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Starting point selection approach for power system model validation using event playback

2020; Institution of Engineering and Technology; Volume: 14; Issue: 19 Linguagem: Inglês

10.1049/iet-gtd.2020.0094

1751-8695

Shahrokh Akhlaghi, Ning Zhou, Hsiao‐Dong Chiang,

Machine Fault Diagnosis Techniques

IET Generation, Transmission & DistributionVolume 14, Issue 19 p. 3972-3982 Special Section: Next Generation of Synchrophasor-based Power System Monitoring, Operation and ControlFree Access Starting point selection approach for power system model validation using event playback Shahrokh Akhlaghi, Corresponding Author Shahrokh Akhlaghi sakhlag1@binghamton.edu orcid.org/0000-0002-7982-1631 Department of Electrical and Computer Engineering, Binghamton University, State University of New York, Binghamton, NY, 13902 USASearch for more papers by this authorNing Zhou, Ning Zhou Department of Electrical and Computer Engineering, Binghamton University, State University of New York, Binghamton, NY, 13902 USASearch for more papers by this authorHsiao-Dong Chiang, Hsiao-Dong Chiang Department of Electrical and Computer Engineering, Cornell University, Ithaca, NY, 14853 USASearch for more papers by this author Shahrokh Akhlaghi, Corresponding Author Shahrokh Akhlaghi sakhlag1@binghamton.edu orcid.org/0000-0002-7982-1631 Department of Electrical and Computer Engineering, Binghamton University, State University of New York, Binghamton, NY, 13902 USASearch for more papers by this authorNing Zhou, Ning Zhou Department of Electrical and Computer Engineering, Binghamton University, State University of New York, Binghamton, NY, 13902 USASearch for more papers by this authorHsiao-Dong Chiang, Hsiao-Dong Chiang Department of Electrical and Computer Engineering, Cornell University, Ithaca, NY, 14853 USASearch for more papers by this author First published: 07 July 2020 https://doi.org/10.1049/iet-gtd.2020.0094Citations: 1AboutSectionsPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinkedInRedditWechat Abstract Model validation is an essential task to determine whether a model can accurately describe the actual behaviours of a power system. Currently, major commercial software tools are equipped with an 'event playback' function to validate dynamic models using testing data from phasor measurement units (PMUs). Due to their limited bandwidth and low sampling rates, PMUs cannot capture the fast-transient dynamics. As such, the playback function using the conventional approach may mistakenly invalidate an accurate model during the high-frequency responses. To overcome the deficiency, a batch state estimation approach is proposed in this study to improve the performance of the 'event playback' function by focusing on low-frequency responses in model validation. The proposed approach consists of three major steps. First, a multi-model adaptive Kalman filter approach is used to estimate the dynamic states of the system. Second, the singular spectrum analysis (SSA) is used to detect the fault clearance time. Finally, the estimated states after the fault are used as the initial states of the 'event playback' to validate the dynamic model during the low-frequency responses. The analytical basis of the proposed method is also provided by showing the existence and uniqueness of the trajectory of the underlying model. The effectiveness of the proposed approach is demonstrated using the PSS/E. Nomenclature sampling interval scaling parameter to control the positions of the sigma rotor angle in radian rotor speed in pu dk measurement innovation CKF cubature Kalman filter DSE dynamic state estimation Dt speed control gain in pu and damper winding flux linkage along d and q axes in pu E(*) expected value Ecomp terminal voltage and transient voltages along d and q axes in pu EKF extended Kalman filter EnKF ensemble Kalman filter Eterm terminal voltage in pu f(*) state transition function Jacobian matrix of the state transition function at step k−1 h(*) measurement function H inertia constant Jacobian matrix of the measurement function at step k id and iq stator currents along d and q axes in pu k time index KA voltage regulator gain KD damping factor KE internal voltage gain KF feedback gain MMAKF multi-model adaptive Kalman filter MSE mean square errors nEnKF total number of samples P state covariance matrix PMU phasor measurement unit pu per-unit Q covariance matrix of the process noise R covariance matrix of the measurement noise RD steady-state gain in pu Sd and Sq saturation along d and q axes SE saturation function SSA singular spectrum analysis and sub-transient time constants in the dq0 frame in seconds T1 governor valve time constant in second T2 high-pressure section time constant in second T3 reheater time constant