Dynamic State Estimation of Power Systems with Uncertainties Based on Robust Adaptive Unscented Kalman Filter

Dongchen Hou; Yonghui Sun; Jianxi Wang; Linchuang Zhang; Sen Wang

网刊加载中。。。

使用Chrome浏览器效果最佳，继续浏览，你可能不会看到最佳的展示效果，

确定继续浏览么?

复制成功，请在其他浏览器进行阅读

Dynamic State Estimation of Power Systems with Uncertainties Based on Robust Adaptive Unscented Kalman Filter PDF

- ORCID：
Dongchen Hou
✉
- ORCID：
Yonghui Sun
✉
- ORCID：
Jianxi Wang
✉
- ORCID：
Linchuang Zhang
✉
- ORCID：
Sen Wang
✉

the College of Energy and Electrical Engineering, Hohai University, Nanjing, 210098, China

Updated：2023-07-25

DOI：10.35833/MPCE.2022.000157

Abstract

In this paper，a robust adaptive unscented Kalman filter (RAUKF) is developed to mitigate the unfavorable effects derived from uncertainties in noise and in the model. To address these issues, a robust M-estimator is first utilized to update the measurement noise covariance. Next, to deal with the effects of model parameter errors while considering the computational complexity and real-time requirements of dynamic state estimation, an adaptive update method is produced. The proposed method is integrated with spherical simplex unscented transformation technology, and then a novel derivative-free filter is proposed to dynamically track the states of the power system against uncertainties. Finally, the effectiveness and robustness of the proposed method are demonstrated through extensive simulation experiments on an IEEE 39-bus test system. Compared with other methods, the proposed method can capture the dynamic characteristics of a synchronous generator more reliably.

Keywords

Dynamic state estimation; Kalman filter; synchronous generator; unscented transformation; robust estimation

I. Introduction

WITH the rapid development of measurement and communication technologies, phasor measurement unit (PMU) technology has gradually matured, and numerous PMU devices have been connected to power systems. The implementation of PMU technology can enhance the real-time system operator identification of network conditions, where the technology can generate measurements by sampling instantaneous waveforms. Currently, wide-area measurement systems (WAMSs), which are supported by PMUs, are widely used in power systems. WAMSs can provide a solid information foundation for real-time monitoring during the dynamic process of the power system, making dynamic state estimation (DSE) possible [

1]-[3]. In addition, with the large-scale integration of distributed energy and the increasing diversification of power demands, the operation of modern power systems has become more complex and changeable, thus presenting higher requirements for power system DSE.

DSE can accurately track the dynamics of system states, thereby playing a critical role in ensuring the safe and stable operation of power systems, particularly for DSE of generators [

4]-[6]. Under these circumstances, Kalman filtering and its variants have received considerable attention because of their application in power systems with nonlinear characteristics [7]-[10]. Using the measurement information of PMUs, a DSE method is proposed in [11] to estimate states and parameters related to electromechanical dynamics, effectively reducing the amount of computation required for estimation. In [12], an extended Kalman filter (EKF) based lateral two-level estimator with a graphics processing unit is investigated to realize DSE using large datasets. In [13], a multi-area DSE method is proposed in an EKF framework, which could successfully coordinate the states of boundary buses in different areas.

However, discarding the higher-order term in the Taylor series leads to linearization errors in the estimation results of the EKF, which can be ignored only under the condition of slight nonlinearity. To circumvent errors in the linearization process of the EKF, an unscented Kalman filter (UKF) method that does not employ Jacobian-based linearization is developed in [

14]. Utilizing

H_{\infty}

filtering theory, [15] develops a UKF-based DSE method to capture the dynamic characteristics of a synchronous generator against uncertainties. In [16], based on derivative-free Kalman filtering, a method is introduced to enhance numerical stability and scalability. A decentralized derivative-free dynamic estimation framework based on a fourth-order synchronous generator model is introduced in [17]. Compared with the EKF, the UKF has obvious advantages in terms of the estimation accuracy of nonlinear systems.

For $n$ -dimensional nonlinear systems, a set of $2 n + 1$ sigma points is selected in the UKF using an unscented transform (UT) to approximate the distribution information of random variables. Although the UT-based derivative-free Kalman filtering is unnecessary for computing the Jacobian matrix, it increases the execution time because of the increasing sigma points. In [

18], a novel sigma point selection method is proposed to obtain statistical information comparable with that of the symmetric point selection method. In [19], a spherical simplex sigma set with

n + 2

points is obtained using a spherical simplex unscented transformation (SSUT). Unlike traditional UT, SSUT can reduce the number of sigma points by placing it directly on the origin or on a hypersphere centered at the origin.

It is noteworthy that the Kalman filter works well only under the assumption that the knowledge of the power system DSE model can be accurately obtained [

20]-[22]. Thus, when unknown changes and disturbances occur in the system, the conventional Kalman filter has limitations. This means that capturing the dynamic characteristics of the power system is difficult when using the assumed model [23]-[25]. Specifically, these uncertainties, such as the non-Gaussian distribution of PMU measurement noise and variations in generator parameters caused by component aging, significantly reduce the performance of DSE. In this context, identifying the actual operating state and formulating appropriate control schemes for system operators are difficult.

To address these problems, based on the robust M-estimator and SSUT, a novel robust adaptive unscented Kalman filter (RAUKF) is developed in this paper that is suitable for generator DSE. The key aspects of this paper are as follows: ① SSUT effectively reduces the sigma points and improves the calculation efficiency with transformation accuracy; ② considering the computational complexity and real-time requirements of DSE, a novel adaptive update method is proposed to adaptively update the model as a whole; and ③ compared with the conventional UKF [

14], unscented

H_{\infty}

filter (UHF) [15], constrained iterated unscented Kalman filter (CIUKF) [26], and square-root central difference Kalman filter (SRCDKF) [27], the proposed RAUKF improves the robustness of the algorithm against model uncertainties with a higher computational efficiency.

