Toward adaptive robust state estimation based on MCC by using the generalized Gaussian density as kernel functions

doi:10.1016/j.ijepes.2015.03.011

International Journal of Electrical Power & Energy Systems

Volume 71, October 2015, Pages 297-304

https://doi.org/10.1016/j.ijepes.2015.03.011 Get rights and content

Highlights

•
A generic formulation for robust state estimator is proposed.
•
The proposed formulation unifies several existing robust state estimator models.
•
A method is proposed to identify the distribution type of measurement noise.
•
An adaptive robust state estimator is proposed for suppressing different noises.

Abstract

In this paper, a generic formulation is proposed for robust state estimation (RSE) based on maximum correntropy criterion (MCC), leading to an adaptive robust state estimator. By using the generalized Gaussian density (GGD) as the kernel function, the proposed formulation theoretically unifies several existing RSE models, each of which is optimal for a specific type of measurement noise and error distribution. As the noise and error distribution is generally unknown ex-ante and time-varying in operation, a statistical learning scheme is proposed to heuristically identify the actual distribution type online. Afterwards, the optimal RSE can be properly selected so as to adapt to the variation of noise and error distribution types. Simulations are carried on a rudimentary 2-bus system and the standard IEEE-118 bus system, illustrating the correctness and effectiveness of the proposed methodology.

Introduction

Power system state estimation (SE) is one of the core functions of modern energy management system (EMS) [1]. It was firstly investigated by Schweppe and Wildes in 1970 [2], and so far various SE models have been proposed, among which the weighted least square (WLS) model is the most popular one. To improve the computational efficiency of WLS, the fast-decoupled SE is proposed in [3].

As conventional WLS estimators are usually sensitive to gross error, much attention has been devoted to suppress the effects of bad data [4], [5], [6]. Thus, various robust state estimation (RSE) models are proposed to achieve unbiased estimation in the presence of various types of gross errors. The M-estimators are the typical of these RSEs, mainly including the weighted least absolute value (WLAV) estimator [7], [8], [9], [10], [11], the quadratic-linear (QL) estimator [12], [13] and the quadratic-constant (QC) estimator [14], [15], etc. Most recently, the maximum normal measurement rate (M NMR) estimator, the maximum exponential square (MES) estimator and the maximum exponential absolute value (MEAV) estimator have been suggested in [16], [17], [18], respectively, showing appealing performance for suppressing gross errors. However, it should be pointed out that the distribution type of measurement noise and error will effect on the performance of these SE methods. To the best of authors’ knowledge, this issue has not been discussed in-depth in literature yet. As the distribution type of measurement noises is unknown ex-ante in real operation, a possible way is to assume a certain distribution type of measurement noise and error, e.g. Gaussian distribution or Laplace distribution, and then to use the most appropriate SE method. Unfortunately, Turkey has pointed out that the actual measurement error distributions are always far away from the assumed types [19]. This indicates that one cannot expect to use only a fixed SE method to deal with all types of measurement noises perfectly, especially when the noise and error distribution is time-varying. To circumvent this problem, it is desirable to develop an adjustable SE method that can adapt to the change of measurement noise and error distribution types.

Recent literature suggests applying information theory, such as minimum information loss principle [20], [21] and the maximum correntropy criterion (MCC) [22], [23], to improve the performance and applicability of RSE methods. From the information-theoretic point of view, the essential difference among these RSE methods is that they use different kernel functions. Similar to the aforementioned problem, each of the information-theoretic RSE methods can theoretically achieve optimal performance only when its kernel function matches its associated type of measurement noise and error distribution. Thus, when the distribution type is unknown or time-varying, a specific method cannot guarantee its optimality on rejecting measurement noise and error.

In [24], a parametric generalized Gaussian probability density function is proposed for RSE. This inspiring the main idea of this paper that uses the parametric generalized Gaussian density (GGD) as a family of kernel functions, further deriving an adaptive RSE for optimally depressing a variety of measurement noise and error with different distribution types. The main contributions of this paper are threefold:

(1)
A generic formulation is proposed for RSE based on MCC by using the parameterized GGD as kernel functions. This formulation unifies several existing RSE models, based on which new RSE models can also be derived by changing the counterpart noise and error distributions of interest.
(2)
A statistical learning scheme is proposed to heuristically identify the actual distribution type of measurement noise and error, guiding to choose the proper RSE model online.
(3)
Based on (1), (2), an adaptive robust state estimator (ARSE) is derived for optimally suppressing wide variety of measurement noise and errors, even when the distribution types are time-varying.

