Dynamic State Estimation of Synchronous Machines Using Iterated Square-Root Cubature Kalman Filter and Synchrophasor Measurements

Power system dynamic state estimation is the first prerequisite for control and stability prediction under transient conditions. For a stable and reliable power system, it is crucial and helpful to have accurate, precise, and up-to-date information on the states of the synchronous machinesrotor angle and rotor speed deviation. This paper proposes an application of the Iterated Square-root Cubature Kalman filter (ISCKF) to estimate these main states of synchronous generators. The ISCKF method consists of two step modification one is the square-root step modification of CKF and the next step is the addition of iterative approach to the square-root CKF method. To demonstrate the performance of the proposed approach during a three-phase short circuit fault, the simulation results, are compared with that of Extended Kalman Filter (EKF), Unscented Kalman Filter (UKF) and Cubature Kalman Filter (CKF). The test systems considered are a single machine infinite bus (SMIB) system, an IEEE 9-bus system and a 19-generator 42-bus test system. The estimation accuracy of the rotor angle using ISCKF method is increased by 6.8–36.54% when compared to that of the CKF method. Similarly, the improvement in accuracy is 4.4–28.57% for estimation of speed deviation.


Introduction
Electromechanical dynamic models are widely employed in power systems to examine transient and smallsignal stability concerns.A dynamic model with appropriate known states may be used to anticipate and display system behaviour with accuracy, and boost power system stability.By utilising synchronised phasor measurement unit (PMU) readings, power system dynamic state estimation (PSDSE) calculates the states (such as rotor speeds and angles) of the system.
Researchers have explored several estimation techniques to investigate PSDSE.Extended Kalman Filter (EKF) was applied to PSDSE using second order model of the synchronous generator [1].Evaluation of an EKF based estimator for power system considering lack of field voltage has been explored [2].Dynamic state estimation (DSE) of synchronous machines in power system was carried out with the help of EKF [3].The generator swings were observed for the three-phase fault case and load variation.This had further scope for identification of coherent group of generators in the system with the PMU measurements and help in adaptive protection of the system.Although EKF is a recursive update variant of Kalman filter that is computationally effective, it works best only in a mild non-linear conditions, as it is order-I Taylor series approximation of nonlinear formulation [4].It is seen to be inadequate and is prone to divergence.The linearization is only feasible if the Jacobian matrix exists.Nonetheless, determining Jacobian matrix might be difficult and error-prone.When matrix is determined by QR decomposition instead of Cholesky decomposition.Thereby, ISCKF eliminates the hazardous situation of the loss of positive definiteness of error covariance matrix in each update step, which could stop the UKF and CKF from running continuously. Newton-Gauss iterative method was incorporated into the measurement update step of the square-root CKF for implementing the proposed ISCKF approach.Thus, improving the estimation accuracy.3.In this paper, ISCKF is designed for its best use in PSDSE.The performance evaluation is carried out by implementing this proposed filter ISCKF to estimate the rotor speeds and angles of synchronous generators in SMIB, IEEE 9-Bus and 19-generator 42-bus test systems.The simulation results with respect to the accuracy in estimation (mean of absolute errors) of rotor speeds and angles are compared with those obtained using the existing DSE algorithms such as EKF, UKF and CKF.
The state estimation modelling using a classical generator model is described in Section 2. The ISCKF algorithm details are specified in Section 3. The Section 4 delves into how simulation results and observations were carried out for the study using the test systems.Section 5 presents conclusion based on the results obtained for the proposed estimator and the other dynamic state estimators considered.

Modeling and Problem Formulations
Rotor angle and rotor speed deviation are estimated using the classical model of synchronous generator.
where  -rotor angle in radian;   -rotor speed deviation, m P -mechanical power and e P -electric air gap power.0  -rated angular frequency; H -inertia constant; and D -damping coefficient.For generalization, (1)   is transformed into a state space model as follows In ( 2) and (3), x, u, y -state, input and output vectors respectively; fc (*) and hc (*) -state transformation and output functions respectively; subscript "c" stands for continuous time formulation; ωc, vc -process and output noise vectors respectively.ωc and vc represent Gaussian white noises with covariance matrices defined by ( 4) and (5) as Q and R. E [*] -statistical expectation.The above expressions are converted to discrete form utilizing modified Euler's method [22] due to the inherent discrete nature of measurements [9].
All filtering algorithms use state estimation model, which is introduced in this section.We know that the simulation model may be different from the estimation model, simulation model is for mimicking system behaviours while an estimation model is for estimating states.

Iterated Square-Root Cubature Kalman Filter
The proposed ISCKF improves the accuracy and numerical stability of the square-root cubature Kalman filter [21] by including a Newton-Gauss iterative technique [23,24] into the measurement update step as elaborated in the algorithm description below.