II. Dynamic Estimation Model

A. Continuous State-space Model

The fourth-order synchronous generator model in the direct- and quadratic-axis (i.e., d- and q-axis) reference frames [

28]-[30] can be described by:

\{\begin{array}{l} \dot{δ} = ω - ω_{0} \\ \dot{ω} = \frac{ω_{0}}{2 H} [T_{m} - T_{e} - \frac{K_{D}}{ω_{0}} (ω - ω_{0})] \\ {\dot{e}}_{q}^{'} = \frac{1}{T_{d 0}^{'}} [E_{f d} - e_{q}^{'} - (x_{d} - x_{d}^{'}) i_{d}] \\ {\dot{e}}_{d}^{'} = \frac{1}{T_{q 0}^{'}} [- e_{d}^{'} + (x_{q} - x_{q}^{'}) i_{q}] \end{array}

(1)

where $δ$ is the rotor angle; $ω$ is the per-unit rotor speed; $ω_{0} = 2 π f$ is the nominal synchronous speed; $H$ and $K_{D}$ are the inertia constant per unit and damping coefficient, respectively; $T_{m}$ is the mechanical torque; $E_{f d}$ is the field voltage; $T_{e}$ is the electrical torque; ${\dot{e}}_{d}^{'}$ and ${\dot{e}}_{q}^{'}$ are the d- and q-axis transient voltages, respectively; $T_{d 0}^{'}$ and $T_{q 0}^{'}$ are the d- and q-axis open-circuit time constants, respectively; $x_{d}$ and $x_{q}$ are the $d$ - and $q$ -axis synchronous reactances, respectively; and $x_{d}^{'}$ and $x_{q}^{'}$ are the $d$ - and $q$ -axis transient reactances, respectively.

For convenience, (1) can be rewritten as:

\{\begin{matrix} \dot{x} = f_{c} (x, u) + w_{c} \\ y = h_{c} (x, u) + v_{c} \end{matrix}

(2)

x = {[\begin{matrix} δ ω e_{q}^{'} e_{d}^{'} \end{matrix}]}^{T}

(3)

u = {[T_{m}^{} E_{f d}^{} i_{R}^{} i_{I}^{}]}^{T}

(4)

y = {[δ ω e_{R}^{} e_{I}^{}]}^{T}

(5)

where $f_{c} (\cdot)$ and $h_{c} (\cdot)$ are the state transition and measurement functions, respectively; $w_{c}$ and $v_{c}$ are the process and measurement noises, respectively; $i_{R}$ and $i_{I}$ are the real and imaginary parts of the stator current, respectively; and $e_{R}$ and $e_{I}$ are the real and imaginary parts of the stator voltage, respectively.

In order to obtain the state transition function $f_{c} (\cdot)$ in (2), the stator current [

23] can be written as:

i_{d} = i_{R} s i n δ - i_{I} c o s δ

(6)

i_{q} = i_{I} s i n δ + i_{R} c o s δ

(7)

where $i_{d}$ and $i_{q}$ are the $d$ - and $q$ -axis currents, respectively.

Similarly, $e_{R}$ and $e_{I}$ can be written as:

e_{R} = (e_{d}^{'} + x_{q}^{'} i_{q}) s i n δ + (e_{q}^{'} - x_{d}^{'} i_{d}) c o s δ

(8)

e_{I} = (e_{q}^{'} - x_{d}^{'} i_{d}) s i n δ - (e_{d}^{'} + x_{q}^{'} i_{q}) c o s δ

(9)

B. Discrete State-space Model

After the model is discretized, the discrete nonlinear differential algebraic equation can be expressed as:

\{\begin{array}{l} x_{k} = f (x_{k - 1}, u_{k - 1}) + w_{k - 1} \\ y_{k} = h (x_{k}, u_{k}) + v_{k} \end{array}

(10)

where $f (\cdot)$ and $h (\cdot)$ are the discrete functions of $f_{c} (\cdot)$ and $h_{c} (\cdot)$ , respectively; and the subscript $k$ represents the time instance. The discrete state transition function can be obtained by applying the modified Euler technique, and the discrete model can be expressed as:

{\tilde{x}}_{k} = x_{k - 1} + f_{c} (x_{k - 1}, u_{k - 1}) Δ t

(11)

\tilde{f} = \frac{f_{c} ({\tilde{x}}_{k}, u_{k}) + f_{c} (x_{k - 1}, u_{k - 1})}{2}

(12)

x_{k} = x_{k - 1} + \tilde{f} Δ t

(13)

y_{k} = h (x_{k}, u_{k}) + v_{k}

(14)

where ${\tilde{x}}_{k}$ is the pre-estimated state; $Δ t$ is the sampling interval; and $\tilde{f}$ is the average derivative between adjacent times.

III. Proposed RAUKF

Based on the robust M-estimator and SSUT, after a novel adaptive update method against uncertainties is derived, a novel RAUKF with a lower computational burden is developed to deal with reduced estimation performance caused by model uncertainties.

A. SSUT-based UKF

The computational efficiency of UT depends on the selected number of sigma points around the available estimate. SSUT can reduce the number of sigma points by placing the sigma point directly on the origin or on a hypersphere centered at the origin. For an $n$ -dimensional system, based on the given statistical information, a set of $n + 2$ sigma points is generated by SSUT. Formally, we have:

χ_{k}^{j} = {\hat{x}}_{k - 1} + \sqrt[]{{\hat{P}}_{x, k - 1}} S_{j} j = 0,1, \dots, n + 1

(15)

where $χ_{k}^{j}$ denotes the sigma points selected by SSUT; ${\hat{x}}_{k - 1}$ and ${\hat{P}}_{x, k - 1}$ are the estimated state vector and corresponding error covariance matrix of the previous time step, respectively; and $S_{j}$ is the $j$ ^th column vector of the matrix $S$ . For an $n$ -dimensional system, the vector sequence of matrix $S$ and the corresponding weights of each sigma point can be expressed as:

\{\begin{array}{l} S_{0}^{1} = [0] \\ S_{1}^{1} = [\begin{matrix} - \frac{1}{\sqrt[]{2 W_{1}}} \end{matrix}] \\ S_{2}^{1} = [\begin{matrix} \frac{1}{\sqrt[]{2 W_{1}}} \end{matrix}] \end{array}

(16)

S_{c}^{n} = \{\begin{array}{l} [\begin{matrix} S_{0}^{n - 1} \\ 0 \end{matrix}] c = 0 \\ [\begin{matrix} S_{c}^{n - 1} \\ - \frac{1}{\sqrt[]{n (n + 1) W_{1}}} \end{matrix}] c = 1,2, \dots, n \\ [\begin{matrix} 0_{l - 1} \\ \frac{l}{\sqrt[]{n (n + 1) W_{1}}} \end{matrix}] c = n + 1 \end{array}

(17)

W_{j} = \frac{1 - W_{0}}{n + 1} j = 1,2, \dots, n + 1

(18)

where $W_{0}$ is the free parameter in which the range of $W_{0}$ is from 0 to 1 [

30]; and

W_{j}

is the weight of each sigma point.

