Data fusion-based distributed Prony analysis

doi:10.1016/j.epsr.2016.10.052

Electric Power Systems Research

Volume 143, February 2017, Pages 634-642

https://doi.org/10.1016/j.epsr.2016.10.052 Get rights and content

Highlights

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Application of Prony algorithm in power system oscillation identification.
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Distributed Prony analysis algorithm using data fusion technique.
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Identification of noise covariance for each channel to facilitate of weighted LSE construction.
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Large-dimension of PMU data by solving least square estimation problems with small sizes in parallel.
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Real-world data is used for validation.

Abstract

This paper presents a distributed Prony analysis algorithm using data fusion approach. This classic approach can be found in Kalman filter's measurement update. Distributed optimization algorithms, e.g., alternating direction method of multipliers (ADMM), suitable for constrained optimization problems, have been proposed in the previous literature to develop distributed architecture. In this article, we show that Prony analysis, a non-constrained least square estimation (LSE) problem, can be solved using the classic data fusion approach. Compared to the iterative distributed optimization algorithms (e.g., ADMM and subgradient methods), data fusion takes only one step. There is no need for iteration and there is no issue related to convergence. This approach leads to a distributed Prony analysis architecture which requires a much less demanding communication system (the bandwidth can be less than 0.1 Hz) compared to the conventional centralized Prony analysis for multiple channels which requires a bandwidth of 30 Hz. The application discussed in this paper is to identify oscillation modes from real-world phasor measurement unit (PMU) data and further reconstruct signals. A key technical challenge to implement Prony analysis for signals from multiple channels is the difficulty to identify the noise characteristics of each channel. In this paper, a method is proposed to identify the noise covariances, which leads to the construction of a weighted least square estimation (WLSE) problem. This problem is solved through a distributed architecture. The effectiveness of the proposed distributed Prony analysis is demonstrated through case study results. The accuracy of the estimation is improved in one order compared with the centralized Prony analysis.

Introduction

A power system is a massive system that can be perturbed by load changes, generator trips, faults or networks changes. Power system oscillations are common issues. To mitigate oscillations, oscillations should be identified and studied in a timely manner. There are two separate approaches to identify power system oscillations. The first approach is based on detailed dynamic models of the system. State-space modeling and the eigenvalue analysis can give the system's oscillation modes [1]. Detailed modeling of a complicated power system is challenging and prone to errors. The second approach is based on measurements to identify oscillation modes. Measurement-based approach has been adopted by control engineers in practice. For example, equivalent system models will be constructed based on the measurements and further control strategies will be developed based on the identified system models.

With phasor measurement unit (PMU) data collected, electromechanical oscillation modes (<2 Hz) can be identified from these measurements at 30 Hz sampling rate. Several measurement-based system identification have been proposed for PMU data-based estimation, such as Kalman filters [2], [3], [4], least square estimation [5], and subspace algorithm [6]. Prony analysis is one of the most common measurement-based identification approaches to identify oscillatory modes. Prony analysis has been introduced by Hauer et al. in power systems in 1990 [7], [8]. The main idea is to directly estimate the frequency, damping and phase of modal components of a measured signal. An extension to Prony analysis is then introduced which allowed multiple signals to be analyzed at the same time resulting [9].

Application of distributed optimization techniques has recently been introduced in system modes identification [10], [11], [12]. For example, in [10], distributed Prony analysis using alternating direction method of multipliers (ADMM) has been combined with centralized Prony method to estimate the slow frequency eigenvalues. Simulation data generated by PST [13] toolbox of IEEE 39-bus system is used to conduct Prony analysis. In the author's previous paper [14], another distributed Prony analysis algorithm using consensus and subgradient update is developed. Distributed Prony analysis presented in the aforementioned papers can be applied to multiple signals from multiple locations collected at the same period of time. These algorithms can handle a large-dimension of PMU data by solving least square estimation (LSE) problems with small sizes in parallel and iteratively. This paper serves as a rebuttal of the above distributed optimization approaches: iterations are not necessary. Indeed, Prony analysis is essentially an LSE problem without any constraints. Prony analysis of multi-channel signals is a multi-objective LSE problem. LSE problems were introduced by Gauss in 1790s. In 1960s, R. Kalman designed an iterative approach for LSE. See [15] for a detailed description. In Kalman filter, measurement update takes one step to find the best estimate given the prior information and current measurement [16]. There is no need of iteration. Compared to Kalman filter-based approach, distributed optimization approaches are not efficient. Kalman filter-based approach has been used in multi-sensor data fusion [17]. In this paper, the approach is named as data fusion approach.

In this paper, the philosophy of data fusion is examined in detail and applied to develop an effective algorithm for distributed Prony analysis. A key technical challenge to implement Prony analysis for signals from multiple channels is the difficulty to identify the noise characteristics of each channel. In this paper, a method is proposed to identify the noise covariances, which leads to the construction of a weighted least square estimation (WLSE) problem. This problem is solved through a distributed architecture. In a nutshell, the contribution of the paper is to implement Kalman filter-based data fusion approach in Prony analysis with multiple channels. This approach has not been seen in Prony analysis. Compared to the other approaches where constrained optimization problems are formulated and further been solved by iterative distributed algorithms, e.g., [14], [12], the proposed approach does not require iterations and has advantages in computation.

