Luke Dosiek

Mode shape estimation algorithms under ambient conditions: A comparative review

This paper provides a comparative review of five existing ambient electromechanical mode shape estimation algorithms, i.e., the Transfer Function (TF), Spectral, Frequency Domain Decomposition (FDD), Channel Matching, and Subspace Methods. It is also shown that the TF Method is a general approach to estimating mode shape and that the Spectral, FDD, and Channel Matching Methods are actually special cases of it. Additionally, some of the variations of the Subspace Method are reviewed and the Numerical algorithm for Subspace State Space System IDentification (N4SID) is implemented. The five algorithms are then compared using data simulated from a 17-machine model of the Western Electricity Coordinating Council (WECC) under ambient conditions with both low and high damping, as well as during the case where ambient data is disrupted by an oscillatory ringdown. The performance of the algorithms is compared using the statistics from Monte Carlo simulations and results from measured WECC data, and a discussion of the practical issues surrounding their implementation, including cases where power system probing is an option, is provided. The paper concludes with some recommendations as to the appropriate use of the various techniques.

A Recursive Maximum Likelihood Estimator for the Online Estimation of Electromechanical Modes With Error Bounds

Summary form only given. Accurate and near real-time estimates of electromechanical modes are of great importance since the modal damping is a key indicator of the stability of the power system. If the estimates of the electromechanical modes are to be useful, knowing the variability in the estimates is critically important. This paper presents a method of directly estimating the variance of each mode estimate in addition to estimating the frequency and damping of each mode in an online setting using a recursive maximum likelihood (RML) estimator. The variance estimates are achieved using two closed-form multidimensional Taylor series approximations, the details of which are fully derived here. The proposed method is validated using a Monte Carlo simulation with a low order model of the Western Electricity Coordinating Council (WECC) power system under both ambient and probing conditions, with multiple modes closely spaced in frequency, and is compared to the regularized robust recursive least squares (R3LS) method. It is also successfully applied to phasor measurement unit (PMU) data collected from the actual WECC system, also under both ambient and probing conditions.

Electromechanical mode shape estimation based on transfer function identification using PMU measurements

Power system mode shapes are a key indication of how dynamic components participate in low-frequency oscillations. Traditionally, mode shapes are calculated from a linearized dynamic model. For large-scale power systems, obtaining accurate dynamic models is very difficult. Therefore, measurement-based mode shape estimation methods have certain advantages, especially for the application of real-time small signal stability monitoring. In this paper, a measurement-based mode shape identification method is proposed. The general relationship between transfer function (TF) and mode shape is derived. As an example, a least square (LS) method is implemented to estimate mode shape using an autoregressive exogenous (ARX) model. The performance of the proposed method is evaluated by Monte-Carlo studies using simulation data from a 17-machine model. The results indicate the validity of the proposed method in estimating mode shapes with reasonably good accuracy.

New algorithms for mode shape estimation using measured data

Two signal-processing approaches for estimating a power systempsilas electromechanical mode-shape properties using time-synchronized phasor measurements are presented. The methods are termed the ldquocommunication methodrdquo and the ldquochannel matching method.rdquo The basic approach for each method is presented along with examples. A comparison is also made with the traditional Welch periodogram averaging approach. The examples include simulation results as well as measured data from the western North American power system.

Estimating electromechanical modes and mode shapes using the multichannel ARMAX model

A method of estimating the electromechanical modes and mode shapes from multiple synchrophasors is presented in this paper. The approach is based upon identifying the transfer function representation of the state space model of the linearized power system through the estimation of a multichannel AutoRegressive Moving Average eXogenous (ARMAX) model. The relationship between the transfer function and the ARMAX model, which is fully derived here, is shown to be general enough that virtually any multichannel ARMAX estimation algorithm may be applied to this approach. The two-stage least squares (2SLS) algorithm is used here for illustrative purposes. The proposed method is validated with the Monte Carlo procedure using simulated data from a reduced-order model of the Western Electricity Coordinating Council (WECC) system. The performance of the new method is compared with existing multichannel mode and mode shape estimation techniques and is seen to provide increased estimation accuracy with a decrease in computation time. The proposed method is further validated with measured data from the WECC system.

