Abstract
High-impedance faults (HIFs) in distribution networks may result in fires or electric shocks. However, considerable difficulties exist in HIF detection due to low-resolution measurements and the considerably weaker time-frequency characteristics. This paper presents a novel HIF detection method using synchronized current information. The method consists of two stages. In the first stage, joint key characteristics of the system are extracted with the minimal system prior knowledge to identify the global optimal micro-phase measurement unit (μPMU) placement. In the second stage, the HIF is detected through a multivariate Jensen-Shannon divergence similarity measurement using high-resolution time-synchronized data in μPMUs in a high-noise environment. l2,1 principal component analysis (PCA), i.e., PCA based on the l2,1 norm, is applied to an extracted system state and fault features derived from different resolution data in both stages. An economic observability index and HIF criteria are employed to evaluate the performance of placement method and to identify HIFs. Simulation results show that the method can reliably detect HIFs with reasonable detection accuracy in noisy environments.
HIGH-IMPEDANCE faults (HIFs) typically occur in distribution networks (4-34.5 kV) [
In previous studies, the transient and nonlinear properties of an HIF have generally been applied in detection using different domain analytical methods [
Although a PMU has very high precision and relatively low cost as compared with a PMU [
By contrast, HIF transient characteristics and nonlinearities vary with grounding surface features such as material type, humidity, and other factors [
In the present work, we propose an improved HIF detection method using PMU data in distribution networks. The primary purpose of the proposed method is to detect a fault based on the difference between the joint characteristic probability density functions of the system state before and during the fault. l2,1 PCA, i.e., PCA based or the l2,1 norm, is applied to extract the joint characteristics. By comparison, the proposed method requires only a few PMUs, which are determined by the optimal PMU placement algorithm. The main contributions of this paper are summarized as follows.
1) HIF detection: fault detection is accomplished by comparing the difference between the joint characteristic probability density functions of the system state before and during the fault. The proposed detection method is based on the difference in the statistical probability distribution of the synchronized data before the fault and fault duration based on the optimized PMU placement.
2) Optimal placement of PMU (OPP): the optimal results identified by l2,1 PCA are globally optimal locations from which measurement data are sensitive to state changes. This implies that using the data from these locations for fault detection may satisfy the time constraint of fault detection. The proposed economic observability index can be used to evaluate PMU placement schemes in practical applications.
3) Challenge of distribution network: combining optimal PMU placement with HIF detection, the proposed method solves the globally optimal problem of PMU locations, avoids the difficulty in selecting various configuration schemes in practical applications, and realizes HIF detection based on the data provided by PMUs at the optimal locations. The proposed method provides a possible implementation for HIF detection using synchronous data probability distribution features.
The remainder of this paper is organized as follows. Section II describes the HIF model and problem formulation. Section III proposes the materials and methods. Section IV presents the case studies. The conclusion is presented in Section V.
The features of HIFs include grounding materials, grounding surface humidity, system operations, and weather conditions, which vary considerably in different situations. Hence, many HIF models have been presented [

Fig. 1 HIF model.
The HIF model is formulated as:
(1) |
Considering a distribution feeder with two and several buses between the , these buses may or may not have laterals. The model of a distribution feeder is obtained by applying Kirchhoff’s law, as shown in

