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A physics‐informed learning technique for fault location of DC microgrids using traveling waves

A physics‐informed learning technique for fault location of DC microgrids using traveling waves INTRODUCTIONDC microgrids have some advantages compared to AC ones including less amount of power loss and easier integration of power sources with DC nature (e.g. photovoltaic (PV) systems and battery energy storage systems (BESS)). However, their protection is more challenging because of the following reasons: (i) Zero‐crossing does not occur on the DC fault currents which creates some challenges for circuit breakers to interrupt them [1, 2]. (ii) Due to the presence of power electronics converters, the DC fault currents incorporate a high magnitude for a short period of time (a couple of milliseconds) which is then attenuated once the protection of power electronic switches operate. To this end, DC microgrids require a fast protection scheme to isolate faults before the internal protection of DC‐DC converters operate to avoid their unwanted outage [3–6].As opposed to the conventional protection of DC systems, in which overcurrent protection, undervoltage protection, rate of change of current, and differential schemes [2–13] are utilized, traveling wave (TW) protection schemes render much faster fault detection and location since high‐frequency TWs propagate at the speed of light [14–16]. In [17–24], TW protection algorithms are proposed for high voltage DC (HVDC) systems. Medium Voltage DC (MVDC) microgrid's protection utilizing TWs is addressed in [3, 25, 26]. In [3], the TW protection algorithm utilizes the waveshape properties and polarity of TWs measured locally at the protection device to detect and locate faults. Then, based on the extracted waveshape features, a look‐up table is created to map each feature to the corresponding fault scenario. The implementation of this approach is challenging since the lookup table requires to include all possible fault scenarios. To tackle this challenge, machine learning techniques can be utilized to intelligently learn the behavior of different fault scenarios in a computationally efficient manner. Wavelet transform (WT) along with artificial neural networks (ANN) are employed for fault detection and classification in MVDC shipboard power systems in [25]. In [26], WT and ANN are deployed to detect faults and classify their type in DC microgrids. In both [25, 26], the proposed algorithms are only able to detect and classify faults and are not able to locate faults along the cables.In [27, 28], the multiresolution analysis (MRA) is employed which calculates the TW's wavelet coefficients for multiple frequency ranges using discrete wavelet transform (DWT). Then, the Parseval's theorem is utilized to compute the wavelet coefficient's energies. In [27], a machine learning (ML) engine is used for fault location while in [28], a curve‐fitting technique is adopted to mathematically identify Parseval energy curves as a function of fault location and then find fault location using measured Parseval energy values and extracted curves. As discussed in these papers, each level of MRA corresponds to a specific frequency range and for each frequency range, a Parseval energy curve can be extracted. The greater number of MRA levels renders a more accurate fault location algorithm. Even though the algorithms in [27] and [28] result in an accurate fault location, they require an extensive number of simulations in order to extract high fidelity Parseval energy curves. To tackle this challenge, this paper proposes a physics‐informed machine learning approach to accommodate the training of the machine learning engine with a limited number of measurements and labeled datasets. If one can extract the Parseval energy curves of a specific cable type and configuration in a simple DC system, then those Parseval energy curves can be used for fault location on the cables in another DC system with similar cable types and configurations. In fact, the Parseval energy curves for cables act as physics‐based constraints that facilitate the generalization of data points required for the training of machine learning algorithms. For a specific cable type and configuration, regardless of the cable's length, the Parseval energy curves will have a similar pattern in any DC system. Only the Parseval energy curve magnitudes will change based on the change of DC system specifications (e.g. architecture, loads, converter's ratings etc.). So, once the Parseval energy curve patterns for a specific cable type and configuration are extracted, those Parseval energy patterns can be used to identify the Parseval energy curves in any DC system regardless of its size, number of integrated converters, number of cables etc. This approach will significantly decrease the time required to run an extensive number of simulations required for extracting high fidelity Parseval energy curves which in turn increases the scalability of MRA‐based TW fault location in DC systems. After the Parseval energy curves are extracted, they are utilized to train a Gaussian Process (GP) estimator. The GP estimator only needs the Parseval energy values of the current measured at the protection device (PD) location to estimate fault location. The proposed algorithm resembles a physics‐informed ML approach where the GP estimator relies on the Physics laws that describe the Parseval energy curve patterns of DC cables. This paper makes the following contributions:1.The impact of fault location on the generated TWs is investigated and it is demonstrated that the so‐called Parseval energy curves that quantitatively relate TWs to the fault location have a similar pattern for a specific cable type regardless of its length and the system in which the cable is utilized.2.Using the inherent features of Parseval energy curves, a physics‐informed learning algorithm for fault location in DC systems is presented which requires a limited number of simulations for creating the training dataset of machine learning algorithm.3.The presented algorithm renders more accurate fault location estimation results with lower number of required simulations for creating the training dataset of machine learning algorithm.The remainder of the paper includes the following sections: Section 2 elaborates the TW fault location theory. Sections 3 discusses the MRA approach and Parseval's Theorem. The fault‐related Parseval energy curves are addressed in Section 4. Our proposed physics‐informed fault location algorithm is elaborated in Section 5 and is verified by simulation results in Section 6. The paper's conclusion is provided in Section 7.TW THEORYIn this section, the preliminaries of TWs are provided. The elaborated TW mathematical formulation will be used in Section 4 to discuss the impact of fault on TWs. A disruption (e.g. fault) in a power system component such as a cable or a line generates the electromagnetic waves that propagate through the power line, these waves are called traveling waves (TWs). The incident TWs, when it reaches a different environment (e.g. at a bus) having different parameters, gets reflected and refracted. The extent to which it gets reflected and refracted at the terminal depends on the characteristic impedances of the line and the adjacent circuit. A fault location or system buses will cause this phenomenon of reflection and refraction to happen. Telegrapher's equations are the coupled differential equations that relate the current i(x,t)$i(x,t)$ and voltage v(x,t)$v(x,t)$ at any point in space and time, TWs are formulated using these equations. The general solutions to these equations in the phasor domain are, [14, 27],1I∼(x,t)=I0+e−γx+I0−eγxV∼(x,t)=V0+e−γx+V0−eγx,\begin{equation} \begin{split} {\begin{cases} \tilde{I}(x,t) = I^{+}_{0}e^{-\gamma x} + I^{-}_{0}e^{\gamma x} \\[8pt] \tilde{V}(x,t) = V^{+}_{0}e^{-\gamma x} + V^{-}_{0}e^{\gamma x} \end{cases}} \end{split}, \end{equation}where propagation constant is γ and distance from the fault point is x. V0+$V^{+}_{0}$, V0−$V^{-}_{0}$, I0+$I^{+}_{0}$, and I0−$I^{-}_{0}$ denote the incident and reflection current and voltage TW components in Laplace form. Now, the current TW equation in the time domain can be defined as,2i(x,t)=i+(x,t)+i−(x,t)i+(x,t)=|I0+|e−αxcos(ωt−βx)i−(x,t)=|I0−|eαxcos(ωt+βx),\begin{equation} \begin{split} {\begin{cases} i(x,t) = i^{+}(x,t) + i^{-}(x,t) \\[4pt] i^{+}(x,t) = |I^{+}_{0}|e^{-\alpha x}\cos {(\omega t - \beta x)} \\[4pt] i^{-}(x,t) = |I^{-}_{0}|e^{\alpha x}\cos {(\omega t + \beta x)} \end{cases}} \end{split}, \end{equation}where in (2), attenuation constant is α and the phase constant is β which gives the propagation constant as,3γ=α+jβ=(R+jωL)(G+jωC),\begin{equation} \gamma = \alpha + j\beta = \sqrt {(R + j\omega L)(G + j\omega C)}, \end{equation}where TW's angular frequency is ω and G, R, C, and L represents the per unit length conductance, resistance, capacitance, and inductance of the transmission line respectively. So, the propagation velocity can be deduced as :4v=ωβ.\begin{equation} v = \frac{\omega }{\beta }. \end{equation}The above equation implies that although larger α makes travelling waves more attenuated, higher frequency traveling waves travel faster.MULTIRESOLUTION ANALYSISIn this section, the process of capturing and quantifying fault TWs are elaborated in detail. To this end, first wavelet transform (WT) is introduced. Then, MRA is deployed to identify the wavelet coefficients corresponding to TWs at different frequency ranges. Finally, Parseval's theorem is introduced for quantifying the captured TWs. The decomposition of a function into a set of wavelets is basically the WT. Over a range of frequency and time domain, continuous WT uses every possible wavelet, but discrete wavelet transform (DWT) uses only a finite set of wavelets. In both frequency and time domain, TWs can be effectively analyzed by utilizing the tool WT [28] where here DWT is used, which can be written as [25],5WD(m,k)=1a0m∑nx[n]gk−nb0a0ma0m.