Abstract
Gaussian assumptions of non-Gaussian noises hinder the improvement of state estimation accuracy. In this paper, an asymmetric generalized Gaussian distribution (AGGD), as a unified representation of various unimodal distributions, is applied to formulate the non-Gaussian forecasting-aided state estimation problem. To address the problem, an improved particle filter is proposed, which integrates a near-optimal AGGD proposal function and an AGGD sampling method into the typical particle filter. The AGGD proposal function can approximate the target distribution of state variables to greatly alleviate particle degeneracy and promote precise estimation, through considering both state transitions and latest measurements. For rapid particle generation from the AGGD proposal function, an efficient inverse cumulative distribution function (CDF) sampling method is employed based on the derived approximation of inverse CDF of AGGD. Numerical simulations are carried out on a modified balanced IEEE 123-bus test system. The results validate that the proposed method outperforms other popular state estimation methods in terms of accuracy and robustness, whether in Gaussian, non-Gaussian, or abnormal measurement errors.
AS a significant tool in power system monitoring and control, forecasting-aided state estimation (FASE, also referred to as dynamic state estimation [
Since the FASE inevitably involves the aforementioned noises, researchers often assume that the noises follow specific families of distributions such as Gaussian distribution and student-t distribution. Although Gaussian distribution is widely used in existing works due to its nice properties in algebra, increasing works suggest that both measurement noise and process noise are non-Gaussian variables [
To develop such a method, particle filter (PF) is selected as a basic method, as it can cope with arbitrary noise distribution theoretically through the Monte Carlo simulation method [
Experiments are performed on a balanced IEEE 123-bus test system with eight distributed generators (DGs) [
1) An asymmetric generalized Gaussian distribution (AGGD) is introduced for the assumptions of noises in FASE, which can characterize many popular statistical distributions.
2) A near-optimal AGGD proposal distribution close to the target distribution is established to alleviate particle degeneracy.
3) An accurate and robust PF method is proposed to address the non-Gaussian FASE problem, which embeds the AGGD proposal distribution and an efficient AGGD sampling method into the typical PF.
The rest of this paper is organized as follows. Section II reviews the related work. Section III presents the formulation of the FASE problem under AGGD assumptions of noises. The proposed GPF is presented in Section IV. Section V presents case study and illustrates the competitive performances of the GPF. Conclusion is drawn in Section VI.
For better performances of FASE methods in non-Gaussian noise environments, two main routes are optimality criteria against noises, and appropriate distribution assumptions of noises. Their pros and cons are shown in
Route | Basic method | Criterion/distribution | Pros (+) and cons (-) |
---|---|---|---|
Robust criteria to filter out non-Gaussian noises | UKF, EKF |
Minimum error entropy [ |
(+) Wide application (-) Heavy time consumption |
Maximum correntropy [ | |||
Cross-correntropy [ | |||
Generalized correntropy [ | |||
Inflatable noise variance [ | |||
CKF |
Square-root embedded cubature rule [ | ||
Non-Gaussian distribution assumptions of noises | UKF, PF |
GMM [ |
(+) Wide application (-) Heavy time consumption |
PF |
Student-t distribution [ |
(+) Time-efficiency (-) Limited application | |
Asymmetric Laplace distribution [ | |||
Generalized Gaussian distribution [ |
(+) Wide application, time-efficiency (-) Particle degeneracy |
The popular Kalman-based filters, i.e., Kalman filter (KF) and their variants, are generally improved through optimality criteria, as they cannot achieve optimal estimation with respect to the minimum mean square error under non-Gaussian noise assumptions [
Non-Gaussian noise assumptions are described by the Gaussian mixture model (GMM) or heavy-tailed statistical distributions. The GMM is a combination of Gaussian distributions to approximate any continuous distribution. It can be applied to improve Kalman-based filters, as well as combined with the PF, e.g., a GMM-based generalized CKF [
This section introduces an AGGD. With noises satisfying the AGGD, the FASE problem is formulated and the typical PF algorithm is described to solve this problem.