Fig. 2 π model of a distribution feeder with two μPMUs.
In
(2) |
where represents the phasor value of the
(3) |
where and are the real and imaginary parts of the , respectively. The PMU dataset can be rewritten as:
(4) |
where denotes the number of recorded variables.
In this paper, synchronized data are used for the HIF detection procedure. As previously mentioned, the fault detection problem is a time-constraint problem. This means that the data for fault detection are sensitive to abrupt fluctuations in the system states. However, finding a benchmark system with PMUs to meet data sensitivity requirements is difficult because most proposed benchmark systems are not designed to detect a fault but rather to estimate the system state [
Because a data-driven method is used to detect an HIF in this study, the following assumptions are made regarding the used data.
Assumption 1: the maximum percentage measurement errors involve assumptions similar to those found in [
1) Substation measurements: 1%. Measurements include the voltage magnitude, current magnitude, and active and reactive power flows at the substation. Although substation measurements have different accuracies according to different time references, the measurements are considered to be highly accurate if a time reference is precise in the substation. Therefore, a 1% error is chosen for the substation measurement.
2) Real measurements: 10%. Measurements include active and reactive power in lines from the intelligent electronic device (IED). Due to the uncertainty of the instrument and the influence of possible signal dynamics, the error of real measurements is set to be 10%.
3) Pseudo-measurements: 50%. Measurements are obtained from historical load data.
4) µPMU measurements: a total error vector of 1% is compliant with the IEEE synchro-phasor standard [
Assumption 2: since a zero-injection bus rarely appears in a distribution grid [
This section proposes two steps for implementing two sub-procedures: ① feature extraction, and ② feature analysis and discrimination, which consist of HIF detection and optimal allocation of PMUs.
Events, e.g., switching, fault, and contingent, in the system cause data fluctuations in the monitored variables such as voltage and current amplitudes. These data fluctuations are often similar to the step changes at the beginning of an event. Because PCA is quite sensitive to step change, it is increasingly used in power systems, e.g., fault detection [
(5) |
where is the objective function, and is an adjustment coefficient, Y is the PCs obtained by PCA, X is the measurement data matrix, P is the loading matrix; and is the norm of the matrix.
Differentiating J(P) with regard to P and setting the result to be zero, we can obtain:
(6) |
where ; and and are the diagonal matrices, with the diagonal element given as:
(7) |
(8) |
Thus, P can be obtained as:
(9) |
Note that P is dependent on and and is thus an unknown variable. An iterative algorithm to solve (9) can then be designed as follows. The l2,1 PCA algorithm is given in
Algorithm 1 : PCA |
---|
Input: the measurement data matrix |
Output: the component analysis |
Set , and initialize , |
Calculate SVD of for obtaining and |
Repeat |
Calculate |
Update and |
|
Until convergence.
|
Remark 1: a sample size of 30 is sufficiently large to enable the probability density function (PDF) of samples to approach a Gaussian distribution [
Remark 2: in this paper, the sensitivity of l2,1 PCA to data changes and the global optimality of the solution are used to identify the nodes sensitive to system state changes to obtain a benchmark system with optimal PMU placement for HIF detection.
Remark 3: since l2,1 PCA can extract joint features from data, making it more sensitive to data fluctuations caused by system state changes as compared with standard PCA, the l2,1 PCA method is also used to extract data features from synchronized measurements to detect the HIF.
For Gaussian data such as the recorded by PMUs, the principal scores obtained by l2,1 PCA obey a Gaussian distribution with respect to central limit theorem. Then, the PDF of the principal scores satisfies:
(10) |
where and are the PDFs of the principal scores for the
(11) |
where DJS and DKL are the JSD and KLD values, respectively; and satisfies a mixture distribution.
For n PMUs, the historical data matrix is structured according to (4). Without loss of generality, the online data matrix structure is consistent with the historical data matrix structure. Suppose and are denoted as the PDFs of the first principal scores obtained under the normal and faulty conditions, respectively. and are the n-dimensional mean vectors of p and q, respectively; and and are the covariance matrix of p and q, respectively. The univariate JSD value in (11) can be rewritten as:
(12) |
where M is the mixture distribution. The covariance matrix is a symmetric position definite matrix. DJS can be rewritten as:
(13) |
where is the trace of the matrix. Since the mean of the distribution is assumed to be unchanged (zero) after the occurrence of an HIF, an HIF will not move the center of the l2,1 PCA model.
(14) |
(15) |
(16) |
When the divergence value is theoretically equal to zero, the two considered PDFs are identical. The references and tested functions are both obtained under normal operational conditions. In practical cases, a low nonzero divergence is always caused by the factors such as random noise/fluctuations, normal switching operations, or data acquisition devices.
In a noisy environment, the variances of the first principal scores and consist of the variance of the principal score without noises and the noise variance, respectively.
(17) |
where and are the variances of the first PC under the fault-free and faulty states, respectively; is the noise variance; and I is a unit matrix of the same size as . Faults result in a change in the distribution network, and the variance changes considerably in the fault direction [
(18) |
where is the variance bias due to the occurrence of an HIF. Based on (11), (13), and (15), the variances in M are obtained as:
(19) |
Thereby, (12) can be further rewritten as:
(20) |
where DJS has a clear relationship with the variance deviation caused by the fault and random noise. Therefore, a threshold must be used to determine whether the system enters an abnormal operating state. Because the threshold selection affects the method performance, the threshold may be set by evaluating the JSD on the PDFs of the first principal scores calculated from the PMU data in this paper. We use a sliding window of a fixed size L, and the first principal score is partitioned into sections using a sliding window. However, a large window size decreases the detection delay and accuracy and causes some essential fault information to be neglected. Thus, a trade-off between accuracy and delay detection is required. During regular operation, the current waveforms in the two subsequent sections are similar. In other words, the first principal scores in the two subsequent sections are also similar. Therefore, we use the average value of the distribution with the first principal score with fault-free states:
(21) |
where K is the total number of partitioned sections. Since the zero-section f0 does not exist, f0 is generally represented by initializing the section. Since the switching event time is usually approximately 2-3 cycles and the duration of the HIF often lasts for 8-10 cycles or more [
(22) |
(23) |
where is the duration. The time criterion is used to avoid false detections derived from normal operations such as switching events.

