Dynamic state estimation of distribution networks with mixed measurements based on Cubature Kalman Filter

The integration of synchronized phasor measurement units (μPMU) has introduced diverse data sources into the distribution network. Each has distinct time scales and precision levels. To enable effective decision-making, it is crucial to amalgamate these measurements efficiently, ensuring both speed and accuracy in distribution network state estimation. This study explores a dynamic state estimation approach for distribution networks, employing Cubature Kalman Filtering (CKF) as the central methodology. By deeply fusing μPMU, AMI and SCADA data and incorporating CKF techniques, it improves accuracy and stability, which is validated on the IEEE 37-node system. This research supports advanced distribution network apps and large-scale data analysis.


Introduction
Electric power distribution network state estimation utilizing redundant measurement info to predict and estimate sstem state is foundational for electric power distribution networks and advanced apps [1] .Traditional SCADA and AMI systems collect steady-state power parameters for dispatch automation [2] .Micro-phasor measurement units (μPMU) offer real-time voltage and current phasor info and wide-area protection.Deeply fusing μPMU, AMI and SCADA data enhances power distribution network observations, aiding app development.Distribution networks are complex with many branches and rely on current measurements [3] .
Dynamic State Estimation (DSE) monitors and estimates system dynamics, including generators, lines, and transformers with real-time data and strong numerical stability.Researchers use decoupling techniques and current phasors to tackle high-dimensional power distribution networks.Fast decoupling methods reduce computational complexity but may have errors [4][5] .
μPMU, AMI, and SCADA data are used to simultaneously boost observability but pose data synchronization challenges [6] .This paper explores dynamic state estimation with Cubature Kalman Filtering (CKF), enhancing accuracy and stability by using μPMU, AMI and SCADA data.This research holds promise in providing vital support for the development of advanced distribution network application software and the analysis of large-scale distribution network data.

Mixed measurement data fusion strategy
Mixed data fusion strategy involves the integration of measurements from diverse sources, taking into account factors such as measurement precision, time intervals and synchronization.To facilitate this fusion process, we establish a reference point for data integration by using specific μPMU measurements.

1) μPMU
Averaging is an excellent solution for high-precision μPMU data fusion.When examining a μPMU measurement recorded at the time point t, we calculate the combined result denoted as    .
1 21 2) SCADA SCADA with comparatively larger measurement error uses the most recent sampled data at each estimation time directly for fusion.We calculate the combined result denoted as    .
Sm tv zz = (2) 3) AMI AMI data has minute-level sampling intervals.It incorporates node load prediction information, combining the latest AMI measurement with node load prediction data at the estimation time for fusion.We calculate the combined result denoted as    . ()

State estimation of mixed measurement data
In a quasi-steady-state operation, slow changes in generators and controllers allow neglecting their dynamics [7] .In this context, system state transitions are driven by smoothing parameters.Thus, quasisteady-state equations encompass measurement and state transition equations, as in Equation ( 4).

μPMU measurement equation
For μPMU measurement nodes, measurement data is presented in polar coordinates which are converted into Cartesian coordinate variables represented by Equation (5).r (P) x (P) r (P) x (P) cos j sin j cos j sin j

Traditional measurement ( SCADA & AMI ) equation
Active power P and reactive power Q can be expressed as: SCADA & AMI measurement can be converted into equivalent injection current.Equivalent current measurement can be converted into Equations ( 8) and ( 9).

( ) ( )
To mitigate phase angle-related errors in traditional current measurements, the square of branch current serves as an equivalent measurement, as depicted in Equation (10).The SCADA & AMI three-phase node voltage measurement value is equivalently transformed into: cal (eq) r(eq) x (eq) SA cal

Mixed measurement equation
The voltage and current phasor measurement data are transformed into coordinate systems.The mixed measurement variable M Z in the Cartesian coordinate system is: According to the hybrid measurement system, the specific expression of the hybrid measurement equation () M  h is as follows: ( ) ( )