AGGD is a family of unimodal distributions to provide modeling of generic noise statistics [
(1) |
(2) |
where is the Gamma function; is the mode, i.e., the value that is most likely to be sampled; and is the shape parameter that represents the exponential rate of decay. This paper considers the range of as that is general enough to hold different sharpness of noise distributions [
The states of power systems usually refer to voltage magnitudes and phase angles of buses. Although we hardly obtain their true values because of the limited noisy measurements, FASE can provide an optimal estimate of states, according to state transition characteristics, measurements, and noise statistics. Concretely, the state transitions give a priori knowledge of system process trends. The measurements are used to acquire the likelihood that the estimates are equal to the true states. They include the data received from measurement devices such as phasor measurement unit (PMU) and supervisory control and data acquisition (SCADA) system, and the pseudo-measurements generated by algorithms. Noises are inevitably involved in state transitions and measurements, which should also be considered into the FASE.
Based on the above, the non-Gaussian FASE problem can be formulated. Given a power system with nodes, we denote its state variables at timestep as , where and represent the voltage phase angle and magnitude of node at timestep , respectively. To describe the process trends of , a discrete-time first-order Markov model is applied [
(3) |
where is a nonlinear state-transition function. For convenience, it is typically linearized by the Holt-Winters double exponential smoothing. More details are given in [
(4) |
where is the nonlinear measurement function described in [
The priori PDF in (3) and the likelihood PDF in (4) provide priori and likelihood statistics of , respectively. According to the Bayesian theory [
(5) |
(6) |
With the initial PDF known, can be obtained by recursively calculating (5) and (6). We use it to calculate the optimal state estimate via the maximum posterior probability estimator [
(7) |
In practice, the integrals in (5) and (6) usually cannot be calculated analytically. To cope with the problem, the PF is proposed to approximate using the Monte Carlo simulation method [
(8) |
where is the weight of the th particle at timestep . Finally, the posterior PDF is given by:
(9) |
where is the standard Dirac delta function. In terms of (7) and (9), the optimal estimate of is:
(10) |
Typically, the proposal PDF is set equal to the priori PDF [

Fig. 1 An example of one-dimensional particle degeneracy in typical PF.
Moreover, the priori distribution in this paper follows an AGGD rather than a Gaussian distribution to make sampling difficult. To overcome the two issues, we should construct a near-optimal AGGD proposal PDF that holds both priori knowledge and likelihood information [
In this section, a GPF method is proposed to solve the non-Gaussian FASE problem, which combines a near-optimal proposal PDF and the inverse CDF sampling method with the typical PF. Its block diagram is illustrated in

Fig. 2 Block diagram of GPF.
Referring to [
(11) |
where is a constant; and is the unnormalized optimum proposal PDF. Although is the best choice to generate particles, we hardly acquire its exact expression. Therefore, an accessible near-optimal proposal PDF needs to be built.
In general, the likelihood PDF is more peaked compared with the priori PDF, due to the high precision of measurement devices but high volatility of state transitions. This infers that the sharpness of the optimum proposal PDF is similar to . Accordingly, we define an effective likelihood area, i.e., , to judge whether the particles can be regarded as the feasible states, where is a fixed parameter to limit the scale of . In this paper, we set . Inspired by the GMapping algorithm in [
First, points named candidate particles are sampled from the priori PDF . Under condition of , the most effective candidate particle is selected by:
(12) |
The solution to the optimization problem in (12), i.e., , can be obtained through the traversal of likelihood values of all candidate particles, as shown in
Algorithm 1 : find the most effective candidate particle at timestep |
---|
Input: candidate particles , measurements , likelihood PDF , and scale parameter of Output: the most effective candidate particle 1: 2: for do 3: //Calculate the likelihood value of 4: 5: //Judge whether is the most effective candidate particle in 6: if and then 7: , 8: end if 9: end for |
Since the traversal process only needs to calculate , there is no increase in the computational complexity compared with the weight calculation in (8) of typical PF. Next, we generate effective particles around to provide a numerical approximation of . Finally, according to (11), the effective particles can be utilized to obtain a near-optimal AGGD proposal , i.e.,
(13) |
(14) |
(15) |
(16) |
(17) |
(18) |
Each shape parameter can be estimated according to the relationship between the Kurtosis () and [
(19) |
Considering the heavy computation of , we simply approximate to the shape parameter of , since the proposal PDF is based on .