Fig. 3 HIF detection procedure.
Remark 4: it is evident that the proposed method can also be applied to the detection of low-impedance faults (LIFs). LIFs are easily detected and can trigger the protection device. The fault detection procedure presented in
Some operation state changes are often mistaken as disturbances at the early stage, thereby increasing the chance of false identification of sensitive locations. The monitored variable is reconstructed in the state change direction to determine sensitive locations, i.e.,
(24) |
where zi is the reconstructed variable; x is the measurement vector of variable xi; is the direction of the state change and the
(25) |
where is the
(26) |
where is the covariance matrix of ; and is the chi-square distribution with degree of freedom.
For the
(27) |
where is the weight coefficient corresponding to the significance of the monitored variables and ; and and are the combined index and threshold for the
To facilitate a comparison of the comprehensive sensitivity of each node, the relative contribution index (RCI) is defined as:
(28) |
where is the upper control limit of each bus, given in the following form:
(29) |
where is the mean of . If the relative contribution is more than one, we can select it as the bus responsible for placing the PMU. The optimal placement identification procedure is illustrated in

Fig. 4 Optimal placement identification procedure.
Remark 5: the l2,1 PCA method is applied in fault detection and deployment, but the processed data sources are different. The data analyzed by fault detection derive from the measurement data, and the data for deployment derive from the IED or other measuring devices.
The proposed method is tested on an IEEE 34-node system, which is a three-phase unbalanced test system [