in seconds TA voltage regulator time constants in seconds and open circuit time constants in the dq0 frame in seconds Te electric air-gap torque in pu TE internal voltage time constants in seconds TF feedback integrator time constants in sec Tm or Pm mechanical torque in pu TP turbine power state TR transducer filter time constants in seconds uk known input vectors UKF unscented Kalman filter VF feedback voltage integrator vk measurement noise VO governor valve opening state VR voltage regulator Vref control input VT voltage transducer in pu wk process noise and sub-transient reactance along d and q axes in pu and transient reactance along d and q axes in pu xd and xq synchronous reactance along d and q axes in pu xk state vectors xl leakage reactance in pu zk measurement vectors 1 Introduction To efficiently and reliably operate the power grid, it is of critical importance to maintain high-quality dynamic models. Following the blackout on 14 August 2003 in the Northeastern United States and Eastern Canada, North American Electric Reliability Corporation (NERC) and U.S. Canada Power System Outage Task Force recommended to periodically validate power grid models [1]. Model validation can be defined as a procedure for substantiating 'that a computerised model within its domain of applicability possesses a satisfactory range of accuracy consistent with the intended application of the model' [2]. The purpose of the model validation is to determine whether a model can appropriately represent the underlying system phenomena. One of the most well-known techniques in model validation is event validation, which compares the measured system responses with the simulated model responses during a dynamic event to determine whether the model is adequate. The first step of event validation is to align the power-flow simulation results with measured nodal voltage magnitudes and angles to ensure the adequacy of the power-flow model in the steady state. Then, the simulated dynamic responses are compared with the actual system responses measured by phasor measurement units (PMUs) [3]. When the model of the subsystem is accurate, the simulation responses should match the actual system responses. Therefore, when significant mismatches between simulation and measurement are observed, the model is considered inadequate. In power systems, models used by event validation can be categorised into system-wide models and subsystem models. In the former one, the model of the whole system needs to be set up for simulation. Evidently, the accuracy of all components is effectual and therefore needs to be modelled. Also, the huge number of components and their various operating conditions make this process time-consuming and unproductive [4]. To overcome the difficulties, the hybrid dynamic simulation is proposed in [5] to focus on the questioned subsystem and truncate the remaining parts. 'Event playback' is one of the applications of the hybrid dynamic simulation [6-9], which injects PMU measurements as the inputs of the subsystem model at its boundary bus to perform the simulation. Many commercial software programs such as GE PSLF™, Siemens PTI PSS/E™, PowerWorld™ Simulator and EPRI PPPD are equipped with the 'event playback' function [10]. During model validation studies using PMU measurements [11], the authors observed a deficiency in using the conventional 'event playback' function, i.e. significant mismatches were still observed even when an accurate model was used for 'event playback'. After carefully analysing the data and models, the authors found that the mismatches were caused by the limited frequency bandwidth of PMUs instead of model inaccuracy. It was found that PMUs with sampling rates of 30–60 samples/s cannot effectively capture the fast-transient dynamics of the power grid especially during the severe changes of system operating conditions such as faults. Even if an accurate model is used, the limited bandwidth of PMU can still cause significant mismatches between the measured system responses and simulated responses from the 'event playback'. The problem gets worse when the subsystem model under study is closer to the faults during the initial sub-transient responses. Qiu et al. [12] encountered the similar issue when validating the dynamic model of a coal plant in PJM and called for 'in-depth investigation' of the impacts of PMU sampling rates on model validation. Also, the same issue was identified by the engineers in the Independent System Operator New England [13] referred to as PMU data limitation. Therefore, it is essential to address the negative impact of PMU's limited bandwidth and low sampling rates on the 'event playback' function because it can mistakenly invalidate an accurate model. To overcome the deficiency of conventional 'event