The inverse CDF sampling is a random sampling method applicable to arbitrary PDF with its inverse CDF known. It is more accurate and convenient than other methods such as the rejection sampling and the Markov chain Monte Carlo sampling [
Consider the AGGD expression under condition of in (1), i.e.,
(20) |
Around (), we expand to a second-order Taylor series expansion and ignore its third-order Taylor remainder. The quadratic expansion can substitute back into (20) to obtain:
(21) |
where represents the Gaussian PDF with mean and variance . determines the Taylor expansion point and () controls the expansion scale. is a formula about ε, and its derivation process is the same as that of in (A12) in Appendix A. Based on this, the domain of is divided into multiple continuous but non-overlapping intervals, i.e., . In , is proportional to a Gaussian PDF. Similarly, the approximation of is given in Appendix A, where the subscript represents the case of .
According to (21), the CDF of AGGD can be approximately expressed as:
(22) |
where and are constants; and is the CDF of . We denote the inverse of as , and then the inverse of is given by:
(23) |
Finally, the inverse CDF sampling for AGGD is as follows.
1) Determine appropriate intervals and .
2) Randomly sample a point from the uniform distribution , i.e., .
3) Calculate in (23) to get a sample of AGGD.
With embedding the AGGD proposal PDF and the AGGD sampling method into the typical PF, we propose a GPF method to the non-Gaussian FASE problem. It is assumed that the initial state , the priori PDF and the likelihood PDF are known and independent [
(24) |
Finally, the optimal estimate of is given by:
(25) |
In addition, an adaptive resampling technique [
Algorithm 2 : GPF at timestep |
---|
Input: weighted particles , measurements , priori PDF , likelihood PDF , and effective particle number Output: , 1: //Prediction step 2: for do 3: 4: end for 5: //Estimate the effective likelihood 6: Find the most effective candidate particle in (12) 7: if is nonexistent then 8: , , 9: else 10: for do 11: , 12: end for 13: //Obtain AGGD proposal distribution 14: Calculate , , , and in (13), (16), (17), and (18), respectively 15: Calculate and in (14) and (15), respectively 16: is set to be the of 17: Obtain 18: //Generate new weighted particles 19: for do 20: 21: end for 22: 23: , 24: end if 25: //Adaptive resampling 26: if then 27: 28: for do 29: 30: end for 31: 32: for do 33: 34: if and then 35: , 36: end if 37: end for 38: , , 39: end if |
In this section, numerical simulations are conducted on a modified balanced IEEE 123-bus test system with eight DGs, to demonstrate the accuracy and robustness of the proposed GPF. All experiments are run on a computer with Intel-i5-10400F CPU and 16 GB memory.
The balanced IEEE 123-bus network presented in [
The details of eight DGs in the modified balanced IEEE 123-bus test system for simulating power outputs of PVs and wind farms are illustrated in
Node | DG type | Capacity (MVA) | Power output distribution |
---|---|---|---|
15 | PV | 0.2 | |
25 | PV | 0.4 | |
35 | PV | 0.6 | |
68 | PV | 0.6 | |
88 | PV | 0.4 | |
98 | Wind farm | 1.0 | (wind speed) |
105 | PV | 0.4 | |
114 | PV | 0.2 |
Based on the above settings, we simulate dynamic power flows of the test system over 100 timesteps. The voltage magnitudes and phase angles in the outcomes are considered as the true states. For comparison purposes, seven benchmarks are chosen, including one traditional state estimation method, i.e., weighted least squares (WLS) [
(26) |
The RMSEs of voltage magnitudes and phase angles are denoted as and , respectively.
It is assumed that PMUs are set in the test system for the acquisition of voltage phasors. Their locations are referred to the result in [
1) Case 1: PMU measurements are contaminated by the additive Gaussian noise with the variance , , , while the noise PDF of SCADA measurements is with the variance .
