Abnormal event detection is a crucial aspect of real-time situation awareness in the power system control room. Detecting, locating, and diagnosing changes in system operating conditions enable independent system operators to respond to abnormal events promptly. Power systems experience numerous types of disruptions; transmission line outage, in particular, is extensively studied due to its frequent occurrence and potentially severe consequence. Outages can happen due to system faults, power grid component degradation, adverse weather conditions, or vandalism.
The popularization of Phasor Measurement Units (PMUs) makes real-time power system analytics possible. PMUs are devices installed at substations (buses) capable of recording high-fidelity GPS time-synchronized phasors, i.e., physical quantities with both magnitude and phase. Much work has been done recently by utilizing the high-reporting rate of PMUs and, in some approaches, augmenting it with first-principle models to detect line outages. There are two challenges to an effective detection scheme. Firstly, the signal-to-noise ratio varies significantly for different outages. Outages at lines with relatively small power flow create minimal and short-lived disruptions. This scenario is prevalent for a realistically sized power network where most single-line outages can hardly be noticed. Secondly, only a limited number of PMUs can be deployed on a network due to economic and engineering feasibility constraints, leaving some substations unobservable. Therefore, the allocation of PMUs, in terms of the number and location, impacts the detection scheme’s effectiveness. It remains a challenge to design a detection scheme robust to the location of the PMUs and outages.
Ii Related Works
With the above challenges, it may be advantageous to exploit system transient dynamics following an outage. Transient dynamics are the evolution of the system algebraic variables, e.g., bus voltage phase angle and magnitude, and state variables, e.g., generator rotor speed and angle, between two quasi-steady states. It is the result of the synchronous machines reacting to the power imbalance created by the outage. These dynamics provide direct and accurate characterizations of the network’s disruption.
Most state-of-the-art works on outage detection can be grouped by the type of dynamics considered in their formulations. The first group models power systems based on the quasi-steady state assumption where no dynamics are considered [1, 2]. However, transient dynamics can often last up to several seconds and are non-negligible. Therefore, this approach may not be adequate at describing the actual system behavior. The second group relaxes the quasi-steady state assumption and attempts to account for the post-outage transient dynamics. Using participation factor matrices, Rovatsos modeled the evolution of voltage phase angles . Similarly, using voltage angles, a generalized likelihood ratio-based detection scheme was developed using the alternating current (AC) power flow model 
. Monitoring is also done on low-dimensional subspaces derived from PMU measurements using, for example, principal component analysis (PCA)
, and hidden Markov model (HMM). Jamei proposed to monitor the correlation matrix obtained from adjacent bus voltage and current phasors ; Hosur and Duan proposed to monitor that of the observation matrix obtained during outage-free operation . These methods rely on system algebraic variables, e.g., bus voltage and current. However, generator state variables can better characterize the system’s transient response to the power imbalance created by the outage.
The third group models the power system as a dynamical system, utilizing both the measurable algebraic variables and hidden generator state variables. Using the swing equation, Pan formulated outage diagnosis as a sparse recovery problem solved by an optimization algorithm . Similarly, using the swing equation, a visual observer network is constructed to monitor line admittance changes by a parameter identification method 
. Both works focus on the outage diagnosis problem, i.e., localization and parameter estimation. However, a systematic detection scheme is the prerequisite for such tasks and needs to be developed.
In general, statistical monitoring of dynamical systems, e.g., power systems, is mainly investigated through two approaches. One is the model-free approach, where no knowledge about the underlying process is assumed. Various data-driven subspace identification methods are employed to track system states, e.g., PCA , and canonical variate analysis 
. On the other hand, when an analytical model that can sufficiently characterize the system’s dynamic behavior is available, a model-based approach is usually considered. This approach builds on state estimation through various filtering techniques, such as Kalman filters and particle filters (PFs) 
. Monitoring schemes are then formulated using signals defined by Kalman innovation vectors, i.e., residual monitoring, or measurement likelihoods in the PF’s case .