\begin{equation} W_D(m,k) = \frac{1}{\sqrt {a^{m}_{0}}}\sum _{n} x[n]g{\left[ \frac{k-nb_{0}a^{m}_{0}}{a^{m}_{0}} \right]}. \end{equation}In the above equation (5), the mother wavelet is denoted as g[], in which the function m(a0m,nb0a0m)$m(a^{m}_{0}, nb_{0}a^{m}_{0})$ is utilized to fit scaling and time‐shifting features with parameters a0 and b0, respectively.In this study, MRA [29, 30] is used to formulate wavelet coefficients of a TW signal for a wide range of frequencies. As shown in Figure 1, a number of low‐pass and high‐pass filters as well as decimators are utilized to implement MRA. Figure 1 defines each level of frequency range while assuming fs$f_s$ as the initial sampling frequency of DWT. ai[n]$a_{i}[n]$ and di[n]$d_{i}[n]$ are the outputs of low pass and high pass filters for each level. For constructing the next decomposition level's wavelet coefficients, the scaling coefficients or the outputs of the low pass filter at each level are used as the inputs of the next level. The high‐frequency behavior of the TW is preferable as the wavelet coefficients [25], so the detail wavelet coefficients are the output of the high pass filter. According to [31, 32] up‐sampling the coefficients by the factor of 2 is needed with the help of a reverse filter to reconstruct the original signal. With the help of the reconstruction process, one can obtain detail wavelet coefficients with a better time resolution and the same sampling rate as the original signal. This can be achieved by applying inverse filters of HHPF,i$H_{HPF,i}$ and GLPF,i$G_{LPF,i}$.1FIGUREMRA block diagramIn this paper, we have utilized the DWT tool in PSCAD/EMTDC [31] to implement MRA. The block diagram of this tool is illustrated in Figure 3. This tool allows for resampling of the signal at a sampling frequency different from the simulation time step. The tool utilizes an anti‐aliasing filter to filter out high frequency noise. The cut‐off frequency of the filter is adjusted based on the selected sampling frequency. The sampled data is stored in a data buffer and then is pushed into the wavelet decomposition and reconstruction stages. The detail wavelet coefficients will be generated at the same sampling rate as the original signal2FIGUREFault location impact on the Parseval energy value3FIGUREMRA processing steps using PSCADAccording to [25, 28], Parseval's theorem relates the wavelet coefficients to the measured signal energy spectrum if the mother wavelet and scaling function in (5) form an orthonormal basis. The energy of each identified wavelet coefficient is calculated using Parseval's theorem to understand the wavelet coefficients generated by MRA. The Parseval energy of a specific MRA output, at mth time step after an initial time t0 is calculated as [25, 28],6EPRS,i(m)=∑j=1mdi2(t0+jΔt),\begin{equation} E_{PRS,i}(m) = \sum _{j=1}^{m} d_i^2(t_0 + j \Delta t), \end{equation}where di(t)$d_i(t)$ is MRA's output for the ith decomposition level or frequency range at the time t; Δt$\Delta t$ is the time step used in DWT. This paper uses EPRS,i(m)$E_{PRS,i}(m)$ values to implement the physics‐informed learning technique for fault location of DC microgrids.FAULT‐RELATED PARSEVAL ENERGY CURVESFor a specific frequency range, a Parseval energy curve describes the Parseval energy value of current or voltage measurements at the protection device for different fault locations along a cable. A wavelet coefficient's Parseval's energy is analogous to the fault signal's spectrum of energy. A sample of Parseval's energy curve that illustrates the relation between variation in Parseval's energy at various fault locations is depicted in Figure 2. In Figure 2, each point denotes a separate analysis of Parseval energy value corresponding to a fault location. This curve is for a long DC cable on which bolted pole to pole (PP) faults are simulated at every 25 m of its length and a sampling frequency of 1 MHz is used for MRA. This figure shows the recorded level 1 Parseval's energy value of measured current at the cable terminal which corresponds to a frequency range from 250 kHz to 500 kHz. The pattern observed here is because of the inherent behavior of TW signals. The incident TW current experienced due to a fault can be written as,7i+(x,t)=|I0+|e−αxcos(ωt−βx),\begin{equation} i^{+}(x,t) = |I^{+}_{0}|e^{-\alpha x}\cos {(\omega t - \beta x)}, \end{equation}where x is the distance from the fault location. On the other hand, from (7), one can extract a function relating the incident TW current at the sensor location and fault location assuming faults at different locations along the cable. Thence, the incident TW current at the cable terminal and fault location xf$x_f$ can be formulated as,8i+(xf,t)=|I0+|e−αxfcos(ωt−βxf).\begin{equation} i^{+}(x_f,t) = |I^{+}_{0}|e^{-\alpha x_f}\cos {(\omega t - \beta x_f)}. \end{equation}Equation (8) implies that a combined sinusoidal and exponential behavior can be observed for the relationship between the incident TW current at the cable terminal where the sensor is located and the fault location. Some local peaks can be noticed because of the presence of cosine function in the equation. These local peaks at a specific time t1 are observed at a distance,9xLocalPeak=(ωt1−2nπ)/β,n=0,1,…\begin{equation} x_{Local Peak} = (\omega t_1 - 2n\pi )/\beta , n = 0,1,\ldots \end{equation}The Parseval energy is the sum of the square of MRA outputs (i.e. acts as an integrator) and an MRA output represents the magnitude of TW for a specific range of frequency which is extracted by applying DWT to (8). Equations (8) and (9) show that the pattern of the Parseval energy curves depends on α and β which according to (3) are functions of the cable parameters (per unit length conductance, resistance, capacitance, and inductance). Equation (9) describes that the incident TW current's local peaks (See Figure 2) at the protection device location are a function of TW angular frequency ω and take place periodically. β also corresponds to ω based on (3). Therefore, one can conclude that with the increase in ω, as it is in the numerator, more and more local peaks can be seen. Since the Parseval energy curve patterns highly depend on the cable per unit length parameters, the same patterns can be observed for a specific cable type regardless of the system in which the cable is deployed.FAULT LOCATION ALGORITHMThe overall diagram of fault detection and location algorithm is illustrated in Figure 4. As seen in this figure for each type of fault (i.e. bolted or resistive PP or PG faults), one different Gaussian Process (GP) regression engine is used to estimate fault location. For fault type classification, this paper utilizes the same algorithm that the authors proposed in their previous work in [27]. It should be noted that the fault type and its transition resistance affect the peak value of reflected and refracted TWs and therefore the Parseval energy curves. As seen in Figure 4, for each fault type and transition resistance value, a separate Gaussian Process algorithm is required to be trained. The fault type identification algorithm which is adopted from the previous work of the team in [27] can identify the type of a fault and its resistance based on the Parseval energy values of current, pole to pole (PP) voltage, and positive pole to ground (PPG) voltage measured at the protection device location. The fault type includes bolted and resistive pole to pole (PP) and pole to ground (PG) faults. The resistive faults include different values of transition resistance that can range from 1 to 100 Ω depending on the DC microgrid conditions like voltage level or geographical location. The algorithm in [27] utilizes the fact that the Parseval energies of the positive or negative pole to ground voltage are significantly higher than the Parseval energy of PP voltage to distinguish PP faults from PG ones. In order to effectively identify the fault's transition resistance, we have utilized Support Vector Machine (SVM) similar to the approach proposed in [27]. To train the SVM classifier, multiple bolted and resistive faults are simulated at different locations of cable (e.g. every 25 m) in a simulation software package (e.g. PSCAD/EMTDC). The SVM classifier is trained using the labeled Parseval energy values of current, pole to pole (PP) voltage, and positive pole to ground (PPG) voltage measured at the protection device location. The output of the SVM classifier is the fault resistance value. Once the fault type and its transition resistance are identified, the corresponding GP regression engine is selected for fault location. Each Gaussian Process is trained separately using the proposed physics‐informed algorithm in this paper.4FIGUREFault location algorithmThe hypothesis of this work is that if one can extract the Parseval energy curves of a specific cable type and configuration in a simple DC system, then those Parseval energy curves can be used for fault location on the cables in another DC system with similar cable types and configurations regardless of its size, number of integrated converters, number of cables etc. Conventionally, when a DC microgrid is constructed, the same cable type and configuration are utilized at the different locations of the microgrid. For a specific cable type and configuration, regardless of the cable's length, the Parseval energy curves will have a similar pattern in any DC system. Only the Parseval energy curve magnitudes will change based on the change of the DC system specifications (e.g. architecture, loads, converter's ratings etc.). In the proposed approach, a protection device utilizing the proposed fault location algorithm is placed on every cable segment in the system.For training the GP algorithm, we need the six levels of Parseval energy values of current measured at the protection device for multiple fault locations along a cable to train the GP algorithm. We are proposing to have these Parseval