2) Case 2: noise PDFs of PMU and SCADA measurements are and , respectively. Since and are more concentrated near zero than and , respectively, the measurements in Case 2 are more precise than those in Case 1.
3) Case 3: the measurements of Case 2 are taken as the original measurements, and five nodes, i.e., nodes , , , , and , are randomly selected to generate outliers in their associated measurements. It is assumed that the errors between the outliers and true values are 5%-10%. Each original measurement associated with the five nodes has a 20% probability of being replaced with an outlier. Thus, the measurements in Case 3 have bad data.
In this case, each estimate of WLS is obtained after iterations. For particle-based filters, the particle number is usually determined via trial and error [
Method | Process noise | RMSEV (1 | RMSEθ (1 | Runtime (s) |
---|---|---|---|---|
WLS | 3.36 | 4.41 | 0.72 | |
EKF | 3.17 | 4.30 | 0.97 | |
CKF | 2.30 | 3.48 | 1.14 | |
UKF | 2.25 | 3.33 | 1.03 | |
GCL-UKF | 2.17 | 3.26 | 10.61 | |
PF | 2.08 | 3.20 | 2.79 | |
PF | 2.01 | 3.16 | 2.80 | |
UPF | 1.98 | 3.14 | 3.63 | |
UPF | 1.94 | 3.12 | 3.63 | |
GPF | 1.70 | 3.09 | 3.26 | |
GPF | 1.64 | 3.06 | 3.26 |
To demonstrate affect of the AGGD sampling method on real-time of particle-based filters, i.e., GPF, UPF, and PF, we compare the runtime of these methods under Gaussian and AGGD process noises. As shown in
The RMSEs and runtime of the benchmarks and the proposed GPF are shown in
Within the three particle-based filters, the PF has the worst estimation accuracy but the shortest runtime, as its proposal PDF is the readily accessible process noise PDF that only contains priori knowledge. The and of the proposed GPF are 15.5% and 2.0% smaller than those of the UPF, respectively, and the runtime of it is 10.2% less than that of the UPF. This is inferred that the GPF can construct a proposal PDF closer to the target PDF in a shorter time, to efficiently achieve accurate estimation. However, the runtime of particle-based filters cannot satisfy the real-time estimation requirement in power systems, i.e., provide an optimal state estimate within 2 s [
The measurement noise distributions in Case 2 are and . Since the WLS and the Kalman-based filters are unavailable to non-Gaussian FASE model, their measurement noise distributions are approximated as and , respectively. Other parameters are the same as those in Case 1.
The RMSEs of state estimation methods under non-Gaussian measurement noise are provided in
Method | Process noise | RMSEV (1 | RMSEθ (1 |
---|---|---|---|
WLS | 3.32 | 4.37 | |
EKF | 3.13 | 4.23 | |
CKF | 2.26 | 3.41 | |
UKF | 2.22 | 3.30 | |
GCL-UKF | 2.09 | 3.18 | |
PF | 2.00 | 3.12 | |
PF | 1.95 | 3.10 | |
UPF | 1.87 | 3.05 | |
UPF | 1.83 | 3.03 | |
GPF | 1.47 | 3.00 | |
GPF | 1.42 | 2.93 |
The accuracy and runtime of GPF with different effective particle numbers are obtained, as shown in
K | RMSEV (1 | RMSEθ (1 | Runtime (s) |
---|---|---|---|
50 | 1.71 | 3.11 | 1.38 |
100 | 1.64 | 3.07 | 1.61 |
150 | 1.56 | 3.03 | 1.74 |
200 | 1.51 | 2.98 | 1.96 |
250 | 1.47 | 2.96 | 2.04 |
300 | 1.45 | 2.95 | 2.23 |
350 | 1.42 | 2.93 | 2.30 |
400 | 1.39 | 2.92 | 2.45 |

Fig. 3 Effect of effective particle number. (a) . (b) . (c) . (d) .