There is limited work on line outage detection considering generator dynamics in a partially observed network to the best of the authors’ knowledge. No work brings together state and algebraic variable information for systematic outage detection. In this work, we track system transient dynamics through nonlinear state estimation via a PF. A statistical change detection scheme is constructed by monitoring the PF-predicted output’s compatibility with the expected normal-operation measurement. When an outage happens, a significant reduction in the compatibility triggers an outage alarm. This work has three main contributions: 1) A novel unified detection framework that incorporates both generator bus dynamics and load bus power changes is proposed. 2) The framework also facilitates post-outage diagnostic work through a breakdown of the monitored system signals. 3) Extensive simulation studies on IEEE test system demonstrate the robustness of the proposed framework against partial PMU deployment and the effectiveness against other state-of-the-art outage detection methods.
The rest of this paper is organized as follows. A unified outage detection scheme based on nonlinear power system dynamics is formulated in Section III and Section IV. Section V describes the PF-based online state estimation necessary for tracking generator dynamics. The proposed scheme’s effectiveness and advantages are presented in Section VI using simulation studies. Section VII is the conclusion.
Iii Power System State-Space Modelling
In this section, we detail a power system model that captures both the generator dynamics and load bus power flow information in a unified framework. Consider a power system with generator buses where , load buses where , and transmission lines where . The power system is a hybrid dynamical system described by a differential-algebraic model. The second-order generator model, also known as the swing equation , is used in this work111Although the swing equation is used here to model generator rotor dynamics, high-order and more complex models, such as the two-axis model, can be used to develop the detection scheme proposed in this work similarly.. For every generator bus , their states are modeled as the differential variables, i.e., where is the rotor angular position in radians with respect to a synchronously rotating reference, and is the rotor angular velocity in radians/second. The differential equations governing their dynamics are
where is the derivative of with respect to . is the synchronous rotor angular velocity such that where is the known synchronous frequency. , and denote the mechanical power input, the inertia constant, and the damping factor, respectively. They are assumed known and constant for the duration of this study. The inputs for the model are the generated active power, i.e., . Under the classical generator model assumptions, the synchronous machine is represented by a constant internal voltage behind its direct axis transient reactance . Therefore, the active power at generator is
where is the generator bus nodal voltage phase angle. The transient reactance is assumed known and constant, whereas a method will be presented later to adaptively infer the parameter E with online data. Also, denote where
is assumed to be a zero-mean Gaussian variable with a known variance representing the random fluctuations in electricity load on the bus as well as process noise.
The outputs of the system model are nodal voltage magnitudes and phase angles which PMUs can measure. More importantly, the algebraic output and generator states have to satisfy an active power balance constraint. The constraint stipulates that the net active power at a bus is the difference between the active power supplied to it by the generator and the load consumed, i.e.,
for , subject to a random demand fluctuation as mentioned above. is the load on bus , is the nodal net active power and
following the alternating current (AC) power flow equation where are elements of the bus admittance matrix. Note that for load buses in (3). The total active power generated and load demand of the network are assumed to be balanced as well. This relationship will be the basis for our unified outage detection scheme described in the next section.
We define the discrete counterparts of the system model via a first-order difference discretization by Euler’s formula, i.e., let for , and . For PMU devices with a sampling frequency of 30 Hz, s. Thus, the continuous system of a generator bus can be approximated by
where for notational brevity, and
Taking a derivative with respect to time on both sides of (3) and rearranging the terms, we obtain
relating the changes in bus load to the changes in active power generated and transferred from the bus. The discretized relationship is then
where and similarly for the other two terms. Writing the whole system in vector form, we also define
where represents the random load fluctuations and measurement error. The error is assumed to be a zero-mean Gaussian variable with covariance . The net active power change vector, i.e., , is organized such that the top entries correspond to the generator buses.
Through (9), the active power changes in both generator and load buses can be monitored. In comparison, detection schemes developed in previous works focus on monitoring changes in net active power, , through direct current (DC), e.g., , or AC, e.g., , power flow equations. Their formulations can be considered as special cases of the proposed unified framework when no generator information is available, e.g., no PMUs are installed on generator buses. However, as shown in simulation studies, having generator power output information helps to detect certain outages when net active power changes are not significant enough to trigger an alarm.
In this SSM, the generator states are not directly observable, and their dynamics are governed by the state transition function as in (5). The output can be computed from PMU measurements as well as generator states and is governed by the output function as in (9). Note that is a nonlinear function of the system states. Therefore, the power system is a nonlinear dynamical system. As the process is stochastic due to random load fluctuations and measurement errors, the states and outputs can be expressed in a probabilistic way. In particular, denote the state transition density and output density as and , respectively, where and
are probability density functions (PDFs). An important consequence of the SSM is the conditional independence of the states and output due to the Markovian structure. In particular, given, is independent of all other previous states; similarly given , is independent of all other previous states.
