We address the challenge of accurately tracking the state evolution of distribution networks hosting grid-following photovoltaic (PV) inverters. The system exhibits inherently multi-time-scale dynamics: millisecond-level fast transients in inverter controllers and minute-level slow variations in network power flow. Existing state estimation methods often treat inverters as static nodes or ignore their detailed control dynamics, leading to inaccuracies under high renewable penetration. This paper proposes a dual-layer dynamic state estimation (DSE) framework that explicitly accounts for inverter dynamics and fuses information across time scales. At the inverter level, we employ an extended Kalman filter (EKF) based on an eighth-order nonlinear model to estimate internal states and reconstruct system frequency. At the network level, we develop an adaptive fading unscented Kalman filter (AFUKF) that uses high-precision pseudo-measurements generated from inverter-side estimates. The framework is validated on a modified IEEE 33-bus distribution system with multiple PV units. Results demonstrate superior tracking performance, robustness to model uncertainties, and computational efficiency suitable for real-time applications.
1. Introduction
With the global push toward carbon neutrality, distributed PV systems are increasingly integrated into distribution networks. The dynamic behavior of these systems is dominated by power electronic interfaces—specifically, types of solar inverters such as grid-following and grid-forming topologies. Grid-following inverters, the most common type, rely on phase-locked loops (PLL) for synchronization and exhibit fast internal dynamics (e.g., DC-link voltage regulation, current control loops) that evolve on millisecond scales. In contrast, the overall distribution network state (bus voltages, angles) changes on second-to-minute scales. This time-scale disparity necessitates a multi-rate estimation approach.
Traditional static state estimation (SE) using weighted least squares (WLS) cannot track transient phenomena. Dynamic state estimation (DSE) using Kalman filters has been applied to power systems, but most works either ignore inverter dynamics or treat them as simple PQ nodes. Recent studies have begun incorporating inverter models into DSE, using adaptive cubature Kalman filters (CKF) or unscented Kalman filters (UKF). However, they often assume a single estimation rate, leading to either heavy computation (if run at fast rate) or loss of fast dynamics (if run at slow rate). Our work fills this gap by proposing a truly multi-time-scale estimator that respects the physical time constants of both subsystems.
2. Problem Formulation
2.1 Inverter Dynamic Model
We consider a three-phase grid-following inverter with an LCL filter and a decoupled dual-loop controller (outer DC voltage loop, inner current loop) plus PLL. The eight state variables are:
$$
\mathbf{x}_{\text{inv}} = [U_{\text{DC}}, X_{U_{\text{DC}}}, i_d, X_{i_d}, i_q, X_{i_q}, \theta_{\text{pll}}, X_{\text{pll}}]^\mathrm{T}
$$
The continuous-time state equations are:
$$
\begin{aligned}
\frac{dU_{\text{DC}}}{dt} &= \frac{1}{C}(I_{\text{in}} – I_{\text{de}})\\
\frac{dX_{U_{\text{DC}}}}{dt} &= U_{\text{DC,ref}} – U_{\text{DC}}\\
\frac{di_d}{dt} &= \frac{1}{L}[K_{p,id}(i_{d,\text{ref}} – i_d) + K_{I,id}X_{i_d}]\\
\frac{dX_{i_d}}{dt} &= i_{d,\text{ref}} – i_d\\
\frac{di_q}{dt} &= \frac{1}{L}[K_{p,iq}(i_{q,\text{ref}} – i_q) + K_{I,iq}X_{i_q}]\\
\frac{dX_{i_q}}{dt} &= i_{q,\text{ref}} – i_q\\
\frac{d\theta_{\text{pll}}}{dt} &= K_{p,\text{pll}}u_q + K_{I,\text{pll}}X_{\text{pll}} + \omega_0\\
\frac{dX_{\text{pll}}}{dt} &= u_q
\end{aligned}
$$
where all symbols have standard meanings (see detailed derivation in the original paper). The system is discretized using a Taylor series expansion to obtain the state transition matrix \(\mathbf{F}_k\) and input matrix \(\boldsymbol{\phi}_k\).
The measurement vector comprises DC voltage, three-phase AC currents, and PLL angle:
$$
\mathbf{z}_{\text{inv}} = [U_{\text{DC}}, i_a, i_b, i_c, \theta_{\text{pll}}]^\mathrm{T}
$$
The measurement equations are nonlinear functions of the states, e.g., \(i_a = \sin\theta_{\text{pll}} i_d + \cos\theta_{\text{pll}} i_q + i_0\).
