An Elliptic Model-Based Fast Estimation Method of Positive and Negative Sequence Amplitudes for Unbalanced Grid Voltages

Abstract

In this paper, a novel elliptic algebraic model-based positive- and negative-sequence amplitude estimation method is proposed for unbalanced grid voltages. By exploiting the intrinsic elliptic trajectory of unbalanced voltage vectors in the $\alpha \beta$ stationary reference frame, an explicit algebraic relationship between the sequence amplitudes and the elliptic geometric parameters is established. Consequently, the conventional sequence decomposition problem is reformulated as an elliptic parameter identification problem. Based on the proposed elliptic algebraic framework, an online parameter identification scheme is developed to estimate the elliptic parameters and reconstruct the sequence amplitudes. A Lyapunov-based global asymptotic stability analysis is also presented to verify the convergence property of the proposed identification framework. Experimental results further validate the effectiveness of the proposed method.

1. Introduction

In modern power systems, three-phase voltage unbalance has become increasingly prominent, introducing significant negative-sequence components. These components arise from three-phase unbalance in power generation, transmission, distribution, and utilization, and are further intensified by the high penetration of renewable energy and extensive deployment of power electronic converters [1,2,3]. The amplitudes of negative-sequence components directly reflect the degree of system unbalance and serve as a key criterion for initiating mitigation measures such as reactive compensation and load adjustment [4,5,6,7].

Rapid and accurate estimation of the negative-sequence amplitude is essential for mitigating unbalance, ensuring grid security, and maintaining power quality. In the presence of negative-sequence components, the output of the synchronous reference frame phase-locked loop (SRF-PLL) exhibits pronounced double-fundamental-frequency ripple in phase and frequency signals [8,9]. This ripple can propagate and amplify through control stages, degrading power quality and threatening system stability. Although reducing PLL bandwidth suppresses these ripples, it also slows dynamic response, creating a trade-off between speed and noise immunity [10,11].

To address the impact of voltage unbalance on synchronous signal extraction, various improved approaches have been proposed. Notably, the decoupled double synchronous reference frame PLL (DDSRF-PLL) separately tracks positive- and negative-sequence components using two rotating reference frames with feedforward compensation, achieving effective decoupling under steady unbalanced conditions. However, it assumes a known or slowly varying grid frequency; rapid frequency changes degrade its decoupling performance and increase transient phase errors [12,13,14]. The dual second-order generalized integrator PLL (DSOGI-PLL) adaptively tracks frequency variations using orthogonal signal generation, achieving accurate phase locking under unbalanced conditions. However, the feedback cross-coupling between its frequency and phase control loops causes dynamic lag and overshoot during frequency fluctuations, limiting its effectiveness in fast-response fault scenarios [15,16].

To further improve the accuracy and dynamic performance of sequence component separation, various techniques have been widely investigated. Delayed signal cancellation (DSC) [17,18,19] and its cascaded variants (CDSC) [20,21,22] are commonly employed to extract positive- and negative-sequence components as well as multiple harmonics, yet they are plagued by inherent delays, frequency sensitivity, and increased complexity in harmonic suppression. Complex-coefficient filters (CCFs) have also been applied to achieve rapid and accurate extraction of fundamental sequence components without symmetrical component methods or excessive coordinate transformations, while enabling precise harmonic estimation [23,24,25,26]. Additionally, adaptive notch filter (ANF)-based state observers offer a simple and computationally efficient time-domain solution for grid synchronization and sequence component extraction, but are limited by constrained dynamic response, high parameter sensitivity, and inadequate high-order harmonic rejection under distorted grid conditions [27,28,29].

Recently, observer-based approaches have attracted attention for grid synchronization and sequence-component estimation. Adaptive observers, sliding-mode observers, and Kalman-filter-based estimators can directly estimate positive- and negative-sequence components without relying on conventional PLL structures or frequency-locked loops [30,31,32]. These methods generally exhibit good dynamic performance, robustness under unbalanced and fault conditions. However, they often require careful observer-gain tuning and involve relatively high computational complexity, particularly for covariance-matrix updating in Kalman-filter-based implementations.