Each sigma point is transformed using a transition function to predict the state. The pre-estimated state ${\tilde{x}}_{k}$ and its corresponding error covariance ${\tilde{P}}_{x, k}$ are calculated by:

{\tilde{χ}}_{k}^{j} = f (χ_{k}^{j})

(19)

{\tilde{x}}_{k} = \sum_{j = 0}^{n + 1} W_{j} {\tilde{χ}}_{k}^{j}

(20)

{\tilde{P}}_{x, k} = \sum_{j = 0}^{n + 1} W_{j} ({\tilde{χ}}_{k}^{j} - {\tilde{x}}_{k}) ({\tilde{χ}}_{k}^{j} - {\tilde{x}}_{k})^{T} + Q_{k}

(21)

where ${\tilde{χ}}_{k}^{j}$ indicates the sigma points through state function.

The sigma points are transformed using the measurement function. The predicted output ${\tilde{y}}_{k}$ and its corresponding error covariance ${\tilde{P}}_{y, k}$ are derived as:

{\tilde{y}}_{k} = \sum_{j = 0}^{n + 1} W_{j} η_{k}^{j}

(22)

η_{k}^{j} = h ({\tilde{χ}}_{k}^{j})

(23)

{\tilde{P}}_{y, k} = \sum_{j = 0}^{n + 1} W_{j} (η_{k}^{j} - {\tilde{y}}_{k}) (η_{k}^{j} - {\tilde{y}}_{k})^{T} + R_{k}

(24)

where $R_{k}$ is the covariance matrix of measurement noise.

The cross-covariance matrix is derived as:

P_{x y, k} = \sum_{j = 0}^{n + 1} W_{j} ({\tilde{χ}}_{k}^{j} - {\tilde{x}}_{k}) (η_{k}^{j} - {\tilde{y}}_{k})^{T}

(25)

The filtered state can be obtained by:

K_{k} = P_{x y, k} {\tilde{P}}_{y, k}^{^{- 1}}

(26)

{\hat{x}}_{k} = {\tilde{x}}_{k} + K_{k} (y_{k} - {\tilde{y}}_{k})

(27)

where $K_{k}$ is the Kalman gain.

B. Measurement Noise Estimator

As a nonlinear filtering technique, when the conventional UKF is applied to a power system, the mean and covariance matrices of noise are assumed to be known and to satisfy the Gaussian distribution. Nevertheless, in an actual scenario, noise statistics mostly follow a non-Gaussian distribution with a long tail, which means that meeting the assumption of a Gaussian distribution is difficult. In addition, because of the system operating conditions, aging of equipment components, channel noise, and other factors, the noise statistics may change and deviate from the a priori statistical characteristics [

31]. The uncertainty of the statistical characteristics means that the measurement noise covariance is inconsistent with the actual error in the measurement prediction process. These problems limit the practicability of UKF in power system state estimation to a certain extent.

To address the gross error caused by unknown or inaccurate a prior statistics of noise, robust M-estimation theory is combined with the conventional UKF to detect the effects of abnormal observation errors on state estimation [

32], [33]. This enables the UKF to adapt to the statistical characteristics of measurement noise. Because the measurement information affects only the measurement and prediction processes, (24) can be expressed as:

{\bar{P}}_{y, k} = \sum_{j = 0}^{n + 1} W_{j} (η_{k}^{j} - {\tilde{y}}_{k}) (η_{k}^{j} - {\tilde{y}}_{k})^{T} + {\bar{R}}_{k}

(28)

{\bar{R}}_{k} = {\bar{P}}^{- 1}

(29)

where $\bar{P}$ is the equally weighted matrix. The cost function can then be defined as:

J (x_{k}) = \sum_{i = 0}^{n + 1} ρ (r_{i}^{'})

(30)

where $r_{i}^{'} = r_{i} / σ_{r i}$ is the $i$ ^th component of the residual vector, $r_{i} = (y_{k} - {\tilde{y}}_{k})_{i, i}$ is the residual component, and $σ_{r i} = ({\tilde{P}}_{y, k})_{i, i}$ is the mean square error of $r_{i}$ ; and $ρ (\cdot)$ is the Huber function, which can be expressed as:

ρ (r_{i}^{'}) = \{\begin{array}{l} \frac{1}{2} {r_{i}^{'}}^{2} |r_{i}^{'}| \leq c \\ c |r_{i}^{'}| - \frac{1}{2} c^{2} |r_{i}^{'}| > c \end{array}

(31)

where $c$ is a constant, which is selected to be in the range of 1.3 to 2.

Next, $φ (r_{i}^{'}) = \partial ρ (r_{i}^{'}) / \partial (r_{i}^{'})$ is defined, and the minimization cost function in (30) can be expressed as:

\sum_{i = 0}^{n + 1} φ (r_{i}^{'}) \frac{\partial r_{i}^{'}}{\partial x_{i}} i = 0,1, \dots, n + 1

(32)

From $ψ (r_{i}^{'}) = φ (r_{i}^{'}) / r_{i}^{'}$ , we can obtain:

ψ (r_{i}^{'}) = \{\begin{array}{l} 1 |r_{i}^{'}| \leq c \\ \frac{c}{|r_{i}^{'}|} |r_{i}^{'}| > c \end{array}

(33)

$\bar{P}$ can be expressed as:

{\bar{P}}_{i, i} = \{\begin{array}{l} \frac{1}{σ_{i, i}} |r_{i}^{'}| \leq c \\ \frac{c}{σ_{i, i} | r_{i}^{'} |} |r_{i}^{'}| > c \end{array}

(34)

{\bar{P}}_{i, j} = \{\begin{array}{l} \frac{1}{σ_{i, j}} |r_{i}^{'}| \leq c, |r_{j}^{'}| \leq c \\ \frac{c}{σ_{i, j} m a x (|r_{i}^{'}|, |r_{j}^{'}|)} |r_{i}^{'}| > c, |r_{j}^{'}| > c \end{array}

(35)

where $σ_{i, i}$ and $σ_{i, j}$ are the diagonal and off-diagonal elements of the original $R_{k}$ , respectively. Because the measurement error variance is a diagonal matrix, the off-diagonal elements of the equivalent weight matrix are directly taken as zero.