The rest of this paper is organized as follows. Three RSE models are shortly reviewed in section ‘Three existing robust state estimation models’. Section ‘A generic RSE formulation’ presents a generic RSE formulation. A statistical learning scheme is proposed to identify the distribution type of measurement noise and error and an ARSE model for measurement errors is proposed in section ‘Estimating the distribution type’. The overall algorithm is also presented in section ‘Estimating the distribution type’. Case studies on a rudimentary 2-bus system and standard IEEE 118-bus system are given in section ‘An illustrative example. Finally, conclusions are drawn in section ‘Conclusion’.

Section snippets

Measurement equations

For SE, the relationship between measurements and state variables is described by the measurement equations $z = h (x) + ε$ where z is the m-dimensional measurement vector (the measurements), usually including the line power flows, bus power injections, bus voltage magnitudes and line current flow magnitudes, etc.; N the number of buses, x the n-dimensional state vector (n = 2N − 1); $h : R^{n} \to R^{m}$ the nonlinear vector function mapping the state vector to the measurement vector; $ε$ the m-dimensional measurement

A generic RSE formulation based on MCC using GGD

Refs. [22], [23] propose the RSE model based on MCC using the Gaussian density function as the kernel. However, the actual measurement noise and error may not follow the Gaussian distribution. In [24], it is revealed that the generalized Gaussian density (GGD) is a more suitable candidate of kernel functions used in density estimation. This motivates us to use the GGD as the kernel function in MCC for RSE.

The GGD is defined as follows [24]: $G_{σ} (ε) = \frac{τ}{2 σ Γ (1 / τ)} \exp (- {(\frac{|ε - μ|}{σ})}^{τ})$ where $G_{σ} (ε)$ is the density

Estimating the distribution type

To estimate the distribution type of measurement noise and error, it is required to determine the shape parameter defined in the generic RSE model (8).

ARSE based on the error distribution type estimation

Fig. 3 illustrates the proposed procedure of ARSE that includes two diagrams: the online estimation of distribution types of noise and error using historical data and the real-time RSE calculation using current measurements. These two diagrams are processed separately. As soon as the distribution type is estimated, the corresponding RSE method can be determined and employed in the real-time state estimation. Detail of the procedure is given in Fig. 4.

An illustrative example

In this subsection, the performance of the proposed ARSE will be tested on a rudimentary 2-bus system and IEEE bus systems. The algorithm is coded in JAVA and performed on a Laptop with Intel(R) Core(TM) i5, 2.60 GHz processor and 4 GB RAM.

Conclusion

In this paper, a general RSE model and an ARSE methodology are proposed. The presented general model can unify many existing RSE models in theory and can also deduce some new RSE models, while the proposed ARSE can switch among different RSE models corresponding to different distribution types of measurement noise and error, thereby adapting for various distribution types of noise and error. Numerical experiments illustrate the correctness and effectiveness of the proposed approach.

It is worth

Acknowledgements

This work was supported in part by the National High Technology Research and Development Program (2012AA050208) and in part by National Natural Science Foundation of China (51407069).

References (26)

K. Kokkinakis et al.
Exponent parameter estimation for generalized Gaussian probability density functions with application to speech modeling
Signal Process
(2005)
A. Abur et al.
Power system state estimation: theory and implementation
(2004)
F.C. Schweppe et al.
Power system static state estimation, part I–III
IEEE Trans Power App Syst
(1970)
A. Garcia et al.
Fast decoupled state estimation and bad data processing
IEEE Trans Power App Syst
(1979)
A. Monticelli et al.
Multiple bad data identification for state estimation by combinatorial optimization
IEEE Trans Power Deliv
(1986)
B.M. Zhang et al.
A linear recursive bad data identification method with real-time application to power system state estimation
IEEE Trans Power Syst
(1992)
H.M. Merrill et al.
Bad data suppression in power system static state estimation
IEEE Trans Power Syst
(1971)
M.R. Irving et al.
Power system state estimation using linear programming
IEE Proc
(1978)
W.W. Kotiuga et al.
Bad data rejection properties of weighted least absolute value techniques applied to static state estimation
IEEE Trans Power App Syst
(1982)
A. Abur
A bad data identification method for linear programming state estimation
IEEE Trans Power Syst
(1990)

R.A. Jabr et al.