Fig. 5 Single-line illustration of IEEE 34-node system.
Bus No. | Active power (MW) | Reactive power (Mvar) |
---|---|---|
890 | 0.235 | 0.127 |
840 | 0.124 | 0.021 |
The maximum error of the DG outputs is 3%. It is assumed that the measurement noise obeys Gaussian distributions. In the detection cases, PMUs obtain voltage and current phasor measurements for HIF detection. In most HIF detection results, the ground resistances are between 100 and 2 [
Component () | Value range |
---|---|
Rp, Rn | 500-5000 |
Lp, Ln | 20-80 |
PSCAD/EMTDC is used to obtain the magnitude and phase angle of the voltage and current, active and reactive power flows, and injected active and reactive power on each bus, and MATLAB 2016(a) is applied to analyze the measurement data.
Compared with the improved PCA (IPCA) [
No. of PC | l2,1 PCA | IPCA | ||
---|---|---|---|---|
Var. (%) | Cum var. (%) | Var. (%) | Cum var. (%) | |
1 | 94.56 | 94.56 | 88.08 | 88.08 |
2 | 3.62 | 98.18 | 7.70 | 95.78 |
3 | 1.34 | 99.52 | 3.52 | 99.30 |
4 | 0.35 | 99.87 | 0.51 | 99.81 |
5 | 0.12 | 99.99 | 0.08 | 99.89 |
Table III shows that the sum of the first two explained variances is higher than that of the IPCA (98.18% versus 95.78%), and the cumulative variances of the five PCs are 6.48%, 2.4%, 0.22%, 0.06%, and 0.1% higher than those of the IPCA, respectively. The information in the first PCs is more than that in the IPCA. This is why the proposed method has joint feature extraction characteristics that are more sensitive to an operation state change resulting from DG access. To achieve a trade-off between system observability and device cost, the economic observability index (EOI) is defined as:
(30) |
where is the system observability redundancy for every $100; is the cost of , which is approximately $3500 for every PMU [
Method | No. of PMU | SORI | EOI |
---|---|---|---|
Proposed method | 12 | 42 | 1.000 |
Reference [ | 14 | 38 | 0.776 |
Reference [ | 12 | 42 | 1.000 |
Reference [ | 13 | 40 | 0.879 |
Table IV shows that 12 PMUs are acquired by the proposed method, which are fewer than the numbers reported in [
This subsection uses the IEEE 34-node system as a case to show fault detection using measurements from PMUs on the representative buses, as shown in Table V and by the red dots in
PMU No. | Bus No. | PMU No. | Bus No. | PMU No. | Bus No. |
---|---|---|---|---|---|
1 | 800 | 5 | 820 | 9 | 890 |
2 | 802 | 6 | 824 | 10 | 858 |
3 | 808 | 7 | 854 | 11 | 844 |
4 | 850 | 8 | 888 | 12 | 836 |
Fault detection performance is evaluated for transition impedances of 2 k and 5 k for a single-phase-to-ground fault. The measurement noise is assumed to follow a Gaussian distribution. The SNRs are 60 dB, 30 dB, and 10 dB, respectively. In this paper, the size of window L is 6, that is, one bin for every six sampling points, which is an empirical value. The total vector error of the PMU is 1% [
A single-phase-to-ground occurs in phase B between the 82

Fig. 6 Fault phase current.
The fault current exhibits distinct HIF characteristics, and the fault current magnitude is 1% of the load current, as shown in

Fig. 7 Fault detection results for and .

Fig. 8 Fault detection results for when .

Fig. 9 Fault detection results for . (a) . (b) .

Fig. 10 Fault detection results for dB. (a) . (b) .
The proposed method can detect HIFs in both scenarios, as shown in
The fault inception angle (FIA) affects the fault voltage/current magnitude, which in turn affects the fault detection results. When the most severe situation, i.e., is considered,

Fig. 11 Fault detection results for and (a) . (b) .
In addition, the cumulative time of the continuous limit violation exceeds the time limit. Although the monitoring result of the 85
Nevertheless, the monitoring results fluctuate near the threshold under the influence of noise. An HIF can be determined by calculating the cumulative time of the continuous limit violation.

Fig. 12 Detection probability against different SNRs and different earthing impedances.
Because the switching of loads and capacitor banks produces features similar to those from high-impedance waveforms, the performance of the proposed method is tested during capacitor switching (CS) and load switching (LS). The first scenario is to switch on the one-phase capacitor bank installed at the 80

Fig. 13 Capacitor and load models. (a) Capacitor model. (b) Linear load model. (c) Nonlinear load model.
The capacitor bank and two load models are similar to those in [