C. Model and Process Noise Modification

For various reasons, operators are likely to consider erroneous data in power system analysis. It should be noted that the generation unit model is subject to uncertainties. Under stressed conditions, over-excitation limiters may have a certain effect on the excitation voltage [

34]. In addition, measuring the field current and voltage in brushless excitation systems is difficult for PMUs [9], [10]. The error of the dynamic model established for the generation unit affects the estimation of all state parameter components.

Therefore, to deal with the effects of the model parameter error, considering the computational complexity and real-time requirements of DSE, an adaptive update method is derived to handle the effects of the model error. The specific method is to modify ${\tilde{P}}_{x, k}$ using the adaptive regulator $α_{k}$ , whereby:

{\tilde{P}}_{x, k} = α_{k}^{- 1} [\sum_{j = 0}^{n + 1} W_{j} ({\tilde{χ}}_{k}^{j} - {\tilde{x}}_{k}) ({\tilde{χ}}_{k}^{j} - {\tilde{x}}_{k})^{T} + Q_{k}]

(36)

The following theorem can then be presented.

Theorem 1: ${\hat{P}}_{y, k}$ is the matrix obtained after new measurement information is introduced, $P_{y, k}$ is the matrix calculated by adaptive filtering, and ${\tilde{P}}_{y, k}$ is the matrix obtained by the covariance propagation law. Then, the adaptive factor satisfies:

P_{y, k} = {\hat{P}}_{y, k}

(37)

The adaptive factor is:

α_{k} = \frac{t r ({\tilde{P}}_{y, k} - {\bar{R}}_{k})}{t r ({\hat{P}}_{y, k} - {\bar{R}}_{k})}

(38)

where $t r (\cdot)$ represents the trace function.

Proof: the one-step prediction error ${\hat{x}}_{k}$ and filtered residual $r_{k}$ are expressed as:

{\hat{x}}_{k} = x_{k} - {\tilde{x}}_{k}

(39)

r_{k} = y_{k} - h ({\hat{x}}_{k}) = y_{k} - h (x_{k} - {\tilde{x}}_{k})

(40)

Based on the Taylor formula, a first-order Taylor expansion of $h (x_{k} - {\tilde{x}}_{k})$ is performed:

h (x_{k} - {\tilde{x}}_{k}) \approx h (x_{k}) - {\frac{\partial h}{\partial x}|}_{x = {\tilde{x}}_{k}} {\tilde{x}}_{k} = h (x_{k}) - D_{k} {\tilde{x}}_{k}

(41)

Substituting (41) into (40) yields:

r_{k} = y_{k} - h (x_{k}) + D_{k} {\tilde{x}}_{k} = v_{k} + D_{k} {\tilde{x}}_{k}

(42)

The residual covariance matrix obtained by the propagation law can be expressed as:

\begin{array}{l} {\tilde{P}}_{y, k} = E (r_{k} r_{k}^{T}) = E ((v_{k} + D_{k} {\tilde{x}}_{k}) (v_{k} + D_{k} {\tilde{x}}_{k})^{T}) = \\ E (D_{k} {\tilde{x}}_{k} {\tilde{x}}_{k}^{T} D_{k}^{T}) + E (v_{k} v_{k}^{T}) = D_{k} {\tilde{P}}_{x, k} D_{k}^{T} + {\bar{R}}_{k} \end{array}

(43)

In adaptive filtering, the covariance matrix ${\tilde{P}}_{x, k}$ is modified adaptively by the adaptive factor $α_{k}^{- 1}$ , and the theoretical residual covariance matrix is obtained as:

P_{y, k} = α_{k}^{- 1} D_{k} {\tilde{P}}_{x, k} D_{k}^{T} + {\bar{R}}_{k}

(44)

Based on the equivalence relationship in (37), we can obtain:

{\hat{P}}_{y, k} = P_{y, k} = α_{k}^{- 1} D_{k} {\tilde{P}}_{x, k} D_{k}^{T} + {\bar{R}}_{k}

(45)

α_{k} ({\hat{P}}_{y, k} - {\bar{R}}_{k}) = D_{k} P_{x, k} D_{k}^{T} = {\tilde{P}}_{y, k} - {\bar{R}}_{k}

(46)

The optimal adaptive factor can then be obtained by taking the trace of the matrix and the migration transformation. It is noteworthy that the adaptive factor of the adaptive filtering algorithm in practical applications is usually less than or equal to 1. The adaptive factor can be modified as:

α_{k} = \{\begin{array}{l} 1 t r ({\tilde{P}}_{y, k}) \geq t r ({\hat{P}}_{y, k}) \\ \frac{t r ({\tilde{P}}_{y, k} - {\bar{R}}_{k})}{t r ({\hat{P}}_{y, k} - {\bar{R}}_{k})} t r ({\tilde{P}}_{y, k}) < t r ({\hat{P}}_{y, k}) \end{array}

(47)

Omitting the measurement noise variance terms of the numerator and denominator in (47), the approximate expression of the optimal adaptive factor is:

α_{k} \approx \{\begin{array}{l} 1 t r ({\tilde{P}}_{y, k}) \geq t r ({\hat{P}}_{y, k}) \\ \frac{t r ({\tilde{P}}_{y, k})}{t r ({\hat{P}}_{y, k})} t r ({\tilde{P}}_{y, k}) < t r ({\hat{P}}_{y, k}) \end{array}

(48)

${\hat{P}}_{y, k}$ can be obtained by estimating the residual vector at the current time.

t r (P_{y, k}) = t r (r_{k} r_{k}^{T}) = r_{k} r_{k}^{T}

(49)

For convenience, the proposed estimation method can be presented as Algorithm 1.