energy values for faults at every 25 m of the cable. For example, for a cable with length of 1000 m, we need the Parseval energy values for 40 different fault locations to train the GP algorithm. However, gathering these Parseval energy values for an extensive number of fault locations can be a very time consuming effort. It should be noted that extracting the Parseval energy curves from PSCAD simulations of a simple DC system is relatively fast due to the system size. However, running each PSCAD simulation of a large‐scale DC microgrid can take several hours since TWs are required to be captured at small simulation time steps with frequency‐dependent cable models to be used in the simulation model. To tackle this issue, the following algorithm is proposed in which we just need to gather the actual Parseval energy values for a limited number of fault locations and estimate the Parseval energy values of other fault locations that are required to train the GP algorithm. The proposed fault location algorithm is summarized as follows:Step1:$Step 1:$ On a simple DC system, the Parseval energy values for the current at the protection device location are calculated for faults at different locations of the line/cable (e.g. faults at every 25 m of the line/cable length) using PSCAD/EMTDC. This paper uses at least six MRA frequency levels to incorporate an adequate portion of the frequency spectrum for extracting fault current features as the fault location changes.Step2:$Step 2:$ We use the patterns of the extracted Parseval energy curves from the simple DC system to identify the Parseval energy curves for a specific protection device in the DC microgrid under study. To this end, we need Parseval energy values for faults at some sample locations (lk$l_k$) along the cable protected by the protection device in the DC microgrid under study. The Parseval energy values of faults at these sample locations will be used to tune the magnitude of the Parseval energy curves. Our goal here is to construct Parseval energy curves with 25 m fault location resolution. To this end, the Parseval energy values of fault locations at every 25 m (ln$l_n$) are constructed by the Parseval energy values of the closest sample location using10EPRS,i,ln=EPRS,i,lkEPRS,i,ln′EPRS,i,lk′,\begin{equation} E_{PRS,i,l_n} = \frac{E_{PRS,i,l_k}E^{^{\prime }}_{PRS,i,l_n}}{E^{^{\prime }}_{PRS,i,l_k}}, \end{equation}where EPRS,i,ln′$E^{^{\prime }}_{PRS,i,l_n}$ and EPRS,i,lk′$E^{^{\prime }}_{PRS,i,l_k}$ are the ith MRA level's Parseval energy values captured from simple DC system at faults at locations ln$l_n$ and lk$l_k$. EPRS,i,ln′$E^{^{\prime }}_{PRS,i,l_n}$ and EPRS,i,lk′$E^{^{\prime }}_{PRS,i,l_k}$ are the ith MRA level's Parseval energy values captured from DC microgrid under test for faults at locations ln$l_n$ and lk$l_k$. It should be noted that the Parseval energy values of the sample locations can be either gathered from the field measurements or from the simulation model of the DC microgrid under study in PSCAD.Step3:$Step 3:$ Once the Parseval energy curves with 25 m fault location resolution are constructed for the protection device in the DC microgrid under study, the Parseval energy values will be used to train a GP regression engine. The Gaussian Process regression model training procedure is shown in Figure 5. Once the GP engine is trained, one can use it to find fault location for any new fault scenarios by feeding the measured Parseval energy values at the protection device location to the GP engine.RemarkFor each frequency range, the Parseval energy is calculated right after the arrival of the first TW incident. It should be noted that the TWs' speed corresponds to the frequency range. Therefore, with a higher sampling frequency of MRA, one can accommodate higher frequency ranges in the algorithm and detect the first incident of TWs faster. We have selected 1 MHz of sampling frequency in his paper to make its implementation feasible on commercially available microprocessors. With the recent advancements in signal processing and measurement technologies, high‐frequency data sampling and measurement can be easily accommodated for the implementation of the proposed scheme. In fact, existing commercial TW relays are able to perform very high frequency (in the order of MHz) measurements [33–35].5FIGUREGaussian process regression model training procedureSIMULATION RESULTSConsider the simple DC system in Figure 6a. The nominal pole‐to‐pole voltage of this system is 750 V. This system is modeled in PSCAD/EMTDC. The resistance of the load is 10 Ω. The configuration of the cable under study is shown in Figure 6b. The length of the cable is 3000 m. The frequency‐dependent distributed parameter model available in PSCAD/EMTDC is used for modeling the cable. The cable is placed 1 m under the ground. The core conductor resistivity is 2×10−8$2\times 10^{-8}$ Ωm; the sheath resistivity is 30×10−8$30\times 10^{-8}$ Ωm. To generate Parseval energy curves for the cable shown in Figure 6a, PP and pole‐to‐ground (PG) faults are applied at every 25 m of the cable. The sampling frequency of MRA is equal to 1 MHz. The DC microgrid under study is shown in Figure 7. The proposed approach is used to create the fault location algorithm at protection device R25, R52, R56, R65, R26, and R62 in this microgrid system. This microgrid is supplying six nanogrids (NG). Each NG includes a load, a PV system, a BESS, and DC‐DC converters to integrate NG into the rest of the microgrid. In each NG, the BESS size is 3 kW and 6 kWh and the PV size is 5 kW. The load of each NG is equal to 5 kW. It is assumed that all cables in this microgrid are of the same cable type shown in Figure 6b but with different length values. In this study, the sampling frequency of DWT is 1 MHz. Daubechies (db8) is used as the mother wavelet. According to [25, 27], db mother wavelets are promising candidates that facilitate fast and accurate MRA by incorporating enough vanishing points for accounting for the salient features of waveforms, accommodating sharp cutoff frequencies to minimize the amount of energy leakage to the next decomposition level, and being orthonormal.6FIGURE(a) The simple DC test system; (b) the configuration of cable7FIGUREDC microgrid under studyTo show that same Parseval energy patterns can be observed for a specific type of cable that used in two different systems, the Parseval energy values are captured for some bolted PP fault scenarios along the cable from node 2 to node 5 (at R25) of the mesh microgrid shown in Figure 7. For a similar type of fault, the Parseval energy curves gathered from the simple DC system in Figure 6a are also plotted to compare their patterns against the Parseval energy values captured from the mesh DC microgrid under study in Figure 7. These comparisons are provided in Figures 8–13. As seen, similar patterns can be observed for the Parseval energy curves of a specific cable type regardless of the system in which the cable is deployed. The Parseval energy is the sum of the square of MRA outputs (i.e. acts as an integrator) and an MRA output represents the magnitude of TW for a specific range of frequency which is extracted by applying DWT to (8). Equation (8) shows that the pattern of the Parseval energy curves depends the cable parameters (per unit length conductance, resistance, capacitance, and inductance). It should be noted that the Parseval energy curves of the simple DC microgrid system have different scales and magnitudes compared to the Parseval energy curves gathered from the mesh DC microgrid. The scale of Parseval energy curve depends on |I0+|$|I^{+}_{0}|$ in (8) which in turn depends on different factors like the specifications and ratings of converters. The local minimum and maximum peaks on the curves can be found according to (9) which shows that these minimum and maximum peaks are a function of TW angular frequency ω and take place periodically. β also corresponds to ω based on (3). Therefore, one can conclude that with the increase in ω, as it is in the numerator, more and more local peaks can be seen. This behavior can be observed in Figures 8–13 where the number of local peaks decrease for the Parseval energy curves of lower frequency ranges. In conclusion, a specific cable type renders similar Parseval energy curve pattern in different systems, which is the hypothesis of the proposed algorithm.8FIGURELevel 1 Parseval energy curve pattern comparison between (a) the simple DC microgrid system and (b) mesh DC microgrid under study at R259FIGURELevel 2 Parseval energy curve pattern comparison between (a) the simple DC microgrid system and (b) mesh DC microgrid under study at R2510FIGURELevel 3 Parseval energy curve pattern comparison between (a) the simple DC microgrid system and (b) mesh DC microgrid under study at R2511FIGURELevel 4 Parseval energy curve pattern comparison between (a) the simple DC microgrid system and (b) mesh DC microgrid under study at R2512FIGURELevel 5 Parseval energy curve pattern comparison between (a) the simple DC microgrid system and (b) mesh DC microgrid under study at R2513FIGURELevel 6 Parseval energy curve pattern comparison between (a) the simple DC microgrid system and (b) mesh DC microgrid under study at R25In addition to the actual Parseval energy curves for the cable from node 2 to node 5 of mesh microgrid system, Figures 8b, 9b, 10b, 11b, 12b and 13b illustrate the estimated Parseval energy curves based on the algorithm discussed in Section 5. In these figures, the red dots describe the Parseval energy values that are captured from simulation of Mesh microgrid system and are utilized to estimate the Parseval energy values for faults at other locations of the cable. The estimated Parseval energy values will be later used to train the GP fault location estimation engine.We have applied the proposed technique on the loop highlighted in red in Figure 7 (i.e. protection device R25, R52, R56, R65, R26, and R62). At these protection devices, we