playback' approaches, a batch state estimation approach is proposed in this paper to focus on low-frequency responses of a subsystem. By starting the playback function after a fault is cleared and its high-frequency responses die out, the proposed validation approach focuses on the low-frequency bandwidth of the dynamic model which can be well covered by PMU measurements. In contrast, conventional 'event playback' approaches start the playback function before the fault and cover not only the low-frequency responses but also the high-frequency sub-transient responses which cannot be covered by the bandwidth of PMUs. As a result, when the fault responses include significant high-frequency sub-transient responses, the conventional 'event playback' approaches may fail because of PMU's limited bandwidth whereas the proposed approach can more efficiently use the PMU data and successfully validate dynamic models over the low-frequency responses. To start the playback function after a fault is cleared and its high-frequency responses die out, the fault needs to be detected and the dynamic states of the system at that point are required to implement the proposed approach. Therefore, the proposed approach consists of the following three major steps: (i) A multi-model adaptive Kalman filtering (MMAKF) approach is used to accurately estimate the dynamic states. (ii) The singular spectrum analysis (SSA) is proposed to detect the fault clearance time. (iii) The estimated states after the fault are used as the initial states of the 'event playback' to validate the dynamic model during the low-frequency responses, which can be effectively captured by PMUs. The rest of this paper is organised as follows. Section 2 reviews the 'event playback' and the subsystem model. The proposed MMAKF and fault detection approaches are introduced in Section 3. A case study is presented in Section 4. Conclusions are drawn in Section 5. 2 Event playback and model under study This section reviews the conventional 'event playback' approach and the subsystem models it uses for model validation studies. 2.1 Review on event playback As it is shown in Fig. 1, the 'event playback' approach can be used to validate the model accuracy of a subsystem, which can be decoupled from the external grid at the point of common coupling by a PMU. First, the bus voltage magnitude, phase angle, along with frequency and active and reactive power recorded by the PMU during an actual dynamic event in the system (such as line/generator tripping) are recorded. Then, the recorded inputs, i.e. the voltage magnitude and frequency/phase angle measured by the PMU, are injected into the subsystem model to perform dynamic simulation using the 'event playback' function of simulation software. The injected PMU measurements in 'event playback' are called 'play-in signals'. Finally, the simulated responses, usually active and reactive power, are compared with measured responses from PMUs [7, 8]. If the difference between the simulated responses and measured responses is sufficiently small, the model is considered accurate. Otherwise, the model is considered inaccurate. Fig. 1Open in figure viewerPowerPoint Model validation with 'event playback' Note that the conventional 'event playback' approach starts its simulation during the steady-state responses before the dynamic responses. Thus, all the derivatives in the differential equations of the subsystem dynamic models can be set to 0 to determine the initial states to start the simulation. Also, all the sub-transient, transient and steady-state responses, which include all the bands of frequency responses, are compared to determine the model accuracy. Four methods, which have been proposed to playback the measurement signals, are summarised as follows: (i) the fast response generator method, which injects measured voltage magnitude and phase angle/frequency into a generator model [14], (ii) the phase-shifting transformer method, which uses measured voltage magnitude and phase angle as inputs [15], (iii) the variable impedance method, which models the truncated section of the system using a time-varying impedance [16, 17], (iv) the forced power injection method, which injects measured active and reactive powers into subsystem models [17]. These methods use different inputs and outputs as well as implementation methods to validate subsystem models. 