In light of the above, a novel elliptic algebraic model-based positive- and negative-sequence amplitude estimation method is proposed for unbalanced grid voltages. By exploiting the intrinsic elliptic trajectory of unbalanced voltage vectors in the $\alpha \beta$ stationary reference frame, an explicit algebraic relationship between the sequence amplitudes and the elliptic geometric parameters is established. Consequently, the conventional sequence decomposition problem is reformulated as an elliptic parameter identification problem. Based on the proposed elliptic algebraic framework, an online parameter identification mechanism is developed to achieve fast and accurate sequence-amplitude extraction without requiring additional frequency estimation loops or complex decoupling structures. Lyapunov-based analysis is further conducted to verify the convergence property of the proposed identification framework. Experimental results further validate the effectiveness of the proposed method.

The paper is structured as follows: Section 2 presents the estimation model and fast method, Section 3 conducts Lyapunov-based stability analysis, Section 4 provides comparative experimental validation, and Section 5 concludes the paper.

2. Elliptic Model-Based Fast Estimation Method for Sequence Amplitudes

This section establishes a simple elliptic algebraic model for unbalanced grid voltages in the $\alpha \beta$ stationary reference frame. By introducing intermediate parameters, the model is simplified into a linear form. An online parameter identification scheme is then adopted to estimate the model parameters and the positive- and negative-sequence amplitudes.

Firstly, under voltage unbalance in the grid, for three-phase three-wire systems, the zero-sequence component can be neglected due to the lack of a return path. Considering only the fundamental frequency, the three-phase grid voltage can be decomposed into the sum of positive- and negative-sequence components, expressed as follows:

$$\mathit{u}\left(t\right)={\mathit{u}}^{+}\left(t\right)+{\mathit{u}}^{-}\left(t\right)=\left(\begin{array}{c}{v}_{p}cos(\omega t+{\delta }_p)\\ v_{p}cos(\omega t+{\delta }_p-{120}^{\circ })\\ v_{p}cos(\omega t+{\delta }_p+{120}^{\circ })\end{array}\right)+\left(\begin{array}{c}{v}_{n}cos(\omega t+{\delta }_n)\\ v_{n}cos(\omega t+{\delta }_n+{120}^{\circ })\\ v_{n}cos(\omega t+{\delta }_n-{120}^{\circ })\end{array}\right)$$

(1)

where $\omega$ is the fundamental angular frequency of the grid, $v_p$ and ${\delta }_p$ denote the amplitude and initial phase of the positive-sequence component, and $v_n$ and ${\delta }_n$ denote those of the negative-sequence component.

It is assumed that the positive- and negative-sequence components have the same initial phase angle, i.e., ${\delta }_p={\delta }_n=\delta$. After discretizing the three-phase grid voltage signals with a sampling period $T_s$ and transforming them into the $\alpha \beta$ stationary reference frame via Clarke transformation, the components can be expressed as

$$\begin{array}{cc}\hfill & \left\{\begin{array}{c}{u}_{\alpha }^{+}\left(k\right)=v_{p}cos(\omega k{T}_s+\delta )\hfill \\ u_{\beta }^{+}\left(k\right)=v_{p}sin(\omega k{T}_s+\delta )\hfill \end{array}\right.\hfill \\ \hfill & \left\{\begin{array}{c}{u}_{\alpha }^{-}\left(k\right)=v_{n}cos(\omega k{T}_s+\delta )\hfill \\ u_{\beta }^{-}\left(k\right)=-v_{n}sin(\omega k{T}_s+\delta )\hfill \end{array}\right.\hfill \end{array}$$

(2)

where $u_{\alpha }^{+}\left(k\right)$ and $u_{\beta }^{+}\left(k\right)$ are the $\alpha$-axis and $\beta$-axis components of the positive-sequence voltage in the $\alpha \beta$ frame, respectively; and $u_{\alpha }^{-}\left(k\right)$ and $u_{\beta }^{-}\left(k\right)$ are the corresponding components of the negative-sequence voltage.