Iv EWMA-Based Outage Detection Scheme
We propose a system-wide detection scheme that utilizes the output of the SSM detailed in the previous section. Under an outage-free scenario, the active power generated, transmitted, and consumed in the network are expected to be balanced with only small random load demand fluctuations. Therefore, the distribution of the system output is the basis for the proposed outage detection scheme:
When a line trips in the power grid, there are two ways that the above relationship will be violated. First, as the system topology changes, the outage-free AC power flow equation (4) used to compute the net active power is no longer valid. In particular, the admittance corresponding to the tripped line becomes zero; the bus admittance matrix changes to a new one that reflects the post-outage system topology. Thus in (11) does not represent the actual net active power changes anymore. Second, line outage events trigger a period of transient re-balancing in the system where generators respond to the power imbalance caused by the outage. The immediately affected buses also experience an abrupt change in the net active power due to the outage. As a combination of these effects, the relationship of (11) will be violated. For example, using data simulated from the IEEE 39-bus test system, Fig. 1 shows the contrast between the signals from a normal system and that with an outage at the 3rd second.
Therefore, the early outage detection problem is formulated as a multivariate process monitoring problem. The multivariate signal’s deviation, , from the expected distribution indicates an abnormal event, in this case, an outage. For its robustness to non-Gaussian data and superior performance on small to median shifts, the multivariate exponentially weighted moving average (MEWMA) control chart, initially developed by , is adopted for the detection task. In particular, with system outputs computed from PMU measurements and estimated generator states, , an intermediate quantity that captures not only current but also past signal information is constructed, i.e.,
where is a pre-defined smoothing parameter that determines the extent of reliance on past-information and , . The statistic under monitoring is then constructed similar to that of a Hotelling’s statistic:
where the covariance matrix is
An outage alarm is then triggered when the monitoring statistic crosses a pre-determined threshold, , chosen to satisfy a certain sensitivity requirement:
Here is the stopping time of the proposed outage detection scheme. The difference between and the onset time of the outage is the detection delay. The proposed scheme’s prime objective is to minimize the detection delay should an outage happen at an a priori unknown location.
A common way to quantify the detection scheme’s sensitivity is through the so-called average run length to a false alarm (), i.e., the number of samples required to produce a false alarm when the system is outage-free. MEWMA-type control chart allows system operators to specify an appropriate sensitivity level by selecting and . Charts with lower values of
are generally more robust against non-Gaussian distributions and have better detection performance for small to medium shifts. Given and a false alarm constraint , the detection threshold can be determined by solving an integral equation of Theorem 2 in 222
The equation can be solved using various numerical algorithms or Markov chain approximation, and this process can be done offline. Interested readers can refer to for a detailed description of the computation procedure required.. The selection of the parameter values and their impact on the detection scheme will be presented in the simulation studies section.
Many power systems have to work with a limited number of PMUs, i.e., some buses are not equipped with a PMU. The detection scheme proposed here is also applicable in this case since the signal under monitoring, , can be adjusted to include only buses with PMUs. In particular, can include those generator buses with PMUs. can be calculated for load buses with fully observable neighbor buses. The impact of an unobserved neighbor bus on the computation of the bus net active power would be an unknown term, , in the AC power flow equation since the neighbor bus’ and are not available. While this impact can be mitigated through a careful selection of the PMU locations, unlike  and , the proposed detection scheme is effective when most generator buses are monitored, a result corroborated by the simulation studies in this work, e.g., see Fig. 6. Also, the number of generator buses is typically much smaller than the total number of buses.
V Generator State Estimation via Particle Filtering
In the previous section, a unified framework of real-time system monitoring utilizing post-outage transient dynamics computed from state and algebraic variables, i.e., active power generated and net active power injection, is described. The premise of the unified framework is the availability of accurate state and algebraic variables data. While algebraic variables can be measured by PMUs, generator states are not directly observable. This section shows how the hidden states could be reliably estimated online using a particle filter.