2.2 Distribution Network Model
For the grid side, the state vector consists of all bus voltage magnitudes and angles except the slack bus angle:
$$
\mathbf{x}_{\text{grid}} = [V_1, V_2, \ldots, V_N, \delta_1, \delta_2, \ldots, \delta_{N-1}]^\mathrm{T}
$$
The system evolution is difficult to model analytically due to load variations and distributed generation. We adopt the Holt’s two-parameter exponential smoothing model for state prediction:
$$
\begin{aligned}
\tilde{\mathbf{x}}_{\text{grid},k|k-1} &= \mathbf{a}_{k-1} + \mathbf{b}_{k-1}\\
\mathbf{a}_k &= \alpha\mathbf{x}_{\text{grid},k|k} + (1-\alpha)(\mathbf{a}_{k-1} + \mathbf{b}_{k-1})\\
\mathbf{b}_k &= \beta(\mathbf{a}_k – \mathbf{a}_{k-1}) + (1-\beta)\mathbf{b}_{k-1}
\end{aligned}
$$
The measurement vector includes conventional SCADA data: voltage magnitudes, nodal injections, and branch power flows. The measurement function is the standard AC power flow equations.
3. Proposed Multi-Time-Scale Estimation Framework
3.1 Inverter-Level EKF
At the inverter level, we run an EKF at a fast sampling rate (e.g., 0.303 ms) to track the high-frequency dynamics. The prediction step uses the linearized discrete model:
$$
\begin{aligned}
\tilde{\mathbf{x}}_{\text{inv},k|k-1} &= \mathbf{F}_{k-1}\hat{\mathbf{x}}_{\text{inv},k-1|k-1} + \boldsymbol{\phi}_{k-1}\mathbf{u}_{k-1}\\
\mathbf{P}_k^- &= \mathbf{F}_{k-1}\mathbf{P}_{k-1}^+\mathbf{F}_{k-1}^\mathrm{T} + \mathbf{Q}_{\text{inv},k}
\end{aligned}
$$
The update step with measurements provides filtered states. The EKF effectively suppresses measurement noise and yields accurate estimates of the PLL angle \(\hat{\theta}_{\text{pll}}\). From this filtered phase angle, we reconstruct the instantaneous system frequency via backward difference and Savitzky-Golay smoothing:
$$
f_{k’} = \frac{\hat{\theta}_{\text{pll},k’} – \hat{\theta}_{\text{pll},k’-1}}{2\pi \Delta t_{\text{inv}}}
$$
This enables tracking of grid frequency even during disturbances (load step, faults), as demonstrated in our simulations.
3.2 Grid-Level AFUKF
At the network level, we run an AFUKF at a slower rate (e.g., every 15 minutes, synced with SCADA). The UKF uses sigma points to handle nonlinearity without Jacobian calculations. To address model mismatch due to the simplified Holt’s model and time-varying measurement noise (especially after injecting pseudo-measurements whose error distribution deviates from Gaussian), we incorporate two fading factors \(\rho_{x,k}\) and \(\rho_{z,k}\) that adaptively scale the predicted measurement covariance and measurement noise covariance:
$$
\begin{aligned}
\mathbf{P}_{yy,k} &= \rho_{x,k}\mathbf{P}_{\tilde{y}\tilde{y},k} + \rho_{z,k}\mathbf{R}_{\text{net},k}\\
\text{Tr}(\hat{\mathbf{P}}_{e,k}) &= \rho_{x,k}\text{Tr}(\mathbf{P}_{\tilde{y}\tilde{y},k}) + \rho_{z,k}\text{Tr}(\mathbf{R}_{\text{net},k})
\end{aligned}
$$
The fading factors are estimated online via a recursive least-squares (RLS) module. The process noise covariance \(\mathbf{Q}_{\text{net},k}\) is also adaptively estimated to capture changes in load and solar generation.
3.3 Boundary Information Fusion
To couple the two estimators, we construct pseudo-measurements of the inverter-interfaced bus injection power using the inverter’s filtered current and the grid’s predicted voltage:
$$
\mathbf{S}_{\text{pcc},k’} = \mathbf{S}_{\text{net},k’} + \tilde{V}_{\text{net},k’|k’-1}^{-}\hat{I}_{\text{inv},k’} e^{-j\hat{\theta}_{\text{inv},k’}}
$$
These pseudo-measurements have higher confidence than raw SCADA data; we assign them a smaller variance \(\sigma_{\text{pseudo}}^2 = \varepsilon \sigma_{\text{SCADA}}^2\) with \(\varepsilon = 0.5\). The fusion scheme switches between “multi-time-scale filtering” (using pseudo-measurements) and “direct fusion” (when inverter power is negligible, e.g., at night) based on a threshold \(\delta\).