Fig. 14 Detection results of CS and LS. (a) CS. (b) Linear LS. (c) Nonlinear LS.
This paper proposes an HIF detection method using synchronized data divergence based on globally optimal placement. The proposed method consists of optimal placement and HIF detection. First, an optimal placement algorithm is proposed to obtain the globally optimal location for fulfilling fault detection time constraints. Second, HIF detection is presented based on the probability distribution divergence of the synchronous data before and during a fault. The performance of the proposed method is validated using simulated data from an IEEE 34-node test feeder system. The results reveal that, through feature extraction and synchronous data divergence discrimination, sensitive nodes could be captured to determine the optimal location and accurately detect HIFs using multivariate JSD combined with l2,1 PCA in high-noise environments. The probabilities of fault detection are 92.12% (500 ), 83.28% (2 ), and 73.93% (5 ) for , respectively, and are greater than 99% under . The method can accurately detect faults in high-noise environments and is robust to transient resistance, the fault inception angle, and noise.
Renewable energy sources with different access locations, access capacities, and capacity ratios affect fault detection. Suppose that the effect of renewable energy on fault detection is the noise. As shown in the simulation verification, the noise may play a supporting role in enhancing HIF detection. Therefore, in future research, we intend to use the noise-assisted method to increase the sensitivity of HIF detection under reliability constraints. Another main task is to identify fault lines and phrases and locate the fault based on data probabilistic statistics.
References
A. Ghaderi, H. L. Ginn, and H. A. Mohammadpour, “High impedance fault detection: a review,” Electric Power Systems Research, vol. 143, pp. 376-388, Feb. 2017. [Baidu Scholar]
M. Adamiak, C. Wester, M. Thakur et al. (2015, Jan.). High impedance fault detection on distribution feeders. [Online]. Available: https://www.aer.gov.au/system/files/Mr%20Marcus%20Steel%20-%20Attachment%20B%20to%20submission%20on%20Ergon%20Energy%20application%20for%20ring%20fencing%20waiver%20-%20December%202015.pdf [Baidu Scholar]
B. D. Russell and C. L. Benner, “Arcing fault detection for distribution feeders: security assessment in long term field trials,” IEEE Transactions on Power Delivery, vol. 10, no. 2, pp. 676-683, Apr. 1995. [Baidu Scholar]
D. P. S. Gomes, C. Ozansoy, A. Ulhaq et al., “The effectiveness of different sampling rates in vegetation high-impedance fault classification,” Electric Power Systems Research, vol. 174, pp. 1-10, May 2019. [Baidu Scholar]
X. Wang, J. Gao, X. Wei et al., “High impedance fault detection method based on variational mode decomposition and Teager-Kaiser energy operators for distribution network,” IEEE Transactions on Smart Grid, vol. 10, no. 6, pp. 6041-6054, Nov. 2019. [Baidu Scholar]
Q. Cui, K. El-Arroudi, and Y. Weng, “A feature selection method for high impedance fault detection,” IEEE Transactions on Power Delivery, vol. 34, no. 3, pp. 1203-1215, Jun. 2019. [Baidu Scholar]
B. Wang, J. Geng, and X. Dong, “High-impedance fault detection based on nonlinear voltage-current characteristic profile identification,” IEEE Transactions on Smart Grid, vol. 9, no. 4, pp. 3783-3791, Dec. 2018. [Baidu Scholar]
Q. Cui and Y. Weng, “Enhance high impedance fault detection and location accuracy via [Baidu Scholar]
,” IEEE Transactions on Smart Grid, vol. 11, no. 1, pp. 797-809, Jul. 2020. [Baidu Scholar]
S. Kantra, H. A. Adbelsalam, and E. B. Makram, “Application of PMU to detect high impedance fault using statistical analysis,” in Proceedings of 2016 IEEE PES General Meeting (PESGM), Boston, USA, Nov. 2016, pp. 1-5. [Baidu Scholar]
M. Farajollahi, A. Shahsavari, E. M. Stewart et al., “Locating the source of events in power distribution systems using micro-PMU data,” IEEE Transactions on Power Systems, vol. 33, no. 6, pp. 6343-6354, May 2018. [Baidu Scholar]