Remark 1: in general, in UKF methods [

14]-[17], a set of

2 n + 1

sigma points is obtained using the symmetric point selection method. Notably, the computational efficiency of UT is related to the sigma point selection strategy and depends on the selected number of sigma points around the available estimate. The SSUT is utilized to improve computational efficiency by reducing the number of sigma points while ensuring transformation accuracy [19], [30].

Algorithm 1 : RAUKF
1: Initialization: initialize all parameters at time $k$
2: Input: $u_{k}$ , $y_{k}$ , and $N_{t}$ , where $N_{t}$ is the total number of time steps
3: while $k = 0$ to $N_{t}$ do
4: Step 1: generate spherical simplex sigma points by (12)-(15)
5: Step 2: state prediction
6: ${\tilde{χ}}_{k}^{j} = f (χ_{k}^{j})$
7: ${\tilde{x}}_{k} = \sum_{j = 0}^{n + 1} W_{j} {\tilde{χ}}_{k}^{j}$
8: ${\tilde{P}}_{x, k} = \sum_{j = 0}^{n + 1} W_{j} ({\tilde{χ}}_{k}^{j} - {\tilde{x}}_{k}) ({\tilde{χ}}_{k}^{j} - {\tilde{x}}_{k})^{T} + Q_{k}$
9: Step 3: measurement prediction
10: $η_{k}^{j} = h ({\tilde{χ}}_{k}^{j})$
11: ${\tilde{y}}_{k} = \sum_{j = 0}^{n + 1} W_{j} η_{k}^{j}$
12: ${\tilde{P}}_{y, k} = \sum_{j = 0}^{n + 1} W_{j} (η_{k}^{j} - {\tilde{y}}_{k}) (η_{k}^{j} - {\tilde{y}}_{k})^{T} + R_{k}$
13: Step 4: complete the correction of measurement noise using (25)-(32)
14: ${\bar{P}}_{y, k} = \sum_{j = 0}^{n + 1} W_{j} (η_{k}^{j} - {\tilde{y}}_{k}) (η_{k}^{j} - {\tilde{y}}_{k})^{T} + {\bar{R}}_{k}$
15: Step 5: calculate the adaptive factor using (45) and (46) and adaptively correct the model error and process noise as a whole
16: if $t r ({\tilde{P}}_{y, k}) \geq t r ({\hat{P}}_{y, k})$
17: ${\tilde{P}}_{x, k} = \sum_{j = 0}^{n + 1} W_{j} ({\tilde{χ}}_{k}^{j} - {\tilde{x}}_{k}) ({\tilde{χ}}_{k}^{j} - {\tilde{x}}_{k})^{T} + Q_{k}$
18: else
19: ${\tilde{P}}_{x, k} = α_{k}^{- 1} [\sum_{j = 0}^{n + 1} W_{j} ({\tilde{χ}}_{k}^{j} - {\tilde{x}}_{k}) ({\tilde{χ}}_{k}^{j} - {\tilde{x}}_{k})^{T} + Q_{k}]$
20: end if
21: Step 6: update the measurement using (22)-(24)
22: $K_{k} = P_{x y, k} {\tilde{P}}_{y, k}^{- 1}$
23: ${\hat{x}}_{k} = {\tilde{x}}_{k} + K_{k} (y_{k} - {\tilde{y}}_{k})$
24: Step 7: output ${\hat{x}}_{k}$ and ${\tilde{P}}_{x, k}$ , and update the time instance
25: end while

Remark 2: to address reduced estimation performance when the statistical feature of measurement noise deviates from the assumptions, an M-estimator is constructed based on a spherical simplex UT-based UKF. In addition, considering the computational complexity and real-time requirements of DSE, we propose a novel adaptive update method for adaptive modification of the model as a whole. The proposed method inherits the advantages of the conventional UKF and effectively improves the robustness of state estimation against uncertainties while ensuring the real-time demand of DSE.

IV. Numerical Results

To verify the performance of the proposed RAUKF, extensive simulations are conducted on a New England 10-generator 39-bus system, as shown in Fig. 1. Detailed parameters can be found in [

35]. PSCAD/EMTDC software is used for transient stability simulations to derive the truth states and measurements. To simulate the operation of the system, we assume that the three-phase fault of bus 16 occurs when

t = 0.5

s, and the fault is cleared when

t = 0.7

s. Here, the fault impedance is 0.001 p.u.. We set the total simulation time to 10 s, and the sampling interval

Δ t

is set to be 0.2 s [36]-[38]. The steady-state value of the generator is selected as the initial value of the state variable. Due to space constraints, only the estimation results of generator 8 (G8) are considered.

Fig. 1 New England 10-generator 39-bus system.

In an actual power system, because of the interference during the transmission process and the change in system operation, prior knowledge of noise is difficult to acquire. In addition, component aging and change in operating temperature also lead to deviations in the model. Therefore, considering the reduced state estimation performance derived from uncertainties, comparative experiments are set as follows.

Case 1: the UKF [

14], UHF [15], CIUKF [26], SRCDKF [27], and proposed RAUKF are compared and discussed with uncertain noise statistical characteristics.

Case 2: the aforementioned methods are analyzed and discussed with uncertain non-Gaussian noises derived from long-tailed and Gaussian distributions.

Case 3: the discussed methods are implemented with an uncertain model derived from parameter deviation in the state-space model.

Monte Carlo simulations of $N_{m c} = 200$ are run for these cases. The mean absolute error (MAE) and average estimation error index $E_{x}$ are used to assess the estimation performance of the algorithms:

M A E = \frac{1}{N_{m c}} \sum_{j = 1}^{N_{m c}} \frac{1}{N_{s}} \sum_{i = 1}^{N_{s}} |{\hat{x}}_{i, k} - x_{i, k}|

(50)

E_{x} = \frac{1}{N_{m c}} \sum_{j = 1}^{N_{m c}} \sqrt[]{\frac{1}{N_{t}} \sum_{k = 1}^{N_{t}} ({\hat{x}}_{i, k} - x_{i, k})^{2}}

(51)

where $N_{m c}$ is the number of Monte Carlo simulations; $N_{s}$ is the number of states; and ${\hat{x}}_{i, k}$ and $x_{i, k}$ are the estimated and true values of the state at each time instance, respectively. and $N_{t}$ is the total number of time steps.