Online state estimation typically involves the inference of the posterior distribution of the hidden states given a collection of output measurements , denoted by . For systems with nonlinear dynamics and possibly non-Gaussian noises, e.g., power system, the posterior distribution is intractable and cannot be computed in closed form. Extended and unscented Kalman filters have been extensively studied to address the above problem, e.g., [13, 22]. However, these methods’ effectiveness becomes questionable when the underlying nonlinearity is substantial or when the posterior distribution is not well-approximated by Gaussian distribution. Instead, PF is increasingly used for this task, e.g., , as it handles nonlinearity well and accommodates noise of any distribution with an affordable computational cost . PFs belong to the family of sequential Monte Carlo methods where Monte Carlo samples approximate complex posterior distributions, and the distribution information is preserved beyond mean and covariance.
In particular, PF approximates by samples, called particles, obtained via an importance sampling procedure. Each particle is assigned an importance weight proportional to its likelihood of being sampled from the posterior distribution333This type of PF is also known as the bootstrap filter first proposed in . The idea is to use the state transition density as the importance distribution in the importance sampling step. More sophisticated algorithms, such as the guided and auxiliary particle filter could be implemented in the same detection framework proposed here. However, these algorithms are, in general, more difficult to use and interpret. For details, readers can refer to .. PF proceeds in a recursive prediction-correction framework. Assuming at time we have the particles and weights obtained from the previous time step, , where is the number of particles, the posterior distribution at time is approximated by weighted Dirac delta functions as
where is the Dirac delta function, and the weights are normalized such that . The algorithm starts by propagating particles from time to time through the state transition function in (5), i.e., the prediction step. That means, new particles are sampled from the state transition density . The predicted states then have a prior distribution approximated by
When the new measurement arrives, the prior distribution is corrected by updating the particles’ weights proportional to their conditional output likelihood to obtain the posterior distribution as
where . The intuitive interpretation is that the particles are reweighted based on their compatibility with the actual system measurement. The approximation of the posterior distribution by these particle-weight pairs is consistent as at a standard Monte Carlo rate of
guaranteed by the Central Limit Theorem.
A well-known problem of PF is that the weights will become highly degenerate overtime. In particular, the density approximation will be concentrated on a few particles, and all the other particles carry effectively zero weight. A common way to evaluate the extent of this degeneracy is by using the so-called Effective Sample Size (ESS) criterion :
In the extreme case where one particle has the weight of 1 and all others of 0, ESS will be 1. On the other hand, ESS is when every particles has an equal weight of
. A resampling move can be used to solve the degeneracy problem where particles with higher weights are duplicated and others removed, thus focusing computational efforts on regions of higher probability. The systematic resampling method is used in our PF as it usually outperforms other resampling algorithms. When ESS falls below a threshold, typically , we resample particles from the existing ones. The number of offspring, , is assigned to each particle such that . The systematic sampling proceeds as follows to select . A random number
is drawn from the uniform distribution. Then we obtain a series of ordered numbers by for . is the number of where by convention. Finally, resampled particles are each assigned an equal weight before a new round of prediction-correction recursion begins. The detailed PF algorithm with the resampling move is summarized in Algorithm 1.
In this work, it is assumed that the system parameters in the power system SSM are known and static; therefore, the PF’s state estimation is reliable. In real-world applications, these parameters may be known but slow-varying due to factors like system degradation. While parameter estimation in a non-linear system is generally a difficult problem and outside the scope of this paper, there is a natural extension from the particle filtering framework that can tackle the problem. An online expectation maximization (EM) algorithm based on the particles can be implemented to learn the parameters as data arrives sequentially in real-time. The EM algorithm is an iterative optimization method that finds the maximum likelihood estimates of the parameters in problems where hidden variables are present. It can be reformulated to perform the estimation online using the so-called sequential Monte Carlo forward smoothing framework when the complete-data density, i.e.,where denote the set of unknown parameters, is from the exponential family .
Vi Simulation Study
Vi-a Simulation Setting
The proposed PF-based outage detection scheme is tested on the IEEE 39-bus 10-machine New England system 
. System transient responses after an outage are simulated using the open-source dynamic simulation platform COSMIC. A third-order machine model and AC power flow equations are used. The simulation results are assumed to be the true generator states, and corrupted measurements are synthesized from the noise-free simulation data. Ten PMUs are assumed to be installed at bus 19, 20, 22, 23, 25, 33, 34, 35, 36, and 37, covering five generator buses and their connected load buses. Their sampling frequency is assumed to be 30 samples per second. Each simulation runs for 10 seconds, and the outage happens at the 3rd second. A line outage is detected if the monitoring statistic crosses the detection threshold by the end of the simulation. The global constants are Hz and p.u.. For the SSM defined in this study, state function noise
are assumed to be uncorrelated and homogeneous with a standard deviation ofin (5). Output function error are assumed to follow a zero-mean Gaussian distribution with a standard deviation of in (9).