have gathered the Parseval energy values for faults at every 100 m fault location. Then, using the proposed method, the high‐resolution Parseval energy curves for these protection devices are constructed with a resolution of 25 m fault location. These Parseval energy values are then used to train a GP estimator. In order to test the performance of the GP estimator, we have used Parseval energy values of faults at 50 m, 150 m, 250 m, 350 m etc. We have created a separate GP estimator for each fault type (i.e. bolted and resistive pole‐to‐pole (PP) and pole‐to‐ground (PG) faults). For resistive faults, the fault transition resistance is equal to 5 Ω. The testing results including Mean Absolute Percentage Error (MAPE) and Mean Absolute Error (MAE) for different types of faults are summarized in Table 1. Each protection device is responsible for fault location on its protected cable. The fault detection is performed within 1 ms after the fault is applied.1TABLEFault location estimation errors in the DC microgrid under studyBolted PPResistive PPBolted PGResistive PGProtection devicesMAPEMAE (m)MAPEMAE (m)MAPEMAE (m)MAPEMAE (m)R250.51%3.930.91%7.432.96%16.621.02%8.04R521.16%9.320.80%7.983.95%20.240.80%6.72R560.90%4.450.83%5.731.64%11.311.03%6.61R651.08%5.550.90%5.992.23%15.701.24%7.25R260.41%4.080.75%6.570.91%8.270.85%7.35R620.64%4.880.66%6.681.45%15.170.85%8.02GP model parametersAccording to [36], a GP represents a set of any finite number of random variables that have a joint Gaussian distribution. A GP regression tool finds the solution by incorporating the GP‐based latent variables {f(xi)|i=1,…,n}$\lbrace f({{\mathbf {x}}_{\mathbf {i}}})|i=1,\ldots ,n\rbrace$ and the basis functions h according to y=h(x)Tγ+f(x)$y=h{{\text{(}\mathbf {x}\text{)}}^{T}}\gamma +f\text{(}\mathbf {x}\text{)}$, with f(x)$f\text{(}\mathbf {x}\text{)}$ formed by a GP with zero mean and covariance function of k(x,x′)$k\text{(}\mathbf {x},{{\mathbf {x}}^{^{\prime }}}\text{)}$ (the covariance functions are represented by a set of Kernel hyperparameters). A sample solution y can be formulated as11P(yi|f(xi),xi)∼N(yi|h(xi)Tγ+f(xi),σ2).\begin{equation} P({y}_{i}|f({\mathbf{x}}_{i}),{\mathbf{x}}_{i})\sim N{({y}_{i}|h({\mathbf{x}}_{i})}^{T}\gamma +f({\mathbf{x}}_{i}),{\sigma}^{2}). \end{equation}A set of n observations (inputs and outputs) comprises the training set. In the training set, the input includes a set of six Parseval energy values calculated for the current measured at the protection device location, and the output is the fault location. In this paper, we have utilized the Gaussian Process Regression tool in MATLAB [37] to train the GP engines. In total, for six protection device locations and four fault types, twenty four GP engines are modeled. For all of them, the exponential Kernel function is used. The γ and σ of the trained models are summarized in Table 2.2TABLEGaussian process parameters (exponential Kernel function is used)ProtectionBolted PPResistive PPBolted PGResistive PGDevicesγσγσγσγσR25909.45.301082.75.3656.65.31103.85.3R521197.15.310808.81244.65.31089.17.3R56731.23.3742.13.3718.612.2749.310.3R65739.53.3746.23.3691.327.5752.37.5R261061.36.41272.96.410826.41298.46.4R621236.26.41286.96.4810.96.41293.96.4To highlight the performance of proposed physics‐informed machine learning fault location algorithm against the curve fitting algorithm in [28], these two algorithms are compared in Figure 14. This comparison is performed on protection device R25 of the mesh DC microgrid in Figure 7. It is assumed that both algorithms only use the Parseval energy values of faults at every 100 m of the cable to train their algorithms. In the curve fitting approach, these Parseval energy values are used to identify the best curves fitting the data for all six MRA levels. As seen in these figures, the Physics‐informed algorithm renders much more accurate fault locations. The curve fitting algorithm's MAPE values for bolted and resistive PP and PG faults at R25 are 29.25%, 11.06%, 21.04%, and 25.07%, respectively.14FIGUREA comparison of fault location prediction between physics‐informed machine learning and curve fitting tool of mesh DC microgrid under study at R25 for (a) bolted PP faults (physics‐informed approach's MAPE = 0.51% and curve fitting approach's MAPE = 29.25%); (b) resistive PP faults (physics‐informed approach's MAPE = 0.91% and curve fitting approach's MAPE = 11.06%); (c) bolted PG faults (physics‐informed approach's MAPE = 2.96% and curve fitting approach's MAPE = 21.04%); (d) resistive PG faults (physics‐informed approach's MAPE = 1.02% and curve fitting approach's MAPE = 25.07%)Verification of the performance of proposed algorithm for different fault resistance valuesHerein, the proposed algorithm is applied to protection device R25 in Figure 7 considering PG faults with fault transition resistance values of 1, 5, 10, 50, and 100 Ω. The accuracy of the proposed method for different fault transition resistance values is summarized in Table 3. In general, the fault transition resistance value has an impact on the magnitude of the Parseval energy curves, and their magnitude decreases as the fault resistance value increases. This behavior is shown in Figure 15.3TABLEFault location estimation errors for different fault resistance values in the DC microgrid under studyFault ResistanceR25Values (Ω)MAPEMAE (m)11.45%11.4751.02%8.04100.98%7.91500.68%6.631000.97%8.7415FIGURELevel 1 Parseval energy curve of the Simple DC Microgrid for various fault resistances when PG fault occurs.Comparison of the proposed algorithm against the existing algorithms in the literatureTable 4 compares the MAE of our proposed algorithm's fault location estimation against the fault location errors provided in [17]. Table 5 compares this paper's proposed algorithm against the existing DC microgrid TW protection techniques in the literature. This paper compares these algorithms considering the network type, line models, methodology, required data, required number of simulations and training data points to create the model, type of faults considered, and accuracy. The comparison shows that the proposed approach renders a very high accuracy while requiring a low number of data points for training the machine learning algorithm. This is the major advantage of our approach compared to the existing approaches in the literature.4TABLEComparison between Bolted PG MAE values of this paper with [17]RelayThis paperReference [3]Bolted PG MAE (m)Bolted PG error (m)R2516.621R5220.2434R568.2724R6515.1722R2611.3114R6215.7225TABLEComparison between approach of this paper and existing approaches in the literatureApproach (author, year)Network typeLine ModelFault location levelMethodologyRequired data for modelNumber of simulations and training data points required to create the modelType of FaultAccuracyThis paperDC microgridFrequency dependentDistance on the cablePhysics‐informed machine learning and DWTFault data for different locations on the cableLowBolted and Resistive PP and PGHigh (more than 96%)Ref. [[3]] (Saleh et al., 2020)DC microgridFrequency dependentDistance on the cableNumerical analysis on waveshape featuresFault data for different locations on the cableHighBolted and Resistive PP and PG (*Fault location results are only provided for bolted PG)High (percentage is not provided)Ref. [[23]] (Li et al., 2014)DC shipboard power systemNot applicableFaulted busANN and DWTFault data for different buses in the systemMedium (Only bus faults simulation data is required)Bolted PPHigh (more than 96%)Ref. [[24]] (Jayamaha et al., 2019)DC microgridFrequency dependentFaulted busANN and DWTFault data for different buses in the systemMedium (Only bus faults simulation data is required)Bolted and Resistive PP and PGHigh (more than 96%)CONCLUSIONThis paper creates a physics‐informed machine learning approach for fault location in DC microgrid by capturing high‐frequency TWs. TWs are extracted by the so‐called multiresolution analysis which identifies the TW's wavelet coefficients for multiple frequency ranges. This paper deploys Parseval's theorem to find the energy of wavelet coefficients as a quantitative metric for describing TWs. Once the Parseval energy curves for a specific cable are extracted, they can be utilized to locate faults along with that cable regardless of the DC system in which the cable is deployed. The fault location algorithm uses Parseval energy curves to train a Gaussian Process estimator. With the Parseval energy values of measured current at the protection device location as the inputs, the Gaussian Process estimator is able to estimate fault locations with high accuracy. The fault location algorithm is verified on a DC microgrid system simulated in PSCAD/EMTDC.ACKNOWLEDGEMENTSThis material is based upon work supported by the Laboratory Directed Research and Development program at Sandia National Laboratories, the U.S. Department of Energy's Office of Energy Efficiency and Renewable Energy (EERE) under Solar Energy Technologies Office (SETO) Agreement Number 36533, and National Science Foundation EPSCoR Cooperative Agreement OIA‐1757207. Sandia National Laboratories is a multimission laboratory managed and operated by National Technology & Engineering Solutions of Sandia, LLC, a wholly owned subsidiary of Honeywell International Inc., for the U.S. Department of Energy's National Nuclear Security Administration under contract DE‐NA0003525. This paper describes objective technical results and analysis. Any subjective views or opinions that might be expressed in the paper do not necessarily represent the views of the U.S. Department of Energy or the United States Government.CONFLICT OF INTERESTThe authors have declared no conflict of interest.DATA AVAILABILITY STATEMENTThe data that support the findings of this study are available from the corresponding author upon reasonable request.REFERENCESAugustine, S., Quiroz, J.E., Reno, M.J., Brahma, S.: DC Microgrid Protection: Review and Challenges, Report No. SAND2018‐8853. Office of Scientific and Technical Information, Oak Ridge, TN (2018)Chandra, A., Singh, G., Pant, V.: Protection techniques for dc microgrid‐a review. Electr. Power Syst. Res. 187, 106439 (2020)Saleh, K.A., Hooshyar, A., El Saadany, E.F.: Ultra‐high‐speed traveling‐wave‐based protection scheme for medium‐voltage dc microgrids. IEEE Trans. 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A physics‐informed learning technique for fault location of DC microgrids using traveling waves

