Multi-Timescale Dynamic State Estimation for Distribution Networks with Photovoltaic Inverters

In the context of achieving carbon neutrality, the integration of distributed photovoltaic (PV) systems, particularly through various types of solar inverter topologies, is reshaping the dynamic behavior of modern distribution networks. The system evolution now exhibits pronounced multi-timescale characteristics, encompassing millisecond-level fast fluctuations within the inverter’s internal control loops and minute-level slow variations of the grid-wide power flow. To accurately track these intertwined dynamics, I propose a dual-layer, multi-rate collaborative dynamic state estimation (DSE) framework. The core idea is to decouple the estimation problem into two distinct models: a high-speed Extended Kalman Filter (EKF) for the inverter-side and an Adaptive Fading Unscented Kalman Filter (AFUKF) for the network-side. This approach leverages high-fidelity pseudo-measurements generated from the inverter’s filtered states to enhance the accuracy of the grid-wide estimation, effectively addressing the challenge of model mismatch and noise uncertainty introduced by power electronic interfaces.

The accurate tracking of a grid-following PV inverter’s internal states is critical. I selected eight state variables to fully describe its dynamics: $x_{inv} = [U_{DC}, X_{UDC}, i_d, X_{id}, i_q, X_{iq}, \theta_{pll}, X_{pll}]$. This vector includes both physical quantities (DC-link voltage $U_{DC}$, dq-axis currents $i_d, i_q$, and phase-locked loop angle $\theta_{pll}$) and the integral states of the PI controllers ($X_{UDC}, X_{id}, X_{iq}, X_{pll}$). The continuous-time differential equations governing these states are derived from the standard double-loop control structure. For instance, the $d$-axis current dynamics are described by:

where $L$ is the filter inductance, and $K_{p,id}$ and $K_{I,id}$ are the proportional and integral gains of the current controller. The measurement vector for the inverter side is $z_{inv} = [U_{DC}, i_a, i_b, i_c, \theta_{pll}]^T$, where the three-phase currents $i_a, i_b, i_c$ are functions of the dq-axis currents and the PLL angle. A Taylor series discretization method is used to form the state transition matrix $F_k$, enabling the recursive prediction and update steps of the EKF. The excellent tracking performance of this EKF for the inverter’s fast dynamics is evidenced by the ability to reconstruct the system frequency from the filtered PLL angle, $\hat{\theta}_{pll}$, using a backward difference method:

This frequency tracking capability was validated across multiple scenarios, including steady-state operation, sudden load spikes, and three-phase faults.

For the distribution network side, which evolves at a much slower rate, a Holt’s two-parameter exponential smoothing model is employed to predict the state evolution. The state vector $x_{grid} = [U_1, …, U_N, \delta_1, …, \delta_{N-1}]^T$ consists of all node voltage magnitudes and phase angles. The measurement vector comprises standard SCADA data and branch power flows. Pseudo-measurements for the PV-connected node are constructed by fusing the inverter’s filtered current output, $\hat{I}_{inv}$, with the distribution network’s predicted voltage, $\tilde{U}_{net}$:

These pseudo-measurements inherit the high accuracy of the inverter-side EKF and are assigned a smaller variance ($\sigma^2_{pseudo} = \epsilon \sigma^2_{SCADA}$) to reflect their higher reliability. This fusion strategy is a key contribution, as it directly improves the quality of boundary information.

To handle the non-Gaussian noise characteristics resulting from the inverter-side filtering, I implemented an Adaptive Fading Unscented Kalman Filter (AFUKF) for the distribution network. The core innovation of AFUKF is its ability to correct model mismatch through two fading factors, $\rho_{x,k}$ and $\rho_{z,k}$, which scale the predicted measurement covariance $P_{\tilde{y}\tilde{y},k}$ and the measurement noise covariance $R_{net,k}$, respectively. The relationship is established via the trace of the actual innovation covariance $P_{e,k}$:

A recursive least squares algorithm is used to estimate these fading factors online. The AFUKF algorithm is summarized in the table below, highlighting its key steps.

The experimental setup was based on a modified IEEE 33-node test system with four PV inverters. Appendix A Table A1 presents the key parameters of the specific types of solar inverter used, including PI gains and filter components, with 1% Gaussian noise introduced to simulate realistic conditions. The entire simulation was run for a 24-hour period using a Gaussian-like irradiation profile shown in Appendix A Fig. A3. The multi-timescale strategy was validated by comparing the “multi-timescale filtering strategy” (using inverter-filtered data for pseudo-measurements) against a “direct fusion strategy” (using raw SCADA measurements). The results at Node 22, a PV-connected node, demonstrate the superior performance of the proposed strategy.

The results above clearly demonstrate the superiority of the AFUKF algorithm. When using the multi-timescale fusion strategy, the AFUKF achieves a 73.9% improvement in voltage magnitude accuracy and an 86.6% improvement in phase angle accuracy over the standard EKF. A further comparison shows that the multi-timescale fusion strategy consistently outperforms the direct fusion strategy across all algorithms, with the most significant improvement observed in the more sensitive phase angle estimates.

To further test the robustness of the proposed framework, I evaluated its performance under increasing PV penetration levels by varying the rated power and the number of integrated units. The results are summarized in the following tables.

These results show that the proposed algorithm maintains high accuracy as PV power increases. Furthermore, while increasing the number of PV nodes leads to a gradual increase in overall error due to more complex power flows, the multi-timescale fusion strategy consistently provides better or comparable accuracy. The benefit is most pronounced when the pseudo-measurements form a significant portion of the total measurement set, highlighting the value of high-quality boundary information. Finally, a computational performance analysis confirmed that the processing time for both the inverter-side EKF (0.191 ms) and the grid-side AFUKF (1.01 s) is well within their respective sampling intervals (0.303 ms and 15 minutes), satisfying the real-time operation requirements for distribution system monitoring.