The rest of the paper is as follows. Section 2 describes the fundamentals of Prony analysis. Section 3 describes the centralized multi-channel Prony analysis. Section 4 presents data fusion and distributed Prony analysis. Section 5 further examines the relationship of data fusion based Prony analysis and multi-channel Prony analysis. Section 6 presents the case study results. Section 7 concludes this paper.

Section snippets

Fundamentals of Prony analysis

Consider a Linear-Time Invariant (LTI) system with the initial state of x(t₀)=x₀ at the time t₀, if the input is removed from the system, the dynamic system model can be represented as [18]: $\dot{x} (t) = Ax (t)$ $y (t) = Cx (t)$ where $y \in ℝ$ is defined as the output of the system, $x \in ℝ^{n}$ is the state of the system, $A \in ℝ^{n \times n}$ and $C \in ℝ^{1 \times n}$ are system matrices. The order of the system is defined by n. If the λ_i, p_i, and q_i are the ith eigenvalue, the corresponding right eigenvector, and left eigenvectors of A respectively,

Centralized multi-channel Prony analysis

The conventional way to find the vector a from multiple signals has been documented in [9]. A brief description is offered as follows. Suppose that there are m channels of PMU data taken from the same period of time. For each channel of the PMU data, it is possible to formulate the D matrix and Y vector. They will be notated as D_i and Y_i for the ith channel. a can be found by the following estimation problem. $[\begin{matrix} D_{1} \\ D_{2} \\ ⋮ \\ D_{m} \end{matrix}] a = [\begin{matrix} Y_{1} \\ Y_{2} \\ ⋮ \\ Y_{m} \end{matrix}]$ The m channels can be from one single PMU or from multiple PMUs. For

Data fusion

Data fusion can be explained by the following simple example. Assume the state x to be estimated has prior information $x_{1} \in ℝ^{n}$ from the first measurement and $x_{2} \in ℝ^{n}$ from the second measurement. These two measurements have different accuracy and the covariance matrices are assumed to be Σ₁ and Σ₂. In order to find the best estimation, we conduct the following multi-objective weighted least square estimation (LSE). $\min_{x} {(x - x_{1})}^{T} Σ_{1}^{- 1} (x - x_{1}) + {(x - x_{2})}^{T} Σ_{2}^{- 1} (x - x_{2})$

Suppose $x \in ℝ^{2}$ , we can then plot x₁ and x₂ in

Comparison of multi-channel Prony analysis and data fusion Prony analysis

In this section, the relationship between the two methods is examined. First, the objective functions of the least square estimation problems are examined.

For multi-channel Prony analysis, the objective function can be written as follows. $Multi - channel Prony : \min_{a} \sum_{i = 1}^{m} ∥ D_{i} a - Y_{i} ∥^{2}$

As an LSE problem, the underlying assumption of the above objective function is that the noises of all channels are with the same covariance. Solving the above minimization problem is the same as solving the following

Case study

In this section, a case study of real-world PMU data is carried out. 17 frequency deviation signals from PMU measurements are given in Fig. 3. Three scenarios are studied:

1.
Centralized multi-channel Prony analysis using (17). The reconstructed signals are shown in Fig. 5. The reconstructed signals match poorly with the PMU measurements presented in Fig. 3.
2.
Data fusion assuming the same covariance. Reconstructed signals are shown in Fig. 6. The reconstructed signals have an overall better matching

Conclusion

This paper presents distributed Prony analysis based on classic Kalman filter data fusion approach. The approach is much more efficient compared to many distributed Prony analysis algorithms in the literature. The paper presents the in-depth analysis of data fusion and the relationship of multi-channel Prony analysis and distributed Prony analysis. Real-world PMU measurements are used in case studies to demonstrate the effectiveness of the distributed Prony analysis method. The proposed

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View full text

Data fusion-based distributed Prony analysis

Highlights

Abstract

Introduction

Section snippets

Fundamentals of Prony analysis

Centralized multi-channel Prony analysis

Data fusion

Comparison of multi-channel Prony analysis and data fusion Prony analysis

Case study

Conclusion

Electr. Power Syst. Res.

Electr. Power Syst. Res.

Electr. Power Syst. Res.

Identification and tracking of harmonic sources in a power system using a Kalman filter

IEEE Trans. Power Deliv.

Least squares estimation-based synchronous generator parameter estimation using PMU data

Initial results in power system identification from injected probing signals using a subspace method

Initial results in Prony analysis of power system response signals

IEEE Trans. Power Syst.

Application of Prony analysis to the determination of modal content and equivalent models for measured power system response

IEEE Trans. Power Syst.

Making Prony analysis more accurate using multiple signals

IEEE Trans. Power Syst.