Vi-B Illustrative Outage Detection Example
To illustrate the working of the detection scheme, line 18 outage is used as an example. Fig. 2 shows a typical performance of the particle filter used to estimate generator bus states. The rotor angular speed, , can be accurately tracked while the rotor angular position, , has some biases after the outage. This is acceptable since the focus is on capturing the abnormal changes, i.e., and in turn , in response to the outage rather than accurate state estimations.
One significant advantage of the proposed detection scheme is the ability to break down the output signals and pinpoint the components leading to an early outage detection. Fig. 3 shows such a breakdown for line 18 outage. The upper two components are the generator bus information, and the lower-left one is the load bus information. They register different signal strength levels depending on the outage location, e.g., the magnitude of initial shock and the transient oscillation duration. The proposed scheme can detect outages as long as one of them picks up significant changes. It is clear in this case that the signals from monitored load buses do not contribute meaningfully to the outage detection. Instead, the changes in generated active power and net power injection on generator buses display significant abnormal fluctuations, leading to the outage detection.
The typical progression of the monitoring statistic, , computed via MEWMA from the output signals is shown in Fig. 4. Before the outage, the statistic remains close to zero. After the outage at the 3rd second, it increases rapidly and crosses the threshold. Thus, the scheme raises an outage alarm, and no detection delay is incurred.
Vi-C Results and Discussion
This section shows the effectiveness of the proposed unified scheme using average performance computed from 1000 random simulations of each line outage. The performance comparison with other state-of-the-art methods is also presented.
Vi-C1 Detection Rate
Fig. 5 presents the empirical likelihood of detection for all 35 simulated line outages, which is the percentage of successful detection over 1000 simulations. For both small and large values of , the detection scheme can detect 28 out of 35 outages over 90% of the time. In some cases, it can be seen that larger values of tend to have a better detection rate, i.e., line 8, 13, 15, and 26. The reason is that these line outages produce more severe initial shock relative to their after-outage oscillation. Hence, larger values of help to capture the immediate shock. Also, a small group of outages is challenging to detect regardless of the value, i.e., line 2, 6, 19, 35, and 36. Diagnostic inspection of these cases’ output signals reveals that they generally produce weak system disturbances, especially from the generators, hence often not triggering an outage alarm. The weak disturbance might be explained by the fact that these lines are connected to buses that serve zero or small loads.
Vi-C2 Detection Delay
The empirical distribution of the detection delays is presented in Fig. 6. The figure shows the results of the proposed scheme with different values and the detection scheme based on AC power flow equations from . Intuitively, the scheme is faster at detecting outages when the area under the curve towards the left of the figure is larger. In this case, the proposed scheme has a much higher chance of detecting outages with zero detection delay than the AC scheme. The best-performing scheme () also detects most outages within 0.2 seconds, whereas the AC scheme detects most outages within 1 second.
Vi-C3 Effect of Outage Location Relative to the PMUs
In our previous work, significant variations of average detection delays for outages at different lines relative to the PMU locations can be observed . Fig. 7 shows a comparison of detection delays for outage lines with at least one PMU connected to it versus those with no PMU nearby. Since only ten buses are equipped with PMUs, most lines belong to the second group444All line outages not displayed can be detected with zero mean detection delay, except for line 35 and 36, which are often undetected.. While outages at line 11 and 19 are often detected with 0.1-second delay, most outages are detected immediately regardless of the relative position to the PMUs. Line 11 connects to the slack bus, and its outage creates a minimal disturbance in all three output channels. This result demonstrates the spatial advantage of the proposed method and its robustness to the outage locations.