Y. Zhang, X. Wang, J. He et al., “A transfer learning-based high impedance fault detection method under cloud-edge collaboration framework,” IEEE Access, vol. 8, pp. 165099-165110, Sept. 2020. [Baidu Scholar]
M. Pignati, L. Zanni, P. Romano et al., “Fault detection and faulted line identification in active distribution networks using synchrophasors-based real-time state estimation,” IEEE Transactions on Power Delivery, vol. 32, no. 1, pp. 381-392, Feb. 2017. [Baidu Scholar]
N. M. Manousakis, G. N. Korres, and P. S. Georgilakis, “Taxonomy of PMU placement methodologies,” IEEE Transactions on Power Systems, vol. 27, no. 2, pp. 1070-1077, May 2012. [Baidu Scholar]
A. T. Langeroudi and M. M. A. Abdelaziz, “Preventative high impedance fault detection using distribution system state estimation,” Electric Power Systems Research, vol. 186, pp. 1-11, May 2020. [Baidu Scholar]
B. C. de Oliveira, J. L. R. Pereira, I. D. de Melo et al., “A new methodology for high impedance fault detection, classification and location using PMUs,” in Proceedings of 2018 Simposio Brasileiro de Sistemas Eletricos (SBSE), Niteroi, Brazil, May 2018, pp. 1-6. [Baidu Scholar]
J. J. G. Ledesma, K. B. do Nascimento, L. R. Araujo et al., “A two-level ANN-based method using synchronized measurements to locate high-impedance fault in distribution systems,” Electric Power Systems Research, vol. 186, p. 106576, Jul. 2020. [Baidu Scholar]
M. Wei, F. Shi, H. Zhang et al., “High impedance arc fault detection based on the harmonic randomness and waveform distortion in the distribution system,” IEEE Transactions on Power Delivery, vol. 35, no. 2, pp. 837-850, Apr. 2020. [Baidu Scholar]
S. Nezamzadeh-Ejieh and I. Sadeghkhani, “HIF detection in distribution networks based on Kullback-Leibler divergence,” IET Generation, Transmission & Distribution, vol. 14, no. 1, pp. 29-36, Nov. 2020. [Baidu Scholar]
X. Zhang, C. Delpha, and D. Diallo, “Incipient fault detection and estimation based on Jensen-Shannon divergence in a data-driven approach,” Signal Processing, vol. 169, p. 107410, Apr. 2020. [Baidu Scholar]
X. Zhang, C. Delpha, and D. Diallo, “Jensen-Shannon divergence for non-divergence incipient crack detection and estimation,” IEEE Access, vol. 8, pp. 116148-116162, Jun. 2020. [Baidu Scholar]
X. Zhang and C. Delpha, “Improved incipient fault detection using Jensen-Shannon divergence and KPCA,” in Proceedings of Prognostics and Health Management Conference (PHM-Besançon), Besancon, France, Jul. 2020, pp. 241-246. [Baidu Scholar]
N. Zamanan and J. Sykulski, “The evolution of high impedance fault modeling,” in Proceedings of 2014 16th International Conference on Harmonics and Quality of Power (ICHQP), Bucharest, Romania, May 2014, pp. 1-5. [Baidu Scholar]
D. I. Jeering and J. R. Linders, “Ground resistance – revisited,” IEEE Transactions on Power Delivery, vol. 4, pp. 949-956, Apr. 1989. [Baidu Scholar]
A. E. Emanuel, D. Cyganski, J. A. Orr et al., “High impedance fault arcing on sandy soil in 15 kV distribution feeders: contributions to the evaluation of the low frequency spectrum,” IEEE Transactions on Power Delivery, vol. 5, no. 2, pp. 676-686, Apr. 1990. [Baidu Scholar]
J. Liu, J. Tang, F. Ponci et al., “Trade-offs in PMU deployment for state estimation in active distribution grids,” IEEE Transactions on Smart Grid, vol. 3, no. 2, pp. 915-924, Jun. 2012. [Baidu Scholar]
IEEE Standard for Synchrophasor Measurements for Power Systems Amendment 1: Modification of Selected Performance Requirements. IEEE Std. C37.118.1a-2014 (Amendment to IEEE Std.C37.118.1-2011), Mar. 2014. [Baidu Scholar]
T. Li, Y. Li, and X. Chen, “Fault diagnosis with wavelet packet transform and principal component analysis for multi-terminal hybrid HVDC network,” Journal of Modern Power Systems and Clean Energy, vol. 9, no. 6, pp. 1312-1326, Nov. 2021. [Baidu Scholar]
Y. Zhang, X. Wang, Y. Luo et al., “A CNN based transfer learning method for high impedance fault detection,” in Proceedings of 2020 IEEE PES General Meeting (PESGM), Montreal, Canada, Dec. 2020, pp. 1-5. [Baidu Scholar]