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Abstract

INTRODUCTIONDC microgrids have some advantages compared to AC ones including less amount of power loss and easier integration of power sources with DC nature (e.g. photovoltaic (PV) systems and battery energy storage systems (BESS)). However, their protection is more challenging because of the following reasons: (i) Zero‐crossing does not occur on the DC fault currents which creates some challenges for circuit breakers to interrupt them [1, 2]. (ii) Due to the presence of power electronics converters, the DC fault currents incorporate a high magnitude for a short period of time (a couple of milliseconds) which is then attenuated once the protection of power electronic switches operate. To this end, DC microgrids require a fast protection scheme to isolate faults before the internal protection of DC‐DC converters operate to avoid their unwanted outage [3–6].As opposed to the conventional protection of DC systems, in which overcurrent protection, undervoltage protection, rate of change of current, and differential schemes [2–13] are utilized, traveling wave (TW) protection schemes render much faster fault detection and location since high‐frequency TWs propagate at the speed of light [14–16]. In [17–24], TW protection algorithms are proposed for high voltage DC (HVDC) systems. Medium Voltage DC (MVDC) microgrid's protection utilizing TWs is addressed in [3, 25, 26]. In [3], the TW protection algorithm utilizes the waveshape properties and polarity of TWs measured locally at the protection device to detect and locate faults. Then, based on the extracted waveshape features, a look‐up table is created to map each feature to the corresponding fault scenario. The implementation of this approach is challenging since the lookup table requires to include all possible fault scenarios. To tackle this challenge, machine learning techniques can be utilized to intelligently learn the behavior of different fault scenarios in a computationally efficient manner. Wavelet transform (WT) along with artificial neural networks (ANN) are employed for fault detection and classification in MVDC shipboard power systems in [25]. In [26], WT and ANN are deployed to detect faults and classify their type in DC microgrids. In both [25, 26], the proposed algorithms are only able to detect and classify faults and are not able to locate faults along the cables.In [27, 28], the multiresolution analysis (MRA) is employed which calculates the TW's wavelet coefficients for multiple frequency ranges using discrete wavelet transform (DWT). Then, the Parseval's theorem is utilized to compute the wavelet coefficient's energies. In [27], a machine learning (ML) engine is used for fault location while in [28], a curve‐fitting technique is adopted to mathematically identify Parseval energy curves as a function of fault location and then find fault location using measured Parseval energy values and extracted curves. As discussed in these papers, each level of MRA corresponds to a specific frequency range and for each frequency range, a Parseval energy curve can be extracted. The greater number of MRA levels renders a more accurate fault location algorithm. Even though the algorithms in [27] and [28] result in an accurate fault location, they require an extensive number of simulations in order to extract high fidelity Parseval energy curves. To tackle this challenge, this paper proposes a physics‐informed machine learning approach to accommodate the training of the machine learning engine with a limited number of measurements and labeled datasets. If one can extract the Parseval energy curves of a specific cable type and configuration in a simple DC system, then those Parseval energy curves can be used for fault location on the cables in another DC system with similar cable types and configurations. In fact, the Parseval energy curves for cables act as physics‐based constraints that facilitate the generalization of data points required for the training of machine learning algorithms. For a specific cable type and configuration, regardless of the cable's length, the Parseval energy curves will have a similar pattern in any DC system. Only the Parseval energy curve magnitudes will change based on the change of DC system specifications (e.g. architecture, loads, converter's ratings etc.). So, once the Parseval energy curve patterns for a specific cable type and configuration are extracted, those Parseval energy patterns can be used to identify the Parseval energy curves in any DC system regardless of its size, number of integrated converters, number of cables etc. This approach will significantly decrease the time required to run an extensive number of simulations required for extracting high fidelity Parseval energy curves which in turn increases the scalability of MRA‐based TW fault location in DC systems. After the Parseval energy curves are extracted, they are utilized to train a Gaussian Process (GP) estimator. The GP estimator only needs the Parseval energy values of the current measured at the protection device (PD) location to estimate fault location. The proposed algorithm resembles a physics‐informed ML approach where the GP estimator relies on the Physics laws that describe the Parseval energy curve patterns of DC cables. This paper makes the following contributions:1.The impact of fault location on the generated TWs is investigated and it is demonstrated that the so‐called Parseval energy curves that quantitatively relate TWs to the fault location have a similar pattern for a specific cable type regardless of its length and the system in which the cable is utilized.2.Using the inherent features of Parseval energy curves, a physics‐informed learning algorithm for fault location in DC systems is presented which requires a limited number of simulations for creating the training dataset of machine learning algorithm.3.The presented algorithm renders more accurate fault location estimation results with lower number of required simulations for creating the training dataset of machine learning algorithm.The remainder of the paper includes the following sections: Section 2 elaborates the TW fault location theory. Sections 3 discusses the MRA approach and Parseval's Theorem. The fault‐related Parseval energy curves are addressed in Section 4. Our proposed physics‐informed fault location algorithm is elaborated in Section 5 and is verified by simulation results in Section 6. The paper's conclusion is provided in Section 7.TW THEORYIn this section, the preliminaries of TWs are provided. The elaborated TW mathematical formulation will be used in Section 4 to discuss the impact of fault on TWs. A disruption (e.g. fault) in a power system component such as a cable or a line generates the electromagnetic waves that propagate through the power line, these waves are called traveling waves (TWs). The incident TWs, when it reaches a different environment (e.g. at a bus) having different parameters, gets reflected and refracted. The extent to which it gets reflected and refracted at the terminal depends on the characteristic impedances of the line and the adjacent circuit. A fault location or system buses will cause this phenomenon of reflection and refraction to happen. Telegrapher's equations are the coupled differential equations that relate the current i(x,t)$i(x,t)$ and voltage v(x,t)$v(x,t)$ at any point in space and time, TWs are formulated using these equations. The general solutions to these equations in the phasor domain are, [14, 27],1I∼(x,t)=I0+e−γx+I0−eγxV∼(x,t)=V0+e−γx+V0−eγx,\begin{equation} \begin{split} {\begin{cases} \tilde{I}(x,t) = I^{+}_{0}e^{-\gamma x} + I^{-}_{0}e^{\gamma x} \\[8pt] \tilde{V}(x,t) = V^{+}_{0}e^{-\gamma x} + V^{-}_{0}e^{\gamma x} \end{cases}} \end{split}, \end{equation}where propagation constant is γ and distance from the fault point is x. V0+$V^{+}_{0}$, V0−$V^{-}_{0}$, I0+$I^{+}_{0}$, and I0−$I^{-}_{0}$ denote the incident and reflection current and voltage TW components in Laplace form. Now, the current TW equation in the time domain can be defined as,2i(x,t)=i+(x,t)+i−(x,t)i+(x,t)=|I0+|e−αxcos(ωt−βx)i−(x,t)=|I0−|eαxcos(ωt+βx),\begin{equation} \begin{split} {\begin{cases} i(x,t) = i^{+}(x,t) + i^{-}(x,t) \\[4pt] i^{+}(x,t) = |I^{+}_{0}|e^{-\alpha x}\cos {(\omega t - \beta x)} \\[4pt] i^{-}(x,t) = |I^{-}_{0}|e^{\alpha x}\cos {(\omega t + \beta x)} \end{cases}} \end{split}, \end{equation}where in (2), attenuation constant is α and the phase constant is β which gives the propagation constant as,3γ=α+jβ=(R+jωL)(G+jωC),\begin{equation} \gamma = \alpha + j\beta = \sqrt {(R + j\omega L)(G + j\omega C)}, \end{equation}where TW's angular frequency is ω and G, R, C, and L represents the per unit length conductance, resistance, capacitance, and inductance of the transmission line respectively. So, the propagation velocity can be deduced as :4v=ωβ.\begin{equation} v = \frac{\omega }{\beta }. \end{equation}The above equation implies that although larger α makes travelling waves more attenuated, higher frequency traveling waves travel faster.MULTIRESOLUTION ANALYSISIn this section, the process of capturing and quantifying fault TWs are elaborated in detail. To this end, first wavelet transform (WT) is introduced. Then, MRA is deployed to identify the wavelet coefficients corresponding to TWs at different frequency ranges. Finally, Parseval's theorem is introduced for quantifying the captured TWs. The decomposition of a function into a set of wavelets is basically the WT. Over a range of frequency and time domain, continuous WT uses every possible wavelet, but discrete wavelet transform (DWT) uses only a finite set of wavelets. In both frequency and time domain, TWs can be effectively analyzed by utilizing the tool WT [28] where here DWT is used, which can be written as [25],5WD(m,k)=1a0m∑nx[n]gk−nb0a0ma0m.