By combining and simplifying the expressions of the positive- and negative-sequence components (2) in the $\alpha \beta$ stationary reference frame, the following relationships can be obtained:

$$(v_p+v_n)cos(\omega k{T}_s+\delta )=u_{\alpha }\left(k\right),(v_p-v_n)sin(\omega k{T}_s+\delta )=u_{\beta }\left(k\right)$$

(3)

Using trigonometric identities, the following elliptic algebraic relationship is derived:

$${\left({\displaystyle \frac{u_{\alpha }\left(k\right)}{v_p+v_n}}\right)}^2+{\left({\displaystyle \frac{u_{\beta }\left(k\right)}{v_p-v_n}}\right)}^2=1$$

(4)

To simplify the model and facilitate the subsequent parameter estimation of $v_p$ and $v_n$, two intermediate parameters are defined as

$$\begin{array}{c}\hfill a={\left({\displaystyle \frac{1}{v_p+v_n}}\right)}^2,b={\left({\displaystyle \frac{1}{v_p-v_n}}\right)}^2\end{array}$$

(5)

By substituting the above-defined parameters a and b into (4), the original elliptic relationship can be rearranged into a standard linear equation, constructed as follows:

$$y={\mathit{\theta }}^T\mathit{x}$$

(6)

In this linear equation, the constant term y has a true value of 1, and the coefficient vector $\mathit{\theta }$ and input vector $\mathit{x}$ are defined, respectively, as

$$\begin{array}{c}\hfill \mathit{\theta }={(a,b)}^T,\mathit{x}={(u_{\alpha }^2\left(k\right),u_{\beta }^2\left(k\right))}^T\end{array}$$

(7)

Given the dynamic characteristics of grid voltages, an online parameter identification mechanism is employed to dynamically track the time-varying parameter vector $\mathit{\theta }$. The parameter update law of the proposed method is designed as follows:

$$\begin{array}{c}\hfill \widehat{\mathit{\theta }}\left(k\right)=\widehat{\mathit{\theta }}(k-1)+{\displaystyle \frac{\mathit{P}(k-1)\mathit{x}\left(k\right)}{\lambda +{\mathit{x}}^T\left(k\right)\mathit{P}(k-1)\mathit{x}\left(k\right)}}e\left(k\right)\end{array}$$

(8)

where $\widehat{\mathit{\theta }}\left(k\right)$ is the estimated value of the parameter vector $\mathit{\theta }$ and expressed as $\widehat{\mathit{\theta }}\left(k\right)={(\widehat{a}\left(k\right),\widehat{b}\left(k\right))}^T$, and $\widehat{a}\left(k\right)$ and $\widehat{b}\left(k\right)$ are the estimates of intermediate parameters a and b. $\lambda$ is a design parameter, with a range of $\lambda \in (0,1)$. Furthermore, $e\left(k\right)$ is the estimation error defined as

$$e\left(k\right)=y\left(k\right)-{\mathit{x}}^T\left(k\right)\widehat{\mathit{\theta }}(k-1)$$

(9)

Matrix $\mathit{P}$ is updated as

$$\mathit{P}\left(k\right)={\displaystyle \frac{1}{\lambda }}\left(\mathit{P}(k-1)-{\displaystyle \frac{\mathit{P}(k-1)\mathit{x}\left(k\right)}{\lambda +{\mathit{x}}^T\left(k\right)\mathit{P}(k-1)\mathit{x}\left(k\right)}}{\mathit{x}}^T\left(k\right)\mathit{P}(k-1)\right)$$

(10)

and $\mathit{P}$ is initialized as $\mathit{P}\left(0\right)=\alpha \mathit{I}$ with $\alpha >0$.

To simplify the analytical process, the intermediate variable $\mathit{A}\left(k\right)$ is defined as

$$\mathit{A}\left(k\right)={\displaystyle \frac{\mathit{P}(k-1)\mathit{x}\left(k\right)}{\lambda +{\mathit{x}}^T\left(k\right)\mathit{P}(k-1)\mathit{x}\left(k\right)}}$$

(11)

After obtaining the estimated value of $\mathit{\theta }$, the estimates of the positive- and negative-sequence amplitudes, ${\widehat{v}}_p\left(k\right)$ and ${\widehat{v}}_n\left(k\right)$, can be derived as follows:

$${\widehat{v}}_p\left(k\right)={\displaystyle \frac{\sqrt{\frac{1}{\widehat{a}\left(k\right)}}+\sqrt{\frac{1}{\widehat{b}\left(k\right)}}}{2}},{\widehat{v}}_n\left(k\right)={\displaystyle \frac{\sqrt{\frac{1}{\widehat{a}\left(k\right)}}-\sqrt{\frac{1}{\widehat{b}\left(k\right)}}}{2}}$$