Using the abnormal measurements, we obtain the RMSEs of the benchmarks and the proposed GPF, where the noise distribution assumptions and other parameters of each method are the same as those in Case 2. Compared with the estimation errors under normal non-Gaussian measurements, the values of WLS, EKF, CKF, UKF, GCL-UKF, PF, UPF, and GPF, as shown in
Method | Process noise | RMSEV (1 | RMSEθ (1 |
---|---|---|---|
WLS | 3.89 | 5.16 | |
EKF | 3.59 | 5.07 | |
CKF | 2.52 | 3.96 | |
UKF | 2.44 | 3.79 | |
GCL-UKF | 2.24 | 3.55 | |
PF | 2.17 | 3.51 | |
UPF | 1.98 | 3.36 | |
GPF | 1.48 | 3.14 |

Fig. 4 Voltage magnitude errors of abnormal SCADA measurements and estimates at node 30.

Fig. 5 Voltage phase angle errors of abnormal PMU measurements and estimates at node 55.
This paper proposes to use a generic distribution assumption of noise, i.e., AGGD, to formulate the non-Gaussian FASE problem. To solve this problem, a novel GPF is presented, which improves the typical PF with a near-optimal AGGD proposal PDF and the corresponding rapid sampling method. Experiments are implemented on a balanced IEEE 123-bus test system with DGs. The results demonstrate that the proposed GPF is the most accurate and robust compared with the seven benchmarks, i.e., WLS, EKF, CKF, UKF, GCL-UKF, PF, and UPF, under conditions of the measurements with Gaussian noise, non-Gaussian noise, and outliers. Nevertheless, its computation is intense in the cases of large-scale power systems, and accuracy becomes a little worse under abnormal measurements. In future, we will study the distributed implementation and the bad data detection of the GPF to improve its real-time performance and robustness. Another focus of our future studies is the parameter estimation of AGGD to accurately characterize real non-Gaussian noises based on historical data.
Appendix
Appendix A will present the detailed derivation of the Gaussian piecewise approximation of AGGD.
Around (), in (20) is expanded to a second-order Taylor series expansion, i.e.,
(A1) |
where determines the Taylor expansion point; () controls the expansion scale; and are the first derivative and the second derivative of , respectively; is the third-order Taylor remainder, which is equal to zero in the case of . We ignore the remainder and merge the similar terms in (A1) to obtain:
(A2) |
(A3) |
(A4) |
(A5) |
The formula in (A2) can be rewritten as:
(A6) |
(A7) |
Substituting (A6) back into (20), we can obtain:
(A8) |
Accordingly, in is proportional to a Gaussian PDF.
The derivation of approximating AGGD under is the same as the above. Consequently, the AGGD under condition of can be expressed as:
(A9) |
(A10) |
(A11) |
(A12) |
(A13) |
(A14) |
(A15) |
where ; and .
References
M. B. D. C. Filho and J. C. S. D. Souza, “Forecasting-aided state estimation ‒ Part I: panorama,” IEEE Transactions on Power Systems, vol. 24, no. 4, pp. 1667-1677, Sept. 2009. [Baidu Scholar]
A. Primadianto and C. Lu, “A review on distribution system state estimation,” IEEE Transactions on Power Systems, vol. 32, no. 5, pp. 3875-3883, Dec. 2017. [Baidu Scholar]
J. Zhao, A. Gomez-Exposito, M. Netto et al., “Power system dynamic state estimation: motivations, definitions, methodologies, and future work,” IEEE Transactions on Power Systems, vol. 34, no. 4, pp. 3188-3198, Jul. 2019. [Baidu Scholar]
S. Wang, J. Zhao, Z. Huang et al., “Assessing Gaussian assumption of pmu measurement error using field data,” IEEE Transactions on Power Delivery, vol. 33, no. 6, pp. 3233-3236, Dec. 2019. [Baidu Scholar]