\begin{equation} W_D(m,k) = \frac{1}{\sqrt {a^{m}_{0}}}\sum _{n} x[n]g{\left[ \frac{k-nb_{0}a^{m}_{0}}{a^{m}_{0}} \right]}. \end{equation}In the above equation (5), the mother wavelet is denoted as g[], in which the function m(a0m,nb0a0m)$m(a^{m}_{0}, nb_{0}a^{m}_{0})$ is utilized to fit scaling and time‐shifting features with parameters a0 and b0, respectively.In this study, MRA [29, 30] is used to formulate wavelet coefficients of a TW signal for a wide range of frequencies. As shown in Figure 1, a number of low‐pass and high‐pass filters as well as decimators are utilized to implement MRA. Figure 1 defines each level of frequency range while assuming fs$f_s$ as the initial sampling frequency of DWT. ai[n]$a_{i}[n]$ and di[n]$d_{i}[n]$ are the outputs of low pass and high pass filters for each level. For constructing the next decomposition level's wavelet coefficients, the scaling coefficients or the outputs of the low pass filter at each level are used as the inputs of the next level. The high‐frequency behavior of the TW is preferable as the wavelet coefficients [25], so the detail wavelet coefficients are the output of the high pass filter. According to [31, 32] up‐sampling the coefficients by the factor of 2 is needed with the help of a reverse filter to reconstruct the original signal. With the help of the reconstruction process, one can obtain detail wavelet coefficients with a better time resolution and the same sampling rate as the original signal. This can be achieved by applying inverse filters of HHPF,i$H_{HPF,i}$ and GLPF,i$G_{LPF,i}$.1FIGUREMRA block diagramIn this paper, we have utilized the DWT tool in PSCAD/EMTDC [31] to implement MRA. The block diagram of this tool is illustrated in Figure 3. This tool allows for resampling of the signal at a sampling frequency different from the simulation time step. The tool utilizes an anti‐aliasing filter to filter out high frequency noise. The cut‐off frequency of the filter is adjusted based on the selected sampling frequency. The sampled data is stored in a data buffer and then is pushed into the wavelet decomposition and reconstruction stages. The detail wavelet coefficients will be generated at the same sampling rate as the original signal2FIGUREFault location impact on the Parseval energy value3FIGUREMRA processing steps using PSCADAccording to [25, 28], Parseval's theorem relates the wavelet coefficients to the measured signal energy spectrum if the mother wavelet and scaling function in (5) form an orthonormal basis. The energy of each identified wavelet coefficient is calculated using Parseval's theorem to understand the wavelet coefficients generated by MRA. The Parseval energy of a specific MRA output, at mth time step after an initial time t0 is calculated as [25, 28],6EPRS,i(m)=∑j=1mdi2(t0+jΔt),\begin{equation} E_{PRS,i}(m) = \sum _{j=1}^{m} d_i^2(t_0 + j \Delta t), \end{equation}where di(t)$d_i(t)$ is MRA's output for the ith decomposition level or frequency range at the time t; Δt$\Delta t$ is the time step used in DWT. This paper uses EPRS,i(m)$E_{PRS,i}(m)$ values to implement the physics‐informed learning technique for fault location of DC microgrids.FAULT‐RELATED PARSEVAL ENERGY CURVESFor a specific frequency range, a Parseval energy curve describes the Parseval energy value of current or voltage measurements at the protection device for different fault locations along a cable. A wavelet coefficient's Parseval's energy is analogous to the fault signal's spectrum of energy. A sample of Parseval's energy curve that illustrates the relation between variation in Parseval's energy at various fault locations is depicted in Figure 2. In Figure 2, each point denotes a separate analysis of Parseval energy value corresponding to a fault location. This curve is for a long DC cable on which bolted pole to pole (PP) faults are simulated at every 25 m of its length and a sampling frequency of 1 MHz is used for MRA. This figure shows the recorded level 1 Parseval's energy value of measured current at the cable terminal which corresponds to a frequency range from 250 kHz to 500 kHz. The pattern observed here is because of the inherent behavior of TW signals. The incident TW current experienced due to a fault can be written as,7i+(x,t)=|I0+|e−αxcos(ωt−βx),\begin{equation} i^{+}(x,t) = |I^{+}_{0}|e^{-\alpha x}\cos {(\omega t - \beta x)}, \end{equation}where x is the distance from the fault location. On the other hand, from (7), one can extract a function relating the incident TW current at the sensor location and fault location assuming faults at different locations along the cable. Thence, the incident TW current at the cable terminal and fault location xf$x_f$ can be formulated as,8i+(xf,t)=|I0+|e−αxfcos(ωt−βxf).\begin{equation} i^{+}(x_f,t) = |I^{+}_{0}|e^{-\alpha x_f}\cos {(\omega t - \beta x_f)}. \end{equation}Equation (8) implies that a combined sinusoidal and exponential behavior can be observed for the relationship between the incident TW current at the cable terminal where the sensor is located and the fault location. Some local peaks can be noticed because of the presence of cosine function in the equation. These local peaks at a specific time t1 are observed at a distance,9xLocalPeak=(ωt1−2nπ)/β,n=0,1,…\begin{equation} x_{Local Peak} = (\omega t_1 - 2n\pi )/\beta , n = 0,1,\ldots \end{equation}The Parseval energy is the sum of the square of MRA outputs (i.e. acts as an integrator) and an MRA output represents the magnitude of TW for a specific range of frequency which is extracted by applying DWT to (8). Equations (8) and (9) show that the pattern of the Parseval energy curves depends on α and β which according to (3) are functions of the cable parameters (per unit length conductance, resistance, capacitance, and inductance). Equation (9) describes that the incident TW current's local peaks (See Figure 2) at the protection device location are a function of TW angular frequency ω and take place periodically. β also corresponds to ω based on (3). Therefore, one can conclude that with the increase in ω, as it is in the numerator, more and more local peaks can be seen. Since the Parseval energy curve patterns highly depend on the cable per unit length parameters, the same patterns can be observed for a specific cable type regardless of the system in which the cable is deployed.FAULT LOCATION ALGORITHMThe overall diagram of fault detection and location algorithm is illustrated in Figure 4. As seen in this figure for each type of fault (i.e. bolted or resistive PP or PG faults), one different Gaussian Process (GP) regression engine is used to estimate fault location. For fault type classification, this paper utilizes the same algorithm that the authors proposed in their previous work in [27]. It should be noted that the fault type and its transition resistance affect the peak value of reflected and refracted TWs and therefore the Parseval energy curves. As seen in Figure 4, for each fault type and transition resistance value, a separate Gaussian Process algorithm is required to be trained. The fault type identification algorithm which is adopted from the previous work of the team in [27] can identify the type of a fault and its resistance based on the Parseval energy values of current, pole to pole (PP) voltage, and positive pole to ground (PPG) voltage measured at the protection device location. The fault type includes bolted and resistive pole to pole (PP) and pole to ground (PG) faults. The resistive faults include different values of transition resistance that can range from 1 to 100 Ω depending on the DC microgrid conditions like voltage level or geographical location. The algorithm in [27] utilizes the fact that the Parseval energies of the positive or negative pole to ground voltage are significantly higher than the Parseval energy of PP voltage to distinguish PP faults from PG ones. In order to effectively identify the fault's transition resistance, we have utilized Support Vector Machine (SVM) similar to the approach proposed in [27]. To train the SVM classifier, multiple bolted and resistive faults are simulated at different locations of cable (e.g. every 25 m) in a simulation software package (e.g. PSCAD/EMTDC). The SVM classifier is trained using the labeled Parseval energy values of current, pole to pole (PP) voltage, and positive pole to ground (PPG) voltage measured at the protection device location. The output of the SVM classifier is the fault resistance value. Once the fault type and its transition resistance are identified, the corresponding GP regression engine is selected for fault location. Each Gaussian Process is trained separately using the proposed physics‐informed algorithm in this paper.4FIGUREFault location algorithmThe hypothesis of this work is that if one can extract the Parseval energy curves of a specific cable type and configuration in a simple DC system, then those Parseval energy curves can be used for fault location on the cables in another DC system with similar cable types and configurations regardless of its size, number of integrated converters, number of cables etc. Conventionally, when a DC microgrid is constructed, the same cable type and configuration are utilized at the different locations of the microgrid. For a specific cable type and configuration, regardless of the cable's length, the Parseval energy curves will have a similar pattern in any DC system. Only the Parseval energy curve magnitudes will change based on the change of the DC system specifications (e.g. architecture, loads, converter's ratings etc.). In the proposed approach, a protection device