(12)

3. Lyapunov-Based Convergence Analysis

Based on the proposed elliptic model and the associated online parameter adaptation mechanism in Section 2, this section provides a rigorous proof of the global asymptotic stability of the proposed parameter estimation method. A Lyapunov-based global asymptotic stability analysis is employed to analyze the convergence properties of the estimation error, demonstrating that the algorithm converges to the true parameter values.

First, the Sherman–Morrison formula is applied to obtain the inverse of matrix $\mathit{P}$:

$$\begin{array}{cc}\hfill \mathbf{R}\left(k\right)& ={\left[\lambda {\mathit{P}}^{-1}(k-1)+\mathit{x}\left(k\right){\mathit{x}}^T\left(k\right)\right]}^{-1}\hfill \\ \hfill & ={\displaystyle \frac{1}{\lambda }}\mathit{P}(k-1)-{\displaystyle \frac{1}{\lambda }}\mathit{P}(k-1)\mathit{x}\left(k\right){\left(1+{\mathit{x}}^T\left(k\right){\displaystyle \frac{1}{\lambda }}\mathit{P}(k-1)\mathit{x}\left(k\right)\right)}^{-1}{\mathit{x}}^T\left(k\right){\displaystyle \frac{1}{\lambda }}\mathit{P}(k-1)\hfill \\ \hfill & ={\displaystyle \frac{1}{\lambda }}\left[\mathit{P}(k-1)-{\displaystyle \frac{\mathit{P}(k-1)\mathit{x}\left(k\right){\mathit{x}}^T\left(k\right)\mathit{P}(k-1)}{\lambda +{\mathit{x}}^T\left(k\right)\mathit{P}(k-1)\mathit{x}\left(k\right)}}\right]\hfill \\ \hfill & =\mathit{P}\left(k\right)\hfill \end{array}$$

That is,

$${\mathit{P}}^{-1}\left(k\right)=\lambda {\mathit{P}}^{-1}(k-1)+\mathit{x}\left(k\right){\mathit{x}}^T\left(k\right)$$

(13)

and the following is obtained:

$$\begin{array}{cc}\hfill \mathit{P}\left(k\right){\mathit{P}}^{-1}(k-1)& ={\displaystyle \frac{1}{\lambda }}\left[\mathit{P}(k-1)-\mathit{A}\left(k\right){\mathit{x}}^T\left(k\right)\mathit{P}(k-1)\right]{\mathit{P}}^{-1}(k-1)\hfill \\ \hfill & ={\displaystyle \frac{1}{\lambda }}\mathit{I}-{\displaystyle \frac{1}{\lambda }}\mathit{A}\left(k\right){\mathit{x}}^T\left(k\right)={\displaystyle \frac{1}{\lambda }}\left[\mathit{I}-\mathit{A}\left(k\right){\mathit{x}}^T\left(k\right)\right]\hfill \end{array}$$

(14)

Define the parameter estimation error $\tilde{\mathit{\theta }}\left(k\right)$ as

$$\tilde{\mathit{\theta }}\left(k\right)=\widehat{\mathit{\theta }}\left(k\right)-\mathit{\theta }$$

(15)

Substituting (8) and (14) into the above definition of the estimation error, the following expression is derived through algebraic manipulation:

$$\begin{array}{cc}\hfill \tilde{\mathit{\theta }}\left(k\right)& =\widehat{\mathit{\theta }}(k-1)+\mathit{A}\left(k\right)e\left(k\right)-\mathit{\theta }\hfill \\ \hfill & =\tilde{\mathit{\theta }}(k-1)+\mathit{A}\left(k\right)e\left(k\right)\hfill \\ \hfill & =\tilde{\mathit{\theta }}(k-1)+\mathit{A}\left(k\right){\mathit{x}}^T\left(k\right)(-\tilde{\mathit{\theta }}(k-1))\hfill \\ \hfill & =\left(\mathit{I}-\mathit{A}\left(k\right){\mathit{x}}^T\left(k\right)\right)\tilde{\mathit{\theta }}(k-1)\hfill \\ \hfill & =\lambda \mathit{P}\left(k\right){\mathit{P}}^{-1}(k-1)\tilde{\mathit{\theta }}(k-1)\hfill \end{array}$$