2.2 Subsystem models under study The dynamic model of a synchronous machine for the 'event playback' is based on the NERC suggested models [18]. Following the suggestion, a sixth-order GENTPJ synchronous machine model, a fourth-order IEEET1 exciter model and a second-order TGOV1 turbine-governor model in the local d–q reference frame shown in (1) are used in this paper. In (1), and are the rotor angle (in radian) and rotor speed in per unit (pu); are the transient and sub-transient voltages along d and q axes; is the synchronous speed; Te is the electric air-gap torque in pu; and parameter H is the inertia and KD is the damping coefficient; Parameters are the transient and sub-transient rotor time constants in the dq0 frame. Variables xd and xq are the synchronous reactance along d and q axes, respectively; are the transient and sub-transient reactance along d and q axes, respectively; and id and iq are the stator currents along d and q axes. The IEEET1 exciter states, VT, VR, Efd and VF, are the voltage transducer, voltage regulator, internal field voltage and feedback voltage integrator in pu; Ecomp is the terminal voltage and Vref is the known voltage control input. TR, TA, TE and TF are the transducer filter and voltage regulator, internal voltage and feedback integrator time constants in seconds; KA, KE and KF are the voltage regulator, internal voltage and feedback gains in pu and SE is a saturation function; The TGOV1 governor states, VO and TP, are the governor valve opening and turbine power state variables; T1, T2 and T3 are the governor valve, high-pressure section and reheater time constants in seconds; RD and Dt are the steady-state and speed control gain in pu; and Tm is the mechanical torque in pu (1a, 1b, 1c, 1d, 1e, 1f) (1g, 1h, 1i, 1j) (1k, 1l, 1m) To facilitate the notation, (1) is transformed into general differential-algebraic equations as (2). Here, subscript c indicates that the models are in continuous-time form. Symbols x(t), u(t) and y(t) are the dynamic states, inputs (i.e. play-in signals) and measurement outputs (i.e. system responses), respectively (2a, 2b) To perform the 'event playback', model (2) is discretised into the discrete-time form shown in (3) with a time step of using the modified Euler method [19-33]. Here, subscript k is the time index. Symbols xk, uk and yk are the state, input and measurement output, respectively. Functions and are the state transition and measurement function, respectively. is the Jacobian matrix of at step k − 1, and is the Jacobian matrix of at step k. In (3a), vectors wk and vk are the process and measurement noise, respectively. Their mean and variance are denoted by (5). Here, symbol represents the expected value. Symbols Qk and Rk are the covariance matrices of the process noise and measurement noise, respectively, at step k. Readers may refer to [34] for more details on how to determine the values of Qk and Rk. Assume that u and y are measured. Measured y is denoted as z. The model quality can be assessed by calculating the mismatch between simulated yk and its corresponding measurement zk (3a) (3b) (4) (5a) (5b) Note that the sampling interval plays a critical role in the 'event playback' because it influences the discretising errors during the integration procedure of simulation. In addition, the sampling interval of PMU determines for uk. Even though the effective sampling rate can be increased through interpolation, the interpolated input signal still cannot capture the high-frequency components that were initially missing, which can introduce simulation errors into the conventional 'event playback' approach. As it will be shown in Section 4, the authors have observed significant mismatch between the simulated and actual system responses even when an accurate model was used. The observation suggests the deficiency of the conventional 'event playback' approach in validating dynamic models when the high-frequency components in the system responses are not considered properly. 3 Dynamic state estimation and fault detection approaches To overcome the deficiency of the conventional 'event playback' approach, the authors propose a batch state estimation approach to focus on low-frequency components of dynamic responses by starting the playback function after the fast transient dynamic responses die out. The proposed approach starts the 'event playback' after the fault is cleared in the system. To do so, the procedure of the proposed approach consists of the following three steps: (i) The MMAKF approach is used to estimate the dynamic states of the system using PMU data so that the dynamic simulation can be started during a transient state. Five Bayesian-based filtering approaches, i.e. EKF, UKF, CKF, EnKF and smoother, are run concurrently to estimate the dynamic states of the system. Then, at each time step, probability indexes, which quantify the probability of each estimation filter, are determined using hypothesis testing, based on the measurement innovation. Finally, using the best-fix approach, the estimated states from the EKF, UKF, CKF, EnKF and smoother approaches with the highest probability index are selected to determine the a posteriori estimated states. The goal is to achieve high accuracy