4. Simulation Results
4.1 Setup
We use the IEEE 33-bus test system with PV inverters at buses 18, 22, 25, and 33. The inverter parameters are given in Table 1. Measurements are simulated by adding Gaussian noise to true values from a detailed Simulink model. The inverter-side sampling interval is 0.303 ms; the grid-side interval is 15 minutes. We evaluate performance using root mean square error (RMSE).
| Parameter | True Value | Simulated Value |
|---|---|---|
| \(K_{p,id}\) (pu) | 0.3 | 0.303 |
| \(K_{p,iq}\) (pu) | 0.3 | 0.303 |
| \(K_{p,pll}\) (pu) | 180 | 181.8 |
| \(L\) (mH) | 13.909 | 14.048 |
| \(K_{i,id}\) (s\(^{-1}\)) | 20 | 20.15 |
| \(K_{i,iq}\) (s\(^{-1}\)) | 20 | 20.15 |
| \(K_{i,pll}\) (s\(^{-1}\)) | 3200 | 3168 |
| \(C\) (F) | 0.0543 | 0.0538 |
4.2 Inverter State Estimation
We test the inverter EKF under steady-state operation (1.5 s). The process and measurement noise covariance matrices are:
$$
\begin{aligned}
\mathbf{Q}_{\text{inv}} &= \text{diag}(10^{-2},10^{-2},10^{-3},10^{-3},10^{-3},10^{-3},10^{-5},10^{-7})\\
\mathbf{R}_{\text{inv}} &= \text{diag}(4\times10^{-6},3.6\times10^{-5},3.6\times10^{-5},3.6\times10^{-5},4.9\times10^{-5})
\end{aligned}
$$
The estimated DC voltage, d-q currents, and PLL angle closely follow the true values. The frequency reconstruction from \(\hat{\theta}_{\text{pll}}\) is accurate under normal conditions, a 150% load step, and a single-phase-to-ground fault. Table 2 summarizes the average RMSE over the internal states.
| State | Unit | RMSE |
|---|---|---|
| \(U_{\text{DC}}\) | p.u. | 0.0032 |
| \(i_d\) | p.u. | 0.0018 |
| \(i_q\) | p.u. | 0.0015 |
| \(\theta_{\text{pll}}\) | rad | 0.0009 |
4.3 Grid State Estimation
We simulate a 24-hour PV profile (Gaussian-shaped irradiance, peak at noon). The performance of our proposed multi-time-scale filtering strategy (MTS) is compared with direct fusion of raw SCADA measurements. Table 3 shows the average RMSE over all buses during daytime (high solar period) for five different Kalman filters: EKF, ENKF, CKF, UKF, and our AFUKF. Both strategies are applied; the MTS strategy uses pseudo-measurements from inverter EKF.
| Algorithm | Voltage Magnitude (p.u.) | Voltage Angle (rad) | ||
|---|---|---|---|---|
| MTS Fusion | Direct Fusion | MTS Fusion | Direct Fusion | |
| EKF | 0.001108 | 0.001110 | 0.001841 | 0.001962 |
| ENKF | 0.001038 | 0.001179 | 0.001702 | 0.001751 |
| CKF | 0.001060 | 0.001084 | 0.001372 | 0.001429 |
| UKF | 0.001008 | 0.001051 | 0.001632 | 0.001709 |
| AFUKF | 0.000289 | 0.000315 | 0.000246 | 0.000268 |
Across all methods, the MTS fusion significantly reduces RMSE compared to direct fusion, verifying that high-quality inverter pseudo-measurements improve global estimation. The AFUKF outperforms all other filters, achieving 71.3% lower voltage magnitude error and 84.9% lower angle error than the UKF baseline. This demonstrates the effectiveness of the adaptive fading mechanism in handling non-Gaussian measurement errors and model mismatch.
4.4 Robustness to Penetration Level
We vary the rated PV capacity from 10% to 40% of the base load, keeping the number of PV nodes fixed (4 inverters). Table 4 shows that the RMSE remains nearly constant, confirming that the proposed method is robust to increasing renewable share.
| Rated Power Increase (%) | Voltage Magnitude RMSE (p.u.) | Voltage Angle RMSE (rad) |
|---|---|---|
| 10 | 0.000262 | 0.000274 |
| 20 | 0.000270 | 0.000295 |
| 30 | 0.000282 | 0.000228 |
| 40 | 0.000278 | 0.000218 |
4.5 Computational Performance
The inverter-side EKF computation took about 0.191 ms per step (sampling 0.303 ms), and the grid-side AFUKF took about 1.01 s per step (sampling 15 min). Both satisfy real-time requirements. The dual-rate architecture prevents unnecessary fast computation on the grid side while maintaining high update rate for inverter dynamics.
5. Conclusion
We have presented a multi-time-scale dynamic state estimation framework that explicitly models the fast dynamics of grid-following types of solar inverters (and can be extended to other types of solar inverters like grid-forming) and the slow dynamics of the distribution network. The inverter-level EKF provides high-fidelity estimates of internal states and frequency, while the grid-level AFUKF adaptively handles model uncertainties and measurement anomalies. Boundary pseudo-measurements seamlessly integrate the two scales. Extensive simulations on the IEEE 33-bus system demonstrate that the proposed method significantly improves estimation accuracy over conventional single-rate filters, is robust to high penetration and disturbances, and meets real-time computation constraints. Future work will extend the framework to grid-forming inverter models and consider communication delays.