(16)

To analyze the stability of the adaptive estimation system, a Lyapunov function is constructed:

$$V(\tilde{\mathit{\theta }},k)={\tilde{\mathit{\theta }}}^T\left(k\right){\mathit{P}}^{-1}\left(k\right)\tilde{\mathit{\theta }}\left(k\right)$$

(17)

This function is positive definite, as guaranteed by the symmetric positive-definite property of ${\mathit{P}}^{-1}\left(k\right)$. Combined with (16), the variation in the Lyapunov function $\Delta V$ is analyzed as follows:

$$\begin{array}{cc}\hfill \Delta V& ={\tilde{\mathit{\theta }}}^T(k+1){\mathit{P}}^{-1}(k+1)\tilde{\mathit{\theta }}(k+1)-{\tilde{\mathit{\theta }}}^T\left(k\right){\mathit{P}}^{-1}\left(k\right)\tilde{\mathit{\theta }}\left(k\right)\hfill \\ \hfill & ={\lambda }^2{\tilde{\mathit{\theta }}}^T\left(k\right){\mathit{P}}^{-1}\left(k\right)\mathit{P}(k+1){\mathit{P}}^{-1}(k+1)\mathit{P}(k+1){\mathit{P}}^{-1}\left(k\right)\tilde{\mathit{\theta }}\left(k\right)-{\tilde{\mathit{\theta }}}^T\left(k\right){\mathit{P}}^{-1}\left(k\right)\tilde{\mathit{\theta }}\left(k\right)\hfill \\ \hfill & ={\lambda }^2{\tilde{\mathit{\theta }}}^T\left(k\right){\mathit{P}}^{-1}\left(k\right)\mathit{P}(k+1){\mathit{P}}^{-1}\left(k\right)\tilde{\mathit{\theta }}\left(k\right)-{\tilde{\mathit{\theta }}}^T\left(k\right){\mathit{P}}^{-1}\left(k\right)\tilde{\mathit{\theta }}\left(k\right)\hfill \\ \hfill & ={\tilde{\mathit{\theta }}}^T\left(k\right)\left[{\lambda }^2{\mathit{P}}^{-1}\left(k\right)\mathit{P}(k+1){\mathit{P}}^{-1}\left(k\right)-{\mathit{P}}^{-1}\left(k\right)\right]\tilde{\mathit{\theta }}\left(k\right)\hfill \end{array}$$

(18)

Let $\mathit{M}\left(k\right)={\lambda }^2{\mathit{P}}^{-1}\left(k\right)\mathit{P}(k+1){\mathit{P}}^{-1}\left(k\right)-{\mathit{P}}^{-1}\left(k\right)$. Using the result from (14), this matrix is simplified to

$$\begin{array}{cc}\hfill \mathit{M}\left(k\right)& ={\lambda }^2{\mathit{P}}^{-1}\left(k\right)\left[{\displaystyle \frac{1}{\lambda }}\left(\mathit{I}-\mathit{A}(k+1){\mathit{x}}^T(k+1)\right)\right]-{\mathit{P}}^{-1}\left(k\right)\hfill \\ \hfill & =(\lambda -1){\mathit{P}}^{-1}\left(k\right)-{\displaystyle \frac{\lambda {\mathit{P}}^{-1}\left(k\right)\mathit{P}\left(k\right)\mathit{x}(k+1){\mathit{x}}^T(k+1)}{\lambda +{\mathit{x}}^T(k+1)\mathit{P}\left(k\right)\mathit{x}(k+1)}}\hfill \\ \hfill & =(\lambda -1){\mathit{P}}^{-1}\left(k\right)-{\displaystyle \frac{\lambda \mathit{x}(k+1){\mathit{x}}^T(k+1)}{\lambda +{\mathit{x}}^T(k+1)\mathit{P}\left(k\right)\mathit{x}(k+1)}}\hfill \end{array}$$

(19)