and robustness in estimating dynamic states. The details of the proposed MMAKF approach are presented in Section 3.1. (ii) The fast-transient responses incurred by faults are detected using an algorithm based on the SSA so that the initial sub-transient responses dominated by high-frequency responses can be skipped. The main idea of the SSA is performing the eigenvalue decomposition (EVD) of the trajectory matrix obtained from the original time series with subsequent reconstruction of the series. In turn, fault occurrence and changes in the operating conditions can be determined. The details of the proposed change-point detection approach are presented in Section 3.2. (iii) The 'event playback' is started after the high-frequency components die out in system responses. The simulation is initialised using the estimated dynamic states from the MMAKF. The simulated responses are compared with measurement responses to determine whether or not the model is adequate in describing system responses within the frequency bands covered by PMU measurements. The proposed 'event playback' approach is different from the conventional one in that it starts simulation after the high-frequency components die out. Thus, the model validation can focus on the low-frequency bands covered by the PMU measurements. In contrast, the conventional 'event playback' approach starts the simulation at the steady-state and includes the dynamic responses over all the frequency bands. Note that when the system's dynamic responses exceed the bandwidth of PMU measurement, the conventional approach may mistakenly invalidate an adequate model because of the limited bandwidth of the measurements. 3.1 MMAKF approach This subsection gives an overview of the MMAKF approach, which is used to estimate the dynamic states of synchronous machines and shown in Fig. 2. Readers may refer to [35] for more details on the MMAKF. Fig. 2Open in figure viewerPowerPoint MMAKF approach to DSE of synchronous machines As one of the most widely used Bayesian-based filtering approaches, the Kalman filter (KF) was introduced for the first time in 1960 by Rudolf Kalman [35]. For a linear system, the KF provides unbiased minimum variance estimates of its states through a recursive approach. The KF not only is successful in linear systems but also has been adapted to non-linear systems. These non-linear versions of the KF include but are not limited to the EKF, UKF, CKF and EnKF. The major difference among these non-linear versions is their approaches to propagating the mean and covariance of the dynamic states. The EKF [19, 36] approximates a non-linear function using its first-order Taylor expansion. The mean and covariance of states are propagated using Jacobian matrices. Yet, when applied to a highly non-linear system, the EKF can perform poorly and may even diverge. The UKF proposed by Julier and Uhlmann in 1997 [20] can achieve the second or third order of accuracy of the Taylor expansion. Therefore, the UKF may provide more accurate estimation than the EKF when dealing with a highly non-linear system. In addition, the UKF has the same order of computational complexity as the EKF. The UKF propagates the mean and covariance of states using a deterministic-sampling approach to pass the sigma points through the non-linear system [21]. The EnKF introduced by Evensen in 1994 [22] propagates the mean and covariance of states using a Monte-Carlo sampling approach. In the EnKF, the distribution of the states is represented by a collection of samples, referred to as ensembles. The EnKF was mainly motivated by a system with a large number of states. In a large system, the EnKF may require less computation time than the EKF and UKF. Also, the EnKF can overcome the unbounded error growth problem of the EKF [23]. The CKF introduced by Arasaratnam and Haykin in 2009 [24] uses the spherical-radial cubature rule to approximate the Gaussian-weighting integrals. Different from the UKF, the CKF uses the cubature rule, which leads to an even number (i.e. 2n points, where n is the state dimension) of equally weighted cubature points. In contrast, the UKF leads to an odd number (i.e. 2n + 1 points) of sigma points which are the result of 'unscented' transformation. As a smoother estimates the dynamic states using all the available data (before and after time step k), it has the advantage of providing state estimates with lower error-covariance than the KF which uses only data up to the moment of estimated states (before the time step k). In this paper, the Rauch, Tung and Striebel (RTS) fixed-interval smoother [25, 26] is used to estimate the dynamic states. To accomplish this task, two filters are used: the forward-time filter and the backward-time filter. All the above KFs are derived based on the additive Gaussian noise