Vi-C4 Comparison with Other Methods
The proposed method’s performance is also compared with three other methods in Table I. The chosen outages are, in general, more difficult to detect. The first method for comparison is based on the DC power flow model from  and the second based on subspace identification from . Both of them are tested using a full PMU deployment. The third is based on the AC power flow model where 10 PMUs are assumed to be installed . The detection thresholds of all schemes are selected by satisfying a false alarm constraint of 1 in 30 days. The average detection delays and their standard deviations are computed from 1000 random simulations; A dash means a missed detection.
The DC method missed half of the presented outages, despite with a full PMU deployment. The subspace and AC method can detect all outages, however, they incurred longer detection delays on average compared to the unified scheme. This might be because only algebraic variable information are used in their monitoring. The AC scheme comes close to the proposed scheme in terms of coverage and delay. However, the proposed scheme is still faster and, as noted before, improves substantially on the robustness to relative locations to available PMUs.
|Average Detection Delay1|
|Line||DC - full||Subspace - full||AC||Unified|
|2||1.165 (0.006)||2.822 (1.924)||0.283 (0.263)||0.012 (0.183)|
|6||–||3.060 (2.011)||0.246 (0.129)||0.052 (0.463)|
|11||0 (0)||3.048 (1.969)||0.602 (0.205)||0.058 (0.077)|
|15||0 (0)||2.634 (1.850)||0.005 (0.034)||0 (0)|
|19||–||2.836 (2.018)||0.335 (0.378)||0.160 (0.315)|
|26||–||2.850 (1.958)||0.385 (0.228)||0 (0)|
1 standard deviation appears in ().
We have proposed a unified framework of online transmission line outage detection. The framework utilizes information from both generator machine states and load bus algebraic variables. The signals are obtained through nonlinear state estimation of particle filters and direct measurements of PMUs. They are effectively used for outage monitoring and detection by MEWMA control charts while meeting a particular false alarm criterion. The approach is shown to be quicker at detecting outages and more robust to a priori unknown outage locations under a limited PMU deployment through an extensive simulation study. Further research can be done to improve the detection scheme’s effectiveness by investigating the optimal installation location of limited PMUs given a network of power stations. Also, it is observed that a group of lines is consistently challenging to detect regardless of the detection schemes or parameter designs used. More work needs to be done in this area so that these detection blind spots could be reduced.
This work was supported in part by the Future Resilient Systems program at the Singapore-ETH Centre, which was established collaboratively between ETH Zurich and in part by Singapore’s National Research Foundation (FI 370074011) under its Campus for Research Excellence and Technological Enterprise program.
-  J. E. Tate and T. J. Overbye, “Line outage detection using phasor angle measurements,” IEEE Transactions on Power Systems, vol. 23, no. 4, pp. 1644–1652, 2008.
-  Y. C. Chen, T. Banerjee, A. D. Dominguez-Garcia, and V. V. Veeravalli, “Quickest line outage detection and identification,” IEEE Transactions on Power Systems, vol. 31, no. 1, pp. 749–758, 2016.
-  G. Rovatsos, X. Jiang, A. D. Dominguez-Garcia, and V. V. Veeravalli, “Statistical power system line outage detection under transient dynamics,” IEEE Transactions on Signal Processing, vol. 65, no. 11, pp. 2787–2797, 2017.
-  X. Yang, N. Chen, and C. Zhai, “A control chart approach to power system line outage detection under transient dynamics,” IEEE Transactions on Power Systems, vol. 36, no. 1, pp. 127–135, 2021.
-  L. Xie, Y. Chen, and P. R. Kumar, “Dimensionality reduction of synchrophasor data for early event detection: Linearized analysis,” IEEE Transactions on Power Systems, vol. 29, no. 6, pp. 2784–2794, 2014.
Q. Huang, L. Shao, and N. Li, “Dynamic detection of transmission line outages using Hidden Markov Models,”IEEE Transactions on Power Systems, vol. 31, no. 3, pp. 2026–2033, 2016.
M. Jamei, A. Scaglione, C. Roberts, E. Stewart, S. Peisert, C. McParland, and A. McEachern, “Anomaly detection using optimally placed µPMU sensors in distribution grids,”IEEE Transactions on Power Systems, vol. 33, no. 4, pp. 3611–3623, 2017.
-  S. Hosur and D. Duan, “Subspace-driven output-only based change-point detection in power systems,” IEEE Transactions on Power Systems, vol. 34, no. 2, pp. 1068–1076, 2019.