Algorithm
The algorithm of the proposed ISCKF is described in the following steps.Time Update: 1. Evaluate the cubature points (i = 1, 2, ... m) where m = 2nx and nx is the number of states.When k = 1 where f(X) is given in (2). 3. Estimate predicted state 4. Estimate the square-root factor of the predicted error covariance where , 1 Measurement Update: where = 0, 1, ... (internal iteration count) for ISCKF and when = 0, ( 3. Estimate the predicted measurement 4. Estimate the square-root of the innovation covariance matrix where ( )   , j R k S denotes a square-root factor of ( ) where 8. Estimate the square-root factor of the corresponding error covariance 9. Make ( 1) Return to Measurement Update Step 1 and end for = .
The flowchart of the algorithm to show how the estimation and filtering process takes place using the actual data and ISCKF algorithm is illustrated in Figure 1.

Simulations
To demonstrate the validity and reliability of the proposed algorithm, in this section ISCKF is implemented for DSE for the test systems considered in the subsequent subsections.The estimator algorithms ISCKF, CKF, UKF and EKF are coded in MATLAB© 2015a platform and run on Windows 10 with Intel core i5 with 8GB RAM.
Some initial parameters are defined as follows: the covariance of state process noise Q is a diagonal matrix [10 −6 , 10 −2 ], the covariance of the measurement noise R = 0.001, the initial states 0   x , initial covariance 0 = 10 −3 × I.Here I is the identity matrix of size nx × nx. of the internal iteration number N for ISCKF is set as 5.All the simulation data for state estimation is generated at 25 samples/s to mimic the PMU sampling rate.Adding Volume 2 Issue 1|2023| 111

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more measurements increases the efficient sampling rate, consequently minimising linearisation errors [25].The sampling rate is enhanced from 25 to 1000 samples per second by utilising linear interpolation to insert extra pseudo measurements for every two measurement instants.As a consequence of the interpolation, the sampling time interval Δt is reduced from 40ms to 1ms.For simulating the practical PMU measurements, 5% of measurement noise added to the voltage and current phasors.The measurement noise includes the noises introduced by PMU device, current transformers and potential transformers.

SMIB
A compact equivalent model of a generic power system that consists of a single generator coupled to an infinite bus via parallel transmission lines and a transformer is simulated in PowerWorld Simulator [26].The PowerWorld schematic of SMIB is given in Figure 2. The specified values used for the generator parameters are provided in Appendix: Table 5 with Pbase = 100 MVA.The simulation scenario is a symmetrical permanent three-phase-to-ground bolted short circuit fault added to bus-3 at 1s and cleared by removing the line 1-3 and line 2-3 at 1.3s.Assuming the PMU is placed at the main generator.The simulation data of the main generator and bus is taken from the PowerWorld Simulator [26].The simulation results for the main states, rotor angle and rotor speed deviation of the main generator estimated using ISCKF are compared with EKF, UKF and CKF are presented in Figure 3 and Figure 4 respectively.
From the Figures 3b and 4b, ISCKF is giving least error.The mean of absolute errors in the states and the estimation time per iteration are listed in the Table 1.From these results, it is observed that by using ISCKF, the accuracy in rotor angle estimation has increased by 6.82% and in speed deviation the estimation accuracy has increased by 4.44% that of CKF.The percentage increase in accuracy is calculated as follows: Here, MAE (mean of absolute errors) is either with respect to rotor angle or rotor speed deviation of all the generators.And the estimation time per iteration is within the time next set of measurements arrive, that is 1ms for all the 4 types of estimators.

IEEE 9-Bus System
The case study is carried out on IEEE 9-bus system shown in Figure 5.The classical model as described in Section 2, is used to represent all generators in this system.To analyse the transient stability of a system with many machines, the initial conditions of the generators and buses must be determined.The power flow solution helps to determine the steady state voltages and angles of the buses.Together with the given values used for the generator parameters (Refer Appendix: Table 6), these steady state values are utilised as initial conditions for the simulation in Power World simulator.The fault scenario considered for this case is a balanced three-phase to ground fault on Bus-8 at t = 1s which is cleared at t = 1.1s by removing the lines 7-8 and line 9-8.The PMU sampling rate of 25 samples per second is used in simulation data and linear interpolation is used to increase the effective sampling rate to 1000 samples per second.
For comparison of the ISCKF with EKF, UKF and CKF the estimation results of rotor angle, speed deviation and the corresponding error plots at generator bus-2 are given in Figure 6 and Figure 7.The mean of absolute errors in the estimated states and the estimation time per iteration and the estimation time per iteration are listed in Table 2.   From the results and the mean of absolute errors, it can be observed that the MAE in rotor angle estimation using ISCKF is 0.0019 rad.Therefore, there is an increased accuracy of 13.64% in rotor angle estimation with respect to that obtained using CKF.Similarly, the MAE in speed deviation using ISCKF is 0.0088 p.u., implies increased accuracy of 9.28% in speed deviation estimation when compared to that obtained using CKF.ISCKF is proved to be more accurate for the state estimation in the multi-machine system as well.And the estimation time per iteration is within the time next set of measurements arrive, that is 1ms for all the 4 types of estimators.