For any nonzero vector $\mathit{z}$, the corresponding quadratic form of $\mathit{M}\left(k\right)$ satisfies

$${\mathit{z}}^T\mathit{M}\left(k\right)\mathit{z}=(\lambda -1){\mathit{z}}^T{\mathit{P}}^{-1}\left(k\right)\mathit{z}-{\displaystyle \frac{\lambda {\left({\mathit{z}}^T\mathit{x}(k+1)\right)}^2}{\lambda +{\mathit{x}}^T(k+1)\mathit{P}\left(k\right)\mathit{x}(k+1)}}$$

Since $\lambda \in (0,1)$ and ${\mathit{P}}^{-1}\left(k\right)$ is symmetric positive definite,

$$(\lambda -1){\mathit{z}}^T{\mathit{P}}^{-1}\left(k\right)\mathit{z}<0,\forall \mathit{z}\ne 0$$

Moreover,

$$-{\displaystyle \frac{\lambda {\left({\mathit{z}}^T\mathit{x}(k+1)\right)}^2}{\lambda +{\mathit{x}}^T(k+1)\mathit{P}\left(k\right)\mathit{x}(k+1)}}\le 0$$

Therefore,

$${\mathit{z}}^T\mathit{M}\left(k\right)\mathit{z}<0,\forall \mathit{z}\ne 0$$

which proves that $\mathit{M}\left(k\right)$ is negative definite.

Hence,

$$\Delta V={\tilde{\mathit{\theta }}}^T\left(k\right)\mathit{M}\left(k\right)\tilde{\mathit{\theta }}\left(k\right)<0,\forall \tilde{\mathit{\theta }}\left(k\right)\ne 0$$

Consequently, the Lyapunov function decreases monotonically, and the proposed adaptive estimation system is asymptotically stable.

4. Experiment Results

In this section, the effectiveness and performance of the proposed method are further evaluated through experiments. The proposed method is implemented on the YXSPACE-SP2000 processing platform with a sampling frequency of 20 kHz. This platform is based on the TMS320F28377 digital signal processor operating at 200 MHz, which is equipped with multi-channel synchronous ADC/DAC modules, PWM outputs, and encoder interfaces to support real-time validation of control algorithms. The method parameters are set as follows: the forgetting factor is $\lambda =0.99$; the DDSRF-PLL gains are $k_p=92,k_i=4232$; and the filter cutoff frequency is $\zeta =50$ Hz. The experimental conditions for four test cases are summarized in

Table 1. Experimental conditions.

Cases	Experimental Conditions
Case A	Start: balanced 1.0 p.u. (no negative-sequence). At 0.1 s: positive-sequence step $-0.2$ p.u. + negative-sequence $+0.1$ p.u.
Case B	Start: balanced 1.0 p.u. (no negative-sequence). At 0.1 s: inject $0.3$ p.u. negative-sequence. At 0.2 s: negative-sequence step $-0.2$ p.u.
Case C	Start: $1.0$ p.u. positive-sequence + $0.3$ p.u. negative-sequence. At 0.1 s: $+3\mathrm{Hz}$ frequency step. At 0.2 s: step to $60\mathrm{Hz}$.
Case D	Start: $1.0$ p.u. positive-sequence + $0.3$ p.u. negative-sequence. At 0.1 s: $\pi /6$ phase jump.

.

Figure 1 shows the experimental results under Case A. Both methods achieve a similar settling time of approximately 15 ms for positive-sequence amplitude tracking. However, the DDSRF-PLL exhibits a 15% undershoot, whereas the proposed method shows nearly zero undershoot. For negative-sequence amplitude tracking, the proposed method settles within 25 ms, while the DDSRF-PLL requires 27 ms. In addition, the proposed method maintains an almost overshoot-free response, whereas the DDSRF-PLL exhibits a 47% overshoot. These results indicate that, under voltage unbalance conditions, the proposed approach effectively and rapidly tracks both positive- and negative-sequence amplitudes.