-  W. Pan, Y. Yuan, H. Sandberg, J. Gonçalves, and G. B. Stan, “Online fault diagnosis for nonlinear power systems,” Automatica, vol. 55, pp. 27–36, 2015.
-  C. Yang, Z. H. Guan, Z. W. Liu, J. Chen, M. Chi, and G. L. Zheng, “Wide-area multiple line-outages detection in power complex networks,” International Journal of Electrical Power and Energy Systems, vol. 79, pp. 132–141, 2016.
-  S. W. Choi and I.-B. Lee, “Nonlinear dynamic process monitoring based on dynamic kernel pca,” Chemical engineering science, vol. 59, no. 24, pp. 5897–5908, 2004.
P.-E. P. Odiowei and Y. Cao, “Nonlinear dynamic process monitoring using canonical variate analysis and kernel density estimations,”IEEE Transactions on Industrial Informatics, vol. 6, no. 1, pp. 36–45, 2009.
-  J. Zhao, M. Netto, and L. Mili, “A robust iterated extended kalman filter for power system dynamic state estimation,” IEEE Transactions on Power Systems, vol. 32, no. 4, pp. 3205–3216, 2017.
-  Y. Cui and R. Kavasseri, “A particle filter for dynamic state estimation in multi-machine systems with detailed models,” IEEE Transactions on Power Systems, vol. 30, no. 6, pp. 3377–3385, 2015.
-  N. Chen and S. Zhou, “Detectability study for statistical monitoring of multivariate dynamic processes,” IIE Transactions, vol. 41, no. 7, pp. 593–604, 2009.
-  V. Kadirkamanathan, P. Li, M. H. Jaward, and S. G. Fabri, “Particle filtering-based fault detection in non-linear stochastic systems,” International Journal of Systems Science, vol. 33, no. 4, pp. 259–265, 2002.
-  P. Kundur, N. J. Balu, and M. G. Lauby, Power system stability and control. McGraw-Hill New York, 1994.
-  C. A. Lowry, W. H. Woodall, C. W. Champ, and S. E. Rigdon, “A multivariate exponentially weighted moving average control chart,” Technometrics, vol. 34, no. 1, pp. 46–53, 1992.
-  D. C. Montgomery, Introduction to Statistical Quality Control. John Wiley & Sons, 2007.
-  S. E. Rigdon, “An integral equation for the in-control average run length of a multivariate exponentially weighted moving average control chart,” Journal of Statistical computation and simulation, vol. 52, no. 4, pp. 351–365, 1995.
-  S. Knoth, “Arl numerics for mewma charts,” Journal of Quality Technology, vol. 49, no. 1, pp. 78–89, 2017.
-  S. Wang, W. Gao, and A. P. Meliopoulos, “An alternative method for power system dynamic state estimation based on unscented transform,” IEEE Transactions on Power Systems, vol. 27, no. 2, pp. 942–950, 2012.
-  O. Cappé, S. J. Godsill, and E. Moulines, “An overview of existing methods and recent advances in sequential monte carlo,” Proceedings of the IEEE, vol. 95, no. 5, pp. 899–924, 2007.
-  N. J. Gordon, D. J. Salmond, and A. F. Smith, “Novel approach to nonlinear/non-gaussian Bayesian state estimation,” IEE Proceedings, Part F: Radar and Signal Processing, vol. 140, no. 2, pp. 107–113, 1993.
A. Doucet and A. M. Johansen, “A tutorial on particle filtering and smoothing:
Fifteen years later,”
Handbook of nonlinear filtering, vol. 12, no. 656-704, p. 3, 2009.
-  J. S. Liu, Monte Carlo strategies in scientific computing. Springer Science & Business Media, 2008.
-  S. Yildirim, S. S. Singh, and A. Doucet, “An online expectation–maximization algorithm for changepoint models,” Journal of Computational and Graphical Statistics, vol. 22, no. 4, pp. 906–926, 2013.
-  T. Athay, R. Podmore, and S. Virmani, “A practical method for the direct analysis of transient stability,” IEEE Transactions on Power Apparatus and Systems, no. 2, pp. 573–584, 1979.
-  J. Song, E. Cotilla-Sanchez, G. Ghanavati, and P. D. H. Hines, “Dynamic modeling of cascading failure in power systems,” IEEE Transactions on Power Systems, vol. 31, no. 3, pp. 2085–2095, 2016.