utilizing the proposed fault location algorithm is placed on every cable segment in the system.For training the GP algorithm, we need the six levels of Parseval energy values of current measured at the protection device for multiple fault locations along a cable to train the GP algorithm. We are proposing to have these Parseval energy values for faults at every 25 m of the cable. For example, for a cable with length of 1000 m, we need the Parseval energy values for 40 different fault locations to train the GP algorithm. However, gathering these Parseval energy values for an extensive number of fault locations can be a very time consuming effort. It should be noted that extracting the Parseval energy curves from PSCAD simulations of a simple DC system is relatively fast due to the system size. However, running each PSCAD simulation of a large‐scale DC microgrid can take several hours since TWs are required to be captured at small simulation time steps with frequency‐dependent cable models to be used in the simulation model. To tackle this issue, the following algorithm is proposed in which we just need to gather the actual Parseval energy values for a limited number of fault locations and estimate the Parseval energy values of other fault locations that are required to train the GP algorithm. The proposed fault location algorithm is summarized as follows:Step1:$Step 1:$ On a simple DC system, the Parseval energy values for the current at the protection device location are calculated for faults at different locations of the line/cable (e.g. faults at every 25 m of the line/cable length) using PSCAD/EMTDC. This paper uses at least six MRA frequency levels to incorporate an adequate portion of the frequency spectrum for extracting fault current features as the fault location changes.Step2:$Step 2:$ We use the patterns of the extracted Parseval energy curves from the simple DC system to identify the Parseval energy curves for a specific protection device in the DC microgrid under study. To this end, we need Parseval energy values for faults at some sample locations (lk$l_k$) along the cable protected by the protection device in the DC microgrid under study. The Parseval energy values of faults at these sample locations will be used to tune the magnitude of the Parseval energy curves. Our goal here is to construct Parseval energy curves with 25 m fault location resolution. To this end, the Parseval energy values of fault locations at every 25 m (ln$l_n$) are constructed by the Parseval energy values of the closest sample location using10EPRS,i,ln=EPRS,i,lkEPRS,i,ln′EPRS,i,lk′,\begin{equation} E_{PRS,i,l_n} = \frac{E_{PRS,i,l_k}E^{^{\prime }}_{PRS,i,l_n}}{E^{^{\prime }}_{PRS,i,l_k}}, \end{equation}where EPRS,i,ln′$E^{^{\prime }}_{PRS,i,l_n}$ and EPRS,i,lk′$E^{^{\prime }}_{PRS,i,l_k}$ are the ith MRA level's Parseval energy values captured from simple DC system at faults at locations ln$l_n$ and lk$l_k$. EPRS,i,ln′$E^{^{\prime }}_{PRS,i,l_n}$ and EPRS,i,lk′$E^{^{\prime }}_{PRS,i,l_k}$ are the ith MRA level's Parseval energy values captured from DC microgrid under test for faults at locations ln$l_n$ and lk$l_k$. It should be noted that the Parseval energy values of the sample locations can be either gathered from the field measurements or from the simulation model of the DC microgrid under study in PSCAD.Step3:$Step 3:$ Once the Parseval energy curves with 25 m fault location resolution are constructed for the protection device in the DC microgrid under study, the Parseval energy values will be used to train a GP regression engine. The Gaussian Process regression model training procedure is shown in Figure 5. Once the GP engine is trained, one can use it to find fault location for any new fault scenarios by feeding the measured Parseval energy values at the protection device location to the GP engine.RemarkFor each frequency range, the Parseval energy is calculated right after the arrival of the first TW incident. It should be noted that the TWs' speed corresponds to the frequency range. Therefore, with a higher sampling frequency of MRA, one can accommodate higher frequency ranges in the algorithm and detect the first incident of TWs faster. We have selected 1 MHz of sampling frequency in his paper to make its implementation feasible on commercially available microprocessors. With the recent advancements in signal processing and measurement technologies, high‐frequency data sampling and measurement can be easily accommodated for the implementation of the proposed scheme. In fact, existing commercial TW relays are able to perform very high frequency (in the order of MHz) measurements [33–35].5FIGUREGaussian process regression model training procedureSIMULATION RESULTSConsider the simple DC system in Figure 6a. The nominal pole‐to‐pole voltage of this system is 750 V. This system is modeled in PSCAD/EMTDC. The resistance of the load is 10 Ω. The configuration of the cable under study is shown in Figure 6b. The length of the cable is 3000 m. The frequency‐dependent distributed parameter model available in PSCAD/EMTDC is used for modeling the cable. The cable is placed 1 m under the ground. The core conductor resistivity is 2×10−8$2\times 10^{-8}$ Ωm; the sheath resistivity is 30×10−8$30\times 10^{-8}$ Ωm. To generate Parseval energy curves for the cable shown in Figure 6a, PP and pole‐to‐ground (PG) faults are applied at every 25 m of the cable. The sampling frequency of MRA is equal to 1 MHz. The DC microgrid under study is shown in Figure 7. The proposed approach is used to create the fault location algorithm at protection device R25, R52, R56, R65, R26, and R62 in this microgrid system. This microgrid is supplying six nanogrids (NG). Each NG includes a load, a PV system, a BESS, and DC‐DC converters to integrate NG into the rest of the microgrid. In each NG, the BESS size is 3 kW and 6 kWh and the PV size is 5 kW. The load of each NG is equal to 5 kW. It is assumed that all cables in this microgrid are of the same cable type shown in Figure 6b but with different length values. In this study, the sampling frequency of DWT is 1 MHz. Daubechies (db8) is used as the mother wavelet. According to [25, 27], db mother wavelets are promising candidates that facilitate fast and accurate MRA by incorporating enough vanishing points for accounting for the salient features of waveforms, accommodating sharp cutoff frequencies to minimize the amount of energy leakage to the next decomposition level, and being orthonormal.6FIGURE(a) The simple DC test system; (b) the configuration of cable7FIGUREDC microgrid under studyTo show that same Parseval energy patterns can be observed for a specific type of cable that used in two different systems, the Parseval energy values are captured for some bolted PP fault scenarios along the cable from node 2 to node 5 (at R25) of the mesh microgrid shown in Figure 7. For a similar type of fault, the Parseval energy curves gathered from the simple DC system in Figure 6a are also plotted to compare their patterns against the Parseval energy values captured from the mesh DC microgrid under study in Figure 7. These comparisons are provided in Figures 8–13. As seen, similar patterns can be observed for the Parseval energy curves of a specific cable type regardless of the system in which the cable is deployed. The Parseval energy is the sum of the square of MRA outputs (i.e. acts as an integrator) and an MRA output represents the magnitude of TW for a specific range of frequency which is extracted by applying DWT to (8). Equation (8) shows that the pattern of the Parseval energy curves depends the cable parameters (per unit length conductance, resistance, capacitance, and inductance). It should be noted that the Parseval energy curves of the simple DC microgrid system have different scales and magnitudes compared to the Parseval energy curves gathered from the mesh DC microgrid. The scale of Parseval energy curve depends on |I0+|$|I^{+}_{0}|$ in (8) which in turn depends on different factors like the specifications and ratings of converters. The local minimum and maximum peaks on the curves can be found according to (9) which shows that these minimum and maximum peaks are a function of TW angular frequency ω and take place periodically. β also corresponds to ω based on (3). Therefore, one can conclude that with the increase in ω, as it is in the numerator, more and more local peaks can be seen. This behavior can be observed in Figures 8–13 where the number of local peaks decrease for the Parseval energy curves of lower frequency ranges. In conclusion, a specific cable type renders similar Parseval energy curve pattern in different systems, which is the hypothesis of the proposed algorithm.8FIGURELevel 1 Parseval energy curve pattern comparison between (a) the simple DC microgrid system and (b) mesh DC microgrid under study at R259FIGURELevel 2 Parseval energy curve pattern comparison between (a) the simple DC microgrid system and (b) mesh DC microgrid under study at R2510FIGURELevel 3 Parseval energy curve pattern comparison between (a) the simple DC microgrid system and (b) mesh DC microgrid under study at R2511FIGURELevel 4 Parseval energy curve pattern comparison between (a) the simple DC microgrid system and (b) mesh DC microgrid under study at R2512FIGURELevel 5 Parseval energy curve pattern comparison between (a) the simple DC microgrid system and (b) mesh DC microgrid under study at R2513FIGURELevel 6 Parseval energy curve pattern comparison between (a) the simple DC microgrid system and (b) mesh DC microgrid under study at R25In addition to the actual Parseval energy curves for the cable from node 2 to node 5 of mesh microgrid system, Figures 8b, 9b, 10b, 11b, 12b and 13b illustrate the estimated Parseval energy curves based on the algorithm discussed in Section 5. In these figures, the red dots describe the Parseval energy values that are captured from simulation of Mesh microgrid system and are utilized to estimate the Parseval energy values