assumption for both states and measurements. However, the distribution of the measurement noise may not always obey this assumption. Recent research in [26] suggests that the errors in PMU measurements such as voltage phasors tend to follow non-Gaussian distributions with long tails such as a Laplace distribution. Readers may refer to [27] for more detailed information. Using the multi-model adaptive estimation algorithm introduced by Hanlon and Maybeck [29], the proposed MMAKF approach estimates the dynamic states by combining the states estimated from five Bayesian estimation approaches through the following three steps: Step (i). Multi-model filter: In this step, five Bayesian-based KFs (i.e. EKF, UKF, CKF, EnKF and smoother) run concurrently to estimate states () at step k. In addition, measurement innovation () and its corresponding covariance (Sk) are estimated. Step (ii). Hypothesis testing: In this step, hypothesis testing is used to quantify the probability of each estimation filter. First, the likelihood function associated with the ith measurement innovation at kth step is estimated using (6) where (7) Symbol m is the number of measurement vector components, and i is the index of a filter. Then, the probability index for a particular hypothesis can be defined as (8) by normalising (6) (8) In (8), M is the number of filters being used. The probability indexes sum to unity as (9). Note that in (8), if a filter performs well, its measurement innovation will have small value (close to 0) [29]. Therefore, the filter with the smallest measurement innovation is assigned the largest probability (9) Step (iii). Adaptive estimation: In this step, the estimated states with the highest probability are selected. To perform the multiple-hypothesis testing, three commonly used approaches are the best-fix, weighted fix and multi-hypothesis filtering [25]. The best-fix approach accepts the estimated states with the highest probability score at each step of time and rejects the others. This approach is simple and can be effective when one hypothesis is clearly dominant on most iterations. As such, the best-fix approach is used in this paper as Fig. 2 shows. In this step, the measurement innovation associated with each filter is utilised to calculate the probability indexes corresponding to each filter (i.e. ). Here, stands for the probability index corresponding to the EKF and so on. To estimate the states using the five Bayesian filters, the a posteriori estimated states corresponding to the highest probability index are selected. In the following description, the '(−)' in superscript indicates the a priori estimate, and the '( + )' in superscripts indicates the a posteriori estimate. Symbols x and P are the mean and covariance of the states, respectively. The implemented algorithms are briefly reviewed as follows. (i) EKF Table 1. Two-step procedure of EKF EKF initialisation: (10a) EKF prediction step: (11a) (11b) EKF correction step: (12a) (12b) (12c) Table 2. UKF estimation method UKF prediction: (13a) (13b) (13c) (13d) UKF correction: (14a) (14b) (14c) The two-step EKF procedure is summarised in Table 1 [19]. (ii) UKF The UKF estimation method is summarised in Table 2 [20]. where and are 2n + 1 sigma points and their corresponding weights, respectively. is a scaling parameter that controls the positions of the sigma points. (iii) EnKF Table 3. EnKF estimation method EnKF prediction: (15a) (15b) (15c) EnKF correction: (16a) (16b) (16c) (16d) The EnKF uses samples (also known as ensembles) to represent and propagate the probability distributions of the states. By using a large number of samples, the probability density can be approximated with higher accuracy. The EnKF can be summarised in Table 3 [22]. Here nEnKF is the total number of samples, which are used to represent the distribution. The variable is a sample generated according to Qk−1 to simulate process noise. Symbol stands for the samples of a posteriori states. Note that using (16b) and (16c), the covariance matrix Pk does not need to be expressively calculated. (iv) CKF Table 4. CKF estimation method CKF prediction: (17a) (17b) (17c) (17d) (17e) CKF correction: (18a) (18b) (18c) (18d) (18e) (18f) (18g) (18h) The CKF uses a cubature rule to equally weight a set of points to approximate the Gaussian distribution. Then, these points are propagated through the non-linear functions f and h to reconstruct the mean and covariance. The CKF estimation method is summarised in Table 4 [24]. Here ei is a vector with dimension of n, whose ith element is one and others are zeros. is the 2n equally weighted cubature points. (v) RTS fixed-interval smoother The RTS fixed-interval smoother consists of the four steps shown in Table 5 [26]. Table 5. RTS fixed-interval smoother Forward propagation: , k = 1,