Basic Principles of CKF
In complex networks with numerous nodes and high dimensions, the Cubature Kalman Filter (CKF) surpasses traditional Kalman filters in accuracy and stability.CKF employs the third-order spherical radial cubature law for Gaussian-weighted integration, approximating the integral term for probabilistic filtering in multiple directions, generating 4n equally weighted cubature points for 2n-dimensional state variables on a centered sphere [8][9] .It uniformly propagates the probability distribution across state space directions, comprehensively capturing correlations and uncertainties among variables.Compared to traditional Kalman filtering, CKF excels in handling nonlinear systems and high-dimensional state estimation, providing robust, high-precision, and stable state estimation.
The nonlinear state equation can be represented by using a product integral expression in conjunction with a Gaussian probability density, which is presented as follows: The expression for cubature point

State prediction and correction
The computation of the cubature point involves utilizing the prior state variable estimates and the error covariance matrix, as depicted in Equations ( 16) and ( 17).
The cubature point of the predicted state values with equal weights propagated through the state equation is shown in Equations ( 18) and ( 19).
The predicted state values are used to obtain the state prediction step.The predicted cubature point is calculated equal weight measurement by using the following equation: The measured predicted value H is shown in Equation (21): Utilizing the outcomes derived from those two phases, we proceed to forecast both the covariance matrix and the cross-covariance matrix relating the state variable x to the measurement z.The calculation equation is provided as follows: The state variable undergoes a filtering and correction process utilizing the measurement data   recorded at time t.The Kalman gain   is determined by using the Kalman filtering approach outlined in Equation (24).The final state estimation and the associated covariance of error correlation are computed through Equations ( 25) and (26).
In CKF execution, no Jacobian matrix computation is required.It approximates posterior probability distribution with cubature points.CKF introduces cubature points from a spherical symmetric distribution with parameters not impacting cubature point weights.This grants CKF notable advantages in numerical stability and adaptability.

Basic parameters
A simulation analysis was performed by using the IEEE 37-node system in Figure 1, operating at a 4.8 kV voltage level and 1 MV•A apparent power base.Two photovoltaic (PV) systems are PV1 and PV2.Each rate is at 0.2 MW and connects to nodes 04 and 08.In Figure 1, blue rectangles mark μPMU measurement nodes and red triangles indicate AMI and SCADA measurement nodes.

Comprehensive Error Analysis
In the example system, a thorough error analysis of M-DSE is performed, comparing CKF, Unscented Kalman Filter (UKF) and Extended Kalman Filter (EKF).After applying these filters, Figures 2 (a Figure 2 presents the root mean square error (RMSE) values for the current where CKF achieves an RMSE of 0.23%.In contrast, the RMSE values for UKF and EKF are 0.44% and 0.79% respectively.These findings demonstrate the superior performance of CKF compared to UKF and EKF in this context.Figure 3 compares RMSE in polar coordinates using different filters.The average amplitude RMSE is 0.18% for CKF, 0.51% for UKF and 0.82% for EKF.Likewise, the average phase angle RMSE is 0.27% for CKF, 0.90% for UKF and 1.62% for EKF.These results highlight CKF's superior accuracy and numerical stability compared to the other methods.

Conclusion
In this study, a hybrid measurement dynamic state estimation method based on CKF was proposed for uPMU, SCADA and AMI mixed measurement data.The main conclusions are as follows: • The fusion of mixed measurement data was achieved, considering the differences in time scales, synchronicity and measurement accuracy of different measurement data sources.• By employing Cubature Kalman Filtering to predict the state equation and measurement data in the mixed measurement equation, numerical stability in state prediction can be improved while ensuring filtering effectiveness.• Compared to UKF and EKF algorithms, CKF demonstrates higher estimation accuracy and faster computation speed, enhancing real-time state estimation in the presence of mixed data.In the future, leveraging CKF and mixed data fusion techniques, it is anticipated that more real-time advanced applications for distribution networks can be developed, providing support for big data analytics in distribution grid management.
) and (b) display the real and imaginary parts of phase-A current at node 27.Figures2 (c) and (d) show the root mean square errors (RMSE) of phase-A current at each sampling time.

Figure 2 .
Figure 2. State estimation of branch current.

Figure 3 .
Figure 3.Comparison of RMSE under different filters.