMI chooses the correct path can also be calculated in the same way, and expressed as P_diag(L_diag < L₁, L_diag < L_diag). The value of P₁ and P_diag in various SNRs is shown in figure 8(b). It can be observed that P₁ is greater than P_diag, and with the increase of SNR, both P₁ and P_diag increase. Assuming that the correct path is uniformly distributed in these three paths, the accuracy of SPAMI in the element unit is obtained and shown in figure 8(b).

Figure  8.  (a) Element unit of the weighted directed graph of SPAMI. Power densities at points indicate that the horizontal path is the correct path. (b) The probability that SPAMI chooses the correct path at various SNRs when the correct path is the horizontal path (diamond) and the path passing by diagonal edges (square), respectively. The total probability is marked by circles.

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5.2   Global uncertainty

Compared with the accuracy in the element unit, the global uncertainty between the result of SPAMI and actual mode frequency is more significant in the application. The randomness of the evolution of the mode and noise can both influence the error of SPAMI. It introduces great complexity to the uncertainty analysis. Here, the Monte Carlo method is applied to solve this problem. In each test of Monte Carlo, the spectrum of the test signal is combined with a sample of random continuous modes and random noises, and SPAMI is performed on the test signal to work out the shortest path. The uncertainty is defined as the standard deviation of the difference between the shortest path and the actual mode frequency.

Ten thousand tests are performed at a fixed SNR. Figure 9 shows the results of the test signal spectrum and the results of SPAMI in one test at various SNRs. The frequency evolution of mode becomes more difficult to observe directly in the spectrum as SNR decreases, as shown in the first row of figure 9. However, SPAMI can obtain superior estimations of the frequency evolution with low SNR, as shown in the second row of figure 9. Actual mode frequencies are indicated by solid lines, and the results of SPAMI in one test are indicated by dashed lines. With the increase in SNR, the deviation between the result of SPAMI and the actual mode frequency is smaller. Shaded areas in figure 9 indicate the standard deviation σ and the 99.7% confidence boundary (3σ boundary) of the result of SPAMI obtained from ten thousand tests. The frequencies of general MHD modes in J-TEXT range from kHz to tens of kHz, the requirement for accuracy of the mode frequency estimation in practice is around 0.5–1 kHz. SPAMI meets the requirement when the SNR exceeds 0.5.

Figure  9.  Power spectra of the noised signal (panels in the first row) and the corresponding mode frequency evolutions (panels in the second row). The true mode frequencies (solid line) and estimations of SPAMI (dashed line) are compared using the reference of the standard deviation (blue shaded area) and the 99.7% confidence boundary (red shaded area).

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The dependence of the standard deviation of SPAMI results on SNR is shown in figure 10. When SNR is less than 0.6, the standard deviation is relatively high and drops significantly with the increase in SNR. The standard deviation gradually approaches 0 with the increase in SNR when SNR is more than 0.6. This demonstrates the benefit of SPAMI for low SNR signals in mode frequency estimation.

Figure  10.  Dependence of the standard deviation of SPAMI results on SNR.

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6.   Conclusions

In conclusion, we propose a novel method called SPAMI for identifying the evolution of MHD modes from low SNR signals. SPAMI generates a weighted directed graph from the signal spectrum and then obtains the mode evolution by solving the shortest path. It is demonstrated that SPAMI effectively identifies the evolution of an unlocked tearing mode on the J-TEXT tokamak. This method's feasibility is demonstrated by its ability to maintain acceptable levels of uncertainty, even when the signal-to-noise ratio is as low as 0.5. Furthermore, with appropriate parameters, SPAMI can be applied to not only MHD modes but also to any signals with continuous modes. SPAMI has low time complexity and can potentially be applied in quasi-real-time mode identification.