A channel matching approach for estimating electromechanical mode shape and coherence

A method of estimating a power systems electromechanical mode shape and coherence from time synchronized phasor measurements is presented. The approach uses a parametric estimate of the transfer function (TF) between signals at different buses throughout the power system. The relationship between the TF and mode shape and coherence is reviewed. A non-causal autoregressive exogenous (ARX) model is used in a least squares (LS) minimization to estimate the TF parameters and to estimate the magnitude squared coherence function. The method is applied to both a simulated system and measured data from the western North American power system and is compared to the traditional Welch periodogram averaging approach.

Weighted update method for spectral mode shape estimation from PMU measurements

A method of tracking a power systems electromechanical mode shape and coherence over time is presented. It uses an algorithm of weighted-updates to efficiently compute an existing spectral estimation algorithm. The advantage of this over other mode shape estimators is that it overcomes the assumption that the system being estimated is stationary. The relationship between spectral densities and mode shape and coherence is reviewed. The method is applied to both a simulated system and measured data from the western North American power system. It is shown to accurately track the mode shape and coherence over time in the presence of a major system configuration change.

An improved bootstrap method for electromechanical mode estimation using multivariate probability distributions

Electromechanical modes must be estimated with a high level of accuracy in order for the estimates to be useful in helping to ensure reliable power system operation. To assess the accuracy of modal estimates, one would ideally use a Monte Carlo approach where several independent experiments would be performed on an unchanging system. However, in reality a power system is constantly changing, making Monte Carlo tests impractical. Therefore, bootstrapping has been applied to the task of estimating the accuracy of mode estimators. The previously proposed bootstrapping methods involved resampling residuals obtained from various algorithms, from which resampled mode estimates were obtained using a computationally intensive method that includes filtering and the reapplication of the modal estimation algorithm for each bootstrap. This paper proposes a more efficient method of bootstrapping by directly resampling the parameter estimates of an algorithm through the estimation of a multivariate probability distribution. The proposed method is compared with the old method using both simulated and measured data and is shown to retain the accuracy of the old method while significantly reducing the computation time.

Overview of System Identification for Power Systems from Measured Responses

Abstract Large interconnected power systems are arguably some of the most complicated man-made systems to understand and to characterize. The scale of the problem is immense, involving large numbers of generators, controllers, and transmission lines covering millions of square kilometers. Measurement technology has reached a point where Phasor Measurement Units (PMUs) are being widely installed in power systems all over the world. These devices provide time synchronized (via GPS) phasor measurements from throughout the power grid to Phasor Data Concentrators (PDCs) at power system control centers. These time series can be used to better characterize the system and hopefully, in the long term, to better control the system. This paper presents a tutorial on estimating power system characteristics from measured responses. About a given operating point, power system low-frequency dynamics are well modeled as a high-order, multi-input, multi-output linear system. Of primary interest is the estimation of the inter-area electromechanical modes of the system. These inter-area modes involve generators from one area of the system oscillating against generators in another area of the system. The modes are characterized by their frequency, damping, and shape. In August 1996, the western United States experienced a massive wide spread black out caused by an unstable inter-area mode, involving generators in the north swinging against generators in the south. This paper overviews the problem and examines several methods of estimating the electromechanical modes under different signal conditions. Several real-world examples are given for estimating the electromechanical modes from ambient, transient, or probing situations. When the system is probed, more general state-space and transfer-function models are estimated. Probing a power system with known inputs is challenging and is discussed in this paper. Estimation performance issues are also discussed.

The empirical transfer function estimate method for determining mode shape

A very direct method for estimating a power systems electromechanical mode shape is presented. The method develops an empirical transfer function estimate (ETFE) from measurement data acquired during low-level periodic probing of a power system to obtain a relationship between system outputs distributed throughout the power system. This frequency response is then used to obtain mode shape estimates. The method is applied to a system model and results are compared with true values. Monte Carlo simulations are performed to assess the performance of the method. The algorithm is also applied to Phasor Measurement Unit (PMU) data from Western Electricity Coordinating Council (WECC) system wide probing tests and comparisons with the known configuration of the power system are provided.