Iteratively reweighted least-squares implementation of the WLAV state estimation method

IEE Proc Gen Transm Distrib

(2004)

H. Wei et al.

An interior method for power system weighted nonlinear L1 norm static state estimation

IEEE Trans Power Syst

(1998)

E. Handshin et al.

Bad data analysis for power state estimation

IEEE Trans Power App Syst

(1975)

Cited by (23)

Assessment on power systems non-deterministic state estimation algorithms
2023, Electric Power Systems Research
Citation Excerpt :
Among them, they use kernels to calculate which distribution best fits the error. For example; Bayesian estimators, where the prior PDF is calculated using kernel distribution fitting [24,71], or Maximum Correntropy Criterion (MCC) based on Information Theory [129–133]. This article studies the SE algorithms used at transmission and distribution levels separately since they are generally operated by different control centers.
Power systems are operated in a deterministic way. However, the increase in uncertainties (caused by measurement and communication errors or the absence of complete knowledge of the measured quantity) requires algorithms capable of assessing randomness in real-time monitoring and operation of energy management systems. Among them, state estimation is a fundamental element that provides complete and reliable information about the current network state. This paper offers a review of non-deterministic algorithms, used to assess the randomness caused by the uncertainties in power system state estimation. Depending on the uncertainty modeling method, different static and dynamic non-deterministic state estimation algorithms for transmission and distribution systems are reviewed.
Also, four of the most relevant algorithms are implemented and results are compared for distribution networks. The relevance of such models, as well as their motivation, are addressed.
An adaptive hybrid dynamic state estimation method of the medium-voltage DC integrated power system with pulse load
2022, International Journal of Electrical Power and Energy Systems
Citation Excerpt :
In recent years, scholars such as Vladimir Terzija, Junbo Zhao, Bikash Pal, Ali Abur, Innocent Kamwa and Sassan Goleijani have done a lot of work on the state estimation. Their researches refer to the DSE robustness [8–11], unknown input [12–13], error detection [14], constraints [15,16], parameter estimation [17–18] and hybrid measurement [19–21]. In [22,23], a technical summary was offered about the status quo of DSE and more relevant contents about DSE could be found.
When a great change periodically takes place in power of the periodic pulse load in the medium-voltage DC integrated power system in a short time, it will lead the system to stay in a periodic electromagnetic (EM) transient state. Its time constant is far less than that of the traditional land power system which is in an electromechanical transient state, so the convergence of the filter and accuracy of the estimation will be affected. This problem is similar to the estimation of a maneuvering target in tracking, but the methods used for estimating the maneuvering target cannot be directly applied to estimate the EM transient of the system. For this reason this paper proposes an adaptive estimation method. Based on the high accuracy of unscented Kalman filter and reliability of Hybrid extended Kalman filter under EM transient conditions, the method can ensure the accuracy of estimation and reliability of the algorithm by adaptively adjusting the estimation method according to changes in the system operating conditions. The method is proved to be better than the existing method in terms of estimative effect through experiment and also proved to be reliable by the consistency test on the filter.
Practical state estimation model and algorithm based on kernel density theory
2022, International Journal of Electrical Power and Energy Systems
Citation Excerpt :
The robust state estimator based on MEAV was proposed in [26–28]. The model of MEAV is continuous and differentiable due to the use of an equivalent transformation which eliminates the absolute symbol in the objective function. [29] proposed an adaptive robust state estimation merging the MNMR, MEAV and MES model.
This paper presented a practical state estimation model and algorithm based on the kernel density theory (PKDE), which owns better performance than the existing methods. First, a PKDE was presented, which transforms the calculation of the kernel bandwidth of each measurement to that of each state variable. Therefore, the number of kernel bandwidths required to be determined is greatly reduced as the number of state variables are much less than that of the measurements. The feasibility of the PKDE model is supported by a proposition which is strictly proved. Second, an approach for choosing the appropriate kernel bandwidth was presented through the integration of the optimal bandwidth method based on the rule of thumb, the initial bandwidth selection method and the correcting bandwidth selection of the state variables. Third, PKDE’s robustness to bad data is further enhanced by a strategy to correct the measurements according to the hidden component of the measurement error. The simulation results finally verify the effectiveness of the proposed models and algorithms.
Modeling and application of ship density based on ship scale conversion and grid
2021, Ocean Engineering
Citation Excerpt :
Theoretically, under the condition of independent distribution of the same sample, kernel estimation method has the properties of asymptotic normality, asymptotic unbiasedness and strong consistent, hence, reliable density estimation results can be obtained by using any kernel function. Some literatures use the parametric generalized Gaussian (Chen et al., 2015) or Laplace (Chen et al., 2017) as kernel functions to derive the robust state estimator. To verify the influence of different kernel functions on ship density estimation, four common kernel functions are selected for experimental comparison to select the optimal kernel function.
To explore the regularity of ship density and ensure the safety of maritime traffic, a ship density model based on ship scale conversion and grid is established by considering the impact of ship size on traffic safety. A non-parametric kernel density estimation (KDE) method is used to obtain the probability density function of the ship density by selecting the appropriate kernel function and determining the optimal bandwidth. To process and store Automatic Identification System (AIS) data efficiently, a grid-based storage method of AIS data is proposed in this paper, and the relationship between grid width and sampling interval is established. The experiments result proves that the grid-based AIS data storage method saves 64% of storage space compared with the original AIS data. The ship density model based on ship scale conversion can reflect the impact of ship size on the maritime traffic. The distribution regularity of ship density can be obtained through the probability density curve of ship density and statistical characteristics. The research results show that the proposed method can provide theoretical support for the safety supervision of the maritime authority. It can also be recommended for the evaluation of VTS centers in vessel traffic management.
Information Theoretic Generalized State Estimation in power systems
2020, Electric Power Systems Research
Citation Excerpt :
This contrasts with conventional LS-based estimators, which require further iterative steps to remove suspicious measurements from the data set as part of the SE procedure. After the seminal publication [14], several research efforts have attempted to explore this approach, by proposing alternative implementations [15], employing a slightly distinct terminology [16], or exploiting distinct correntropy definitions based on a generalized version of the Gaussian density function [17]. All of them, however, are concerned with state estimation under the assumption of a known topology.
This paper introduces an Information Theoretic approach for Generalized State Estimation, aiming at achieving reliable topology and state variables co-estimation results, even in the presence of both topology errors and gross measurements. Attention is focused on the final bad data processing stage in which only relevant parts of the power network are represented at the bus-section level. The proposed generalized strategy applied at physical level relies on the superior outlier rejection properties of state estimators based on Maximum Correntropy, a concept borrowed from Information Theoretical Learning. A single objective function unifies the treatment of analog measurements and topology data, leading to an algorithm that does not require re-estimation runs for bad data suppression, and is simpler and more efficient than previously proposed co-estimation methods. Case studies conducted for distinct test-systems are presented, including various types of topology errors and simultaneous occurrence of topology and gross measurement errors. The results suggest that the proposed information-theoretic co-estimation algorithm is able to successfully provide bad data-free real-time network models even in the presence of multiple topology errors, simultaneous gross measurements and inaccurate topology information. Finally, additional tests confirm its superior computational performance as compared with other co-estimation algorithms.
Output control of linear time-invariant systems under input and output disturbances
2020, IFAC-PapersOnLine
The novel control algorithm for linear time-invariant systems under disturbances and measurement noises is proposed. The designed control law, based on the noise and disturbance estimation, ensures the accuracy in steady state depending on the disturbance, only one component of noise vector and its first derivatives. Sufficient conditions in terms of linear matrix inequality (LMI) providing stability of the closed-loop system are obtained. The simulations show efficiency of the proposed method compared with existing ones.

