Inverter Output Impedance Estimation in Power Networks: A Variable Direction Forgetting Recursive-Least-Square Algorithm Based Approach

TL;DR

This work addresses non-invasive estimation of inverter-to-grid output impedance by recasting the measurement problem in a rotating $d$-$q$ frame using a secondary, low-bandwidth PLL to create a favorable frequency separation. It introduces a band-pass preconditioning step and a novel VDF-RLS estimator that selectively forgets data along informative directions, enabling rapid adaptation to impedance changes while remaining stable without persistent excitation. Compared with CF-RLS and Kalman filtering, the proposed approach significantly reduces estimation error in low-excitation scenarios (up to about 3x improvement) and is robust to inverter-induced noise. The combined preconditioning and VDF-RLS framework yields accurate, non-invasive line-impedance estimates suitable for real-time grid-active control of inverter-based resources.

Abstract

As inverter-based loads and energy sources become increasingly prevalent, accurate estimation of line impedance between inverters and the grid is essential for optimizing performance and enhancing control strategies. This paper presents a non-invasive method for estimating output-line impedance using measurements local to the inverter. It provides a specific method for signal conditioning of signals measured at the inverter, which makes the measured data better suited to estimation algorithms. An algorithm based on the Variable Direction Forgetting Recursive Least Squares (VDF-RLS) method is introduced, which leverages these conditioned signals for precise impedance estimation. The signal conditioning process transforms measurements into the direct-quadrature (dq) coordinate frame, where the rotating frame frequency is determined to facilitate a simpler and more accurate estimation. This frequency is implemented using a secondary Phase-Locked Loop (PLL) to attenuate grid voltage measurement variations. By isolating the variation-sensitive q-axis and relying solely on the less sensitive d-axis, the method further minimizes the impact of variations. The VDF-RLS estimation method achieves rapid adaptation while ensuring stability in the absence of persistent excitation by selectively discarding outdated data during updates. Proposed conditioning and estimation methods are non-invasive; estimations are solely done using measured outputs, and no signal is injected into the power network. Simulation results demonstrate a significant improvement in impedance estimation stability, particularly in low-excitation conditions, where the VDF-RLS method achieves more than three time lower error compared to existing approaches such as constant forgetting RLS and the Kalman filter.

Figures (6)

• Figure 1: Grid model: (a) Grid with a complex structure, and (b) Thevenin equivalent model from the inverter, where R and L represent the equivalent resistance and inductance of the Thevenin equivalent model.
• Figure 2: Power setpoints: Between $t=0$ and $t=20$ seconds, the setpoints change every 2 seconds, providing excitation. After $t=20$ seconds, no further excitation is applied.
• Figure 3: Line impedance estimation results using different algorithms. Impedance changes at $t=10$ seconds, while estimation begins at $t=2$ seconds to allow the PLL to lock onto the frequency.
• Figure 4: (a) Estimation result comparison between using only d axis and both d,q axis dynamics (b) noise related term ($BPF(V_g^{d,q})$)
• Figure 5: Estimation results using different frequency as a source of rotating frame