T. Ahmad and N. Senroy, “Statistical characterization of PMU error for robust WAMS based analytics,” IEEE Transactions on Power Systems, vol. 35, no. 2, pp. 920-928, Mar. 2019. [Baidu Scholar]
A. de Santis, A. Germani, and M. Raimondi, “Optimal quadratic filtering of linear discrete-time non-Gaussian systems,” IEEE Transactions on Automatic Control, vol. 40, no. 7, pp. 1274-1278, Jul. 1995. [Baidu Scholar]
L. Dang, B. Chen, S. Wang et al., “Robust power system state estimation with minimum error entropy unscented Kalman filter,” IEEE Transactions on Instrumentation and Measurement, vol. 69, no. 11, pp. 8797-8808, Jun. 2019. [Baidu Scholar]
J. A. D. Massignan, J. B. A. London, and V. Miranda, “Tracking power system state evolution with maximum-correntropy-based extended Kalman filter,” Journal of Modern Power Systems and Clean Energy, vol. 8, no. 4, pp. 616-626, Jul. 2020. [Baidu Scholar]
T. Ahmad and N. Senroy, “An information theoretic approach to power-substation level dynamic state estimation with non-Gaussian noise,” IEEE Transactions on Power Systems, vol. 35, no. 2, pp. 1642-1645, Jan. 2020. [Baidu Scholar]
W. Ma, J. Qiu, X. Liu et al., “Unscented Kalman filter with generalized correntropy loss for robust power system forecasting-aided state estimation,” IEEE Transactions on Industrial Informatics, vol. 15, no. 11, pp. 1642-1645, May 2019. [Baidu Scholar]
T. Chen, Y. Cao, X. Chen et al., “A distributed maximum-likelihood-based state estimation approach for power systems,” IEEE Transactions on Instrumentation and Measurement, vol. 70, pp. 1-10, Sept. 2020. [Baidu Scholar]
Z. Jin, J. Zhao, S. Chakrabarti et al., “A hybrid robust forecasting-aided state estimator considering bimodal Gaussian mixture measurement errors,” International Journal of Electrical Power & Energy Systems, vol. 120, pp. 105962, Sept. 2020. [Baidu Scholar]
D. Xu, C. Shen, and F. Shen, “A robust particle filtering algorithm with non-gaussian measurement noise using student-t distribution,” IEEE Signal Processing Letters, vol. 21, no. 1, pp. 30-34, Jan. 2014. [Baidu Scholar]
M. S. Arulampalam, S. Maskell, N. Gordon et al., “A tutorial on particle filters for online nonlinear/non-Gaussian Bayesian tracking,” IEEE Transactions on Signal Processing, vol. 50, no. 2, pp. 174-188, Feb. 2002. [Baidu Scholar]
G. Grisetti, C. Stachniss, and W. Burgard, “Improved techniques for grid mapping with Rao-Blackwellized particle filters,” IEEE Transactions on Robotics, vol. 23, no. 1, pp. 34-46, Feb. 2007. [Baidu Scholar]
A. Tesei and C. S. Regazzoni, “The asymmetric generalized Gaussian function: a new HOS-based model for generic noise PDFs,” in Proceedings of 8th Workshop on Statistical Signal and Array Processing, Corfu, Greece, Jun. 1996, pp. 210-213. [Baidu Scholar]
U. Andersson and S. Godsill, “Optimum kernel particle filter for asymmetric Laplace noise in multivariate models,” in Proceedings of 2020 IEEE 23rd International Conference on Information Fusion, Rustenburg, South Africa, Jul. 2020, pp. 1-4. [Baidu Scholar]
Y. Chai, L. Guo, C. Wang et al., “Network partition and voltage coordination control for distribution networks with high penetration of distributed PV units,” IEEE Transactions on Power Systems, vol. 33, no. 3, pp. 3396-3407, May 2018. [Baidu Scholar]
P. Risbud, N. Gatsis, and A. Taha, “Multi-period power system state estimation with PMUs under GPS spoofing attacks,” Journal of Modern Power Systems and Clean Energy, vol. 8, no. 4, pp. 597-606, Jul. 2020. [Baidu Scholar]
S. S. Yu, J. Guo, T. K. Chau et al., “An unscented particle filtering approach to decentralized dynamic state estimation for DFIG wind turbines in multi-area power systems,” IEEE Transactions on Power Systems, vol. 35, no. 4, pp. 2670-2682, Jul. 2020. [Baidu Scholar]