View all citing articles on Scopus

View full text

Toward adaptive robust state estimation based on MCC by using the generalized Gaussian density as kernel functions

Highlights

Abstract

Introduction

Section snippets

Measurement equations

A generic RSE formulation based on MCC using GGD

Estimating the distribution type

ARSE based on the error distribution type estimation

An illustrative example

Conclusion

Acknowledgements

Signal Process

Power system state estimation: theory and implementation

Power system static state estimation, part I–III

IEEE Trans Power App Syst

Fast decoupled state estimation and bad data processing

IEEE Trans Power App Syst

Multiple bad data identification for state estimation by combinatorial optimization

IEEE Trans Power Deliv

A linear recursive bad data identification method with real-time application to power system state estimation

IEEE Trans Power Syst

Bad data suppression in power system static state estimation

IEEE Trans Power Syst

Power system state estimation using linear programming

IEE Proc

Bad data rejection properties of weighted least absolute value techniques applied to static state estimation

IEEE Trans Power App Syst

A bad data identification method for linear programming state estimation

IEEE Trans Power Syst

Iteratively reweighted least-squares implementation of the WLAV state estimation method

IEE Proc Gen Transm Distrib

An interior method for power system weighted nonlinear L1 norm static state estimation

IEEE Trans Power Syst

Bad data analysis for power state estimation

IEEE Trans Power App Syst