19-Generator 42-bus Test System
The 19-generator 42-bus test system data is generated using the existing script files "case19l" and "s_simu" in psdat and pstv2 folders of Power System toolbox version 2 [27].MATLAB© R2015a platform is used to run and generate the data for state estimation.The system details utilized from "case19l" script file are provided in Appendix: Table 7.The simulation scenario is a three-phase fault applies at bus-30 at 1.0s and is cleared by removing the line at 1.1s.For simulation, the sampling time is taken as typical PMU reporting rate 25 samples per second linearly interpolated to 1000 samples per second as considered in previous system simulations.As a consequence of the interpolation, the sampling time interval Δt for the input and measurement data is reduced from 40ms to 1ms.From the tabulated errors in the Table 3, it is observed that by using ISCKF, the accuracy in the rotor angle estimations is increased by 29.41%and the accuracy in the speed deviation estimation has increased by 22.95% than that obtained by CKF.But it has been observed that the estimation time per iteration is exceeding 1ms for ISCKF, CKF and UKF estimators (See Table 3).That is, the estimation is not being performed before the arrival of next set of measurements.For this larger system, as the estimation process involves large number of variables, a trade-off has to be made between the samples per second of measurement data and the number of variables in the estimation process.The only part which can be modified is samples per second of measurement data, by using down-sampling technique.Hence, 250 samples per second is chosen, to make the estimation process feasible within the arrival time of next set of measurement, by interpolating 25 samples per second to 250 samples per second.Also, the number of internal iterations count N of ISCKF has been reduced to 3 instead of 5 (which has been used in smaller systems discussed in previous sub-sections).Now, the results after these changes are noted in Table 4 showing that the estimation time per iteration is within 4ms for all types of estimators.Thus, can be implemented for real-time online DSE.Further if the measurement data are considered at the rate 250 samples per second (See Table 4), the accuracy of state estimations by ISCKF is the best among all.The accuracy in the rotor angle estimations by ISCKF is increased by 36.54%than that of CKF.While the accuracy in the speed deviation estimation by ISCKF has increased by 28.57% than that of CKF.The estimation results obtained at generator bus-6 with 250 samples per second input and measurement data is shown in Figure 8 and Figure 9.The mean of the absolute errors in estimation of states at all the generator buses using different filters is tabulated in the Table 4.

Journal of Electronics and Electrical Engineering
When the results of Tables 1, 2 and 3 are compared, it can be observed that with increase in number of generator buses from 1 to 19 in the system, though the computational time increases almost linearly, the MAE is almost the same and hence estimation accuracy is consistently good.
Since the proposed method has repetitive iterations within the state update step, the computational time is higher than the other methods as observed in Tables 1, 2 and 3.But the accuracy is improved significantly and hence is advantageous.The increase in estimation time for the first two test system SMIB system and IEEE 9bus system is well within the arrival time of next set of data and hence does not adversely affect the estimator performance.

Figure 1 .
Figure 1.The flowchart for the general flow of logic in dynamic state estimation using Iterated square-root Kalman filter.

Figure 3 .
Figure 3.The estimated rotor angle and error plots at main generator bus for SMIB.

Figure 4 .
Figure 4.The estimated speed deviation and error plots for SMIB.

Figure 6 .
Figure 6.The estimated rotor angle and absolute error plots at generator bus-2 for IEEE 9-bus system.

Figure 7 .
Figure 7.The estimated speed deviation and absolute error plots at generator bus-2 for IEEE 9-bus system.
(a) Rotor angle plots (b) Error in rotor angle

Figure 8 .
Figure 8.The rotor angle and error plots at generator bus-6 in 19-gen 42-bus test system.

Table 1 .
Mean of Absolute Errors (MAE) in the Estimated States using SMIB test system.
(a) Rotor angle plots (b) Error in rotor angle

Table 2 .
Mean of Absolute Errors (MAE) in the Estimated States at all generator buses in IEEE 9-bus system.

Table 3 .
Mean of Absolute Error (MAE) in the Estimated States of all generator buses of 42-bus test system (Measurement data: 1000 samples/s).

Table 4 .
Mean of Absolute Error (MAE) in the Estimated States of all generator buses of 42-bus test system (Measurement data: 250 samples/s).

Table 5 .
Specified parameter values used for the simulated synchronous generator model for SMIB test system.
m P 1

Table 6 .
Specified parameter values used for the simulated synchronous generator models for IEEE 9-bus system.

Table 7 .
Specified parameter values used for the simulated synchronous generator models for 19-generator 42-bus test system.