Figure 2 shows the experimental results under Case B. As shown in Figure 2b, both methods achieve a settling time of approximately 13 ms following the disturbance at 0.1 s, while the proposed method exhibits a slightly smaller maximum tracking error of 0.066 V compared with 0.084 V for the DDSRF-PLL. After the disturbance at 0.2 s, the proposed method settles within 9 ms, whereas the DDSRF-PLL requires 12 ms, and the corresponding maximum tracking errors are 0.032 V and 0.055 V, respectively. Figure 2c shows that the proposed method achieves a settling time of 24 ms with nearly zero overshoot/undershoot, whereas the DDSRF-PLL exhibits a 10.7% overshoot at 0.1 s and a 9% undershoot at 0.2 s. These results demonstrate the improved transient performance of the proposed method in negative-sequence amplitude tracking.

Figure 3 shows the experimental results under Case C. As shown in Figure 3b, following the 3 Hz frequency step at 0.1 s, the DDSRF-PLL exhibits a settling time of 15 ms and a maximum tracking error of 0.022 V, whereas the proposed method shows no observable transient response. After the frequency is changed to 60 Hz at 0.2 s, the DDSRF-PLL requires 42 ms to settle and exhibits a maximum tracking error of 0.066 V, while the proposed method remains virtually unaffected by the frequency variation. Figure 3c shows similar results for negative-sequence amplitude tracking. The DDSRF-PLL exhibits settling times of 44 ms and 55 ms, with maximum tracking errors of 0.019 V and 0.075 V under the two frequency disturbances, respectively, whereas the proposed method maintains accurate amplitude estimation throughout the entire process. These results demonstrate the strong robustness of the proposed method against frequency variations.

Figure 4 shows the experimental results under Case D. As shown in Figure 4b, following the phase jump at 0.1 s, the DDSRF-PLL exhibits a settling time of 29 ms and a maximum tracking error of 0.171 V, whereas the proposed method shows no observable transient response. Figure 4c shows similar results for negative-sequence amplitude tracking. The DDSRF-PLL requires 56 ms to reach steady state and exhibits a maximum tracking error of 0.212 V, while the proposed method remains virtually unaffected by the phase disturbance and maintains accurate amplitude estimation throughout the entire process. These experimental results demonstrate the effectiveness of the proposed method in estimating positive- and negative-sequence amplitudes under phase-jump conditions.

Table 2. Settling time comparison under different test cases (ms).

Case	Method	$\mathit{t}=0.1\mathbf{s}$		$\mathit{t}=0.2\mathbf{s}$
		${\mathit{v}}_{\mathit{P}}$	${\mathit{v}}_{\mathit{n}}$	${\mathit{v}}_{\mathit{P}}$	${\mathit{v}}_{\mathit{n}}$
A	DDSRF	15	27	—	—
	Proposed	15	25	—	—
B	DDSRF	13	24	12	24
	Proposed	13	24	9	24
C	DDSRF	15	44	42	55
	Proposed	0	0	0	0
D	DDSRF	29	56	—	—
	Proposed	0	0	—	—

Note: “—” indicates no corresponding transient disturbance under this working condition.

summarizes the settling-time performance of the two methods. Under voltage-unbalance conditions, the proposed method achieves comparable or slightly shorter settling times than the DDSRF-PLL. Under frequency and phase disturbances, the proposed method exhibits negligible transient deviation, whereas the DDSRF-PLL requires settling times ranging from 15 ms to 56 ms depending on the disturbance scenario. These results further confirm the fast dynamic response and robustness of the proposed method.

5. Conclusions

In this paper, a novel elliptic algebraic model-based positive- and negative-sequence amplitude estimation method is proposed for unbalanced grid conditions. By exploiting the elliptic trajectory of unbalanced voltages in the $\alpha \beta$ stationary reference frame, the sequence-amplitude estimation problem is transformed into an elliptic parameter identification problem, eliminating the need for additional frequency estimation loops or complex decoupling structures. Lyapunov-based analysis verifies the asymptotic convergence of the proposed identification framework.

Experimental results demonstrate that the proposed method achieves fast dynamic response, high estimation accuracy, and strong robustness under voltage-unbalance, frequency-variation, and phase-jump conditions. Compared with the DDSRF-PLL, it achieves comparable or shorter settling times, significantly reduced transient oscillations, and zero steady-state tracking error. Future work will extend the proposed framework to harmonic-distorted and more general unbalanced grid scenarios.