J. Zhang, G. Welch, G. Bishop et al., “A two-stage Kalman filter approach for robust and real-time power system state estimation,” IEEE Transactions on Sustainable Energy, vol. 5, no. 2, pp. 2670-2682, Apr. 2014. [Baidu Scholar]
X. Zhang, “A novel cubature Kalman filter for nonlinear state estimation,” in Proceedings of 52nd IEEE Conference on Decision and Control, Florence, Italy, Dec. 2013, pp. 7797-7802. [Baidu Scholar]
A. Zia, T. Kirubarajan, J. P. Reilly et al., “An EM algorithm for nonlinear state estimation with model uncertainties,” IEEE Transactions on Signal Processing, vol. 56, no. 3, pp. 921-936, Mar. 2008. [Baidu Scholar]
Z. Xiao, D. Xiao, V. Havyarimana et al., “Toward accurate vehicle state estimation under non-Gaussian noises,” IEEE Internet of Things Journal, vol. 6, no. 6, pp. 921-936, Dec. 2019. [Baidu Scholar]
M. O. M. Mahmoud, M. Jaidane-Saidane, J. Souissi et al., “Modeling of the load duration curve using the asymmetric generalized Gaussian distribution: case of the Tunisian power system,” in Proceedings of the 10th International Conference on Probablistic Methods Applied to Power Systems, Rincon, USA, Aug. 2008, pp. 1-7. [Baidu Scholar]
J. Yang, W. Zhang, and F. Guo,“Dynamic state estimation for power networks by distributed unscented information filter,” IEEE Transactions on Smart Grid, vol. 11, no. 3, pp. 2162-2171, May 2020. [Baidu Scholar]
O. Hlinka, F. Hlawatsch, and P. M. Djuric, “Consensus-based distributed particle filtering with distributed proposal adaptation,” IEEE Transactions on Signal Processing, vol. 62, no. 12, pp. 3029-3041, Jun. 2014. [Baidu Scholar]
S. Kotz, T. Kozubowski, and K. Podgorski, “The Laplace distribution and generalizations: a revisit with applications to communications, economics, engineering, and finance,” in Springer Science and Business Media, Boston: Birkhäuser, 2001, pp. 22-29. [Baidu Scholar]
S. Yin and X. Zhu, “Intelligent particle filter and its application to fault detection of nonlinear system,” IEEE Transactions on Industrial Electronics, vol. 62, no. 6, pp. 3852-3861, Jun. 2015. [Baidu Scholar]
L. Shi, C. Wang, L. Yao et al., “Optimal power flow solution incorporating wind power,” IEEE Systems Journal, vol. 6, no. 2, pp. 233-241, Jun. 2012. [Baidu Scholar]
A. Baharvandi, J. Aghaei, A. Nikoobakht et al., “Linearized hybrid stochastic/robust scheduling of active distribution networks encompassing PVs,” IEEE Transactions on Smart Grid, vol. 11, no. 1, pp. 357-367, Jan. 2020. [Baidu Scholar]
K. Chauhan and R. Sodhi, “Placement of distribution-level phasor measurements for topological observability and monitoring of active distribution networks,” IEEE Transactions on Instrumentation and Measurement, vol. 69, no. 6, pp. 3451-3460, Jun. 2020. [Baidu Scholar]
P. A. Pegoraro, A. Angioni, M. Andrea et al., “Bayesian approach for distribution system state estimation with non-Gaussian uncertainty models,” IEEE Transactions on Instrumentation and Measurement, vol. 66, no. 11, pp. 2957-2966, Aug. 2017. [Baidu Scholar]
N. Kashyap, S. Werner, and Y. Huang, “Decentralized PMU-assisted power system state estimation with reduced interarea communication,” IEEE Journal of Selected Topics in Signal Processing, vol. 12, no. 4, pp. 607-616, Aug. 2018. [Baidu Scholar]
J. H. Kotecha and P. M. Djuric, “Gaussian sum particle filtering,” IEEE Transactions on Signal Processing, vol. 51, no. 10, pp. 2602-2612, Oct. 2003. [Baidu Scholar]
A. Monticelli, “Electric power system state estimation,” Proceedings of the IEEE, vol. 88, no. 2, pp. 262-282, Feb. 2000. [Baidu Scholar]