A. Uncertain Noise Statistical Characteristics

Current DSE is typically conducted when the prior statistical characteristics of noise are known. In an actual power system, the statistical distribution of noise may change based on the operating conditions, and state estimation will be affected. To simulate the effects of this uncertainty on the estimation performance of the algorithm, the prior information of the noise statistics is assumed to be unknown. The covariance matrices of the actual noise are set to be $Q_{k} = 10^{- 6} I_{4 \times 4}$ and $R_{k} = 10^{- 5} I_{4 \times 4}$ . Based on the assumption that the prior statistical distribution deviates from the true distribution, the initial covariance matrices of the noise are set to be $Q_{k} = 10^{- 5} I_{4 \times 4}$ and $R_{k} = 10^{- 4} I_{4 \times 4}$ .

To evaluate the performance capabilities of the aforementioned methods against these uncertainties, all methods are used to track the state changes of G8. The state estimation results of each considered method are shown in Figs. 2 and 3. The results of MAE and average estimation error $E_{x}$ are presented in Fig. 4 and Table I, respectively.

Fig. 2 Estimation results of rotor angles and speeds for G8 with Case 1. (a) Estimation results of rotor angles. (b) Estimation results of rotor speeds.

Fig. 3 Estimation results of q- and $d$ -axis transient voltages for G8 with Case 1. (a) Estimation results of q-axis transient voltages. (b) Estimation results of d-axis transient voltages.

Fig. 4 MAE of all methods for G8 with Case 1.

Table I Average Estimation Error Results for Case 1

Method	Average estimation error
Method	$δ$	$ω$	$e_{q}^{'}$	$e_{d}^{'}$
UKF	0.04338	0.00066	0.01675	0.03647
UHF	0.01539	0.00051	0.01264	0.01846
RAUKF	0.00297	0.00007	0.00383	0.00446
CIUKF	0.00759	0.00010	0.00452	0.00945
SRCDKF	0.00390	0.00016	0.00582	0.01251

As the simulation results show, only RAUKF could accurately track the dynamic changes of state variables, whereas the estimation results of other algorithms deviate from the actual state, particularly for UKF and UHF. When the noise covariance matrix deviates from this assumption, the performance of the UKF deteriorates, particularly for the estimation results of the transient voltage along the local $d$ axis. Although the UHF could constrain the estimation error to a certain extent, it could not dynamically correct the mismatched initial covariance matrix estimation, where a poor effect is the result. With the SRCDKF, central differential sigma points could be used to obtain a higher estimation accuracy. In addition, better estimation results could be obtained through continuous iterations of the CIUKF. By contrast, the proposed RAUKF could adaptively correct the noise and shows strong robustness under noise uncertainty. For instance, RAUKF had the smallest MAE and average SE error among all the discussed methods.

B. Uncertain Non-Gaussian Noises

In an actual power system, the measurement noise of the PMU often follows a non-Gaussian distribution with a long tail. It is noteworthy that the Laplace distribution is one of the most widely used non-Gaussian distributions for describing thick-tailed distributions in many signal processing problems. To analyze the effectiveness of state estimation methods under unknown non-Gaussian noise, the noise covariance matrices are assumed to follow a Gaussian-Laplace noise mixture model, where the level of non-Gaussian contamination is 2%.

Figures 5-7 show the estimation results for the generator state variables. The MAE and average estimation error index results for each method are presented in Fig. 7 and Table II. As indicated, the estimation effect of the RAUKF is better than that of the others. As expected, due to the lack of adaptive updates to the estimation error covariance matrix, the UKF and SRCDKF could not bind the uncertainties. In addition, as shown in Fig. 6, the estimation performance of the UHF is better than those of both the UKF and SRCDKF because the uncertainties are limited to a certain extent. Nevertheless, in UHF, the estimation error covariance could not be updated adaptively according to the changes in the operating conditions, resulting in poor estimation effect. In the CIUKF, the estimation accuracy is effectively improved by sacrificing computational efficiency. Compared with other methods, the adaptive factor is used in the RAUKF to modify the estimation error covariance to a certain extent to obtain better estimation results and at a faster rate.

Fig. 5 Estimation results of rotor angles and speeds for G8 with Case 2. (a) Estimation results of rotor angles. (b) Estimation results of rotor speeds.

Fig. 6 Estimation results of q- and d-axis transient voltages for G8 with Case 2. (a) Estimation results of q-axis transient voltages. (b) Estimation results of d-axis transient voltages.

Fig. 7 MAE of all methods for G8 with Case 2.

Table II Average Estimation Error Index Results for Case 2

Method	Average estimation error
Method	$δ$	$ω$	$e_{q}^{'}$	$e_{d}^{'}$
UKF	0.01204	0.00016	0.00597	0.01288
UHF	0.01102	0.00013	0.00408	0.00821
RAUKF	0.00311	0.00001	0.00396	0.00457
CIUKF	0.00365	0.00001	0.00311	0.00454
SRCDKF	0.00250	0.00016	0.00597	0.01287

C. Uncertain State Model Parameters

Because of different operating environments, the performances of equipment parts may degenerate to various degrees and thus may be difficult to maintain. In such a situation, some parameters that remain unchanged by default may change over time, resulting in deviation from the model. To analyze the effectiveness of state estimation algorithms under uncertain state model parameters, the deviations of the transient reactance along the $d$ and $q$ axes are assumed to be 20%, which can be simulated using a Gaussian random variable.

Comparative experiments with the parameter deviation in the state-space model are depicted in Figs. 8 and 9. The evaluation indices of the algorithms are shown in Fig. 10 and Table III. As might be expected, the tracking effects of the UKF and SRCDKF are severely affected by model uncertainty, making it difficult for the UKF and SRCDKF to track the state accurately.

Fig. 8 Estimation results of rotor angles and speeds for G8 with Case 3. (a) Estimation results of rotor angles. (b) Estimation results of rotor speeds.

Fig. 9 Estimation results of $q$ - and $d$ -axis transient voltages for G8 with Case 3. (a) Estimation results of q-axis transient voltages. (b) Estimation results of d-axis transient voltages.