for faults at other locations of the cable. The estimated Parseval energy values will be later used to train the GP fault location estimation engine.We have applied the proposed technique on the loop highlighted in red in Figure 7 (i.e. protection device R25, R52, R56, R65, R26, and R62). At these protection devices, we have gathered the Parseval energy values for faults at every 100 m fault location. Then, using the proposed method, the high‐resolution Parseval energy curves for these protection devices are constructed with a resolution of 25 m fault location. These Parseval energy values are then used to train a GP estimator. In order to test the performance of the GP estimator, we have used Parseval energy values of faults at 50 m, 150 m, 250 m, 350 m etc. We have created a separate GP estimator for each fault type (i.e. bolted and resistive pole‐to‐pole (PP) and pole‐to‐ground (PG) faults). For resistive faults, the fault transition resistance is equal to 5 Ω. The testing results including Mean Absolute Percentage Error (MAPE) and Mean Absolute Error (MAE) for different types of faults are summarized in Table 1. Each protection device is responsible for fault location on its protected cable. The fault detection is performed within 1 ms after the fault is applied.1TABLEFault location estimation errors in the DC microgrid under studyBolted PPResistive PPBolted PGResistive PGProtection devicesMAPEMAE (m)MAPEMAE (m)MAPEMAE (m)MAPEMAE (m)R250.51%3.930.91%7.432.96%16.621.02%8.04R521.16%9.320.80%7.983.95%20.240.80%6.72R560.90%4.450.83%5.731.64%11.311.03%6.61R651.08%5.550.90%5.992.23%15.701.24%7.25R260.41%4.080.75%6.570.91%8.270.85%7.35R620.64%4.880.66%6.681.45%15.170.85%8.02GP model parametersAccording to [36], a GP represents a set of any finite number of random variables that have a joint Gaussian distribution. A GP regression tool finds the solution by incorporating the GP‐based latent variables {f(xi)|i=1,…,n}$\lbrace f({{\mathbf {x}}_{\mathbf {i}}})|i=1,\ldots ,n\rbrace$ and the basis functions h according to y=h(x)Tγ+f(x)$y=h{{\text{(}\mathbf {x}\text{)}}^{T}}\gamma +f\text{(}\mathbf {x}\text{)}$, with f(x)$f\text{(}\mathbf {x}\text{)}$ formed by a GP with zero mean and covariance function of k(x,x′)$k\text{(}\mathbf {x},{{\mathbf {x}}^{^{\prime }}}\text{)}$ (the covariance functions are represented by a set of Kernel hyperparameters). A sample solution y can be formulated as11P(yi|f(xi),xi)∼N(yi|h(xi)Tγ+f(xi),σ2).\begin{equation} P({y}_{i}|f({\mathbf{x}}_{i}),{\mathbf{x}}_{i})\sim N{({y}_{i}|h({\mathbf{x}}_{i})}^{T}\gamma +f({\mathbf{x}}_{i}),{\sigma}^{2}). \end{equation}A set of n observations (inputs and outputs) comprises the training set. In the training set, the input includes a set of six Parseval energy values calculated for the current measured at the protection device location, and the output is the fault location. In this paper, we have utilized the Gaussian Process Regression tool in MATLAB [37] to train the GP engines. In total, for six protection device locations and four fault types, twenty four GP engines are modeled. For all of them, the exponential Kernel function is used. The γ and σ of the trained models are summarized in Table 2.2TABLEGaussian process parameters (exponential Kernel function is used)ProtectionBolted PPResistive PPBolted PGResistive PGDevicesγσγσγσγσR25909.45.301082.75.3656.65.31103.85.3R521197.15.310808.81244.65.31089.17.3R56731.23.3742.13.3718.612.2749.310.3R65739.53.3746.23.3691.327.5752.37.5R261061.36.41272.96.410826.41298.46.4R621236.26.41286.96.4810.96.41293.96.4To highlight the performance of proposed physics‐informed machine learning fault location algorithm against the curve fitting algorithm in [28], these two algorithms are compared in Figure 14. This comparison is performed on protection device R25 of the mesh DC microgrid in Figure 7. It is assumed that both algorithms only use the Parseval energy values of faults at every 100 m of the cable to train their algorithms. In the curve fitting approach, these Parseval energy values are used to identify the best curves fitting the data for all six MRA levels. As seen in these figures, the Physics‐informed algorithm renders much more accurate fault locations. The curve fitting algorithm's MAPE values for bolted and resistive PP and PG faults at R25 are 29.25%, 11.06%, 21.04%, and 25.07%, respectively.14FIGUREA comparison of fault location prediction between physics‐informed machine learning and curve fitting tool of mesh DC microgrid under study at R25 for (a) bolted PP faults (physics‐informed approach's MAPE = 0.51% and curve fitting approach's MAPE = 29.25%); (b) resistive PP faults (physics‐informed approach's MAPE = 0.91% and curve fitting approach's MAPE = 11.06%); (c) bolted PG faults (physics‐informed approach's MAPE = 2.96% and curve fitting approach's MAPE = 21.04%); (d) resistive PG faults (physics‐informed approach's MAPE = 1.02% and curve fitting approach's MAPE = 25.07%)Verification of the performance of proposed algorithm for different fault resistance valuesHerein, the proposed algorithm is applied to protection device R25 in Figure 7 considering PG faults with fault transition resistance values of 1, 5, 10, 50, and 100 Ω. The accuracy of the proposed method for different fault transition resistance values is summarized in Table 3. In general, the fault transition resistance value has an impact on the magnitude of the Parseval energy curves, and their magnitude decreases as the fault resistance value increases. This behavior is shown in Figure 15.3TABLEFault location estimation errors for different fault resistance values in the DC microgrid under studyFault ResistanceR25Values (Ω)MAPEMAE (m)11.45%11.4751.02%8.04100.98%7.91500.68%6.631000.97%8.7415FIGURELevel 1 Parseval energy curve of the Simple DC Microgrid for various fault resistances when PG fault occurs.Comparison of the proposed algorithm against the existing algorithms in the literatureTable 4 compares the MAE of our proposed algorithm's fault location estimation against the fault location errors provided in [17]. Table 5 compares this paper's proposed algorithm against the existing DC microgrid TW protection techniques in the literature. This paper compares these algorithms considering the network type, line models, methodology, required data, required number of simulations and training data points to create the model, type of faults considered, and accuracy. The comparison shows that the proposed approach renders a very high accuracy while requiring a low number of data points for training the machine learning algorithm. This is the major advantage of our approach compared to the existing approaches in the literature.4TABLEComparison between Bolted PG MAE values of this paper with [17]RelayThis paperReference [3]Bolted PG MAE (m)Bolted PG error (m)R2516.621R5220.2434R568.2724R6515.1722R2611.3114R6215.7225TABLEComparison between approach of this paper and existing approaches in the literatureApproach (author, year)Network typeLine ModelFault location levelMethodologyRequired data for modelNumber of simulations and training data points required to create the modelType of FaultAccuracyThis paperDC microgridFrequency dependentDistance on the cablePhysics‐informed machine learning and DWTFault data for different locations on the cableLowBolted and Resistive PP and PGHigh (more than 96%)Ref. [[3]] (Saleh et al., 2020)DC microgridFrequency dependentDistance on the cableNumerical analysis on waveshape featuresFault data for different locations on the cableHighBolted and Resistive PP and PG (*Fault location results are only provided for bolted PG)High (percentage is not provided)Ref. [[23]] (Li et al., 2014)DC shipboard power systemNot applicableFaulted busANN and DWTFault data for different buses in the systemMedium (Only bus faults simulation data is required)Bolted PPHigh (more than 96%)Ref. [[24]] (Jayamaha et al., 2019)DC microgridFrequency dependentFaulted busANN and DWTFault data for different buses in the systemMedium (Only bus faults simulation data is required)Bolted and Resistive PP and PGHigh (more than 96%)CONCLUSIONThis paper creates a physics‐informed machine learning approach for fault location in DC microgrid by capturing high‐frequency TWs. TWs are extracted by the so‐called multiresolution analysis which identifies the TW's wavelet coefficients for multiple frequency ranges. This paper deploys Parseval's theorem to find the energy of wavelet coefficients as a quantitative metric for describing TWs. Once the Parseval energy curves for a specific cable are extracted, they can be utilized to locate faults along with that cable regardless of the DC system in which the cable is deployed. The fault location algorithm uses Parseval energy curves to train a Gaussian Process estimator. With the Parseval energy values of measured current at the protection device location as the inputs, the Gaussian Process estimator is able to estimate fault locations with high accuracy. The fault location algorithm is verified on a DC microgrid system simulated in PSCAD/EMTDC.ACKNOWLEDGEMENTSThis material is based upon work supported by the Laboratory Directed Research and Development program at Sandia National Laboratories, the U.S. Department of Energy's Office of Energy Efficiency and Renewable Energy (EERE) under Solar Energy Technologies Office (SETO) Agreement Number 36533, and National Science Foundation EPSCoR Cooperative Agreement OIA‐1757207. Sandia National Laboratories is a multimission laboratory managed and operated by National Technology & Engineering Solutions of Sandia, LLC, a wholly owned subsidiary of Honeywell International Inc., for the U.S. Department of Energy's National Nuclear Security Administration under contract DE‐NA0003525. This paper describes objective technical results and analysis. 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"IET Generation, Transmission & Distribution" – Wiley

Published: Dec 1, 2022

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