Z. Zhang, X. Zhou, X. Wang et al., “A novel diagnosis and location method of short-circuit grounding high-impedance fault for a mesh topology constant current remote power supply system in cabled underwater information networks,” IEEE Access, vol. 7, pp. 121457-121471, Aug. 2019. [Baidu Scholar]
H. Shu, Z. Gong, and X. Tian, “Fault-section identification for hybrid distribution lines based on principal component analysis and Euclidean distance,” CSEE Journal of Power and Energy Systems, vol. 7, no. 3, pp. 591-603, May 2021. [Baidu Scholar]
T. Yi, Y. Xie, H. Zhang et al., “Insulation fault diagnosis of disconnecting switches based on wavelet packet transform and PCA-IPSO-SVM of electric fields,” IEEE Access, vol. 8, pp. 176676-176690, Sept. 2020. [Baidu Scholar]
I. T. Jolliffe, “A note on the use of principal component in regression,” Applied Statistics, vol. 31, no. 3, pp. 300-303, Mar. 1982. [Baidu Scholar]
H. Zou, T. Hastie, and R. Tibshirani, “Sparse principal component analysis,” Journal of Computational and Graphical Statistics, vol. 15, no. 2, pp. 265-286, Jan. 2006. [Baidu Scholar]
Y. Liu, G. Zhang, and B. Xu, “Compressive spare principal component analysis for process supervisory monitoring and fault detection,” Journal of Process Control, vol. 50, pp. 1-10, Feb. 2017. [Baidu Scholar]
H. Yue and S. Qin, “Reconstruction-based fault identification using a combined index,” Industrial & Engineering Chemistry Research, vol. 40, no. 20, pp. 4403-4414, Aug. 2001. [Baidu Scholar]
B. K. Chaitanya, A. Yadav, and M. Pazoki, “An intelligent detection of high-impedance faults for distribution lines integrated with distributed generators,” IEEE Systems Journal, vol. 14, no. 1, pp. 1-10, May 2019. [Baidu Scholar]
X. Wang, G. Song, J. Gao et al., “High impedance fault detection method based on improved complete ensemble empirical mode decomposition for DC distribution network,” International Journal of Electrical Power & Energy Systems, vol. 107, pp. 538-556, May 2019. [Baidu Scholar]
IEEE PES Distribution System Analysis Subcommittee’s Distribution Test Feeder Working Group. (2014, Nov.). IEEE 34 bus Test Feeder. [Online]. Available: http://ewh.ieee.org/soc/pes/dsacom/testfeeders/ [Baidu Scholar]
X. Wang, W. Liu, Z. Liang, et al., “Faulty feeder detection based on the integrated inner product under high impedance fault for small resistance to ground systems,” International Journal of Electrical Power & Energy Systems, vol. 140, p. 108078, Feb. 2022. [Baidu Scholar]
L. Zhang, Y, Liu, L. Li et al., “Identifying critical deployment locations of measurement units based on principal component subspace in active distribution grids,” Electrical Power Automation Equipment, vol. 37, no. 11, pp. 54-58, Mar. 2017. [Baidu Scholar]
A. von Meier, E. Stewart, A. McEachern et al., “Precision micro-synchrophasors for distribution systems: a summary of applications,” IEEE Transactions on Smart Grid, vol. 8, no. 6, pp. 2926-2936, Nov. 2017. [Baidu Scholar]
D. Dua, S. Dambhare, R. K. Gajbhiye et al., “Optimal multistage scheduling of PMU placement: an ILP approach,” IEEE Transactions on Power Delivery, vol. 23, no. 4, pp. 1812-1820, Oct. 2008. [Baidu Scholar]
A. Tahabilder, P. K. Ghosh, S. Chatterjee et al., “Distribution system monitoring by using micro-PMU in graph-theoretic way,” in Proceedings of 2017 4th International Conference on Advances in Electrical Engineering (ICAEE), Dhaka, Bangladesh, Sept. 2017, pp. 159-163. [Baidu Scholar]
E. Jamil, M. Rihan, and M. A. Anees, “Towards optimal placement of phasor measurement units for smart distribution systems,” in Proceedings of 2014 6th IEEE Power India International Conference (PIICON), Delhi, India, Dec. 2014, pp. 1-6. [Baidu Scholar]
W. C. Santos, F. V. Lopes, N. S. D. Brito et al., “High-impedance fault identification on distribution networks,” IEEE Transactions on Power Delivery, vol. 32, no. 1, pp. 23-32, Feb. 2017. [Baidu Scholar]