Fig. 10 MAE of all methods for G8 with Case 3.

Table III Average Estimation Error Index Results for Case 3

Method	Average estimation error
Method	$δ$	$ω$	$e_{q}^{'}$	$e_{d}^{'}$
UKF	0.01008	0.00016	0.03859	0.00955
UHF	0.00894	0.00013	0.02783	0.01187
RAUKF	0.00308	0.00001	0.00383	0.00523
CIUKF	0.00308	0.00001	0.00371	0.00492
SRCDKF	0.00248	0.00016	0.03866	0.00955

In addition, the effects of model parameter uncertainty could be attenuated to a certain extent in the UHF, but its accuracy is much lower than that of the RAUKF. In the CIUKF, the estimation error caused by variations in parameters could be dealt with effectively by iteration. By comparison, the estimation covariance matrix in the RAUKF could be updated by adding adaptive factors to obtain better estimation results when the model parameters change. As shown by the estimation results in Fig. 9, the UKF, UHF, and SRCDKF deviate significantly from the true values, which reflect only the general trend of the state change, particularly for the transient voltage along the $q$ axis.

For DSE, the computational efficiency is another major factor that must be considered. Thus, to compare the computation time of the algorithms considered for Cases 1-3, simulations are implemented on an Intel CPU i7-7700 system with 8 GB RAM in a MATLAB environment. It should be noted that the computation time shown in Table IV is implemented in MATLAB without being fully optimized, and the computational efficiency could be further improved using C-based code. The computation time of the methods considered for Cases 1-3 is shown in Table IV. Through continuous iterations to approach the true value of the state, CIUKF exhibits good estimation precision with Cases 1-3. However, its computational efficiency is significantly less than that of the proposed RAUKF. Because the fourth-order generator model is used in the simulation, the number of sigma points collected by the SSUT is one-third less than that of the conventional UT. In general, during state and measurement predictions, the calculation efficiency of the error covariances is affected by the number of sigma points, where the SSUT saves approximately one-third of the computational load compared with that of the conventional UT. Therefore, in this paper, although model modification increases the computational load of the method to a certain extent, its computational efficiency of RAUKF is slightly higher than that of the UKF.

Table IV Computation Time of Methods for Cases 1-3

Case	Computation time (ms)
Case	UKF	UHF	RAUKF	CIUKF	SRCDKF
Case 1	3731	4146	3580	11357	3759
Case 2	3632	4231	3462	11283	3611
Case 3	3681	4064	3532	11208	3658

V. Conclusion

In this paper, an adaptive update method based on a robust M-estimator is developed to address uncertainties in power systems. To improve the estimation efficiency and accuracy, the noise covariance matrix of the spherical simplex UT-based UKF is modified to deal with the effects of abnormal errors of measurement quantity in measurement prediction using a Huber-based robust M-estimator. Accordingly, an adaptive factor is used to modify the state prediction error covariance matrix, which can deal with the effects of model and noise errors on state prediction. Compared with other methods, the effectiveness of the RAUKF is demonstrated under model uncertainties.

Results show that the proposed adaptive update method could effectively bind uncertainties caused by unknown noise statistical characteristics and non-Gaussian noise. The proposed method remains effective when the values of generator model parameters are either inaccurate or incorrect. In future research, we will extend our method to bind the uncertainty caused by cyber-attacks to detect outliers.

References

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. 2020. [Baidu Scholar]

M. Amin, A. Al-Durra, and W. Elmannai, “Experimental validation of high-performance HIL-based real-time PMU model for WAMS,” IEEE Transactions on Industry Applications, vol. 56, no. 3, pp. 2382-2392, Jun. 2020. [Baidu Scholar]

R. Xiao, G. Wang, X. Hiao et al., “Dynamic state estimation of medium-voltage DC integrated power system with pulse load,” Journal of Modern Power Systems and Clean Energy, vol. 8, no. 4, pp. 689-698, Jul. 2020. [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]

M. Rostami and S. Lotfifard, “Distributed dynamic state estimation of power systems,” IEEE Transactions on Industrial Informatics, vol. 14, no. 8, pp. 3395-3404, Aug. 2018. [Baidu Scholar]

Y. Wang, Y. Sun, V. Dinavahi et al., “Adaptive robust cubature Kalman filter for power system dynamic state estimation against outliers,” IEEE Access, vol. 7, pp. 105872-105881, Jul. 2019. [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]

Y. Wang, Y. Sun, and V. Dinavahi, “Robust forecasting-aided state estimation for power system against uncertainties,” IEEE Transactions on Power Systems, vol. 35, no. 1, pp. 691-702, Jan. 2020. [Baidu Scholar]

E. Ghahremani and I. Kamwa, “Dynamic state estimation in power system by applying the extended Kalman filter with unknown inputs sto phasor measurements,” IEEE Transactions on Power Systems, vol. 26, no. 4, pp. 2556-2566, Nov. 2011. [Baidu Scholar]

S. Wang, W. Gao, and A. S. 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, May 2012. [Baidu Scholar]

L. Fan and Y. Wehbe, “Extended Kalman filtering based real-time dynamic state and parameter estimation using PMU data,” Electric Power Systems Research, vol. 103, pp. 168-177, Oct. 2013. [Baidu Scholar]

H. Karimipour and V. Dinavahi, “Extended Kalman filter-based parallel dynamic state estimation,” IEEE Transactions on Smart Grid, vol. 6, no. 3, pp. 1539-1549, May 2015. [Baidu Scholar]

C. Wang, Z. Qin, Y. Hou et al., “Multi-area dynamic state estimation with PMU measurements by an equality constrained extended Kalman filter,” IEEE Transactions on Smart Grid, vol. 9, no. 2, pp. 900-910, Mar. 2018. [Baidu Scholar]

G. Valverde and V. Terzija, “Unscented Kalman filter for power system dynamic state estimation,” IET Generation, Transmission & Distribution, vol. 5, no. 1, pp. 29-37, Jan. 2011. [Baidu Scholar]

W. Li and Y. Jia, “H-infinity filtering for a class of nonlinear discrete-time systems based on unscented transform,” Signal Processing, vol. 90, no. 12, pp. 3301-3307, Dec. 2010. [Baidu Scholar]

J. Qi, K. Sun, J. Wang et al., “Dynamic state estimation for multi-machine power system by unscented Kalman filter with enhanced numerical stability,” IEEE Transactions on Smart Grid, vol. 9, no. 2, pp. 1184-1196, Mar. 2018. [Baidu Scholar]

G. Anagnostou and B. C. Pal, “Derivative-free Kalman filtering based approaches to dynamic state estimation for power systems with unknown inputs,” IEEE Transactions on Power Systems, vol. 33, no. 1, pp. 116-130, Jan. 2018. [Baidu Scholar]

S. Julier and J. Uhlmann, “Reduced sigma point filters for the propagation of means and covariances through nonlinear transformations,” in Proceedings of the 2002 American Control Conference, Anchorage, USA, Nov. 2002, pp. 887-892. [Baidu Scholar]

S. Julier, “The spherical simplex unscented transformation,” in Proceedings of the 2003 American Control Conference, Denver, USA, Nov. 2003, pp. 2430-2434. [Baidu Scholar]

A. Tummala and R. Inapakurthi, “A two-stage Kalman filter for cyber-attack detection in automatic generation control system,” Journal of Modern Power Systems and Clean Energy, vol. 10, no. 1, pp. 50-59, Jan. 2022. [Baidu Scholar]

W. Wang, C. K. Tse, and S. Wang, “Dynamic state estimation of power systems by p-norm nonlinear Kalman filter,” IEEE Transactions on Circuits and Systems I: Regular Papers, vol. 67, no. 5, pp. 1715-1728, May 2020. [Baidu Scholar]

C. Wu, X. Li, W. Pan et al., “Zero-sum game-based optimal secure control under actuator attacks,” IEEE Transactions on Automatic Control, vol. 66, no. 8, pp. 3773-3780, Aug. 2021. [Baidu Scholar]

Y. Wang, Y. Sun, V. Dinavahi et al., “Robust dynamic state estimation of power systems with model uncertainties based on adaptive unscented H-infinity filter,” IET Generation, Transmission & Distribution, vol. 13, no. 12, pp. 2455-2463, Jun. 2019. [Baidu Scholar]

J. Ye and X. Yu, “Detection and estimation of false data injection attacks for load frequency control systems,” Journal of Modern Power Systems and Clean Energy, vol. 10, no. 4, pp. 861-870, Jul. 2022. [Baidu Scholar]

Y. Wang, Y. Sun, Z. Wei et al., “Parameters estimation of electromechanical oscillation with incomplete measurement information,” IEEE Transactions on Power Systems, vol. 33, no. 5, pp. 5016-5028, Sept. 2018. [Baidu Scholar]

A. Rouhani and A. Abur, “Constrained iterated unscented Kalman filter for dynamic state and parameter estimation,” IEEE Transactions on Power Systems, vol. 33, no. 3, pp. 2404-2414, May 2018. [Baidu Scholar]

X. Yu, X. Fan, T. K. Chau et al., “Square-root sigma-point filtering approach to state estimation for wind turbine generators in interconnected energy systems,” IEEE Systems Journal, vol. 15, no. 2, pp. 1557-1566, Jun. 2021. [Baidu Scholar]

N. Zhou, D. Meng, and S. Lu, “Estimation of the dynamic states of synchronous machines using an extended particle filter,” IEEE Transactions on Power Systems, vol. 28, no. 4, pp. 4152-4161, Nov. 2013. [Baidu Scholar]

S. Akhlaghi, N. Zhou, and Z. Huang, “A multi-step adaptive interpolation approach to mitigating the impact of nonlinearity on dynamic state estimation,” IEEE Transactions on Smart Grid, vol. 9, no. 4, pp. 3102-3111, Jul. 2018. [Baidu Scholar]

T. Joseph, B. Tyagi, and V. Kumar, “Dynamic state estimation of generators using spherical simplex unscented transform-based unbiased minimum variance filter,” IET Generation, Transmission & Distribution, vol. 14, no. 15, pp. 2997-3005, Aug. 2020. [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. 2018. [Baidu Scholar]

Y. Li, J. Li, J. Qi et al., “Robust cubature Kalman filter for dynamic state estimation of synchronous machines under unknown measurement noise statistics,” IEEE Access, vol. 7, pp. 29139-29148, Feb. 2019. [Baidu Scholar]

G. Wang, X. Xu, and T. Zhang, “M-M estimation-based robust cubature Kalman filter for INS/GPS integrated navigation system,” IEEE Transactions on Instrumentation and Measurement, vol. 70, pp. 1-11, Sept. 2020. [Baidu Scholar]

G. Anagnostou and B. C. Pal, “Impact of overexcitation limiters on the power system stability margin under stressed conditions,” IEEE Transactions on Power Systems, vol. 31, no. 3, pp. 2327-2337, May 2016. [Baidu Scholar]

C. Canizares, T. Fernandes, E. Geraldi et al., “Benchmark models for the analysis and control of small-signal oscillatory dynamics in power systems,” IEEE Transactions on Power Systems, vol. 32, no. 1, pp. 715-722, Jan. 2017. [Baidu Scholar]

G. Tian, Y. Gu, D. Shi et al., “Neural-network-based power system state estimation with extended observability,” Journal of Modern Power Systems and Clean Energy, vol. 9, no. 5, pp. 1043-1053, Sept. 2021. [Baidu Scholar]

Y. Li, L. He, F. Liu et al., “Flexible voltage control strategy considering distributed energy storages for DC distribution network,” IEEE Transactions on Smart Grid, vol. 10, no. 1, pp. 163-172, Jan. 2019. [Baidu Scholar]

C. Wu, W. Yao, W. Pan et al., “Secure control for cyber-physical systems under malicious attacks,” IEEE Transactions on Control of Network Systems, vol. 9, no. 2, pp. 775-788, Jun. 2022. [Baidu Scholar]

Address:No.19 Chengxin Avenue, Jiangning District, Nanjing 211106, China

E-mail: mpce@alljournals.cn

Tel:86-25-81093060

Fax:86-25-81093040

Home

Introduction

Editorial Board

For Author

Call For Papers

APC

Sponsor & Publisher